Well-Posedness and S abili y Analysis o a Landscape
E olu ion Model
Julie Bina d1
· Pie e Degond2
· Pascal Noble1
Recei ed: 17 No embe 2022 / Accep ed: 4 No embe 2023 /
Abs ac
In his pape , we s udy a sys em o pa ial di e en ial equa ions modeling he
e olu ion o a landscape in o de o desc ibe he mechanisms o pa e n o ma ions.
A g ound su ace is e oded by he low o wa e o e i , by ei he sedimen a ion o
dilu ion. We conside a model, composed o h ee e olu ion equa ions: one on he
ele a ion o he g ound su ace, one on he luid heigh and one on he concen a ion
o sedimen s in he luid laye . We i s es ablish he well-posedness o he sys em in
sho ime and unde he assump ion ha he ini ial luid heigh does no anish.
Then, we ocus on pa e n o ma ion in he case o a ilm low o e an inclined
e odible plane. Fo ha pu pose, we ca y ou a spec al s abili y analysis o cons an
s a e solu ions in o de o de e mine ins abili y condi ions and iden i y a mechanism
o pa e n o ma ions. These pa e ns, which a e ills and gullies, a e he s a ing
poin o he o ma ion o i e s and alleys in landscapes. Finally, we ca y ou some
nume ical simula ions o he ull sys em in o de o alida e he spec al ins abili y
scena io, and de e mine he esul ing pa e ns.
Keywo ds Pa e n o ma ion · Ins abili ies · Landscape e olu ion model · E osion by
wa e · S eam incision law · Sedimen a ion
Communica ed by Jose A. Ca illo.
BJulie Bina d
[email p o ec ed]
Pie e Degond
[email p o ec ed] - oulouse.
Pascal Noble
[email p o ec ed]
1Ins i u de Ma héma iques de Toulouse, UMR 5219, CNRS, INSA, Uni e si é de Toulouse,
31077 Toulouse, F ance
2Ins i u de Ma héma iques de Toulouse, UMR 5219, CNRS UPS, Uni e si é de Toulouse, 31062
Toulouse Cedex 9, F ance
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Ma hema ics Subjec Classi ica ion 35M30 ·35Q86 ·35B35 ·35B36 ·35A01 ·86-10
1 In oduc ion
The modeling o landscape e olu ion unde he e ec o wa e low has ecei ed
inc easing a en ion hese las decades. Se e al physical phenomena mus be consid-
e ed in o de o desc ibe he e osion and sedimen a ion p ocesses, which occu when
wa e lows o e an e odible su ace. The e osion is he emo al o sedimen s om he
soil by a luid, and he sedimen a ion is he in e se p ocess, when he sedimen s se le
on he su ace. Heu is ically, he in ensi y o he e osion p ocess is s ongly ela ed o
he low a e o he luid, he e osion a e is highe when he low a e is highe . An o he
p ocess ha a ec s landscape e olu ion is he c eep e ec , which ends o smoo h he
bo om su ace on imescales ha a e much la ge han he ones in ol ed in sedimen
anspo . This e ec has mul iple causes, such as g a i a ional o ces ac ing on he
soil, and co osion and dila a ion caused by physical and chemical ac o s. I can be
desc ibed as a simple di usion p ocess o he soil. The c eep e ec was in oduced in
1892, in Da is (1892). I was used in Gilbe (1909) o explain he con exi y o hill ops
p o ile. La e , in 1963 a de i a ion o he soil c eep e ec as a limi o a s ochas ic
p ocess had been done in Culling (1963). The s ochas ic e ec models soil pa icles
which ollow a andom mo ion, unde he cons ain o g a i y. In his a icle, Culling
w i es “Soil c eep—so g adual as o be impe cep ible—appea s o be he esul o he
pe sis en e ec o molecula and mac omolecula o ces ending o displace he soil
pa icles,” and he desc ibes he soil low as a quasi- iscous luid.
The desc ip ion o he geological p ocesses ha occu in landscape e olu ion was
ini ia ed in Gilbe (1877). In his book, he au ho gi es he undamen al p inciples
o landscape e olu ion and explains why he p o iles o s eam beds a e conca e
upwa ds. This conca i y p ope y had been illus a ed la e in Culling (1960), wi h a
ma hema ical desc ip ion o he e osion o a slope. The slope e olu ion is desc ibed
wi h a eac ion di usion equa ion, he exponen ial sou ce e m ep esen ing he e osion
a e. The landscapes e ol e mainly in unc ion o hese wo compe i i e ac o s: c eep
on he hill ops and s eam incision on he lowe slopes. Indeed, in he uppe slopes he
wa e low is weak and dispe si e; hus, he c eep e ec p edomina es. Downwa d,
he e ec o shea s ess o he low becomes dominan , and bedload anspo p ocess
inc eases. This leads o he o ma ion o s eam beds and alley, he p o iles being
conca e. Since hen, he complexi y o he landscape e olu ion models inc eased: See
Chen e al. (2014a,b) o a e iew o hese models.
The p ocess o e osion and anspo o sedimen s can be desc ibed by wo di e en
laws, he allu ial anspo law o he s eam incision law, which depends on he na u e
o he soil. When he soil is co e ed by allu iums, he sedimen s a e di ec ly anspo ed
by wa e lux, and he amoun o sedimen s mo ed by he wa e lux in a gi en ime
qs ollows a law ha depends on he wa e discha ge q. This anspo discha ge law
is desc ibed, o example, in Smi h and B e he on (1972) and is gi en by
qs=kqm|∇z|n(1)
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whe e k,mand na e cons an s. This is called he anspo limi ed case.
On he o he side, he s eam incision law is used when he su ace is bed ock,
because in his case he sedimen anspo is limi ed by he esis ance o he bed ock
o he shea s ess caused by he wa e lux. This is modeled by an e osion sou ce e m
in he equa ion o su ace heigh e olu ion, as desc ibed in Howa d and Ke by (1983).
The sedimen s emo ed om he soil by he e osion a e supposed o be dissol ed in
wa e and hus a e anspo ed by he wa e lux. This s eam incision law is desc ibed
in Sec .2. This case ha we s udy in his pape is called he de achmen limi ed case.
In his pape , we s udy a sys em o pa ial di e en ial equa ions desc ibing he
e osion o he soil by wa e , which had been p oposed in Chen e al. (2014a), Sec ion
4. I includes he modeling o he wa e low, he e osion o he su ace by wa e ,
he anspo and deposi ion o sedimen s, and he c eep e ec . The cha ac e is ic
luid eloci y is usually much la ge han he e osion a e: In o de o desc ibe pa e n
o ma ion, we only conside la ge-scale luc ua ions o he luid eloci y. As a esul ,
we assume ha he luid eloci y is p opo ional o he g adien o he ee su ace
ele a ion. The o he physical p inciples aken in accoun a e:
•The conse a ion law o wa e and sedimen s dissol ed in wa e ,
•The s eam incision law: he e osion g ows wi h he wa e speed and he wa e
heigh ,
•The sedimen a ion a e is p opo ional o he concen a ion o sedimen s in wa e ,
•The c eep e ec : he soil is subjec o a di usion p ocess.
O he e ec s, such as in il a ion, ege a ion, wind o ice o ma ion, a e neglec ed.
Mo eo e , we suppose ha he bo om su ace is cons i u ed by one ype o sedimen s.
The sys em s udied in his a icle models he ime e olu ion o he soil and o he luid.
This sys em is composed by he ollowing h ee pa ial di e en ial equa ions:
⎧
⎨
⎩
∂ h−di (h∇(h+z)) = ( ,x),
∂ z=Kz+sc −ehm| |n,
∂ (ch)+di (ch ) =ehm| |n−sc.
(2)
The i s equa ion is a mass conse a ion law o he luid and desc ibes he e olu ion
o he luid heigh , deno ed by h, and he luid is anspo ed a speed := −∇(h+z).
The e m is a sou ce e m ep esen ing an ex e io sou ce o wa e , like he ain.
The second equa ion models he ime e olu ion o he bo om opog aphy (deno ed
by z). The cons an Kis he cons an o c eep, whe eas eand sa e, espec i ely, he
cons an s o he incision law and he sedimen a ion a e. As in Howa d and Ke by
(1983), we will suppose ha he incision law depends on he powe o he no m o
he wa e eloci y and on he powe o he wa e heigh . Thanks o his hypo hesis,
as long as he wa e heigh and eloci y do no anish, he e osion a e is posi i e.
Howe e , in p ac ice he e osion s a s i he wa e eloci y is high enough o b eak
he cohesion o he soil. Thus, a h eshold e ec could be in oduced in he model.
This e ec could be easily added o he nume ical scheme o he model, bu i would
inc ease signi ican ly he di icul y o i s ma hema ical s udy. The e o e, we choose
o igno e he h eshold e ec in he e osion p ocess. This app oxima ion is jus i ied
in he egime ha we s udy, because he wa e heigh and wa e eloci y a e no close
20 Page 4 o 55
Fig. 1 Ske ch o he ma hema ical amewo k o s abili y analysis: The wa e is lowing on a il ed plane
o ze o. The las equa ion is he conse a ion equa ion o he sedimen s dissol ed in
he luid. This concen a ion, deno ed by c, is he a e age o he concen a ion o e
he heigh o he luid and hus is gi en in g am pe squa e me e . The igh -hand side
is he sou ce e m, which ep esen s he exchange o sedimen s be ween he g ound
and he luid, caused by he e osion and he sedimen a ion. The model is se in wo
dimensions, he a iables a e he ime ∈R+and he posi ion (x,y), which belongs
o a domain ⊂R2.
