A corrector for the Sverdrup solution for a domain with islands

AcorrectorfortheSverdrupsolution for a domain with islands D. Bresch•, F. Guill´en-Gonzalez••, M.A. Rodr´ıguez-Bellido•• •Laboratoire de Math´ematiques Appliqu´ees (UMR6620), Universit´e Blaise Pascal, 63177 Aubi`ere cedex, France. e-mail: Didier.Bresc[email protected]clermont.fr •• Dpto. de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Espagne. e-mail: [email protected], [email protected] Abstract In this paper we look at the influence of the Coriolis force on the quasi-geostrophic equations on a domain with islands. Weprove that asymptotically we obtain the solution of the Sverdrup equation with homogeneous Dirichlet conditions on the inward boundary plus a corrector function which takes into account the presence of the islands. This work is motivated by the fact that in oceanography most ofthe surfaces are not simply connected. This is the case for example for the North Pacific with the Japanese islands. At our knowledge,in all the previous mathematical works, just simply connected domains have been considered. Finally we will give some simple numerical simulations related to the Stommel model to see the importance of the corrector. Keywords. Ocean circulation, asymptotic model, singular perturbations, islands. AMS subjects classification. 35Q30, 35B40, 76D05. 1 1Introduction We consider Ω⊂IR 2the surface of a water extension that has some islands, which is the case of North Pacific Ocean with the Japanese islands, for instance. All the models used to obtain the Sverdrup relation through an asymptotic analysis consider the case of a simply connected domain, which implies Dirichlet homogeneous boundary conditions in the initial model. Here, we consider the case of a bidimensional domain Ωwith an island. The case of several islands can be treated in the same way. A simple model, called the quasi-geostrophic equation with one layer (of constant depth), allows us to describe roughly the stream intensification on the West coasts. We consider the case of an island, that means the domain of figure 1 given by Ω=Ω1\Ω2with Ωi⊂IR 2simply connected, Ω2⊂⊂ Ω1,andΓ i=∂Ωi.Themodel related to the stream function Ψcan be described as: (1)              E∆2Ψ−µ∆Ψ +ε∇⊥Ψ.∇∆Ψ +a.∇Ψ=∇⊥.fin Ω, Ψ=0onΓ 1, Ψ=cE,µ,εon Γ2, ∇Ψ.n=0on∂Ω, with the compatibility condition (2) %Γ2 (E∇(∆Ψ) + f⊥).n=0, where E,µ yεare small positive constants, ∇⊥=(−∂y,∂ x), a= (−1,0), f⊥=(−f2,f 1)andnis the exterior unit normal of the boundary ∂Ω. We remark that ∇⊥.corresponds to the curl operator. To get this compatibility condition, we use %Γ2 ∇Ψ.n=%Γ2 a.n=%Γ2 ∆Ψ∇⊥Ψ.n=0. First and last equalities are obtained thanks to the boundary conditions (in particular ∇Ψ=0on∂Ω, since ∇Ψ.τ=∇Ψ.n=0on∂Ω), and the second one due to the fact that Γ2is a closed curve (is a Jordan curve or a simple closed curve). The reader interested by compatibility conditions on fluid mechanics problems is referred for instance to [7] and references cited therein. 