The influence of magma o cean
crystallization on man tle dynamics
In v estigating the Martian and lunar magma o ceans
v orgelegt v on
Dipl.-Ing.
Maxime Olivier Maurice
v on der F akultät I I - Mathematik und Naturwissensc haften
der T ec hnisc hen Univ ersität Berlin
zur Erlangung des akademisc hen Grades
Doktor der Ingenieurwissensc haften
-Dr.-Ing.-
genehmigte Dissertation
Promotionsaussc h uss:
V orsitzender: Prof. Dr. Holger Stark
Gutac h ter: Prof. Dr. Dieter Breitsc h w erdt
Gutac h ter: Prof. Dr. Bernhard Stein b erger
Gutac h ter: Dr. Nicola T osi
T ag der wissensc haftlic hen A ussprac he: No v em b er 16 th , 2020
Berlin 2020

Zusammenfassung
V or allem die frühen Prozesse in der En t wic klung terrestrisc her Planeten prägen die
gesam te w eitere Planetenen t wic klung und sind daher für unser V erständnis der Ent wic klung
terrestrisc her Planeten v on b esonderer Bedeutung. Neb en der Differenzierung in einen
metallisc hen Kern und einen silikatisc hen Man tel spielt b ei vielen terrestrisc hen Planeten auch
die A uskristallisation eines globalen Magmaozeans eine wic htige Rolle, da dieser Prozess zu
einer w eiteren Differenzierung des Silikatman tels führt und damit die Anfangsb edingungen
für spätere thermisc he und dynamisc he Prozesse im festen Man tel festlegt. Dab ei wird
t ypisc herweise angenommen, dass die Kristallisation des Magmaozeans so sc hnell ist, dass sie
b ereits abgesc hlossen ist b ev or K on v ektion im festen Silikatman tel einsetzt. Daher wurden
die W ec hselwirkungen zwisc hen diesen Prozessen bisher nic h t explizit un tersuc h t. Allerdings
k önn ten die Bildung einer dic h ten, opak en A tmosphäre durc h Ausgasung des Magmaozeans
o der die Bildung einer stabilen festen Kruste an der Ob erfläc he des Magmaozeans das
Abkühlen des Magmaozeans stark v erlangsamen. In diesem F all kann K on v ektion im festen
T eil des Silikatman tels b ereits einsetzen b ev or der Magmaozean v ollständig auskristallisiert ist,
w as durc h die W ec hselwirkung b eider Prozesse zu einem v eränderten Misc h ungsverhalten
des Man tels und thermisc hen R üc kk opplungsmec hanismen führt. In dieser Doktorarb eit
w erden Man telk on v ektionsmo delle eingesetzt, um die A uswirkung des frühen Einsetzens v on
K on v ektion in einem erstarrenden Silikatman tel zu un tersuc hen. Ein allgemeines Mo dell v on
gleic hzeitiger A uskristallisation des Magmaozeans und K onv ektion des festen Man tels wird
für den F all eines marsähnlic hen Planeten v orgestellt. Es wird gezeigt, dass der frühe Beginn
v on Man telk on v ektion v or der v ollständigen V erfestigung des Marsman tels w arsc heinlic h ist
und zu einer ausgeprägteren Homogenisierung des Man tels führen kann. Dieses neue Mo dell
bietet eine möglic he geo dynamisc he Grundlage für eine dauerhafte Aktivität des Marsman tels,
auf die Spuren v on jungem V ulkanism us hindeuten. In einem näc hsten Sc hritt wird dieses
Mo dell auf den Mond angewendet, für w elc hen die Existenz eines frühen Magmaozeans und die
Bildung einer auf dem Magma sc h wimmenden primären Kruste am b esten dokumen tiert ist. Es
wird gezeigt, dass thermisc he W ec hselwirkungen zwisc hen dem kristallisierenden Magmaozean
und dem k on vektierenden festen Man tel so wie die Bildung einer Kruste zu einer v erlängerten
Erstarrung des Mondman tels führen, w elc he in Üb ereinstimm ung mit geo c hronologisc hen
Sc hätzungen für die Altersspanne der Mondkruste bis zu 200 Millionen Jahren andauern
kann. Durc h die K opplung der Sim ulation der thermischen En t wic klung des Mondes mit
Mo dellen, w elc he die F raktionierung v on Spurenelemen ten und den Zerfall radioaktiv er Isotop e
b esc hreib en, kann die Isotop ensystematik v on Mondgesteinen und c hemisc hen Komponenten
des Mondman tels repro duziert w erden und somit das Alter des Mondes auf 4.40 - 4.45 Ga
eingegrenzt w erden. Ansc hließend wird der Einfluss eines frühen Beginns der Man telk on v ektion

auf die langfristige Man teldynamik des Mondes un tersuc ht. Insbesondere wird gezeigt, dass
dieser Prozess zu einer homogeneren V erteilung von Ti-haltigem Material im Mondman tel
führt – einer not w endigen V orraussetzung zur Bildung Ti-reic her Marebasalte.
iv

Abstract
The early stages of terrestrial planets ev olution pla y a crucial role in determining their long-
term dev elopmen t. In particular, the differen tiation b et w een an iron core and a silicate man tle
is lik ely follo w ed b y the crystallization of a primordial magma o cean, controlling the initial
conditions for solid-man tle thermal and dynamical ev olution. Because it w as though t that
magma o cean crystallization w as m uc h faster than the onset of solid-man tle dynamics, the
in terpla y b et w een these t w o pro cesses has not b een considered so far. Ho w ever, the outgassing
of a thic k opaque atmosphere from a crystallizing magma o cean, or the formation of a solid
flotation lid at the surface can strongly slo w do wn the solidifiation of the man tle. In this case,
solid-state dynamics can set in b efore the end of the magma o cean’s lifetime, inducing p eculiar
regimes of man tle mixing and thermal feedbac k mec hanisms. In this thesis, w e use man tle
con v ection to ols to study the effects of the early onset of solid-state dynamics in a solidifying
man tle. A general mo del of sim ultaneous magma o cean solidification and man tle con v ection
is in tro duced for a case based on the Martian magma o cean. W e sho w that for realistic
parameters, it is lik ely that Mars underw en t man tle con v ection and mixing b efore the complete
solidification of its magma o cean. This new paradigm sheds new ligh t on the p ossibilit y for the
Martian man tle to sustain long-term man tle activit y , as suggested by traces of late v olcanism.
W e then apply this mo del to the case of the Mo on, where the existence of a primordial magma
o cean, solidifying b elo w an insulating flotation crust, is b est do cumen ted. W e show that
the thermal feedbac ks resulting from sim ultaneous magma o cean solidification and man tle
con v ection lead to a solidification of the lunar man tle extending o v er up to 200 millions of
y ears, in agreemen t with geo c hronological estimates for the lunar crust age span. By coupling
our thermal ev olution sim ulations with a trace-elemen t fractionation and radio-isotop es deca y
mo del, w e repro duce the isotope systematics inherited from the lunar magma o cean, and use
it to constrain the age of the Mo on forming even t to 4.40 to 4.45 Ga. Finally , as for Mars,
w e in v estigate the influence of the early onset of man tle con v ection on the long-term man tle
dynamics of the Mo on. In particular, w e sho w that it enhances the mixing of Ti-bearing
material throughout the lunar man tle. This mec hanism is necessary to explain a class of
high-Ti samples in the lunar mar e basalts.

This thesis is dedicated to m y late grandfather R iton .

A ckno wledgements
I w ould lik e to thank first m y sup ervisor Dr. Nicola T osi, for his dedication, his supp ort,
and his trust. I am deeply grateful that the fate of PhD studen ts left b y themselv es with no
direction to follo w remains for me only something I ha ve heard of, y et nev er exp erienced. I
also thank, ob viously , m y family , which is and has alw a ys b een unshakable ground for me.
I am thankful to the p eople I ha v e b een w orking with, who help ed shaping m y mind and
metho ds as a researc her, and who significan tly con trbuted to the w ork achiev ed in this thesis,
b eing directly in v olv ed or pro viding insigh tful conv ersations: Prof. Dr. Doris Breuer, Dr.
Ana-Catalina Plesa, Dr. Sabrina Sc h winger, Dr. Christian Hüttig and Prof. Dr. Thorsten
Kleine. A sp ecial thanks to the latter for pro viding funding whic h help ed me finishing writing
this thesis. I thank all m y colleagues and friends from DLR, in particular Nasia, Giggio, Hugo
and Mic ka. A great thanks to all m y friends, from all places and all times: those from the
b eginnings in Dourdan and Lomener, those from T oulouse, who are no w ev erywhere, and those
from Berlin and who ha v e b een a ma jor part of m y life these last y ears. Finally , I thank F red
and Jam y , and all p eople who con tributed to creating the TV sho w ”C’est pas sorcier”, who
most surely had a great influence on a whole generation of frenc h-sp eaking scien tists.

T able of Contents
Title P age i
Zusammenfassung iii
Abstract v
List of Figures xv
List of T ables xxv
1 In tro duction 1
1.1 The formation of the solar system . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 The collapse of the protosolar nebula . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Building up the planets . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 . 1 . 3 C h r o n o l o g y .................................. 7
1.2 T errestrial b o dies differen tiation . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Core-man tle differen tiation . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Silicate differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1 . 3 M a g m a o c e a n s .................................... 1 3
1.3.1 Chemical evidences of magma o ceans: the case of the Mo on and Mars . 13
1.3.2 The physics of magma o cean crystallization . . . . . . . . . . . . . . . . 14
1.3.3 Influence of magma o cean crystallization on planetary evolution . . . . 20
2 Early onset of con v ection and man tle o v erturn: the case of Mars 25
2.1 Constrain ts and problematics ab out the Martian magma o cean . . . . . . . . . 25
2 . 2 P h y s i c a l m o d e l .................................... 2 6
2.2.1 F ractional crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.2 Batc h crystallization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 . 2 . 3 C u m u l a t e s d y n a m i c s ............................. 2 8
2 . 2 . 4 P a r a m e t e r s p a c e ............................... 3 0
2 . 3 N u m e r i c a l s e t - u p ................................... 3 1
2.3.1 Domain mehsing and time-ev olving geometry of the cum ulates . . . . . 31
2.3.2 Initial and b oundary conditions . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.3 Ly apuno v exp onent and shrinking factor . . . . . . . . . . . . . . . . . . 33
2 . 4 R e s u l t s ......................................... 3 4
2.4.1 Early onset of man tle con v ection . . . . . . . . . . . . . . . . . . . . . . 34
xi

T ABLE OF CONTENTS
2 . 4 . 2 M i x i n g o f t h e m a n t l e ............................. 3 7
2.4.3 Correlation b etw een early or late onset of con v ection and mixing . . . . 40
2 . 5 D i s c u s s i o n ....................................... 4 2
2.5.1 Application to Mars’ man tle . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5.2 A more realistic ev olution of man tle solidification . . . . . . . . . . . . . 44
2.5.3 Application to other terrestrial b odies . . . . . . . . . . . . . . . . . . . 45
3 Gro wth of a flotation lid: the case of the Mo on 49
3 . 1 T h e l u n a r m a g m a o c e a n ............................... 4 9
3.1.1 Existing mo dels for the lunar magma o cean solidification . . . . . . . . 50
3.1.2 Unconsidered effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3 . 2 C r y s t a l l i z a t i o n m o d e l ................................. 5 2
3.2.1 Initial depth of the magma o cean . . . . . . . . . . . . . . . . . . . . . . 53
3.2.2 Re-melting of the cum ulates and the crust . . . . . . . . . . . . . . . . . 54
3.3 Thermal evolution mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 . 3 . 1 T h e c o r e .................................... 5 6
3 . 3 . 2 T h e s o l i d c u m u l a t e s ............................. 5 6
3 . 3 . 3 M a g m a o c e a n ................................. 5 8
3 . 3 . 4 T h e fl o t a t i o n c r u s t .............................. 5 9
3 . 3 . 5 I n i t i a l s e t - u p ................................. 6 0
3.4 Thermal mo del b enc hmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.1 Comparison with existing mo dels . . . . . . . . . . . . . . . . . . . . . . 61
3.4.2 Comparison b etw een differen t effects included in the thermal mo del . . 62
3.4.3 Heat-piping and mass conserv ation . . . . . . . . . . . . . . . . . . . . . 63
3.4.4 Con v ergence tests for the con vection sim ulations . . . . . . . . . . . . . 64
3.5 Application to the lunar magma o cean . . . . . . . . . . . . . . . . . . . . . . . 66
3.5.1 Choice of the parameter space . . . . . . . . . . . . . . . . . . . . . . . 66
3.5.2 Effect of k crust : purely diffusiv e cum ulates . . . . . . . . . . . . . . . . . 67
3.5.3 Effect of η ref : solid-state of con v ection in the cum ulates and heat-piping 68
3.5.4 T ransien t remelting of the cum ulates and the crust . . . . . . . . . . . . 70
3.5.5 P artitioning of the heat-pro ducing elemen ts in the crust . . . . . . . . . 70
3.5.6 Influence of the crystallization mo del . . . . . . . . . . . . . . . . . . . . 71
4 Repro ducing the isotopic record of the Mo on 73
4.1 The KREEP trace elemen ts signature . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Isotop es fractionation during silicate differen tiation . . . . . . . . . . . . . . . . 74
4.2.1 Isotop es fractionation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.2 Isotop e age dating of a single sample . . . . . . . . . . . . . . . . . . . . 75
4.2.3 Isotop e age dating of the fractionation ev en t . . . . . . . . . . . . . . . 77
4.2.4 Application to the lunar magma o cean . . . . . . . . . . . . . . . . . . . 78
4.3 Coupled fractionation and radioactiv e deca y mo del . . . . . . . . . . . . . . . . 80
4.3.1 Ev olution of the magma o cean’s isotopic ratios . . . . . . . . . . . . . . 80
4.3.2 Cum ulates whole-ro c k isotopic signature . . . . . . . . . . . . . . . . . . 81
4.4 Re-in terpreting the isotopic data of KREEP samples . . . . . . . . . . . . . . . 82
xii

T ABLE OF CONTENTS
4.4.1 Mon te-Carlo sim ulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.4.2 Matc hing KREEP isotopic signature . . . . . . . . . . . . . . . . . . . . 82
4.4.3 Matc hing Kal009 isotopic signature . . . . . . . . . . . . . . . . . . . . . 84
4 . 5 S e n s i t i v i t y s t u d y ................................... 8 4
4.5.1 Sensitivit y to the partition co efficien ts . . . . . . . . . . . . . . . . . . . 84
4.5.2 Sensitivit y to the magma o cean solidification’s time-series . . . . . . . . 86
4.5.3 Sensitivit y to KREEP comp osition: the case of the lunar zircons . . . . 88
5 Mixing the lunar man tle: the fate of the ilmenite-b earing cum ulates 91
5.1 The ilmenite-b earing cum ulates . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.1.1 Mineralogical mo dels and exp erimen tal studies . . . . . . . . . . . . . . 92
5.1.2 The origin of Ti in mar e b a s a l t s ...................... 9 3
5.2 En training the IBCs in to the man tle . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2.1 The imp ortance of IBC o v erturn . . . . . . . . . . . . . . . . . . . . . . 95
5.2.2 P ost-magma o cean o v erturn . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2.3 Ov erturn during magma o cean crystallization . . . . . . . . . . . . . . . 97
5.3 IBC ov erturn sim ulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.1 3 Lay ers man tle mo del . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.2 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5 . 3 . 3 P a r a m e t e r s p a c e ............................... 1 0 0
5.3.4 Quan tification of the IBC o v erturn . . . . . . . . . . . . . . . . . . . . . 100
5 . 3 . 5 I B C m o b i l i z a t i o n ...............................1 0 1
5.3.6 Thermal coupling b etw een IBC o v erturn and magma o cean solidification 106
5.4 Long-term evolution and volcanism of the Mo on . . . . . . . . . . . . . . . . . 107
5.4.1 Presen t-da y thermal state of the Mo on . . . . . . . . . . . . . . . . . . . 107
5.4.2 Re-melting of the IBC and pro duction of high-Ti mare basalts . . . . . 108
6 Conclusions and Outlo oks 113
References 117
App endix A App endix A: Man tle dynamics 131
A . 1 T h e r m a l c o n v e c t i o n ..................................1 3 1
A.1.1 Boussinesq appro ximation . . . . . . . . . . . . . . . . . . . . . . . . . . 131
A.1.2 General form of conserv ation equations . . . . . . . . . . . . . . . . . . 131
A.1.3 Conserv ation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A.1.4 Conserv ation of momen tum . . . . . . . . . . . . . . . . . . . . . . . . . 133
A.1.5 Conserv ation of thermal energy . . . . . . . . . . . . . . . . . . . . . . . 134
A . 2 R h e o l o g y ....................................... 1 3 5
A.3 Conserv ation of comp osition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A . 4 C o r e c o o l i n g ...................................... 1 3 6
App endix B App endix B: Numerical implemen tation 137
B.1 Non-dimensionalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
B . 2 F i n i t e v o l u m e m e t h o d ................................ 1 3 8
xiii

T ABLE OF CONTENTS
B . 2 . 1 I m p l e m e n t a t i o n ................................ 1 3 8
B . 2 . 2 D o m a i n m e s h i n g ............................... 1 3 9
B.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
B.3 P articles in cell metho d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
xiv

List of Figures
1.1
Protoplaneatry disks observ ed b y the SPHERE instrument on ESO’s V ery Large
T elescop e. Some exhibit circular gap suggesting the orbits of already formed
planets. A dapted from h ttps://www.eso.org/public/images/eso1811a/ . . . . . 2
1.2
Sk etc h of the successiv e stages of a sp eculativ e scenario for solar system
formation. a) A dense core forms in a gian t molecular cloud’s parcel, previously
isolated p ossibly b y the sho c k w a v e of a neigh b oring sup erno v a. b) The
gra vitational collapse of this parcel results, due to the conserv ation of angular
momen tum, in the formation of a flattened disk with contin uously infalling
material hitting a compression sho ck fron t (transparen t turquoise en v elop). Dust
settles in the mid-plane of the disk and gas (turquoise clouds) is compressed
to w ards the cen ter, forming the proto-star. c) The pre-main sequence proto-
star sw eeps out nebular gas while planetesimals are forming and, b ey ond the
sno wline, gian t cores already accreted and cleared their surroundings of gas and
dust. d) The proto-star en ters the main sequence b y igniting h ydrogene burning.
Gas gian ts migration has cleared the dust and scattered planetesimals b ey ond
∼
1
.
5 A U, lea ving a limited reserv oir for the prolonged oligarc hic gro wth of
t e r r e s t r i a l p l a n e t s . .................................. 5
1.3
Gro wth path w a y of the planets. The arro ws represen t inferred gro wth
mec hanism, the streaming instability b eing one candidate to o v ercome the
meter-sized barrier. The p ebble accretion and gian t impacts roughly corresp ond
to the runa w a y and oligarchic gro wth phases, resp ectiv ely . . . . . . . . . . . . . 6
1.4
Solar system formation c hronology . While for CAIs and c hondrules as w ell as
for Earth and Mars, the time span represen t isotopic dating estimates (Pb-Pb
age estimates for the former (Bouvier & W adh w a, 2010; Kruijer, Kleine, & Borg,
2020) and Hf-W age of core formation for the latter (Kleine & W alk er, 2017)),
the timespan indicated for Jupiter’s formation accoun ts for the assumption
that it formed b efore clearing of the nebular gas. The ”isotop es” time spans
at the b ottom corresp ond to the system’s half-life, ho w ev er the system is still
activ e (i.e. still pro duces energy or can still b e used for age dating) during
ab out 7 half-liv es. Isotop e systems whose deca y con tributes to the early heating
of planets are represen ted in red while systems only used for age dating are
represen ted in grey . The black axis indicates time after CAIs formation in Myr
while the red axis indicates time b efore presen t, in Gyr (Ga). . . . . . . . . . . 7
xv

LIST OF FIGURES
1.5
Heating induced b y radion uclides (grey dashed lines) at v arious times (after
CAIs formation), accretion (solid black lines) for v arious impactor-to-target
mass ratios (
γ
), and core-man tle differen tiation (red dotted lines) for differen t
terrestrial b o dies. The x axis represents the bo dy’s mass in unit Earth’s mass
(
M e
) and the y axis the resulting temp erature increase. In the yello w area, the
b o dy’s man tle is heated by more than 1000 K, sufficien tly to cause a global
magma o cean. The heating pro vided b y core-man tle differen tiation do es not
only dep ends on the b o dy’s size but also its in ternal structure, therefore it is
not represen ted as a function of the
x
-axis, butgiv en for three differen t cases.
F urthermore, it is computed considering an instan taneous differen tiation ev en t,
thereb y significan tly ov erestimating the resulting heating. Adapted from R ubie,
Nimmo, and Melosh (2015b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6
Sk etc h of the mec hanism of
182 Hf
–
182
W dating system of the core-man tle
differen tiation. F or an early episo de of core-man tle differen tiation (top line), the
separation b et w een Hf (squares) going preferen tially in to the silicate man tle, and
W (circles) going preferen tially in to the iron core o ccurs while some unstable
Hf still exists, resulting in increase of radiogenic W in the man tle, hence a high
presen t-da y
182
W
/ 184
W v alue. If the core-man tle differen tiation o ccurs after all
unstable Hf has deca y ed (b ottom line), no deviation of the
182
W
/ 184
W ratio
from the bulk comp osition o ccurs. . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7
Melting curv es (blue for solidus and green for liquidus) for fertile KLB-1
p eridotite plotted on the pressure range relev an t for the Martian man tle. Magma
o cean’s adiabatic temp erature profiles for three differen t p oten tial temp eratures
(ligh t green for 2200 K, orange for 2000 K and red for 1800 K) are plotted do wn
to the R CMF, where the liquid-lik e b eha viour, i.e. the magma o cean stops.
F rom Herzb erg, Raterron, and Zhang (2000). . . . . . . . . . . . . . . . . . . . 15
1.8
F ractional crystallization of a b ottom-up crystallizing melt reserv oir. The
crystallization sequence corresp onds to the evolution of the mineralogy of
the crystals that form as the magma o cean solidifies (i.e. as the temp erature,
pressure and comp osition conditions evolv e). Eac h crystallized la y er corresp onds
to the crystallization sequence. Because the bulk of the cum ulates pile and
the magma o cean are not in equilibrium, their resp ectiv e melting curv es differ.
Therefore the crystallization temp erature, i.e. the temp erature retained b y each
new solid cum ulates la y er, is more relev an t. . . . . . . . . . . . . . . . . . . . . 17
1.9 Equilibrium (batc h) crystallization of a b ottom-up crystallizing melt reserv oir.
Note the co existence of differen t comp ositions (colors) in the en trained crystals,
and the m uc h less significan t chemical la y ering that results compared with
Figure 1.8 (for pure equilibrium crystallization, no c hemical la yering w ould b e
e x p e c t e d ) ........................................ 1 8
1.10
Time series of the ev olution of the p oten tial temp erature of an Earth-lik e
full-man tle magma o cean radiating directly to space as a blac k b o dy (red
curv e) or co oling b elo w a progressiv ely outgassed atmosphere (blue curv e). The
atmosphere is treated as a simple grey transmitter. F rom (Nik olaou et al., 2019)
18
xvi

LIST OF FIGURES
1.11
Radial distribution of the concen tration of a t ypical incompatible elemen t in an
Earth-lik e man tle. The partitioning b eha viour is go v erned b y a single partition
co efficien t (
K
) of v alue 0.5 (red curv e) or 0.1 (blue curve). Both curv es are
normalized to the same bulk v alue, taken in to consideration a 3D spherically
symmetrical geometry of the man tle. . . . . . . . . . . . . . . . . . . . . . . . . 19
1.12
Magma o cean crystallization inherited temp erature (a) and densit y (b) profiles
in the Martian man tle (blac k curves), and post mantle o v erturn densit y profile
(red curv es). V arious profiles are studied: an initially linear unstable profile
(solid curv es), the crystallization profile computed in (Elkins-T an ton, Zaranek,
P armen tier, & Hess, 2005b) (dashed curv es) and the same profile including
phase transition, from (Plesa, T osi, & Breuer, 2014) (mixed curv es). The general
in v ersion of the densit y gradient induced b y the man tle o v erturn is visible in
all case, ho w ev er in the t w o latter cases the o v erturn tak es place b elo w a stiff
stagnan t-lid, and the upp ermost unstable densit y stratification is conserv ed. . . 22
1.13
LLSVPs as retriev ed b y seismic tomograph y (Bull, McNamara, & Ritsema,
2009). The upp er panel represents the isosurface corresp onding to an S-w a v es
v elo cit y decrease of 0.6% and the lo w er panel reprsen ts the S-wa v es v elo cit y
distribution at 2750 km-depth. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1
Sc hematic diagram illustrating the fractional (a) and batc h (b) crystallization
scenarios. The solidification fron t lies at the in tersection b etw een the magma
o cean temp erature and the solidus in the fractional case, and the temp erature
corresp onding to the RCMF in the batc h case. Ov er a time step ∆
t
, the fully
solid man tle (fractional case) or the m ush region (batc h case) grow up w ards
b y ∆
r
= ∆
tD MO
(0)
/t MO
, where
D MO
(0) is the initial thic kness of the magma
o cean and t MO the prescrib ed solidification time of the magma o cean. . . . . . 28
2.2
Ev olution of the finite v olume grid b et w een t wo time steps. A t time
t
(left
panel), the radius of the solidification fron t (
R SF
) is suc h that all grid shells
ab o v e the
i th
shell (included) are deactiv ated (set as ghost cells, represen ted
in grey while activ e cells are represen ted in black), while at time
t
+
dt
(righ t
panel), the ascension of the solidification fron t results in the activ ation of the
cells of the i th s h e l l . ................................. 3 2
2.3
Ev olution of the comp osition field for a fractional crystallization scenario with
Ra
= 10
10
,
B
= 1
.
8 , and
t MO
= 1 Myr. The four panels sho w snapshots
and corresp onding laterally-a v eraged profiles of temp erature (red line) and
comp osition (blue line) at the b eginning of the sim ulation (a), and after 0.7
Myr (b), 1 Myr (c), and 10 Myr (d). The blac k dashed and dotted lines denote
the solidus temp erature and the p osition of the solidification fron t, resp ectiv ely . 35
xvii

LIST OF FIGURES
2.4
Time-series of the b ottom Nusselt n um b er for all combinations of parameters.
P anels a–i refer to fractional crystallization cases (
B >
0 ) and j–l to the batc h
crystallization cases (
B
= 0 ). All sim ulations are run for 11 Myr. The red
line in panel h (
t MO
= 1 Myr and
B
= 1
.
8 ) corresp onds to the mo del sho wn
in Figure 2.3. F or batc h mo dels, only sim ulations with
Ra ref
of 10
7
ha v e b een
p erformed. The dotted line marks the end of the crystallization for eac h case in
p a n e l s a – l . ....................................... 3 6
2.5
Semi-log plots of the time-series of the shrinking factor for all com binations of
parameters. Similar to Figure 2.4, the dotted line marks the end of the magma
o c e a n s o l i d i fi c a t i o n . .................................. 3 8
2.6
Histograms of the distribution of the shrinking factor after 11 Myr for all
sim ulations. The v ertical axis represen ts (in log scale) the p ercen tage of tracers
ha ving a shrinking factor around the v alue indicated on the x -axis. . . . . . . . 39
2.7
Snapshots of the comp osition field (a, d, g), integrated shrinking factor (b, e,
f ) and laterally a v eraged profiles of comp osition, temp erature, and in tegrated
shrinking factor (c, f, i) after 11 Myr for sim ulations assuming fractional
crystallization with
B
= 1
.
8 ,
Ra
= 10
10
, and differen t solidification durations
(
t MO
= 0
.
1 Myr in a–c,
t MO
= 1 Myr in d–f, and
t MO
= 10 Myr in g–i). The
in tegrated shrinking factor is a mapping of the final v alue of the shrinking factor
of each tracer, plotted at the initial p osition of the tracer. . . . . . . . . . . . . 41
2.8
A v erage shrinking factor after 11 Myr for all sim ulations as a function of
t MO
and Ra ref for the three differen t v alues of B The color-scale is linear. . . . . . . 42
3.1
Time-series of the temp erature of the magma o cean, solidifying b elo w a lid of
constan t thic kness (40 km, dashed curv es) or a gro wing lid (final thic kness 40
km, solid curv e). The n um b ers asso ciated with eac h curv e indicate the depth at
whic h plagio clase starts to crystallize (from (Elkins-T an ton, Burgess, & Yin,
2 0 1 1 ) ) ......................................... 5 1
3.2
Depth profiles of the crystallization temp eratures (a) and the internal heating
distributions (b) for an initially 1350-km-deep (i.e. whole-man tle, red curv es),
1000-km-deep (blue curv es), and 500-km-deep (green curv es) lunar magma
o cean. The dashed lines at the b ottom of the crystallization temp erature
profiles corresp ond to the accretion-like temperature profile assumed in the
unmolten pristine lo w er mantle. The crystallization temp erature profile used in
Elkins-T an ton, Burgess, and Yin (2011) is sho w in grey for comparison. . . . . 54
3.3
The thermal ev olution mo del consists in four coupled reserv oirs: the core, the
gro wing cum ulates, the shrinking magma o cean and the gro wing flotation crust. 55
3.4
T emp erature profile at the b eginning of the sim ulations. Note that the surface
temp erature extends b ey onds the left b order of the plot, do wn to 250 K. . . . . 61
xviii

LIST OF FIGURES
3.5
(a) Time ev olution of the heat fluxes and sources for a reference case, accoun ting
for heat-piping with
k crust
= 2 W/m/K,
η ref
= 10
21
P a s, and (b) corresp onding
ev olution of the temp erature (red) and magma o cean’s top and b ottom
b oundaries (blac k). (c) Time evolution of the heat fluxes and sources for
a siilar but purely diffusiv e case, i.e. with
k crust
= 2 W/m/K but neglecting
con v ection and heat-piping, and (d) corresp onding ev olution of the temp erature
(red) and magma o cean’s top and b ottom b oundaries (blac k). . . . . . . . . . . 63
3.6 Time-series of the depth of the magma o cean (solid curv es) and crust (dahsed
curv es) b ottom, for purely conductiv e cum ulates with
k crust
= 2 W/m/K. The
blac k curv es corresp ond to a case where laten t heat effect, in ternal heating and
conductiv e heating b y the cum ulates w ere accoun ted for in the energy budget
of the magma o cean, the blue curv es corresp ond to a similar case where the
conductiv e heat flux w as neglected, the red curv e to a case similar to the first
one where inernal heating w as neglected and the green curv es to a case similar
to the first one where the effect of laten t heat w as neglected. The v ertical
dotted lines mark the end of the magma o cean solidification for the cases of
c o r r e s p o n d i n g c o l o r . ................................. 6 4
3.7
Time-serie of the ratio of cum ulativ e mass of the extracted melt (
∫ t
0 dM HP
)
to instan taneous magma o cean mass (
M MO
(
t
) ), computed for the three cases
presen ted in Section 3.5.3. In order for the comparison b et ween the case to be
easier, the time is normalized (
t
= 0 corresp onds to the b eginning of the magma
o cean solidification and t = 1 t o i t s e n d ) . ...................... 6 4
3.8
Con v ergence tests made on 2D cylindrical grids, with a radial resolution of 132
shells (blue), 191 shells (green), 220 shells (red), and 249 shells (magen ta). . . . 65
3.9
Comparison b et w een a 3D sim ulation with 132 radial shells (red), a 2D sim ulation
with 249 radial shells with the actual lunar core radius (blue), and a 2D
sim ulation with 249 radial shells but a core radius rescaled according to (v an
Kek en, 2001) (green). As in Figure 3.8, panel a) represen ts the ev olution of the
b ottom of the crust and magma o cean, panel b) that of the conductiv e heat
flux at the top of the cum ulates, and panel c) that of the heat-piping flux. . . . 66
3.10
Time-series of the depths of the b ottom of the magma o cean (lo wer curv es)
and of the b ottom of the crust (upp er curv es) for v arious v alues of the thermal
c o n d u c t i v i t y o f t h e c r u s t . .............................. 6 7
3.11
(a) Snapshot of the temp erature field in the cum ulate after 100 Myr for our
fiducial case using
k crust
= 2 W/m/K and
η ref
= 10
21
P a s (blue curv e in
Figure 3.12). Con v ection in the cum ulates o ccurs while the magma o cean
(y ello w area) is still solidifying and the plagio clase crust (grey area) still gro wing.
(b) Corresp onding laterally-a v eraged temp erature profile (blue line). In hot
up w ellings, the temp erature exceeds the solidus (red dashed line in panel b),
causing partial melting (pale areas in panel a) that results in heat-piping. . . . 68
xix

LIST OF FIGURES
3.12
Time-series of the depths of the b ottom of the magma o cean (lo w er curv es) and
of the b ottom of the crust (upp er curv es) for v arious v alues of the reference
viscosit y in the solid cum ulates and
k crust
= 2 W/m/K (colored curv es) and
for the purely conductiv e case for the same v alue of
k crust
(blac k dashed curv e,
iden tical as that on Figure 3.10). . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.13
Time-series of the depth of the magma o cean and the crust (a), and of the
b ottom conductiv e (b) and heat-piping (c) fluxes for t w o cases using similar
parameters (
k crust
= 2 W/m/K and
η ref
= 10
19
P a s), one allo wing the b ottom of
the crust to remelt (this is the case presen ted on figure 3.12 for the corresp onding
reference viscosit y), and the other where re-heating of the magma o cean is not
asso ciated with re-melting of the crust . . . . . . . . . . . . . . . . . . . . . . . 70
3.14
Time-series of the b ottom of th e crust (upp er curves) and of the magma o cean
(lo w er curves) for the three differen t initial depths of the magma o cean presen ted
in section 3.2.1, when b oth the the conductiv e heat flux from the cum ulates and
in ternal heating in the magma o cean are tak en in to consideration (a), when only
the former is accoun ted for (b), when only the later (c), or when b oth effects
are discarded (d). Note that radioactive heating has no noticeable effect, and
conductiv e heating only a limited one on the ev olution of a initally 500 km-deep
magma o cean (blue curv es). This is due to the limited mantle fractionation
o c c u r i n g i n t h i s c a s e . ................................. 7 2
4.1
T race elemen ts signature of KREEP , normalized to c hondritic comp osition.
F rom (Shearer et al., 2006) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2
Radial profiles of the paren t-to-stable daugh ter sp ecies ratio ( [
P
]
/
[
D s
] , (a)) and
radiogenic-to-stable daugh ter sp ecies ratio ( [
D r
]
/
[
D s
] , (b)) in the monomineralic
cum ulates formed from the crystallization of a magma o cean (solid lines) and
the bulk v alue of this ratio in the whole man tle (dashed lines), at t wo differen t
p oin ts in time: righ t after the magma o cean solidification (blue) and after t w o
half-liv es of the system
τ 1 / 2
(red). Both the paren t and the daugh ter sp ecies
are incompatible as it is usually the case. The partition co efficien ts of the
paren t and the daugh ter sp ecies are resp ectiv ely:
K P
= 0
.
5 and
K D
= 0
.
2
(the daugh ter sp ecies is th us more incompatible than the paren t one). The
bulk initial v alue of [
P
]
/
[
D s
] is 1.1 (blue dashed line on panel a), and that
of [
D r
]
/
[
D s
] is 0.1 (blue line on panel b) and is not affected b y fractionation
(the solid and dashed blue lines are sup erp osed). After t w o half-liv es of the
paren t sp ecies, radioactiv e deca y has afftected b oth [
P
]
/
[
D s
] and [
D r
]
/
[
D s
] in
the cumulates, as shown b y the red curv es in b oth panels. . . . . . . . . . . . . 76
4.3
T ypical iso c hron plot: t w o differen t minerals forming a sample are represen ted
b y the square and the circle rep ectiv ely . A t the time of crystallization (
t crys
)
they align follo wing a horizon tal line corresp onding to the bulk sample v alue
of [
D r
]
/
[
D s
] at this time. As time passes [
P
]
/
[
D s
] decreases causing the circle
and square to drift left w ards, while [
D r
]
/
[
D s
] increases accordingly causing an
up w ard drift. Ov erall, the iso c hrons remain aligned following a straigh t line,
with slop e exp ( ( t crys − t ) ln(2) /τ 1 / 2 ) − 1 . ..................... 7 7
xx

