SeMA Jou nal
h ps://doi.o g/10.1007/s40324-025-00382-y
Lag angian desc ip o s in geophysical lows: a su ey
Jezabel Cu belo1,2,3
Recei ed: 8 Sep embe 2024 / Accep ed: 18 Feb ua y 2025
© The Au ho (s) 2025
Abs ac
This su ey ocuses on he applica ion o Lag angian desc ip o s o e eal he geome y o
phase space s uc u es ha de e mine anspo in dynamical sys ems. We p esen di e se
o mula ions o he me hod and examine a ious applica ions o Lag angian desc ip o s in
geophysical luids,suchasa mosphe ic lowsandoceaniccu en s.Theme hodo Lag angian
Desc ip o s has p o en o be a powe ul ool o cha ac e izing anspo and mixing in hese
con ex s, demons a ing how hese ools ha e enhanced ou unde s anding o complex luid
dynamics in c i ical en i onmen s.
Keywo ds Lag angian desc ip o s ·Lag angian cohe en s uc u es ·Phase space ·
Geophysical lows
Ma hema ics Subjec Classi ica ion 37-02 ·86-02 ·37N10 ·37C10 ·37C60 ·76F20 ·
76F25 ·76U60 ·86A05 ·86A10
1 In oduc ion
In dynamical sys ems heo y, unde s anding he global and long- e m beha io o complex
sys ems o en hinges on iden i ying he s uc u es ha o ganize hei dynamics [1]. These
dynamical s uc u es, such as pe iodic o bi s, in a ian mani olds, homoclinic o bi s, and
in a ian o i, se e as he backbone o cons uc ing a comp ehensi e pic u e o he sys em’s
e olu ion. By pe o ming local analyses, such as linea iza ion and he compu a ion o no mal
o ms a ound hese s uc u es, esea che s can begin o piece oge he how hey in luence he
b oade dynamics.
In luid dynamics, a ull unde s anding o any ansi ional luid low equi es he iden i ica-
ion and analysis o unde lying Cohe en S uc u es ha go e n he sys em’s beha io [2–5].
Acco ding o [6], cohe en s uc u es a e la ge-scale, connec ed egions o luid cha ac e -
ized by phase-co ela ed o ici y o o he mac oscopic quan i ies. These s uc u es appea
in a wide a ay o luid low p oblems, and hei iden i ica ion and accu a e compu a ion a e
BJezabel Cu belo
[email p o ec ed]
1Depa amen de Ma emà iques, Uni e si a Poli ècnica de Ca alunya, A da. Diagonal, 647, 08028
Ba celona, Ca alunya, Spain
2IMTech, Ins i u e o Ma hema ics o UPC-Ba celonaTech, 08028 Ba celona, Ca alunya, Spain
3Cen e de Rece ca Ma emà ica, Bella e a, Spain
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J. Cu belo
c ucial because hey play a cen al ole in he global anspo o mass, hea , and momen-
um. Despi e he inhe en complexi y o u bulen luid lows, which o en sel -o ganize in o
cohe en pa e ns o a ying spa io- empo al complexi y, he e is a p essing ques ion: Is i
possible o accu a ely cap u e he complica ed beha io o luid low using a ini e numbe
o cohe en s uc u es wi h known in e ac ion ules?
In he s udy o luid dynamics, wo undamen al app oaches a e used o desc ibe he
mo ion o luid: he Eule ian and Lag angian desc ip ions. The Eule ian pe spec i e ocuses
on speci ic loca ions wi hin he luid domain, obse ing how he luid p ope ies change o e
ime a hese ixed poin s. This app oach is o en used in adi ional luid mechanics, whe e
he eloci y ield and o he quan i ies a e measu ed as unc ions o posi ion and ime.
