Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics

MATHEMATICAL JUSTIFICATION OF THE HYDROSTATIC APPROXIMATION IN THE PRIMITIVE EQUATIONS OF GEOPHYSICAL FLUID DYNAMICS∗ PASCAL AZ´ ERAD†AND FRANCISCO GUILL´ EN‡ SIAM J. MATH. ANAL.c 2001 Society for Industrial and Applied Mathematics Vol. 33, No. 4, pp. 847–859 Abstract. Geophysical fluids all exhibit a common feature: their aspect ratio (depth to horizontal width) is very small. This leads to an asymptotic model widely used in meteorology, oceanography, and limnology, namely the hydrostatic approximation of the time-dependent incompressible Navier–Stokes equations. It relies on the hypothesis that pressure increases linearly in the vertical direction. In the following, we prove a convergence and existence theorem for this model by means of anisotropic estimates and a new time-compactness criterium. Key words. Navier–Stokes equations, shallow domains, geophysical fluid dynamics, hydrostatic approximation, singular perturbation, compactness criterium, asymptotic analysis AMS subject classifications. 35Q30, 35B40, 76D05, 34C35 PII. S0036141000375962 1. Introduction. Atmospheric flow in meteorology, water flow in oceanography, and limnology are all described by the Navier–Stokes equations. Due to the fact that the aspect ratio =characteristic depth characteristic width is very small in most geophysical domains, asymptotic models have been used; see, e.g., [9, 15, 22]. One such model is the primitive equations model; see, e.g., [11, 12], wherein the unknown flow variables are velocity, pressure, temperature, and salinity (in the case of an ocean). Besides, most geophysical fluids are stratified (i.e., density is a known function of the temperature (and salinity, if any)) and have a free surface. We shall not investigate these features in this paper, leaving it, rather, for forthcoming work. Instead we shall focus on the assumption that the pressure is hydrostatic, i.e., increases linearly with respect to the depth, as in the static case. This law agrees well with experiment (as first observed by Blaise Pascal around 1650; see [14])) and is frequently taken as a hypothesis in geophysical fluid dynamics. We justify this assumption by means of asymptotic analysis (taking as the small parameter). Our derivation is made possible by the use of anisotropic eddy viscosities, namely ν= (νx,ν y,ν z), relying on the fact that the ratio between the horizontal and vertical scales leads to very different sizes for the horizontal and vertical eddies (see [9, 15]). Specifically, if we assume that ν=(ν1,ν 2, 2ν3) with νi= O(1) for i=1,2,3, then we will see that weak solutions of the Navier–Stokes equations converge to a weak solution of a limit problem with hydrostatic pressure. ∗Received by the editors July 26, 2000; accepted for publication (in revised form) August 22, 2001; published electronically December 18, 2001. This work was supported by the fonds Franco-Espagnol D.R.E.I.F. (ref. UC 815) and the project MAR98-0486 C.I.C.Y.T. (Espa˜na). http://www.siam.org/journals/sima/33-4/37596.html †Laboratoire de Mod´elisation, Analyse Non Lin´eaire et Optimisation, Universit´e de Perpignan, 52 av. de Villeneuve, F-66860 Perpignan cedex, France ([email protected]). ‡Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, c/ tarfia s/n Universidad de Sevilla, 41012 Sevilla, Spain ([email protected]). 847 Downloaded 05/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 848 PASCAL AZ´ ERAD AND FRANCISCO GUILL´ EN The stationary case has already been studied (see [4] for the linear problem and [5] for the nonlinear one), whereas the linear time-dependent case was solved in [1]. The main task of this paper is then to solve the nonlinear time-dependent case. Our result was announced in [2], whereas numerical simulations stemming from it were discussed in [3]. Fluid flow in thin domains (flat, curved, and with various boundary conditions) has been extensively studied; see [7, 13, 16, 20, 21]. In these works, an isotropic viscosity is used, and the depth is constant. By averaging along the vertical direction, two-dimensional (2D) limit models are obtained, together with existence and global regularity results. Our approach is different, because we neither eliminate the vertical velocity by averaging nor assume the depth of the domain to be constant. By making use of different