LIST OF FIGURES
4.4
Time ev olution of
εD r
in the origin reserv oir (blac k dashed line) and t w o samples
(red and blue segmen ts). The presen t-da y v alues of [
P
]
/
[
D s
] and [
D r
]
/
[
D s
] ha v e
b een measured for the t w o samples, and their resp ectiv e crystallization times
(
t 1
and
t 2
) ha v e b een determined. This allo ws us to dra w b oth red and blue
line. The v alues of those lines at their resp ectiv e sample formation times (
t 1
for the red line and
t 2
for the blue one) define t w o distinct p oin ts, hence a new
line, whic h represen ts the isotopic evolution of the origin reserv oir. Finally , the
time at whic h the blac k line intercepts the
x
axis corresp onds to the time at
whic h the reserv oir had a v alue of [
D r
]
/
[
D s
] similar to that of the bulk man tle,
corresp onding to the isolation time of the reserv oir. . . . . . . . . . . . . . . . . 78
4.5
Isotopic deviation from the bulk silicate Mo on (considered c hondritic, CHUR =
c hondritic homogeneous uniform reserv oir) against time for a set of lunar samples
represen tativ e of KREEP isotopic signature. Note that contrary to Figure 4.4,
the deviation is negativ e for b oth Hf and Nd due to a higher incompatibilit y of
their parent sp ecies throughout the lunar man tle. . . . . . . . . . . . . . . . . . 79
4.6
Global mo del w orkflo w. Input parameters are indicated in green, in termediate
results in blue, and final results in red (the main results,
t 0
and
t KREEP
are
framed b y a dashed line). Note that
t 0
and the partition co efficien ts are iterated
to fit the paren t-to-daugh ter ratio of the KREEP inferred b y Gaffney and Borg
(2014). Chapter 3 is concerned with the upp er part of the workflo w, while the
presen t c hapter deals with the lo op in the lo w er half, the righ t b o x (”Coupled
fractionation and radioactiv e deca y mo del”) b eing describ ed in sections 4.3.1
and 4.3.2, and the ”Fitting to KREEP” one in section 4.4.1. . . . . . . . . . . . 83
4.7
(a)
ε 143 Nd
time-series for the b est matching cases after fitting to KREEP
data p oin ts. Eac h grey curv e corresp onds to one sim ulation, the ligh t part
corresp onding to the magma o cean ev olution prior to KREEP formation and the
dark part to the deca y line of the asso ciated KREEP reserv oir. The orange p oin ts
are the KREEP samples from Gaffney and Borg (2014) (G14) (circle for basalts
and diamonds for Mg-suite ro c ks) and the orange line is the KREEP mo del
line deriv ed b y the same study . (b) Similar to (a) for
ε 176 Hf
. (c) Histograms of
t 0
(red) and
t KREEP
(blue) for the same set of cases. The orange shaded area
corresp onds to the KREEP age span inferred from the t w o isotopic systems b y
Gaffney and Borg (2014) and the blac k dashed line to the age of F AN 60025
from Borg, Connelly, Bo y et, and Carlson (2011). The turquoise histograms
corresp ond to
t 0
and
t KREEP
for the cases presen ted whose cum ulates whole-ro c k
Lu-Hf systematics are also consisten t with the formation of Kal009. . . . . . . . 85
xxi

LIST OF FIGURES
4.8
(a) Radial distribution of the whole-ro ck
ε 176 Hf
in the solid cum ulates pile at
4.369 Ga (age estimated for Kal 009 (Snap e et al., 2018)) for all cases matc hing
KREEP isotopic ratios (red en v elop e). The initial
ε 176 Hf
v alue for Kal 009
(Sok ol et al., 2008) is highligh ted in blue (within tw o standard deviations). (b)
ε 176 Hf
ev olution of the p oten tial source reserv oir of Kal 009 for all cases selected
b y the KREEP fitting pro cess. The cases that cross the uncertain t y ellipsoid of
Kal 009 (blac k p oin t) are plotted in turquoise, while the cases that do not are
plotted in grey . (c)
t 0
distribution, as plotted in Figure 4.7, with the same color
c o d e a s f o r ( b ) . .................................... 8 6
4.9
Lu mineral/melt partition co efficien ts of cases presen ted in Figure 4.7. The
orange normal distributions corresp ond to the means and standard deviations
listed in table 4.2 are rep orted in orange. . . . . . . . . . . . . . . . . . . . . . 86
4.10 Same as Figure 4.9, but for Hf, the daugh ter sp ecies of the Lu-Hf system. . . . 87
4.11
As in Figure 4.7, but using time-series based on a 1000 km-deep magma o cean
with
η ref
= 10
21
P a s and
k crust
= 1.5 (left panels), and 4 W/m/K (middle
panels), and a whole-man tle magma o cean with the same reference viscosit y
and k crust = 2 W/m/K (righ t panels). . . . . . . . . . . . . . . . . . . . . . . . 88
4.12
(a) Results of the fit to the KREEP signature from zircon samples of T a ylor,
McKeegan, and Harrison (2009). The grey hatc hed area corresp onds to the
first 40 Myr after the CAI formation time, during whic h the Mo on forming
impact has b een sho wn to b e highly unlik ely (Jacobson et al., 2014). As in
Figure 4.7, the ligh t grey curv es corresp ond to the ev olution of
ε 176 Hf
in the
magma o cean prior to KREEP formation and the dark grey lines to the deca y
of KREEP . The thic k blac k line corresp onds to the ev olution of the CHUR
reserv oir (corresp onding to the bulk silicate Mo on comp osition) and the thin
blue line to T a ylor, McKeegan, and Harrison (2009)’s KREEP mo del. (b) Mo on
formation (red histograms) and KREEP isolation times (blue histograms) for
the cases presen ted in panel a. (c) Results of the fit to the KREEP signature
from Barb oni et al. (2017). Note that the mo del’s deca y line from (Barb oni
et al., 2017) (dotted blac k line) assumes a v alue of
176 Lu / 177 Hf
= 0 , resulting
in the steep est slop e p ossible. Fitting a straight line through the cloud of data
p oin ts pro vides the blue line, to whic h our mo dels are fitted (grey lines). (d)
Similar to (b), for the case fitted to Barb oni et al. (2017)’s data. . . . . . . . . 89
5.1
Lunar surface Ti con ten t map deriv ed from the Kaguy a mid-infrared data
(Otak e, Oh tak e, & Hirata, 2012). There is a clear correlation b et w een Ti con ten t
and the Pro celarum KREEP T errane on the nearside. Figure from Kato et al.
( 2 0 1 7 ) .......................................... 9 2
xxii

LIST OF FIGURES
5.2
Comparison b et w een the crystallization sequences from Sn yder, T a ylor, and
Neal (1992), Elkins-T an ton, Burgess, and Yin (2011), Charlier, Gro v e, Nam ur,
and Holtz (2018), Rapp and Drap er (2018), Lin, T ronc he, Steenstra, and v an
W estrenen (2016) and from that used in the presen t w ork. The minerals’
names are the follo wing: Ol=olivine, Px=p yroxene, Op x=orthop yro xene,
Cp x=clinop yro xene, Pl=plagio clase, Pig=pigeonite, A ug=augite, Sp=spinel,
Sph=sphene, Qz=quarz, Ilm=ilmenite and Sil=silimanite. Note that in our
mo del ilmenite app ears v ery late, but the earlier formed clinop yro xene is already
enric hed in Ti. The fraction of eac h mineral crystallizing out of eac h la y er is
prop ortional to the width of its color-co ded b o x. The y-axis is the precen tage
of v olume of magma o cean crystallized. Since the initial depth of the magma
o cean v aries from one mo del to the other (from 400 km for Sn yder, T a ylor, and
Neal (1992) to the whole man tle for Rapp and Drap er (2018)) and plagio clase
do es not settle at the b ottom but floats to form the crust, these diagram should
not b e in terpreted as the final man tle la y ering resulting from the magma o cean
c r y s t a l l z a t i o n . ..................................... 9 4
5.3
Densit y profiles in the upp ermost 200 kilometers of the Mo on considered b y
the differen t studies that sim ulated the ov erturn of IBC. Both dense (i.e. thin)
and dilluted (i.e. thic k) IBC la y er end-members are represen ted when relev an t,
the former with solid lines and the latter with dashed lines (Note that Zhao,
de V ries, v an den Berg, Jacobs, and v an W estrenen (2019) also ran a simulation
with no densit y anomaly in the IBC). The depth is indicated without accoun ting
for the o v erlying plagio clase crust. . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.4
Densit y (a) and heat pro duction (b) profiles in the upp er 140 kilometers of the
Mo on. The top 42 km corresp ond to the plagio clase crust. The red dashed lines
indicate the a v eraged v alues used in the implemen tation and listed in T able 5.1.
Due to the strong densit y oscillations, a v eraging o v er the IBC la y er is sensitive
to the a v eraging metho d. Both lo w and high IBC densit y end-mem b ers are th us
r e p r e s e n t e d . ...................................... 9 9
5.5
Time-series of the IBC o v erturn for an activ ation energy of 335 kJ/mol (a)
and of 150 kJ/mol (b),
η ref
= 10
19
P a s (red lines),
η ref
= 10
20
P a s (blue
lines),
η ref
= 10
21
P a s (green lines), with (solid lines) and without (dashed
lines) magma o cean solidification. The time series for cases without a magma
o cean are offset b y the time coresp onding to the duration of the magma ocean
crystallization for the same set-up. The step-like appearance of the curves in (b)
for
E ∗
= 150 kJ/mol and
η ref
= 10
20
and 10
21
P a s is due to resolution effects,
but do es not affect the final v alue of
ψ IBC
b ecause it only app ears when the IBC
in one grid shell completely o v erturns, but is damp ed when the mobilization is
less efficient (e.g. blue curv e after 200 Myr). . . . . . . . . . . . . . . . . . . . . 102
xxiii

LIST OF FIGURES
5.6
Snapshots of the IBC con ten t in the cumulates at four differen t times: initial state
(leftmost column), onset of IBC o v erturn (middle-left column), in termediate
state (middle-righ t column) and end of the magma o cean solidification (righ tmost
column), for three cases featuring
E ∗
= 150 kJ/mol:
η ref
= 10
21
P a s (top ro w),
η ref
= 10
20
P a s (cen ter ro w), and
η ref
= 10
19
P a s (b ottom column). The
magma o cean is represen ted in y ello w and the crust in grey . T wo close-ups
illustrate the small-scale mixing of IBC for the case
η ref
= 10
20
. Streamlines
are indicated in white on the close-ups. . . . . . . . . . . . . . . . . . . . . . . . 103
5.7
Similar to Figure 5.5, (a) for
η ref
= 10
20
P a s,
E ∗
= 335 kJ/mol and
ρ IBC
= 3775
kg/m
3
(green curv es),
E ∗
= 335 kJ/mol and
ρ IBC
= 3995 kg/m
3
(blue curv es)
and
E ∗
= 150 kJ/mol and
ρ IBC
= 3775 kg/m
3
(red curv es), and (b) for
η ref
= 10
21
P a s,
E ∗
= 335 kJ/mol, ∆
η
= 10 (red curv es) and ∆
η
= 100 (blue
curv es), and for
E ∗
= 150 kJ/mol, ∆
η
= 10 (green curv es) and ∆
η
= 100 (cy an
c u r v e s ) . ........................................ 1 0 5
5.8
Time-series of the magma o cean and a v erage cum ualtes temp eratures (a), of the
crust’s and magma o cean’s bottom depths (b), and of the v arious heat fluxes at
pla y (c) for the case: η ref = 10 19 P a s and E ∗ = 150 kJ/mol. . . . . . . . . . . . 106
5.9
Time-series of the magma o cean and a v erage cum ualtes temp eratures (a), of the
crust’s and magma o cean’s bottom depths (b), and of the v arious heat fluxes at
pla y (c) for the case: η ref = 10 21 P a s and E ∗ = 150 kJ/mol. . . . . . . . . . . . 107
5.10
Presen t-da y temp erature profile for b oth high and low viscosit y end-mem b er
cases (
η ref
= 10
21
P a s and
E ∗
= 335 kJ/mol, and
η ref
= 10
19
P a s and
E ∗
= 150
kJ/mol, resp ectiv ely). The blac k dashed line is the presen t-da y temp erature
profile from (Ziethe, Seiferlin, & Hiesinger, 2009) and the color shaded areas
corresp ond to estimates from (Zhang, P armen tier, & Liang, 2013) (orange,
diagonally hatc hed), (Laneuville, Wieczorek, Breuer, & T osi, 2013) (blue),
(Khan, McLennan, T a ylor, & Connolly, 2006) (green) and (Gagnepain-Beyneix,
Lognonné, Chenet, Lom bardi, & Sp ohn, 2006) (red, horizon tally hatc hed). . . . 108
5.11
IBC melting curv es (from W y att (1977)) plotted o v er the depth range
corresp onding to the m ultiple saturation p oin ts of mar e basalt (e.g. Elkins-
T an ton, v an Orman, Hager, & Gro v e, 2002), and p ost-magma ocean (solid
curv es) and presen t-day (dashed curv es) thermal states of the cases
η ref
= 10
19
P a s (red curv es),
η ref
= 10
20
P a s (blue curv es), and
η ref
= 10
21
P a s (green
curv es), all with E ∗ = 150 k J / m o l . ......................... 1 1 0
5.12
Thic kness of a surface la yer equiv alen t to the v olume of IBC melt pro duced o v er
the whole ev olution of the Mo on, for an activ ation energy of 335 kJ/mol (a)
a n d 1 5 0 k J / m o l ( b ) . .................................1 1 1
B.1
Discretization of the conserv ation equation o v er a control v olume
V i
(on a
2-D mesh). On a colo cated grid as is used in GAIA (see Section B.2.2), all
parameters (here
q i
and
s i
) are defined at the cen ter of eac h con trol v olume.
Sev eral sc hemes exist to compute the flux through the b oundary ( φ ). . . . . . . 139
xxiv

List of T ables
2.1
V alues of th quantities in use. Note that the reference Ra yleigh n um b er
corresp onds to the Ra yleigh n um b er computed for the whole man tle (see Section
2.3.1). Study parameters are indicated in red . . . . . . . . . . . . . . . . . . . 30
3.1
Non-dimensional initial con tribution to the total heat budget and deca y constan t
for eac h of the heat pro ducing sp ecies accoun ted for. The
λ i
are indicated in
unit Ga − 1 . ....................................... 5 7
3.2
Ph ysical quan tities used. V alues computed from our crystallization mo del are
highligh ted in blue and mo del parameters in red. . . . . . . . . . . . . . . . . . 61
4.1
Ages and
ε X
v alues of the KREEP samples used to select the fractionation
mo dels. The first three samples are KREEP basalts and the fiv e last are Mg-
suite ro c ks. The last line corresp onds to the basalt sample Kal 009 used to
refine the fit. References:
a
(Gaffney & Borg, 2014),
b
(Carlson & Lugmair, 1979),
c
(Shih, Nyquist, Bansal, & Wiesmann, 1992),
d
(Nyquist, Shih, Reese, & Irving,
2009),
e
(Borg, Gaffney, & Shearer, 2014),
f
(Snap e et al., 2018),
g
(Sok ol et al.,
2 0 0 8 ) .......................................... 7 9
4.2
P artition co efficien ts for th eminerals in the cum ulate sequence. References:
a
McDade, Blundy, and W o o d, 2003,
b
Nash and Crecraft., 1985,
c
Klemme,
Gun ther, Pro watk e, and Zac k, 2006,
d
Pro w atke and S., 2006,
e
Pro w atke and S.,
2 0 0 4 .......................................... 8 2
4.3
Mo on formation time (
t 0
) and KREEP isotlation time (
t KREEP
) spans for all
v alues of the reference viscosit y in v estigated, with our fiducial v alue of the
crustal thermal conductivit y:
k crust
= 2 W/m/K, our fiducial v alue of the
reference viscosit y ( 10 21 P a s) app earing in the second line f rom the top. . . . . 88
5.1 Initial depth extension and a v erage densit y of the three lay ers of the mo del. . . 98
B.1
Non-dimensionalization of the ph ysical parameters c haracterizing the man tle.
The non-dimensional quan tities in the left column are calculated with the
expressions in the righ t column, in v olving dimensional quan tities. . . . . . . . . 138
xxv

1
Intro duction
Lo oking for what made the Earth a fav ourable place for life to thriv e is the outstanding
motiv ation that is driving to da y’s planetary sciences. The ev olution of terrestrial planets,
whic h is the framew ork of the presen t work, is only the ”emerged part of the iceb erg” . It can
b e considered as the study of the pro ximate cause for habitabilit y , the ultimate causes for it
in v olving an ov erwhelming range of domains, from astroph ysics to biology , in terwinding with
formidable complexit y . The early stages of planets formation, in particular, are thought to be
decisiv e in understanding their habitabilit y . The scop e of this thesis is the early dynamical and
thermal ev olution of the man tle of terrestrial b o dies, fo cussing Mars and the Mo on. Sp ecifically ,
w e will sho w ho w the crystallization of a primordial magma o cean dramatically affects the
long-term man tle ev olution of these b o dies.
Of course, when it comes to studying sp ecific features of the early solar system, one can
only rely on partial and v ery indirect observ ations, leaving m uc h ro om for interpretation
and unkno wn. This is when mo dels en ter the pla y . Building up on decades of n umerical
breakthroughs, it is no w p ossible to sim ulate complex pro cesses that w ould otherwise remain
hop elessly obscured. Unfortunately , n umerical mo dels are still far from b eing able to faithfully
address differen t ph ysical problems at the same time, as it is the case when dealing with
suc h questions as planetary ev olution. This thesis will put in ligh t some of these limitations.
The art of the mo deller is hence to reac h the righ t compromises, to accept simplifications of
the non-crucial issues in order to b etter tac kle the heart of the problem – and to k eep these
concessions in mind, to question them again in the future.
In man y asp ects, magma o ceans em b o dy these difficulties. They no w app ear as natural
consequences of the pro cess of planets formation, as explained in the t w o first sections of this
in tro duction. Being long extinct, their existence remains in essence sp eculativ e, although
evidences ha v e b een stac king up since the theory started in the early 70’s with the analysis of
the first lunar samples from the Ap ollo missions. The b eginning of the third section pro vides a
brief review of these evidences. Finally , the extremes in v olved in the ph ysics of magma o ceans
(temp erature, pressure, length scale etc.) almost completely prohibit lab oratory exp erimen ts,
and mak e n umerical simulations incredibly c hallenging. The ”compromises” men tioned ab o v e,
for the study of magma o ceans, are in tro duced in the third section, whic h closes with a first
1

1. In tro duction
Figure 1.1:
Protoplaneatry disks observ ed b y the SPHERE instrumen t on ESO’s V ery Large
T elescop e. Some exhibit circular gap suggesting the orbits of already formed planets. A dapted
from h ttps://www.eso.org/public/images/eso1811a/
insigh t in what is the bulk of this w ork: the influence of magma o ceans on the long-term
ev olution of terrestrial b o dies.
1.1 The formation of the solar system
The study of the solar system formation faced, un til recen tly , t w o ma jor obstacles: the
imp ossibilit y to directly observ e long extinct pro cesses, and the uniqueness of the ob ject under
scrutin y , that prev en ts from using reliable comparisons or statistical considerations. Ph ysical
mo dels w ere hence privileged to infer the history of the solar system, assisted b y mineralogical,
p etrological and chemical analysis of the meteoritical record as w ell as a list of remarkable
features of the solar system lik e the dic hotomy betw een small terrestrial planets and gas ice
gian ts, the coplanarit y and lo w eccen tricity of the planets’ orbits or the v ariabilit y in their spins.
The dev elopmen t of p o w erful observ ation to ols and tec hniques, whic h enabled the observ ation
of the first protoplanetary disks (Figure 1.1), allo w ed for a substan tial leap forw ard, helping to
pro v e or dispro v e some of the established ideas.
1.1.1 The collapse of the protosolar nebula
Planets formation is a b y-pro duct of the formation of stars. Not all stars – p ossibly only few
of them – host a planetary system. Nev ertheless it has b een kno wn since the first observ ation
of an exoplanet (Ma y or & Queloz, 1995) that it is not a feature unique to the solar system.
2

1.1 The formation of the solar system
Understanding the ph ysics of stellar formation is hence an imp ortan t step to w ard a theory of
planetary formation.
The solar system formed ab out 4.57 billions of y ears ago (see section 1.1.3 for a discussion
ab out the meaning of this age), from the gra vitational collapse of a parcel of a gian t molecular
cloud (GMC), arguably isolated b y the sho c k w a ve of a supernov a. This collapse of a dense
core in this cloud (of densit y
∼
10
5
particles/
cm 3
, the a v erage cloud’s densit y ranging from
10
2
to 10
3
particles/
cm 3
(T a ylor, 2001)), resulted in the formation of an opaque, disk-shap ed
nebula (due to the conserv ation of angular momentum) composed of dust and gas, cradle of the
future planets, called the protoplanetry disk. Figure 1.1 sho ws six examples of protoplanetary
disks observ ations.
The infalling matter from the collapsing GMC parcel heated up the disk b y compressiv e
heating, while within the disk, mass flo w o ccured cen terw ard, feeding the gro wing protosun.
The gra vitational collapse of the solar nebula lasts ab out 10
5
y ears (T a ylor, 2001). This
duration is p o orly constrained b ecause early stellar accretion rates cannot b e observ ed for
y oung extrasolar systems due to the optical thic kness of the nebula at this stage. When
the mass of the protosun reac hed
∼
0
.
2 solar masses, deuterium fusion is ignited (Larson,
2003), resulting in the increased glowing and activit y . Strong p olar outflo w and radiation
pressure limited the further gro wth of the protosun, while equatorial stellar wind sw ept out
the remaining gas from the protoplanetary disk. When stars b ecome observ able, and un til
h ydrogen com bustion is ignited (i.e. en tering the main sequence of their liv es), they are said to
b e in the pre-main sequence, a phase estimated to last a few tens of milions of y ears (Kunitomo,
Guillot, Ida, & T ak euc hi, 2018).
Ho w long it tak es for a pre-main sequence star to clear up the gas from its protoplanetary
disk is v ery imp ortan t for planetary formation as explained b elo w, and remains p o orly
constrained. Observ ations from Haisc h, Lada, and Lada (2001) suggest an a verage v alue of 3
Myr. The initial mass of the nebula pla ys a decisiv e role, in particular regarding the early
accretion of gian t planets. Y et, it also remains p o orly constrained, b ecause the ratio of escaping
to accreted matter throughout nebula collapse is unkno wn.
1.1.2 Building up the planets
The formation of the planets starts once the protoplanetary disk has formed and the densit y of
gas and dust (settling in the mid-plane) is high enough. A strong constrain t on the planetary
formation scenario is placed b y the observ ed dichotom y b et ween gaseous/icy planets and
terrestrial ones: the former formed at a time and place in the nebula where gas was still
presen t, while the later formed in a gas-depleted en vironmen t. The main gas-depleting ev en t is
asso ciated with the in tense stellar activit y during the pre-main sequence phase. Th us Saturn
and esp ecially Jupiter (assumed to form first in order to b e able to accrete more gas) m ust
ha v e formed b efore the pre-main sequence activit y of the protosun cleared the nebular gas,
while all terrestrial planets m ust ha v e formed later.
T w o differen t mec hanism ha ve been prop osed for the gro wth of planets from nebular dust
particles and gas: the gra vitational instabilit y , prop osing that planets form essen tially the same
w a y the protosun formed, i.e. b y gra vitational collapse of nebular material, and solid accretion,
in whic h planets gro w by collisional accretion of the surrounding nebular material. The former
3

1. In tro duction
mec hanism yields an app ealingly short formation time scale for gas gian t (P aardek o op er &
Johansen, 2018). It fails ho w ev er to predict the observ ed v ariet y in planets spins and obliquities.
It is th us unlik ely to b e the exclusiv e mec hanism in v olv ed in planetary formation of the solar
system, although it might b e a go od candidate for the formation of large exoplanets with
large orbits compared to those found in the solar system (T a ylor, 2001; Nimmo, Kretke,
Ida, Matsum ura, & Kleine, 2018). Solid accretion is hence the fa v oured mec hanism to form
terrestrial planets in the inner solar system. A h ybrid scenario has also b een proposed for the
gas gian ts, where their solid cores form through solid accretion un til reaching about 10 Earth
masses, when they b ecome massive enough to trigger a gra vitational instabilit y and accrete a
gaseous en v elop e (T a ylor, 2001; Nimmo et al., 2018).
Core accretion has pro v ed a realistic scenario capable of forming the gas gian ts on short
time scales, but it is still not clear wh y they could form faster than smaller inner terrestrial
planets. One solution is that, if the gas giants formed beyond the ”sno w line” (the distance
from the Sun at whic h v olatiles are stable in solid state, corresp onding to 2-3 astronomical units
(A U) for w ater and 20 A U for carb on mono xide), m uch more solid matter w as a v ailable for
accretion since v olatiles w ere under ice form (Lammer & Blanc, 2018; Nimmo et al., 2018). The
early stages of solar system and planetary formation are presen ted in Figure 1.2. A detailed
gro wth path wa y for terrestrial planets (and p oten tially gas gian ts’ cores) ev olution, represen ted
in Figure 1.3, go es through distinct phases corresp onding to increasing sizes and c haracterized
b y differen t accretion and orbital dynamics regimes. Nimmo et al. (2018) distinguishes four
stages:
•
Dust particles (
∼
1
µ
m), already presen t in the primordial molecular cloud, are efficiently
coupled to the surrounding nebular gas and stic k together up on colliding with one
another.
•
P ebbles (
∼
1 mm-10 cm) start to decouple from the gas, inducing a drag b ecause the
nebular gas is rotating at a sub-Keplerian v elo cit y . Collisional stic king b ecomes inefficien t
(Blum, 2018).
•
Planetesimals (
∼
10-100 km) are completely decoupled from nebular gas. They efficiently
accrete remaining p ebbles due to their high gravitational cross section, enhanced b y
the gas drag exp erienced by the pebbles (a pro cess termed ”p ebble accretion”). They
undergo a ”runa w a y gro wth” phase whic h leads to the increasingly fast gro wth of a few
of them.
•
Planetary em bry os (>1000 km) are big enough to clear the disk around their orbit, when
gas is still remaining. They gro w via accretion of surrounding planetesimals during
the ”oligarc hic gro wth” phase. The absence of surrounding gas allows for increased
eccen tricit y resu lting in crossing orbits, causing the gian t impacts phase.
As of to day , no consensus has b een met on a mec hanism resp onsible for gro wth b et w een
the cen timeter and the kilometer scale (Nimmo et al., 2018). This problem is referred to as
the ”meter-sized barrier” . Sev eral h yp otheses ha ve been prop osed, lik e the so-called streaming
instabilit y (Y oudin & Go odman, 2005; Johansen et al., 2007) where ”clumps” of p ebbles form
in the disk’s mid-plane via turbulen t sh uffling of matter and trigger gra vitational collapse,
forming directly planetesimals from p ebble-sized ob jects.
4

1.1 The formation of the solar system
Figure 1.2:
Sk etc h of the
successiv e stages of a sp ec-
ulativ e scenario for solar
system formation. a) A
dense core forms in a gian t
molecular cloud’s parcel,
previously isolated p ossi-
bly b y the sho c k w a v e of a
neigh b oring sup erno v a. b)
The gra vitational collapse
of this parcel results, due
to the conserv ation of an-
gular momen tum, in the
formation of a flattened
disk with con tin uously in-
falling material hitting a
compression sho c k fron t
(transparen t turquoise en-
v elop). Dust settles in
the mid-plane of the disk
and gas (turquoise clouds)
is compressed to w ards the
cen ter, forming the proto-
star. c) The pre-main se-
quence proto-star sw eeps
out nebular gas while plan-
etesimals are forming and,
b ey ond the sno wline, giant
cores already accreted and
cleared their surroundings
of gas and dust. d) The
proto-star en ters the main
sequence b y igniting h ydro-
gene burning. Gas gian ts
migration has cleared the
dust and scattered plan-
etesimals b ey ond
∼
1
.
5
A U, lea ving a limited reser-
v oir for the prolonged oli-
garc hic gro wth of terres-
trial planets.
5

1. In tro duction
10 -6 10 -3 10 0 10 3 10 6
Dust
P ebbles
Planetesima ls
Planetar y embryos
Collision stickin g
R una way gr owth
Oligar ch ic gr owth
meter -sized
bar rier
size [m]

Figure 1.3:
Gro wth path w a y of the planets. The arro ws represen t inferred gro wth mec hanism,
the streaming instabilit y b eing one candidate to o v ercome the meter-sized barrier. The p ebble
accretion and gian t impacts roughly corresp ond to the runa w a y and oligarc hic gro wth phases,
resp ectiv ely .
The rapid formation of gas gian ts also has an influence on the comparativ ely slo w er
forming terrestrial planets. Dep ending on whether a massiv e b o dy has efficien tly accreted
the surrounding nebular gas and th us cleared its o wn orbit, it can exp erience tw o t yp es of
alteration of its semi-ma jor axis, termed ”migrations” (W ard, 1997). Because of the mass of
the migrating planet, all smaller b o dies whose orbit gets crossed are scattered, causing b oth a
depletion of the terrestrial planets building blo c ks and an enhannced c hance of collision of
the already formed planetary em bry os b y the scattered planetesimals. S uc h migrations of the
gas gian t w ou ld ha v e strongly mo dified the protoplanetary disk, at least o v er the radius span
through whic h they o ccurred, and probably also further.
The so-called ”Nice mo del” introduced the idea that giant planet migrations shap ed the early
solar system, in particular allo wing for the existence of Neptune and trans-neptunian ob jects
orbits (T siganis, Gomes, Morbidelli, & Levison, 2005), for the presence of the T ro jan asteroids
(Morbidelli, Levison, T siganis, & Gomes, 2005), and pro viding a dynamical explanation
for a late surge in small b o dies impacting on terrestrial planets, the so-called ”late heavy
b om bardmen t” (see Section 1.1.3) (Gomes, T siganis, Levison, & Morbidelli, 2005). In a scenario
called the ”gran tac k” prop osed b y W alsh, Morbidelli, Ra ymond, O’Brien, and Mandell (2011),
Jupiter migrated in w ards un til reac hing a semi-ma jor axis of 1.5 A U, when it was caugh t up
b y Saturn’s in w ard migration. The t w o planets entered a mean-motion resonance that caused
them to migrate bac kw ards to their presen t-day orbits. This scenario was sho wn to explain
the small size of Mars b y clearing up its accretionary reserv oir, as w ell as the con ten t of the
asteroid b elt in carb on-ric h (C type) asteroids (only S type asteroids b eing exp ected to form
at the orbit of the asteroid b elt).
6

1.1 The formation of the solar system
Although new findings question the v alidit y of the link b et w een gas gian ts migrations and
the late hea vy b om bardmen t as in tro duced in the Nice mo del (Morbidelli et al., 2018; Zh u
et al., 2019), it remains a p oten tially influen tial mec hanism to shap e the protoplanetary disk.
1.1.3 Chronology
Isotop e age dating metho ds pro vide the most reliable wa y to deriv e an absolute age for
planetary materials. The refractory Calcium- Aluminium-ric h inclusions (CAIs) con tained in
c hondrites (the most pristine meteorites) constitute the oldest dated solar system material.
Their a v erage Pb-Pb age pro vides what is usually considered as the age of the solar system:
4568
.
2 Ma (Bouvier & W adh w a, 2010). Chondrules, another class of inclusion, which represen ts
the main comp onen t of the c hondrites (the most pristine class of meteorites), is also estimated
to ha v e formed within a few milion years after this date (Kruijer, Kleine, & Borg, 2020). This
material m ust ha v e formed within the dense and hot conditions found in the protoplanetary
disk and th us p ostdates the initial nebular collapse. Using CAIs age as an anc hor p oin t, w e
can attempt a sp eculative c hronology for the ev olution of the solar nebula (Figure 1.4).
0 5 10 50 100
time [Myr ]
150
time [Ga]
4.5182 4.4682 4.4182
Sun
Nebula Planets
pr e-ma in sequen ce main sequ ence
nebular
gas
CAIs
chondr ules
Mars
Jupiter
Giant impa cts
nebular collapse
Isotopes
26 Al- 26 Mg
182 Hf - 182 W
146 Sm- 142 Nd
60 F e- 60 Ni
4.5582
4.5632 4.5682
Eart h
Moon

Figure 1.4:
Solar system formation c hronology . While for CAIs and c hondrules as w ell as for
Earth and Mars, the time span represen t isotopic dating estimates (Pb-Pb age estimates for the
former (Bouvier & W adh w a, 2010; Kruijer, Kleine, & Borg, 2020) and Hf-W age of core formation
for the latter (Kleine & W alker, 2017)), the timespan indicated for Jupiter’s formation accoun ts for
the assumption that it formed b efore clearing of the nebular gas. The ”isotop es” time spans at the
b ottom corresp ond to the system’s half-life, ho wev er the system is still active (i.e. still pro duces
energy or can still b e used for age dating) during ab out 7 half-liv es. Isotop e systems whose deca y
con tributes to the early heating of planets are represen ted in red while systems only used for age
dating are represen ted in grey . The blac k axis indicates time after CAIs formation in Myr while
the red axis indicates time b efore present, in Gyr (Ga).
7

1. In tro duction
Mo dels of gra vitational collapse and accretion pro vide rough time scales for the nebular
and protosun ev olution, as w ell as for planetary gro wth, as sho wn in Figure 1.4. The initial
gro wth of the sun w as lik ely fairly rapid, leading to deuterium com bustion ignition within
ab out 10
5
y ears of the onset of the GMC collapse (T a ylor, 2001). The intense stellar activit y
asso ciated with the pre-main sequence stage m ust ha v e cleared the nebular gas in
∼
3 Myr
(Haisc h et al., 2001), placing a strong constrain t on the planetary gro wth c hronology (see
Section 1.1.2). The protosun ev en tually settled in the main sequence within 10 Myr.
The timing of planetary em bry os accretion is less constrained, esp ecially due to the p o or
understanding of the meter-size barrier. Ho w ev er, as men tioned ab o v e, correlation with nebular
ev olution helps infer a c hronology . On the one hand, the gas giants m ust ha v e formed in a
gas-ric h nebular en vironmen t, i.e. within
∼
3 Myr (Haisc h et al., 2001). On the other hand,
terrestrial planets, lac king a primordial gas en velope, formed in a gas-depleted environmen t,
i.e. after ∼ 3 Myr.
The last phase of terrestrial planets’ gro wth, the so-called oligarc hic gro wth phase, w as
lik ely protracted and spanned 10 to 100 Myr b ecause it w as driv en b y collisions b et w een large
planetary em bry os whose crossings among the emp ying disk b ecame more and more scarce.
The age of the core-man tle differen tiation (for which an absolute age can be d eriv ed based on
isotop e dating metho ds) can b e used as a pro xy for the end of planetary formation. Indeed,
ma jor core formation even ts are asso ciated with large-scale impacts that c harcterize the end of
the oligarc hic gro wth phase. The Hf-W isotopic system is used to infer the age of core-man tle
differen tiation. Com bined with a coupled accretion and core differen tiation mo del, Kleine and
W alk er (2017) found a core formation time of
∼
10 Myr after CAIs formation for Mars, and of
∼ 34 Myr for the Earth (see Section 1.2.1.2).
The end of the oligarc hic gro wth of planetary embry os, the gian t impact phase, w as
c haracterized b y an in tense b om bardmen t of the y oung planets. Although the Earth has long
lost an y trace of this episo de due to the con tin uous recycling of its surface b y plate tectonics,
less activ e terrestrial b o dies still bear the scars of this temp etuous episo de. The ev en t that
created the abundan t crater record on the Mo on, Mars and Mercury is called the late hea vy
b om bardmen t (LHB). The disco v ery that most Ap ollo lunar samples could b e dated b et w een
3.95 and 3.75 Ga Bottk e and Norman (2017) suggested that, rather than a smo oth, con tinuous
exhaustion of the remaining planetesimals (a scenario called ”accretion tail”), the LHB o ccured
o v er this reduced span of 200 Myr, dela y ed b y ab out 500 Myr from the formation of the solar
system (a scenario that w as called the ”terminal cataclysm”). Ho w ev er, although the differen t
Ap ollo landing sites corresp ond to as man y different impact basins, it is not clear whether the
ro c k samples ultimately originate from those basins. Sev eral landing sites could ha v e b een
co v ered b y ejecta blank ets asso ciated with the formation of another basin, th us questioning
the exhaustiv eness of our lunar sample record and hence the significance of the measured age
regarding the LHB.
Another age dating metho d used to constrain the LHB is the crater counting method.
F rom statistics on the density of craters as w ell as their stratigraphic p osition, a relative
c hronology of planetary surfaces can b e deriv ed. Impactors flux time-series ha v e b een deriv ed
from the crater record of the Mo on (Neukum, Iv ano v, & Hartmann, 2001). Ho w ev er, the
deriv ed c hronologies provide only relativ e times, and man y studies ha v e tried to anchor them to
8

1.2 T errestrial b o dies differen tiation
an absolute time p oint, with div erging results (Kamata et al., 2015; Conrad, Nimmo, F assett,
& Kamata, 2018; Morbidelli, Marc hi, Bottk e, & Kring, 2012b; Morbidelli et al., 2018; Zh u
et al., 2019). The gian t planet migration w as inferred to trigger the terminal cataclysm long
after the end of planetary formation, ho w ever this scenario requires some fine-tuning of the
mo dels, and some recen t studies suggest that the simpler ”accretion tail” scenario in whic h the
impactor flux decreased constan tly through time migh t also b e compatible with observ ations
(Morbidelli et al., 2012b; Morbidelli et al., 2018).
1.2 T errestrial b o dies differen tiation
Mass acquisition, b e it through solid accretion or gra vitational collapse, is only one asp ect
of planetary formation. The re-organization of the inner parts of a planet (mostly driv en b y
p oten tial gra vitational energy decrease), from a largely homogeneous configuration (exp ected
in the case of accretion gro wth), resulting in the formation of distinct reserv oirs, is the other
main asp ect of planetary formation. This pro cess is referred to as ”differentiation” . The most
comp elling evidence of the heterogeneity of terrestrial planets’ in teriors is pro vided b y the
v alue of their momen t of inertia factors, i.e. the normalized p olar momen t of inertia
I / M R 2
,
where
I
is the p olar moment of inertia,
M
the mass of the planet and
R
its radius, whic h
measures their inner mass distribution. The momen t of inertia factor of a homogeneous sphere
is 0.4, a lo w er v alue indicating an in w ard concen tration of mass, i.e. a gra vitationally stable
configuration. Mercury’s v alue of the momen t of inertia factor is 0
.
346
±
0
.
0014 (Margot
et al., 2012), V en us’ v alue is
∼
0
.
33 (R ubie et al., 2015a), the Earth’s v alue is 0
.
3307
±
0
.
0
(Williams, 1994), and Mars’ v alue is 0
.
3662
±
0
.
0017 (F olkner, 1997). All terrestrial planets
hence significan tly deviate from a homogeneous sphere. Suc h a configuration is ac hiev ed b y
separation of an iron-ric h, dense core from a ligh ter silicate man tle (core-man tle differentiation).
Differen tiation o ccurs further within the silicate man tle (silicate differen tiation), via differen t
mec hanisms, and is probably initiated b y an episo de of large-scale melting, referred to as
a ”magma o cean” . The study of the influence of suc h magma o ceans and esp ecially their
solidifications is the aim of this w ork. This section briefly presents other asp ects of planetary
differen tiation and the follo wing one in tro duces the concept of magma o cean.
1.2.1 Core-man tle differen tiation
1.2.1.1 Mec hanism of core-man tle differen tiation
The differen tiation of a terrestrial b o dy b et w een an iron core and a silicate man tle is usually
considered to mark the end of its formation. Differen tiation o ccurs on all bo dies massiv e
enough for their o wn gra vit y to induce buo y ancy forces that o v ercome friction resistance and
ac hiev e separation b et w een elemen ts of different densities, namely iron and silicates. The
critical size for a ro c ky b ody at which differen tiation b ecomes effectiv e is
∼
10 km (T a ylor,
2001). It is th us lik ely that large-sized planetesimals and planetary em bry os inv olv ed in the
late stages of accretion w ere differen tiated.
Iron-silicate separation in v olv es v arious pro cesses that are far from b eing p erfectly
understo o d, and are v ery difficult to repro duce in lab oratory due to the extreme conditions of
pressure and temp erature as well as large length- and time scales at pla y . Size is not the only
9