In con as , he Lag angian desc ip ion ollows indi idual luid pa icles as hey mo e
h ough space and ime, acking hei ajec o ies and how hei p ope ies e ol e. This
pe spec i e is pa icula ly use ul o unde s anding he anspo and mixing o subs ances
wi hin he luid, as i p o ides a di ec iew o how pa icles and ace s a e ad ec ed by
he low. In la ge-scale geophysical lows, whe e ad ec ion- he anspo o ace s by he
luid-is o en he dominan p ocess, he Lag angian app oach becomes essen ial o cap u ing
he beha io o hese sys ems. Geome ic s uc u es in phase space, as pionee ed by Poinca è
[7], ha e been c ucial in cha ac e izing he global beha io o dynamical sys ems. This
pe spec i e o e s insigh s in o he e olu ion o di e en ajec o y classes and has also
been highly in o ma i e o unde s anding luid anspo and mixing, especially gi en he
o malequi alencebe ween heequa ions o incomp essible luidmo ion(wi hou molecula
di usion) and Hamil on’s equa ions, whe e he s eam unc ion ac s as he Hamil onian, and
he physical space co esponds o he phase space [8].
When ad ec ion is he p ima y d i e , as is o en he case in geophysical lows, Lag angian
Cohe en S uc u es (LCSs) (concep in oduced by [9]) become he c i ical ini e- ime s uc-
u es ha dic a e hede o ma iono he luid and,consequen ly, hee olu iono anyad ec i e
ace ields. These s uc u es include a ac ing/ epelling hype bolic LCSs, which a e ma e-
ial lines ha e ol e wi h he low, ac ing as egions ha maximally a ac / epulsion luid.
A ound hese LCSs, he luid is s e ched in one di ec ion and comp essed in he pe pen-
dicula di ec ion, leading o he o ma ion o ilamen ous ace pa e ns. The e o e, LCSs
se e as key ma e ial su aces ha shape global anspo , ac ing as ba ie s ha in luence
he mo emen and mixing o subs ances in he luid.
The iden i ica ion and s udy o cohe en s uc u es, pa icula ly LCSs, ha e become majo
challenges in geophysical luid dynamics and a e inc easingly ele an in applied ma h-
ema ics, physics, and enginee ing. This is especially ue gi en hei signi ican ole in
unde s anding and p edic ing na u al phenomena [9–12] such as clima e p ocesses-like mon-
soons [13], a mosphe ic i e s [14], opical cyclones[15], sudden s a osphe ic wa mings
[16–19], and ozone deple ion [19–21]—and hei impo ance in analyzing en i onmen al
disas e s, including oil spills [22–27], anspo pollu ion [27–29], olcanic e up ions [30,
31] os dus plumes [32]. This challenge has led o he de elopmen o powe ul ma hema ical
ools o iden i ying and analyzing hese s uc u es, which o e a mo e insigh ul iew in o
he unde lying dynamics o luid lows. Se e al ecen e iew a icles [33–37] o e comp e-
hensi e desc ip ions and e e ences o he mul i ude o me hods p oposed o LCS de ec ion.
Apa om Lag angian Desc ip o s, o he me hods a e Fini e ime Lyapuno exponen s [38,
39], ini e size Lyapuno exponen s [40,41], clus e ing algo i hms [42,43]; spec al colou -
ing echniques [44], me hods based on he idea o complexi y o isola ed ajec o ies[45], line
in eg al con olu ion [46,47], Dis inguished hype bolic ajec o ies [48], encoun e olumes
[49], a ia ional LCSs me hods [50], Lag angian a e aged o ici y de ia ion (LAVD) [51],
s ochas ic sensi i i y [52], cohe en se de ec ion [53–55].
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Lag angian desc ip o s in geophysical lows: a su ey
This e iew a icle p o ides a comp ehensi e o e iew o he Lag angian Desc ip o
me hod o s udying and de ec ing Lag angian Cohe en S uc u es (LCS). I does no engage
in compa a i e analyses o a ious de ec ion and analysis me hods, as hese ha e been ho -
oughly discussed in he exis ing li e a u e (e.g., [37]). Ins ead, he ocus is on su eying
he li e a u e on Lag angian Desc ip o s, highligh ing he di e se o mula ions o he me h-
ods (de ailed in Sec .2) and emphasizing hei ex ensi e applica ions in geophysical lows
Sec .3. Special a en ion is gi en o hei use in a mosphe ic (Sec .3.1) and oceanic s udies
(Sec .3.2), demons a ing how hese ools ha e enhanced ou unde s anding o complex luid
dynamics in hese c i ical en i onmen s.