In Chen e al. (2014a), he au ho s e iewed a ious landscape e olu ion models
and ocused on sys em (2). They lis ed some open ma hema ical p oblems like local
exis ence in ime, egula i y o solu ions o s abili y. They pe o med a nume ical
s udy on he e osion o a Gaussian shaped hill o demons a e he abili y o he model
o exhibi pa e n o ma ion and s udied he in luence o some pa ame e s on he
complexi y o hese pa e ns. In Leb un e al. (2018), he au ho s pe o med nume ical
simula ions in a mo e p ac ical con ex whe e he ini ial opog aphy is a eal one like
he ones ound in La Reunion o Madei a islands: They ook a pa icula ca e o he
isualiza ion o landscape e olu ion h ough a pa icula colo iza ion p ocess. The
simula ions show he capaci y o he model o desc ibe a ealis ic e olu ion in ime
o he i e s and gullies on hese landscapes, p o ided i s pa ame e s a e p ope ly
chosen. No e ha he de achmen limi hypo hesis is alid o bed ock i e s and
mus be app op ia ely modi ied o allu ial i e s, whe e sedimen s o m a laye o
loose ma e ial. In his la e case, o he classes o models may be conside ed like he
shallow wa e equa ions coupled wi h Exne equa ions o he anspo o sedimen s:
See Fe nández-Nie o e al. (2017) o a o mal de i a ion o hese models om bilaye -
ype models (one laye o he luid and one laye o he sedimen ) and Fe nandez-
Nie o e al. (2014); Escalan e e al. (2021) o a ious nume ical s a egies o pe o m
simula ions o bedload sedimen anspo . The aim o his pape is o explo e he
mechanism o pa e n o ma ion in he soil caused by wa e low and o p esen he
o m and he equency o appa i ion o hese pa e ns. In landscapes, hese pa e ns
a e channels, bed i e s, alley. We will ocus on a simple amewo k whe e he ini ial
bo om su ace is a il ed plane. This su ace is co e ed by a laye o wa e , he wa e
low om he op o he plane. The si ua ion is p esen ed in Fig. 1, whe e he il ed
plane is seen om he side.
123
Page 5 o 55 20
Al hough i is a less ealis ic amewo k han he landscape e olu ion desc ibed in
Leb un e al. (2018) whe e eal ini ial opog aphies a e conside ed, i is su icien o
s udy he eme gence o pa e ns on he su ace. No e ha expe imen s o e osion on a
la su aces we e conduc ed in Gué in e al. (2020) wi h a hin ilm low which e odes
blocks o plas e o sal .
The main ou comes o his pape a e he ollowing:
•We es ablish he well-posedness o Sys em (2) when he ini ial luid heigh does
no anish.
•We s udy he spec al s abili y o cons an s a es when he bo om is an
inclined plane and cha ac e ize igo ously he spec al ins abili y h ough a non-
dimensional numbe , he so-called channeliza ion index.
•We pe o m di ec nume ical simula ions o Sys em (2) o ealis ic expe imen al
da a and on imescales associa ed wi h he e osion phenomenon. We ackle he
p oblem o se e e CFL es ic ions due o e y sho imescales associa ed wi h
he luid low.
We p o ide he ea e a sho desc ip ion o ou main esul s. Fi s , we p o e he
well-posedness cha ac e o Sys em (2) locally in ime unde some assump ions on
he da a and he pa ame e s. We p o e he exis ence o solu ions in sui able Sobole
spaces. In o de o simpli y no a ions, we will deno e o any T>0 and Ha Hilbe
space
L2
T(H):= L2((0,T);H);CT(H):= C((0,T);H)
he space o unc ions u: → u( ,.) ∈H ha a e, espec i ely, L2in eg able o
con inuous in ime on he in e al (0,T). We p o e he ollowing esul :
Theo em 1.1 Le m >0,n >3o n =2,K >0, and T0>0. Le us ix wo cons an s
luid heigh s h e >hmin >0. Suppose ha he ini ial da a h0,z
0,c
0sa is y
h0−h e ,z0∈Hk+1(R2), c0∈Hk,h0(x)≥2hmin ∀x∈R2.
wi h k =3. Suppose ha ∈L2
T0(Hk),Kh
min −||h0||2
L∞≥0. Then he e exis s
0<T<T0such ha Sys em (2) admi s a unique solu ion (h,z,c)wi h
h−h e ,z∈L2
T(Hk+2)∩CT(Hk+1), c∈CT(Hk).
The p oo o Theo em 1.1 is based on ene gy es ima es and on a ixed poin a gu-
men . The p oblem o he well-posedness o his sys em was aised in he conclusion
o Chen e al. (2014a). We p o e ha i is well posed locally in ime and unde he
assump ion ha he luid heigh does no anish. Wi h his assump ion, he equa ions
on wa e heigh and wa e concen a ion a e pa abolic, and his p ope y is used in he
p oo , o he ene gy es ima es. This alida es he model in he si ua ion whe e he
luid heigh is close o a cons an , hence, a om anishing. This will be he case in
ou s abili y s udy and ou nume ical simula ions. The case o anishing luid heigh
is mo e in ol ed as he model becomes degene a ed pa abolic and he egula izing

20 Page 6 o 55
e ec on he luid heigh , and hus on he luid eloci y, is los : one has o conside
weake classes o solu ions. Mo e p ecisely, d opping he zdependence, he equa ion
on h eads
∂ h=hh+|∇h|2.
In B ändle and Vázquez (2005), he au ho s p o e he well-posedness o his equa ion
wi hin he class o bounded, con inuous and nonnega i e iscosi y solu ions. This is
beyond he scope o his pape o couple his ype o solu ions wi h he o he equa ions
o he model, and we lea e his p oblem open o u u e s udies. An o he in e es ing
ques ion is he nume ical simula ion o he model in he p esence o d y a eas: This
will be ca ied ou in a o hcoming wo k.
The second pa o he pape is dedica ed o pa e n o ma ion when he ini ial
opog aphy is an inclined plane and he bo om su ace is weakly e oded. Ou aim is
o iden i y ins abili y mechanisms ha could explain he o ma ion o pa e ns. Fo ha
pu pose, we linea ize Sys em (2) a ound a cons an s a e and we s udy he condi ions o
spec al ins abili y and he na u e o he ins abili ies. We expec ha when he sys em is
spec ally uns able o some pa ame e s and wa e ec o s, a sligh pe u ba ion o he
sys em wi h hese uns able modes will g ow up and lead o he o ma ion o pa e ns
in he soil.
S abili y s udies ha e been done p e iously, o o he landscape e olu ion models.
The pape s Smi h and B e he on (1972) and Loewenhe z (1991) analyze a model
o wo equa ions, whe e he wa e is supposed o be a equilib ium. This model has
s eady solu ions o which he soil heigh can be conca e in some a eas and con ex in
o he a eas, depending on he sedimen discha ge law. They show ha he linea ized
sys em is s able in he con ex pa s and uns able in he conca e pa s, wi h a s onge
ins abili y in he ans e se di ec ion. In hei model, hey use he sedimen anspo
law; hus, i is no he same amewo k as in he model (2).
No e ha in a couple o ecen pape s Anand e al. (2020) and Bone i e al. (2020),
he ollowing sys em o 2 PDEs was conside ed:
∂ z=Kz−e(h
H)m|∇z|n+U,∂
h=di (h 0∇z
|∇z|)+R,(3)
whe e R,Uand V0a e cons an s. A nume ical scheme is designed in Anand e al.
(2020) o (3) whe e he ime de i a i e o he luid heigh is neglec ed wi h es cases
whe e he ini ial bo om opog aphy is py amidal. I is ound ha a channeliza ion
index
CI=em+n
KnU1−n
d i es he o ma ion o channels: The numbe o channels and hei b anching inc ease
wi h CI. This analysis is comple ed by a spec al s abili y analysis o a spa ially non-
homogeneous s eady s a e whe e he opog aphy is a hillslope which is di ided in he
middle. I is ound nume ically ha he e exis s a c i ical C0
Isuch ha he s eady s a e
is s able i CI≤C0
Iand uns able o he wise.
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Page 7 o 55 20
The spec al s udy ca ied ou in his pape is new and ou analysis p o ides some
explana ions o he o ma ion o pa e ns in landscapes. The appea ance o channels
on he la plane is indeed he ini ial s age o de elopmen o he o ma ion o alley
and i e s in landscapes. The s abili y o he sys em depends on he pa ame e s; in
pa icula , he cons an o c eep Kplays an impo an ole in his s udy. We show ha
he e is a c i ical alue ¯
Ksuch ha i K≥¯
K, and i ano he condi ion on pa ame e s
is sa is ied, hen he sys em is spec ally s able a all equencies. I K<¯
K, hen he e
exis some wa e numbe s and ec o s o which he sys em is spec ally uns able.
Mo eo e , he ins abili ies g ow as he wa e ec o s o he pe u ba ions poin s in he
di ec ion ans e se o he low, which explains he o ma ion o gullies and channels
aligned wi h he di ec ion o he luid low. We hen eco e quali a i ely he esul s
o Bone i e al. (2020).
Finally, ou s abili y s udy is comple ed by di ec nume ical simula ions, which
illus a e he appea ance o pa e ns o he nonlinea sys em. The space and ime
scales o he model can ake a la ge ange o alues, depending o he en i onmen .
On eal landscapes, he domain size can be measu ed in kilome e s, wi h a e y slow
e osion a e, in he o de o 20 −200 millime e s pe housand yea . In he expe ience
on sal and plas e made in Gué in e al. (2020), he domain has a size o he o de o en
cen ime e s, whe eas he e osion speed is a ound one millime e pe hou (so, much
as e han in eal landscape) and he luid eloci y is 1 m pe second. Pa ame e s
chosen in he nume ical simula ions a e based on hese expe imen s. As in Anand
e al. (2020); Bone i e al. (2020), we ha e obse ed ha dec easing K( espec i ely,
inc easing he channeliza ion index CI) ein o ces he channeliza ion p ocess. No e
ha we ha e ocused he e on he o ma ion o channels: Unlike simula ions made in
Leb un e al. (2018) whe e he ini ial s a e is a ma u ed landscape, we s a om a
simple s a e and he landscape e ol es by himsel in he simula ions.
The pape is o ganized as ollows: Fi s , in Sec . 2, we desc ibe he model and
he associa ed sys em o equa ions. Then, Sec .3is de o ed o he p oo o he well-
posedness cha ac e o he sys em in sho ime, (see Theo em 1.1). Nex , in Sec .4we
ca y ou a spec al s abili y analysis o he Sys em (2) linea ized abou a s a iona y
solu ion. These s abili y esul s a e compa ed o di ec nume ical simula ions o he
nonlinea sys em (2) in Sec .4.2. Finally, Sec . 6d aws a b ie conclusion o he pape
and p o ides some u u e pe spec i es.
2 The Landscape E olu ion Model
In his sec ion, we in oduce he landscape e olu ion model conside ed in his pape .
This is a sys em o h ee pa ial di e en ial equa ions o he luid heigh h, he bo om
opog aphy zand he sedimen concen a ion c.
E olu ion o opog aphy.
The e olu ion o he bo om opog aphy zis gi en by:
∂ z=Kz−E+S.