1. In tro duction
decisiv e factor con trolling differen tiation. T emp erature, whic h directly affects the ph ysical
state of matter, is a crucial parameter. Indeed, differen tiation is largely facilitated in a liquid
b o dy . A core-man tle differen tiation o ccurring via solid-state creep w ould last more than 100
Myr (Stev enson, 1990) and is th us considered inefficien t to pro duce the core on the times cales
inferred (see Section 1.2.1.2). Therefore iron, at least, whic h has a lo w er melting temp erature
than silicates, is assumed to ha v e b een liquid during differen tiation. R ubie, Nimmo, and
Melosh (2015b) distinguishes b etw een three main differen tiation mec hanisms, dep ending to
the ph ysical state of silicates and the initial distribution of (liquid) iron:
•
F or solid silicates, differen tiation can o ccur via p ercolation of iron through silicate grains
b oundaries. The efficiency of this pro cess is con trolled b y the dyhedral angle in the v eins,
and is lik ely inefficien t at lo w pressures (<25 GPa), but could b e enhanced b y shear
stress in the silicates.
•
F or solid silicates also, sinking of large-scale (1-10 km) iron diapirs, or iron diking through
brittle crac ks are efficien t mec hanisms, largely help ed b y the high viscosit y con trast
b et w een liquid iron and solid silicates.
•
Liquid silicates are most lik ely undergoing (turbulen t) con v ection. Iron-silicate separation
is then con trolled b y the comp etition b etw een en trainmen t and settling of iron drops,
whic h is in turn a function of the in tensity of con v ection and the c haracteristic size of
iron drops. It is lik ely the most efficien t mec hanism for iron-silicate differen tiation.
The initial thermal state of a planet is con trolled b y three main heat sources: 1) accretional
heating (whic h turns kinetic energy in to heat), 2) decay of energetic, short-liv ed radion uclides
(
26 Al
and
60 F e
), and 3) gra vitational p oten tial energy reduction resulting from differen tiation.
The first pro vides massiv e heating in the case of impacts of large b o dies and when the target
and impactor ha v e similar sizes, a configuration most lik ely to b e met to w ards the end of
planetary formation (during oligarc hic gro wth of planetary embry os). Ho w ev er, how w ell the
heat is distributed throughout the b o dy is not w ell constrained, and it could b e largely lo calized.
The second is indep enden t of the b ody’s size and is entirely con trolled b y the timing of the
planet’s formation. It is initially largely dominated b y
26 Al
, whose short half-life (
∼
730000
y ears) allo ws efficien t heating only during the first few million y ears, thereb y pro viding a
relev an t mechanism for planetesimals differen tiation. Finally , the conv ersion of gra vitational
p oten tial energy in to heat can induce a p ositiv e feedbac k, where temp erature increases as
core-differen tiation pro ceeds, whic h could lead to runa w ay differen tiation, and ev en tually to
large-scale melting of the silicate man tle, i.e. a magma o cean. The relativ e imp ortance of
heating resulting from these three mec hanisms is represen ted in Figure 1.5 as a function of the
c haracteristic size of the b o dy .
1.2.1.2 Dating the core-man tle differen tiation
Core-man tle differen tiation represents the most ancien t planetary-scale ev en t that can b e dated
on terrestrial b o dies, thereb y pro viding v aluable constrain ts on the timing of planets formation
pro cess. The c hronometry pro vided b y the short-liv ed Hf-W isotopic system is the most widely
used metho d to estimate the age of core-mantle differen tiation.
182 Hf
is unstable and deca ys in
10

1.2 T errestrial b o dies differen tiation
E a rt h
Ma rs
Mo o n
0 Myr
1 Myr
2 Myr
3 Myr
4 Myr

Figure 1.5:
Heating induced b y radion uclides (grey dashed lines) at v arious times (after CAIs
formation), accretion (solid blac k lines) for v arious impactor-to-target mass ratios (
γ
), and core-
man tle differen tiation (red dotted lines) for differen t terrestrial b o dies. The x axis represen ts the
b o dy’s mass in unit Earth’s mass (
M e
) and the y axis the resulting temp erature increase. In
the y ello w area, the b o dy’s mantle is heated b y more than 1000 K, sufficien tly to cause a global
magma o cean. The heating pro vided b y core-man tle differentiation does not only dep ends on
the b o dy’s size but also its in ternal structure, therefore it is not represen ted as a function of the
x
-axis, butgiv en for three differen t cases. F urthermore, it is computed considering an instan taneous
differen tiation ev en t, thereb y significan tly o v erestimating the resulting heating. A dapted from
R ubie, Nimmo, and Melosh (2015b).
182
W with a half-life of 8.9 Myr (Kleine & W alk er, 2017). F urthermore, W is siderophile while
Hf is lithophile, meaning that the former partitions preferen tially in to iron and the latter in to
silicates during the formation of the core. Th us the core-man tle differen tiation systematically
results in an increase of the Hf/W ratio in the silicate man tle compared to the bulk planetary
comp osition. If this o ccurs while the system is still activ e (i.e. within a few half-liv es from the
CAI formation time), some
182 Hf
en ters the man tle and deca ys in to
182
W , thereb y raising the
relativ e quan tit y of this isotop e compared to other non-radiogenic W isotop es (
184
W b eing
usually tak en as the reference). If core-man tle differen tiation p ostdates the extinction of the
Hf-W system, then the ratio
182
W
/ 184
W in the silicate man tle is not affected and remains
similar to that of the bulk comp osition. Figure 1.6 illustrates these t w o p ossibilities.
Computing the age of the core-man tle differen tiation based on Hf–W systematics in v olv es
kno wing (i.e. in general making assumptions on) the conditions of the differen tation pro cess.
The simplest mo dels are the so-called one-stage mo dels, where a single, instan taneous
differen tiation ev en t is assumed. Based on the measured isotop es systematics, it is then
p ossible to infer the date of this ev en t. Assumptions also ha v e to b e made on the partitioning
of Hf and W b etw een iron and silicates, usually considered as a simple c hemical equilibrium.
In particular, the reaction constan t of this equilibrium (i.e. the partition co efficien t of Hf or W
b et w een iron and silicates) dep ends on the pressure at whic h equilibration is ac hiev ed, that
is, practically , the depth of the magma o cean. Using suc h mo dels, Kleine and W alk er (2017)
calculated a core-man tle differen tiation age of 4
.
1
±
2
.
7 Myr after CAI formation for Mars,
and 34 ± 3 Myr after CAI formation for the Earth.
11

1. In tro duction
Early di ff er entiation:
( 182 W/ 184 W) mantle =6/2
( 182 W/ 184 W) bulk =8/8
Late di ff er entiation:
( 182 W/ 184 W) mantle =2/2
( 182 W/ 184 W) bulk =8/8
time
t~0 (C AI) t~7 τ 1/2
Undi ff er entiated
Mantle
Cor e
182 Hf
182 W
184 W

Figure 1.6:
Sk etc h of the mec hanism of
182 Hf
–
182
W dating system of the core-man tle
differen tiation. F or an early episo de of core-mantle differen tiation (top line), the separation b et w een
Hf (squares) going preferen tially in to the silicate man tle, and W (circles) going preferen tially into
the iron core o ccurs while some unstable Hf still exists, resulting in increase of radiogenic W in
the man tle, hence a high presen t-da y
182
W
/ 184
W v alue. If the core-man tle differen tiation o ccurs
after all unstable Hf has deca y ed (b ottom line), no deviation of the
182
W
/ 184
W ratio from the
bulk comp osition o ccurs.
Ho w ev er, as also stressed b y Kleine and W alker (2017), assuming a single, instantaneous
core-forming ev en t is likely too simplistic to grasp the complexit y of core-man tle differen tiation,
esp ecially in the case of short-liv ed isotopic systems whose half-life ma y b e shorter than
geoph ysical pro cesses at pla y during the early stages of a planet. The c hronologies deriv ed from
these mo dels only pro vide an upp er bound for the age of planets’ core-mantle differen tiation.
Using a more sophisticated mo del where successiv e gian t impacts act as as man y core-forming
ev en ts with asso ciated equilibration conditions (impacts b eing tak en from exp onen tially
decreasing impactor flux time-series computed b y accretion mo dels), Dauphas and P ourmand
(2011) prop osed that Mars accreted >90% of its mass within 6 Myr after the CAI formation
(the differen tiation ev en ts asso ciated with eac h impact b eing then considered instan taneous).
Similarly , Jacobsen (2005) estimated that the Earth accreted >95% of its mass in 30 Myr.
Ho w ever, the stochastic nature of gian t impacts mak es the in terpretation of the Hf–W system
a highly non-unique problem. In particular, the most dramatic impact ev en t that o ccured
on the Earth is assumed to ha v e resulted in the formation of b oth the Mo on and the end of
the core formation, t w o even ts whic h are indep endently estimated to date past 100 Myr after
CAIs formation.
F urthermore, the lik ely differen tiation of early-formed planetesimals due to heating of
26 Al
suggests that iron-silicate equilibration w as far from complete during planetary core formation
ev en ts. Indeed, a homogeneous distribution of small-scale iron drops (from an undifferen tiated
impactor) equilibrates m uc h more efficien tly than one fully-formed core, esp ecially when
deliv ered in to a liquid magma o cean where it sinks o v er a v ery short time. This suggests
that treating Hf and W partitioning as a simple equilibrium at the b ottom of the magma
o cean migh t b e to o simple an assumption, casting further doubts on the accuracy of these
mo del ages. In spite of these ca v eats, the Hf-W has b een extensiv ely used to estimate the
c hronology of planetary formation. It is also in tro duced here as an illustration of the use of
isotopic c hronometers.
12

1.3 Magma o ceans
1.2.2 Silicate differen tiation
Once the silicate and iron reserv oirs are isolated from eac h other, they k eep on ev olving. The
high temp erature at the core-mantle boundary (CMB) (the core b eing heated b y in ternal heat
sources as w ell as the gra vitational p oten tial energy release asso ciated with its formation)
induces con v ection in the man tle. This enhances the heat-flux at the CMB and in turn, the
co oling of the core, un til the formation of a solid inner-core. F urthermore, man tle conv ection
induces pressure-release melting, resulting in v olcanism. V olcanic activit y extracts basaltic
material on to the surface, progressiv ely forming a crust. But b efore all these pro cesses of silicate
differen tiation tak e place, core-man tle differen tiation is lik ely to result in large-scale melting of
the silicate reserv oir. Th us the long-term ev olution of the man tle starts with the solidification
of the magma o cean. The ph ysical pro cesses in v olv ed in magma o cean solidification, the time
scales at pla y as w ell as the c hemical comp osition of the bulk silicate part of the b o dy are
decisiv e parameters regarding its long-term thermal and dynamical ev olution.
1.3 Magma o ceans
The magma o cean concept go es b ey ond the framew ork of planetary formation. In to da y’s
solar system, the Jo vian mo on Io is though t to host an in ternal magma o cean, supp orted b y
heating induced b y in tense tidal dissipation (Khurana et al., 2011). Out of the solar system,
exoplanets ha v e b een observ ed, lik e 55 Cancri E (Demory et al., 2016), whose close-in orbit
and gra vitationally lo c k ed spin-orbit configuration are lik ely to cause a p erp etual magma
o cean on the substellar hemisphere. Induction heating has also b een h yp othesized to result in
magma o ceans in the exoplanetary system TRAPPIST- 1 (Kisly ak o v a et al., 2017). If these
examples offer the p ossibility to study extan t magma o ceans, those in v olv ed in the formation of
terrestrial b o dies in our solar system are long extinct, and only left scarce traces. In particular,
plate tectonics on the Earth has erased an y hin t of a past magma o cean at the surface. The
absence of samples from Mercury and V en us prev en ts from p erforming c hemical analyses
that w ould pro vide the most v aluable indications for the existence of an ancien t magma
o cean. Nev ertheless, the isotopic v ariet y in Martian meteorites hints at an extensiv e silicate
differen tiation that could ha ve been caused by the crystallization of a primordial magma o cean
(Jargoutz, 1991). Y et it is the Mo on that sho ws the most comp elling traces of a past magma
o cean.
1.3.1 Chemical evidences of magma o ceans: the case of the Mo on and Mars
If mo dels of core-man tle differen tiation strongly suggest massiv e melting of the silicate man tle
early on, the magma o cean concept w as really recognized after the Ap ollo missions from the late
60’s and early 70’s brough t bac k lunar ro c k samples to the Earth. Mineralogical and isotopic
analysis of these samples rev ealed a large v ariet y in their nature, suggesting that some of them
formed through the crystallization of a deep magma o cean (W arren, 1985). In particular, the
lunar crust w as found to b e comp osed for a large part of anorthosite, a plagio clase feldspar
(an Al- and Ca-ric h mineral) whose presence at the surface of the Mo on is b est explained
b y buo y an t separation of crystals from a molten silicate la y er at least sev eral h undreds of
kilometers thic k (W o o d, Dic k ey, Marvin, & P o w ell, 1970; Smith et al., 1970). Another class of
13

1. In tro duction
material, particularly enric hed in p otassium (K), rare-Earth elemen ts (REE) and phosphorus
(P), w as iden tified and called ”KREEP” . These elemen ts are termed incompatible b ecause they
do not en ter ro c k forming minerals’ crystals and preferen tially remain in the liquid during
solidification. Th us the most plausible mec hanism to create suc h an enric hed reserv oir is the
progressiv e solidification of a large part of the lunar man tle, KREEP represen ting its final
cum ulates (Carlson & Lugmair, 1988; Shih, Nyquist, Bansal, & Wiesmann, 1992; Edm unson,
Borg, Nyquist, & Asmerom, 2009; Gaffney & Borg, 2014; Borg, Gaffney, & Shearer, 2014).
The Martian meteoritic sample record has also b een found to sho w comp elling traces of
silicate differen tiation, in terpreted as resulting from the crystallization of a magma o cean.
Martian meteorites fall roughly in to three main groups: the Shergottites, the Nakhlites and
the Chassignites, collectiv ely referred to as SNC meteorites. The large geo c hemical v ariabilit y
among SNC meteorites is though t to b e due to a large-scale silicate differen tiation ev en t,
i.e. the crystallization of a magma o cean, rather than to a primordial heterogeneit y of Mars’
building blo c ks (Jargoutz, 1991). Using the short-lived isotopic systems
182 Hf
-
182
W (the
differen t degree of incompatibilit y of Hf and W mak es it usable to study silicate differentiation
as w ell as core-man tle differentiation) and
146 Sm
-
142 Nd
(half-life 103 Myr), F oley et al. (2005)
sho w ed th at the Shergottites and Nakhlites formed from t w o distinct reserv oirs, that b ecame
isolated at 4525
+19
− 21
millions of y ears ago (Ma) and more than 4542 Ma, resp ectiv ely . Using
similar tec hniques, but a more elab orate formation mo del, Debaille, Brandon, Yin, and
Jacobsen (2007) prop osed a protracted differen tiation of the Martian man tle (o v er
∼
100 Myr).
Relying on b oth long- and short-lived Sm-Nd systems, th us limiting the mo del’s dep endence
on v arious assumptions, Borg, Brennecka, and Symes (2016) found a later formation age for
the Shergottites source reserv oir, of 4504
±
6 Ma. Since suc h a late crystallization of the
Martian magma o cean is not consisten t with thermal ev olution mo dels (Elkins-T an ton, 2008),
they suggested that Shergottites isotop e systematics were reset b y a late gian t impact. Using
the same cross-systematics as F oley et al. (2005), Kruijer and Kleine (2017) prop osed that
the magma o cean crystallization and subsequent crust formation o ccurred b etw een 20 and
40 Myr after CAIs formation (i.e b etw een 4547 and 4527 Ma). Finally , studying
176 Lu
-
176 Hf
systematics in crustal zircons, Bouvier et al. (2018) concluded that the Martian magma o cean
crystallization m ust ha v e b een completed within 20 Myr after CAIs formation (i.e. b efore
4547 Ma). Although these study disagree on the solidification time scale, they do agree on the
necessit y of a large-scale melting ev en t to create distinct reserv oirs suc h as those from whic h
the differen t groups of Martian meteorites originate.
1.3.2 The ph ysics of magma o cean crystallization
The ph ysics go v erning the dynamics of liquid magmas as w ell as the c hemistry in v olved in
their crystallization are b ey ond the scop e of this thesis, and they shall here b e considered
using a parametrized approac h. This section justifies this approac h and offers an insigh t in to
the v arious pro cesses at pla y and the assumptions underlying our mo del.
1.3.2.1 Melt fraction and melting curv es
”Magma o cean” is a lo osely defined term, which can designate a m ush y , partially molten
la y er as w ell as a completely fluid region whose viscosit y can b e as lo w as that of liquid
14

1.3 Magma o ceans
1400 1600 1800 2000 2200 2400 2600
T e m p e r a t u r e [ K ]
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
P r e s s u r e [ G P a ]
solidus
liquidus
R C M F = 0 . 4
T
p o t = 2200 K
T
p o t = 2000 K
T
p o t = 1800 K
0
200
400
600
800
1000
1200
1400
1600
D e p t h [ k m ]

Figure 1.7:
Melting curv es (blue for solidus and green for liquidus) for fertile KLB-1 p eridotite
plotted on the pressure range relev an t for the Martian man tle. Magma o cean’s adiabatic temperature
profiles for three differen t p oten tial temp eratures (light green for 2200 K, orange for 2000 K and
red for 1800 K) are plotted do wn to the R CMF, where the liquid-lik e b eha viour, i.e. the magma
o cean stops. F rom Herzb erg, Raterron, and Zhang (2000).
w ater under ro om conditions. An essen tial parameter to prop erly describ e a magma o cean
is th us its melt fraction, i.e. the v olumetric ratio of melt to solid. Under the assumption of
c hemical equilibrium, mo dels exist (e.g. Gibbs free energy minimizers), based on lab oratory
melting exp erimen ts, that compute the melt fraction as a function of mineralogy , temp erature,
pressure, o xydation state and p ossibly further parameters. In a first order approac h ho w ev er,
the complexit y of melting and freezing pro cesses is usually reduced to considering a set of
t w o melting curves: the solidus, at whic h the ro c k is en tirely solid (melt fraction is zero), and
the liquidus at whic h the ro c k is completely liquid (melt fraction is one). T ypical solidus and
liquidus for the Martian man tle are represen ted on a pressure-temp erature diagram in Figure
1.7. The melt fraction (noted
ϕ
) is often assumed to gro w linearly with temp erature b et w een
the solidus and the liquidus:
ϕ = T − T sol
T liq − T sol
, (1.1)
where
T
is the lo cal temp erature and
T sol
and
T liq
are the solidus and liquidus temp e rature,
resp ectiv ely .
In order for the thermo-dynamical state of the magma o cean to b e con trolled b y a turbulen t
con v ection regime, the crystal lattice has to b e disrupted o v er large length scales, whic h
happ ens at a certain threshold v alue of the melt fraction, referred to as the rheologically critical
melt fraction (R CMF) (Costa, Caricc hi, & Bagdassaro v, 2009). The v alue of the R CMF has
b een exp erimen tally measured for div erse mineralogies, and v aries widely b et w een 0.1 and 0.6
(K ohlstedt, Bai, W ang, & Mei, 2000; Costa et al., 2009). In Figure 1.7, the v alue
ϕ R CMF
= 0
.
4 ,
often used in magma o cean studies (Solomatov, 2007) is plotted as the blac k dotted line.
1.3.2.2 A diabatic temp erature distribution and b ottom-up crystallization
In a turbulen tly con v ecting magma o cean, matter is efficien tly mixed on large length scales
(Solomato v, 2007), so that temp erature is homogenized, and only dep ends on lithostatic
pressure. This results in an adiabatic-lik e temp erature distribution, c haracterized b y the
15

1. In tro duction
p oten tial temp erature, i.e. the temp erature that the ro c k parcel w ould ha ve if it w ere
adiabatically brough t to the surface of the magma o cean. Note that the p oten tial temp erature
differs from the true surface temp erature of the magma o cean due to thermal b oundary
la y er effects that are not grasp ed b y the adiabatic temp erature parametrization of the bulk
con v ecting magma o cean. If represen ted on the same plot as the melting curv es (see Figure
1.7), the depth gradien t of the adiabatic profiles app ears lo w er than that of the melting curv es.
This means that up on co oling from a fully molten state, the temp erature en ters the partially
molten domain first at the b ottom of the mantle. As a result, the crystals form first at depth.
If they settle at the b ottom of the magma o cean (if they are denser than the surrounding
melts) rather than remaining en trained in the turbulen t flow, bottom-up crystallization of the
man tle is induced.
1.3.2.3 Crystal settling or en trainmen t and crystallization scenario
When the temp erature reac hes the liquidus, crystals start to n ucleate. Whether those crystals
settle or remain en trained dep ends mainly on the con v ectiv e v elo cities, on the crystals sizes and
on the densit y difference b et w een crystals and melt. While the former can b e appro ximated
b y scaling la ws based on the the depth of the magma o cean and the ph ysical prop erties of the
magma, the latter remain p o orly constrained, esp ecially b ecause the n ucleation and crystal
gro wth pro cesses under magma o cean-lik e conditions are v ery difficult to study . Based on
lab oratory exp erimen ts, Solomato v (2007) suggested that, dep ending on the turbulen t regime,
the threshold size for crystal en trainmen t v aries b et w een 1 mm and 10 cm. Ho w ev er, b onding
within crystals migh t efficien tly prev en t en trainmen t, and is not captured b y suc h exp erimen ts.
If the crystals efficien tly settle, then the forming solid la y ers b ecome progressiv ely c hemically
isolated from the magma o cean, which is in equilibrium only with the top of the cum ulates
pile. The compaction of the cum ulates pile (i.e. the extraction of its interstitial melt) has b een
sho wn to pla y an imp ortan t role in setting the c hemistry of the man tle (Hier-Ma jumder &
Hirsc hmann, 2017; Miy azaki & K orenaga, 2019). If the crystals are en trained b y con v ectiv e
flo ws, the temp erature con tin ues to decrease un til the rheological transition preven ts further
con v ection to tak e place. The resulting mineralogy of the solid man tle is lik ely to differ
significan tly b et w een those t w o cases since in the first case, successiv e differen tiated la y ers
are formed (a pro cess referred to as ”fractional crystallization”), while in the second case, all
crystals formed un til a giv en lay er reac hes the R CMF are sampled in this la y er (a pro cess
referred to as ”equilibrium” or ”batc h” crystallization). Figures 1.8 and 1.9 illustrate these
t w o crystallization scenarios.
1.3.2.4 Heat flo w and solidification time scale
In a magma o cean, the heat-flux is dominated b y turbulen t con vection. T urbulen t heat-
fluxes are parametrized in a similar fashion as turbulen t v elo cities, essen tially based on the
temp erature con trast in the thermal b oundary la yer at the surface and the viscosit y of the
magma o cean. Because the viscosit y of molten silicates can b e very small (do wn to
∼
0
.
003
P a s (R ubie et al., 2015b)), the turbulent heat-flux reac hes v ery high v alues. Solomato v
(2007) giv es an estimate of the turbulen t heat-flux out of a magma o cean of ∼ 10 6 W/ m 2 . If
the surface of the magma o cean is free, then the turbulent heat-flux in the magma ocean is
16

1.3 Magma o ceans
Figure 1.8:
F ractional crystallization of a b ottom-up crystallizing melt reserv oir. The
crystallization sequence corresp onds to the evolution of the mineralogy of the crystals that
form as the magma o cean solidifies (i.e. as the temp erature, pressure and comp osition conditions
ev olv e). Each crystallized la yer corresponds to the crystallization sequence. Because the bulk of
the cum ulates pile and the magma o cean are not in equilibrium, their resp ective melting curv es
differ. Therefore the crystallization temp erature, i.e. the temp erature retained b y eac h new solid
cum ulates la y er, is more relev ant.
equilibrated with the radiativ e heat-flux at its surface, whic h also reac hes very high v alues,
resulting in a rapid crystallization of the magma o cean.
Ho w ever, sev eral pro cesses can induce a heat-flux bottleneck at the surface of the magma
o cean. The gro wth of a secondary atmosphere is one: while terrestrial planets either nev er
accreted a primordial H/He atmosphere (if they formed after the nebular disk w as sw ept a wa y)
or lost it, a secondary atmosphere consisting of v olatile elemen ts (mostly w ater and carb on
dio xide) can form during the solidification of a large scale magma o cean (Ab e, 1997). V olatile
elemen ts are usually incompatible and accum ulate in the shrinking magma o cean during its
crystallization. The c hemical equilibrium at the surface of the magma o cean ensures a balance
b et w een the v olatiles’ con tent in the melt and the their pressure just abov e the surface, so that
magma o cean solidification is asso ciated with v olatiles outgassing that pro duces a secondary
atmopshere. When this atmosphere gro ws thic k enough, the radiativ e heat-flux at the surface
is hindered, and the magma o cean solidfication can b e dramatically slo w ed do wn. Sim ulating
coupled magma o cean crystallization and secondary atmosphere outgassing, Elkins-T an ton
(2008), Hamano, Ab e, and Genda (2013), Lebrun et al. (2013), Hier-Ma jumder and Hirsc hmann
(2017), Salv ador et al. (2017), Nikolaou et al . (2019) found solidification time scales reac hing
sev eral milions of y ears. Figure 1.10 shows the difference betw een the solidification of an
Earth-lik e magma o cean with and without outgassing of a secondary atmosphere. This effect
17

1. In tro duction
Figure 1.9:
Equilibrium (batc h) crystallization of a b ottom-up crystallizing melt reserv oir.
Note the co existence of differen t comp ositions (colors) in the entrained crystals, and the m uc h
less significan t c hemical la y ering that results compared with Figure 1.8 (for pure equilibrium
crystallization, no c hemical la y ering w ould b e exp ected).
can ha v e a strong influence on the crystallization scenario, a protracted solidification facilitating
crystal settlemen t and th us fractional crystallization.
1 0 0 1 0 1 1 0 2 1 0 3 1 0 4 1 0 5
time [Myr]
2000
2500
3000
3500
4000
temperature [K]
black body
grey atmosphere

Figure 1.10:
Time series of the ev olution of the p oten tial temp erature of an Earth-lik e full-man tle
magma o cean radiating directly to space as a blac k b o dy (red curv e) or cooling b elo w a progressiv ely
outgassed atmosphere (blue curv e). The atmosphere is treated as a simple grey transmitter. F rom
(Nik olaou et al., 2019)
18

1.3 Magma o ceans
0 5 10 15 20
composition [non-dim]
1.0
1.2
1.4
1.6
1.8
2.0
radius [non-dim]
K
= 0 . 1
K
= 0 . 5

Figure 1.11:
Radial distribution of the concen tration of a t ypical incompatible elemen t in an
Earth-lik e man tle. The partitioning b eha viour is go verned b y a single partition co efficien t (
K
) of
v alue 0.5 (red curv e) or 0.1 (blue curv e). Both curves are normalized to the same bulk v alue, tak en
in to consideration a 3D spherically symmetrical geometry of the man tle.
If no atmosphere is presen t and the surface is in con tact with cold space, it is lik ely
that a thin la y er of magma will quench, as observ ed in free surface melt p onds on some
Earth’s v olcano es. Ho w ever, b ecause of its mobilit y , it is unlik ely that suc h a quenc hed
crust significan tly prolongs the duration of the magma o cean solidification (P erera, Jac kson,
Elkins-T an ton, & Asphaug, 2018). In the case of the Mo o n, as mentioned in Section 1.3.1,
a solid flotation crust forms and gro ws up to a thic kness of 45 km (Wieczorek et al., 2013),
forming an insulating lid through whic h heat has to b e conducted in order for the magma
o cean to further co ol do wn. This significan tly prolongs the lifetime of the lunar magma o cean,
as dev elop ed in Chapter 3.
1.3.2.5 F ractionation
As previously men tioned, some c hemical elements lik e v olatiles are incompatible, whic h means
that, up on solidification, they tend to remain preferen tially in the liquid phase. There are
sev eral reasons for that. T race elemen ts ma y not fit in crystal lattice sites, either b ecause
they ha v e to o big an ionic radius or b ecause they ha v e to o high a cathionic c harge. As
in tro duced in Section 1.2.1.2 for Hf and W isotop es, the incompatible b eha viour of elemen ts
is usually parametrized b y a partition co efficien t. The distribution of incompatible elemen ts
during the fractional crystallization of a magma o ce an is illustrated in Figure 1.11 for a simple
mo del where eac h newly crystallized la y er is in equilibrium with the remaining magma o cean,
follo wing the equation:
[ X ] solid = K [ X ] liquid , (1.2)
where [
X
]
solid
is the concen tration of sp ecies
X
in the solid,
K
the partition co efficien t and
[
X
]
liquid
the concen tration of
X
in the liquid. This pro cess is in general referred to as the
fractionation pro cess, and is an essen tial mec hanism related to magma o cean solidification, in
creating c hemical heterogeneites in the man tle.
19

1. In tro duction
1.3.2.6 Magma o ceans on large terrestrial b o dies
F or large terrestrial b o dies, lik e the Earth, the wide pressure range in the man tle ma y add
complexit y to this mo del. In particular, ab o v e a giv en pressure (
∼
75 GP a in the terrestrial
man tle (Boukaré, Ricard, & Fiquet, 2015)), molten silicates are denser than the surrounding
ro c k, thus prev en ting crystal settling, or even resulting in a top-do wn crystallization. Y et
Miy azaki and K orenaga (2019) claims that melt cannot b e denser than its corresp onding
solid solely due to a higher compressibilit y , but that it can o ccur when iron is preferen tially
partitioned in to it. A c hange in the ev olution of the crystallization can also happen if the
crystallization temp erature’s gradien t b ecomes less steep than the adiabatic temperature
gradien t. This w as suggested to b e the case for Earth’s lo w er man tle-lik e pressures on the base
of first-principle sim ulations (de K ok er, Karki, & Stixrude, 2013), resulting in a middle-out
crystallization of a solid cum ulates sandwic hed b et w een a surface and a basal magma o cean.
Ho w ev er no exp erimen tal measurement ha v e b een able to reproduce these results so far. It
seems nev ertheless p ossible that large terrestrial b o dies migh t host deep magma o ceans, buried
at the b ottom of the mantle. Such basal oceans would b e thermally insulated and could b e
sustained o v er billions of y ears (Labrosse, Hernlund, & Coltice, 2007; Laneuville, Hernlund,
Labrosse, & Gutten b erg, 2018).
1.3.3 Influence of magma o cean crystallization on planetary ev olution
1.3.3.1 Man tle o v erturn
Once the magma o cean reac hes the rheological transition, it conserv es its crystallizing
temp erature un til heat diffusion or solid-state con v ection b ecome effectiv e. Assuming that
the c haracteristic time scale for either of these pro cesses is m uc h longer than that for magma
o cean solidification (whic h is in general a righ t assumption if no particular insulating effect is
considered), the whole man tle retains its crystallization temp erature, whic h increases with
depth (see the blac k dashed line on Figure 1.7). The p ositiv e thermal expansivit y of silicate
ro c ks th us induces a decrease of densit y with increasing depth. If the temp erature gradien t
is higher than the adiabatic gradien t (whic h is generally the case for the crystallization
temp erature gradien t), the increase in densit y with depth cannot b e comp ensated b y lithostatic
compression and a gra vitational instabilit y app ears (the Ra yleigh-Bénard instabilit y). This
instabilit y triggers solid-state con vection in the man tle, a pro cess referred to as the thermal
o v erturn of the man tle.
Moreo v er, the incompatible b eha viour of iron results in its progressiv e enric hmen t in the
magma o cean up on solidification, and in turn, in the late crystallized cum ulates (i.e. the
shallo w ones, under the assumption of b ottom-up crystallization). Since iron-ric h cum ulates
are denser, this effect also induces a negativ e densit y gradien t with increasing depth, which
causes a Ra yleigh-T a ylor instabilit y configuration, a pro cess referred to as the comp ositional
o v erturn of the man tle.
The fractionational crystallization of the magma o cean generally leads to a configuration
whic h is prone to b oth thermal and comp ositional o v erturn. The resulting onset of man tle
con v ection is usually simply referred to as the man tle o v erturn. While the stable configuration
ac hiev ed after a thermal ov erturn can in general b e easily destabilized b y b ottom (due to secular
20

1.3 Magma o ceans
core co oling) or internal (due to radioactiv e deca y of heat-pro ducing elements) heating of the
man tle, it is not necessarily the case for the comp ositional o v erturn. In fact, the comp etition
b et w een the gra vitationally stable configuration follo wing a comp ositional o v erturn and the
engine for thermal con v ection (mentioned abov e) can lead to a broad v ariet y of con v ectiv e
patterns as sho wn b y T osi, Plesa, and Breuer (2013). Dep ending mainly on the densit y con trast
inherited from fractional crystallization of the magma o cean, but also on the viscosit y of the
man tle and its rate of in ternal heating as w ell as the yield stress that man tle material can
supp ort, v arious regimes app ear relev an t for the differen t terrestrial b o dies, as describ ed b elo w.
1.3.3.2 Mars
Elkins-T an ton, Zaranek, P armen tier, and Hess (2005b) computed the crystallization sequence
of a b ottom-up solidifying Martian magma o cean that underw ent fractional crystallization, and
deduced the p ost-solidification temp erature and densit y profiles in the man tle (Figure 1.12).
Using this set-up as the initial condition for man tle con v ection simulations, they computed
the ev olution of the Martian man tle ov erturn o v er a time range of 150 Myr. They found it to
b e in agreemen t with the onset of an early dynamo on Mars (Mittelholz, Johnson, F einberg,
Langlais, & Phillips, 2020) p o w ered b y the heat-flux at the core-man tle b oundary , enhanced
during the man tle o verturn. The extraction of secondary melts formed in the hot rising
plumes during the o v erturn is also compatible with the formation of a 50 km-thic k basaltic
crust. Finally the stable p ost- o v erturn configuration reached inhibits further con v ectiv e mixing
and th us allo ws for the long-term preserv ation of the isolated, isotopically distinct reserv oirs
from whic h the SNC meteorites lik ely originate (see Section 1.3.1). Ho w ever, running similar
sim ulations with a more realistic rheology , Plesa, T osi, and Breuer (2014) show ed that, in
absence of an external ev en t that would w eak en the primordial crust, the ov erturn o ccurs
b elo w a stagnan t lid where the densest (iron-ric h) material as w ell as the heat-pro ducing
elemen ts remain trapp ed. This stagnan t lid is not mobilized b ecause of its stiffness, caused b y
the strong temp erature dep endence of the viscosit y . In their sim ulations, con v ection and th us
v olcanism halts within less than one billion y ears b ecause of the depletion in in ternal heat
pro duction, whic h is strongly at o dds with evidences for recen t (
∼
100 Ma) v olcanism on Mars
(Neukum et al., 2001) as w ell as with the y oung eruption age of Shergottites (
∼
180 Ma). The
implied existence of a dense crust, trapp ed in the stiff stagnan t lid is also problematic when
compared with the lo w v alues inferred from gra vit y measuremen ts (P auer & Breuer, 2009).
Ho w ev er, Plesa et al. (2014) also sho wed that if some surface y ielding mec hanism (e.g.
large scale impact) w ould allo w the outermost, densest cum ulates to o v erturn and settle
at the core-man tle b oundary , the gravitationally stable stratification thus ac hiev ed w ould
also prev en t long-term thermal con v ection. Ev en if the o v erturned cumulates are enric hed
in heat-pro ducing elemen ts (whose enric hmen t correlate with enric hment in dense material
follo wing their similarly incompatible b eha viour), this la y er could reac h large degree of partial
melting, but not b e destabilized and rise again. While this w ould explain the preserv ation
of isotopically distinct man tle reserv oirs, it is hardly reconcilable with evidences for recen t
v olcanism on Mars. The interpla y b et w een the solidification of the magma o cean and the
long-term man tle dynamics of Mars is the sub ject of Chapter 2, where this p oint will be
discussed more in-depth.
21