2 Me hod
The Lag angian desc ip o s (LD) is a dynamical sys em ool which is able o highligh
geome ical objec s in phase po ai s o dynamical sys ems wi h a gene al ime dependence.
The concep o Lag angian desc ip o s was i s in oduced by [56], whe e i was applied
o oceanic lows wi h gene al ime dependence, using al ime ic da a se s. Howe e , in he
wo k by [57] he gene al me hodology o cons uc ing Lag angian desc ip o s was ully
p esen ed. In ha s udy, a “heu is ic a gumen ” was discussed, explaining he e ec i eness o
his me hod in e ealing geome ical s uc u es wi hin he phase space o a dynamical sys em.
I was demons a ed ha Lag angian desc ip o s could p o ide a comp ehensi e dynamical
pic u e o geome ic s uc u es o lows wi h a bi a y ime dependence. No ably, his ool
is capable o iden i ying he p ima y “o ganizing cen e s” in he low, such as hype bolic
ajec o ies along wi h hei s able and uns able mani olds, as well as ellip ic egions.
In his sec ion, we will e iew he p ecise meaning o he e m “Lag angian desc ip o ,”
explo e he heu is ic jus i ica ion o hei e ec i eness, and discuss he p ocess o compu ing
hem along wi h he a ious heo ies de eloped a ound hem.
Neglec ing di usion, he mo ion o passi e luid pa icles is go e ned by he o dina y
di e en ial equa ion
˙
x= (x, ). (1)
In his equa ion, (x, ) ep esen s a su icien ly smoo h, ime-dependen eloci y ield,
whe e x∈Xis he s a e, ∈Ris ime, and X⊂Rn(n=2,3,...) is a compac subse
ep esen ing hephysicaldomain.Lag angian pa icle ajec o iesa e hesolu ionsx( )o his
di e en ial equa ion, wi h cohe ence desc ibed by he beha io o g oups o hese Lag angian
ajec o ies.
The Lag angian desc ip o o unc ion Mis de ined as he Euclidean a c leng h o a cu e
in phase space, ep esen ing he pa h o a ajec o y go e ned by Eq. (1):
M(x0, 0,τ)= 0+τ
0−τ
(x( ), )d ,(2)
whe e ·is he Euclidean no m, and (x1( ), x2( ),...,xn( )) a e he componen s o he
ajec o y x( )in Rn. The ajec o y s a s om he poin x0a ime = 0and is obse ed
o e he ime in e al [ 0−τ, 0+τ]. The e o e, i is clea ha M=M(x0, 0,τ), i.e.,
depends on he s a ing poin (x0, 0)and he ime in e al gi en by τ.
Fo example, in Fig.1we analyze he linea saddle which eloci y ield gi en by:
˙x=λx,
˙y=−λy,λ>0,(3)
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J. Cu belo
Fig. 1 (Le panels) Mmap o he Linea saddle poin (Eq. (3)) wi h λ=1andτ=1andτ=10. The da k
colo ep esen s sho ajec o ies, while he ligh colo ep esen s long ajec o ies. I τis su icien ly la ge,
he no m o he g adien ∇ Mhighligh s he s able and uns able mani olds associa ed wi h he hype bolic
poin ( igh panels)
The low gene a ed by his eloci y ield is:
x( ,x0)=x0eλ ,(4)
y( ,y0)=y0e−λ ,λ>0.(5)
whe e he o igin, (x,y)=(0,0), is a hype bolic ixed poin wi h s able and uns able
mani olds gi en by:
Ws(0,0)={(x,y)∈R2|x=0,y= 0},(6)
Wu(0,0)={(x,y)∈R2|y=0,x= 0}.(7)
Figu e1a shows ha o small τ,Mis smoo h, and o inc easing τ(panel b), he plo
displays he mani olds by means o la ge alues o he ∇ M( igh panel).