2 We note that such model is used to describe vertically averaged flows in a three dimensional flat domain in terms of the stream function Ψ associated to the mean velocity field u=(−∂yΨ,∂ xΨ). The purpose of this work is to perform the asymptotic analysis when E,µ,εconverge to 0. In a first step (Theorem 1), we assume that ε=0 (that is to say, the linear case). The study of a such linear equation is interesting from a pedagogical point of view, see for instance [9] and [12]. We obtain at the limit the Sverdrup solution with a corrector which takes into account the presence of the island. In a second step (Theorem 2) we show how to extend the result to the nonlinear case that means the case ε$= 0. In the last section, we present a simple simulation on the Stommel model with an island. We see the influence of the island. An asymptotic study on the same 4th order model in the nonlinear case (ε$= 0) was made in [2] and over a Stommel type model (2nd order for E=ε= 0) in [3] for a field atangent to the boundary and which tends locally to a field atransversal to the boundary. The case of model (1) with E=ε= 0, homogeneous boundary condition and a=(−1,0) has been largely studied because there are many physical applications modelled by this kind of equation. All the previous works are only related to a simply connected domain Ω. We note that the configuration of the domain implies the presence of characteristic boundary layers (North, South), free boundary layers (issued from the south and the north of the island), Stommel or Munk layers (Western part of the domain). We did not study here the associated boundary layers correctors necessary to obtain better convergence results since we are only interested by the main order. This will be done in a forthcoming work related to the study of characteristic boundaries. The non-stationary case in a simply connected domain is studied in [5] where they build the western boundary layers and they obtain an approximate solution. They assume that ∇⊥.fvanishes in a neightbourhood of the North and the South Parts of the boundary. It allows them to not study the characteristic boundary layers which appearfor general data. Here we consider the stationary version of the quasigeostrophic equations. This may be seen as the study of the long time behavior of the flow. We have not the time derivative to obtain better convergence results as in [5]. 3 2InternalconvergenceandclosetoEast coasts. Here, we will prove the following result Theorem 1 Let Ωbe a domain of C2class with ∂Ω=Γ 1∪Γ2where Γ1∩Γ2=∅defined as in figure 1. Let f∈H1(Ω)2such that ∇⊥.f∈ H2(Ω).LetΨbe a solution in H4(Ω)of (1) with ε=0and (2).For all neighbourhood V=V−∪VI,II of Γ−∪ΓI,II,whereΓ−={x∈∂Ω: nx≤0}and ΓI,II =ΩI∩ΩII,withΩI,ΩII as in figure 2, one has Ψ→Ψ+c1ΩII weakly in L2(Ω)and strongly in L2(Ω\V), ∂xΨ$∂ xΨweakly in L2(Ω\V−) where Ψis the solution in L2(Ω)∩H2(Ω\ΓI,II)of the Sverdrup equation    −∂xΨ=∇⊥·fin Ω, Ψ=0on Γ+, and cis computed by the equality c=−&Γ− 2Ψnx+&Γ2f⊥.n &Γ+ 2nx with Γ+ 2={x∈Γ2:nx>0},Γ− 2={x∈Γ2:nx<0}and 1ΩII the characteristic function of the sub-domain ΩII.+, Remark. We assume the same kind of regularity on fthan in [7] to obtain our result. More precisely, we assume ∇⊥.f∈H2(Ω) and they assume that ∇⊥.f∈W1,∞(0,T;H2(Ω)).