Aco ec o o heS e d upsolu ion
o a domain wi h islands
D. B esch•, F. Guill´en-Gonzalez••, M.A. Rod ´ıguez-Bellido••
•Labo a oi e de Ma h´ema iques Appliqu´ees (UMR6620), Uni e si ´e
Blaise Pascal, 63177 Aubi`e e cedex, F ance.
e-mail: Didie .B esc[email p o ec ed]cle mon .
•• Dp o. de Ecuaciones Di e enciales y An´alisis Num´e ico,
Uni e sidad de Se illa,
Ap do. 1160, 41080 Se illa, Espagne.
e-mail: guillen@nume .us.es, angeles@nume .us.es
Abs ac
In his pape we look a he in luence o he Co iolis o ce on
he quasi-geos ophic equa ions on a domain wi h islands. Wep o e
ha asymp o ically we ob ain he solu ion o he S e d up equa ion
wi h homogeneous Di ichle condi ions on he inwa d bounda y plus a
co ec o unc ion which akes in o accoun he p esence o he islands.
This wo k is mo i a ed by he ac ha in oceanog aphy mos o he
su aces a e no simply connec ed. This is he case o example o
he No h Paci ic wi h he Japanese islands. A ou knowledge,in
all he p e ious ma hema ical wo ks, jus simply connec ed domains
ha e been conside ed. Finally we will gi e some simple nume ical
simula ions ela ed o he S ommel model o see he impo ance o
he co ec o .
Keywo ds. Ocean ci cula ion, asymp o ic model, singula pe u ba-
ions, islands.
AMS subjec s classi ica ion. 35Q30, 35B40, 76D05.
1
1In oduc ion
We conside Ω⊂IR 2 he su ace o a wa e ex ension ha has some
islands, which is he case o No h Paci ic Ocean wi h he Japanese is-
lands, o ins ance. All he models used o ob ain he S e d up ela ion
h ough an asymp o ic analysis conside he case o a simply connec ed
domain, which implies Di ichle homogeneous bounda y condi ions in
he ini ial model. He e, we conside he case o a bidimensional do-
main Ωwi h an island. The case o se e al islands can be ea ed in
he same way. A simple model, called he quasi-geos ophic equa ion
wi h one laye (o cons an dep h), allows us o desc ibe oughly he
s eam in ensi ica ion on he Wes coas s. We conside he case o an
island, ha means he domain o igu e 1 gi en by Ω=Ω1 Ω2wi h
Ωi⊂IR 2simply connec ed, Ω2⊂⊂ Ω1,andΓ
i=∂Ωi.Themodel
ela ed o he s eam unc ion Ψcan be desc ibed as:
(1) 