A his poin , i is help ul o explain why Lag angian desc ip o s a e e ec i e o e ealing
geome ic s uc u es in he phase space o (1). O e he ime in e al o in e es [ 0−τ, 0+τ],
ajec o ies wi h ini ial condi ions ha emain close h oughou his pe iod a e expec ed o
ha e simila M alues. Howe e , a he bounda ies be ween egions whe e ajec o ies beha e
di e en ly, he a c-leng hs o ajec o ies s a ing on ei he side o he bounda y a e expec ed
o di e . These bounda ies, o singula ea u es, a e indica ed by ab up changes, meaning
he de i a i e o he Lag angian desc ip o s ac oss hese bounda ies is discon inuous. In a
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Lag angian desc ip o s in geophysical lows: a su ey
Fig. 2 Maps o M(a)and∇ M(b) a 600 K on Janua y 29, 2008, using τ=20 days. The yellow a eas in
panel (a) co espond o ini ial condi ions wi h long ajec o ies du ing he co esponding ime in e al, while
da k blue a eas co espond o sho ajec o ies. In panel (b), he black lines indica e la ge alues o ∇ M,
highligh ing he singula ea u es o he unc ion M, which app oxima es he loca ions o he mani olds
hype bolic egion o phase space, hese bounda ies co espond o he s able and uns able
mani olds o hype bolic ajec o ies (see Fig.1). These ea u es a e easy o isualize when
plo ing he esul s o he me hod and a e easy o highligh by conside ing he g adien o
he M unc ion [16,58]. See an example in Fig.2,whichshowsamapo Mand he no m
o i s g adien using τ=20 in an a mosphe ic da ase . Fu he mo e, he e is a igo ous
ma hema ical connec ion be ween hese “singula ea u es,” which a ise om he discon i-
nui y o he Lag angian desc ip o s o i s de i a i e, and he s able and uns able mani olds
o hype bolic poin s [57]. This connec ion was i s es ablished o wo-dimensional lows
in [59], ex ended o h ee-dimensional sys ems in [60] o o no mally hype bolic in a ian
mani olds in Hamil onian sys ems wi h wo o mo e deg ees o eedom [61–63].
The (2) in eg al can be di ided in o wo pa s:
M(x0, 0,τ) =Mback(x0, 0,τ)+M wd(x0, 0,τ)
= 0
0−τ
(x( ), )d + 0+τ
0
(x( ), )d ,(8)
whe e he o wa d and backwa d ime in eg a ion componen s can be used o de ec he s able
and uns able mani olds sepa a ely.
The a c leng h o a ajec o y segmen , de ined by (2), is calcula ed by in eg a ing he
Euclidean eloci y along he backwa d and o wa d ajec o y o e ime. Howe e , his
app oach can also be applied o o he posi i e scala unc ions ha ep esen in insic physical
o geome ical p ope ies o ajec o ies, in eg a ed along he ajec o y o e he ime in e al
( 0−τ, 0+τ), wi h he only equi emen being ha hese in eg als a e well-de ined.
The e o e, a gene al o m o he Lag angian Desc ip o s can be exp essed as
MF(x0, 0,τ)= 0+τ
0−τ
|F(x( ), )|d ,(9)
whe e |F(x, )|is a bounded, posi i e scala unc ion depending on x0, 0 ep esen ing he
physical o geome ical p ope y o he eloci y ield o in e es . Fo ins ance, Mancho e
al. [57] p esen s some examples o bounded, posi i e scala unc ions o de ine MF,such
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J. Cu belo
Table 1 Di e en o mula ions o he Lag angian desc ip o s ha exis in he li e a u e
Func ion F(x, )No m Some e e ences
Veloci y: (x, )Euclidean no m [57]
Veloci y: (x, )p-no m o p-quasi no m [57,59]
Accele a ion: a(x, )Euclidean no m [57]
Time de i a i e o accele a ion: (da/d ) Euclidean no m [57]
Func ion o cu a u e Euclidean no m [57]
Vo ici y: ∇× (x, )Modulus [65]
The a cleng h o a ajec o y p ojec ed on he con igu a ion space [66]
as accele a ion (a), he modulus o he ime de i a i e o accele a ion (da/d ), o he
modulus o combina ions o ,a,andda/d .
Ano he possibili y in he gene aliza ion o his me hods is conside o he no m ins ead o
he euclidean one, i.e. whe e he posi i e scala unc ion accumula ed along he ajec o ies
o he sys em is he p-no m o he ec o ield ha de e mines he low [57,59], i.e.