1. In tro duction
Figure 1.12:
Magma o cean crystallization inherited temp erature (a) and densit y (b) profiles in
the Martian man tle (blac k curv es), and post mantle o verturn densit y profile (red curv es). V arious
profiles are studied: an initially linear unstable profile (solid curves), the crystallization profile
computed in (Elkins-T an ton, Zaranek, P armentier, & Hess, 2005b) (dashed curv es) and the same
profile including phase transition, from (Plesa, T osi, & Breuer, 2014) (mixed curv es). The general
in v ersion of the densit y gradien t induced by the man tle ov erturn is visible in all case, how ev er in
the t w o latter cases the o verturn tak es place b elo w a stiff stagnant-lid, and the upp ermost unstable
densit y stratification is conserv ed.
1.3.3.3 The Earth
As for Mars, the terrestrial man tle is though t to b e heterogeneous, in particular on the accoun t
of seismic w a ves propagation measuremen ts that suggest the existence of domains of anomalous
seismic v elo cities. Such anomalies can originate from temperature and/or comp osition as w ell
as partial melting. The most prominen t seismic features of the lo w er man tle are t w o large, lo w
shear v elo cit y pro vinces (LLSVPs) lo cated one b eneath Africa and one beneath the Pacific
(Hernlund & Houser, 2007; Garnero & McNamara, 2008) (see Figure 1.13). These pro vinces are
asso ciated with a degree-2 geoid feature suggesting that they are denser than the surrounding
man tle (Ishii & T romp, 1999). They are theorized to consist in thermo c hemical anomalies
b ecause a purely thermal one would fail to pro duce the observ ed an ti-correlation b et w een
the S- and P-w a v es v elo cities. Their origin has b een sough t in the sub duction of o ceanic
crust but also in the long-term preserv ation of anomalies inherited from a large degree of
fractionation in the man tle asso ciated with an episo de of magma o cean solidification (Labrosse
et al., 2007; Ballmer, Lourenço, Hirose, Caracas, & Nom ura, 2017; Laneuville et al., 2018).
Another imp ortant feature, although of more mo dest scale (
∼
10 km thic k), are ultra lo w
v elo cit y zones (UL VZs), where b oth P- and S-wa v es’ v elo cities are more significan tly reduced,
b y up to 10 to 30 % resp ectiv ely (Thorne & Garnero, 2004). Such a decrease in seismic w a v es
v elo cit y could b e due to chemical anomalies induced b y equilibration with the core, but also
to some degree of partial melting. As suc h, they could represen t the remnan ts of a long-liv ed
basal magma o cean (Labrosse et al., 2007; Laneuville et al., 2018) (see Section 1.3.2.6).
Heterogeneities in the terrestrial man tle are also manifested b y isotopic v ariations b et w een
the differen t sorts of terrestrial magmas, namely mid-o cean rige basalts (MORBs), pro duced
at the div erging o ceanic ridges b y decompression melting of upp er-man tle material, and o cean
island basalts (OIBs), pro duced by in traplate (hotsp ot) v olcanism, due to partial melting in
plumes whose origin in the man tle is still unresolv ed (Zindler & Hart, 1986; Hofmann, 1997).
MORBs, though t to b e represen tativ e of the upp er man tle, exhibit a trace-elemen t depleted
22

1.3 Magma o ceans
Figure 1.13:
LLSVPs as retriev ed b y seismic tomograph y (Bull, McNamara, & Ritsema, 2009).
The upp er panel represen ts the isosurface corresp onding to an S-w a v es v elo city decrease of 0.6%
and the lo w er panel reprsen ts the S-w av es velocity distribution at 2750 km-depth.
comp osition compared to the bulk silicate Earth (assumed c hondritic). This observ ation led
Bo y et and Carlson (2006) to propose that the terrestrial mantle separated early on into an
early depleted reserv oir (EDR) and an early enric hed reserv oir (EER), the former evolving
in to the presen t-day upper man tle. In this scenario, the EER remains largely unsampled, and
could b e sequestered in the low er man tle. OIBs exhibit a more enric hed comp osition, and
they could originate, at least partially , f rom the EER (Coltice, Moreira, Hernlund, & Labrosse,
2011; P eters, Carlson, Da y, & Horan, 2018). The source of man tle plumes that pro duce OIBs
is still sp eculative, and some of them ha ve been attributed to UL VZs (Y uan & Romao vicz,
2017) or to LLSVPs (Ballmer, Sc h umac her, Lekic, Thomas, & Ito, 2016). The latter, if they
are stable piles of dense material, are a lik ely candidate for the EER.
Although not the unique candidate, the crystallization of a magma o cean offers an app ealing
explanation for b oth seismic and geo c hemical man tle features. As p oin ted out in Section
1.3.2.6, the solidification of the terrestrial magma o cean might not ha v e o ccurred from the
b ottom-up, as on Mars. In fact, t w o scenarios seem plausible: 1) in the case of a b ottom-up
crystallization follo w ed by man tle o v erturn, the dense o v erturned material migh t ha v e gathered
to form piles at the b ottom of the mantle (Ballmer et al., 2017; Bro wn, Elkins-T an ton, &
W alk er, 2014), observed today as the LLSVPs. 2) If the magma o cean solidified from the
middle-out, then a long-liv ed basal magma o cean lik ely surviv ed for billions of y ears (Labrosse
et al., 2007; Laneuville et al., 2018), and the UL VZs migh t b e remnan ts of it.
23

1. In tro duction
1.3.3.4 The Mo on
Although the Mo on is muc h smaller than Mars and the Earth, the solidification of its
magma o cean lik ely induced a gra vitationally unstable configuration as w ell. In particular,
crystallization mo dels predict the existence of a dense lay er enric hed in the Ti-o xide ilmenite
among the late cum ulates, referred to as the ilmenite-b earing cum ulates (IBC). Hess and
P armen tier (1995) prop osed a lunar man tle o ver turn triggered b y the Ra yleigh-T a ylor instabilit y
due to the sinking of the IBC. In this mo del, a dense lay er of IBC settles at the core-man tle
b oundary . Because it is also enric hed in heat-pro ducing elements, this la y er heats up and,
con trary to Mars where the exp ected densit y con trast is prohibitiv ely high, migh t b ecome
unstable and form rising plumes. This mec hanism has b een prop osed to explain differen t
features of the lunar history , like the existence of an early core dynamo (Stegman, Jellinek,
Zatman, Baumgardner, & Richards, 2003), recorded in natural remnan t magnetization of lunar
samples (W eiss & Tik o o, 2014), or the hemispherical asymmetry in the mare basalts (Zhong,
P armen tier, & Zub er, 2000). Ho w ever, all these mo dels strongly dep end on a succession of
estimations, based largely on scaling la ws and p o orly constrained parameters. More realistic
dynamical studies of the lunar IBC o v erturn hav e b een led b y Y u et al. (2019), Zhao, de V ries,
v an den Berg, Jacobs, and v an W estrenen (2019) and Li et al. (2019), and this problematic is
discussed in depth in Chapter 5.
24

2
Ea rly onset of convection and mantle
overturn: the case of Ma rs
This c hapter is largely based on the published pap er: Maurice, M., T osi, N., Samuel, H., Plesa,
A.-C., Hüttig, C. and Breuer, D. (2017). Onset of solid-state mantle con v ection and mixing
during magma o cean solidification. Journal of Ge ophysic al R ese ar ch: Planets , 122
2.1 Constrain ts and problematics ab out the Martian magma
o cean
The early p ost-magma o cean in terior evolution computed b y Elkins-T an ton et al. (2005b) is
compatible with sev eral Martian features. It is ho w ever hardly reconciliable with some other
observ ations. In particular, the comp ositional gradien t inferred from p etrological mo delling
(Elkins-T an ton, Hess, & Parmen tier, 2005a) is sufficiently large to completely suppress an y
form of thermal or thermo c hemical man tle con v ection after the cum ulate o v erturn (T osi et al.,
2013; Plesa et al., 2014). This, in turn, strongly limits the time span o v er whic h partial melt
is generated, whic h, under these conditions, can only o ccur o v er the first few h undred million
y ears of ev olution (Plesa et al., 2014). This picture is difficult to reconcile with Mars’ large
v olcanic pro vinces and long-lasting volcanic activity , whic h are lik ely the pro ducts of billions of
y ears of vigorous man tle conv ection (e.g., Li & Kiefer, 2007; O’Neill, Lenardic, Moresi, T orsvik,
& Lee, 2007; Grott & Breuer, 2010; Sekhar & King, 2014).
One p ossibilit y that ma y help resolving this issue is that Mars’ magma o cean underw en t
a com bination of batc h and fractional crystallization, similar to what has b een prop osed
b y Solomato v (2000) for the batc h crystallization of the Earth’s lo w er man tle follo wed b y
the fractional crystallization of the upp er man tle. In this case, the comp ositional gradien t
established at the end of solidification w ould not b e as strong as in the purely fractional case.
This w ould facilitate the onset of p ost-o v erturn man tle con vection (T osi et al., 2013) and y et
allo w for the long-term preserv ation of primordial comp ositional heterogeneities as suggested
b y the isotopic signatures of the SNC meteorites (e.g., F oley et al., 2005).
25

2. Early onset of con v ection and man tle o v erturn: the case of Mars
F ractional crystallization of the entire magma ocean relies on the hypothesis that the
timescale of man tle solidification is shorter than the timescale for the onset of the con v ectiv e
cum ulates o v erturn. Highly efficien t heat loss due to turbulen t con v ection in the magma
o cean can indeed ensure that the en tire man tle crystallizes v ery rapidly with a time scale
of the order of 10
3
y ears (Mon teux, Andrault, & Sam uel, 2016), unless a thic k atmosphere
generated b y v olatile degassing builds up (e.g., Ab e & Matsui, 1985; Zahnle, Kasting, &
P ollac k, 1988; Ab e, 1997). In this case, the time o v er which the solidification fron t reac hes
the surface b ecomes of the order of 10
5 −
10
6
y ears (Zahnle et al., 2007; Lebrun et al., 2013;
Salv ador et al., 2017; Nik olaou et al., 2019). The magma o cean solidification time is defined
here as the time during whic h a liquid-lik e, turbulen tly con v ecting la y er can b e main tained
at the surface of the planet, allo wing for efficien t heat extraction. Over suc h protracted time
scales, newly formed solid cum ulates with high, near-solidus temp eratures and hence lo w
viscosit y , p ossibly further reduced b y the presence of residual melt, can b egin o v erturning
b efore the solidification front has reac hed the surface. solid-state con v ection can start mixing
comp ositional heterogeneities while solidification is still taking place, th us effectiv ely reducing
the final comp ositional gradient induced b y fractional crystallization.
In this c hapter, w e in v estigate the conditions for the initiation of solid-state thermo-c hemical
con v ection during the b ottom-up solidification of the Martian magma o cean. W e treat the t w o
end-mem b er scenarios of batc h and fractional crystallization and demonstrate that in b oth
cases, for the co oling rates that can b e exp ected in the presence of an insulating atmosphere,
thermal and comp ositional mixing via solid-state conv ection can start so on after the formation
of the first solid man tle cum ulates and b efore the end of magma o cean crystallization.
2.2 Ph ysical mo del
W e consider a Mars-like bo dy whose man tle is initially completely molten except for a thin
la y er atop the CMB that is assumed to hav e already solidified. The magma o cean temperature
follo ws the temp erature reac hed at the solidification fron t, whic h dep ends on the c hosen
crystallization scenario (see Sections 2.2.1 and 2.2.2 and Figure 2.1). F or simplicit y , and since
w e are not aiming at repro ducing the ph ysics of the liquid magma o cean itself, w e assume that
the thermal ev olution of the magma o cean results in a rise of the solidification front linear in
time, an assumption that w e discuss in Section 2.5.2. The time needed to ac hiev e complete
solidification of the magma o cean (
t MO
) is one of the in v estigated parameters. F or the solidus
and liquidus curv es, w e use parametrizations prop osed by Herzberg, Raterron, and Zhang
(2000) and Zhang and Herzb erg (1994) resp ectiv ely , whic h are based on melting exp erimen ts
conducted on fertile KLB-1 p eridotite. A ccording to these t w o studies, solidus and liquidus
temp eratures (resp ectiv ely
T sol
and
T liq
) can b e expressed as a function of pressure
p
as follo ws:
T sol = 1400 + 149 . 5 p − 9 . 64 p 2 + 0 . 313 p 3 − 0 . 0039 p 4 , (2.1)
T liq = 1977 + 64 . 1 p − 3 . 92 p 2 + 0 . 141 p 3 − 0 . 0015 p 4 , (2.2)
where
T sol
and
T liq
are expressed in K and
p
in GP a. Note that, in the expression of the
melting curv es, w e shall mak e the assumption that the pressure is equal to the lithostatic
26

2.2 Ph ysical mo del
pressure only (Equations
(2.1)
and
(2.2)
can then b e written as function of depth). These are
the melting curv es represen ted in Figure 1.7.
In order to assess the consequences of magma o cean solidification on the dynamics of the
solid man tle, w e treat the t w o end-mem b er scenarios of fractional and batc h crystallization
(Solomato v, 2007), whose ph ysical description is pro vided in Section 1.3.2.3. The follo wing
sections describ es their implementation in the model.
2.2.1 F ractional crystallization
In the fractional crystallization scenario, w e assume that residual melt cannot remain em b edded
the solid matrix and is extracted in to the o v erlying liquid, lea ving b eneath a differen tiated
solid man tle. In this case, the temp erature of the solidification fron t simply coincides with
the solidus (Figure 2.1a). Up on solidification, iron is preferen tially partitioned in to the liquid
phase (e.g., Elkins-T anton et al., 2005a), whic h causes the enric hmen t of the magma o cean
and, in turn, of the latest (i.e., shallow est) solidified la y ers. As a consequence, an unstable
comp ositional profile forms, with progressiv ely denser cum ulates that o ccup y shallo w er and
shallo w er man tle la y ers.
This scenario is qualitativ ely similar to the one considered in previous studies (e.g. Elkins-
T an ton et al., 2005a; T osi et al., 2013; Plesa et al., 2014; Sc hein b erg, Elkins-T an ton, &
Zhong, 2014). The fundamen tal difference here is that instead of assuming the en tire unstable
comp ositional profile to b e built b efore an y solid-man tle o v erturn can take place, w e mo del its
formation allo wing for the p ossibilit y that solid-state con v ection sets in b efore the solidification
fron t has reac hed the surface.
It m ust b e also noted that, b ecause of c hemical partitioning, the actual solidus of the
crystallizing cum ulates should ev olve as the solidification proceeds. A t the b eginning, when
the most mafic materials crystallize, it corresp onds to the liquidus of the bulk man tle, while at
the end, when the most felsic materials crystallize, it corresp onds to its solidus. Ho w ev er, for
simplicit y , w e consider the crystallization temp erature to b e equal to the solidus of the bulk
man tle, whic h, in turn, translates in to a low er temp erature, and hence a higher viscosit y , at
the solidification fron t for the deep est cum ulates.
2.2.2 Batc h crystallization
As an alternativ e to the fractional crystallization scenario, w e consider a simple mo del of batc h
crystallization where the melt remains en tirely em b edded in the matrix un til solidification
is completed, th us leading to a homogeneous man tle comp osition. This situation could
b e encoun tered if, for example, the man tle solidified rapidly , o v er a time shorter than the
c haracteristic time scale for melt p ercolation and crystal segregation, or if the melt w as
neutrally buo y an t.
Bet w een the liquidus and solidus, the v olumetric melt fraction
ϕ
decreases linearly with
temp erature from 1 to 0, as giv en b y Equation
(1.1)
. Exp eriments sho w that the rheology of
t w o-phase crystal-melt mixtures presents a sharp transition betw een liquid-lik e and solid-lik e
b eha viour around a certain rheologically critical melt fraction (R CMF) (e.g., Costa et al.,
2009). The RCMF iden tifies the critical v alue of
ϕ
b elo w whic h newly-formed crystals build
an in terconnected net work that effectiv ely deforms via solid-state creep, whic h can b e treated
27

2. Early onset of con v ection and man tle o v erturn: the case of Mars
b y solving the equations of solid man tle con v ection. In the batc h crystallization case, the
temp erature of the solidification fron t is no longer the solidus as in the fractional case, but the
temp erature
T R CMF
corresp onding to the R CMF (Figure 2.1b). The partially molten region
where the temp erature lies b et w een
T sol
and
T R CMF
, the so-called m ush domain (Solomato v,
2007), is th us treated as part of the solid man tle with a viscosit y , whic h, b esides b eing dep enden t
on temp erature and pressure, also dep ends on the melt fraction (see Section A.2 and equations
((A.23)) and ((A.24))).
Note that in the fractional crystallization scenario, the end of magma o cean solidification
corresp onds to the time when the solidification fron t reac hes the surface, whic h means the
time when the surface temp erature reac hes the solidus in the fractional crystallization case, or
the RMCF temp erature in the batch crystallization case.
F ractional crystallization
Batc h crystallization
Δ r
Magma o cean
Mush region
Liquid us
T R CMF
Solidus
T emp erature
Solid cum ulates
Solidi fi cation fron t
t t+ Δ t
Δ r
Radius
T emp erature
Radius
T emp erature
a)
b)

Figure 2.1:
Sc hematic diagram illustrating the fractional (a) and batc h (b) crystallization
scenarios. The solidification fron t lies at the in tersection betw een the magma o cean temp erature
and the solidus in the fractional case, and the temp erature corresp onding to the R CMF in the
batc h case. Ov er a time step ∆
t
, the fully solid man tle (fractional case) or the m ush region (batc h
case) gro w up w ards b y ∆
r
= ∆
tD MO
(0)
/t MO
, where
D MO
(0) is the initial thic kness of the magma
o cean and t MO the prescrib ed solidification time of the magma o cean.
2.2.3 Cum ulates dynamics
T o sim ulate the dynamics of the solid cum ulates during magma o cean solidification, w e solv e the
non-dimensional conserv ation equations of mass, momen tum, thermal energy , and comp osition
28

2.2 Ph ysical mo del
under the Boussinesq appro ximation, using the con v ection co de GAIA, as introduced in
App endices A1 and A2. These equations read (see Section B.1):

∇ ·  v = 0 , (2.3)

∇ · [ η ( 
∇  v + ( 
∇  v ) T )] − 
∇ p + Ra( T − B C )  e z = 
0 , (2.4)
∂ T
∂ t +  v · 
∇ T − ∇ 2 T = 0 , (2.5)
∂ C
∂ t +  v · 
∇ C = 0 , (2.6)
where
 v
is the v elo cit y ,
η
the dynamic viscosit y ,
p
the dynamic pressure,
Ra
the thermal
Ra yleigh n um b er,
T
the temp erature,
B
the buo y ancy ratio,
C
the comp ositional densit y
field, and
 e z
the unit v ector in radial direction p oin ting out w ards. All material parameters
(listed in T able 2.1) are assumed to b e constan t with the exception of the densit y , whic h,
according to the Boussinesq appro ximation, v aries only when it app ears in the buo y ancy
terms through the Ra yleigh n umber and buoy ancy ratio (Equation
(2.4)
), and of the viscosit y ,
whic h is assumed to dep end on temp erature and depth according to the Arrhenius la w for
diffusion creep (see Section A.2). The Ra yleigh n um b er describ es the ratio of buo y ancy forces
due to thermal expansion and con traction to the stabilizing effects of heat and momen tum
diffusion, while the buo y ancy ratio measures the imp ortance of densit y contrasts associated
with v ariations in comp osition (when accoun ted for) with resp ect to densit y con trasts due to
v ariations in temp erature. Note that in our sim ulations, b oth the initial temp erature and the
initial comp ositional gradient act in fa v our of the onset of con v ection rather than comp eting
against eac h other. It is only after the first o v erturn has tak en place that comp ositional
buo y ancy acts against thermal con vection. Also note that in Equation
(2.5)
w e neglected
b oth in ternal heating and viscous dissipation. On the one hand, in the case of fractional
crystallization, heat-pro ducing elemen ts, b eing highly incompatible, w ould b e strongly enric hed
initially in the liquid phase and, to w ard the end of solidification, in the shallo w solid cumulates
(see Plesa et al., 2014; Sc hein b erg et al., 2014, and references therein). As a consequence,
the bulk of the solid man tle is lik ely depleted in heat sources during crystallization. On the
other hand, in the case of batc h crystallization, tests conducted with and without in ternal
heating did not evidence significan t differences in the onset time of con v ection, which is largely
con trolled b y the v ery lo w viscosit y caused b y the high temp erature and b y the presence of
in terstitial melts (Section 2.4.1).
In the fractional crystallization case, we use the non-dimensional field
C
to mo del the
ev olving comp osition of the solid cum ulates assuming, for simplicit y , that
C
increases linearly
from the CMB (where
C
= 0 ), to the surface (where
C
= 1 ) (T osi et al., 2013). The buo y ancy
ratio
B
is used as mo del parameter to con trol the strength of the comp ositional gradien t
(Section 2.2.4). In the case of batc h crystallization, for temp eratures b et w een the
T R CMF
and
T sol
, w e tak e in to accoun t the supplementary effect of the presence of melt on the viscosit y
through an additional exp onen tial term (see Section A.2). In b oth crystallization scenarios,
the temp erature in the solid cumulates is buffered to its initial v alue, i.e. the solidus for the
fractional crystallization scenario and the R CMF temp erature for the batc h crystallization
scenario. The melt that w ould b e generated b y hot up w elling cum ulates whose temp erature
29

2. Early onset of con v ection and man tle o v erturn: the case of Mars
rises ab o v e these thresholds is assumed to b e extracted instantaneously in to the o v erlying
liquid magma o cean. Also note that laten t heat effects are not explicitly included but their
exp ected influence is discussed in Section 2.5.2.
Quan tit y V alue Unit
R p 3400 km
R c 1700 km
D ref 1700 km
g 3.7 m/s 2
D MO (0) 1530 km
t MO 0.1, 1, 10 Myr
T ref 2503 K
p ref 22 GP a
E ∗ 3 . 35 × 10 5 J/mol
V ∗ 4 × 10 − 6 m 3 /mol
α m 21 –
ϕ R CMF 0.3 –
R 8.314 J/(kg mol)
c p 1200 J/kg/K
κ 10 − 6 1/m 2
α 3 × 10 − 5 K − 1
ρ ref 3500 kg/m 3
c p,c 840 J/kg/K
ρ c 7200 kg/m 3
∆ T 2283 K
T s 220 K
Ra ref 10 8 , 10 9 , 10 10 P a s
B 0.2, 1, 1.8 –
L 5 × 10 5 kJ/mol
T able 2.1:
V alues of th quan tities in use. Note that the reference Rayleigh n umber corresp onds
to the Ra yleigh n um ber computed for the whole mantle (see Section 2.3.1). Study parameters are
indicated in red
2.2.4 P arameter space
W e in v estigate the onset of solid-state con v ection for the t w o crystallization scenarios b y
v arying the follo wing parameters: the reference Rayleigh n um b er
Ra ref
, the magma o cean
solidification duration
t MO
, and, in case of fractional crystallization, the buo y ancy ratio
B
.
In the case of batc h crystallization, w e assume a homogeneous man tle with no comp ositional
heterogeneities, i.e. with
B
= 0 (see Section 2.2.2). The reference Ra yleigh n um b er Ra
ref
,
computed for the whole man tle thic kness and the reference viscosit y
η ref
, is defined at the
depth of the CMB (
p ref
= 22 GP a), and at the temp erature of 2503 K, i.e. the temp erature
of the R CMF at the CMB. W e v aried Ra
ref
b et w een 10
8
, 10
9
and 10
10
for the fractional
crystallization case, and set it to 10
7
for the batc h crystallization case (see Section 2.4.1).
These v alues corresp ond to reference viscosities of
η ref
= 4
.
4
×
10
20
, 4
.
4
×
10
19
, and 4
.
4
×
10
18
P a s, resp ectively , in the fractional crystallization case, and 4
.
4
×
10
21
P a s in the batc h
crystallization case. At the more con v en tional reference pressure and temp erature of 3 GP a
and 1600 K often emplo y ed in studies of man tle conv ection and thermal ev olution (e.g. T osi
et al., 2013; Plesa et al., 2014), the ab o v e reference viscosities w ould b e 2
.
4
×
10
22
, 2
.
4
×
10
21
,
and 2
.
4
×
10
20
P a s, resp ectiv ely in the fractional crystallization case, and 2
.
4
×
10
23
P a s in the
batc h crystallization case, yielding Ra yleigh num b ers of 1
.
23
×
10
5
, 1
.
23
×
10
6
and 1
.
23
×
10
7
resp ectiv ely for the fractional crystallization case, and 1
.
23
×
10
4
for the batc h crystallization
case. Note that
η ref
can b e formally calculated b y c hanging the pre-exp onen tial term of the
30

2.3 Numerical set-up
olivine flo w la w for diffusion creep using different v alues of the grain size and/or of the w ater
con ten t (Hirth & Kohlstedt, 2003). The v alues that w e used ha ve b een obtained b y setting the
grain size to 1 cm and v arying the w ater con ten t b et w een 1, 10, and 100 ppm. F urthermore, in
the case of batc h crystallization, the presence of in terstitial melt further decreases the effectiv e
viscosit y (see Equation ((A.24)) and Section 2.4.1).
The magma o cean solidification duration (
t MO
), defined as the time necessary for the
solidification fron t to propagate from the CMB to the surface, is v aried b et w een 0.1, 1,
and 10 Myr. These v alues are represen tativ e of relativ ely long-liv ed magma o ceans whose
crystallization is slo w ed do wn b y the blank eting effect of a gro wing H
2
O-CO
2
atmosphere
generated up on degassing of greenhouse v olatiles (Ab e, 1997; Hamano et al., 2013; Lebrun
et al., 2013; Salv ador et al., 2017; Nik olaou et al., 2019).
In the case of fractional crystallization, w e test three differen t v alues of the buo y ancy ratio:
B
= 0
.
2 , 1, and 1.8, corresp onding to maxim um comp ositional densit y con trasts ∆
ρ
b et w een
the surface and the CMB of 47.8, 239 and 430 kg/m 3 , resp ectively .
2.3 Numerical set-up
Con v ection in the cum ulates is computed using GAIA, as in tro duced in Appendix B. This
section in tro duces features sp ecific to the presen t case.
2.3.1 Domain mehsing and time-ev olving geometry of the cum ulates
Because of the v ery high computational cost of using the particle-in-cell metho d (to adv ect
the comp osition field) on a 3D grid, w e sim ulate the solid cum ulates b y a 2D quarter cylinder,
with an inner-to-outer radius ratio of 1:2, as appropriate for a Mars-lik e planet. Discrepancies
b et w een 2D and 3D con v ection can app ear in steady state (Guerrero, Lo wman, Desc hamp s,
& T ac kley, 2018), but are not exp ected to b e significan t for suc h transien t pro cesses as the
man tle o v erturn (see Section 3.4.4 for a longer discussion). The domain is meshed using a
pro jected grid with equally spaced radial shells. The resolution is chosen according to the
v alue of the reference Ra yleigh n umber: 100 radial shells of 237 cells eac h for
Ra ref
= 10
8
,
150 radial shells of 355 cells eac h for
Ra ref
= 10
9
and 200 radial shells of 473 cells eac h for
Ra ref = 10 10 .
The time-dep endence of the domain’s size (i.e. the cum ulates’ gro wth) is tak en into accoun t
b y successiv ely ”switc hing on” shells of con trol v olumes when the solidification fron t reac hes
their depth (see Fig. 2.2). When it still b elongs to the magma o cean’s domain, a con trol v olu me
exists as a ghost cell. When it is switc hed on, it receiv es the corresp onding crystallization
temp erature as initial condition for equation
(B.3)
and the corresp onding comp osition when
equation (A.25) is solv ed.
The non-con tinous nature of this implemen tation is balanced by adopting a sufficien tly
high resolution and a sufficien tly small time stepping. Th us, at eac h time step, the b oundary
of the n umerical domain (the upp er faces of the cells from the upp ermost activ e shell) are the
closest p ossible to the actual depth of the solidification fron t.
31

2. Early onset of con v ection and man tle o v erturn: the case of Mars
Figure 2.2:
Ev olution of the finite v olume grid b et w een t w o time steps. A t time
t
(left panel),
the radius of the solidification fron t (
R SF
) is suc h that all grid shells ab o v e the
i th
shell (included)
are deactiv ated (set as ghost cells, represen ted in grey while activ e cells are represen ted in blac k),
while at time
t
+
dt
(righ t panel), the ascension of the solidification fron t results in the activ ation
of the cells of the i th shell.
Note that, due to the time evolution of the geometry , the Ra yleigh n um b er scaling Equation
(2.4)
is time-dep enden t. It is obtained b y m ultiplying the reference Ra yleigh n um b er b y the
cub e of the relative thic kness of the solid cum ulates at eac h time step.
In order to reliably capture the onset of con v ection, a high temp oral resolution is also
essen tial (e.g. K orenaga & Jordan, 2003). W e c hose the time step ∆
t
according to the Couran t-
F riedric h-Levy (CFL) criterion, i.e. ∆
t
=
λδ min / | v | max
, where
δ min
is the minim um spatial
resolution in the radial or lateral direction,
| v | max
is the maxim um v elo cit y in the domain, and
λ
is the CFL n um b er that w e set to 0.5 in all sim ulations. In addition, w e limited the actual
v alue of ∆
t
b y defining a maxim um non-dimensional time step ∆
t max
= 10
− 8
(corresp onding
to
∼
900 y ears) that w as used whenev er the CFL criterion w ould result in a larger time step.
F or a represen tativ e run that leads to an early onset of con v ection with Ra
ref
= 10
9
,
t MO
= 10 Myr, and
B
= 1 , w e carried out resolution tests b y v arying b oth the spatial and
temp oral resolution of the simulation. T ests conducted using 100 to 250 radial shells and
setting ∆
t max
b et w een 10
− 7
and 10
− 9
did not evidence significan t differences in terms either
of the onset time of solid-state con v ection, or of its characteristic planform. Sp ecifically , the
time for the onset of con v ection v aries b y up to 0.04 Myr up on v arying the temp oral resolution
and 0.06 Myr up on v arying the spatial resolution within the ab o ve ranges.
2.3.2 Initial and b oundary conditions
The sim ulations start whith a thin initial solid la yer already existing atop the CMB, so that the
actual solution domain is presen t from the b eginning. In all sim ulations, w e used a shell with
an initial thic kness of 170 km (i.e. the initial depth of the magma o cean is
D MO
(0) = 1530
km) that mimics this first solid cum ulate la y er. The subsequen t b ottom-up crystallization of
the magma o cean is mo delled b y progressively increasing the size of the solid domain (see
Section 2.3.1) at a predefined rate according to the parameter t MO and considering a growth
linear in time (Section 2.2.4) un til the solidification fron t reaches the planetary surface. The
upp er and lo w er b oundaries of the gro wing partial cylinder shell ha v e free-slip conditions and
a prescrib ed temp erature. The temp erature of the upp er b oundary ev olv es according to the
p osition of the solidification fron t, follo wing the crystallization temp erature, i.e. either the
profile of
T sol
or
T R CMF
, dep ending on the crystallization scenario (Sections 2.2.1 and 2.2.2).
32

2.3 Numerical set-up
The CMB temp erature
T CMB
corresp onds initially to the CMB temp erature of the solidus (in
fractional crystallization cases) or of the R CMF (in batc h crystallization cases) and ev olves
follo wing Equation (A.26).
The initial temp erature distribution in the thin pre-existing solid domain follo ws the profiles
of
T sol
in the fractional crystallization case, or of
T R CMF
in the batc h crystallization case,
b eneath the rising solidification fron t. Note that for short lifetimes of the magma o cean and
relativ ely small reference Ra yleigh num b ers, the solidification fron t will likely reac h the surface
b efore the onset of solid-state conv ection. In that case, the initial temp erature distribution is
the en tire solidus (for fractional crystallization cases) or the en tire profile of
T R CMF
(for batc h
crystallization cases). Note also that the surface temp erature is set to its presen t-day v alue as
so on as the solidification fron t reac hes the surface. After the cessation of the liquid magma
o cean phase, in fact, a very high surface temp erature cannot b e sustained. Indeed, at least
for the Earth, V alley et al. (2005) sho w ed that there is evidence for a cold surface as early as
4.4 Ga. Therefore, for a sim ulation time
t>t MO
, the top b oundary has a temp erature
T s
of
220 K. Because of the strong temp erature dep endence of the viscosit y , this causes the rapid
gro wth of a stagnan t lid since no plastic yielding is tak en in to accoun t. T o initiate con vection,
a random p erturbation with an (non-dimensional) amplitude of 10
− 3
is added to the initial
temp erature profile (corresp onding to anomalies of ∼ 1 K).
Finally , in the fractional crystallization case, an unstable comp ositional profile increasing
linearly from the CMB (where
C
= 0 ) up w ards to the surface (where
C
= 1 ) (T osi et al., 2013)
is progressiv ely built up as the magma o cean solidifies (see Figure 2.3).
2.3.3 Ly apuno v exp onen t and shrinking factor
In the fractional crystallization cases, a primordial heterogeneit y is created in the man tle up on
magma o cean solidification. One of the goals of this study is to quan tify the degree of man tle
mixing ac hiv ed dep ending on the v alues of the parameters. T o do so w e compute the Ly apuno v
exp onen t in the con v ecting domain (F arnetani & Sam uel, 2003). Due to the action of the lo cal
v elo cit y field and its corresp on ding v elo cit y gradien t tensor (

∇  v
), eac h man tle parcel stretc hes
and shrinks along preferen tial directions. In the frame of the Boussinesq appro ximation, these
t w o directions are p erp endicular. F ollo wing F arnetani and Samuel (2003), in a t wo-dimensional
domain, the maxim um strain along the stretc hing and shrinking directions are deriv ed from the
Lagrangian time-in tegration of the follo wing equation for the entries of a 2
×
2 linear op erator
matrix:
d 

M
dt = 
∇  v × 

M , (2.7)
sub ject to the initial condition :


M
(
t
= 0) =


I 2
, where


I 2
is the second order iden tit y matrix.
Along a flo w streamline,


M
relates the v ectors connecting t w o small, initially close, fluid parcels
from
 r 0
=
 r
(
t
= 0) to
 r
(
t
) =


M  r 0
. The maximum amoun t of stretc hing and shrinking at a
giv en time and lo cation corresp ond to
σ +
and
σ −
, the maxim um and minim um eigenv alues of


M , resp ectiv ely . The incompressibilit y constrain t (Equation (B.1)) implies that σ − = 1 /σ + .
W e c haracterize the efficiency of mixing for eac h mo del b y monitoring
σ −
, the minim um
shrinking factor (as explained b elo w, a small shrinking factor means a high degree mixing).
This quan tit y is initially equal to unit y . F or a v elo cit y field generating deformation,
σ −
33

2. Early onset of con v ection and man tle o v erturn: the case of Mars
decreases to w ards a minim um asymptotic v alue of 0. Low v alues of
σ −
indicate stronger
shortening of man tle parcels with larger asp ect ratio, fa v oring mixing and homogenization
with the surroundings via c hemical diffusion (Olson, Y uen, & Balsinger, 1984; Gurnis, 1986;
Ottino, 1989). Equiv alen tly , this quan tit y expresses the amoun t by whic h the initial size of a
man tle parcel has b een reduced as a result of the deformation along its tra jectory (Samuel,
Aleksandro v, & Deo, 2011; Sam uel & King, 2014). In other w ords, for a man tle parcel of
giv en initial size
δ 0
, its dimension in the direction of maxim um shrinking at time
t
is just
δ
=
δ 0 σ −
, implying that the smaller
σ −
, the more efficien t mixing. Maxim um shrinking
factors are trac k ed along as many trajectories as grid cells, whose initial lo cations are spread
o v er the computational domain according to a random homogeneous distribution. Equation
(2.7)
is then in tegrated forw ard in time using a second-order predictor-corrector R unge-Kutta
metho d with bi-linear in terp olation of v elo cit y comp onen ts lo cated at the four nearest grid
p oin ts. With resp ect to studies in whic h con v ectiv e mixing is discussed on the basis of the
distribution of tracers (e.g. O’Neill, Debaille, & Griffin, 2013; T osi et al., 2013), the calculation
of the shrinking factor has the adv an tage of allo wing us to c haracterize with a single n um b er
the mixing prop erties of each of the analyzed cases (see Section 2.4.2).
2.4 Results
2.4.1 Early onset of man tle con v ection
The onset of con v ection during the magma o cean solidification results from a comp etition
b et w een the timescale of the first con v ectiv e o v erturn and the timescale of solidification (
t MO
),
the former b eing influenced b y the reference Ra yleigh n um b er
Ra
and, in the case of fractional
crystallization, also b y the buo yancy ratio
B
(see Section B.1 and T able B.1). F rom now on,
w e will sp eak of ”early” onset of con v ection if this o ccurs b efore the end of the solidification
of the magma o cean, and of ”late” onset if con v ection b egins only afterw ards, resulting in a
global-scale o v erturn o ccurring underneath a stagnan t lid.
Figure 2.3 sho ws the ev olution of a represen tative case c haracterized b y an early onset
of con v ection. In this example, w e considered a fractional crystallization scenario with the
follo wing parameters:
Ra ref
= 10
10
,
B
= 1
.
8 and
t MO
= 1 Myr. Figure 2.3a sho ws the
initial conditions. The semi-transparen t la y er denotes the parametrized magma o cean region
(i.e. the region where the conserv ation equations are not solv ed); the opaque lay er, whic h
is colour-co ded according to the comp osition field and gro ws radially outw ards with time,
represen ts the solid cum ulates with an initial temp erature (red line) set at the solidus (blac k
dashed line), and an initial comp osition set to a linear unstable gradien t (blue line). After 0.7
Myr the first o v erturn has already taken place (Figure 2.3b). Comp ositionally dense material
accum ulates near the CMB, while ligh ter material is driv en b elo w the solidification fron t b y hot
up w ellings. A t 1 Myr, the solidification is completed. By this time, the initial comp ositional
gradien t is significan tly reduced as sho wn b y the comp osition profile in Figure 2.3c, indicating
a relativ ely w ell mixed in terior. F or the subsequent ev olution, the surface temp erature is set to
its presen t-da y v alue, whic h quic kly creates a highly viscous stagnan t lid in absence of yielding
(whic h is not included in the mo del), slo wly delaminated b y dripping do wn w ellings caused b y
the underlying con v ection. A t 10 Myr most of the densest cum ulates that formed near the
34