+, Proof. Existence. We use the linearity of the problem and the uniqueness of solution for the equation (1) with ε= 0. In this way, we decompose the unknown of the problem (1), Ψ, into (3) Ψ=Ψ1+cE,µ Ψ2 4 where Ψ1and Ψ2are respectively strong solutions (in H4(Ω)) of        E∆2Ψ1−µ∆Ψ1−∂xΨ1=∇⊥.fin Ω, Ψ1=0on∂Ω, ∇Ψ1.n=0on∂Ω, and              E∆2Ψ2−µ∆Ψ2−∂xΨ2= 0 in Ω, Ψ2=0onΓ 1, Ψ2=1onΓ 2. ∇Ψ2.n=0on∂Ω, The existence and uniqueness of the solutions Ψ1and Ψ2in H4(Ω) is a classic result, cf. [6]. The constant cE,µ can be determined by the compatibility condition (2) as, (4) cE,µ =−&Γ2(E∇∆Ψ1+f⊥).n &Γ2E∇∆Ψ2.n. Remark that &Γ2E∇∆Ψ2.n$= 0, because if we multiply the equation for Ψ2by Ψ2and we integrate by parts, we obtain E%Ω|∆Ψ2|2+µ%Ω|∇Ψ2|2+E%Γ2 ∇∆Ψ2·n=0 and this would imply that Ψ2= 0 if we impose that &Γ2E∇∆Ψ2.n= 0. Convergence. From expression (3), we have to observe the convergence for the different terms Ψ1,Ψ 2et cE,µ,whenE,µ →0. The convergence of cE,µ needs to introduce a function θbecause we will only know the weak convergence in L2(Ω) for Ψ1and Ψ2in the whole Ω. For the sake of simplicity, we will not remark the dependency from Eand µin Ψ1 and Ψ2. i) Convergence for Ψ1.As f∈H1(Ω), ∇⊥·f∈H2(Ω) and Ωis given by figure 1, using the results obtained in [2] and [4], we get: Ψ1→Ψ1weakly in L2(Ω) and strongly in L2(Ω\V), ∂xΨ1$∂ xΨ1weakly in L2(Ω\V−) 5 where Ψ1is the solution in L2(Ω) for the following Sverdrup ”homogeneous” problem:    −∂xΨ1=∇⊥·fin Ω, Ψ1=0onΓ +. Existence and uniqueness of a solution for the previous problem is done in [1]. Remark that Ψ1is smooth in Ω\V, more precisely Ψ1∈ H2(Ω\V). We review quickly the main steps used in [2] for the reader’s convenience. These steps allow to establish the convergence from Ψ1through Ψ1. Multiplying the equation verified by Ψ1by Ψ1ex, we obtain -Ψ1-L2(Ω)≤C where Cis independent from Eand µ. Then, multiplying by Ψ1we get the estimate E-∆Ψ1-2 L2(Ω)+µ-∇Ψ1-2 (L2(Ω))2≤C1 where C1is independent from Eand µ. These estimates allow us to get the weak limit in L2for a subsequence that tends to a solution of the Sverdrup relation, that is to say, without boundary conditions. To prove the weak convergence of ∂xΨ1through ∂xΨ1in L2(Ω\V−), we only have to test the equation satisfied by Ψ1against (∂xΨ1)ηwhere η∈C 2(Ω), η= 0 in V−and η≥0 in Ω. We obtain an uniform estimate for %Ω|∂xΨ1|2η. This gives the strong convergence of Ψ1in L2(Ω\V). The idea is the same as in [7] for a simply connected domain and the linear case, and as in [4] for the nonlinear case. This last argument, use the boundary condition ∇Ψ1.n=0surΓ +strongly. ii) Convergence for Ψ2.Now, we focus on the problem for Ψ2. First, we lift the boundary condition to study an homogeneous problem as we heve done for Ψ1. More concretely, we consider ξ∈C 4(Ω) such that ξ= 0 on a neighbourhood of Γ1and ξ= 1 on a neighbourhood of Γ2. If we take Ψ2=' Ψ2+ξthen ' Ψ2verifies          E∆2' Ψ2−µ∆' Ψ2−∂x' Ψ2=−E∆2ξ+µ∆ξ+∂xξin Ω, ' Ψ2=0on∂Ω, ∇' Ψ2.n=0on∂Ω, 6 The reasoning will finish in the same way that for Ψ1, i.e. ' Ψ2$' Ψ2weakly in L2(Ω) and strongly in L2(Ω\V), ∂x' Ψ2$∂ x' Ψ2weakly in L2(Ω\V−) where ' Ψ2is the solution in L2(Ω) of :    −∂x' Ψ2=∂xξin Ω, ' Ψ2=0onΓ +. Therefore Ψ2→Ψ2=' Ψ2+ξweakly in L2(Ω), strongly in L2(Ω\V) and ∂xΨ2→∂xΨ2weakly in L2(Ω\V−)whereΨ2is the solution in L2(Ω) of        −∂xΨ2= 0 in Ω, Ψ2=0onΓ + 1 Ψ2=1onΓ + 2 that is to say Ψ2=1 ΩII . The characteristic line crossing trough the extremal points of an island divide the domain in two subregions denoted as ΩIand ΩII. Then it appears a boundary layer along ΓI,II. iii) Convergence for the constant cE,µ.Let θ∈H2(Ω) be such that θ= 0 on a neighbourhood of Γ1and θ= 1 on a neighbourhood of Γ2. From the equation verified by Ψ1, we obtain ∇.