Mp(x0, 0,τ)= 0+τ
0−τ
n
i=1
|˙xi( ;x0)|pd ,
whe e p∈(0,1]is a eely chosen pa ame e , and he o e do symbol ep esen s he de i a-
i e wi h espec o ime. A heo e ical ma hema ical amewo k o his al e na i e de ini ion
is p o ided in [59].
O e he yea s, new o mula ions o he me hod ha e been de eloped o enhance he
isualiza ion o phase space s uc u es using Lag angian Desc ip o s (LD). Fo ins ance,
some o mula ions in he li e a u e conside a ying alues o τ(see [64]) (unlike he classical
Lag angiandesc ip o (2)whe eτis hesame o allini ial poin sx0), allowing hein eg a ion
o s op once a ajec o y exi s a speci ic phase space egion, e ealing only he ele an
s uc u es. I is also possible o de ine Lag angian desc ip o s based on he de ini ion o
cu a u e [57], which is an in insic p ope y o cu es:
Mκ(x0, 0,τ)= 0+τ
0−τ
1
|κ|+εd ,(10)
whe e ε>0 is in oduced o a oid singula i ies, and he cu a u e is de ined as a posi i e
quan i y ha combines bo h eloci y and accele a ion,
κ=( · )(a·a)−( ·a)2
( · )3/2.
Some examples o |F(x, )|, no ms, and o mula ions a e summa ized in Table 1.
To summa ize, his me hodology assigns a posi i e numbe M o each ini ial condi ion in
phase space. This numbe is compu ed by in eg a ing a p ede ined posi i e unc ion along he
ajec o y as he sys em e ol es bo h o wa d and backwa d o e a speci ied ime in e al.
While he posi i e unc ion used o de ine di e en ypes o Lag angian Desc ip o s (LD) can
ha e geome ical o physical signi icance, his is no essen ial o he me hod’s applica ion
[63].
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Lag angian desc ip o s in geophysical lows: a su ey
And, in a mo e gene al case, Lag angian desc ip o s encompass a amily o me hods based
on a e ages along ajec o ies [37]:
A(x0, 0,τ)=1
| 0−τ|τ
0
g(F(x( ), )d (11)
whe e he A- ield is an a e age along he Lag angian ajec o ies x( ) ha s a a x0a ime
0. The choice o he unc ion gand he obse able Fde e mines wha exac ly is being
a e aged. Fo example, F= and g(x)=x eco e he classical lag angian desc ip o
(2) bu aking g(x)=x,A(x0, 0,τ)is he a e age lag angian eloci y [67]. Taking
g◦F=| 0−τ||ω(x( ), )−¯ω( 0)|,(12)
whe e ωis he o ici y and ¯ωis i s a e age o e he en i e domain a each ime, he e o e
(11) becomes he Lag angian A e aged Vo ici y De ia ion (LAVD) p oposed by [51].
Al e na i e app oaches in ol e in eg a ing he ins an aneous eigen ec o ield o he a i-
a ional equa ion [68] o employing a se o unc ions o quan i y a ajec o y’s e godici y
de ec [45].
The o mula ions p esen ed in his sec ion sha e he unde lying p inciple ha ajec o ies
wi hin cohe en egions display consis en beha io , e lec ed in simila a e aged alues. In
con as , ajec o ies ac oss di e en cohe en egions exhibi dis inc cha ac e is ics. The
LD app oach o e s signi ican imp o emen s o e o he echniques and has been e ec i ely
applied o geophysical low anspo , including oceanic and a mosphe ic dynamics, which
will be e iewed in de ail in he nex wo sec ions. Addi ionally, LDs ha e ound applica ions
in o he a eas, such as magne ohyd odynamics [69], ansi ion s a e heo y in chemis y
[70–73], in he s udy o ca dio ascula lows [74] o billia d dynamics [75] onamea ew.
