a Xi :2409.08714 1 [ma h.AP] 13 Sep 2024
Mean low eloci ies and mass anspo o
Equa o ially- apped wa e wa es wi h an unde lying cu en
Da id Hen y and Sil ia Sas e-Gómez
Abs ac
In his pape we p esen an analysis o he mean low eloci ies, and ela ed mass
anspo , which a e induced by ce ain Equa o ially- apped wa e wa es. In pa -
icula , we examine a ecen ly-de i ed exac and explici solu ion o he geophysical
go e ning equa ions in he β−plane app oxima ion a he Equa o which inco po a es
a cons an unde lying cu en .
1 In oduc ion
The ques ion o de e mining he luid d i induced by he p opaga ion o su ace wa e
wa es is a ascina ing issue and, despi e pionee ing wo k on his subjec being ins iga ed by
S okes as a back as he mid-1800’s, i is s ill a highly cu ious and pe plexing ma e a e en
he mos undamen al le el. Fo ins ance, in he se ing o pe iodic su ace g a i y wa e
wa es S okes demons a ed by way o app oxima ions [39] ha luid pa icles expe ience a
(mean) o wa d d i o he o de o ǫ2, whe e ǫ ela es o he wa e s eepness. This d i is
in a mean sense, whe eby an a e age is aken o e he wa e pe iod, and i is an inhe en ly
nonlinea phenomenon wi h ega d o he o de o wa e ampli ude. The sub le ies o hese
d i p ope ies may be illus a ed by conside ing he classical assump ion ha o pe iodic
i o a ional wa e mo ion i was assumed, a he linea le el, ha luid pa icles ollow closed
ajec o ies [30], whe eas acco ding o he S okes d i phenomenon a he o de o expansion
ǫ2i is implied ha a leas some pa icle pa hs a e non-closed. I is no ewo hy ha , wi h
ega d o pa icle ajec o ies o pe iodic i o a ional wa e wa es, i was ecen ly p o en by
a ious me hods ha all pa icle pa hs h oughou he luid domain a e indeed non-closed
o lows induced by a wide- ange o g a i y (and capilla y-g a i y) wa es, bo h in he
app oxima e linea egime and o exac solu ions o he ully nonlinea go e ning equa ions
[6, 7, 8, 12, 15, 21, 22, 27, 33]. In ecen decades, ollowing he wo k o Longue -Higgins,
he s udy o mean d i eloci ies induced by su ace wa e mo ion was placed on a i me
heo e ical oo ing o a b oad ange o luid mo ions [1, 3, 31, 32]. I was obse ed ha
key ea u es o he mean luid d i eloci y, o so-called S okes’ d i eloci y, could be
cha ac e ised in e ms o he mean Eule ian low eloci y and he mean Lag angian low
eloci y, whe eby: Lag ange = Eule + S okes. In spi e o ecen p og ess, de e mining
he mean luid low eloci ies emains a highly complex and in ica e issue om bo h a
heo e ical, and expe imen al [37, 41], iewpoin .
1
In his pape we p esen an analysis o he mean low eloci ies, and ela ed mass
anspo , induced by ce ain Equa o ially- apped wa e wa es. In pa icula , we exam-
ine a ecen ly-de i ed [23] exac solu ion o he geophysical go e ning equa ions in he
β−plane app oxima ion [14, 16, 17] a he Equa o . The o m o his solu ion is explici in
e ms o Lag angian a iables, and a bene i inhe en in employing he Lag angian ame-
wo k is ha luid kinema ics may o en be desc ibed explici ly and wi h ( ela i e) ease,
[2, 4, 5, 9, 10, 11, 13, 18, 19, 20, 23, 25, 26, 28, 38, 40]. A signi ican complica ing ac-
o o he analysis unde aken in his pape , pa icula ly wi h ega d o de e mining he
mean Eule ian low eloci y and subsequen ly he S okes d i eloci y, is he p esence o
a cons an unde lying cu en e m in he solu ion gi en in [23]. This is in spi e o he
unde lying cu en assuming a ela i ely simple mani es a ion in he Lag angian o mula-
ion o he solu ion, along he lines o he unde lying cu en e ms which we e in oduced
by Mollo-Ch is ensen [36] in o Ge s ne - ype solu ions in an a emp o model billows and
a ious o he complica ing e ec s in bo h a mosphe ic and oceanog aphic si ua ions. We
also no e ha i is well es ablished ha cu en s play a i al ole in Equa o ial dynamics
[9, 11, 14, 16, 29], and in e es ingly a ans e se Equa o ial cu en can be inco po a ed in o
a Ge s ne -like exac solu ion in he Equa o ial −plane o mula ion, c . [24]. The pape
is concluded wi h a b ie discussion o some mass- anspo p ope ies o hese Equa o ially
apped wa es.
