A corrector for the Sverdrup solution for a domain with islands

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