E∆2Ψ−µ∆Ψ +ε∇⊥Ψ.∇∆Ψ +a.∇Ψ=∇⊥. in Ω,
Ψ=0onΓ
1,
Ψ=cE,µ,εon Γ2,
∇Ψ.n=0on∂Ω,
wi h he compa ibili y condi ion
(2) %Γ2
(E∇(∆Ψ) + ⊥).n=0,
whe e E,µ yεa e small posi i e cons an s, ∇⊥=(−∂y,∂
x), a=
(−1,0), ⊥=(− 2,
1)andnis he ex e io uni no mal o he bound-
a y ∂Ω. We ema k ha ∇⊥.co esponds o he cu l ope a o .
To ge his compa ibili y condi ion, we use
%Γ2
∇Ψ.n=%Γ2
a.n=%Γ2
∆Ψ∇⊥Ψ.n=0.
Fi s and las equali ies a e ob ained hanks o he bounda y condi-
ions (in pa icula ∇Ψ=0on∂Ω, since ∇Ψ.τ=∇Ψ.n=0on∂Ω),
and he second one due o he ac ha Γ2is a closed cu e (is a Jo dan
cu e o a simple closed cu e). The eade in e es ed by compa ibili y
condi ions on luid mechanics p oblems is e e ed o ins ance o [7]
and e e ences ci ed he ein.
2
We no e ha such model is used o desc ibe e ically a e aged lows
in a h ee dimensional la domain in e ms o he s eam unc ion Ψ
associa ed o he mean eloci y ield u=(−∂yΨ,∂
xΨ).
The pu pose o his wo k is o pe o m he asymp o ic analysis when
E,µ,εcon e ge o 0. In a i s s ep (Theo em 1), we assume ha ε=0
( ha is o say, he linea case). The s udy o a such linea equa ion is
in e es ing om a pedagogical poin o iew, see o ins ance [9] and
[12]. We ob ain a he limi he S e d up solu ion wi h a co ec o
which akes in o accoun he p esence o he island. In a second s ep
(Theo em 2) we show how o ex end he esul o he nonlinea case
ha means he case ε$= 0. In he las sec ion, we p esen a simple
simula ion on he S ommel model wi h an island. We see he in luence
o he island.
An asymp o ic s udy on he same 4 h o de model in he nonlinea
case (ε$= 0) was made in [2] and o e a S ommel ype model (2nd
o de o E=ε= 0) in [3] o a ield a angen o he bounda y and
which ends locally o a ield a ans e sal o he bounda y. The case
o model (1) wi h E=ε= 0, homogeneous bounda y condi ion and
a=(−1,0) has been la gely s udied because he e a e many physical
applica ions modelled by his kind o equa ion. All he p e ious wo ks
a e only ela ed o a simply connec ed domain Ω.
We no e ha he con igu a ion o he domain implies he p esence
o cha ac e is ic bounda y laye s (No h, Sou h), ee bounda y lay-
e s (issued om he sou h and he no h o he island), S ommel o
Munk laye s (Wes e n pa o he domain). We did no s udy he e he
associa ed bounda y laye s co ec o s necessa y o ob ain be e con-
e gence esul s since we a e only in e es ed by he main o de . This
will be done in a o hcoming wo k ela ed o he s udy o cha ac e is ic
bounda ies.
The non-s a iona y case in a simply connec ed domain is s udied in
[5] whe e hey build he wes e n bounda y laye s and hey ob ain an
app oxima e solu ion. They assume ha ∇⊥. anishes in a neigh -
bou hood o he No h and he Sou h Pa s o he bounda y. I allows
hem o no s udy he cha ac e is ic bounda y laye s which appea o
gene al da a. He e we conside he s a iona y e sion o he quasi-
geos ophic equa ions. This may be seen as he s udy o he long ime
beha io o he low. We ha e no he ime de i a i e o ob ain be e
con e gence esul s as in [5].
3
2In e nalcon e genceandclose oEas
coas s.
He e, we will p o e he ollowing esul
Theo em 1 Le Ωbe a domain o C2class wi h ∂Ω=Γ
1∪Γ2whe e
Γ1∩Γ2=∅de ined as in igu e 1. Le ∈H1(Ω)2such ha ∇⊥. ∈
H2(Ω).Le Ψbe a solu ion in H4(Ω)o (1) wi h ε=0and (2).Fo
all neighbou hood V=V−∪VI,II o Γ−∪ΓI,II,whe eΓ−={x∈∂Ω:
nx≤0}and ΓI,II =ΩI∩ΩII,wi hΩI,ΩII as in igu e 2, one has
Ψ→Ψ+c1ΩII weakly in L2(Ω)and s ongly in L2(Ω V),
∂xΨ$∂
xΨweakly in L2(Ω V−)
whe e Ψis he solu ion in L2(Ω)∩H2(Ω ΓI,II)o he S e d up equa ion



−∂xΨ=∇⊥· in Ω,
Ψ=0on Γ+,
and cis compu ed by he equali y
c=−&Γ−
2Ψnx+&Γ2 ⊥.n
&Γ+
2nx
wi h Γ+
2={x∈Γ2:nx>0},Γ−
2={x∈Γ2:nx<0}and 1ΩII he
cha ac e is ic unc ion o he sub-domain ΩII.+,
Rema k. We assume he same kind o egula i y on han in [7] o
ob ain ou esul . Mo e p ecisely, we assume ∇⊥. ∈H2(Ω) and hey
assume ha ∇⊥. ∈W1,∞(0,T;H2(Ω)).+,
P oo .
Exis ence. We use he linea i y o he p oblem and he uniqueness o
solu ion o he equa ion (1) wi h ε= 0. In his way, we decompose
he unknown o he p oblem (1), Ψ, in o
(3) Ψ=Ψ1+cE,µ Ψ2
4
whe e Ψ1and Ψ2a e espec i ely s ong solu ions (in H4(Ω)) o