2 The Equa o ially apped wa e solu ion
2.1 Go e ning equa ions
We conside geophysical wa es in he Equa o ial egion, whe e we assume ha he ea h is a
pe ec sphe e o adius R= 6378 km, and wo k in a e e ence ame o a ing wi h he ea h
whose o igin is ixed a he ea h’s su ace, wi h he {x, y, z}-coo dina e ame chosen so
ha he x-axis is poin ing ho izon ally due eas ( he zonal di ec ion), he y-axis is due no h
(me idional di ec ion), and he z-axis is poin ing e ically upwa ds and pe pendicula o
he ea h’s su ace. The go e ning equa ions o geophysical ocean wa es a e gi en by
u +uux+ uy+wuz+ 2Ωwcos Φ −2Ω sin Φ = −1
ρPx(2.1a)
+u x+ y+w z+ 2Ωusin Φ = −1
ρPy(2.1b)
w +uwx+ wy+wwz−2Ωucos Φ = −1
ρPz−g, (2.1c)
oge he wi h he mass conse a ion equa ion
ρ +uρx+ ρy+wρz= 0 (2.2a)
and he equa ion o incomp essibili y
ux+ y+wz= 0.(2.2b)
2
He e Φ ep esen s he la i ude, (u, , w)is he luid eloci y, Ω = 73.10−6 ad/s is he
(cons an ) o a ional speed o ea h [16], g= 9.8m/s−2is he g a i a ional cons an , ρ
is he wa e densi y, and Pis he p essu e. We a e in e es ed in Equa o ial wa es, ha
is, geophysical ocean wa es in a egion which is wi hin 2ola i ude o he Equa o . Since
he la i ude is small, we may use he app oxima ions sin Φ ≈Φ, and cos Φ ≈1, and hus
linea ising he Co iolis o ce leads o he β-plane app oxima ion o equa ions (2.1) gi en by
u +uux+ uy+wuz+ 2Ωw−βy =−1
ρPx
+u x+ y+w z+βyu =−1
ρPy
w +uwx+ wy+wwz−2Ωu=−1
ρPz−g,
(2.2c)
whe e β= 2Ω/R = 2.28·10−11 m−1s−1. The ele an bounda y condi ions a e he kinema ic
bounda y condi ions
w=η +uηx+ ηyon z=η(x, y, ),(2.2d)
P=Pa m on z=η(x, y, ),(2.2e)
whe e Pa m is he (cons an ) a mosphe ic p essu e, and η(x, y, )is he ee su ace. The
bounda y condi ion (2.2d) s a es ha all he pa icles in he su ace will s ay in he su ace
o all ime , and he bounda y condi ion (2.2e) decouples he wa e low om he mo ion
o he ai abo e. Finally, we assume he wa e o be in ini ely deep, wi h he low con e ging
apidly wi h dep h o a uni o m zonal cu en , ha is,
(u, , w)→(−c0,0,0) as z→ −∞.(2.2 )
The se o equa ions (2.2) comp ises he go e ning equa ions o he β−plane app oxima ion
o geophysical ocean wa es wi h a cons an unde lying cu en .
2.2 Exac solu ion
In his sec ion we b ie ly desc ibe he exac solu ion o he β-plane go e ning equa ions
(2.2) which was p esen ed in [23]. This solu ion p esc ibes a h ee-dimensional eas wa d-
p opaga ing s eady geophysical wa e in he p esence o a cons an unde lying cu en o
magni ude |c0|. The wa e-like e m is pe iodic in he zonal di ec ion and i has a cons an
phasespeed c > 0. Fu he mo e, he wa e is Equa o ially apped, exhibi ing a s ong ex-
ponen ial decay away om he Equa o . Equa o ially apped wa es which a e symme ic
abou he Equa o and p opaga e eas wa d a e known o exis , and hey a e ega ded as
an impo an ac o in a possible explana ion o he El Niño phenomenon (c . [14, 16, 17]).