2.1 Impo ance o he elec ion o ime in eg a ion
The choice o τ, he ime in e al o e which he ajec o y is in eg a ed, plays a c ucial ole
in he e ec i eness and accu acy o Lag angian Desc ip o s (LD). This pa ame e di ec ly
in luences he esolu ion o he unde lying dynamical s uc u es and he abili y o cap u e
ele an ea u es o he sys em. A well-chosen τallows he LD o e ec i ely e eal s able and
uns able mani olds, hype bolic s uc u es, and o he impo an ea u es o he phase space. I
τis oo small, he me hod may no cap u e enough o he ajec o y’s e olu ion o e eal hese
s uc u es, leading o an incomple e o blu ed ep esen a ion o he dynamics. Con e sely, i
τis oo la ge, he in eg a ion may a e age ou ine de ails, po en ially missing local s uc u es
o in oducing noise om un ela ed dynamics.
Inc easing he τ alues adds mo e Lag angian de ail o he igu es, e ealing s uc u es
ha may no be isible wi h smalle τ. Howe e , hese di e ences o en in ol e ilamen ous
s uc u es ha a e di icul o ollow and compa e. The e o e, ca e ul conside a ion o he
choice o τbased on he speci ic dynamical sys em and he phenomena o in e es is essen ial
o he success ul applica ion o his me hod. Fo example, in geophysical lows like ocean
cu en s o a mosphe ic dynamics, τmigh need o ma ch he ime scales o ele an physical
p ocesses, such as he li espan o a cyclone o he ime i akes o a o a ion a ound he Ea h.
In o he ields, such as chemical eac ion dynamics, τshould be chosen o e lec he ime
scale o he ansi ion s a es o in e es .
Fu he mo e, he alue o τalso a ec s he compu a ional cos . La ge τ alues equi e
longe in eg a ion imes, inc easing he compu a ional load. The e o e, i is impo an o
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J. Cu belo
selec τsuch ha i balances he need o de ailed esolu ion o phase space s uc u es wi h
he a ailable compu a ional esou ces.
2.2 Compu a ion o Lag angian desc ip o s in da ase
The gene al p ocedu e o compu e Lag angian desc ip o s u ilizing any o he di e en o -
mula ions desc ibed in he Sec .2 ypically begins wi h calcula ing a scala o ec o ield
o e he phase space, ollowed by he ex ac ion o key ea u es such as idges (locally max-
imizing cu es), alleys (locally minimizing cu es), o cu es wi h he highes g adien o
he M- ield. In speci ic cases, such as wi h he Lag angian A e aged Vo ici y De ia ion
(LAVD), his also includes iden i ying ou e mos con ex closed con ou s [51,76]. Ano he
app oach could in ol es iden i ying cohe en egions as a eas cha ac e ized by pla eaus o
nea ly cons an M alues. Al hough he connec ion be ween hese cohe en s uc u es and
he idges o pla eaus o Mis no always igo ously de ined, in e es ing co espondences
ha e been obse ed in applica ion and o a ious low scena ios.
While he ask o compu ing Lag angian desc ip o s ac oss di e en o mula ions is
ela i ely s aigh o wa d, ools like he open-sou ce Py hon so wa e package [77]ha e
simpli ied heanalysis.Thisso wa eo e smodules ailo ed o examiningphasespaces uc-
u es in bo h con inuous and disc e e wo-dimensional nonlinea sys ems, accommoda ing
bo h de e minis ic and s ochas ic se ings h ough he use o Lag angian desc ip o s.
Ne e heless, ce ain adap a ions a e necessa y when ex ending hese compu a ions o
h ee-dimensional sys ems o when applying hem in geog aphic coo dina e sys ems, as we
will elabo a e in he ollowing sec ions.
2.2.1 Geog aphical coo dina es
The compu a ion o he Lag angian Desc ip o s (LD) unc ion Min geog aphical coo dina es
(la i ude, longi ude, and heigh ) o he analysis o 3D low o e he Ea h equi es a se ies o
me hodical s eps. These s eps ensu e ha he da a is p ocessed and in e pola ed co ec ly o
accoun o he complexi ies o sphe ical coo dina es and he la ge da ase s ypically in ol ed
in geophysical s udies. Below is an ou line o he p ocess ha we ha e ollowed in p e ious
s udies [16,17,19,78,79], as desc ibed in [58], wi h addi ional de ails p o ided o cla i y.