2.4 Results
T emp erature
Comp osition
Solidus
Solidi fi cation
fron t
T
i
m
e

(
M
y
r
)
0
0.7
1.0
10
0 1
a)
b)
c)
d)
Comp osition

Figure 2.3:
Ev olution of the comp osition field for a fractional crystallization scenario with
Ra
= 10
10
,
B
= 1
.
8 , and
t MO
= 1 Myr. The four panels sho w snapshots and corresp onding
laterally-a v eraged profiles of temp erature (red line) and comp osition (blue line) at the beginning of
the sim ulation (a), and after 0.7 Myr (b), 1 Myr (c), and 10 Myr (d). The blac k dashed and dotted
lines denote the solidus temp erature and the p osition of the solidification fron t, resp ectiv ely .
surface ha v e either sunk en at the CMB or hav e b een mixed b y mantle con v ection with the
consequence that the effectiv e buo y ancy ratio of the conv ecting part of the man tle is m uc h
lo w er than in the case of a whole man tle o v erturn (Figure 2.3d).
In order to in v estigate ho w the onset of conv ection is influenced b y the v arious mo del
parameters, w e follo wed the ev olution of the Nusselt n um b er calculated at the b ottom of the
domain (
Nu b ot
). Defined as the ratio of the heat-flux through the top b oundary la y er to the
steady-state conductiv e heat flux for a similar geometry , the Nusselt n um b er can b e effectiv ely
used to trac k the onset of finite-amplitude con v ection. As long as heat is transp orted by
conduction,
Nu b ot
will b e equal to one, while it will b ecome higher as so on as con v ection sets
in. Figure 2.4 sho ws time-series of Nu b ot for all sim ulations o v er a time in terv al of 11 Myr.
An early onset of man tle con vection is only possible if the solidification duration of
the magma o cean is sufficien tly long in comparison with the con v ectiv e ov erturn timescale.
Therefore the fast solidifying cases (small v alues of
t MO
) generally result in a late onset of
con v ection. This is the case, in fact, for all the runs with
t MO
= 0
.
1 Myr, regardless of other
parameters (
Ra ref
and
B
). With
t MO
= 1 Myr, ho w ev er, in the fractional crystallization cases,
early onset of con v ection o ccurs for
Ra ref ≥
10
10
and, with
t MO
= 10 Myr, for
Ra ref ≥
10
9
.
While increasing the buo y ancy ratio sligh tly shifts the time of onset of conv ection to w ards
earlier v alues, it do es not hav e a first order influence as the v alue of
Ra ref
do es. This reflects the
fact that our uncertain t y in the densit y con trast spans not more that one order of magnitude,
whereas the uncertain t y in the viscosit y co v ers three orders of magnitude.
35

2. Early onset of con v ection and man tle o v erturn: the case of Mars
0 3 6 9 1 2
0
1 5
3 0
4 5
N u b o t
a
)
t
M O = 0
.
1 M y r
0 3 6 9 1 2
0
1 5
3 0
4 5
b
)
t
M O = 1 M y r
0 3 6 9 1 2
0
1 5
3 0
4 5
c
)
B
= 0
.
2
t
M O = 1 0 M y r
R a r e f = 1 0 1 0
R a r e f = 1 0 9
R a r e f = 1 0 8
0 3 6 9 1 2
0
1 5
3 0
4 5
N u b o t
d
)
0 3 6 9 1 2
0
1 5
3 0
4 5
e
)
0 3 6 9 1 2
0
1 5
3 0
4 5
B
= 1
f
)
0 3 6 9 1 2
0
1 5
3 0
4 5
N u b o t
g
)
0 3 6 9 1 2
0
1 5
3 0
4 5
h
)
0 3 6 9 1 2
0
1 5
3 0
4 5
B
= 1
.
8
i
)
0 3 6 9 1 2
T i m e ( M y r )
0
.
0
1
.
5
3
.
0
4
.
5
N u b o t
j
)
0 3 6 9 1 2
T i m e ( M y r )
0
.
0
1
.
5
3
.
0
4
.
5
k
)
0 3 6 9 1 2
T i m e ( M y r )
0
.
0
1
.
5
3
.
0
4
.
5
B a t c h
l
)
R a r e f = 1 0 7

Figure 2.4:
Time-series of the b ottom Nusselt num b er for all com binations of parameters. P anels
a–i refer to fractional crystallization cases (
B >
0 ) and j–l to the batc h crystallization cases (
B
= 0 ).
All sim ulations are run for 11 Myr. The red line in panel h (
t MO
= 1 Myr and
B
= 1
.
8 ) corresp onds
to the mo del sho wn in Figure 2.3. F or batc h mo dels, only sim ulations with
Ra ref
of 10
7
ha v e b een
p erformed. The dotted line marks the end of the crystallization for eac h case in panels a–l.
The con v ectiv e phase alw a ys b egins with an intense o v erturn driv en b y the thermal and
comp ositional densit y con trasts a v ailable at the time of the onset. This first o v erturn is then
follo w ed b y a less in tense phase during whic h thermal con v ection acts against the stabilizing
effect of the o v erturned cum ulates. This b eha vior can b e recognized in the time-series of
Nu b ot
sho wn in Figure 2.4. On the one hand, when the o v erturn tak es place only after the whole
man tle has solidified, e.g. as in Figure 2.4a, d, and g, for
t MO
= 0
.
1 Myr,
Nu b ot
initially gro ws
rapidly b ecause of the sudden co oling of the deep man tle due to the sinking of shallo w, cold
cum ulates. After reac hing the first p eak, it decreases since the dense o v erturned cum ulates
tend to hinder con v ection and heat extraction from the core. On the other hand, in those
cases where an early onset is observ ed (see e.g. Figure 2.4c, f, and i), the first o v erturn results
in the temp orary formation of a stably stratified la y er atop the CMB. If
Ra ref
is sufficien tly
36

2.4 Results
large, this la y er can b e then quic kly destabilized b y thermal con v ection, leading to a secondary
gro wth of Nu b ot (Figures 2.4c, 2.4f, and 2.4i).
The batc h crystallization cases exhibit v ery differen t conv ectiv e regimes, mainly due to
their melt fraction-dep enden t rheology and their purely thermal con v ection. Despite the lo w er
v alue of the reference Ra yleigh n um b er (
Ra ref
= 10
7
), the effectiv e Ra yleigh n um b er is actually
v ery large b oth b ecause of the higher initial temp erature (
T R CMF
instead of
T sol
used in the
fractional cases) and of the additional effect of in terstitial melt on the rheology (Equation
(A.24)
), yielding a maxim um reac hed Ra yleigh n um b er of 5
.
45
×
10
9
. While the time for the
onset of con v ection is comparable to that of the fractional crystallization cases with a similar
effectiv e Ra yleigh num b er (Figures 2.4j, k, and l), the ev olution of
Nu b ot
differs considerably .
The initial o v erturn is purely thermal, hence less abrupt, and do es not preven t subsequen t
con v ection as it is not follo wed b y the formation of a stable comp ositional gradien t. The
strong decrease of the viscosit y due to its dep endence on the melt fraction concentrates the
deformation in v ery thin, rapidly ascending plumes, while cold do wn w ellings, whose viscosit y
is only con trolled b y temp erature, are slo w and diffused. As a consequence, basal heat is
extracted rather inefficien tly through these v ery thin channels, ultimately resulting in m uc h
lo w er v alues of Nu b ot .
2.4.2 Mixing of the man tle
Whether or not solid-state con v ection starts during the magma o cean phase has a critical
influence on the degree of mixing of comp ositional heterogeneities achiev ed immediately after
solidification, but also o v er the long-term ev olution of the in terior. Assessing the degree of
man tle mixing at the end of magma o cean solidification is imp ortan t b ecause early-formed
comp ositional heterogeneities can strongly affect the subsequent thermal ev olution of the
man tle. T o measure the degree of mixing of the solid mantle as a function of time, w e
computed for all sim ulations the ev olution of the shrinking factor as describ ed in Section 2.3.3.
Time-series for the a v eraged v alue of the shrinking factor are displa y ed in Figure 2.5.
A late onset of con v ection leads to a sharp decrease in the shrinking factor. This feature is
due to the initial whole-man tle o v erturn, whic h causes the slop e of the time-series to become
increasingly steep er as
Ra ref
and
B
(and th us the effectiv e Rayleigh n um b er) b ecome larger.
The subsequen t con v ection causes a further decrease of the shrinking factor tow ards a quasi-
asymptotic v alue, which also depends on
Ra ref
and
B
. In the cases c haracterized b y an early
onset of con v ection (Figures 2.5c, f, and i), the initial o v erturn is still mark ed by a sharp
decrease in the shrinking factor, whose amplitude is also con trolled b y the Ra yleigh num b er.
As in the previous cases, the slop e of the time-series is steep er for higher
Ra ref
and
B
. After
the first o v erturn, the shrinking factor further decreases approximately linearly with time (the
curv es app ear b en t due to the semi-log axes of the plot) un til the end of solidification due to
the progressiv e mixing of the dense, newly crystallized la y ers. In this phase, mixing accelerates
when, after the initial o v erturn, the first dep osited dense la y ers are progressiv ely re-en trained
b y the flo w. The shrinking factor, ev en tually , ev olv es as b efore to w ards a slo wly decreasing
v alue.
Mixing is a t ypically heterogeneous pro cess in the man tle. F or example, the shallo w est
la y ers of the mantle are rapidly exp osed to the lo w surface temp erature and form a conductiv e
37

2. Early onset of con v ection and man tle o v erturn: the case of Mars
0 3 6 9 1 2
0
.
1
1
.
0 0
S h r i n k i n g f a c t o r
a
)
t
M O = 0
.
1 M y r
0 3 6 9 1 2
b
)
t
M O = 1 M y r
0 3 6 9 1 2
B
= 0
.
2
c
)
t
M O = 1 0 M y r
R a r e f = 1 0 1 0
R a r e f = 1 0 9
R a r e f = 1 0 8
0 3 6 9 1 2
0
.
1
1
.
0 0
S h r i n k i n g f a c t o r
d
)
0 3 6 9 1 2
e
)
0 3 6 9 1 2
B
= 1
f
)
0 3 6 9 1 2
T i m e ( M y r )
0
.
1
1
.
0 0
S h r i n k i n g f a c t o r
g
)
0 3 6 9 1 2
T i m e ( M y r )
h
)
0 3 6 9 1 2
T i m e ( M y r )
B
= 1
.
8
i
)

Figure 2.5:
Semi-log plots of the time-series of the shrinking factor for all com binations of
parameters. Similar to Figure 2.4, the dotted line marks the end of the magma o cean solidification.
stagnan t lid that is to o viscous to exp erience significan t deformation. Figure 2.6 sho ws the
distribution of the shrinking factor across the solid man tle for all sim ulations after 11 Myr. A
homogeneously high shrinking factor, close to one, indicates that con v ection has not set in
b efore the end of the represen ted time span (although it starts later, as indicated in Section
2.4.3). The cases for whic h mixing w as mostly efficien t ha v e shrinking factors close to 0 (see
e.g. red histograms in Figures 2.6f and i).
The presence of a cluster of high shrinking factor v alues in all sim ulations is asso ciated
with the formation of the stagnan t lid. It can b e b est seen in the cases presen ting a late
onset of con v ection (for instance, all cases with
Ra ref
= 10
8
), as previously discussed b y Plesa
et al. (2014). Nonetheless, a stagnan t lid also forms in the cases exhibiting an early onset of
con v ection, and its thic kness decreases with increasing
B
. Indeed a high v alue of
B
destabilizes
the upp er b oundary la y er b y increasing its comp ositional densit y gradien t.
In general, a high v alue of
Ra
generates small-scale con v ectiv e patterns and an increased
time dep endence of the flow that enhances man tle mixing, causing the shrinking factor to
cluster around v alues close to 0. A lo w
Ra
leads instead to longer w a v elength con v ection that
ultimately results in a large spread of the shrinking factor v alues. When
B
is large enough,
cases c haracterized b y a late whole-man tle o v erturn result in a dense la y er at the b ottom of the
man tle that do es not tak e part in the subsequen t mixing, limiting the con vection to the upp er
la y ers b elo w the stagnan t lid, pro vided that
Ra
is high enough to sustain it. This con v ecting
38

2.4 Results
0 0
.
2 5 0
.
5 0
.
7 5 1
0
.
0 1
0
.
1
1
1 0
1 0 0
P r o p o r t i o n ( % )
a
)
t
M O = 0
.
1 M y r
R a r e f = 1 0 1 0
R a r e f = 1 0 9
R a r e f = 1 0 8
0 0
.
2 5 0
.
5 0
.
7 5 1
0
.
0 1
0
.
1
1
1 0
1 0 0
B
= 0
.
2
b
)
t
M O = 1 M y r
0 0
.
2 5 0
.
5 0
.
7 5 1
0
.
0 1
0
.
1
1
1 0
1 0 0
c
)
t
M O = 1 0 M y r
0 0
.
2 5 0
.
5 0
.
7 5 1
0
.
0 1
0
.
1
1
1 0
1 0 0
P r o p o r t i o n ( % )
d
)
0 0
.
2 5 0
.
5 0
.
7 5 1
0
.
0 1
0
.
1
1
1 0
1 0 0
B
= 1
e
)
0 0
.
2 5 0
.
5 0
.
7 5 1
0
.
0 1
0
.
1
1
1 0
1 0 0
f
)
0 0
.
2 5 0
.
5 0
.
7 5 1
S h r i n k i n g f a c t o r
0
.
0 1
0
.
1
1
1 0
1 0 0
P r o p o r t i o n ( % )
g
)
0 0
.
2 5 0
.
5 0
.
7 5 1
S h r i n k i n g f a c t o r
0
.
0 1
0
.
1
1
1 0
1 0 0
B
= 1
.
8
h
)
0 0
.
2 5 0
.
5 0
.
7 5 1
S h r i n k i n g f a c t o r
0
.
0 1
0
.
1
1
1 0
1 0 0
i
)

Figure 2.6:
Histograms of the distribution of the shrinking factor after 11 Myr for all sim ulations.
The v ertical axis represen ts (in log scale) the percentage of tracers ha ving a shrinking factor around
the v alue indicated on the x -axis.
zone exp eriences prolonged mixing, while the b ottom la y er exp eriences just limited shrinking
during the o v erturn, resulting in a significan t spread in the v alues of the shrinking factor (see
red histogram on Figure 2.6g).
The heterogeneit y of solid-state mixing is illustrated in Figure 2.7, where we compare
sim ulations with differen t solidification times for the case
B
= 1
.
8 and
Ra ref
= 10
10
. The
figure sho ws, after 11 Myr, snapshots of the comp osition field (left column), of the in tegrated
shrinking factor (middle column), and laterally a v eraged profiles (righ t column) of temp erature
(red line), comp osition (blue line), and in tegrated shrinking factor (green line, log-axis). The
latter represen ts the final v alue of the shrinking factor mapp ed according to the initial p osition
of the tracers. A region with a high (resp ectiv ely lo w) v alue of the in tegrated shrinking factor
indicates that man tle material initially lo cated in that sp ecific region has undergone a po or
(resp ectiv ely in tense) mixing.
The case with
t MO
= 0
.
1 Myr (top line in Figure 2.7) is c haracterized b y a late onset of
con v ection, with a large-scale o v erturn taking place underneath a stagnan t lid. This global
o v erturn results in a comp ositionally la y ered structure as sho wn in Figure 2.7a and b y the blue
profile of Figure 2.7c. The upp ermost lay ers lo cated b eneath the stagnan t lid sink to the CMB
39

2. Early onset of con v ection and man tle o v erturn: the case of Mars
and remain there. These la y ers exp erience th us considerably less shrinking than other parts of
the man tle that are in volv ed in the subsequen t con v ection. These p o orly mixed cum ulates
originate from the region mark ed b y the t w o dark blue areas in Figure 2.7b corresp onding to the
lo cal maxim um of the green profile in Figure 2.7c lo cated at a non-dimensional radius b et w een
1.50 and 1.75. Note that after the initial o v erturn, thermal con v ection is not immediately
suppressed b y the stable comp ositional gradien t, but con tin ues b elo w the stagnan t lid, causing
this to thin from its initial thic kness of
∼
25% of the man tle’s depth to its final thic kness as
seen in Figures 2.7a and 2.7c. It is also w orth noting the presence of a rather small region
c haracterized b y a v ery lo w v alue of the in tegrated shrinking factor (red patc h on Figure 2.7b).
T racers initially lo cated in this region w ere fi rst adv ected in an up w elling plume that turned
later in to a do wn w elling, thereb y ep xp eriencing extensiv e shear, translating in to a significan t
shrinking o v er the analyzed time in terv al.
The second case, with
t MO
= 1 Myr (middle line in Figure 2.7) exhibits an early onset of
con v ection. Ho w ev er, the initial o v erturn o ccurs near the end of the crystallization, making it
qualitativ ely similar to the previous case. Y et an early onset promotes mixing (Figure 2.7d)
and, as a consequence, this case is c haracterized b y a generally lo w er v alue of the in tegrated
shrinking factor (i.e., more efficien t mixing) throughout the man tle (Figure 2.7f ).
The last case, with
t MO
= 10 Myr (b ottom line in Figure 2.7) presen ts a v ery different
mixing pattern. Due to the long-liv ed magma o cean, the first o v erturn o ccurs early during
solidification and causes a first la y er lo cated at mid-man tle depth to sink to the CMB where it
remains unmixed for some time (see blue strip e at mid-man tle depth in Figure 2.7h and lo cal
maxim um of the green profile in Figure 2.7i), while the initially underlying material is entrained
b y vigorous thermal con vection. By the end of the solidification, the densit y heterogeneit y has
b een progressiv ely erased b y con v ectiv e mixing (see blue profile on Figure 2.7i), and ev en the
material that accum ulated at the CMB after the first o verturn is re-en trained in the bulk of
the man tle b y con v ection (no dense la y er is presen t atop the CMB on Figure 2.7g), while the
newly crystallized upp ermost lay ers of the stagnan t lid are progressiv ely en trained b y large
do wn w ellings.
2.4.3 Correlation b et w een early or late onset of con v ection and mixing
The v arious influences of the main parameters on the final state of the man tle are not
straigh tforw ard to define b ecause the trends are not necessarily monotonic and some effects
exhibit couplings. Figure 2.8 presen ts the final v alue (after 11 Myr) of the a v erage shrinking
factor for all sim ulations co v ering the en tire parameter space in v estigated. Eac h diagram is
divided in t w o domains according to the time of onset of con v ection in the sim ulations (early
or late onset). As exp ected, the reference Ra yleigh n umber has a clear monotonic influ ence
on the onset of con v ection and on the degree of mixing (Sam uel et al., 2011; Sam uel & T osi,
2012). Ho w ev er, the dep endence of the final v alue of the shrinking factor on the buo y ancy
ratio
B
and on the solidification time
t MO
is more complex. In particular, t w o opp osite effects
of t MO can b e identified:
1.
When con v ection starts only after complete solidification (”late onset” in Figure 2.8), its
onset o ccurs earlier for lo w er v alues of
t MO
b ecause the full thermal and comp ositional
densit y con trasts across the man tle b ecome a v ailable earlier, i.e. the final v alue of the
40

2.4 Results
Figure 2.7:
Snapshots of the comp osition field (a, d, g), in tegrated shrinking factor (b, e, f ) and
laterally a v eraged profiles of comp osition, temp erature, and in tegrated shrinking factor (c, f, i)
after 11 Myr for sim ulations assuming fractional crystallization with
B
= 1
.
8 ,
Ra
= 10
10
, and
differen t solidification durations (
t MO
= 0
.
1 Myr in a–c,
t MO
= 1 Myr in d–f, and
t MO
= 10 Myr
in g–i). The in tegrated shrinking factor is a mapping of the final v alue of the shrinking factor of
eac h tracer, plotted at the initial p osition of the tracer.
effectiv e Ra yleigh num b er is reac hed so oner. As a consequence, con v ection has more time
to mix the man tle, so that, b y the end of the sim ulation time, a higher lev el of mixing
(i.e. a smaller shrinking factor) is ac hiev ed, which is further enhanced b y increasing the
buo y ancy ratio.
2.
When con v ection starts b efore the end of solidification (”early onset” in Figure 2.8), the
mixing of the solid cum ulates is more efficien t during solidification than after it b ecause
the con tin uous formation of denser and denser material at the top of the solid cum ulates
acts as a supplemen tary source of buo yancy . Mixing during solidification also erases
efficien tly comp ositional heterogeneities, av oiding the formation of a stably stratified
structure that w ould hinder subsequen t thermal conv ection. Therefore, in these cases, a
b etter mixing is achiev ed for b oth a longer t MO and a higher B .
41

2. Early onset of con v ection and man tle o v erturn: the case of Mars
The influence of
t MO
is con trolled b y b oth
Ra ref
and
B
. F or lo w v alues of
Ra ref
, the first effect
discussed ab o v e is dominan t b ecause an early onset of con v ection is less lik ely , and a higher
degree of mixing is ac hiev ed for lo w er
t MO
(compare the upp er rows of the three diagrams in
Figure 2.8). F or high v alues of
Ra ref
, the second effect b ecomes dominan t and the opp osite
trend is observ ed (compare the b ottom ro ws in Figure 2.8). F or in termediate v alues of
Ra ref
,
the transition from the first to the second effect o ccurs up on increasing
B
(compare the middle
ro ws in Figure 2.8). Note that in case of early onset, although high v alues of
B
enhance mixing
at the end of solidification as sho wn in Figure 2.8, they also increase the absolute densit y
con trasts. As a consequence, cases with a high v alue of
B
still tend to end up with a stronger
stable comp ositional gradien t in the bulk con v ecting man tle than cases with a lo w v alue of
B
.
t
M O = 0
.
1 M y r
t
M O = 1 M y r
t
M O = 1 0 M y r
B
= 0
.
2
R a r e f = 1 0 8
R a r e f = 1 0 9
R a r e f = 1 0 1 0
l a t e o n s e t
e a r l y o n s e t
t
M O = 0
.
1 M y r
t
M O = 1 M y r
t
M O = 1 0 M y r
B
= 1
R a r e f = 1 0 8
R a r e f = 1 0 9
R a r e f = 1 0 1 0
l a t e o n s e t
e a r l y o n s e t
t
M O = 0
.
1 M y r
t
M O = 1 M y r
t
M O = 1 0 M y r
B
= 1
.
8
R a r e f = 1 0 8
R a r e f = 1 0 9
R a r e f = 1 0 1 0
l a t e o n s e t
e a r l y o n s e t
0 1
S t r o n g m i x i n g W e a k m i x i n g
S h r i n k i n g f a c t o r a f t e r 1 1 M y r

Figure 2.8:
A verage shrinking factor after 11 Myr for all sim ulations as a function of
t MO
and
Ra ref for the three different v alues of B The color-scale is linear.
2.5 Discussion
2.5.1 Application to Mars’ man tle
2.5.1.1 Plausible parameter range for Mars
The reference Ra yleigh n umber for the early Martian man tle is p o orly constrained, particularly
b ecause of the large uncertainties in the man tle rheology . Among the n umerous factors
that affect the reference viscosit y , whic h en ters the Ra yleigh num b er, the w ater con ten t is
particularly critical b ecause of the wide range of v alues adv o cated in the literature (e.g. R uedas,
T ac kley, & Solomon, 2013; Breuer, Plesa, T osi, & Grott, 2016). F rom the analysis of the SNC
meteorites, w ater concen trations in the man tle source regions ha v e b een estimated to range
from 55 to 160 ppm for the depleted shergottites and from 120 to 220 ppm for the enric h ed
ones (W atson, Hutc heon, Epstein, & Stolp er, 1994; Leshin, 2000; McCubbin et al., 2012). If
these ranges are also represen tativ e for the solid cumulates that formed during magma ocean
solidification, the w ater concen tration of the early Martian mantle ma y ha v e b een ev en higher
than the maxim um v alue of assumed here of 100 ppm, for whic h w e ha v e
Ra ref
= 10
10
. T esting
the effects of higher w ater concen trations, i.e. of higher reference Ra yleigh n um b ers has pro v en
42

2.5 Discussion
computationally c hallenging. Y et, increasing
Ra ref
b y lo w ering the reference viscosit y w ould
simply imply an earlier onset of con v ection and more efficien t man tle mixing. F or comparison,
note that Elkins-T an ton et al. (2005b) used
Ra ref
= 3
×
10
9
, alb eit in the con text of iso viscous
sim ulations.
Note also that w e ha v e not considered the effects of stress-dep enden t, non-Newtonian
viscosit y . This kind of rheology , whic h is often mimic k ed by simply using a Newtonian viscosit y
similar to the diffusion creep regime, with a reduced activ ation energy (e.g. Sc hein b erg et al.,
2014), could b e relev an t for Mars and ha v e an imp ortan t influence on the onset time of
con v ection and on man tle mixing. In particular, the first ov erturn is lik ely to induce large
shear stresses that in turn could reduce the viscosit y ev en further. This is an imp ortan t issue
that should b e addressed in future studies.
The buo y ancy ratio ultimately depends on the amount of iron presen t in the man tle and
on ho w it is partitioned b et w een the melt and the solid cum ulates as the man tle solidifies
under the assumption that fractional crystallization o ccurs. In a recent review, T aylor (2013)
suggested an iron o xide con ten t of around 18 wt.% for the bulk silicate Mars. In the case
of a linear comp ositional gradien t (completely iron-depleted at the b ottom), this abundance
w ould result in a buo y ancy ratio 0.86, assuming that iron replaces magnesium o xide (whic h
is in excess compared to iron o xide) in the solid cum ulates as crystallization pro ceeds. The
crystallization sequence prop osed by Elkins-T an ton et al. (2005b) results in a v ery large
B
of 3.34 (but with a more realistic densit y gradien t). This v alue w as emplo y ed by Plesa et al.
(2014) to sim ulate a whole-man tle ov erturn and to sho w that, with suc h a large buo y ancy ratio,
p ost-o v erturn man tle con v ection and melting would b e strongly inhibited. The v alues of
B
that w e tested in this study (
B ≤
1
.
8 ) are lo w er than the v alue predicted b y Elkins-T anton
et al. (2005b), but close to the v alues based on T a ylor (2013) with a simple linear profile.
In our analysis of the early onset of solid-state con v ection, the time scale of man tle
solidification
t MO
pla ys a cen tral role. F or this, the initial v olatile con ten t of the planet is
of fundamen tal imp ortance as it can determine the emergence of an insulating atmosphere
generated up on magma o cean solidification and degassing. Along with w ater, the
CO 2
con ten t
of the man tle is also essen tial b ecause of its greenhouse p oten tial as a gas. Debaille et al. (2007)
suggest a long-liv ed Martian magma o cean with a lifetime in excess of 100–200 Myr based
on Sm-Nd systematics of shergottites. This time scale, how ev er, is significan tly longer than
that determined using coupled in terior-atmosphere mo dels of man tle solidification. Assuming
0.6 wt.% con ten t of
CO 2
and no w ater, Elkins-T an ton (2008) found that a 2000 km deep
magma o cean on Mars w ould solidify in less than 3 Myr, while Lebrun et al. (2013) obtained
solidification times of ab out 1 Myr assuming 420 ppm H
2
O and 140 ppm CO
2
. A recen t
study b y Bouvier et al. (2018) based on isotop e age dating of zircons from Martian meteorites
prop osed that the Martian magma o cean w as crystallized within the first 20 Myr of the solar
system ev olution. These estimates are thus in line with our c hoice of t MO ≤ 10 Myr.
In summary , despite significan t uncertain ties in the parameters that go vern the early
dynamics of solid man tle cum ulates, it seems lik ely that the Martian man tle was at least
partly mixed b y the time the magma o cean solidified, whic h calls in to question the scenario of
a whole-man tle o v erturn that has b een often used as a starting condition for sim ulations of
in terior ev olution.
43

2. Early onset of con v ection and man tle o v erturn: the case of Mars
2.5.1.2 Martian geo c hemical reserv oirs
The isotopic analysis of the Martian meteorites rev eals large v ariations in lithophile radiogenic
isotop e systems suc h as
87
Rb
− 86
Sr,
147 , 146
Sm
− 143 , 142
Nd, and
176
Lu
− 176
Hf, and suggests the
existence of three to four distinct c hemical reserv oirs (e.g., Jargoutz, 1991; F oley et al.,
2005). The range of crystallization ages determined for these samples, from 4.5 Ga to 160 Ma
(Nakam ura, Unruh, T atsumoto, & Hutc hinson, 1982b; Nakam ura, K ogi, & Kagami, 1982a;
Nyquit et al., 2001; Huma yun et al., 2013; Agee et al., 2013), indicates that their source-
reserv oirs ha ve b een preserv ed o v er the en tire thermo-c hemical ev olution of Mars. T o explain
suc h systems, the scenario of a global cum ulate ov erturn has b een in v ok ed in previous studies
b ecause of its ability to create a stable densit y configuration, whic h easily allo ws the isotopic
signature of v arious reserv oirs to b e preserv ed un til presen t-day (Elkins-T anton et al., 2005b;
Elkins-T an ton et al., 2005a). Ho w ev er, the predicted comp ositional densit y gradien t follo wing
the crystallization and global o v erturn of the magma o cean, alb eit b eing able to form and
main tain distinct reserv oirs ov er the en tire thermo-c hemical ev olution of the planet, is to o
large to allo w subsequen t thermo-c hemical con v ection, with the consequence that sampling of
the v arious mantle sources through partial melting is unlikely (T osi et al., 2013; Plesa et al.,
2014). The crystallization of the liquid magma o cean is a highly dynamical pro cess and, as
sho wn in Section 2.4.3, the onset of solid-state con v ection w ell b efore complete solidification
ma y strongly reduce c hemical heterogeneities. In fact, a more suitable scenario w ould require
smaller densit y gradien ts to b e present at the end of the crystallization phase, so that reserv oirs
can b e sampled b y subsequen tly formed man tle plumes and partial melting. This configuration
can b e ac hiev ed through progressiv e mixing of the solid cum ulate due to an early onset of
con v ection.
It should b e noted that Ogaw a and Y anagisaw a (2011) and Plesa and Breuer (2014) prop osed
differen tiation due to secondary melting in an initially hot and homogeneous p ost-magma o cean
man tle as an alternativ e wa y to generate – and p ossibly preserv e – large-scale comp ositional
heterogeneities. This mec hanism w ould apply in the case of a magma o cean undergoing batc h
crystallization. Ho w ever, in the case of fractional crystallization, and particularly if the man tle
exp erienced an early onset of con v ection and hence partial mixing, it is not straigh tforw ard
to predict to whic h exten t primordial comp ositional anomalies generated b y magma o cean
fractionation w ould affect the subsequen t differen tiation asso ciated with partial melting of the
solidified man tle.
2.5.2 A more realistic ev olution of man tle solidification
The linear unstable densit y profile that w as used in this and previous works (Zaranek &
P armen tier, 2004; T osi et al., 2013; Elkins-T an ton et al., 2005b) is only a crude appro ximation
of the densit y profile obtained with mo dels of fractional crystallization (e.g. Elkins-T an ton,
P armen tier, & Hess, 2003; Elkins-T an ton et al., 2005b). Its actual shap e influences b oth the
onset time of con v ection b ecause it affects directly the ev olution of the effectiv e buo yancy
forces term, and the degree of mixing ac hiev ed at the end of solidification since it determines
the lo cal densit y gradien t in the upp er b oundary la yer, where newly crystallized material is
lo cated. Realistic profiles are t ypically c haracterized b y a relativ ely flat gradient in the deep
man tle that gro ws steep er and steep er close to the surface, as that represen ted in Figure 1.11.
44

2.5 Discussion
Since w e ha ve demonstrated that the buo y ancy ratio has only a small influence on the onset of
con v ection (Figure 2.4), for given
Ra ref
and
t MO
– the o v erturn b eing mainly thermally driv en
– the use of a more realistic crystallization sequence w ould only result in a sligh tly dela y ed
onset time.
A constan t co oling rate of the magma o cean is also only a first-order appro ximation of
the solidification ev olution. A v olumetrically constan t solidification rate w ould result in a
slo w er solidification of the shallow er la y ers with resp ect to the deep er ones. F urthermore, the
heat flo w through a turbulen t liquid magma o cean scales with the Ra yleigh n um b er of the
magma o cean (
Ra MO ∼
10
20 −
10
30
) to an exp onen t v arying b et w een 1
/
3 and 2
/
7 (Solomato v,
2007). Th us heat is less efficien tly extracted as the magma o cean b ecomes shallo w er, i.e. as its
Ra yleigh n umber decreases. If no atmosphere is formed during the solidification, the radiativ e
heat loss at the surface of the planet decreases with the surface temp erature of the planet
to the fourth p o w er, while if a blank eting atmosphere is outgassed, the heat escap e ma y b e
considerably slo w ed do wn (Lebrun et al., 2013; Nik olaou et al., 2019). The evolution of the
differen t heat-fluxes all act in fa v our of a decrease of the solidification rate with time, i.e. a
slo w er solidification of the shallo wer la y ers. A more realistic solidification ev olution is th us
lik ely c haracterized b y a low er man tle solidifying rapidly , p ossibly via batch crystallization,
and an upp er mantle solidifying m uc h more slo wly , probably via fractional crystallization.
The time-ev olution of the solidification fron t would also be influenced by the release of
laten t heat during crystallization. Since the crystallization time is imp osed and not mo deled
self-consisten tly , laten t heat effects are implicitly incorp orated in this parametrization. The
influence of laten t heat consumption or release on solid-state con vection is not relev an t for
the fractional crystallization cases where only subsolidus temp eratures are considered in the
cum ulates. F or the batc h crystallization cases, instead, laten t heat effects w ould hav e to
b e considered in the m ush domain. F or a sp ecific case with
t MO
= 10
Myr
, w e carried out
an additional sim ulation including laten t heat, whic h w e to ok it in to accoun t b y adding the
term
− L∂ ϕ/∂ t
to the righ t hand side of the thermal energy equation (
(2.5)
). The addition of
laten t heat acts as a lo cal heat source up on solidification, whic h tends to equalize temp erature
differences and, in turn, to sligh tly hinder the gro wth of the imp osed initial p erturbation. As
a consequence, w e observ ed an onset of con v ection dela y ed by about 10% with resp ect to the
corresp onding case, where latent heat w as neglected.
2.5.3 Application to other terrestrial b o dies
The terrestrial b o dies of the inner solar system including (and in particular) the Mo on are
though t to ha v e exp erienced magma o cean episo des (Elkins-T an ton, 2012). Ho w ev er, these
b o dies ha v e largely differen t prop erties, whic h lik ely resulted in different solidification scenarios.
Belo w w e discuss ho w the ma jor terrestrial b o dies w ould fit in the parameter space that w e
explored.
The reference Ra yleigh n um b er is mainly con trolled by the size of the bo dy , while the
buo y ancy ratio, indicativ e of its man tle’s iron con ten t, migh t increase with distance from the
Sun as a result of the accretion history (T a ylor, 2013). Assessing the initial v olatile con tent
is extremely sp eculativ e, therefore the lifetime of the magma o cean is probably the least
constrained parameter.
45

2. Early onset of con v ection and man tle o v erturn: the case of Mars
2.5.3.1 Earth
The Earth’s man tle is 1.7 times as deep as Mars’s, whic h increases the reference Ra yleigh
n um b er b y a factor 5. Although it formed from a v olatile-depleted zone of the protoplanetary
disc (Carlson et al., 2014), and the mec hanisms in volv ed in the deliv ery of v olatile on Earth
are not clearly iden tified, it is lik ely that Earth’s water con ten t was deliv ered during planetary
formation (Morbidelli, Lunine, O’Brien, Ra ymond, & W alsh, 2012a), so that it w as already
a v ailable b y the time of the magma o cean solidification. The Earth can th us ha v e dev elop ed a
thic k blank eting atmosphere, able to sustain a long-liv ed magma o cean (Ab e, 1997; Zahnle
et al., 2007; Lebrun et al., 2013; Nik olaou et al., 2019, e.g.). A ccording to our results, a
late magma o cean on Earth is likely to ha v e exp erienced solid cum ulate mixing during its
crystallization, resulting in a roughly homogeneous man tle structure, ev en though the pressure
dep endence of the rheology , more relev ant for the Earth than for Mars because of the higher
pressure range, ma y ha v e caused a less efficient mixing of the lo w er man tle. These findings
are corrob orated b y (Ballmer et al., 2017), who p erformed similar sim ulations but using a
more realistic iron fractionation mo del and a simplified (isoviscous) rheology . As noted in
Section 1.3.2.6, the b ottom-up crystallization scenario migh t not b e applicable to the Earth,
due to either the melting curv e gradien t b ecoming less steep than the adiabat at high pressures
(Stixrude, de K ok er, Sun, Mo okherjee, & Karki, 2009) or the densit y cross-o v er b et w een solid
and liquid due to enric hmen t of melts in iron (Boukaré et al., 2015). Mo delling the latter effect,
Miy azaki and K orenaga (2019) sho w ed that, dep ending on the efficiency of crystal settling
and cum ulates compaction (whic h they found to b e decisiv e for the fate of iron distribution),
sub-man tle scale Ra yleigh-T a ylor instabilit y w as also lik ely to affect the thermal structure of
the man tle, in fa vour of man tle mixing and p oten tially formation of a long-liv ed basal magma
o cean (Labrosse et al., 2007; Laneuville et al., 2018). This scenario still has to b e compared
with presen t-da y p ossible traces of a past magma o cean, in particular with the presence of
large lo w-shear-v elo cit y pro vinces at the core man tle b oundary (e.g. Garnero, McNamara,
& Shim, 2016). In terpreted in the ligh t of p ost magma o cean solidification dynamics, they
could represen t remnan ts of a long-liv ed basal magma o cean, or o v erturned cumulates of a
b ottom-up solidified mantle (Ballmer et al., 2017).
2.5.3.2 V en us
V en us and Earth are similar in man y resp ects; in particular, they ha v e nearly the same size and,
p ossibly , also the same bulk comp osition (F egley, 2005), although their presen t-da y surface
conditions are v ery differen t. The ev olution of a putativ e magma o cean clearly dep end on the
early surface conditions on V en us, whic h are essen tially unconstrained. Nev ertheless, it should
b e noted that, dep ending on its initial w ater con ten t, V en us could ha v e sustained a magma
o cean for more than 100 Myr (Hamano et al., 2013; Nikolaou et al., 2019). This, together
with a high Ra yleigh n um b er comparable with that of the Earth, ma y hav e resulted in highly
efficien t mixing during magma o cean solidification and hence in a w ell homogenized man tle
already at the b eginning of its evolution.
46