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The unc ions E=E( ,x,y)and S=S( ,x,y) ep esen he e osion speed o
he soil and he sedimen a ion speed, espec i ely, wi h ≥0, (x,y)∈R2.
The pa ame e K>0 is a cons an , and he e m Kzmodels he c eep o he
soil. This phenomenon is a slow di usi e mo emen o he soil which occu s a
la ge ime and space scales. This mo emen is caused by se e al p ocesses, such
as he g a i a ional low o he soil, wind, ain splash, expansions and con ac ions
o he soil due o eeze– haw, we –d y and ho –cold cycles, o biological ac i i y. In
su icien ly e oded landscape, he e a e gene ally no many sha p edges, and his c eep
e m, which ends o smoo h he bo om su ace, models his phenomenon. We shall
see ha his e m plays a signi ican ole in he well-posedness o he model. Howe e ,
he c eep e ec is no supposed o be ele an in he o ma ion o pa e ns as i is a
sho ime e ec and should be supposed o be small in compa ison wi h he e osion
and sedimen a ion e ms.
The e osion o he su ace is caused by he shea s ess and he ic ion o he wa e
low. Assuming ha he luid eloci y is cons an ac oss he luid laye , his amoun s
o conside ha he e osion inc eases wi h he (no m o he) wa e eloci y, and wi h
he wa e discha ge Q, as in Howa d and Ke by (1983):
E( ,x,y)=αQμ ν.
Consequen ly, as Q=h , we will suppose ha he e osion speed depends on he
powe o he no m o he wa e eloci y and on he powe o he wa e heigh . Thus,
we se
E( ,x,y)=eh( ,x,y)
Hm| ( ,x,y)|
Vn
,
whe e eis he e osion speed in he condi ions h=Hand | |=Vwi h H,V>0
ha , espec i ely, ep esen a e e ence luid heigh and luid eloci y.
The sedimen a ion occu s when he concen a ion o sedimen s in wa e is high
enough. The sedimen a ion speed inc eases wi h he concen a ion o sedimen in he
luid. Fo he sake o simplici y, we suppose ha his speed is p opo ional o he
concen a ion and we se :
S( ,x,y)=sc( ,x,y)
csa
,
wi h s he speed o sedimen a ion in he e e ence condi ion c=csa . The e o e, he
soil ele a ion e ol es acco ding o he equa ion:
∂ z=Kz−eh( ,x,y)
Hm| ( ,x,y)|
Vn
+sc( ,x,y)
csa
.(4)
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The landscape e olu ion model.
We comple e Equa ion (4) wi h wo e olu ion equa ions o he luid heigh hand
sedimen concen a ion c. The mass conse a ion law o he luid eads
∂ h+di (h ) = ,(5)
whe e is a sou ce e m, modeling an incoming low in a channel o he ain o e
he bo om.
On he o he hand, he mass conse a ion law o he sedimen eads
∂ (hc)+di (ch ) =ρs(E−S), (6)
whe e ρsis he olume ic mass densi y o he sedimen s and is cons an . In o de
o close Sys em (4), (5), (6), we need o w i e an equa ion o he luid eloci y. One
possibili y would be o w i e a shallow wa e - ype model wi h an e olu ion equa ion
o he momen um h . We a he choose he simple closu e
=−μ∇(h+z), (7)
whe e μ>0 is some cha ac e is ic luid eloci y and ∇(h+z)is he g adien o
he luid su ace ele a ion. Sys em (4), (5), (6), (7) is closed, and we shall conside i s
well-posedness in Sec .3.
We a e also in e es ed in he pa e n o ma ion a he su ace o he soil. Fo ha
pu pose, we ha e chosen o explo e he case o wa e lowing down an inclined plane.
This si ua ion was conside ed expe imen ally in Gué in e al. (2020). The domain
⊂R2has leng h Lxand wid h Ly:=[0,Lx]×[0,Ly]. Deno e θ he inclina ion
o he plane. We can decompose he bo om opog aphy zas z( ,x,y)=(Lx−
x) an θ+˜z( ,x,y)whe e ˜zis he e oded heigh o he soil. Thus, he luid eloci y
is w i en as ( ,x,y)=μ( an θ,0)−μ∇(˜z+h). Consequen ly, omi ing he ilde
o e z, Sys em (4), (5), (6), (7) admi s he new o m:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∂ h+μ an θ∂xh=μdi (h∇(h+z)), (8a)
h∂ c+μh an θ∂xc=μh∇(h+z).∇c+ρseh( ,x)
Hm| ( ,x)|
Vn
−ρssc( ,x)
csa
,(8b)
∂ z=Kz−eh( ,x)
Hm| ( ,x)|
Vn
+sc( ,x)
csa
.(8c)
3 Well-Posedness o he Landscape E olu ion Model
In his sec ion, we s udy he exis ence and uniqueness o solu ions o he sys-
em (4), (5), (6), (7), locally in ime.
20 Page 16 o 55
The classical heo y o pa abolic equa ions and o linea anspo equa ions p o-
ides condi ions o ob ain a well-posed sys em o equa ions. Fo he wo pa abolic
equa ions (25a) and (25b), we s a e he ollowing esul (see Che ie and Milani 2012
o mo e de ails):
P oposi ion 3.7 Assume ha h0−h e ,z0∈Hk+1(R2), ,sc
i−ehm
i| i|n∈L2
T(Hk),
hi∈L∞([0,T]×R2)wi h hi≥hmin and ∇hi,∇zi∈L1
T(Hk). Then he e exis s
a unique solu ion (hi+1,zi+1) o he equa ions (25a), (25band hi+1−h e ,zi+1∈
L2
T(Hk+2)∩CT(Hk+1(R2)).
Fo he anspo equa ion (25c), we use heo em 7.2.2inMé i ie (2008) o s a e
he esul :
P oposi ion 3.8 Assume ha c0∈Hk(R2),∇ i∈L∞
T(Hk−1),ehm−1
i| i|n−sci/hi−
ci/hi∈L1
T(Hk)and i∈L1
T(W1,∞). Then he e exis s a unique solu ion ci∈
CT(Hk) o he equa ion (25c).
We p o e by induc ion ha o all i∈N, Sys em (25a) is well posed. Fi s , o i=0,
he unc ions o (hi,zi,zi)a e ime independen and h0≥hmin. Thus, as T0=Tis
ini e, we ha e sc0−ehm
0| 0|n∈L2
T0(Hk),h0∈L∞
T0(R2)and ∇h0,∇z0∈L1
T0(Hk).
Then he e exis s a unique solu ion (h1,z1)∈L2
T1(Hk+2)∩CT1(Hk+1)o (25a)
and (25b) wi h 0 <T1≤T0such ha h1≥hmin. Simila ly, he e exis s c1∈CT1(Hk)
solu ion o (25c). Now, i we assume ha hi−h e ,zi∈L2
T(Hk+2)∩CTi(Hk+1)and
ci∈CTi(Hk) hen, one has hi∈h e +CTi(H2)⊂L∞([0,T]×R2). Mo eo e , we
ha e ∇hi,∇zi∈CTi(Hk)⊂L1
Ti(Hk)and ci∈CTi(Hk)⊂L2
Ti(Hk). Finally, one
has
hm
i| i|nHk≤Chm−3
iL∞hiL∞+∇hiHk+∇ziHkn+3∈L∞
Ti(Hk)⊂L2
Ti(Hk).
Thus, he assump ions o P oposi ion 3.7 a e sa is ied and he e exis s a unique
solu ion hi+1,zi+1o (25a,25b) such ha hi+1−h e ,zi+1∈L2
Ti+1(Hk+2)∩
CTi+1(Hk+1(R2)) wi h 0 <Ti+1≤Tisuch ha hi+1≥hmin. The exis ence o a
solu ion ci+1∈CTi+1(Hk) ollows simila ly.
P oposi ion 3.9 (Uni o m bounds) Deno e
Ei( )=hi−h e L2+∇hiHk+ziHk+1+ciHk.
The e exis s B >0and 0<T∗≤T independen o i ∈Nsuch ha , o all i ∈N
sup
∈[0,T∗]
Ei( )≤B,and hi( ,x)≥hmin,∀x∈R2.
Mo eo e ,
T∗
0∇hi2
Hk+1+∇zi2
Hk+1≤C(T∗)B,∀i∈N∗.

Page 17 o 55 20
P oo We p oceed by induc ion. One has easily E0( )=E(0)≤Band h0≥hmin
since h0,z0,c0a e ime independen . Following he s a egy used o de i e a p io i
es ima es, one can p o e ha o some ε>0, o all ∈[0,T],
d
d
E1( )+ε∇h12
Hk+1+∇z12
Hk+1≤C(ε)( 2
Hk+E0( )α)E1( ), (26)
o some cons an C(ε) depending only on εand h0. By in eg a ing Equa ion (26)
wi h espec o ime, one inds:
E1( )+ε
0∇h12
Hk+1+∇z12
Hk+1
≤E(0)+C(ε) 
0 2
Hk+
0
C(ε)E0(s)αE1(s)ds.
Deno e
B=2E(0)+C(ε) T
0 2
Hk.
We choose ˜
T∗such ha
e˜
T∗C(ε)Bα≤2.
Then, by applying G onwall’s lemma, one ob ains:
E1( )≤B,∀ ∈[0,˜
T∗],
˜
T∗
0∇h12
Hk+1+∇z12
Hk+1≤C(ε)1+ln(2)
εB.
Nex , we ha e
h1=h0+
0
di (h0∇(h1)+h1∇z0)) + .
Thus, h1−h0L∞≤C(˜
B +√B ) o some cons an Cindependen o he p oblem
and ela ed o Sobole injec ions, whe eas ˜
B=B+h e √B. Then, he e exis s
0<T∗≤˜
T∗such ha C(˜
B +√B )≤hmin and we deduce ha
h1≥h0−hmin ≥hmin ∀ ∈[0,T∗].
This p o es he ini ial s ep o i=1. Now assume ha Ei( )≤Band hi≥hmin o
all ∈[0,T∗]. The es ima es on hi+1,zi+1and ci+1a e a di ec consequence o he
20 Page 18 o 55
ene gy es ima e:
d
d
Ei+1( )+ε∇hi+12
Hk+1+∇zi+12
Hk+1≤C(ε)( 2
Hk+Ei( )α)Ei+1( )
(27)
which is p o ed by ollowing he s a egy used o de i e he a p io i es ima es. This
comple es he p oo o he p oposi ion. 