•S ep 1: Da a acquisi ion, con e sion and p epa a ion. The ini ial s ep in ol es down-
loading and p epa ing he necessa y da a om he co esponding da ase . In gene al, o
compu e (2), we only need he eloci y ield. Fluid eloci y da a can come om model
simula ions (based on equa ions and physical models) and/o eanalysis p oduc s, which
e e o da ase s c ea ed by combining pas obse a ional da a wi h nume ical wea he
p edic ion (NWP) models o gene a e a consis en , comp ehensi e, and long- e m eco d
o a mosphe ic and clima e condi ions. Fo ins ance, in a mosphe ic s udies, we could
use da ase s such as he Whole A mosphe e Communi y Clima e Model (WACCM),
de eloped a he Na ional Cen e o A mosphe ic Resea ch (NCAR) and based on
he Communi y Ea h Sys em Model (CESM) wi h s a e-o - he-a chemis y [80,81].
O he sou ces include ERA-5, he i h-gene a ion ECMWF a mosphe ic eanalysis o
he global clima e by he Cope nicus Clima e Change Se ice (C3S) [82], which p o ides
comp ehensi e da a on a ious a mosphe ic a iables, and he NCEP/NCAR eanalysis
[83], p o ided by he NOAA-OAR-ESRL PSL, Boulde , Colo ado, USA, among many
o he s.
In some cases, da a p ep ocessing is equi ed be o e use because hey a e p o ided in
123
Lag angian desc ip o s in geophysical lows: a su ey
di e en e e ence sys ems. Fo ins ance, ERA-5 da a a e p o ided on sigma le els and
need o be con e ed o speci ied heigh le els o analysis. The e a e many da ase s
a ailable, and i is impossible o explain he peculia i ies o each one o hem he e.
•S ep 2: Ve ical eloci y. In geophysical lows, ully compu ing he e ol ing 3D eloci y
ield is challenging. The e ical eloci y, w, which is gene ally much smalle han he
ho izon al eloci ies, is o en es ima ed as a diagnos ic quan i y a he han p ognos ically
sol ed as pa o he equa ions o mo ion like he ho izon al eloci y componen s [82,84].
Consequen ly, i is less eliable. I is he e o e emp ing o igno e win he compu a ion o
Lag angian Cohe en S uc u es, assuming ha he low mo ion occu s in 2D. In pa icu-
la , in a mosphe ic lows, a widely accep ed app oxima ion assumes ha pa cels emain
on isen opic su aces du ing hei ajec o ies [19,79]. This app oxima ion is jus i ied
because, o e he imescale o mos a mosphe ic e en s (τ∼10 days), s a osphe ic
lows a e adiaba ic and ic ionless, meaning ha luid pa icles and hei ajec o ies
a e cons ained o emain on su aces o cons an po en ial empe a u e. The e o e, he
da a is p ep ocessed o use po en ial empe a u e θins ead o p essu e pas he e ical
coo dina e. The po en ial empe a u e θis de ined as he empe a u e ha a pa cel o d y
ai a p essu e pand empe a u e Twould acqui e i i we e expanded o comp essed
adiaba ically o he e e ence p essu e ps=1000 hPa, i.e.,
θ=Tp
psR/cp
whe e Ris he gas cons an o d y ai , and cpis he speci ic hea a cons an p essu e,
wi h R/cp≈2/7 in he a mosphe e.
Once cons ained o an isen opic su ace, we can conside he low as being essen ially
2D and dismiss he e ical componen o wind eloci y.
Howe e , in pa icula cases, e ical eloci y and e ical shea can signi ican ly a ec
he compu a ion o LCS, equi ing a 3D analysis [29]. To wo k in 3D, he e ical eloci y
wis necessa y. The ECMWF p o ides he e ical eloci y ωin Pa/s, wi h nega i e alues
co esponding o upwa d mo ion. To compu e he e ical eloci y in me e s pe second,
we use he hyd os a ic app oxima ion, which assumes ha he ho izon al scale is much
la ge han he e ical one, i.e.,
ω=−ρgw
whe e ρis he densi y, gis g a i y, and wis he e ical eloci y in m/s. The densi y
is ela ed o p essu e pand empe a u e T h ough he equa ion o s a e o ideal gases,
p=RρT, wi h R=287.058 m2/s2K−1.
This p ocedu e compu es w, which is essen ial o ob aining he 3D eloci y ield in he
a mosphe e.