E∆2Ψ1−µ∆Ψ1−∂xΨ1=∇⊥. in Ω,
Ψ1=0on∂Ω,
∇Ψ1.n=0on∂Ω,
and













E∆2Ψ2−µ∆Ψ2−∂xΨ2= 0 in Ω,
Ψ2=0onΓ
1,
Ψ2=1onΓ
2.
∇Ψ2.n=0on∂Ω,
The exis ence and uniqueness o he solu ions Ψ1and Ψ2in H4(Ω) is
a classic esul , c . [6]. The cons an cE,µ can be de e mined by he
compa ibili y condi ion (2) as,
(4) cE,µ =−&Γ2(E∇∆Ψ1+ ⊥).n
&Γ2E∇∆Ψ2.n.
Rema k ha &Γ2E∇∆Ψ2.n$= 0, because i we mul iply he equa ion
o Ψ2by Ψ2and we in eg a e by pa s, we ob ain
E%Ω|∆Ψ2|2+µ%Ω|∇Ψ2|2+E%Γ2
∇∆Ψ2·n=0
and his would imply ha Ψ2= 0 i we impose ha &Γ2E∇∆Ψ2.n=
0.
Con e gence. F om exp ession (3), we ha e o obse e he con e gence
o he di e en e ms Ψ1,Ψ
2e cE,µ,whenE,µ →0. The con e gence
o cE,µ needs o in oduce a unc ion θbecause we will only know he
weak con e gence in L2(Ω) o Ψ1and Ψ2in he whole Ω. Fo he sake
o simplici y, we will no ema k he dependency om Eand µin Ψ1
and Ψ2.
i) Con e gence o Ψ1.As ∈H1(Ω), ∇⊥· ∈H2(Ω) and Ωis gi en
by igu e 1, using he esul s ob ained in [2] and [4], we ge :
Ψ1→Ψ1weakly in L2(Ω) and s ongly in L2(Ω V),
∂xΨ1$∂
xΨ1weakly in L2(Ω V−)
5

whe e Ψ1is he solu ion in L2(Ω) o he ollowing S e d up ”homo-
geneous” p oblem:



−∂xΨ1=∇⊥· in Ω,
Ψ1=0onΓ
+.
Exis ence and uniqueness o a solu ion o he p e ious p oblem is
done in [1]. Rema k ha Ψ1is smoo h in Ω V, mo e p ecisely Ψ1∈
H2(Ω V).
We e iew quickly he main s eps used in [2] o he eade ’s con e-
nience. These s eps allow o es ablish he con e gence om Ψ1 h ough
Ψ1. Mul iplying he equa ion e i ied by Ψ1by Ψ1ex, we ob ain
-Ψ1-L2(Ω)≤C
whe e Cis independen om Eand µ. Then, mul iplying by Ψ1we
ge he es ima e
E-∆Ψ1-2
L2(Ω)+µ-∇Ψ1-2
(L2(Ω))2≤C1
whe e C1is independen om Eand µ. These es ima es allow us o
ge he weak limi in L2 o a subsequence ha ends o a solu ion o
he S e d up ela ion, ha is o say, wi hou bounda y condi ions.
To p o e he weak con e gence o ∂xΨ1 h ough ∂xΨ1in L2(Ω V−),
we only ha e o es he equa ion sa is ied by Ψ1agains (∂xΨ1)ηwhe e
η∈C
2(Ω), η= 0 in V−and η≥0 in Ω. We ob ain an uni o m es ima e
o %Ω|∂xΨ1|2η. This gi es he s ong con e gence o Ψ1in L2(Ω V).
The idea is he same as in [7] o a simply connec ed domain and he
linea case, and as in [4] o he nonlinea case. This las a gumen ,
use he bounda y condi ion ∇Ψ1.n=0su Γ
+s ongly.
ii) Con e gence o Ψ2.Now, we ocus on he p oblem o Ψ2. Fi s ,
we li he bounda y condi ion o s udy an homogeneous p oblem as
we he e done o Ψ1. Mo e conc e ely, we conside ξ∈C
4(Ω) such ha
ξ= 0 on a neighbou hood o Γ1and ξ= 1 on a neighbou hood o Γ2.
I we ake Ψ2='
Ψ2+ξ hen '
Ψ2 e i ies