The solu ion o (2.2) we p esen is o mula ed in he Lag angian amewo k, whe eby he
e olu ion in ime o indi idual luid pa icles is p esc ibed [2]. In his Lag angian o mu-
la ion he Eule ian coo dina es o luid pa icles (x, y, z)a e exp essed as unc ions o he
3
Lag angian labelling a iables (q, , s)∈(R,(−∞, 0),I), and ime , as ollows:
x=q−c0 −1
kek[ − (s)] sin [k(q−c )],(2.3a)
y=s, (2.3b)
z= +1
kek[ − (s)] cos [k(q−c )],(2.3c)
whe e 0<0and kis he wa enumbe de ined by k= 2π/L, and whe e Lis he ( ixed)
wa eleng h. Fo c0>0 he unde lying cu en is ad e se, while o c0<0 he cu en is
ollowing, and we see below ha he sign o he cu en de e mines whe he Iis he eal line
Ro a ini e in e al. The unc ion (s)de e mines he decay o he pa icle oscilla ions in
he la i udinal di ec ion away om he equa o and i is gi en by
(s) = cβ
2γs2,(2.4)
whe e γ:= 2Ωc0+g(>0) is a “modi ied g a i y” e m and we make he (physically eason-
able) assump ion ha c0>−g
2Ω . Fo no a ional con enience le us choose
ξ=k( − (s)) , θ =k(q−c ).
Then he Jacobian ma ix o he ans o ma ion (2.3) is gi en by
∂(x, y, z)
∂(q, s, )=

1−eξcos θ0−eξsin θ
seξsin θ1− seξcos θ
−eξsin θ0 1 + eξcos θ

,(2.5)
which has he ime-independen de e minan 1−e2ξ. Consequen ly he low de ined by
(2.3) is olume p ese ing, ensu ing ha (2.2b) holds in he Eule ian se ing [2]. Since he
solu ion (2.3) is explici in he Lag angian o mula ion, we may immedia ely disce n some
quali a i e p ope ies o he physical luid mo ion. Indeed, a signi ican bene i o wo king
in he Lag angain amewo k is ha he luid kinema ics can o en be desc ibed explici ly
and wi h ela i e ease. In he case abo e we calcula e he eloci y ield di ec ly om (2.3)
o ge
u(q, , s; ) = Dx
D =ceξcos θ−c0,(2.6a)
(q, , s; ) = Dy
D = 0,(2.6b)
w(q, , s; ) = Dz
D =ceξsin θ, (2.6c)
whe e D/D is he ma e ial (o con ec i e) de i a i e wi h espec o Eule ian a iables. Fo
ixed la i udes, ha is o e e y ixed s, he sys em (2.3) desc ibes he low benea h a su ace
wa e p opaga ing eas wa ds a cons an speed cde e mined by he dispe sion ela ion (2.10)
4
below. Addi ionally, o ixed la i udes he ee su ace z=η(x, y, )is ob ained by se ing
= 0(s)in (2.3c), whe e 0(s)< 0is he unique solu ion o
e2k[ (s)−cβ
2γs2]
2k− (s) + c0β
2γs2−e2k 0
2k+ 0= 0,(2.7)
The exis ence o a unique solu ion (s) o (2.7) o |s|>0is equi alen o he condi ion
e2k[ 0−cβ
2γs2]
2k+c0β
2γs2<e2k 0
2k,(2.8)
c . [23] o de ails. Fo c0≤0, i is easy o see ha condi ion (2.8) holds o all s∈R.