P oposi ion 3.10 The sequences (∂ hi)e (∂ zi)a e uni o mly bounded in L2
T∗(Hk).
The sequence (∂ ci)is uni o mly bounded on L2
T∗(Hk−1).
P oo Fo p∈{0,...,k}, we di e en ia e p imes he equa ions (25a), (25b)(25b)
and mul iply i by ∇p∂ hi,∇p∂ ziand ∇p∂ ci, espec i ely, and in eg a e in space.
Using P oposi ion 3.9, we ob ain bounds o ∂ hiL2
T∗(Hk),∂ ziL2
T∗(Hk)and
∂ ciL2
T∗(Hk−1).

3.5 Con e gence o he Sequences
In wha ollows, we deno e T∗=Tin o de o simpli y he no a ions.
P oposi ion 3.11 The sequences (hi)i∈N,(∂ hi)i∈Nand (zi)i∈N,(∂ zi)i∈Na e Cauchy
sequences in L2
T(L2). The sequences (ci)i∈Nand (∂ ci)i∈Na e Cauchy sequences in
CT(L2).
P oo In he ollowing, we deno e he quan i ies o he o m i+1− iby δ i. Fo all
i≥1, he equa ions o δhi,δzi,δcia e w i en as:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∂ δhi−di (hi∇δhi+δhi−1∇hi)−di (hi+1∇δzi+δhi∇zi−1)=0,(28a)
∂ δzi−Kδzi=sδci−ehm
i(| i|n−| i−1|n)+| i−1|n(hm
i−hm
i−1),(28b)
∂ δci+δ i−1∇ci+1+ i−1δ∇ci
=ehm−1
i(| i|n−| i−1|n)+| i−1|n(hm−1
i−hm−1
i−1)
−(s+ )ci
hi−ci−1
hi−1.
(28c)
Bounds on δhi,δziand δci: We mul iply he equa ion (28a)byδhiand in eg a e
o e space:
1dδhi2
L2+R2(hi∇hi+1−hi−1∇hi)∇δhi+R2(∇zihi+1−∇zi−1hi)∇δhi=0.
Page 19 o 55 20
Fi s , on he one hand, we ha e
R2
(hi∇hi+1−hi−1∇hi)∇δhi=R2
hi(∇δhi)2+δhi−1∇hi∇δhi
≥hmin∇δhi2
L2+R2
δhi−1∇hi∇δhi.
On he o he hand, we ha e he es ima e:
R2
(∇zihi+1−∇zi−1hi)∇δhi=1
2R2∇zi∇(δhi)2+R2
δzi−1hi∇δhi
=−1
2R2
zi(δhi)2+R2
δzi−1hi∇δhi
≤Cδhi2
L2+hiL∞δzi−1L2∇δhiL2
≤Cδhi2
L2+1
2hmin δzi−12
L2+hmin
2∇δhi2
L2.
Consequen ly, we ob ain:
d
d δhi2
L2+hmin∇δhi2
L2≤Cδhi2
L2+Cδhi−12
L2+Cδzi−12
L2.(29)
We p oceed simila ly o δzi. By using Equa ion (28b), one inds:
d
d δzi2
L2+2K∇δzi2≤s(δci2
L2+δzi2
L2)+2eR2(hm
i| i|n−hm
i−1| i−1|n)δzi.
In o de o bound he igh -hand e m, we use he inequali y ||x|n−|y|n|≤n|x−
y|max(|x|,|y|)n−1:
hm
i| i|n−hm
i−1| i−1|n=hm
i(| i|n−| i−1|n)+| i−1|n(hm
i−hm
i−1)
≤nhm
i| i− i−1|max( i−1L∞, iL∞)n−1
+m
hmin | n
i−1|hi−hi−1|max(hiL∞,hi−1L∞)m.
Consequen ly, as hi,hi−1and i,
i−1a e uni o mly bounded in L∞([0,T]×R2):
R2
(hm
i| i|n−hm
i−1| i−1|n)δzi≤C( i− i−1L2+hi−hi−1L2)δziL2
≤C(∇δhi−1L2+∇δzi−1L2+δhi−1L2)δziL2
≤C(∇δhi−12
L2+∇δzi−12
L2)+δhi−12
L2+δzi2
L2).
Thus,
20 Page 20 o 55
d
d δzi2
L2+2K∇δzi2
L2≤Cδzi2
L2+C(∇δhi−12
L2+∇δzi−12
L2+δhi−12
L2).
(30)
Finally, o δci, we use he same me hod as be o e wi h he equa ion (28c) and we
ob ain:
1
2
d
d δci2
L2=−(δ i−1∇ci+1+ i∇δci)δci
+e(hm
i| i|n−hm
i−1| i−1|n)δci−(s+ )ci
hi−ci−1
hi−1δci
=−δ i−1∇ci+1δci−1
2di ( i)(δci)2
+e(hm
i| i|n−hm
i−1| i−1|n)δci−(s+ )ci
hi−ci−1
hi−1δci
≤δ i−1L2∇ci+1L∞δciL2+∇ iL∞δci2
L2
+C(∇δhi−12
L2+∇δzi−12
L2+δhi−12
L2)
+s+ H2
hmin ciL2+ci−1L2δciL2.(31)
Then we add he inequali ies (29), (30) and (31) and in eg a e on [0, ]wi h 0 ≤
≤T:
δhi2
L2+δzi2
L2+δci2
L2+hmin 
0∇δhi2
L2d +2K
0∇δzi2
L2d
≤δh02
L2+δz02
L2+δc02
L2+C
0δhi2
L2+δzi2
L2+δci2
L2d
+C
0δhi−12
L2+δzi−12
L2+δci−12
L2d
+C
0
(∇δhi−12
L2+∇δzi−12
L2)d .
Page 21 o 55 20
We apply he G onwall lemma and ob ain, o all ∈[0,T],
δhi2
L2+δzi2
L2+δci2
L2+hmin 
0∇δhi2
L2d
+2K
0∇δzi2
L2d ≤Cδh02
L2+δz02
L2+δc02
L2
+
0δhi−12
L2+δzi−12
L2+δci−12
L2d
+
0
(∇δhi−12
L2+∇δzi−12
L2)d eC .
Wi h his inequali y, we deduce by induc ion on i ha ∀i∈N,
δhi2
L2
T(L2)+δzi2
L2
T(L2)+δci2
L∞
T(L2)
≤CTe
CTi
i!δh02
H1+δz02
H1+δc02
L2.
Consequen ly, he se ies δhi2
L2
T(L2),δzi2
L2
T(L2)and δci2
L∞
T(L2)con-
e ge; hus, he sequences (hi)i∈N,(zi)i∈N,(ci)i∈Na e Cauchy sequences in he
equi ed spaces.
Bounds on ∂ δhi,∂ δziand ∂ δci: Like he es ima es in P oposi ion 3.10,weuse
he sys em (28a) and P oposi ion 3.9 o ob ain he bounds. 
By P oposi ion 3.11, he e exis s h−h e ,z∈L2
T(L2)such ha hi−h e con e ges
o h−h e and zicon e ges o zin L2
T(L2).As(hi−h e )and (zi)a e uni o mly
bounded in L2
T(Hk+2), we ob ain by in e pola ion ha ∀1/2<θ<1, ∀i>j∈N,
T
0hi−hj2
Hθ(k+2)d ≤T
0hi−hj1−θ
L2hi−hjθ
Hk+2d
≤hi−hj1−θ
CT(L2)T
0hi−hjθ
Hk+2d
≤hi−hj1−θ
CT(L2)T2/(2−θ)hi−hjθ
L2
T(Hk+2)
≤hi−hj1−θ
CT(L2)T2/(2−θ)(2C1)θ−→
n→+∞ 0.
The e o e, ∀k/2<s<k, he sequences (hi−h e )and (zi)a e Cauchy sequences in
L2
T(Hs+2); hus, h−h e ,z∈L2
T(Hs+2). Mo eo e , ∂ hicon e ges o ∂ hin L2
T(L2)
and (∂ hi)is uni o mly bounded in L2
T(Hk), simila ly o ∂ z. Consequen ly, ∀s<k,
∂ h,∂ z∈L2
T(Hs); hus, by P oposi ion 3.5,h−h e ,z∈CT(Hs+1). Finally he a
p io i es ima es on hand zallow o conclude ha h−h e ,z∈L2
T(Hk+2)∩CT(Hk+1).
In pa icula , as k+1=4, (hi−h e ),(∇hi),(∇2hi),(∂ hi)and (zi),(∇zi),(∇2hi),
(∂ zi)con e ges in C([0,T]×R2).

20 Page 22 o 55
Now we conside (ci)i∈N. By P oposi ion 3.11, he e exis s c∈CT(L2)limi o
(ci)i∈Nin his space. We know ha (ci)i∈Nis uni o mly bounded in CT(Hk),soby
in e pola ion: ∀0<θ<1, ∀i>j∈N,
sup
∈[0,T]ci−cjHθ(k)≤sup
∈[0,T]ci−cj1−θ
L2sup
∈[0,T]ci−cjθ
Hk
≤ci−cj1−θ
CT(L2)(ciCT(Hk)+cjCT(Hk))θ
≤Cci−cj1−θ
CT(L2)−→
n,m→+∞ 0.
The e o e, (ci)i∈Nis a Cauchy sequence and hus con e ges o cin he space CT(Hs),
o all s<k. Mo eo e , (∂ ci)con e ges o ∂ cin CT(L2)and is uni o mly bounded
in CT(Hk−1), so i con e ges o cin CT(Hs−1). And we conclude by he a p io i
es ima e on c ha c∈CT(Hk). To conclude, (ci)i∈N,(∂ ci)i∈Nand (∇ci)i∈Ncon e ge
in CT(R2); hus, we can ake he limi in he equa ions, and (h,z,c)is solu ions o
Sys em (9a). This concludes he p oo o he exis ence.
3.6 Uniqueness
P oposi ion 3.12 Le (h1,z1,c1)and (h2,z2,c2)be wo solu ions o Sys em (9a)sa -
is ying he hypo heses o Theo em 1.1. Then ∀ ∈[0,T]:
h1( )−h2( )2
L2+z1( )−z2( )2
L2+c1( )−c2( )2
L2
≤h0
1−h0
22
L2+z0
1−z0
22
L2+c0
1−c0
22
L2eCT
In pa icula , when he ini ial condi ions a e he same o bo h solu ions, hese
solu ions a e he same. Consequen ly, his p oposi ion shows he uniqueness o he
solu ion (h,z,c)o he heo em.