E∆2'
Ψ2−µ∆'
Ψ2−∂x'
Ψ2=−E∆2ξ+µ∆ξ+∂xξin Ω,
'
Ψ2=0on∂Ω,
∇'
Ψ2.n=0on∂Ω,
6
The easoning will inish in he same way ha o Ψ1, i.e.
'
Ψ2$'
Ψ2weakly in L2(Ω) and s ongly in L2(Ω V),
∂x'
Ψ2$∂
x'
Ψ2weakly in L2(Ω V−)
whe e '
Ψ2is he solu ion in L2(Ω) o :



−∂x'
Ψ2=∂xξin Ω,
'
Ψ2=0onΓ
+.
The e o e Ψ2→Ψ2='
Ψ2+ξweakly in L2(Ω), s ongly in L2(Ω V)
and ∂xΨ2→∂xΨ2weakly in L2(Ω V−)whe eΨ2is he solu ion in
L2(Ω) o







−∂xΨ2= 0 in Ω,
Ψ2=0onΓ
+
1
Ψ2=1onΓ
+
2
ha is o say Ψ2=1
ΩII .
The cha ac e is ic line c ossing ough he ex emal poin s o an
island di ide he domain in wo sub egions deno ed as ΩIand ΩII.
Then i appea s a bounda y laye along ΓI,II.
iii) Con e gence o he cons an cE,µ.Le θ∈H2(Ω) be such ha
θ= 0 on a neighbou hood o Γ1and θ= 1 on a neighbou hood o Γ2.
F om he equa ion e i ied by Ψ1, we ob ain
∇.(Eθ∇∆Ψ1−µθ∇Ψ1+θaΨ1+θ ⊥)=E∇θ.∇∆Ψ1
−µ∇θ.∇Ψ1+∇θ.aΨ1+∇θ. ⊥
and a=(−1,0). Then, in eg a ing in Ω, and using ha ∇θ=0on
Γ1∪Γ2,∇Ψ1·n=0onΓ,wege
(5) %Γ2
E∇∆Ψ1.n+ ⊥.n=−%ΩE∆θ∆Ψ1
+µ%Ω∇θ.∇Ψ1+%Ω∇θ.aΨ1+%Ω∇θ. ⊥.
We saw ha -Ψ1-L2(Ω)≤C,E-∆Ψ1-2
L2(Ω)≤C,µ-∇Ψ1-2
L2(Ω)≤C
wi h Cindependen om Eand µ.The e o e, o E,µ →0
7
%Γ2
E∇∆Ψ1.n+ ⊥.n→%Ω∇θ.aΨ1+%Ω∇θ. ⊥
(= %Γ−
2
Ψ1nx+%Γ2
⊥.n).
Mo eo e , om he equa ion e i ied by Ψ2,
∇.(Eθ∇∆Ψ2−µθ∇Ψ2+θaΨ2)=E∇θ.∇∆Ψ2−µ∇θ.∇Ψ2+∇θ.aΨ2
he eby
%Γ2
E∇∆Ψ2.n=−%ΩE∆θ∆Ψ2+µ%Ω∇θ.∇Ψ2+%Ω∇θ.aΨ2.
The limi o Ψ2is made as be o e o Ψ1, ob aining
(6) %Γ2
E∇∆Ψ2.n→−%ΩΨ2∂xθ(i.e. %Ω∇θ.aΨ2).
Bu %ΩΨ2∂xθ=%ΩII
∂xθ=%Γ+
2
nx.
Then,
cE,µ →−
&Ω∇θ·aΨ1+∇θ· ⊥
&Γ+
2nx
:= c
The e o e, Ψ=Ψ1+cE,µΨ2→Ψ1+c1ΩII wi h cgi en as be o e.
Obse e ha cdoes no depend on he unc ion θbecause i is he
limi o cE,µ ha is independen om θ. Le us ew i e i in ano he
o m. I we in eg a e by pa s he nume a o o he cons an cand i
we use he S e d up equa ion sa is ied by Ψand he p ope ies o θ,
we ind
c=−&Γ−
2Ψnx+&Γ2 ⊥.n
&Γ+
2nx
.
Finally, collec ing all he p e ious esul s, we inish he p oo o The-
o em 1. +,
Rema k. I we assume o ha e no angen ial o ce on he bounda y