2.5 Discussion
2.5.3.3 Mercury
With a man tle thic kness of only
∼
400 km (Hauc k et al., 2013), Mercury has the thinnest
silicate shell among the terrestrial planets, resulting in a significan tly lo w er Ra yleigh n um b er.
The relativ ely high surface abundance of sulfur (Ev ans et al., 2012), along with the evidence of
p yro clastic v olcanism (Thomas, Rothery, Con w a y, & Anand, 2014) indicate that Mercury ma y
b e relativ ely ric h in v olatiles. Ho w ev er, its pro ximity to the Sun suggests that it should ha v e
formed in a hot region of the protoplanetary disk from v olatile-depleted materials (F egley &
Cameron, 1987), whic h mak es unlikely the outgassing of a significant blank eting atmosphere
able to surviv e escap e pro cesses. A rapid co oling of Mercury’s magma o cean may ha v e
prev en ted crystal-melt separation during crystallization, thus leading to a rather homogeneous
man tle. Nevertheless, spectroscopic measurements of Mercury’s surface materials p erformed
during the MESSENGER mission ha v e rev ealed the existence of comp ositionally distinct
signatures (W eider et al., 2015). As for the Mo on, fractional crystallization of a magma o cean
c haracterized b y a late stage in v olving the flotation of buo y an t crystals – graphite in the case of
Mercury (V ander Kaaden & McCubbin, 2015) – and accompanied b y a whole-man tle o v erturn
seems then to b e a viable scnenario (Brown & Elkins-T anton, 2009; Charlier, Gro v e, & Zub er,
2013). F urthermore, the magma o cean solidification on Mercury has b een sho wn to influence
the distribution of heat-pro ducing elemen ts as a result of their partitioning in exsolv ed sulfides
(Boukaré, P arman, P armentier, & Anzure s, 2019). Although w e did not study the influence of
the initial distribution of heat sources, this lik ely influences the subsequen t man tle dynamics
as w ell.
2.5.3.4 Mo on
In general, based on our results on the onset of con v ection and mixing during magma o cean
solidification, the early lunar mantle is expected to b e less homogeneous compared to the
man tles of Mars, V en us and Earth, essen tially b ecause the effectiv e Ra yleigh n um b er of the
Mo on is smaller. Ho w ever, as will be shown in the next c hapter, w e exp ect the crystallization
of the lunar magma o cean to o ccur o ver a v ery long time (
∼
180 Myr) b ecause of the formation
of an insulating flotation crust. This protracted timescale allo ws for a prolonged p eriod of
mixing. An in-depth analysis of the lunar mantle mixing in relationship with the magma o cean
solidification is pro vided in Chapter 5.
Conclusions
The solidification of a magma o cean during the early stages of the ev olution of terrestrial b o dies
is often assumed to b e a rapid pro cess, o ccuring within thousands of y ears (e.g. Mon teux et al.,
2016) b ecause of the v ery efficien t radiativ e co oling of the hot surface in con tact with cold
space. In case of a rapid solidification, batc h crystallization, whic h results in a homogeneous
man tle, or fractional crystallization follo w ed by a global ov erturn, whic h results in a stably
stratified man tle, can tak e place. Nev ertheless, outgassing of a blank eting atmosphere can
retard heat escap e and sustain a magma o cean for a few million y ears (Elkins-T an ton, 2008;
Lebrun et al., 2013; Nik olaou et al., 2019) up to h undreds of million y ears for b o dies, lik e
V en us, that exp erience a strong insolation (Hamano et al., 2013; Nik olaou et al., 2019). W e
47

2. Early onset of con v ection and man tle o v erturn: the case of Mars
sho w ed that, due to the lo w viscosit y of the hot solid cum ulates (p oten tially further decreased
b y the presence of w ater and residual interstitial melts), a solidification duration of
∼
1 Myr is
sufficien t for solid-state con v ection to b egin during the crystallization of the magma o cean.
Con v ectoin then starts mixing the unstable comp ositional densit y gradient resulting from iron
partitioning during fractional crystallization. F or the longest magma o cean’s lifetime tested
(10 Myr), and for a sufficien tly high reference Ra yleigh n umber (
Ra ref ≳
10
9
), a high lev el of
mixing of the solid cum ulates can b e ac hiev ed b y the end of the solidification. In this situation,
comp ositional heterogeneities are at least partly erased during the early ev olution, with the
consequence that the traditional scenario of a whole-man tle o v erturn b ecomes unlik ely .
48

3
Gro wth of a flotation lid: the case of
the Mo on
This c hapter is largely based on the published pap er: Maurice, M., T osi, N., Sc h winger, S.,
Breuer, D. and Kleine, T. (2020). A long-liv ed magma o cean on a y oung Mo on. Scienc e
A dvanc es 6 , eaba8949
3.1 The lunar magma o cean
The most widely accepted theory ab out the formation of the Mo on, the so-called gian t impact
h yp othesis (Can up & Asphaug, 2001; Asphaug, 2014; Barr, 2016), p osits that the Mo on
accreted from a debris disk formed around the proto-Earth after it w as hit b y a planetary
em bry o. Although some studies ha v e in v estigated the pro cess of lunar ac cretion from the debris
disc (Can up, 2004; P ahlev an & Stev enson, 2007; Lo c k et al., 2018) and its thermal outcomes
(Pritc hard & Stev enson, 2000), this remains too p o orly constrained for mo dels to pro vide a
robust estimate of the initial thermal state of the Mo on. The existence of a global magma
o cean on the early Mo on is nonetheless strongly supp orted b y geo chemical and mineralogical
evidences from the lunar sample record as exp osed in the in tro duction of this thesis.
The results presen ted in the previous c hapter seem to suggest that an early onset of
solid-state con v ection during the solidification of the lunar magma ocean is unlike ly due to
the small size (hence lo w Ra yleigh n um b er) of the Moon. Ho w ev er, due to the formation of
a plagio clase flotation crust at the surface of the magma o cean, its solidification timescale
is lik ely to exceed the maxim um v alue previously in v estigated (10 Myr). Indeed, on one
hand, the radiated heat flux at the liquid surface of a magma o cean can be estimated as
q rad ∼ σ T 4
M O ∼
10
4
to 10
5
W/m
2
(with the Stefan-Boltzmann constan t
σ
= 5
.
67
×
10
− 8
W/m
2
/K
4
and the surface temp erature of the magma o cean
T MO ∼
10
3
K). On the other
hand, the conductiv e heat flux through a flotation crust can b e estimated b y F ourier’s la w:
q cond ∼ k crust
(
T MO − T s
)
D − 1
crust ∼
10
− 2
to 10
− 1
W/m
2
(with the thermal conductivit y of the
crust
k crust ∼
10
0
W/m/K, the temp erature contrast accross the crust
T MO − T s ∼
10
2
to
49

3. Gro wth of a flotation lid: the case of the Mo on
10
3
K –
T s
b eing the temp erature at the surface of the crust – and the thic kness of the crust
D crust ∼ 10 4 m). Th us the presence of a flotation crust induces a decrease of 4 to 5 orders of
magnitude in the efficiency of the magma o cean’s co oling. If one considers that a radiating
magma o cean crystallizes in 10
3
y ears (due to the small size of the Mo on, the lunar magma
o cean is unlik ely to outgas and retain a significan t insulating atmosphere, unlik e Mars in the
previous c hapter), the solidification of a magma o cean b elo w flotation crust w ould tak e up to
10
9
y ears. Although this simple calculation clearly yields an o v erestimate of the crystallization
timescale for the lunar magma o cean (making it comparable to the age of the solar system,
meaning that the Mo on should still host a liquid magma o cean), it pro vides a go o d illustration
of the reduced colling efficien y induced b y the presence of a flotation crust. The presen t c hapter
aims at mo delling it in a robust fashion, and the follo wing one at using this mo del to prop ose
a new in terpretation of the isotopic record of the ln uar samples related to the early ev olution
of the Mo on.
3.1.1 Existing mo dels for the lunar magma o cean solidification
Plagio clase b ecomes a liquidus phase relativ ely late in the crystalliation sequence of the lunar
magma o cean. Th us, only the crystallization of the last few volume ten ths of the lunar
magma o cean o ccurs b elo w a gro wing flotation lid and is effectiv ely protracted. This largely
decorrelates the duration of the lunar magma o cean solidification from its in tial depth.
Solomon and Longhi (1977) w ere among the first to prop ose a self-consisten t thermal
ev olution mo del for the lunar magma o cean, finding that a 200 to 400 km-deep magma o cean
on the Mo on should crystallize in 90 to 180 Myr. Ho w ev er, their consideration of a conductiv e
lid throughout the whole magma o cean lifetime as w ell as an early crystalization of plagio clase
and a thic k final crust (70 km) result in the o v erestimation of the effect of the flotation crust
thermal insulation. Relying on more recen t crustal thic kness estimates and crystallization
sequence, Elkins-T an ton, Burgess, and Yin (2011) found a solidification lasting up to
∼
35
Myr for an initially 1000 km-deep magma o cean, dep ending on the depth of the magma o cean
at the b eginning of plagio clase crystallization (Figure 3.1).
A protracted crystallization of the lunar magma o cean, asso ciated with prolonged crust
formation, has b een prop osed to explain the large span in isotopic age measuremen ts among
Mo on’s crustal samples. Ho w ev er, this span co v ers 200 Myr (Elkins-T an ton et al., 2011),
whic h is m uch longer than the duration predicted b y the recen t thermal ev olution mo del from
Elkins-T an ton et al. (2011), ev en when solidification is slo w ed b y the insulating p o w er of the
flotation crust.
V arious effects that could further prolong the solidification of the lunar magma o cean ha v e
b een in v estigated. Meyer, Elkins-T anton, and Wisdom (2010) studied the coupling b et w een
the Earth-Mo on orbital system and the thermal ev olution of the inner early Mo on, affected
b y tidal heating. Some gian t impact studies (Ćuk & Stew art, 2012; Canup, 2012) explain
the Earth-Mo on isotopic similarities (suggesting the fact that the Mo on deriv es mainly from
terrestrial man tle’s material, whic h is in con tradiction with basic giant impact sim ulations)
with mo dels in which the inital Earth-Moon system’s angular momentum w as m uc h higher
than it is to da y . A substan tial amoun t of angular momen tum needs to b e dissipated to re ac h
the presen t-da y configuration, an d it is ac hiev ed partly through tidal dissipation. During the
50

3.1 The lunar magma o cean
Figure 3.1:
Time-series of the temp erature of the magma o cean, solidifying b elow a lid of constan t
thic kness (40 km, dashed curv es) or a gro wing lid (final thic kness 40 km, solid curv e). The n um b ers
asso ciated with eac h curv e indicate the depth at whic h plagio clase starts to crystallize (from
(Elkins-T an ton, Burgess, & Yin, 2011))
existence of the lunar magma o cean, the flotation crust is decoupled from the solid man tle,
and is prone to tidally-induced deformation and resulting heating. Although in Meyer et al.
(2010)’s mo del the lunar magma o cean crystallization can th us b e extended up to 200 Myr,
it dep ends on highly underconstrained parameters. F urthermore, in a follo w-up study , Tian,
Wisdom, and Elkins-T anton (2017) pointed out a ca v eat in the thermal ev olution mo del of
Mey er et al. (2010) that can lead to significan t o verestimation of the solidification duration.
Tides ma y also induce heating within the liquid magma o cean. Although tidal dissipation
is an increasing function of viscosit y and is hence minimal in liquids, Chen and Nimmo (2016)
prop osed that a resonan t configuration b et w een the obliquit y tides and the flo w pattern in
the magma o cean can result in significan t dissipation. Ho w ev er, the fo cus of the study w as
put on the orbital ev olution of the Earth-Mo on system, and the duration of the magma o cean
solidification merely used as a free parameter.
Finally , P erera et al. (2018) in v estigated the influence of impacting debris on the duration
of the lunar magma o cean solidification. Although the accretion of the Mo on after the gian t
impact is p o orly constrained, remnan t orbiting debris probably survived for millions of y ears,
whic h ev en tually collided with either the Earth or the Mo on. In the latter case, impactors
massiv e enough could p erforate the flotation crust, inducing an influx of kinetic energy that
con tributed in heating the magma o cean. Ho w ev er the net effect is t w ofold since the ”holes”
punctured in the crust increase the magma o cean’s outgoing heat flux.
3.1.2 Unconsidered effects
A common ca v eat to all the mo dels that aimed at sim ulating the solidification of the lunar
magma o cean b elo w a gro wing flotation crust is the use of to o high a v alue for the thermal
conductivit y of the anorthositic crust. The lunar flotation crust is largely anorthositic (still
to da y , the farside highland’s crust is comp osed of 80 to 97% of anorthosite (e.g. Y amamoto
51

3. Gro wth of a flotation lid: the case of the Mo on
et al., 2012)), a mineral that has a lo w thermal conductivit y , of
∼
1
.
5 W/m/K (Branlund
& Hofmeister, 2012). The studies previously men tioned all used v alues around 3.4 W/m/K,
thereb y significan tly o v erestimating heat conduction through the crust.
Moreo v er, as sho wn in the previous c hapter, ev en for a Ra yleigh n um b er corresp onding to
that of the early Mo on, a crystallization of the magma o cean in at least 10 Myr, as predicted b y
thermal ev olution mo dels and lik ely a lo w er b ound, is compatible with the onset of solid-state
con v ection in the cum ulates b efore the end of the solidification. In the previous chapter, the
temp erature field w as buffered to the solidus v alue and the assumption w as made that further
heating w ould b e comp ensated b y secondary melting. The mafic melts th us pro duced are
p ositiv ely buo y an t and are extracted in to the magma o cean, where their heat is efficien tly
ev acuated via effective con v ectiv e and radiativ e transfer. In the case of the lunar magma o cean,
suc h melts w ould en ter the o v erlying magma o cean as well. Y et, due to the insulating crust,
their thermal energy requires m uc h more time to be remov ed. Hence the thermal ev olution
of the con v ecting cum ulates needs to b e link ed with that of the magma o cean, pro viding a
p oten tial heat source to increase the magma o cean’s lifetime. The heating of the magma
o cean b y extraction of hot secondary melts form the cum ulates is referred to in this study as
”heat-piping” .
In this c hapter, w e inv estigate the timing of the lunar magma o cean crystallization when
b oth a realistic v alue of the thermal conductivit y in the crust and the p ossibilit y of an early
onset of con v ection, and its influence on the thermal evolution of the magma ocean, are tak en
in to accoun t. Because this study is fo cused on the Moon, we use a dedicated crystallization
mo del repro ducing the lunar particularities. This mo del is presen ted in the follo wing Section,
while Section 3.3 describ es the coupled thermal evolution model of the Mo on’s in terior.
3.2 Crystallization mo del
The small pressure reac hed at the Mo on’s core-man tle b oundary makes the lunar man tle a
privileged sub ject for exp erimen tal studies, and a w ealth of p etrological results exist that
help constrain the crystallization of the early Mo on. In this w ork w e used the results of a
crystallization mo del for the lunar magma o cean in tro duced in Sc hwinger and Breuer (2018)
and briefly describ ed here. The mo del relies on a com bination of t w o n umerical crystallization
co des (Gibbs free energy minimizers) that are b enc hmark ed against published exp erimen tal
w ork. High pressure crystallization is computed using FXMOTR (Da v enp ort, Longhi, Neal,
Bolster, & Jolliff, 2014) whic h repro duces exp erimen tal results from (Rapp & Drap er, 2018),
and the lo w-pressure part of the crystallization sequence is obtained with alphaMEL TS (Smith
& Asimo w, 2005), whic h pro vides b etter agreemen t with exp erimen tal data from the same
set (Sc h winger & Breuer, 2018). W e assume that the magma o cean is initially 1000 km deep
(see section 3.2.1 for a discussion on the initial depth of the magma o cean), corresp onding to
a temp erature of 2048 K at the b ottom of the magma o cean. The crystallization pro ceeds
b y constan t temp erature decremen t of 1 K. All crystals (other than plagio clase) pro duced
as a result of this co oling are assumed to settle at the b ottom of the magma o cean, thereb y
thic k ening the cum ulates pile (whic h also retains 5 % of trapp ed in terstitial melt). When
plagio clase b ecomes a liquidus phase, its crystals are similarly separated from the magma o cean
52

3.2 Crystallization mo del
but settle at the surface, forming the flotation crust. The thic kness of the plagio clase c rust at
the end of crystallization is 44 km, in go od agreement with estimates based on gra vit y and
top ograph y data (Wieczorek et al., 2013). The bulk silicate Mo on comp osition is from (O’Neill,
1991) for the ma jor elements and from (T a ylor, 1982) for the heat-pro ducing elemen ts (U, Th,
K). The latter are assumed to b e pferfectly incompatible in the solid man tle, and con tained
only in the magma o cean and in the interstitial melt. The o xygen fugacit y is assumed to b e
constan t at one log-10 unit b elo w the iron-wüstite buffer (IW-1).
3.2.1 Initial depth of the magma o cean
The constrains on the initial depth of the lunar magma o cean are scarce, and their in terpretation
sp eculativ e. Previous studies ha v e considered v alues ranging from the whole man tle (i.e. a
1350-km-deep magma o cean (de V ries, v an den Berg, & v an W estrenen, 2010)) to a
∼
200 -km-
deep magma o cean (Solomon & Longhi, 1977). On one side, the lack of con traction features
at the surface of the Mo on has b een used to argue that the lunar man tle could not hav e
b een m uc h hotter in the past than it is to da y , setting an upp er b ound to the magma o cean’s
initial depth around
∼
400 km (Solomon & Longhi, 1977; Shearer et al., 2006). Ho w ev er
the con traction history of the Mo on is lik ely to ha v e b een influenced b y other pro cesses, lik e
burial of heat-pro ducing elemen ts, and its surface expression migh t ha v e b een blurred b y
asymmetrical heating of the crust. One the other side, the thic kness of the plagio clase crust
sets a lo w er b ound to the initial exten t of the lunar magma o cean, b ecause enough plagioclase
needs to b e crystallized out of it. A ccording to Solomon and Longhi (1977), only a magma
o cean initially deep er than
∼
200 km could pro duce a sufficien t crustal thic kness. Inciden tally ,
the thic kness of the crust has also b een used to argue against a whole-man tle magma o cean,
that w ould crystallize to o m uc h plagio clase (Shearer et al., 2006; Charlier, Gro v e, Nam ur, &
Holtz, 2018). Ho w ev er, our crystallization mo del nev er pro duces a crust that is too thick, ev en
for a whole-man tle magma o cean. The existence of a seismic anomaly at
∼
500 km depths
detected b y the Ap ollo seismometer arra y (Wieczorek et al., 2006) has also b een suggested to
record the exten t of the primordial magma o cean. Finally , the presence of a highly enriched
reserv oir b elo w the lunar crust also seems to indicate that a large part of the lunar man tle w as
efficien tly fractionated, arguably through fractional crystallization of a deep magma o cean.
Previous studies on the thermal ev olution of the lunar magma o cean ha v e largely fo cused
on the case of a 1000 km-deep magma o cean (Elkins-T anton et al., 2011; Mey er et al., 2010;
P erera et al., 2018). F rom the thermal ev olution standp oin t, the duration of the magma o cean
crystallization b efore the switc h from a radiativ e co oling regime to the growth of an insulating
flotation crust is negligible, regardless of its initial depth. Ho w ev er, the initial exten t of the
magma o cean influences its crystallization sequence, i.e. its crystallization temp erature, as
w ell as the melting curv es and heat pro ducing elemen t (HPE) distribution in its cum ulates.
In addition to our fiducial mo del of 1000 km-deep magma o cean (our fiducial mo del, b est
suited for comparison with previous studies, for whic h quan tities are giv en in the text, unless
otherwise stated), w e computed the crystallization sequences corresp onding to a magma o cean
initially co v ering the whole mantle, and an shallo w er one, initially 500-km deep. The resulting
crystallization temp erature profiles are presen ted on Figure 3.2, along with their resp ectiv e
HPE distributions.
53

3. Gro wth of a flotation lid: the case of the Mo on
1000 1200 1400 1600 1800 2000
temperature [K]
0
200
400
600
800
1000
1200
depth [km]
Previous studies
Whole-mantle LMO
1000 km LMO
500 km LMO
1 0 8 1 0 7 1 0 6 1 0 5
i n t e r n a l h e a t i n g [ W / m 3 ]
0
200
400
600
800
1000
1200
depth [km]

Figure 3.2:
Depth profiles of the crystallization temp eratures (a) and the in ternal heating
distributions (b) for an initially 1350-km-deep (i.e. whole-man tle, red curv es), 1000-km-deep (blue
curv es), and 500-km-deep (green curv es) lunar magma o cean. The dashed lines at the b ottom
of the crystallization temp erature profiles corresp ond to the accretion-lik e temp erature profile
assumed in the unmolten pristine lo w er man tle. The crystallization temp erature profile used in
Elkins-T an ton, Burgess, and Yin (2011) is sho w in grey for comparison.
The crystallization temp erature is globally only sligh tly affected b y the initial exten t of
the magma o cean, and remains within a range of
±
25 K from one case to the other, with the
exception of the last crystallized cum ulates, whose enric hmen t in lo w temp erature-crystallizing
elemen ts dep ends on the in tegrated fractionation of the whole magma o cean, and th us its
initial exten t. The crystallization temp erature used in previous studies (Elkins-T an ton et al.,
2011; Mey er et al., 2010; P erera et al., 2018), obtained for a 1000 km-deep magma o cean, is
significan tly lo wer in most of the cum ulates, but con v erges to w ard a similar v alue as our mo del
for the late crystallized la y ers. The distribution of HPE is particularly affected b y the in tial
exten t of the magma o cean. The heat budget of the bulk silicate Mo on computed at 4.567 Ga
amoun ts 1.946 TW (T a ylor, 1982), but the shallo w lay ers are enric hed in HPE when the lo w er
man tle is equilibrated, and ev en more when the whole man tle is initially molten. This results
in an increased heat budget in the magma o cean. The influence of the initial exten t of the
magma o cean on our results is discussed in Section 3.5.6.
3.2.2 Re-melting of the cum ulates and the crust
If the heat flo w in to the magma o cean increases sufficien tly , it will heat up, thereb y causing
re-melting of the cum ulates. They start to remelt at the solidus, whic h is sligh tly hotter than
the crystallization temp erature b ecause secondary melting in volv es the isolated cum ulates,
while, during their initial crystallization out of the magma o cean, they w ere in equilibrium with
the bulk liquid, th us mo difying the melting/crystallization conditions. Those t w o reactions
are differen t in essence, and not just the rev erse of one another. Ho w ev er, when the onset
of heat-piping is strong enough to cause a re-heating of the magma o cean, the top of the
cum ulates, whic h is in equilibrium with the magma o cean, is lik ely to remelt at a temp erature
lo w er than its solidus, and whic h is b etter appro ximated b y the crystallization temp erature.
Therefore, up on re-heating of the magma o cean, the cum ulates accordingly re-melt so that the
54

3.3 Thermal ev olution mo del
Figure 3.3:
The thermal ev olution mo del consists in four coupled reserv oirs: the core, the gro wing
cum ulates, the shrinking magma o cean and the gro wing flotation crust.
crystallization fron t alw ays corresponds to th e in tercept b et w een the magma o cean temp erature
and the crystallization temp erature profile.
Similarly , while plagio clase starts to crystallize from the magma o cean when it reac hes a
temp erature of appro ximately 1530 K, once isolated in the crust, its solidus is close to 1800 K.
This is m uc h higher than the highest temp erature reac hed during transient reheating ev en ts of
the lunar magma o cean induced by the onset of heat-piping, suggesting that the crust do es not
re-melt during suc he ev ents. Nev ertheless, melting at the base of the crust (whic h is in con tact
with the magma o cean) is controlled b y an equilibrium b et w een crustal anorthosites and the
bulk of the magma o cean. Therefore it is not straigh tforw ard whether – and if y es, to whic h
exten t – the base of the crust should partially remelt during transien t reheating of the magma
o cean. Our fiducial mo del considers that, up on re-heating of the magma o cean, the crust melts
in the same prop ortion relativ e to the cum ulates as that with whic h it crystallized (noted
f
in
Section 3.3.4). How ev er, in Section 3.5.4 w e sho w that taking this effect in to accoun t or not
do es not alter our results.
3.3 Thermal ev olution mo del
The thermal mo del consists of four thermally-coupled sub-systems: the core, the solid cum ulates,
the magma o cean, and the flotation crust (Figure 3.3). Belo w, w e describ e the equations
go v erning the ev olution of the sub-systems as w ell as their couplings.
55

3. Gro wth of a flotation lid: the case of the Mo on
3.3.1 The core
The core is treated as an isothermal sphere of temp erature
T c
. The equation con trolling the
time ev olution of T c reads (see Section B.2.3):
4
3 π R 3
c ρ c c p,c
dT c
dt = 4 π R 2
c k ∂ T
∂ r ( R c ) , (3.1)
where
c p,c
is the core heat capacit y ,
ρ c
the core densit y ,
R c
the radius of the core and
k
is the
thermal conductivit y in the man tle.
3.3.2 The solid cum ulates
The temp erature of the s olid cum ulates is mo delled in t w o differen t w a ys. When only the effect
of the thermal conductivit y of the crust is addressed, it follo ws the spherically symmetrical
time-dep enden t heat diffusion equation describ ed in Section 3.3.2.1. When the effect of
heat-piping is considered, it is mo delled b y 3D thermal conv ection as describ ed in Section
3.3.2.2.
3.3.2.1 Spherically symmetrical heat diffusion
W e mo del the thermal ev olution of the solid cum ulates b y solving the heat diffusion equation in
a spherically symmetrical shell of constan t inner radius
R c
and outer radius
R MO
, corresp onding
to the radius of the b ottom of the magma o cean:
∂ T
∂ t = 2
r κ ∂ T
∂ r + κ ∂ 2 T
∂ r 2 + h
c p
, (3.2)
where
κ
=
k /
(
ρ ref c p
) is the thermal diffusivit y ,
ρ ref
the man tle densit y ,
c p
the heat capacit y ,
and
h
the in ternal heat pro duction p er unit mass. The initial in ternal heating
h ref
is the
v olume a v erage of the heat pro duction in the solid cum ulates (see Figure 3.2) at the initial
time of the sim ulations (corresp onding to a 110-km-deep magma o cean, see Section 3.3.5). It
is computed from our crystallization sequence accoun ting for the partitioning of the long-liv ed
radion uclides
235
U ,
238
U ,
232 Th
and
40
K , pro viding the follo wing time-dep enden t in ternal
heating:
h ( t ) = ∑
i
h i exp( − λ i t ) h ref , (3.3)
where the subscript
i
refers to either of the four heat-pro ducing elemen ts,
h i
is the non-
dimensional initial con tribution to in ternal heating of the
i th
elemen t (v erifying Σ
i hi
= 1 ),
λ i
the deca y constan t of the
i th
elemen t (
λ i
=
ln
(2)
/τ i
where
τ i
is the half life of sp ecies
i
) and
h ref
the total initial rate of in ternal heating. The sp ecies-sp ecific quan tities are rep orted in
T able 3.1 while
h ref
is giv en in table 3.2. Note that w e do not consider the shift in in ternal
heating due to a late formation of the Mo on (due to the deca y o ccurring betw een the creation
of the solar system and the Mo on-forming ev en t), whic h ho w ev er causes little c hange in the
heat pro vided b y the long-liv ed systems under consideration.
The b oundary conditions for equation
(3.2)
are:
T | r = R c
=
T c
and
T | r = R MO
=
T MO
, where
T MO
is the magma o cean’s temp erature. Equation
(3.2)
is solv ed using an implicit time-
stepping and a first-order cen tral finite-difference sc heme on a 100 p oin ts regular grid. Since
56

3.3 Thermal ev olution mo del
Sp ecies h i λ i
235 U 0.3138 593
238 U 0.1745 93
232 Th 0.1220 30
40 K 0.3897 334
T able 3.1:
Non-dimensional initial con tribution to the total heat budget and deca y constan t for
eac h of the heat pro ducing sp ecies accoun ted for. The λ i are indicated in unit Ga − 1 .
the geometry ev olv es as the crust gro ws, the p ositions of the grid points c hange from one time
step to the other. T o accoun t for this, the temp erature profile (b efore solving equation
(3.2)
)
is in terp olated from the profile at the previous time step on to the new grid p oin ts p ositions.
3.3.2.2 3D thermal con v ection
In order to compute secondary melting caused b y solid-state con v ection in the solid cumulates,
w e use the finite-v olume co de GAIA (Hüttig, T osi, & Mo ore, 2013) to solv e the equations of
thermal con v ection in a 3D spherical shell comp osed of 132 shells con taining 10242 computation
no des eac h (see section 3.4.4 for a discussion ab out the resolution c hoice). The inner radius of
the domain corresp onds to
R c
while the outer radius ev olv es follo wing
R MO
as describ ed in
Chapter 2.
Similar to Chapter 2, w e solv e the conserv ation of mass, momen tum and thermal energy
with the Boussinesq appro ximation whic h read in non-dimensional form (Section B.1):

∇ ·  v = 0 , (3.4)

∇ · [ η ( 
∇  v + 
∇  v T )] − 
∇ p = g ( r )Ra T  e r , (3.5)
∂ T
∂ t +  v · 
∇ T − ∇ 2 T = Ra H
Ra h, (3.6)
where  v is the velocity , η the viscosit y , p the dynamic pressure, g ( r ) the (radially-dep enden t)
gra vit y acceleration,
 e r
the unitary radial v ector, Ra and Ra
H
the Ra yleigh and thermal
Ra yleigh n umbers resp ectiv ely , and
h
the in ternal heating rate (as in tro duced in Equation
(3.3)
,
made non-dimensional b y dividing it b y
h ref
). All these quan tities b eing made non-dimension al,
follo wing Section B.1 (the dimensional v alues are indicated in T able 3.2).
The viscosit y is temp erature- and pressure-dep enden t, follo wing an Arrhenius la w (see
Section A.2). The reference viscosit y (corresp onding to a pressure
p ref
= 3 GP a and temp erature
T ref
= 1600 K) is one of the study parameters (see Section 3.5.1), the activ ation energy is
E ∗
= 335 kJ/mol and the activ ation v olume
V ∗
= 4
×
10
− 6
m
3
/mol, b oth v alues b eing
c haracteristic of olivine diffusion creep (Hirth & K ohlstedt, 2003).
Because of its small core, the gra vit y acceleration in the lunar mantle is not constan t. W e
compute the non-dimensional gra vit y
g
b y considering a constan t densit y
ρ ref
in the man tle,
resulting in:
g ( r ) = 4 / 3 π G
g ref r 2 ( ρ c R 3
c + ρ ref ( r 3 − R 3
c ) ) (3.7)
where
G
is the gra vitational constan t and
g ref
the reference gra vit y (app earing in b oth Ra yleigh
n um b ers), and corresp onds to the gravit y at the surface of the Mo on. The internal heating
as w ell as the thermal b oundary conditions asso ciated to this system are the same as those
57

3. Gro wth of a flotation lid: the case of the Mo on
in tro duced in Section 3.3.2.1. The dynamical b oundary conditions are free-slip at b oth inner
and outer b oundaries.
3.3.2.3 Heat-piping
W e assume that extraction of the secondary melts due to decompression melting in the rising
up w ellings of conv ecting solid cum ulates buffers the maxim um temp erature in the cum ulates to
the solidus. After solving equation
(3.6)
, if the temp erature lo cally exceeds the solidus, a mass
of melt
dM melt
is pro duced, driving the temp erature bac k to the solidus through consumption
of laten t heat of melting:
dM melt = 1
L ρ ref c p dV max ( T − T sol , 0) . (3.8)
where
L
is the laten t heat of melting,
dV
is the elemen t v olume, and
T sol
the lo cal solidus
temp erature. This mass of melt is then extracted at its solidus temp erature, in to the magma
o cean, pro viding a heat source prop ortional to the temperature difference b et w een the magma
o cean and the solidus at the melting p osition. The resulting total energy increment in the
magma o cean reads:
dE HP = ∫ V cum ulates
c p ( T sol − T MO ) dM melt , (3.9)
where V cum ulates is the v olume of cum ulates crystallized th us far (excluding plagio clase).
3.3.3 Magma o cean
The magma o cean is mo delled as an isothermal spherical shell of inner radius
R MO
and outer
radius
R crust
(the radius of the b ottom of the flotation crust). W e solv e the conserv ation of
heat in the magma o cean b y balancing the differen t fluxes in and out of the magma o cean
with its in ternal heating. The heat conserv ation equation for the thermal energy of the magma
o cean reads:
ρ ref c p
d
dt ( ∫ V MO
T MO dV ) = q b ot 4 π R 2
MO − q top 4 π R 2
crust + ρ ref h MO V MO + dE HP
dt + ρ ref L dV MO
dt
+ ρ ref c p 4 π R 2
MO
dR MO
dt T MO − ρ ref c p 4 π R 2
crust
dR crust
dt T MO ,
(3.10)
where
q b ot
and
q top
are the conductiv e heat fluxes at the b ottom and the top of the magma
o cean,
R MO
and
R crust
the radii of the b ottom of the magma o cean and of the crust resp ectiv ely ,
and
T MO
,
h MO
and
V MO
the magma o cean’s temp erature, in ternal heating p er unit mass and
v olume. The left-hand-side term corresp onds to the v ariation in thermal energy of the magma
o cean (due to v ariation in b oth temp erature and v olume, since it is an op en system). The first
t w o terms on the righ t-hand-side corresp ond to the conductiv e heat fluxes at the top of the
cum ulates in to the magma o cean and at the base of the crust out of the magma o cean, and
are computed as:
q b ot
=
− k ∂ T
∂ r ⏐ ⏐ ⏐ ⏐ r = R MO
and
q top
=
− k crust
∂ T
∂ r ⏐ ⏐ ⏐ ⏐ r = R crust
, where
k
and
k crust
are
the thermal conductivit y of the man tle and the crust, resp ectively . The third term on the
righ t-hand-side of equation
(3.10)
corresp onds to the in ternal heating of the magma o cean.
Note that
h MO
dep ends on time, due to the decay of heat-producing elements, and on
R MO
,
58

3.3 Thermal ev olution mo del
due to the enric hmen t in incompatible heat-pro ducing elemen ts in the magma o cean as it
crystallizes. The fourth term is the heat-piping fl ux in tro duced in Section 3.3.2.3. The fifth
term corresp onds to the release of laten t heat up on crystallization (or consumption of laten t
heat up on melting). Finally the t w o last terms corresp ond to the heat flo w due to the mass
flux out of the magma o cean through s ettling of the crystals (mafic crystals for the first term
and plagio clase crystals for the second). This terms app ear b ecause w e consider the heat
conserv ation of an op en system (the crystallizing magma o cean).
The left-hand-side term consists in a time-deriv ativ e of an in tegral, whose b ounds are
time-dep enden t. Applying Leibniz rule to the in tegral yields:
ρ ref c p
d
dt ( ∫ R crust
R MO
4 π T MO r 2 dr ) = ρ ref c p 4 π R 2
crust
dR crust
dt T MO − ρ ref c p 4 π R 2
MO
dR MO
dt T MO
+ ρ ref c p ∫ R crust
R MO
4 π dT MO
dt r 2 dr .
(3.11)
The first and second terms of the righ t-hand-side cancel out with the last t wo terms of
equation (3.10) resp ectiv ely , while the last term is simply ρ ref c p V MO dT MO /dt .
W e can write the laten t heat term in Equation (3.10) as:
ρ ref L dV MO
dt = ρ ref L dV MO
dR MO
dR MO
dT MO
dT MO
dt . (3.12)
Note that
dV MO
dR MO
is not simply
−
4
π R 2
MO
due to the coupled ev olution of
R MO
and
R crust
defining the magma o cean’s b oundaries (see equation
(3.19)
b elo w). F urthermore, b oth the
ev olution of
T MO
and
R MO
are link ed b ecause
T MO
follo ws the radial profile of the crystallization
temp erature T crys (represen ted in Figure 3.2a) as the magma o cean crystallizes:
dT MO
dR MO
= dT crys
dr ⏐ ⏐ ⏐ ⏐ r = R MO
. (3.13)
Inserting equation
(3.13)
in to equation
(3.12)
, w e can write the laten t heat term of equation
(3.10) as:
ρ ref L dV MO
dt = ρ ref L dV MO
dR MO
dT crys
dr ⏐ ⏐ ⏐ ⏐
− 1
r = R MO
dT MO
dt . (3.14)
Finally , inserting equations
(3.14)
and
(3.11)
in to equation
(3.10)
, w e obtain the equation
that con trols the ev olution of the temp erature of the magma o cean:
ρ ref c p V MO
dT MO
dt =
4 π R 2
MO q b ot − 4 π R 2
crust q top + ρ ref h MO V MO + dE HP
dt
1 − L
c p
dV MO
dR MO
dT crys
dr ⏐ ⏐ ⏐ ⏐
− 1
r = R MO
. (3.15)
3.3.4 The flotation crust
The flotation crust is mo delled as a spherically-symmetrical shell of inner radius
R crust
and outer
radius
R M
, whose temp erature distribution is obtained b y solving the spherically symmetrical
59