P oo We i s w i e he equa ions e i ied by δh:= h1−h2,δz:= z1−z1and
δc:= c1−c2:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∂ δh−di (h1∇δh+δh∇h2)−di (h1∇δz+δh∇z2)=0 (32a)
∂ δz−Kδz=sδc−ehm
1(| 1|n−| 2|n)+| 2|n(hm
1−hm
2)(32b)
∂ δc+δ ∇c2+ 2δ∇c=ehm−1
1(| 1|n−| 2|n)+| 2|n(hm−1
1−hm−1
2)
−(s+ )c1
h1−c2
h2(32c)
Then he bound is ob ained wi h a simila p ocess as in he p oo o P oposi ion
3.11.
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4 Spec al S abili y o Cons an S a es
In his sec ion, we conside he low o e a opog aphy ha is an inclined plane a ime
=0. We assume ha =0 and we s udy he spec al s abili y o cons an s a es. We
expec ha ins abili y will p o ide a mechanism o pa e n o ma ion. We i s w i e
Sys em (8a) in a non-dimensional o m and hen linea ize his sys em a ound cons an
s a es. Then we explo e nume ically he s abili y o he sys em. Finally, we ca y ou he
spec al s abili y analysis by using Rou h–Hu wi z heo em: This p o ides necessa y
and su icien condi ions o cons an s a es o be spec ally s able. Howe e , hese
condi ions do no p o ide any insigh on he na u e o he ins abili ies. We comple e
his analysis by an asymp o ic expansion o he spec um a ound he o igin and in he
high- equency egime.
4.1 Non-dimensionaliza ion and Linea iza ion o he Sys em
We w i e Sys em (8a) in a non-dimensional o m in o de o iden i y he impo an
pa ame e s. We in oduce se e al cha ac e is ic quan i ies: Zis a cha ac e is ic e oded
heigh , Ha cha ac e is ic wa e heigh , La cha ac e is ic wa eleng h and Ta cha ac-
e is ic ime. We chose T o be he necessa y ime o e ode he soil o a heigh Z, wi h
an e osion speed e. Thus, T e i ies eT =Z.As =μ an θe1−μ∇(z+h),we ix
he cha ac e is ic wa e eloci y V=μ. We in oduce he dimensionless a iables:
h:= h
H,z=z
Z,
:=
V,c:= c
csa
x:= x
L, := e
Z .
In o de o simpli y he no a ions, we will assume H=Z. Then, d opping he p imes,
Sys em (8a) is w i en as:
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
∂ h+ZV an θ
eL ∂xh=Z2V
eL2di (h∇(h+z)),
h∂ c+ZV an θ
eL h∂xc=ZV
eL2h∇(h+z).∇c+ρs
csa
hm| an θe1−Z
L∇(h+z)|n−s
e
ρs
csa
c,
∂ z=ZK
eL2z−hm| an θe1−Z
L∇(h+z)|n+s
ec.
To simpli y he equa ions, we se Z
L=e
V, and we de ine α:= Z
L=e
V,K:= ZK
eL2=
K
LV . The sys em eads:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
∂ h+ an θ∂xh=αdi (h∇(h+z)),
h∂ c+ an θh∂xc=αh∇(h+z).∇c+ρs
csa
hm| an θe1−α∇(h+z)|n−ρss
csa ec,
∂ z=Kz−hm| an θe1−α∇(h+z))|n+s
ec.
(33)
20 Page 24 o 55
The s a iona y s a es o Equa ion (33) o a la su ace, deno ed by (h,c,z), e i y:
∀ ∈R+,∀(x,y)∈, ⎧
⎪
⎨
⎪
⎩
h( ,x,y)=h>0,
c( ,x,y)=c=e
shm annθ>0,
z( ,x,y)=0.
This means ha he e osion and he deposi ion p ocess equilib a e each o he and he
bo om is no e oded, whe eas he luid heigh is a cons an .
Le (h+h,c+c,z), wi h |h|,|c|,|z|1, be a small pe u ba ion o he con-
s an s a e, and solu ion o Sys em (33). Then, a i s o de , his solu ion e i ies he
ollowing linea sys em:
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
∂ h+ an θ∂xh=αh(h+z)
∂ c+ an θ∂xc=ρs
csa
s
emc
h2h−αnc
h an θ∂x(h+z)−c
h
∂ z=Kz−s
emc
hh−αnc
an θ∂x(h+z)−c(34)
We deno e =(h,c,z)T. Then e i ies he equa ion ∂ =A0 +A1∂x +A2
wi h:
A0=⎡
⎢
⎢
⎢
⎣
000
amc
h−a0
−amc
¯ρs
ah
¯ρs
0
⎤
⎥
⎥
⎥
⎦,A1=⎡
⎢
⎢
⎢
⎣
− an θ00
−αanc
an θ− an θ−αanc
an θ
αanh c
¯ρs an θ0αanh c
¯ρs an θ
⎤
⎥
⎥
⎥
⎦,A2=⎡
⎣
αh0αh
000
00K⎤
⎦,
whe e we ha e deno ed a=sρs
ehcsa
and ¯ρs=ρs
csa .
We apply he Fou ie ans o m in space and he equa ion e i ied by ˆ
, he Fou ie
ans o m in space o ,is:
∂ ˆ
=A0+iξA1−(ξ2+η2)A2ˆ
:= A(ξ, η) ˆ
.
Consequen ly, in o de o s udy he s abili y o he sys em, we ha e o de e mine
he sign o he eal pa o he eigen alues o he ma ix A(ξ, η). These eigen alues a e
deno ed by λi(ξ, η) wi h i∈{1,2,3}and he associa ed eigen ec o s a e deno ed by
Vi(ξ, η). The exp essions o λia e no explici : In he nex subsec ion, we compu e
nume ically he s abili y o he sys em. Then Sec . 4.3 gi es a s abili y esul on he
domain. This esul is comple ed wi h he asymp o ic s udy o he eigen alues a low
(|ξ|2+|η|21) and high (|ξ|2+|η|21) equencies.
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Page 25 o 55 20
Fig. 2 S abili y diag ams in he plane (ξ, η) ⊂R2, o K>0. The sys em is uns able in he ed a ea and
s able in he g een a ea (Colo igu e online)
4.2 Nume ical Explo a ion o S abili y
In his sec ion, we explo e nume ically he s abili y o he sys em (34). Fo ha pu pose,
we ha e compu ed nume ically he h ee eigen alues o he ma ix A(ξ, η), and he
sys em is s able i and only i he eal pa s o hese eigen alues a e nega i e. Since
A(ξ, −η) =A(ξ, η) and A(−ξ,η) =A(ξ, η), he eal pa o he spec um emains
unchanged unde he ans o ma ion ξ→−ξand η→−η. Thus, we examine he
beha io o he sys em in he op igh qua e o he plane. The choice o pa ame e s
o hese compu a ions is he same as in Sec .5(unless o he wise speci ied), see Table 1
o hei alues.
In Fig.2, we ha e p esen ed an illus a ion o he s abili y o he sys em when
K>0, whe e
Ke=5×10−4
3600 m2s−1.
The domain ep esen ed is a bounded subse o he plane (ξ, η) ⊂R2, and he colo
ep esen s he s abili y: The sys em is s able in he g een a ea and uns able in he ed
a ea. We clea ly see he s abilizing e ec o he c eep e ec : When Kis highe , he
s able a ea is la ge . I seems ha he uns able a ea is bounded. This will be con i med
by P oposi ion 4.2.
Then, in Fig.3 he e is no c eep e ec : K=0, and we ep esen a ious s abili y
diag ams o se e al alues o he a io n/m.The alueo mis he same as in Table 1:
m=1.6, and n a ies be ween m/2 and 10m. We obse e ha he beha io o
he sys em changes wi h he a io m/n, bu he uns able a ea always seems o be
unbounded.
When nmas in Fig.3a, he s able a ea is bigge han he uns able a ea, and
he sys em is uns able only o pe u ba ions o ans e se dominan di ec ion. When
n≈m,asinFig.3c, he sys em is s able only o longi udinal pe u ba ions. Finally,
when n<m he sys em seems o be uns able a all equencies, as in Fig.3d.
20 Page 32 o 55
Table 1 Pa ame e s o nume ical simula ions
Leng h o he domain Lx40 cm
Wid h o he domain Ly10 cm
Cha ac e is ic wa e speed V=μ an θ1m/s
Ini ial wa e heigh h00.5mm
Ini ial sedimen concen a ion c0317 g/m3
Exponen o ic ion o e hm1,6
Exponen o ic ion o e n3,2
Densi y o he sedimen s ρs2.17 ×106g/m3
Concen a ion o sa u a ion csa 3.17 ×105g/m3
E osion speed e0.5 mm/hou
Sedimen a ion speed se/2000
Angle o he plane θ39◦
5 Di ec Nume ical Simula ions
In his sec ion, we p esen some nume ical expe imen s o he e osion o a il ed plane.
The pa ame e s o hese expe imen s come om physical da a. The quan i ies Lx,Ly,
V,h,ρs,csa ,eand θa e chosen acco ding o he labo a o y expe imen (Gué in e al.
2020), which e odes a block o sal . The choice o alues o he exponen s mand
nhas been in es iga ed many imes in he li e a u e. The alues a e chosen be ween
0 and 3, wi h an addi ional ela ion n=2m, as explained in Chen e al. (2014b)
(he e he cons an nco esponds o he cons an m+nin he li e a u e). We choose
n=2m, wi h nsu icien ly la ge in o de o obse e he o ma ion o channels in he
simula ions. Indeed, we ound ha he e ec o digging in dep essions is ein o ced
when hese exponen s a e la ge . The choice o pa ame e s is gi en in Table 1.
Once he a io be ween he e osion speed and he wa e speed has been ixed,
we assume ha he ime a ia ions o he wa e heigh and concen a ion a e small;
consequen ly, we neglec hese a ia ions in he simula ion. Indeed, in he simula ion,
he e oded heigh o he soil is o he o de o a millime e ; hus, he cha ac e is ic ime
T=Z/eis o he o de o an hou . The wa e c osses he domain in 0.4 seconds; hus,
he e a e ou o de s o magni ude be ween he cha ac e is ic ime o he wa e low and
ha o e osion. A di ec nume ical simula ion o he ull sys em, wi h ime de i a i es,
would impose a se e e CFL es ic ion: Indeed, he luid eloci y is abou 1m.s−1,
whe eas he e osion a e is a ound 1mm.h−1. Since we a e in e es ed in he e osion
p ocess, he na u al imescale is one hou and a ypical ime s ep would be a minu e.