o he Island ha means ⊥.n= 0, we ind exac ly he cons an Ψl
8
de ined, Equali y (2.8), in [10]. Tha means we ind he e ical a e age
alue o he S e d up s eam unc ion on he eas e n side o he island
c=1
(yn−ys)%yn
ys
Ψ(x+(y),y)dy
whe e x+deno es he g aph o he eas e n pa o he Island, ysand yn
a e espec i ely he mimimum e ical coo dina e (Sou h), he maxi-
mum e ical coo dina e (No h) on he bounda y o he island. +,
3Ano he bounda ycondi ions
I is possible o choose ano he bounda y condi ions di e en o ∇Ψ·
n=0on∂Ω. We e e o [7] o he eade in e es ed in a physical dis-
cussion on he possible bounda y condi ions o he quasi-geos ophic
equa ions (1).
Fo he p oblem (1), changing o ins ance he bounda y condi ion
∇Ψ·n=0on∂Ωby∆Ψ=0on∂Ωand conse ing he Di ichle ype
condi ion on Ψ, we will ob ain essen ially he same esul s o Theo em
1 (excep he weak con e gence in L2(Ω V−) om∂xΨ o∂xΨ ha ,
seemingly, only wo ks i ∇Ψ.n=0onΓ
+).
To ob ain he s ong con e gence in L2(Ω V) h ough he solu ion
Ψ1in L2(Ω) ha anishes on Γ+, we only ha e o ake he di e ence
be ween he equa ion o Ψ1and he equa ion ha e i ies he S e d up
solu ion Ψ1 ha anishes on Γ+. Then es he esul ing equa ion wi h
(Ψ1−Ψ1)Φ, whe e Φis gi en by
Φ(x, y)=%x
gWes (y)η(x",y)dx"
o η∈C
2(Ω), η= 0 in Vand η≥0 in Ω. Recall ha we conside
gEas o C2class.
In his case, we change he compa ibili y condi ion (2) by :
(7) %Γ2
(E∇∆Ψ −µ∇Ψ+ ⊥).n=0.
Acco dingly, i we conside he p oblem (1) wi h ε=0and(7)we
also ob ain exis ence and uniqueness o a solu ion Ψin H4(Ω). This
solu ion is cons uc ed as o (3) eplacing he bounda y condi ions
∇Ψ1.n=∇Ψ2.n=0on∂Ω
9
Γ
Γ
Γ
Γ
Eas
No h
Sou h
Wes
Island Ω
y
x
x=g (y)
Eas
x=g (y)
Wes
Γ1
Γ
2
Ω2
Ω1=ΩUΩ2
Fig. 1: The domain.
Γ
Γ
Γ
Γ
Eas
No h
Sou h
Wes
ΩΙΙ
ΓΙ,ΙΙ
Γ2
+
Γ1
+
Γ2
−
1
Γ−
1
Γ−
1
Γ−
ΩΙΙ
ΩI
ΩI
Ω = U U ΓΙ,ΙΙ
Fig. 2: The subdomains.
16

Fig. 3: The comple e s eam unc ion.
Fig. 4: The s e d up solu ion.
17
Fig. 5: The co ec o .
18