3. Gro wth of a flotation lid: the case of the Mo on
heat diffusion equation, including heat sources:
∂ T
∂ t = 2 κ crust
r
∂ T
∂ r + κ crust
∂ 2 T
∂ r 2 + h crust
c p
, (3.16)
where
κ crust
=
k crust /
(
ρ crust c p
) is the thermal diffusivit y of the crust and
h crust
its heat
pro duction p er unit mass. The time dep endence of the geometry is handled the same w a y as
describ ed for equation
(3.2)
, with the difference that in the case of the crust, the outer radius
is fixed while the inner radius ev olv es with time. Fixed-temp erature b oundary conditions are
prescrib ed at b oth b oundaries:
T s
at the surface and
T MO
at the in terface with the magma
o cean.
The crust starts to gro w from an initial thic kness of 5 km as in Elkins-T an ton et al. (2011).
The time ev olution of
R crust
is obtained b y assuming a constan t ratio
f
of v olume of crystallized
plagio clase p er unit v olume of crystallized magma o cean, and assuming that all plagio clase
crystals formed settle at the top of the magma o cean. The v alue of
f
is imp osed b y the
thic kness of the crust at the end of crystallization. By definition, we ha v e:
f = − dV crust
dV MO
= −
dV crust
dR crust
dR crust
( ∂ V MO
∂ R MO ) R crust
dR MO + ( ∂ V MO
∂ R crust ) R MO
dR crust
, (3.17)
where
( ∂ V MO
∂ R MO ) R crust
is the deriv ativ e of
V MO
with resp ect to
R MO
when
R crust
is k ept
constan t, and in v ersely for
( ∂ V MO
∂ R crust ) R MO
. Noting that
( ∂ V MO
∂ R crust ) R MO
=
− dV crust
dR crust
(with
V crust = 4 π / 3 ( R 3
Mo on − R 3
crust ) the v olme of the crust), w e can write equation (3.17) as:
dR crust = − f
1 − f ( ∂ V MO
∂ R MO ) R crust
( ∂ V MO
∂ R crust ) R MO
dR MO (3.18)
whic h is the ev olution equation for R crust . Equation (3.18) also yields:
dV MO
dR MO
= ( ∂ V MO
∂ R MO ) R MO ( 1 + f
1 − f ) (3.19)
3.3.5 Initial set-up
The thermal ev olution sim ulations start when the crust has already reac hed a thic kness of 5
km (
R crust , 0
= 1735 km). The magma o cean is then 110 km deep (
R MO , 0
= 1630 km) and its
temp erature is
T MO , 0
= 1518 K. The initial temp erature profile in the crust is linear from the
b ottom of the crust at temp erature
T MO , 0
up to the surface at 250 K. The initial temp erature
profile in the cum ulates follo w the crystallization temp erature, with an accretion-lik e decreasing
profile in the pristine lo w er man tle, as represen ted in Figure 3.4.
60

3.4 Thermal mo del b enc hmark
Quan tit y V alue Unit Quan tit y V alue Unit
Core Crust
c p,c 840 J/kg/K k crust 1.5 to 4 W/m/K
ρ c 7200 kg/m 3 κ crust 4 . 6 × 10 − 7 to 1 . 2 × 10 − 6 m 2 /s
R c 390 km ρ crust 2715 kg/ m 3
Cum ulates h crust , 0 1 . 01 × 10 − 10 W/kg
L 500 kJ/kg D crust 43 km
h ref 1 . 43 × 10 − 11 W/kg Reference
G 6 . 67 × 10 11 m 3 /kg/s 2 R Mo on 1740 km
η ref 10 19 to 10 22 P a s D ref 1350 km
E ∗ 335 kJ/mol ∆ T 1930 K
V ∗ 4 × 10 − 6 m 3 /mol T s 250 K
R 8.314 J/mol/K c p 1200 J/kg/K
p ref 3 GP a α 3 . 0 × 10 − 5 1/K
T ref 1600 K ρ ref 3245 kg/m 3
Magma o cean k 4.0 W/m/K
h MO v ariable W/kg κ 10 − 6 m 2 /s
f 0.39 g ref 1 . 62 m/s 2
T able 3.2:
Ph ysical quan tities used. V alues computed from our crystallization mo del are
highligh ted in blue and mo del parameters in red.
1300 1400 1500 1600 1700 1800 1900 2000 2100
temperature [K]
0
200
400
600
800
1000
1200
depth [km]
initial temperature
initial crust
initial magma ocean

Figure 3.4:
T emp erature profile at the b eginning of the simulations. Note that the surface
temp erature extends b ey onds the left b order of the plot, do wn to 250 K.
3.4 Thermal mo del b enc hmark
3.4.1 Comparison with existing mo dels
In order to test our mo del, w e b enc hmark ed it against results from the literature b y repro ducing
the sim ulations p erformed b y Elkins-T anton et al. (2011) and P erera et al. (2018). The
differences in the implemen tation of the crystallization and thermal ev olution mo dels b et w een
the presen t w ork and these studies induce discrepancies that are complicated to constrain
without a full description of these mo dels. Ho w ev er, w e attempted to repro duce their set-up
as close as p ossible. T o do so, w e used the same crystallization temp erature that was used
in these mo dels, noted
T ′
crys
to distinguish it from the crystallization temp erature computed
from our mo del, and whic h reads (Elkins-T anton e t al., 2011):
T ′
crys ( r ) = ( − 1 . 3714 × 10 − 4 ) r 2 − 0 . 1724 r + 1861 − 4 . 4(0 . 2 V liq ( r )+0 . 01) − 1 (3.20)
61

3. Gro wth of a flotation lid: the case of the Mo on
where
T ′
crys
is in K,
r
is the radius in km and
V liq
(
r
) =
V MO
(
R MO
=
r
)
/V MO
(
t
= 0) with
V MO
(
t
= 0) corresp onding to a 1000 km-deep magma o cean. Using a surface temp erature of
25 K, as done b y Elkins-T an ton et al. (2011) and P erera et al. (2018), w e reac h a final v olume
of the magma o cean of 1% (whic h is their criterion defining the end of solidification) in 28
Myr. This is more than the
∼
10 Myr found in Elkins-T an ton et al. (2011) but v ery similar to
the 26 Myr obtained for the same set-up in P erera et al. (2018), where this discrepancy w as
already p oin ted out.
3.4.2 Comparison b et w een differen t effects included in the thermal mo del
In order to ha v e a precise insigh t of the magnitude of the v arious effects mo delled, this section
offers an individual analysis for eac h of them. They can b e iden tified on the righ t-hand-side of
equation
(3.15)
. Some of them are necessary for the ph ysical consistency of the mo del (lik e the
heat flux through the crust, whic h is necessary to ensure that the magma o cean co ols), others
ha v e b een considered in previous studies (lik e the laten t heat term or the in ternal heating of
the magma o cean), while others ha v e not b een considered b efore (lik e the conductiv e heat
flux from the cum ulates and the heat-piping effect). While the presen t w ork fo cuses on the
top heat flux and heat-piping, this section presen ts a comparison of the influence of the other
terms.
3.4.2.1 T ypical heat fluxes and sources ev olution
Figure 3.5 presen ts, for a reference case (
k crust
= 2 W/m/K and
η ref
= 10
21
P a s, see Section
3.5.1), the ev olution of the v arious heat fluxes and sources at pla y , along with the ev olution of
the temp erature of the magma o cean and the depth of its top and b ottom b oundaries. At the
b eginning of the simulations, the conductiv e heat flo w through the crust dominates b ecause
the crust is still v ery thin (5 km), but it decreases rapidly as the crust thic k ens and the magma
o cean co ols do wn. The onset of conv ection in the cum ulates, whic h results in heat-piping, is
iden tified in Figure 3.5a b y the initial p eak in the heat-piping curv e (red line). In Figure 3.5b,
it corresp onds to the reduction in the co oling rate of the magma ocean and in the growth
of the plagio clase crust. Heat-piping then slo wly decreases as the cum ulates co ol do wn and
less melt is pro duced. The in ternal heating diminishes as the magma o cean shrinks, although
preferen tial partitioning of the heat-pro ducing elemen ts in the liquid is tak en in to account.
Radioactiv e deca y of the heat sources pla ys a minor role in the time-v ariations of in ternal
heating, but is still accoun ted for. The conductive heat flux from the cum ulates is minor for
most of the crystallization except to w ards the end, when the top of the cum ulates solidifies
follo wing a steep temp erature gradien t asso ciated with the solidification of lo w crystallization
temp erature minerals. This steep gradien t results in an increased conductiv e flo w from the
cum ulates, and the end of the magma o cean crystallization is con trolled by the balance b et w een
conductiv e heat fluxes from the cum ulates and through the crust.
62

3.4 Thermal mo del b enc hmark
Figure 3.5:
(a) Time ev olution of the heat fluxes and sources for a reference case, accoun ting
for heat-piping with
k crust
= 2 W/m/K,
η ref
= 10
21
P a s, and (b) corresp onding ev olution of the
temp erature (red) and magma o cean’s top and bottom b oundaries (blac k). (c) Time evolution of
the heat fluxes and sources for a siilar but purely diffusiv e case, i.e. with
k crust
= 2 W/m/K but
neglecting con v ection and heat-piping, and (d) corresp onding ev olution of the temp erature (red)
and magma o cean’s top and b ottom b oundaries (blac k).
3.4.2.2 Effect of the v arious heat fluxes and sources
In Figure 3.6, for a case with
k crust
= 2 W/m/K and purely diffusive cum ulates, w e sho w the
effect of remo ving the conductiv e heat flux from the cum ulates, in ternal heating in the magma
o cean, or the effect of latent heat.
The laten t heat release up on crystallization has the strongest effect, prolonging b y a factor
of 2 the duration of the magma o cean solidification when it is included. Due to the steep
gradien t of the crystallization temp erature at the top of the cum ulates (see Figure 3.2) reac hed
at the end of magma o cean solidification, the conductiv e heat flux from the cum ulates b ecomes
imp ortan t to w ards the end of the solidification. This induces a protracted crystallization of
the last few kilometers of the magma o cean, as seen by the longer ”tail” exhibited b y all other
curv es in comparison with the blue one. Finally , accoun ting for in ternal heating in the magma
o cean generally prolongs its solidification b y
∼
20 Myr. Note that the amoun t of heat stored in
the magma o cean dep ends on the v olume fraction of the mantle that underw en t fractional
crystallization, i.e. its initial depth.
3.4.3 Heat-piping and mass conserv ation
Heat-piping is accoun ted for only in the heat conserv ation equation of the magma o cean, while
w e mak e the assumption that the mass of melt added to the magma o cean remains negligible.
The high temp erature difference b et ween the extracted melts and the magma ocean allows
the resulting heating to b e significan t ev en if the v olume of melts extracted remains mo dest.
Nev ertheless, this assumption b ecomes w eak er as the magma o cean shrinks (i.e. as the ratio
of injected melt mass to total magma o cean mass increases). Figure 3.7 presen ts the time
ev olution of the ratio of the cum ulative quan tit y of melt in tro duced in to the magma o cean via
heat-piping un til time
t
, to the instan taneous v olume of the magma o cean at time
t
. This ratio
63

3. Gro wth of a flotation lid: the case of the Mo on
0 20 40 60 80 100 120
t i m e [ M y r ]
0
20
40
60
80
100
120
140
d e p t h [ k m ]
reference
no conduction from the cumulates
no internal heating
no latent heat

Figure 3.6:
Time-series of the depth of the magma o cean (solid curv es) and crust (dahsed curv es)
b ottom, for purely conductiv e cum ulates with
k crust
= 2 W/m/K. The blac k curv es corresp ond to
a case where laten t heat effect, in ternal heating and conductiv e heating b y the cum ulates w ere
accoun ted for in the energy budget of the magma o cean, the blue curv es corresp ond to a similar
case where the conductiv e heat flux w as neglected, the red curv e to a case similar to the first one
where inernal heating w as neglected and the green curv es to a case similar to the first one where
the effect of laten t heat w as neglected. The v ertical dotted lines mark the end of the magma o cean
solidification for the cases of corresp onding color.
remains b elo w 5% un til for 90% of the solidification time in all cases exhibiting heat-piping
presen ted in Section 3.5.3, and b elo w 10% during 95% of the solidification time. This justifies
the appro ximation of neglecting the mass of extracted heat when considering the ev olution of
the magma o cean, supp orting the robustness of our results.
0.0 0.2 0.4 0.6 0.8 1.0
time [non-dim]
0.0
0.1
0.2
0.3
0.4
0.5
t
0
d M
H P /
M
M O (
t
)
r e f = 1 0 2 1 P a s
r e f = 1 0 2 0 P a s
r e f = 1 0 1 9 P a s

Figure 3.7:
Time-serie of the ratio of cum ulativ e mass of the extracted melt (
∫ t
0 dM HP
) to
instan taneous magma o cean mass (
M MO
(
t
) ), computed for the three cases presen ted in Section
3.5.3. In order for the comparison b et w een the case to b e easier, the time is normalized (
t
= 0
corresp onds to the b eginning of the magma o cean solidification and t = 1 to its end).
3.4.4 Con v ergence tests for the con v ection sim ulations
3D con v ection simulations are computationally extremely costly , and can only b e p erformed
using relativ ely coarse grids. In order to choose a resolution that b oth allo ws for a reasonable
computation time and yields a solution that faithfully captures the dynamics of the system
(esp ecially in terms of the top heat flux and the heat-piping flux, whic h are the main outputs
of the con v ection mo del), w e ran con v ergence tests on 2D cylindrical grids ha ving 132, 191,
64

3.4 Thermal mo del b enc hmark
220 and 249 regularly spaced radial p oin ts (with 654, 947, 1091 and 1234 p oin ts p er shell,
resp ectiv ely), using the fiducial v alues for the mo del parameters (see Sections 3.5.2 and 3.5.3).
These resolutions where c hosen to optimize the in terp olation of the final gradien t of the
crystallization temp erature profile, imp ortan t during the final stage of the solidification (see
Section 3.4.2.2). Time-series of the depths of the magma o cean and crust, and of the cum ulates
and heat-piping fluxes are plotted in Figure 3.8. They clearly sho w that lo w ering the resolution
to 132 radial shells do es not result in significant c hanges in the time-series (the final times
agree within 5 Myr, i.e. 2%). W e therefore c hose to run our 3D sim ulations with a radial
resolution of 132 shells. W e also ev aluated the discrepancy b et w een 2D and 3D sim ulations
0 50 100 150 200 250
time [Myr]
0
20
40
60
80
100
depth [km]
(a)
0 50 100 150 200 250
time [Myr]
0.00
0.02
0.04
0.06
0.08
0.10
q b o t [ W / m 2 ]
(b)
132 shells
191 shells
220 shells
249 shells
0 50 100 150 200 250
time [Myr]
0.00
0.02
0.04
0.06
0.08
0.10
q H P [ W / m 2 ]
(c)

Figure 3.8:
Con v ergence tests made on 2D cylindrical grids, with a radial resolution of 132 shells
(blue), 191 shells (green), 220 shells (red), and 249 shells (magen ta).
since some tests (suc h as the thermo-c hemical ones presented in Chapter 5) w ere run on 2D
grids only .
Discrepancies on the heat flo w and temp erature b et w een 2D cylindrical and 3D spherical
sim ulations are w ell do cumen ted in steady-state (Hernlund & T ac kley , 2008), but less
constrained for transien t regimes suc h as those considered here. A common metho d to
mimic a flo w in 3D spherical shells using a 2D cylindrical shell is to re-scale the core radius in
order to reco v er in 2D the 3D ratio of core-man tle b oundary surface to planetary surface (v an
Kek en, 2001). This results in a decrease of the radius of the core, whic h is already quite small
for the Mo on. Using our fiducial set-up, w e compared three cases: a 3D sim ulation using a
resolution of 132 radial shells (i.e. the geometry used for the main results of the study), a 2D
sim ulation with “real” core radius and 249 radial shells, whic h closely matc hes the sim ulation
computed on a 2D grid with 132 shells (Figure 3.8), and another 2D sim ulation using 249
shells, but with a re-scaled core. Figure 3.9 presen ts the time-series corresp onding to these
three cases. The 3D sim ulation (green lines) yields the same onset time of heat-piping as the
2D sim ulation with non-scaled core (blue lines), but a sligh tly shorter magma o cean duration
due to a lo w er heat-piping flux. The 2D sim ulation with a re-scaled core (red lines) exhibits a
later onset of con v ection, but a similar magma o cean lifetime as the non-scaled 2D sim ulation
due to a trade-off b et w een late onset of heat-piping b elo w thic k crust and early onset b elo w a
thin crust describ ed in Section 3.5.3. This suggests that there is no need to rescale the core
size in our 2D runs.
65

3. Gro wth of a flotation lid: the case of the Mo on
0 50 100 150 200 250
time [Myr]
0
20
40
60
80
100
depth [km]
(a)
0 50 100 150 200 250
time [Myr]
0.00
0.02
0.04
0.06
0.08
0.10
q b o t [ W / m 2 ]
(b)
249 shells, 2D, unscaled core
249 shells, 2D, rescaled core
132 shells, 3D
0 50 100 150 200 250
time [Myr]
0.00
0.02
0.04
0.06
0.08
0.10
q H P [ W / m 2 ]
(c)

Figure 3.9:
Comparison b et w een a 3D sim ulation with 132 radial shells (red), a 2D sim ulation
with 249 radial shells with the actual lunar core radius (blue), and a 2D sim ulation with 249 radial
shells but a core radius rescaled according to (v an Kek en, 2001) (green). As in Figure 3.8, panel
a) represen ts the ev olution of the b ottom of the crust and magma o cean, panel b) that of the
conductiv e heat flux at the top of the cum ulates, and panel c) that of the heat-piping flux.
3.5 Application to the lunar magma o cean
3.5.1 Choice of the parameter space
Eac h of the t w o effects in v estigated in this study can b e broadly quantified using one parameter.
Thermal conduction through the crust is naturally c haracterized b y the thermal conductivit y of
the crustal material. The thermal conductivit y of pure anorthosite (1.5 W/m/K) is one of the
lo w est among ro c k-forming minerals (Branlund & Hofmeister, 2012) Although some samples of
the lunar farside crust exhibit a comp osition close to pure anorthosite, it is lik ely that the bulk
of the lunar crust is comp osed of a mix of minerals, including other more conductiv e minerals.
W e therefore v ary the v alue of the effective thermal conductivit y in the crust (
k crust
) from 1.5
W/m/K up to 4 W/m/K, close to the v alue used in previous studies of the solidification of
the lunar magma o cean. As a compromise b et w een a lo w v alue represen ting pure anorthosite,
and a v alue to o high for an anorthosite-ric h material, w e consider
k crust
= 2 W/m/K as our
fiducial v alue.
Heat-piping dep ends on the amoun t of melt pro duced, whic h relates to the intensit y of
con v ection in the cum ulates, whic h in turn is con trolled to the first order b y the v alue of
the viscosit y . Since in this study , w e consider an Arrhenius la w for the rheology of diffusion
creep (see Section A.1.4), the viscosit y dep ends explicitly on the temp erature and pressure
(i.e. depth). Moreo v er, the reference viscosit y (the prefactor in equation
(A.23)
) can b e
further parametrized b y the w ater conten t and the grain size (Karato & W u, 1993; Hirth &
K ohlstedt, 1996). Deformation of the cum ulates b y dislo cation creep rather than diffusion
creep w ould induce further dep endencies (on the strain rate for example, making the rheology
non-Newtonian and hence the n umerical resolution of con v ection computationally m uc h more
c hallenging). Ho w ev er, diffusion creep is kno wn to dominate at high temp eratures and lo w
pressures, conditions relev an t for the early lunar mantle. Therefore, w e restrict this study to
the case of diffusion creep.
Since most of the quan tities on whic h the rheology dep ends are extremely ill-constrained
for the early Mo on (see for instance Hauri et al. (2017) for the w ater con ten t in the Mo on), w e
66

3.5 Application to the lunar magma o cean
c hose to v ary the reference viscosit y o v er a wide range of v alues (from 10
19
P a s up to 10
22
P a s) in order to capture all realistic regimes b et w een extremely w eak and extremely strong
cum ulates. A reference viscosity of
η ref
= 10
21
P a s has b een previously used for the early
Mo on (Li et al., 2019) and is also th e v alue w e c hose for our fiducial case.
3.5.2 Effect of k crust : purely diffusiv e cum ulates
As a first step, w e in v estigate the influence of the thermal conductivity in the crust. T o isolate
this effect, w e consider that no solid-state con v ection o ccurs in the cum ulates (see Section 3.3),
whic h w ould b e the case if the solid man tle w as to o stiff for an early onset of con v ection, as
demonstrated in Chapter 2. W e run sim ulations of the solidification of the lunar magma o cean
for three v alues of
k crust
: 1.5, 2 and 4 W/m/K. The resulting time-series of the depths of the
magma o cean and crust b ottom are presen ted on Figure 3.10. The solid line indicates that, for
k crust
= 4 W/m/K, the lunar magma o cean solidifies within 50 Myr, i.e. 22 Myr longer than
when using the crystallization sequence from Elkins-T an ton et al. (2011) (see Section 3.4.1).
This is in part due to the presence of a v ery strong gradien t at the end of our crystallization
temp erature profile (see Figure 3.2) that corresp onds to the final crystallization of the most
incompatible materials, and whic h induces a protracted tail in the time-series. This effect
is roughly parametrized b y the last term of equation
(3.20)
, but do es not result in suc h a
significan t prolongation of the solidification.
0 25 50 75 100 125 150 175 200
time [Myr]
0
20
40
60
80
100
depth [km]
k
c r u s t = 1 . 5 W / m / K
k
c r u s t = 2 W / m / K
k
c r u s t = 4 W / m / K

Figure 3.10:
Time-series of the depths of the b ottom of the magma o cean (lo wer curv es) and of
the b ottom of the crust (upp er curv es) for v arious v alues of the thermal conductivit y of the crust.
Since the c haracteristic time for heat diffusion scales in v ersely with the thermal diffusivit y ,
whic h is prop ortional to the thermal conductivit y , a doubling of the solidification time b et w een
k crust
= 4 W/m/K and
k crust
= 2 W/m/K is exp ected. Y et the solidification tak es
∼
130 Myr
for
k crust
= 2 W/m/K. This is due to the v arious non-linear effects included in our set-up, in
particular the late imp ortance of the conductiv e heat flux from the cum ulates (see Section
3.4.2.2). While the scaling of the solidification duration with
k crust
applies b efore en tering the
late ”tail” of the time-series, when the conductiv e flux through the crust is largely dominan t,
it breaks do wn when the conductiv e heat flo w from the cum ulates b ecomes an imp ortan t term
in the energy balance of the magma o cean. Thus, although this stage represen ts only 10% of
67

3. Gro wth of a flotation lid: the case of the Mo on
the solidification duration for
k crust
= 4 W/m/K, it reac hes 20% for
k crust
= 2 W/m/K and
more than 30 % for k crust = 1 . 5 W/m/K.
3.5.3 Effect of η ref : solid-state of con v ection in the cum ulates and heat-
piping
A dopting our fiducial v alue of the crustal thermal conductivit y results in a solidificaition time
of the lunar magma o cean of at least 130 Myr. This estimate exceeds b y far the simple guess
based on scaling with the size of the Mo on from the previous chapter. It sho ws that the
insulating effect of the flotation lid dramatically prolongs lunar magma o cean solidification,
suggesting that solid-state con v ection can set in b efore the lunar man tle completely solidifies.
Using this v alue for
k crust
w e run further sim ulations where w e solve the con v ection in the
cum ulates, and adopt v arious v alues of reference viscosit y , on a similar principle as that for
c hapter 2, when the Ra yleigh n um b er w as v aried.
Figure 3.11:
(a) Snapshot of the temp erature field in the cum ulate after 100 Myr for our fiducial
case using
k crust
= 2 W/m/K and
η ref
= 10
21
P a s (blue curv e in Figure 3.12). Con v ection in the
cum ulates o ccurs while the magma o cean (y ellow area) is still solidifying and the plagioclase crust
(grey area) still gro wing. (b) Corresp onding laterally-a v eraged temp erature profile (blue line). In
hot up w ellings, the temp erature exceeds the solidus (red dashed line in panel b), causing partial
melting (pale areas in panel a) that results in heat-piping.
Figure 3.11 represen ts the con vection pattern obtained for the case featuring
η ref
= 10
21
P a
s, 100 Myr after the b eginning of magma o cean solidification. The buffering of the temp erature
at the solidus prev en ts from developping clear usual con v ectiv e features on the 3D snapshot,
but cold do wn welling are iden tifiable on the temp erature slice. Once conv ection has started, a
nearly con tin uous partial melting zone dev elops (ligh t shaded region on the temp erature slice
68

3.5 Application to the lunar magma o cean
in Figure 3.11a), where the a v erage temp erature is buffered at the solidus (where the solid
blue curv e and the dahsed red curv e are the closest in Figure 3.11b). Ab o v e this region, the
cum ulates remain essen tially conductiv e, th us prev en ting from forming a thinner b oundary
la y er at the top, thereby limiting the conductiv e heat flux from the cum ulates in to the magma
o cean. In fact, without this limiting mec hanism, the conductiv e heat flux from the cum ulates
w ould b e high enough to balance the heat flux through the crust when the magma o cean
b ecomes thin, and to sustain a p erp etual magma o cean.
Figure 3.12 presen ts the time-series for the depths of the b ottom of the magma o cean
and of the flotation crust for all cases, as w ell as that for purely conductiv e cum ulates using
the same v alue of
k crust
, sho wn for reference. F or the stiffest cum ulates (i.e. the highest
0 25 50 75 100 125 150 175 200
time [Myr]
0
20
40
60
80
100
120
140
depth [km]
ref =10 22 Pa s
ref =10 21 Pa s
ref =10 20 Pa s
ref =10 19 Pa s
150 180 210
42
44
46

Figure 3.12:
Time-series of the depths of the b ottom of the magma o cean (lo wer curv es) and of
the b ottom of the crust (upp er curv es) for v arious v alues of the reference viscosity in the solid
cum ulates and
k crust
= 2 W/m/K (colored curv es) and for the purely conductiv e case for the same
v alue of k crust (blac k dashed curv e, iden tical as that on Figure 3.10).
v alue of
η ref
), con v ection do es not set in b efore the end of the magma o cean crystallization,
and no supplemen tary heat flux o ccurs: the orange curv e is iden tical to the d ashed blac k
curv e. F or
η ref ≤
10
21
P a s, an early onset of con v ection in the cum ulates tak es place, and
immediately results in an effectiv e heat-piping flux that slo ws the solidification, and can b e
iden tified b y the deviation of the respective curv es from the reference one (blac k dashed). A
lo w er reference viscosity systematically causes an earlier onset of con v ection, as well as a more
in tense initial effect of the heat-piping: for
η ref ≤
10
20
P a s, the ev olution of the solidification is
ev en temp orarily rev ersed, and a transien t thic kening of the magma ocean o ccurs, b ecause the
initial thermal o v erturn of the cumulates induces enough secondary melting for the heat-piping
flux to outgro w the conductiv e flux through the crust.
Nev ertheless, a similar trend is not observ ed for the o v erall duration of the magma o cean
solidification: for all the cases that exhibit an early onset of conv ection and asso ciated heat-
piping, the lifetime of the magma o cean cluster around 180 Myr. This is due to a trade-off
b et w een a late onset of a mild heat-piping effect (for relativ ely high v alues of
η ref
), taking
place b elo w an already thic k crust, making ev en a lo w heat flux addition hard to ev acuate,
and a strong heat-piping o ccurring b elo w a still thin crust (for the lo west v alues of
η ref
), whic h
is m uc h more prone to dissipate ev en high heat flux additions.
69

3. Gro wth of a flotation lid: the case of the Mo on
0 50 100 150 200 250
time [Myr]
0
20
40
60
80
100
depth [km]
(a)
0 50 100 150 200 250
time [Myr]
0.00
0.02
0.04
0.06
0.08
0.10
q b o t [ W / m 2 ]
(b)
crust re-melt
no crust re-melt
0 50 100 150 200 250
time [Myr]
0.00
0.02
0.04
0.06
0.08
0.10
q H P [ W / m 2 ]
(c)

Figure 3.13:
Time-series of the depth of the magma o cean and the crust (a), and of the b ottom
conductiv e (b) and heat-piping (c) fluxes for t w o cases using similar parameters (
k crust
= 2 W/m/K
and
η ref
= 10
19
P a s), one allo wing the b ottom of the crust to remelt (this is the case presented on
figure 3.12 for the corresp onding reference viscosit y), and the other where re-heating of the magma
o cean is not asso ciated with re-melting of the crust
F rom Sections 3.5.2 and 3.5.3, it clearly app ears that b oth the lo w v alue of the thermal
conductivit y in the anrthoisitic crust and the heat-piping asso ciated with early onset of
con v ection in the solid cum ulates cause a significan t prolongation of the magma o cean
solidification, reac hing 180 Myr in our fiducial case.
3.5.4 T ransien t remelting of the cum ulates and the crust
The uncertain t y on the melting b eha viour of the crust can b e in v estigated b y comparing one
of the previous cases with another one where the crust do es not remelt when the magma
o cean heats up. The strongest heating effect is obtained in the case
η ref
= 10
19
P a s. W e th us
c hose this v alue rather than our fiducial one, in order to accen tuate the difference and mak e
the comparison conserv ativ e. In the case where the crust do es not re-melt, the radius of the
crystallization fron t needs to reac h its highest lev el previously reached b efore new crust is
formed, so that the final v olume of the crust remains the same in b oth cases. The underlying
assumption is that, due to the re-melting of mafic cum ulates, plagio clase is not a liquidus
phase for the magma o cean until it reco v ers the comp osition it had when it started to re-heat.
In this case the difference b etw een the time-series is negligible, as seen on Figure 3.13.
3.5.5 P artitioning of the heat-pro ducing elemen ts in the crust
The results presen ted ab o v e rely on the assumption that the crust con tains a homogeneous
con ten t of heat-pro ducing elemen ts, resulting in an initial in ternal heating rate
h crust , 0
. This
means that up on crustal growth, a constan t rate of heat-pro ducing elemen ts en ters the crust,
th us lea ving the heat budget of the magma ocean. Ho w ev er, the partitioning of the heat-
pro ducing elements in to plagio clase is p o orly constrained. In order to assess the imp ortance of
this assumption, w e presen t here a sim ulation where the opp osite end-mem b er assumption is
tak en, i.e. that no heat-pro ducing elemen ts enter the crust. In this case the in ternal heating
rate in the crust is zero (increasing the efficiency of magma o cean co oling via heat conduction
through the crust) and the in ternal heating rate in the magma o cean is increased (which is
exp ected to ha v e the opp osite effect). Since this effect is indep endent of the con v ectiv e regime
70

3.5 Application to the lunar magma o cean
in the cum ulates, w e assume the solid man tle to b e purely conductiv e for the sak e of simplicit y .
In this case, the increase in the total heat budget of the magma o cean pro vides more thermal
energy to the magma o cean. Ho w ev er this is balanced b y the absence of in ternal heating in the
crust, whic h allo ws for higher co oling flux. Still, the net effect is a lengthening of the magma
o cean crystallization duration of ∼ 20 Myr (i.e. 15%).
3.5.6 Influence of the crystallization mo del
As men tioned in Section 3.2.1, the initial depth of the magma o cean has a t w ofold effect. It
affects the heat budget in the magma o cean and the temp erature drop asso ciated with the
crystallization of the last (
∼
5 km-thic k) cum ulates lay er. T o assess the influence of the initial
depth of the magma o cean on the duration of the magma o cean, w e computed the thermal
ev olution asso ciated with the three differen t cases of Section 3.2.1 (1350-km, 1000-km and
500-km-deep magma o ceans) for diffusiv e cum ulates and
k crust
= 2 W/m/K. The con tribution
of either the in ternal heating in the magma o cean and/or that of the conductiv e heat flux
from the cum ulates are successiv ely disabled in order to assess what their resp ectiv e influences
are. The results are presen ted in Figure 3.14. The sup erp osition of effects resulting in a large
difference b et w een the duration of the crystallization of a 1000 km-deep magma o cean and a
whole-man tle magma o cean (that can b e seen on panel a) can b e analyzed on panels b and
c: in ternal heating is imp ortan t during the early phase of the magma o cean crystallization,
con tributing to the bulk in the shap e of the time-series for the whole man tle magma o cean
(on panels a and c), while the conductiv e heat flux from the cum ulate dominates the end of
the time-series, inducing the protracted ”tail” for the whole-man tle magma o cean (on panels
a and b). Both these effects are due to the higher enric hmen t of the whole-man tle magma
o cean in incompatible elemen ts, whic h include b oth heat-producing elements and elemen ts
crystallizing at lo w temp erature. When neither of these effect is accoun ted for in the heat
budget of the magma o cean, the discrepancy is only due to the higher temp erature drop in the
last cum ulates la y ers for the whole mantle magma ocean, and to a less er exten t for the 1000
km-deep magma o cean with equilibrated lo w er man tle (whic h is also induced b y the increased
enric hmen t of the magma o cean in those t w o scenarios). Because of its smaller v olume when
the flotation crust starts to gro w (the magma o cean is then
∼
75-km-deep) as w ell as the
lo w er final thickness of the crust, an initially 500-km-deep magma ocean solidifies significantly
faster than deep er cases. F urthermore, due to the smaller v olume of fractionated man tle, the
prolonged final tail in the time-series asso ciated with incompatible elements crystallization
do es not app ear for the shallo w magma o cean case.
Conclusions
A ccoun ting for the lo w er thermal conductivit y of the Mo on’s anorthosite crust compared to
most minerals prolongs the solidification of the lunar magma o cean up to 130 Myr (for our
fiducial v alue of
k crust
= 2 W/m/K). A ccoun ting also for heat-piping asso ciated with re-melting
of magma o cean cum ulates further extends the lifetime of the magma o cean up to 180 to
200 Myr, with little influence of the reference viscosit y (as long as it is lo w enough to allo w
for an early onset of con v ection in the solid cum ulates), in spite of the broad range of v alues
71

3. Gro wth of a flotation lid: the case of the Mo on
0 50 100 150 200 250
time [Myr]
0
20
40
60
80
100
120
depth [km]
(a)
500 km LMO
1000 km LMO
1350 km LMO
0 50 100 150 200 250
time [Myr]
0
20
40
60
80
100
120
depth [km]
(b)
0 50 100 150 200 250
time [Myr]
0
20
40
60
80
100
120
depth [km]
(c)
0 50 100 150 200 250
time [Myr]
0
20
40
60
80
100
120
depth [km]
(d)

Figure 3.14:
Time-series of the b ottom of the crust (upp er curves) and of the magma ocean
(lo w er curv es) for the three differen t initial depths of the magma o cean presen ted in section 3.2.1,
when b oth the the conductive heat flux from the cum ulates and internal heating in the magma
o cean are taken in to consideration (a), when only the former is accounted for (b), when only the
later (c), or when b oth effects are discarded (d). Note that radioactiv e heating has no noticeable
effect, and conductiv e heating only a limited one on the ev olution of a initally 500 km-deep magma
o cean (blue curv es). This is due to the limited man tle fractionation o ccuring in this case.
in v estigated. Finally , the duration of the lunar magma o cean solidification also dep ends on its
initial depth, but not for the in tuitiv e reasons: only the crystallization of the last 75 to 110
km are significan t in terms of time. The longer solidification of a deep er magma ocean is due
to the more imp ortant fractionation of heat sources in the magma o cean as w ell as the thic k er
crust pro duced.
These results pro vide us with a relativ e time evolution. In order to anc hor this c hronology
in the absolute (solar system) time, it is p ossible to relate it to the formation and isolation of
distinct isotopic reserv oirs from the silicate Mo on that ha v e b een sampled and dated. This is
the ob jectiv e of the next c hapter.
72

4
Rep ro ducing the isotopic reco rd of the
Mo on
This c hapter is largely based on the published pap er: Maurice, M., T osi, N., Sc h winger, S.,
Breuer, D. and Kleine, T. (2020). A long-liv ed magma o cean on a y oung Mo on. Scienc e
A dvanc es 6 , eaba8949
4.1 The KREEP trace elemen ts signature
Figure 4.1:
T race elemen ts signa-
ture of KREEP , normalized to c hon-
dritic comp osition. F rom (Shearer
et al., 2006)
The Mo on is the extraterrestrial b o dy for whic h w e ha ve
the largest catalogue of samples, thanks to the
∼
382
kg of ro cks brough t bac k to Earth b y the Ap ollo and
Luna missions, as w ell as the lunar meteorites (T artèse
et al., 2019). This record has b een extensiv ely used
to p erform mineralogical, p etrological and geo c hemical
analyses in order to constrain the history of the Mo on. It
con tains pristine anorthosites from the flotation crust as
w ell as plutonic ro c ks and basalts. The oldest anorthosites
ha v e b een dated as far bac k in time as
∼
4
.
4 Ga
(Alib ert, Norman, & McCullo c h, 1994). Along with crater
coun ting, age estimates of y oung basaltic flo ws as recent
as
∼
1
.
2 Ga (Hiesinger, Head I II, W olf, Jaumann, &
Neukum, 2003) suggest either a long p erio d of dynamical
activit y in the lunar man tle, or eruptions induced b y
impact ev en ts. One class of samples (con taining b oth
basalts and the oldest lunar plutonic ro c ks) is c haraterized
b y the highest enric hment in incompatible elemen ts, in
particular P otassium (K), rare-Earth elemen ts (REE) and
Phosphorus (P). This c hemical signature is th us referred
73