Howe e , he nume ical ime s ep δ is d i en by a CFL: δ ≤δx/ wa e ≈0.001si
one conside a ypical mesh size δx=1mm ( o a channel o leng h 400mm). This
inc eases he nume ical cos o he scheme. Ins ead, we sol e s a iona y p oblems
a each ime s eps, whe e he ime s ep is de e mined by he e osion imescale. A
compa ison be ween he wo esolu ion me hods is made in Appendix D, and i shows
ha aking he s a iona y equa ion does no a ec he esul s.
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Page 33 o 55 20
Fig. 7 Pe u ba ion o he ini ial
su ace, on he domain
[0,Lx]×[0,Ly].The
ep esen ed quan i y is he
di e ence be ween he la
su ace and he pe u bed soil, in
millime e s. This quan i y is
ze o a yellow poin s and
inc eases when colo is da ke
(Colo igu e online)
The s a iona y equa ions o he wa e heigh and concen a ion o sedimen s in
wa e a e disc e ized wi h a ini e olume me hod. The scheme is gi en in Appendix D.
Equa ion (8b)oncis a linea equa ion, and we disc e ize i wi h an explici Eule
scheme by conside ing i as an e olu ion equa ion wi h espec o he a iable x,as
he speed in his di ec ion does no anish:
∂xc+ y
x
∂yc=ρs
h(E−S). (37)
Equa ion (8a)onhis nonlinea ; hus, i is ha de o disc e ize i . In o de o a oid an
implici disc e iza ion o his equa ion, we made he choice o linea ize he equa ion.
Deno ing by hn he solu ion o he equa ion a ime n, we app oxima e:
di (h∇h)( n)+di (h∇z)( n)≈di (hn−1∇hn)+di (hn∇zn).
As he solu ion a he p e ious ime s ep is known, he igh -hand e m is linea in
hn. Thus, we can disc e ize i wi h a ini e olume scheme in wo dimensions. The
jus i ica ion o he quasi-s a iona y model and he nume ical scheme a e gi en in
Appendix D.
We ha e chosen pe iodic bounda y condi ions in he ans e se di ec ion; he e o e,
in he s abili y analysis he e a e only coun able equencies in his di ec ion. The
equencies a e he ηn=2πn/Lywhe e Lyis he wid h o he domain. The bounda y
condi ions a he op o he domain a e Di ichle condi ion o h,cand z. A he bo om
o he domain, z is p esc ibed by a Di ichle condi ion and we suppose ha wa e lows
eely. Thus, we ix Neumann condi ion o hand c.
In he simula ions, he ini ial su ace is a la il ed plane wi h a small andom
pe u ba ion. This su ace is p esen ed in Fig.7, and i shows a la iew o he wo-
dimensional plane. The colo scale shows he heigh di e ence be ween he ac ual soil
and he la plane. Thus, he yellow a eas a e he less dug pa s.
20 Page 34 o 55 Jou nal o Nonlinea Science (2024) 34 :20
Fig. 8 E oded heigh o he soil in mm. The da kes a eas a e he mos e oded a eas
Fig. 9 E oded heigh o he soil in mm wi h espec o he la su ace. The da kes a eas a e he mos e oded
a eas. The wid h o he channels is la ge o la ge alues o K
5.1 Simula ions Wi hou Sou ce Te m o Wa e
In his pa , we p esen some esul s o simula ions when =0, as in he spec al
s abili y analysis. These simula ions a e compa ed o he heo e ical esul s o s abili y.
Fi s , in Fig.8we p esen s esul s o he simula ion o he sys em, when
K=Ke=5×10−4
3600 m2s−1.
In his case, he sys em is spec ally s able a all equencies {(ξ, ηn);ξ∈R,n∈
N}. The pic u es ep esen he e oded heigh z(T)−z0a ime T=0.25 hou and
T=2 hou s. We obse e ha su ace pe u ba ions a e quickly smoo hed, and end o
disappea . A e 15 minu es, we see in Fig. 8a ha he ampli ude o pe u ba ion has
no dec eased ye , bu hese pe u ba ions a e smoo he han ini ially. A e 2 hou s,
we see in Fig.8b ha he ampli ude o he pe u ba ions has dec eased.
In Figs.9and 10, we p esen , espec i ely, he e oded su ace and he luid heigh
when he sys em is uns able a some equencies. In Fig.9, we can obse e he o ma-
ion o channels in he soil, in he low di ec ion. The wid h o hese channels is la ge
when Kis highe , and hey ake mo e ime o appea . This can be explained by he
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Page 35 o 55 20
Fig. 10 Wa e heigh on he domain, in millime e s. The wa e dep h is g ea e in da ke a eas
Fig. 11 Disc e e equency analysis o he esul o he simula ion, when K=Ke/20 (Fig. 9a) and
K=Ke/50 (Fig.9b). The quan i y ep esen ed is he no m o he disc e e Fou ie ans o m o he signal
in he ans e se di ec ion
s abili y analysis, as discussed below. Figu e10 has some simila i ies wi h Fig.9since
he wa e ends o ill he e oded channels; he wa e dep h is la ge in he channels
and smalle be ween hem.
We quan i y he numbe o channels gene a ed by he simula ions in o de o alida e
he s abili y analysis. Fo ha pu pose, we compu e he disc e e Fou ie ans o m o
he e oded su ace a he end o he simula ion. As he main di ec ion o pe u ba ions
is ans e se o he slope, we ha e calcula ed his Fou ie ans o m in his di ec ion,
a ixed x.InFig.11, we p esen he no m o he disc e e Fou ie ans o m (DFT) o
he esul o he simula ions 9a, 9b in he ans e se di ec ion, calcula ed a he bo om
o he plane ( o x=Lx). The equencies a e he 2πN/Ly, o N∈{0,...,Ny/2}.
When K=Ke/20, we can see in Fig. 11 ha he dominan equency is eached when
N=2 a he bo om o he il ed plane. When K=Ke/50, we ha e con ibu ion
be ween N=2 and N=5 equency
Then, in o de o compa e his obse a ion wi h he s abili y analysis, we compu e
nume ically he equencies which c ea e he mos uns able modes. In Fig.12,we
20 Page 36 o 55
Fig. 12 Maximum o he eal pa o he h ee eigen alues, no malized by he highe one, o (ξ, η) ∈
[0,20]×[0,300]. The mos uns able a eas a e in yellow, and in he black pa s, he h ee eigen alues ha e
a nega i e eal pa ; hus, he sys em is s able (Colo igu e online)
p esen nume ical compu a ions o he eigen alues o he linea ized sys em, depending
on he equencies. The la ges eal pa o he h ee eigen alues is ep esen ed, o
each equencies ξand ηin a bounded domain. This quan i y con ol he s abili y, he
sys em is s able i and only i i is nega i e. Due o he pe iodic bounda y condi ions,
he equencies allowed in he ans e se di ec ion a e he ηn=2πn/Ly, whe e
Lyis he wid h o he domain. In Fig.12, he whi e lines shows hese equencies.
Fo bo h cases (Fig.12a, b), he sys em is uns able and he maximum o ins abili y
is eached a ξ=0, η1=2π/Ly. Thus, he sys em des abilizes in he ans e se
di ec ion, and he mos des abilizing equency has pe iod one. This is o he same
o de o magni ude as compu ed by he DFT o he simula ions, whe e his equency
is (ξ, η) =(0,2×2π/Ly). The di e ence be ween he pe iod o 1 p edic ed by he
s abili y analysis and he pe iods o 2−5 ob ained in he simula ions could come om
he nonlinea e ec s o he model ha a e no aken in accoun in he s abili y analysis.
5.2 Simula ions Including a Sou ce Te m o Wa e
In his pa , we p esen some esul s o simula ions when he sou ce e m is a posi i e
unc ion, o obse e he e ec o ain in he model. In he ollowing simula ions,
K=Ke/20.
Fi s , we conside a cons an in ime and uni o m in space sou ce e m 1=
0.005mm/s. Figu e13 shows he su ace heigh (le igu e) and wa e heigh ( igh
igu e) compu ed by he simula ion, a ime 0.25 hou s. We can obse e ha he wa e
heigh is almos ou imes highe a he bo om o he domain (x=Lx), han a he
op (x=0). As he e osion a e is p opo ional o he powe o he wa e heigh h, he
su ace e odes as e a he bo om o he domain.
Then, Fig.14 shows he esul s o he same simula ion, a ime 2 hou s. The ini-
ial pe u ba ions ha e almos disappea ed, and some ans e se pe u ba ions ha e
de eloped a he bo om o he domain.
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Page 37 o 55 20
Fig. 13 Resul s o he simula ion when = 1, a e 2000 i e a ions
Fig. 14 Resul s o he simula ion when = 1, a e 16000 i e a ions
Fig. 15 Resul s o he simula ion when = 2, a e 14000 i e a ions
Nex , we choose a bigge sou ce e m: 2=0.01m/s, and esul s o his simula ion
a e shown in Fig. 15. We can see ha he e a e bigge pe u ba ions a he bo om
o he domain han wi h he sou ce e m 1. These pe u ba ions a e also no o ally
ans e se: hey undula e a li le bi in he longi udinal di ec ion, which is a di e en
beha io om he case wi hou sou ce e m (Fig. 9b).
Now we ake a cons an non-uni o m sou ce e m:
3(x,y)=0.005 exp −60 ∗(y−Ly/2)2
Ly2.

20 Page 38 o 55
Fig. 16 Resul s o he simula ion when = 3, a e 6000 i e a ions
ha co espond o a posi i e longi udinal band o ain in he middle o he domain
which dec ease exponen ially as when app oaching y=0o y=0.1. A ime
0.75 hou s, he su ace is highly e oded a he bo om o he band (Fig. 16). He e,
he e olu ion o he sys em is d i en by he sou ce e m, and he e osion landscape
depends mainly on his e m.
These simula ion show ha he sou ce e m may quan i a i ely in luence he esul s,
al hough quali a i ely hey look simila o he case wi h no sou ce e m.