Fo c0>0, condi ion (2.8) will hold o es ic ed alues o son a ini e in e al Iwhich
depends on he magni ude |c0|o he cu en . Fo ou p esen pu poses we ema k ha ,
gi en a cu en wi h c0>0, o a unique solu ion o (2.7) o exis i is necessa y ha
c0< ce2k 0,(2.9)
and acco dingly (2.3) ep esen s a dynamically possible solu ion o (2.2). Since c06=c(by
(2.9)) i ollows ha he dispe sion ela ion o he wa e akes he o m
c=pΩ2+kγ −Ω
k=pΩ2+k(2Ωc0+g)−Ω
k>0.(2.10)
We ema k ha i c0=c hen he dispe sion ela ion o he wa e would ake he o m
c=pg/k. Hence, in his si ua ion geophysical Co iolis e ec s ha e no bea ing on he
dispe sion ela ion, which ins ead ma ches ha o he celeb a ed Ge s ne ’s wa e solu ion
[5, 7, 20] o deep-wa e g a i y wa es. This obse a ion leads us o in e ha p ecluding
he case c0=c, as is consis en wi h condi ion (2.9), is na u al in he con ex o geophysical
wa e wa es (c . [23] o de ails on he dispe sion ela ions). Finally, we no e ha a ixed
la i udes s=s∗ he c es and ough le els o he wa e su ace p o ile a e p esc ibed in
e ms o he Lag angian pa ame e s by
z±(s∗) = 0(s∗)±1
kek[ 0(s∗)− (s∗)].
3 Mean eloci ies and S okes d i
In his sec ion we analyse he e ec ha he cons an unde lying cu en has wi h espec o
bo h he mean Lag angian and Eule ian low eloci ies induced by he exac solu ion (2.3).
In [13] i was shown ha in he absence o a cu en , ha is o c0= 0, he mean Lag angian
eloci y is ze o and he mean Eule ian eloci y lows wes wa ds. Hence, in he absence o
he cu en he S okes d i (o mean S okes low eloci y), which which is he di e ence in
he mean Lag angian and Eule ian eloci ies [31, 32], is eas wa ds. He e we show ha he
si ua ion is a mo e complex in he p esence o an unde lying cu en , in pa icula when
de e mining he mean Eule ian eloci y. Th oughou he ollowing conside a ions we ix he
la i ude by se ing s=s∗.
5

3.1 Mean Lag angian low eloci y
The mean Lag angian low eloci y (also known as he mass- anspo eloci y [31]) a a
poin in he luid domain is he mean eloci y o e a wa e pe iod o a ma ked luid pa icle
which o igina es a ha poin . Fo he exac solu ion (2.3) we may calcula e he a e age o
he ho izon al eloci y uin (2.6a) o e a wa e pe iod T=L/c as ollows:
huiL=1
TZT
0
u(q−c , s, )d
=ceξ
TZT
0
cos [k(q−c )] d −1
TZT
0
c0d =−c0,
(3.1)
whe e we ha e used he ac ha he i s in eg al on he le -hand side abo e anishes.
I is immedia ely appa en ha he mean Lag angian low eloci y is ei he wes wa ds o
eas wa ds, depending on whe he he sign o c0is posi i e o nega i e espec i ely. When
c0= 0 he mean Lag angian eloci y is ze o, which concu s wi h he esul o [13], and in
his ligh he o m o he mean Lag angian low eloci y abo e is no pa icula ly su p ising
conside ing he explici manne in which c0appea s in he exp ession o he Lag angian
eloci y (2.6a). We no e ha he exp ession o he mean Lag angian eloci y is independen
o bo h he la i ude s, and he loca ion in he luid domain whe e he luid pa cel o igina es.
3.2 Mean Eule ian low eloci y
When wo king in he Eule ian se ing ma e s a e g ea ly complica ed by he p esence o
he unde lying cu en . The mean Eule ian low eloci y a a ixed-poin in he luid domain
is he Eule ian luid eloci y a ha ixed-poin a e aged o e a wa e pe iod. In he case
o he eloci y ield (2.6) he mean Eule ian low eloci y may be compu ed by aking he
mean o e a wa e pe iod o he ho izon al eloci y (2.6a) a any ixed-dep h benea h he
wa e ough. Le ing z=z−(s∗)deno e he e ical posi ion o he wa e ough le el, we
ix a dep h z=z0< z−(s∗). This ixed dep h z=z0may be cha ac e ised in e ms o
Lag angian a iables, using (2.3c), by he equa ion
z0=R+1
keξ(R)cos θ, (3.2)
whe e we deno e by =R(q−c ;s∗, z0) he unc ional ela ionship induced by ela ion (3.2)
be ween he o he wise independen a iables and q, as ollows om he implici unc ion
heo em. We no e ha a consequence o (3.2) is ha Ris pe iodic in he q− a iable, wi h
pe iod L. Di e en ia ing (3.2) wi h espec o qyields
0 = Rq+Rqeξ(R(q)) cos θ−eξ(R(q)) sin θ,
ha is
Rq=eξsin θ
1 + eξcos θ.(3.3)
6
We no e om (3.3) ha Ris maximised o minimised wi h espec o qwhene e sin θ= 0,
and he e o e o a ixed-dep h z0 he maximal and minimal alues achie ed by Ra e gi en
implici ly by he ela ions
z0=R±1
keξ(R),
whe e he posi i e (nega i e) sign co esponds o he minimal (maximal) alue o R, e-
spec i ely. To compu e he Eule ian mean eloci y huiE(s∗, z0)a la i ude s∗and dep h
z0≤z−(s∗)we examine
c+huiE(s∗, z0) = 1
TZT
0
[c+u(x−c , y, z0)] d .