•S ep 3: T ans o ma ion o ca esian coo dina es. To a oid complica ions a he poles
du ing ajec o y calcula ions in sphe ical coo dina es, he eloci y componen s u, ,
and wa e ans o med in o Ca esian coo dina es using he ollowing equa ions (adap ed
om [85]):
x=wcosλcosφ−usin λ− cos λsin φ,
y=wsin λcos φ+ucosλ− sin λsin φ,
z=wsin φ+ cos φ,
123
J. Cu belo
he ime-e ol ing na u e o luid lows, making hem pa icula ly well-sui ed o s udying
ime-dependen geophysical p ocesses.
One o he p ima y ad an ages o LDs is hei abili y o iden i y Lag angian Cohe -
en S uc u es (LCSs), which a e key o unde s anding ba ie s o anspo and egions o
enhanced mixing in luid lows. These s uc u es a e o en hidden wi hin he complex eloc-
i y ields ypical o geophysical sys ems, ye LDs can e ec i ely e eal hem by compu ing
scala ields ha highligh egions o in e es . This capabili y has been demons a ed ac oss
a wide ange o applica ions, om oceanog aphic s udies o a mosphe ic dynamics, whe e
LDs ha e p o ided new insigh s in o he mechanisms d i ing anspo and mixing. The LD
me hod can also be ex ended o s udy highe -dimensional sys ems.
Howe e , he applica ion o LDs is no wi hou challenges. Fac o s such as he choice
o in eg a ion ime, he quali y and esolu ion o eloci y da a, and he speci ic LD me ic
used can in luence he me hod’s e ec i eness. Addi ionally, while gene ally obus , LDs may
equi e ca e ul uning and alida ion in complex, eal-wo ld scena ios whe e assump ions
may be challenged by ac o s like u bulence, da a spa si y, o non-s a iona y low condi ions.
The e sa ili y and e ec i eness o LDs ha e made hem a c i ical ool in ad ancing
ou unde s anding o luid anspo p ocesses in a ious geophysical con ex s. While he
me hod aces challenges, ongoing esea ch is likely o add ess hese issues, enhancing he
obus ness and applicabili y o LDs in mo e complex scena ios. As he ield o geophysical
luid dynamics con inues o e ol e, LDs a e poised o play an inc easingly impo an ole in
un a eling he in icacies o anspo and mixing in na u al sys ems.
Acknowledgemen s This pape o igina ed om a alk gi en a he “XXVI Cong eso de Ecuaciones Di e -
enciales y Aplicaciones” and he “XVI Cong eso de Ma emá ica Aplicada,” i led “Lag angian Me hods o
Cha ac e ize T anspo and Mixing in Geophysical Flows.” Du ing his e en , I was hono ed o ecei e he
2020 An onio Valle P ize om he Sociedad Española de Ma emá ica Aplicada (SeMA). I would like o hank
SeMA and he o ganize s o he CEDYA con e ence o p o iding such a g ea oppo uni y.
Funding Open Access unding p o ided hanks o he CRUE-CSIC ag eemen wi h Sp inge Na u e.
We acknowledge he suppo o he RyC G an RYC2018-025169, 2020/2021, he Spanish G an s
PID2020-114043GB-I00, PID2021-122954NB-I00, CEX2020-001084-M, CNS2023-144360 unded by
MICIU/AEI/10.13039/501100011033 and “Eu opean Union Nex Gene a ionEU/PRTR” , he Ramón A eces
Founda ion and he “2022 Leona do G an o Resea che s and Cul u al C ea o s”, BBVA Founda ion.
Da a a ailabili y The da a se s used o gene a e igu es 2and 3a e publicly a ailable: ERA5, Cope nicus
Clima e Change Se ice (C3S) ope a ed by ECMWF on behal o he Eu opean Commission. h ps://doi.
o g/10.24381/cds.bd0915c6. They we e ob ained om h ps://cds.clima e.cope nicus.eu/da ase s/ eanalysis-
e a5-p essu e-le els ( egis a ion equi ed).
Decla a ions
Con lic o in e es Au ho has no con lic o in e es o decla e.
Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which
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123
Lag angian desc ip o s in geophysical lows: a su ey
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