6 Conclusion and Pe spec i es
In his pape , we ha e conside ed a model o he e olu ion o landscape subjec
o wa e e osion, in o de o s udy he o ma ion o pa e ns. This model akes in o
accoun he wa e low, he dissol ed sedimen s and he main e ec s o e osion and
sedimen a ion, while emaining simple enough o be s udied bo h heo e ically and
nume ically. We p o ed ha unde ealis ic hypo heses, he sys em is well posed
locally in ime. Then a comple e spec al analysis o a linea iza ion o he sys em
a ound s a iona y solu ions has been pe o med. This analysis highligh s he a ious
beha io s o he sys em depending on he pa ame e s and on he equencies o he
pe u ba ions. A e y impo an pa ame e o he sys em is he cons an o c eep K,
which con ols he s abili y. The sys em can become s able i his cons an is la ge
enough and is uns able i his cons an is oo small. Mo eo e , he ins abili ies in he
sys em appea in he ans e se di ec ion, and his leads o he o ma ion o channels
pa allel o he wa e low.
This analysis is a p elimina y s ep in he s udy o pa e n o ma ion on e odible
su aces. We ha e shown ha an ins abili y mechanism can explain he o ma ion o
pa allel channels a he ea ly s ages o e osion. We plan o pe o m a mo e comple e
pa ame ic s udy o he sys em, in o de o unde s and he a ious beha io s o he
model. This analysis should be comple ed by nume ical simula ion o illus a e hese
beha io s. The e a e se e al in e es ing di ec ion o esea ch. Fi s , he analysis o
his model could be ex ended o mo e gene al landscapes like moun ains o alley. In
hese cases, he well-posedness o he sys em in sho ime can be s ill alid, as long as
he wa e le el and he wa e speed does no anish. A simila s abili y analysis could
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be ca ied ou i s eady s a es exis . One could also conside mo e in ol ed models.
Indeed a lo o ac o s a e no aken in o accoun as wea he , ege a ion, animal and
human ac i i ies. I is be qui e di icul o include hese ac o s in he model, bu a
possible amelio a ion would be o include andomness in he equa ions. A andom
e m could allow o model hese ac o s which luc ua e o e ime. Mo eo e , a non-
cons an sou ce e m ha would model he a ia ions o ain in ime would make
he model mo e ealis ic. Ano he po en ial imp o emen o he model would be o
conside mo e complex laws o he luid eloci y o e en conside shallow wa e - ype
models o he e olu ion o he laye o luids.
Acknowledgemen s PD holds a isi ing p o esso associa ion wi h he Depa men o Ma hema ics, Impe-
ial College London.
A ailabili y o Da a and Ma e ials The code o nume ical simula ions gene a ed du ing he cu en s udy is
a ailable a h ps://gi hub.com/juliebina d/landscape_e ol_ ini e_ olum.
Decla a ions
Con lic o in e es No unding was ecei ed o assis wi h he p epa a ion o his manusc ip .
A P oo o he S abili y Theo em
This sec ion is de o ed o he p oo o Theo em 4.1. Fi s we s a e he Rou h–Hu wi z
c i e ia o complex polynomial o deg ee 3, p o ed in Mo is (1962):
P oposi ion A.1 Le P(X)=X3+(a1+ib1)X2+(a2+ib2)X+a3+ib3be a
polynomial wi h complex coe icien s. Then he oo s o P ha e a posi i e imagina y
pa i and only i he h ee ollowing condi ions a e sa is ied:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
−2=−1a1
0b1>0
4=
1a1a2a3
0b1b2b3
01 a1a2
00 b1b2

>0,−6=−

1a1a2a300
0b1b2b300
01 a1a2a30
00 b1b2b30
00 1 a1a2a3
00 0 b1b2b3

>0
This p oposi ion is di ec ly used o p o e Theo em 4.1.
20 Page 40 o 55
P oo o Theo em 4.1 The cha ac e is ic polynomial o A, deno ed by P,is:
P(X)=
X+iξ an θ+α2h0α2h
−amc
h+αiξanc
an θX+a+iξ an θαiξanc
an θ
amc
¯ρs−αiξanhc
¯ρs an θ−ah
¯ρs
X−αiξanhc
¯ρs an θ+2K

=
X+iξ an θ+α2h0α2h
−amc
h+αiξanc
an θX+a+iξ an θαiξanc
an θ
0hX+iξh an θ
¯ρs
X+2K

.
whe e we ha e deno ed 2=ξ2+η2. In o de o simpli y he compu a ions, we
in oduce he a iables
Y:= X+iξ an θ, ¯
ξ=ξ an θ, ¯2=αh2,¯
Kαh=K,¯ρs¯
h=h.
We also de ine he cons an s N=αanc/ an2θ,M=amc/h. Consequen ly, he
polynomial w i es as:
P(Y)=
Y+¯20¯2
−M+iξNY+ai
¯
ξN
0¯
hY Y +¯2K−i¯
ξ
=−¯
hY i¯
ξNY +¯2M+(Y+a)(Y+¯2)(Y−i¯
ξ+¯2¯
K)
=Y3+"a+¯2(1+¯
K)−i¯
ξ(1+N¯
h)#Y2
+"¯2(a(1+¯
K)−¯
hM +¯
K¯2)−i¯
ξ(a+¯2)#Y
+a¯2(¯2¯
K−i¯
ξ). (38)
The sys em is s able i and only i he oo s o X→ P(X)ha e a nega i e eal pa . As
Yha e he same eal pa as X, his is equi alen o he ac ha he oo s o Y→ P(Y)
ha e a nega i e eal pa . Deno ing iλ:= Y, he sys em is s able i and only i he oo s
o λ→ P(iλ) ha e a posi i e imagina y pa . Thus, we will apply he Rou h–Hu wi z
c i e ia o he polynomial λ→ Q(λ) := iP(iλ) whe e iλ:= Y:
Q(λ) =λ3−"¯
ξ(1+N¯
h)+ia+¯2(1+¯
K)#λ2−"¯2(a(1+¯
K)−¯
hM +¯
K¯2)
−i¯
ξ(a+¯2)#λ
+a¯2(¯
ξ+i¯2¯
K).
Page 41 o 55 20
We deno e:
¯a1=1+N¯
h,
¯
b1=a+¯2(1+¯
K), ¯a2=a(1+¯
K)−¯
hM +¯2¯
K,
¯
b2=a+¯2.
Thus, Q(λ) =λ3−¯
ξ¯a1+i¯
b1)λ2+−¯2¯a2+i¯
ξ¯
b2λ+a¯2(¯
ξ+i¯2¯
K).
The i s condi ion: The de e minan −2w i es as:
−2=−1−¯
ξ¯a1
0−¯
b1=¯
b1=a+¯2(1+¯
K).
Consequen ly, as a>0 and K≥0, he condi ion −2>0 is always sa is ied.
The second condi ion: The de e minan 4is gi en by:
4=
−¯
b1¯
ξ¯
b2¯4a¯
K
1−¯
ξ¯a1−¯2¯a2
0−¯
b1¯
ξ¯
b2=
−¯
b1¯
ξ(¯
b2−¯a1¯
b1)¯2(¯2a¯
K−¯
b1¯a2)
10 0
0−¯
b1¯
ξ¯
b2
=¯2¯
ξ2
¯2¯
b2¯a1¯
b1−¯
b2+¯
b1¯a2¯
b1−¯2a¯
K.
Le =¯
ξ2
¯2∈[0,μ an2θ
eh ], hen one inds ha 4>0∀(ξ, η) ∈(R2)∗i and only
i
¯
b2¯a1¯
b1−¯
b2+¯
b1¯a2¯
b1−¯2a¯
K>0∀ ∈[0,μ an2θ
eh ],∀¯2>0.
The e ms ¯
b1and ¯
b2a e posi i e, and:
¯a1¯
b1−¯
b2=a+¯2(1+¯
K)+N¯
h(a+¯21+¯
K)−a−¯2
=¯2¯
K+N¯
ha+¯2(1+¯
K)>0.
Consequen ly, one has 4>0,∀(ξ, η) ∈(R2)∗i and only i
¯a2¯
b1−¯2a¯
K≥0∀¯2>0.
We compu e:
¯a2¯
b1−¯2a¯
K=a(1+¯
K)−¯
hM +¯2¯
Ka+¯2(1+¯
K)−¯2a¯
K
=aa(1+¯
K)−¯
hM+¯2(1+¯
K)a(1+¯
K)−¯
hM +¯2¯
K.
20 Page 48 o 55
We can suppose, up o eno maliza ion ha V(ξ, η) =O
+∞(1). We calcula e a Taylo
expansion o
λ(ξ, η)
ξ2+η2=0+1(ξ, η) +2(ξ, η) +o(1),
and o
V(ξ, η) =V0+V1(ξ, η) +V2(ξ, η) +o(1
ξ2+η2).
O de 0: When ξ2+η2→+∞, Sys em (43) becomes A2V0=−0V0. I s solu ions
a e he eigen alues and eigen ec o s o −A2,so1
0=−K,2
0=−αhand 3
0=0.
The associa ed eigen ec o s a e
X1=⎡
⎣
αh
0
K−αh⎤
⎦,X2=⎡
⎣
1
0
0⎤
⎦,X3=⎡
⎣
0
1
0⎤
⎦.
When K>0, he e is one eigen alue bi u ca ing om ze o, 3, and wo o he
eigen alues ha e nega i e eal pa s. When K=0, he wo eigen alues 1and 3
a e bi u ca ing om ze o. Consequen ly, we will s udy hese wo cases.
C.1TheCaseK>0
In his pa , we con inue he asymp o ic expansion o he eigen alue 3. The ze o-o de
e ms o he eigen ec o s a e V1
0=X1,V2
0=X2and V3
0=X3.
We calcula e he i s -o de e m o 3. The i s -o de e ms o Sys em (43) when
ξ2+η2→+∞depends on he asymp o ic beha io o iξ.I ξ→+∞, hen he
i s -o de e ms o he sys em a e:
A2V3
1−iξA1
ξ2+η2V3
0=−3
1V3
0.(44)
And i ξis bounded, iξA1and A0a e o he same o de ; hus, he i s -o de e ms
a e:
A2V3
1−iξA1+A0
ξ2+η2V3
0=−3
1V3
0.(45)

Jou nal o Nonlinea Science (2024) 34 :20 Page 49 o 55 20
To compu e 3
1, we mul iply he equa ions by he le eigen ec o o A2associa ed
wi h he eigen alue 0, X∗=X3. Then, a e some compu a ions:
I ξ→+∞,
3
1=−iξ an θ
ξ2+η2,V3
1=1
ξ⎡
⎣
0
1
0⎤
⎦.