=1
LZL
0
[c+u(x−c , y, z0)] dx,
which upon ans o ming, by way o (2.3), o he labelling a iables (q, s, ), and in oking
unc ional pe iodici y wi h espec o he q− a iable, we ge
c+huiE(s∗, z0) = 1
LZL
0
[c+u(q−c , s∗, R(q−c ;s∗, z0))] ∂x
∂q dq.
By di e en ia ing xin (2.3a) wi h espec o q, using (2.6a), and aking in o accoun (3.3),
we ob ain
c+huiE(s, z0) = 1
LZL
0hc+ceξ(R(q)) cos θ−c0ih1−eξ(R(q)) cos θ−eξ(R(q))Rqsin θidq
=1
LZL
0
c1 + eξ(R(q)) cos θ1−e2ξ(R(q))
1 + eξ(R(q)) cos θdq −c0
LZL
0
1−e2ξ(R(q))
1 + eξ(R(q)) cos θdq
=c−c
LZL
0
e2ξ(R(q))dq −c0
LZL
0
1−e2ξ(R(q))
1 + eξ(R(q)) cos θdq.
The e o e he mean Eule ian eloci y is gi en by he ela ion
huiE(s∗, z0) = −c
LZL
0
e2ξ(R(q))dq −c0
LZL
0
1−e2ξ(R(q))
1 + eξ(R(q)) cos (k[q−c ])dq. (3.4)
The p esence o a non-ze o unde lying cu en c0adds a signi ican complica ing ac o o
exp ession (3.4), and in pa icula he sign (and hence di ec ion) o he mean Eule ian e-
loci y is no easily disce nible om he abo e exp ession in gene al. Ne e heless, depending
on he size and di ec ion o he cu en c0, we may ob ain es ima es which de e mine he
di ec ion o he mean Eule ian eloci y ollowing om he inequali ies
ZL
0
1−e2ξ
1 + eξdq ≤ZL
0
1−e2ξ
1 + eξcos θdq ≤ZL
0
1−e2ξ
1−eξdq. (3.5)
7
3.2.1 The case c0>0:
Fi s o all le us s udy he case when c0is posi i e, which ep esen s an unde lying ad e se
cu en in he Lag angian a iables. The second in eg al e m on he igh -hand side o
inequali y (3.4) sa is ies
−c0
LZL
0
1−e2ξ
1−eξdq ≤ −c0
LZL
0
1−e2ξ
1 + eξcos θdq ≤ −c0
LZL
0
1−e2ξ
1 + eξdq. (3.6)
Since 0< c0< ce2k 0< c om (2.9), equa ion (3.6) yields
huiE≤ − c
LZL
0
e2ξdq −c0
LZL
0
1−e2ξ
1 + eξdq ≤ −c0
LZL
0
1 + e3ξ
1 + eξdq < 0.(3.7)
The e o e he mean Eule ian low eloci y is wes wa ds o all admissible alues o c0 o
which (2.9) holds. To ge an idea o he ange o he mean Eule ian low we no e ha
huiE≥ − c
LZL
0
e2ξdq −c0
LZL
0
1−e2ξ
1−eξdq ≥ − c
LZL
0
1−e3ξ
1−eξdq. (3.8)
Hence, since ξ≤kR < k 0<0, we see ha o all la i udes sand dep hs z0< z−(s) he
mean Eule ian low eloci y is in he ange
huiE(s, z0)∈−c1−e3k 0
1−ek 0,0.(3.9)
Tha he mean Eule ian low is wes wa d o an ad e se cu en is no su p ising, since in
he absence o he cu en he mean Eule ian low is wes wa d (c . [13]) and he p esence o
he ad e se cu en e m in (3.4) me ely se es o exace ba e his e ec .