4. Repro ducing the isotopic record of the Mo on
to as ”KREEP”, and its asso ciated source reservoir is called ”urKREEP” . The KREEP-lik e trace
elemen t comp osition could b e due either to a direct origin of the samples from the urKREEP
reserv oir, or b y assimilation of urKREEP material b y melts pro duced in a more depleted
reserv oir. In either case, the presence of urKREEP is required. Suc h an enric hed reserv oir
can only b e formed as a result of a large degree of fractional crystallization of the man tle,
whic h can b e ac hiev ed through the solidification of the lunar magma o cean. The urKREEP
is often considered to b e comp osed of the last
∼
1 % of the magma o cean, corresp onding to
a cum ulate la y er thic kness b et w een 1 and 5 kilometers. Its incompatible elemen ts signature
is plotted in Figure 4.1. Notice the deficit in Eu compared to other elemen ts, whic h is due
to the high compatibilit y of this elemen t in plagio clase. The formation of the flotation crust
drained a large part of the Eu out of the magma o cean, resulting in a deficit in this elemen t in
all reserv oirs formed p osterior to the crust.
Because of its c haracteristic isotopic signature, its high con ten t in trace elemen ts, and
its inferred c hronology (b eing the last cum ulates crystallized from the lunar magma o cean),
KREEP material has b een used to constrain the timing of the lunar magma o cean crystallization
using radio-isotop es c hronometry . Section 4.2 describ es the metho ds used, while Section 4.3
in tro duces a mo del of radioactiv e trace elemen ts fractionation that builds up on the magma
o cen crystallization mo del presen ted in the previous c hapter. Using this coupled fractionation-
thermal ev olution mo del, we sho w in Section 4.4 that a protracted crystallization of the magma
o cean induces a bias in the interpretation of KREEP isotopic data, and prop ose new results
that allo w us to constrain the early c hronology of the Mo on.
4.2 Isotop es fractionation during silicate differen tiation
The isotop e fractionation induced b y silicate differen tiation is similar in principle to that
induced b y core formation (describ ed for the
182 Hf − 182
W system in the in tro duction, see
Section 1.2.1.2). While the latter is controlled b y the difference in partitioning b eha viour of
the paren t and daugh ter sp ecies b et w een iron-liking (siderophile elemen ts) and silicate-liking
(lithophile elemen ts), the former is con trolled by the difference in partitioning b eha viour of the
sp ecies in v olv ed b et w een solid-lo ving (compatible elemen ts) and liquid-lovi ng (incompatible
elemen ts). An incompatible b eha viour is mainly caused b y the inabilit y of the atoms of the
concerned sp ecies to fit inside of the v acan t sites of the crystal lattice. This, in turn, is due
either to their large ionic radius (sp ecies of this class are referred to as ”large-ion lithophile
elemen ts”, and include K, Rb, Cs, Sr, Ba, Pb and Eu) or to their high cathionic c harge (sp ecies
of this class are referred to as ”high field strength elemen ts”, and include Ti, Zr, Hf, Nb and
T a). The partitioning b eha vior of a sp ecies
X
b et w een melt and the crystals of a giv en mineral
is generaly parametrized using a partition co efficient, whic h corresp onds to the equilibrium
constan t of the reaction
X solid ⇆ X melt
. The partition co efficient of
X
b et w een melt and the
crystals of mineral
i
is noted
K i
X
. Assuming c hemical equilibrium, it ob eys an equation similar
to (1.2), written here in a more general form for m ulti-comp onen t equilibrium:
[ X ] i = K i
X [ X ] melt , (4.1)
74

4.2 Isotop es fractionation during silicate differentiation
where [
X
]
i
is the concen tration of
X
in the crystals of mineral
i
and [
X
]
melt
the concen tration
of
X
in the melt.
K i
X
is in general a function of pressure, temp erature and comp osition of the
melt. By definition, a c hemical sp ecies X is incompatible in mineral i if K i
X < 1 .
In the rest of this section, w e consider a to y mo del of the crystallization of a magma o cean
whose cum ulates are comp osed of only one mineral. W e can th us drop the index
i
in the
notation of the partition co efficien t. Letting
P
and
D
b e the paren t and daugh ter sp ecies,
resp ectiv ely , of a giv en isotopic system, for whic h
K P > K D
(i.e. the daugh ter sp ecies is more
incompatble than the paren t one), then the ratio [
P
]
/
[
D
] in the magma o cean decreases as
it solidifies. In turn, the latest crystallized cum ulates inherit a decreased v alue of this ratio
compared to the v alue of the bulk silicate part of the b o dy . This is illustrated b y the blue
curv e in the left panel of Figure 4.2a (the blue dashed line represen ts the bulk comp osition).
4.2.1 Isotop es fractionation
Let’s further note
D r
the pro duct of
P
’s radioactiv e deca y , and
D s
another reference stable
isotop e of
D
. Since
D s
and
D r
are t w o isotop es of the same sp ecies, they hav e the same
partitioning b eha viour, and the ratio [
D r
]
/
[
D s
] is not affected b y the crystallization of the
magma o cean. The v alue of [
D r
]
/
[
D s
] in the forming crystals is similar to that of the magma
o cean at the time of crystallization. Hence, assuming that the crystallization of the magma
o cean is instan taneous (or of negligible duration compared to the half-life of the system) all
cum ulates ev olve from a similar v alue of [
D r
]
/
[
D s
] at the time of the fractionation ev en t. This
ratio then increases due to radiogenic pro duction of D r . This increase dep ends on the initial
quan tit y of
P
in the cum ulates or, equiv alen tly , on the ratio [
P
]
/
[
D s
] . This ratio, con trarily to
[
D r
]
/
[
D s
] , is affected b y fractionation (if
K P 
=
K D
). Hence, as time go es b y , eac h cum ulate
la y er will ev olve a w a y from the bulk v alue of [
D r
]
/
[
D s
] , at a rate dep ending on the la yer’s
initial deviation of [
P
]
/
[
D s
] from the bulk v alue, inherited from the fractionation of
P
and
D
.
This phenomenon is represen ted in Figure 4.2.
By measuring the presen t da y v alues of [
P
]
/
[
D s
] and [
D r
]
/
[
D s
] of a sample, kno wing the
deca y constan t (noted
λ
) of
P
, and assuming the bulk man tle v alues of b oth these ratios, it is
p ossible to compute the time at whic h the source reserv oir of this sample had the same v alue
of [
D r
]
/
[
D s
] as the bulk man tle, whic h is th en considered to b e the time of the differentiation
ev en t. Ho w ev er, the formation and extraction of the sample from its source reserv oir is unlik ely
to lea v e its isotopic comp osition unaffected. A mo del considering only one fractionation even t
is then not sufficien t to trace bac k the initial formation of inner-mantle reserv oirs. In fact,
using a single sample, an y inference of the v alues of [
P
]
/
[
D s
] and [
D r
]
/
[
D s
] bac k in time
further than the latest fractionation ev en t it exp erienced is a degenerate problem. Therefore,
c hronological studies of man tle reserv oirs in v olv e the analysis of m ultiple samples assumed to
share a common source reserv oir.
4.2.2 Isotop e age dating of a single sample
A prerequisite to understanding isotop e age dating of planetary-scale f ractionation ev en ts
(i.e. man tle reserv oirs formation) like magma o cean crystallization is hence the dating of the
crystallization of single samples. Dating a ro c k sample is in general p ossible if it p ossesses
more than one mineral, and con tains a sufficien t amoun t of isotop es to allo w for precise enough
75

4. Repro ducing the isotopic record of the Mo on
0.0 0.5 1.0 1.5 2.0 2.5 3.0
[
P
] / [
D s
]
1.0
1.2
1.4
1.6
1.8
2.0
radius [non-dim]
(a)
t
= 0
t
= 2 1 / 2
0.0 0.5 1.0 1.5 2.0 2.5 3.0
[
D r
] / [
D s
]
1.0
1.2
1.4
1.6
1.8
2.0
radius [non-dim]
(b)

Figure 4.2:
Radial profiles of the paren t-to-stable daugh ter sp ecies ratio ( [
P
]
/
[
D s
] , (a)) and
radiogenic-to-stable daugh ter sp ecies ratio ( [
D r
]
/
[
D s
] , (b)) in the monomineralic cum ulates formed
from the crystallization of a magma o cean (solid lines) and the bulk v alue of this ratio in the whole
man tle (dashed lines), at t w o differen t p oin ts in time: righ t after the magma o cean solidification
(blue) and after t w o half-liv es of the system
τ 1 / 2
(red). Both the paren t and the daugh ter sp ecies
are incompatible as it is usually the case. The partition co efficients of the paren t and the daugh ter
sp ecies are resp ectiv ely:
K P
= 0
.
5 and
K D
= 0
.
2 (the daugh ter sp ecies is th us more incompatible
than the paren t one). The bulk initial v alue of [
P
]
/
[
D s
] is 1.1 (blue dashed line on panel a), and
that of [
D r
]
/
[
D s
] is 0.1 (blue line on panel b) and is not affected b y fractionation (the solid and
dashed blue lines are sup erp osed). After t w o half-lives of the paren t sp ecies, radioactiv e deca y has
afftected b oth [
P
]
/
[
D s
] and [
D r
]
/
[
D s
] in the cum ulates, as sho wn b y the red curv es in b oth panels.
measuremen ts. Because all the minerals presen t in the sample crystallized from a common
magma reserv oir, they retained its [
D r
]
/
[
D s
] v alue at the time of crystallization. Ho w ev er, the
v arious minerals ha v e differen t partition co efficien ts, inducing a spread in their initial v alues of
[
P
]
/
[
D s
] . Th us, if w e represen t th e v alues of the differen t minerals at the crystallization time
on a [
P
]
/
[
D s
] vs [
D r
]
/
[
D s
] diagram, they all plot on a horizon tal line. As time go es b y , eac h
mineral’s v alue of [
D r
]
/
[
D s
] ev olv es as a function of its initial v alue of [
P
]
/
[
D s
] , according to
the equation:
[ D r ] / [ D s ]( t )=[ D r ] / [ D s ]( t crys )+[ P ] / [ D s ]( t crys ) ( exp ( ( t crys − t ) ln(2) /τ 1 / 2 ) − 1 ) . (4.2)
On a [
P
]
/
[
D s
] vs [
D r
]
/
[
D s
] diagram, Equation 4.2 defines a straight line of slope
( exp ( ( t crys − t ) ln(2) /τ 1 / 2 ) − 1 )
, referred to as an ”iso c hron” (Figure 4.3). Hence, fitting
a straigh t line through the data p oin ts represen ting eac h mineral constitutiv e of a sample, and
measuring its slop e, giv es access to the time elapsed since the crystallization of this sample.
This age dating metho d is called the iso c hron metho d.
Other metho ds exist to retrieve the age of a sample, lik e the U-Pb dating tec hnique, that
uses the differen tial deca y of t w o U isotop es, requiring only the initial
235
U
/ 238
U ratio to b e
kno wn. Ho w ev er, an exhaustiv e description of all metho ds used in the isotop e geo chemistry is
b ey ond the scop e of this w ork.
76

4.2 Isotop es fractionation during silicate differentiation
0.00 0.05 0.10 0.15 0.20 0.25
[
P
] / [
D s
]
0.05
0.10
0.15
0.20
0.25
0.30
[
D r
] / [
D s
]
t
=
t
c r y s
t
1 >
t
c r y s
t
2 >
t
1
mineral 1
mineral 2

Figure 4.3:
T ypical iso c hron plot: t w o differen t minerals forming a sample are represen ted b y
the square and the circle rep ectiv ely . At the time of crystallization (
t crys
) they align follo wing
a horizon tal line corresp onding to the bulk sample v alue of [
D r
]
/
[
D s
] at this time. As time
passes [
P
]
/
[
D s
] decreases causing the circle and square to drift left w ards, while [
D r
]
/
[
D s
] increases
accordingly causing an up w ard drift. Overall, the isochrons remain aligned follo wing a straight
line, with slop e exp ( ( t crys − t ) ln(2) /τ 1 / 2 ) − 1 .
4.2.3 Isotop e age dating of the fractionation ev en t
The difficult y of dating a fractionation ev ent pointed out in Secti on 4.2.1 can b e o v ercome
if sev eral samples are a v ailable for one reserv oir. If these samples w ere formed from the
source reserv oir at differen t times, measuring their [
P
]
/
[
D s
] and [
D r
]
/
[
D s
] ratios giv es access
to the corresp onding v alue of [
D r
]
/
[
D s
] in the origin reserv oir at their resp ectiv e times of
crystallization, as represen ted in Figure 4.4. A time-ev olution of the origin reserv oir (blac k
dashed line in Figure 4.4) can then b e inferred without a priori kno wledge of its [
P
]
/
[
D s
]
v alue. Finally , this time-ev olution allo ws for the calculation of the isolation time of the origin
reserv oir, b y matc hing its v alue of [
D r
]
/
[
D s
] with that of the bulk man tle (
x
-axis in Figure
4.4). In this case, b oth the differen tiation time and the corresp onding v alue of [
P
]
/
[
D s
] are
results. This metho d is illustrated in Figure 4.4, where the deviation of [
D r
]
/
[
D s
] from the
bulk v alue is expressed in the notation:
εD r = ( [ D r ] / [ D s ]
[ D r ] / [ D s ] | bulk
− 1 ) × 10 4 . (4.3)
Expressing the deviation in part-p er- 10
4
allo ws to study small v ariations o ccuring o v er a time
short in comparison with the half-life of the system. If the in v estigated time scale is comparable
with the system’s half-life, the deca y lines can no longer b e appro ximated b y straigh t lines
(there real form is an exp onen tial). In practice, this metho d requires that the sample still ha v e
a measurable [
P
]
/
[
D s
] v alue and th us are applied to long-liv ed systems when studying early
solar system pro cesses. Thus the appro ximation of the deca y lines b y straigh t lines is correct
and the ε notation is usual.
77

4. Repro ducing the isotopic record of the Mo on
t
c r y s
t
1
t
2 present-day
time
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
D r

Figure 4.4:
Time ev olution of
εD r
in the origin reserv oir (blac k dashed line) and t w o samples
(red and blue segmen ts). The presen t-da y v alues of [
P
]
/
[
D s
] and [
D r
]
/
[
D s
] ha v e b een measured
for the t w o samples, and their resp ectiv e crystallization times (
t 1
and
t 2
) ha v e b een determined.
This allo ws us to dra w b oth red and blue line. The v alues of those lines at their resp ectiv e sample
formation times (
t 1
for the red line and
t 2
for the blue one) define t w o distinct p oin ts, hence a
new line, whic h represen ts the isotopic ev olution of the origin reserv oir. Finally , the time at which
the blac k line in tercepts the
x
axis corresp onds to the time at whic h the reserv oir had a v alue of
[ D r ] / [ D s ] similar to that of the bulk man tle, corresp onding to the isolation time of the reserv oir.
4.2.4 Application to the lunar magma o cean
Based on this reasoning, Gaffney and Borg (2014) estimated the time of crystallization of
the urKREEP reserv oir, in terpreted as that of the lunar magma o cean crystallization. They
p erformed high precision measurements of the
176 Lu / 177 Hf
and
176 Hf / 177 Hf
ratios on 4 samples
(basalts 15386 and 72275,383 and plutonic samples of the Mg-suite 77215 and 78238) whose
crystallization had b een deteremined by previous studies (Carlson & Lugmair, 1979; Shih
et al., 1992; Carlson, Borg, Gaffney, & Bo y et, 2013; Edm unson et al., 2009). Using in addition
do cumen ted measuremen ts of
147 Sm / 144 Nd
and
143 Nd / 144 Nd
for these same samples as w ell as
for 4 further ones (basalt NW A773, the only lunar meteorite used in this study , and Mg-suites
14304,267, 15455,247 and 76535) (Borg et al., 2009; Borg et al., 2013; Sn yder, T a ylor, & Neal,
1992), they applied the metho d described ab o v e for b oth isotopic c hronometers and found
a concordan t formation time for urKREEP at 4
.
369
±
45 Ga, as sho wn on Figure 4.5. The
isotopic systematics of the samples used b y Gaffney and Borg (2014) are rep orted on table
4.1. Based on the lunar magma o cean thermal ev olution mo del of Elkins-T an ton et al. (2011),
whic h suggesed a rapid crystallization in no more than 35 millions of y ears (see Section 3.1.1),
Gaffney and Borg (2014) argued for a late formation of the Mo on, posterior to 4.4 Ga.
In the ligh t of the results concerning the duration of the lunar magma o cean solidification
exp osed in the previous c hapter, this last conclusion seems questionable. In fact, more than
this conclusion, our results shed ligh t on a ca v eat in the dating metho d exp osed in the previous
section, namely the assumption that the duration of the fractionation ev en t is negligible
compared to the half-life of the isotop e system b eing used. Lu-Hf is the longest-liv ed system
of Gaffney and Borg (2014)’s study , with a half-life of 151.91 Gyr (
τ 1 / 2 , Lu
= 1
.
52
×
10
11
y ears).
W e ha v e sho wn that the lunar magma o cean tak es up to 180 Myr (
t MO
= 1
.
8
×
10
8
y ears)
78

4.2 Isotop es fractionation during silicate differentiation
Name age (Ma) ε 176 Hf ε 143 Nd
15386 3845 ± 77 a − 6 . 47 ± 0 . 38 a − 1 . 51 ± 0 . 335 b
72275,383 4090 ± 70 a − 3 . 65 ± 0 . 39 a − 0 . 62 ± 0 . 63 c
NW A773 2993 ± 32 d - − 4 . 5 ± 0 . 3 d
77215 4288 ± 22 e − 0 . 81 ± 0 . 45 a − 0 . 54 ± 0 . 09 e
78238 4349 ± 19 e − 0 . 26 ± 0 . 48 a − 0 . 27 ± 0 . 74 e
14304,267 4108 ± 40 e - − 1 . 08 ± 0 . 23 e
15455,247 4291 ± 91 e - − 0 . 35 ± 0 . 31 e
76535 4306 ± 10 e - − 0 . 15 ± 0 . 22 e
Kal 009 4369 ± 7 f +12 ± 4 . 6 g -
T able 4.1:
Ages and
ε X
v alues of the KREEP samples used to select the fractionation mo dels.
The first three samples are KREEP basalts and the fiv e last are Mg-suite ro c ks. The last line
corresp onds to the basalt sample Kal 009 used to refine the fit. References:
a
(Gaffney & Borg,
2014),
b
(Carlson & Lugmair, 1979),
c
(Shih, Nyquist, Bansal, & Wiesmann, 1992),
d
(Nyquist, Shih,
Reese, & Irving, 2009),
e
(Borg, Gaffney, & Shearer, 2014),
f
(Snap e et al., 2018),
g
(Sok ol et al.,
2008)
3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5
time [Ma]
10
8
6
4
2
0
2
176 H f
CHUR
Lu-Hf model
KREEP Lu-Hf
2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50
time [Ma]
6
5
4
3
2
1
0
1
143 N d
CHUR
Sm-Nd model
KREEP Sm-Nd

Figure 4.5:
Isotopic deviation from the bulk silicate Mo on (considered c hondritic, CHUR =
c hondritic homogeneous uniform reserv oir) against time for a set of lunar samples represen tativ e of
KREEP isotopic signature. Note that con trary to Figure 4.4, the deviation is negativ e for b oth Hf
and Nd due to a higher incompatibilit y of their paren t sp ecies throughout the lunar man tle.
to solidify un til forming the urKREEP reserv oir. Hence
t MO /τ 1 / 2 , Lu ∼
10
− 3
, which is one
order of magnitude greater than the deviation in part-p er- 10
4
used b y Gaffney and Borg
(2014) to quan tify KREEP’s isotopic deviation from the bulk lunar silicate reserv oir. Th us the
assumption that the duration of the lunar magma o cean solidification is negligible do es not
hold for a crystallization time of 180 Myr.
In order to study the influence of a long-liv ed magma o cean on the primordial isotop e
fractionation, with a particular fo cus on repro ducing the isotopic ratios measured in KREEP
samples, w e implemen ted a mo del of coupled fractionation and radioactiv e deca y , that w e
feeded with the time-series of the lunar magma o cean solidification obtained in our fiducial
case in Chapter 3. W e can th us compute the ev olving isotopic signature of the magma o cean
to trac k that of its last formed cum ulates: the urKREEP reservoi r. The follo wing section
describ es this mo del.
79

4. Repro ducing the isotopic record of the Mo on
4.3 Coupled fractionation and radioactiv e deca y mo del
The ca v eat when considering the fractionation ev en t as instantaneous is that radioactiv e deca y
o ccuring during the magma o cean crystallization is neglected. W e th us need to mo del b oth the
fractionation induced b y the crystallization of the lunar magma o cean and the sim ultaneous
radioactiv e deca y of the paren t sp ecies of b oth Lu-Hf and Sm-Nd systems. Section 4.3.1
describ es ho w, based on the time-series asso ciated with our fiducial case from Chapter 3,
the isotopic ev olution of the magma o cean is computed, accoun ting for b oth these effects.
The ev olution is trac k ed un til the remaining magma o cean reac hes the isotopic signature
of KREEP , defining the urKREEP formation time (
t KREEP
). Section 4.3.2 describ es the
computation of the whole-ro c k isotopic signature of the man tle cum ulates used to further
constrain our results (see Section 4.4.3). The w orkflo w of the com bined thermal ev olution and
fractionation/radioactiv e deca y mo del is presen ted in Figure 4.6.
4.3.1 Ev olution of the magma o cean’s isotopic ratios
F ollo wing Equation 4.1, the c hemical equilibrium of sp ecies
X
(
X
b eing in the present case
either Lu, Hf, Sm or Nd) b et w een the magma o cean and the differen t crystallizing minerals
reads: [
X
]
MO
= 1
/K i
X
[
X
]
i
, where [
X
]
MO
is the concen tration of
X
in the magma o cean,
[
X
]
i
the concen tration of
X
in mineral
i
, and
K i
X
the partition co efficien t of
X
b et w een
melt and mineral
i
, (see T able 4.2 for the list of the partition co efficien ts asso ciated with the
minerals presen t in the crystallization sequence). The minerals in the cum ulates pile as w ell
as plagio clase are assumed to b e in equilibrium with the magma o cean only at the time step
during whic h they crystallize. They then retain their crystallization concen tration in
X
(unless
mo dified b y radioactiv e deca y). The time of crystallization of a giv en la y er is obtained from
the time-series for the fiducial case (see Section 3.5.1) presen ted in Chapter 3, shifted b y the
formation time of the Mo on (
t 0
), treated as a free parameter (see section 4.4.1). The mass
conserv ation of
X
b et w een times
t
and
t
+
dt
(during whic h the magma o cean v olume c hanges
b y ∆
V MO
=
V MO
(
t
+
dt
)
− V MO
(
t
) and the v olume of the cum ulates and the crust (com bined)
b y ∆ V = − ∆ V MO ) reads:
V MO ( t + dt )[ X ] MO ( t + dt )+[(1 − ϕ t )Σ i C i [ X ] i ( t + dt )∆ V + ϕ t [ X ] MO ( t + dt )∆ V ] = V MO ( t )[ X ] MO ( t )
(4.4)
where
ϕ t
is the amoun t of trapp ed melts remaining in the cum ulate pile and
C i
the abundance
of mineral
i
in the la y er of v olume ∆
V
of cum ulates formed during
dt
. The first term on
the left-hand-side represen ts the total amoun t of
X
in the magma o cean at time
t
+
dt
, the
second term on the left hand side is the amoun t of
X
that en ters the differen t minerals of
the newly crystallized cum ulates and crust la yers and the righ t-hand-side represen ts the total
amoun t of
X
in the magma o cean at time
t
. Using the partition co efficien t
K i
X
, [
X
]
i
can b e
replaced in Equation
(4.4)
b y
K i
X
[
X
]
MO
. If w e write successiv ely Equation
(4.4)
for
X
=
P
(the radiogenic paren t sp ecies, i.e.
176 Lu
or
147 Sm
) and for
X
=
D s
(the stable daugh ter
sp ecies, i.e.
177 Hf
or
143 Nd
), and divide the former b y the latter, w e obtain the isotopic ratio
80

4.3 Coupled fractionation and radioactiv e deca y mo del
up dated for fractionation, b efore radioactiv e deca y:
˜
[ P ]
[ D s ] ⏐ ⏐ ⏐ ⏐ ⏐ MO
( t + dt ) = [ P ]
[ D s ] ⏐ ⏐ ⏐ ⏐ MO
( t ) V MO ( t + dt ) + ( (1 − ϕ t )Σ i C i K i
D + ϕ t ) ∆ V
V MO ( t + dt ) + ( (1 − ϕ t )Σ i C i K i
P + ϕ t ) ∆ V , (4.5)
˜
[ D r ]
[ D s ] ⏐ ⏐ ⏐ ⏐ ⏐ MO
( t + dt ) = [ D r ]
[ D s ] ⏐ ⏐ ⏐ ⏐ MO
( t ) . (4.6)
Since the radiogenic and stable daugh ter sp ecies
D r
and
D s
ha v e the same partition co efficien t,
the ratio [
D r
]
/
[
D s
] is not affected b y fractionation. Equation (4.6) represen ts the ”fractionation”
part of the time step. W e then apply radioactiv e deca y:
[ P ]
[ D s ] ⏐ ⏐ ⏐ ⏐ MO
( t + dt ) = ˜
[ P ]
[ D s ] ⏐ ⏐ ⏐ ⏐ ⏐ MO
( t + dt ) exp( − dt ln(2) /τ 1 / 2 ) , (4.7)
[ D r ]
[ D s ] ⏐ ⏐ ⏐ ⏐ MO
( t + dt ) = ˜
[ D r ]
[ D s ] ⏐ ⏐ ⏐ ⏐ ⏐ MO
( t + dt ) + ˜
[ P ]
[ D s ] ⏐ ⏐ ⏐ ⏐ ⏐ MO
( t + dt ) ( 1 − exp( − dt ln(2) /τ 1 / 2 ) ) , (4.8)
where
τ 1 / 2
is the half-life of the paren t sp ecies. The initial magma o cean has a c hondritic
comp osition for trace elemen ts from Bouvier, V ervoort, and Patc hett, 2008. This pro cedure
is iterated using the time-series to link the v olume of the magma o cean (
V MO
) and the
radius of its b ottom (
R MO
) with time, un til
[ P ] / [ D s ] | MO
reac hes the v alue measured in
KREEP samples from Gaffney and Borg (2014) for b oth systems at the same time, i.e.
[
176 Lu
]
/
[
177 Hf
] = 0
.
0153
±
0
.
0033 and [
147 Sm
]
/
[
143 Nd
] = 0
.
1723
±
0
.
0019 (presen t-da y v alues).
If these ratios are nev er reac hed simultaneously , the simulation is discarded. If they do, w e
consider the isotopic ratios of the remaining magma o cean as represen tativ e of the KREEP
ratios. The earliest time at whic h KREEP’s paren t-to-daugh ter ratios are sim ultaneously
matc hed for b oth isotopic systems is considered the time of formation of the urKREEP
reserv oir, and is noted t KREEP .
4.3.2 Cum ulates whole-ro c k isotopic signature
W e also compute the whole-ro c k isotopic ratios in the cumulates pile forming b efore the
gro wth of the flotation crust. W e can th us mak e t w o simplification: first, all minerals forming
settle in the cum ulate pile (i.e.
C plagio clase
= 0 ); second, con trary to the prolonged ev olution
after the gro wth of the crust, radioactiv e deca y of long-liv ed sp ecies can b e neglected (i.e.
[ D r ] / [ D s ] | MO
=
˜
[ D r ] / [ D s ] ⏐ ⏐ ⏐ ⏐ MO
and
[ P ] / [ D s ] | MO
=
˜
[ P ] / [ D s ] ⏐ ⏐ ⏐ ⏐ MO
). The whole-ro c k (subscript
WR) isotopic ratios in the cum ulate pile at the time of crystallization ( t crys ) hence read:
[ P ]
[ D s ] ⏐ ⏐ ⏐ ⏐ WR
( t crys ) = Σ i C i K i
P
Σ i C i K i
D
[ P ]
[ D s ] ⏐ ⏐ ⏐ ⏐ MO
( t crys ) , (4.9)
[ D r ]
[ D s ] ⏐ ⏐ ⏐ ⏐ WR
( t crys ) = [ D r ]
[ D s ] ⏐ ⏐ ⏐ ⏐ MO
( t crys ) . (4.10)
81

4. Repro ducing the isotopic record of the Mo on
Mineral K Lu K Hf K Sm K Nd
olivine 0 . 0089 ± 11 a 0 . 0008 ± 1 a 0 . 00032 ± 4 a 0 . 00008 ± 1 a
clinop yro xene 0 . 74 ± 12 a 0 . 37 ± 10 a 0 . 459 ± 78 a 0 . 273 ± 51 a
orthop yro xene 0 . 181 ± 31 a 0 . 047 ± 17 a 0 . 0251 ± 3 a 0 . 0120 ± 2 a
plagio clase 0 . 092 ± 4 b 0 . 1483 ± 25 b 0 . 1650 ± 11 b 0 . 203 ± 7 b
ilmenite 0 . 0545 ± 76 c 0 . 52 ± 4 c 0 . 0037 ± 7 c 0 . 0055 ± 4 c
spinel 0 . 28 ± 9 c 0 . 65 ± 17 c 0 . 007 ± 4 c 0 . 0037 ± 21 c
quartz 0 . 01424 ± 104 b 0 . 03025 ± 75 b 0 . 0140 ± 2 b 0 . 016 ± 2 b
whitlo c kite 1 . 902 ± 57 d 0 . 01216 ± 90 d 12 . 67 ± 51 d 10 . 18 ± 26 d
sphene 34 . 1 ± 7 e 27 . 90 ± 1 . 49 e 6 . 23 ± 25 e 14 . 01 ± 32 e
T able 4.2:
P artition co efficien ts for th eminerals in the cum ulate sequence. References:
a
McDade,
Blundy, and W o od, 2003,
b
Nash and Crecraft., 1985,
c
Klemme, Gun ther, Pro w atk e, and Zack,
2006, d Pro w atk e and S., 2006, e Pro w atk e and S., 2004
The ratios at an y time t can b e then computed using a simple radioactiv e deca y la w:
[ P ]
[ D s ] ⏐ ⏐ ⏐ ⏐ WR
( t ) = [ P ]
[ D s ] ⏐ ⏐ ⏐ ⏐ WR
( t crys ) exp ( − ( t − t crys ) ln(2) /τ 1 / 2 ) (4.11)
[ D r ]
[ D s ] ⏐ ⏐ ⏐ ⏐ WR
( t ) = [ D r ]
[ D s ] ⏐ ⏐ ⏐ ⏐ WR
( t crys ) + ( [ P ]
[ D s ] ⏐ ⏐ ⏐ ⏐ WR
( t crys ) − [ P ]
[ D s ] ⏐ ⏐ ⏐ ⏐ WR
( t ) ) (4.12)
4.4 Re-in terpreting the isotopic data of KREEP samples
4.4.1 Mon te-Carlo sim ulations
P erforming random sampling of b oth
t 0
(within the first 400 Myr of the solar system, i.e.
as late as 4.167 Ga) and of the melt/ro c k partition co efficien ts of the cum ulates minerals
within the error-bars listed in the literature (see T able 4.2), w e compute 10
4
fractionation
sim ulations. W e select the sim ulations where a reservoir is formed that corresponds to the
KREEP paren t-to-daugh ter ratios. Dra wing the corresp onding deca y lines for b oth systems,
starting at the obtained time
t KREEP
, w e further select the sim ulations for whic h the KREEP
deca y lines cross most data p oin ts used in (Gaffney & Borg, 2014) and listed in T able 4.1.
The 8 differen t samples pro vide a total of 12 data p oin ts (4 ha ving b oth
ε 143 Nd
and
ε 176 Hf
v alues and four ha ving only
ε 143 Nd
v alues). Some data p oin ts are not compatible with the
others b ecause they cannot b e all matc hed with a single line. Hence, the b est fit consists in
matc hing 11 out of 12 data p oin ts. A case (i.e. a Monte-Carlo sim ulation corresp onding to a
set of partition co efficien ts and a v alue of
t 0
, resulting in a formation time
t KREEP
and isotopic
systematics for the KREEP reserv oir) is considered to fit a data p oin t when its KREEP deca y
line crosses the ellipse defined b y the data p oin t’s error bars in a εX vs time diagram.
4.4.2 Matc hing KREEP isotopic signature
F rom our p o ol of 10
4
sim ulations, 775 match 10 out of 12 data p oin ts and 45 matc h 11.
This lo w matc hing is likely due to an o v er-simplified mo delling of the fractionation induced
b y considering constan t partition co efficien ts (see Section 4.5.1 for a discussion ab out the
influence of the partition co efficien ts). Ho w ev er, the large differences in the v alues of partition
co efficien ts b et w een v arious minerals suggest that this simple mo del grasps the main effect
in the fractionation. The cases that reac h the optimal fit (11/12) are sho wn in Figure 4.7 as
82

4.4 Re-in terpreting the isotopic data of KREEP samples
Crys talliz ation mod el
( alphaMEL TS + FXM O TR )
Bulk silic at e Moo n c ompositio n
- Cry st alliz atio n te m pe rat u re
- Met li ng curves
- Cumula tes comp o si ti on
Thermal evolution model
( dif fusing or convec ti ng
(GAIA) cumula tes )
- " #$% &'
- ( $)*
- R MO tim e - se ries
- R crust time - serie s
Coupled fr actionation
and r adioactiv e dec a y
mode l
- Whol e - ro c k [ P] / [ D s ]
- Whol e - ro c k [ D r ]/[ D s ]
- LMO [P ]/[ D s ]
- LMO [D r ]/[ D s ]
- + ,
- Pa r ti t i o n co e f f i c i e n t s
- Tr a c e elements bulk
com p o si ti on
Fit tin g to K RE E P
KREEP isotopic co m p o si tio n
(Gaf fney et al., 2014)
+ -./ /0

Figure 4.6:
Global mo del w orkflo w. Input parameters are indicated in green, in termediate results
in blue, and final results in red (the main results,
t 0
and
t KREEP
are framed b y a dashed line).
Note that
t 0
and the partition co efficien ts are iterated to fit the paren t-to-daugh ter ratio of the
KREEP inferred b y Gaffney and Borg (2014). Chapter 3 is concerned with the upp er part of the
w orkflo w, while the presen t c hapter deals with the lo op in the lo w er half, the righ t b o x (”Coupled
fractionation and radioactiv e deca y mo del”) b eing describ ed in sections 4.3.1 and 4.3.2, and the
”Fitting to KREEP” one in section 4.4.1.
83

4. Repro ducing the isotopic record of the Mo on
time-series of
εD r
where
D s
is either
176 Hf
or
143 Nd
. F rom panels a) and b), it is clear that, at
time of KREEP formation, the magma o cean isotopic comp osition has already largely drifted
a w ay from the CHUR-lik e comp osition assumed for the bulk lunar man tle. The a v erage v alue
of
ε 143 Nd
at KREEP formation is
∼ −
0
.
32 and that of
ε 176 Hf
is
∼ −
1
.
75 . This deviation
mak es the previously used metho d of reading KREEP age as the in tersection of KREEP’s
deca y line with the CHUR deca y line (c haracterized b y an
ε
v alue of 0) obsolete. In fact, the
age of KREEP is an outcome of our sim ulations, and ranges from 4.27 to 4.19 Ga (Figure
4.7c, blue histogram), i.e. 70 to 180 Myr later than inferred b y Gaffney and Borg (2014).
F urthermore, while once isolated, the isotopic evolution of KREEP follo ws a straigh t deca y
line, the earlier ev olution in the magma o cean is influenced by the crystallization sequence, the
asso ciated partition co efficien ts, and the c hronology of the solidification, and do es no longer
plot as a straigh t line, but as a b en t curv e. Hence the in tercept b et w een KREEP deca y line
and the CHUR deca y line do es not pro vide the time of initiation of the silicate fractionation
either, but it underestimates it. W e find a Mo on-forming ev en t b et w een 4.44 and 4.36 Ga
(Figure 4.7c, red histogram), only marginaly ov erlapping with the estimates b y Borg et al.
(2014) based on Gaffney and Borg (2014).
4.4.3 Matc hing Kal009 isotopic signature
A complemen tary information to KREEP isotop es systematics is brough t b y another sample:
the meteorite Kal 009. It represen ts the oldest crystallized non-crustal lunar sample with
a formation age of 4
.
369
±
7 Ga (Snap e et al., 2018). It formed from a mantle reserv oir
c haracterized b y a high
176 Hf / 177 Hf
ratio (
ε 176 Hf
= +12
.
9
±
4
.
6 ). A mo del that succeeded
in repro ducing the KREEP signature ough t to b e considered realistic only if it also succeeds
at creating a man tle reserv oir compatible with Kal 009. W e th us compute the whole-ro c k
Lu-Hf systematics for our ”successful” cases (see Figure 4.8), and found that a p yro xene-ric h
reserv oir lo cated b et w een 200 and 250 km depth o v erlaps with Kal 009’s
ε 176 Hf
v alue. W e
select the cases that pro duce suc h a reserv oir and plot their
t 0
and
t KREEP
as the turquoise
histograms in Figure 4.7. This results in a KREEP formation time b etw een 4.29 and 4.24 Ga
and a Mo on-formation time b et ween 4.44 and 4.40 Ga, i.e. 40 to 80 Myr older than inferred
b y Gaffney and Borg (2014).
4.5 Sensitivit y study
The mo del presen ted here dep ends on v arious parameters, some of whic h are p o orly constrained.
This section in v estigates the impact of the uncertain t y in these parameters on the results.
4.5.1 Sensitivit y to the partition co efficien ts
As men tioned ab o v e, the c hoice of constan t v alues for the partition co efficien ts is lik ely an
o v er-simplification. Mo dels exist for parametrization of partition co efficien ts as a function of
temp erature, pressure, and mineral and melt comp osition (e.g. Sun, Graff, & Liang, 2017),
ho w ev er their use w ould bring the complexit y of the presen t mo del b ey ond the scop e of the
study . Nevertheless, all minerals do not pla y an equiv alen t role. Some ha v e a lo w partition
co efficien t and do not affect m uc h the ev olution of the magma o cean isotop e signature, while
84

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