I ξbounded,
3
1=−iξ an θ
ξ2+η2−sρs
ehcsa
1
ξ2+η2.
When ξis bounded, we al eady compu ed he second-o de e m o 3. When
ξ→+∞, he second-o de e ms o he sys em a e:
A2V3
2−iξA1
ξ2+η2V3
1−A0
ξ2+η2V3
0=−3
1V3
1−3
2V3
0.
Then, one inds:
3
2=− sρs
ehcsa
1
ξ2+η2.
Finally, he asymp o ic expansion o he eigen alues o A(ξ, η) when ξ2+η2→
+∞and K>0is:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
λ1(ξ, η) =−K(ξ2+η2)+o(ξ2+η2)
λ2(ξ, η) =−αh(ξ2+η2)+o(ξ2+η2)
λ3(ξ, η) =−iξ an θ−s¯ρs
eh +o(1)
C.2TheCaseK=0
When K=0, he ze o-o de e ms o he eigen alues 1and 3 anish. Thus, in
o de o p o e he second poin o P oposi ion 4.2, we need o compu e he nex o de
e ms o hese wo eigen alues.
The eigen ec o s associa ed wi h he wo ze o eigen alues o −A2a e
X1=⎡
⎣
αh
0
−αh⎤
⎦,X3=⎡
⎣
0
1
0⎤
⎦.
Thus, he e exis cons an s u1,
1,u3,
3such ha V1
0=u1X1+ 1X3and V3
0=
u3X1+ 3X3. The le eigen ec o s associa ed wi h he ze os eigen alues o −Ca e:
X1
∗=001
,X3
∗=010
.
20 Page 50 o 55
I ξ→+∞, hen he i s -o de e ms o Sys em (43)a e:
A2V3
1−iξA1
ξ2+η2V3
0=−3
1V3
0.(46)
I ξ= 0 is ixed, he i s -o de e ms a e:
A2V3
1−iξA1+A0
ξ2+η2V3
0=−3
1V3
0.(47)
To compu e 1
1and 3
1, we mul iply he equa ions (46) and (47)byX1
∗and X3
∗.
Then, when ξ→+∞, one has:
⎧
⎨
⎩
1
1=0,V1
0=X1,
3
1=−iξ an θ
ξ2+η2,V3
0=X3.
And when ξ= 0 is ixed,
⎧
⎪
⎨
⎪
⎩
1
1=scm
eh ,
3
1=−iξ an θ+s¯ρs
eh 1
ξ2+η2.
When ξ→+∞, he second-o de e ms o Sys em (43)a e:
A2V2−iξA1
ξ2+η2V1−A0
ξ2+η2V0=−1V1−2V0.
Then, one ob ains:
1
2=sc
eh m−ξ2
2n,
3
2=−s¯ρs
eh
1
ξ2+η2.
As a conclusion, he asymp o ic expansion o he eigen alues o A(ξ, η), when
ξ2+η2→+∞and ξ= 0, K=0is:
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
λ1(ξ, η) =sc
eh m−ξ2
2n+o(1),
λ2(ξ, η) =−αh(ξ2+η2)+o(ξ2+η2),
λ3(ξ, η) =−iξ an θ−s¯ρs
eh +o(1).
Finally, when ξ=0 he cha ac e is ic polynomial Pcan be ac o ized:
P(X)=XX2+(a+α2h)X+α2ah 1−mc
¯ρs.
Page 51 o 55 20
Thus, i has a ze o oo , λ1=0, and he wo o he oo s a e
λ2,3=−α2h−a±(a−α2h)2+4α2amhc
¯ρs
2.
Consequen ly, he asymp o ic de elopmen o he eigen alues o he sys em when
η2→+∞and ξ=0, K=0is
⎧
⎪
⎪
⎨
⎪
⎪
⎩
λ1(0,η)=0,
λ2(0,η)=−αhη2+o(η2),
λ3(0,η)=s¯ρs
eh mc
¯ρs−1+o(1).
This achie es he p oo o P oposi ion 4.2.
D Nume ical Scheme
In his appendix, we gi e he nume ical scheme used o he simula ions o he sys em.
S a iona y egime o he wa e
The a io be ween he e osion speed and he luid eloci y is small and we will
suppose ha he ime de i a i es o cand ha e small. In o de o jus i y his assump ion,
we escale he equa ions by in oducing he cha ac e is ic a iables Z he e oded
heigh , L he cha ac e is ic leng h and T=Z/e he cha ac e is ic ime. We de ine he
dimensionless a iables:
h:= h
H,z=z
Z,
:=
V,c:= c
csa
,
x:= x
L, := e
Z .
The e o e, d opping he p imes, Sys em (8a) is w i en as:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
∂ h+1
di (h ) =
e,
∂ (ch)+1
di (ch ) =ρs
csa hm| |n−s
ec,
∂ z=$
Kz−hm| |n+s
ec,
=−μ
V∇(h+z).
He e, we de ined := Le
HV .I 1, as gene ally ρscsa , we can neglec he
e ms ∂ hand ∂ (ch)in he equa ion he ch. Thus, assuming enough egula i y on he
solu ions, we can w i e di (ch ) =h .∇c+cdi (h ) =h .∇c. Mo eo e , we ix
20 Page 52 o 55
V=μ o simplici y. We ob ain:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
di (h ) =$ ,
h .∇c=ρs
csa hm| |n−s
ec,
∂ z=$
Kz−hm| |n+s
ec,
=−∇(h+z).
(48)
Disc e iza ion o he equa ions
The cons ain on ha each ime di (h∇(h+z)) =0 leads o a ully nonlinea
p oblem ha may be ha d o sol e nume ically. Ins ead, we disc e ize his equa ion as
di (hn−1∇hn)+di (hn∇zn)=− ,(49)
whe e hn ep esen s he luid heigh a ime n=nδ , whe eas he su ace heigh a
ime n,zn, has been calcula ed by an explici Eule me hod, using he solu ions a ime
n−1, hn−1and cn−1:∀1≤i≤Nx,1≤j≤Ny,
zn
i,j−zn−1
i,j
d =K
zn−1
i+1,j−2zn−1
i−1,j+zn−1
i,j
dx +K
zn−1
i,j+1−2zn−1
i,j−1+zn−1
i,j
dy −(E−S)n
i,j.
We disc e ize he equa ion on hnby a cen e ed ini e olume scheme, which w i es:
− i,j=hn−1
i+1/2,j(hn
i+1,j−hn
i,j)−hn−1
i−1/2,j(hn
i,j−hn
i−1,j)
dx2
+hn−1
i,j+1/2(hn
i,j+1−hn
i,j)−hn−1
i,j−1/2(hn
i,j−hn
i,j−1)
dy2
+hn
i+1/2,j(zn
i+1,j−zn
i,j)−hn
i−1/2,j(zn
i,j−zn
i−1,j)
dx2
+hn
i,j+1/2(zn
i,j+1−zn
i,j)−hn
i,j−1/2(zn
i,j−zn
i,j−1)
dy2,
whe e hn−1
i+1/2,j=hn−1
i,j+hn−1
i+1,j
2. Finally, Equa ion (37)oncis disc e ized by an upwind
ini e olume scheme, conside ing he a iable xas a ime a iable. The ime dis-
c e iza ion is an explici Eule scheme.
ci+1,j−ci,j
d +max( x[i,j],0)
y[i,j]
ci,j−ci,j−1
dy
+min( x[i,j],0)
y[i,j]
ci,j−ci,j+1
dy =− ρs
hi,j x[i,j](E−S)n
i,j.
Page 53 o 55 20
Fig. 17 E oded heigh o he soil in mm, o K=Ke,a T=2s
Bounda y condi ions
We choose pe iodic bounda y condi ions in he ydi ec ion. The e emains wo
bounda ies: he op and he bo om o he il ed plane. The bounda y condi ions o
he soil heigh a e Neumann condi ions: ∀1≤j≤Ny,
zn
0,j=zn
1,j,zn
Nx,j=zn
Nx+1,j.
The bounda y condi ions o he wa e heigh and o he concen a ion o sedimen s
a e a Di ichle condi ion a he op because he incoming low is ixed and a ee low
Neumann condi ion a he bo om.
hn
0,j=h0,hn
Nx,j=hn
Nx+1,j,
cn
0,j=c0,cn
Nx,j=cn
Nx+1,j.
Compa ison wi h a scheme ha sol es he non-s a iona y sys em
In his pa ag aph, we d op he s a iona y assump ion o he wa e heigh and sedi-
men concen a ion, and compa e he esul s o hose whe e he s a iona y assump ion
is made. We sol e Sys em (8a), wi h a ini e olume scheme. A ully explici scheme
ails o sol e he sys em, because he compu ed solu ion quickly blows up, e en wi h
a ime s ep smalle han he one gi en by he CFL condi ion:
δ ≤δx
wa e
.
The e o e, we use a semi-explici scheme: Linea e ms o he wa e heigh and sed-
imen concen a ion equa ions a e implici , and we explici a pa o he nonlinea
e ms. The disc e iza ion in ime o equa ions on hand cis gi en by:
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
hk+1−hk
δ +μdi (hh∇hk+1+hk+1∇zk+1)=0,
hk+1ck+1−ck
δ +hk+1 k+1.∇ck+1=ehk+1m k+1n−ρssck+1
csa
,
k=−μ∇(hk+zk).
(50)

20 Page 54 o 55
Disc e iza ion in space is done by a ini e olume scheme, he same as o he s a iona y
Sys em (48). The nume ical pa ame e s a e gi en by Table 1, and δ =0.1s.
Figu e 17 shows he esul s o wo simula ions wi h he same pa ame e s, he non-
s a iona y Sys em (8a)inFig.17a and he s a iona y Sys em (48)inFig.17b. A T=
0.25 hou s, we can obse e ha hese wo simula ions a e almos he same. This esul
suppo s he ac ha he app oxima ion o he comple e Sys em (8a) by Sys em (48)
whe e wa e is in a s a iona y egime is alid.
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And ew, A.L., Chu, K.-W.E., Lancas e , P.: De i a i es o eigen alues and eigen ec o s o ma ix unc ions.
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P oc. Na l. Acad. Sci. 117(3), 1375–1382 (2020)
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