(Eθ∇∆Ψ1−µθ∇Ψ1+θaΨ1+θf⊥)=E∇θ.∇∆Ψ1 −µ∇θ.∇Ψ1+∇θ.aΨ1+∇θ.f⊥ and a=(−1,0). Then, integrating in Ω, and using that ∇θ=0on Γ1∪Γ2,∇Ψ1·n=0onΓ,weget (5) %Γ2 E∇∆Ψ1.n+f⊥.n=−%ΩE∆θ∆Ψ1 +µ%Ω∇θ.∇Ψ1+%Ω∇θ.aΨ1+%Ω∇θ.f⊥. We saw that -Ψ1-L2(Ω)≤C,E-∆Ψ1-2 L2(Ω)≤C,µ-∇Ψ1-2 L2(Ω)≤C with Cindependent from Eand µ.Therefore,forE,µ →0 7 %Γ2 E∇∆Ψ1.n+f⊥.n→%Ω∇θ.aΨ1+%Ω∇θ.f⊥ (= %Γ− 2 Ψ1nx+%Γ2 f⊥.n). Moreover, from the equation verified by Ψ2, ∇.(Eθ∇∆Ψ2−µθ∇Ψ2+θaΨ2)=E∇θ.∇∆Ψ2−µ∇θ.∇Ψ2+∇θ.aΨ2 thereby %Γ2 E∇∆Ψ2.n=−%ΩE∆θ∆Ψ2+µ%Ω∇θ.∇Ψ2+%Ω∇θ.aΨ2. The limit for Ψ2is made as before for Ψ1, obtaining (6) %Γ2 E∇∆Ψ2.n→−%ΩΨ2∂xθ(i.e. %Ω∇θ.aΨ2). But %ΩΨ2∂xθ=%ΩII ∂xθ=%Γ+ 2 nx. Then, cE,µ →− &Ω∇θ·aΨ1+∇θ·f⊥ &Γ+ 2nx := c Therefore, Ψ=Ψ1+cE,µΨ2→Ψ1+c1ΩII with cgiven as before. Observe that cdoes not depend on the function θbecause it is the limit for cE,µ that is independent from θ. Let us rewrite it in another form. If we integrate by parts the numerator of the constant cand if we use the Sverdrup equation satisfied by Ψand the properties of θ, we find c=−&Γ− 2Ψnx+&Γ2f⊥.n &Γ+ 2nx . Finally, collecting all the previous results, we finish the proof of Theorem 1. +, Remark. If we assume to have no tangential force on the boundary of the Island that means f⊥.n= 0, we find exactly the constant Ψl 8 defined, Equality (2.8), in [10]. That means we find the vertical average value of the Sverdrup streamfunction on the eastern side of the island c=1 (yn−ys)%yn ys Ψ(x+(y),y)dy where x+denotes the graph of the eastern part of the Island, ysand yn are respectively the mimimum vertical coordinate (South), the maximum vertical coordinate (North) on the boundary of the island. +, 3Anotherboundaryconditions It is possible to choose another boundary conditions different to ∇Ψ· n=0on∂Ω. We refer to [7] for the reader interested in a physical discussion on the possible boundary conditions for the quasi-geostrophic equations (1). For the problem (1), changing for instance the boundary condition ∇Ψ·n=0on∂Ωby∆Ψ=0on∂Ωand conserving the Dirichlet type condition on Ψ, we will obtain essentially the same results of Theorem 1 (except the weak convergence in L2(Ω\V−)from∂xΨto∂xΨthat, seemingly, only works if ∇Ψ.n=0onΓ +). To obtain the strong convergence in L2(Ω\V) through the solution Ψ1in L2(Ω) that vanishes on Γ+, we only have to take the difference between the equation for Ψ1and the equation that verifies the Sverdrup solution Ψ1that vanishes on Γ+. Then test the resulting equation with (Ψ1−Ψ1)Φ, where Φis given by Φ(x, y)=%x gWest(y)η(x",y)dx" for η∈C 2(Ω), η= 0 in Vand η≥0 in Ω. Recall that we consider gEast of C2class. In this case, we change the compatibility condition (2) by : (7) %Γ2 (E∇∆Ψ −µ∇Ψ+f⊥).n=0. Accordingly, if we consider the problem (1) with ε=0and(7)we also obtain existence and uniqueness for a solution Ψin H4(Ω). This solution is constructed as for (3) replacing the boundary conditions ∇Ψ1.n=∇Ψ2.n=0on∂Ω 9 Γ Γ Γ Γ East North South West Island Ω y x x=g (y) East x=g (y) West Γ1 Γ 2 Ω2 Ω1=ΩUΩ2 Fig. 1: The domain. Γ Γ Γ Γ East North South West 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 complete stream function. Fig. 4: The sverdrup solution. 17 Fig. 5: The corrector. 18