3.2.2 The case c0≤0:
The case when c0is nonposi i e, c0≤0, ep esen s an unde lying ollowing cu en . In his
case he in luence ha he cu en has on he mean Eule ian low in (3.4) is complex and
di icul o disce n, and i is no gene ally possible o analy ically de e mine i s e ec di ec ly
om exp ession (3.4). None heless, we can deduce some b oad cha ac e is ics o he low by
wo king as ollows. The mean Eule ian eloci y (3.4) is wes wa ds, ha is huiE(s∗, z0)<0,
i
−c0
LZL
0
1−e2ξ(R(q))
1 + eξ(R(q)) cos θdq ≤ −c0
LZL
0
1−e2ξ(R(q))
1−eξ(R(q)) dq
≤ −c0max
q∈[0,L]
1−e2ξ(R(q))
1−eξ(R(q)) < c min
q∈[0,L]e2ξ(R(q)) ≤c
LZL
0
e2ξ(R(q))dq.
These se ies o inequali ies hold, and acco dingly huiE(s∗, z0)<0, i
c0>−cmin
q∈[0,L]
e2k(R(q;z0)− (s∗))1−ek(R(q;z0)− (s∗))
1−e2k(R(q;z0)− (s∗)) .(3.10)
8
We no e ha in he absence o an unde lying cu en , ha is when c0= 0, condi ion
(3.10) always holds and so he esul ing mean Eule ian eloci y is always in he wes e ly
di ec ion, an obse a ion which acco ds wi h [13]. The mean Eule ian low (3.4) is eas wa ds,
huiE(s∗, z0)>0, i
−c0
LZL
0
1−e2ξ(R(q))
1 + eξ(R(q)) cos θdq ≥ −c0
LZL
0
1−e2ξ(R(q))
1 + eξ(R(q)) dq
≥ −c0min
q∈[0,L]
1−e2ξ(R(q))
1 + eξ(R(q)) > c max
q∈[0,L]e2ξ(R(q)) ≥c
LZL
0
e2ξ(R(q))dq.
These inequali ies hold, and hence huiE(s∗, z0)>0, i
c0<−cmax
q∈[0,L]
e2k(R(q;z0)− (s∗))1 + ek(R(q;z0)− (s∗))
1−e2k(R(q;z0)− (s∗)) .(3.11)
3.3 S okes d i
The S okes d i (o mean S okes) eloci y US(z0)is de ined (c . [1, 13, 31, 32, 37, 39]) by
he ela ion
huiL(z0) = huiE(z0) + US(z0).
We de i e an exp ession o he S okes d i by compu ing
US=huiL− huiE=c
LZL
0
e2ξ(R(q))dq +c0
LZL
0
1−e2ξ(R(q))
1 + eξ(R(q)) cos (k[q−c ])dq −c0.
Fo an ad e se cu en , c0≥0, i ollows om (2.9) ha
US=1
LZL
0ce2ξ(R(q)) −c0dq +c0
LZL
0
1−e2ξ(R(q))
1 + eξ(R(q)) cos (k[q−c ])dq > 0.
The e o e o c0≥0 he S okes d i is eas wa ds h oughou he luid domain. In he case
a ollowing cu en , c0<0, he exp ession o S okes d i is al oge he mo e complica ed
and in ac able. Ne e heless we ema k ha , o c0<0, i he magni ude o he cu en is
such ha (3.11) holds hen he S okes d i mus be wes wa ds.
4 Mass lux
We conclude wi h a b ie discussion o mass- anspo p ope ies o he low (2.6), whe e
we ecall ha huiL, being he mean eloci y o a ma ked pa icle, is some imes called he
mass- anspo eloci y. Fo a non-ze o unde lying cu en , c06= 0, we in ui i ely expec he
o al mass lux below he ee-su ace wa e pas a poin x=x0 ixed in Eule ian coo dina es
o be in ini e. To see his di ec ly we compu e he in eg al
m(x0−c , s) = Zη(x0−c ,s)
−∞
u(x0−c , s, z)dz. (4.1)
9