horizontal and vertical eddy viscosities, we are able to derive a three-dimensional (3D) limit nonlinear model. Let us emphasize that the anisotropic viscosity hypothesis is fundamental for the derivation of the primitive equations: in the stationary case, keeping an isotropic viscosity, the asymptotic model is linear, with vanishing horizontal diffusion; see [6]. The paper is organized as follows. In section 2, we present the physical model and the scaling leading to the primitive equations. We state the main theorem in section 3. The functional setting and weak formulation are described in section 4. In the next section, we state and prove a time-compactness result, which we shall use in the proof of the main theorem in section 6. Finally, in section 7, we comment on the convergence of the pressure and the orders of magnitude of the vertical velocity with respect to the aspect ratio. 2. Equations governing the flow and scaling. Let us consider an incompressible homogeneous fluid filling a thin domain defined by Ω=(x, y, z)∈R3;(x, y)∈ω,−h(x, y)<z<0, where ωis an open bounded Lipschitz domain in R2and h:ω→Ris a nonnegative lipschitzian application, which is arbitrary provided that Ωis lipschitzian. In particular, hmay vanish, contrary to [12, 9], but in order that the domain Ωhas no cusps, the slope must not vanish on the shores.1We denote by Γs=ω×{0}the fluid surface and by Γ b=∂Ω\Γsthe basin bottom. The fluid flow in Ωis generated by the wind traction on the surface Γs, influenced by the Coriolis and centrifugal forces and governed by the Navier–Stokes equations, in which we take different eddy viscosities according to the direction; see [5, 9, 15]. Finally, we take the density as identically equal to one. In a geophysical rotating frame (zpointing upwards, xeast, and ynorth), the initial-boundary value problem reads as follows. Find v=(v1,v 2,v 3) (velocity) and q(pressure), such that ∂tv+(v·∇)v−∆νv+∇q+2w×v=gin Ω×(0,T),(2.1) div v=0 inΩ ×(0,T),(2.2) v=0 onΓ  b×(0,T),(2.3) νz∂zv1=τ1,ν z∂zv2=τ2,v 3=0 onΓ s×(0,T),(2.4) v(·,t= 0) = v0in Ω.(2.5) 1This is a technical hypothesis. One could probably dispense with it due to the specific shape of the domain. Downloaded 05/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php JUSTIFICATION OF THE HYDROSTATIC APPROXIMATION 849 In (2.1), ∇=(∂x,∂ y,∂ z) denotes the gradient vector, and ∆νdenotes the anisotropic Laplacian defined by ∆ν=νx∂2 xx +νy∂2 yy +νz∂2 zz with ν=(νx,ν y,ν z) the eddy kinematic viscosity vector. Moreover, w=f(0,cos(l(y)),sin(l(y))) represents the earth rotation angular speed (fthe module and l(y) the latitude), 2w×vrepresents the Coriolis acceleration (×denotes the cross-product in R3), and grepresents the force due to gravity (which also includes the centrifugal effect). It is well known (cf. [15, p. 18]) that gis a potential, i.e., g=∇ϕ. It is customary to incorporate the gravity potential in the pressure term; thus we set p=q−ϕ. Equation (2.2) represents the incompressibility condition, and (2.3) represents the no-slip condition on the bottom. In (2.4), τi,i=1,2, stand for the horizontal tractions exerted by the wind on the (fixed) surface Γsof the fluid, and w=0onΓ scomes from the rigid lid hypothesis. In (2.5), v0=(v01,v 02,v 03) designates the initial velocity. Remark. We have neglected the earth’s curvature, and hence our analysis is valid only locally, e.g., for lakes; for seas or oceans, spherical coordinates should be used [12], although this can be somewhat cumbersome. As usual in asymptotic analysis, we perform a vertical scaling to make the domain independent of , that is, x=x1,y=x2,z=x 3, so that Ω = (x1,x 2,x 3)∈R3;(x1,x 2)∈ω, −h(x1,x 2)<x 3<0is the new fixed domain. The corresponding kinematic scaling is v1=u 1,v 2=u 2,v 3=u  3,p=p,(2.6) so that u=(u 1,u  2,u  3) is the new unknown velocity and pis the new pressure. It is necessary to scale the mechanical quantities accordingly. First, it is only natural to assume v01 =u01,v 02 =u02, and v03 =u03, where u0idoes not depend on ,i=1,2,3. Next we assume νx=ν1,νy=ν2, and νz=2·ν3, where ν1,ν 2,ν 3 are constants. As mentioned in the introduction, in oceanography the vertical eddy viscosity is usually very small compared to the horizontal one. We refer to [5] for a mathematical discussion of this assumption, and here we content ourselves with one heuristic comment. Basically, a kinematic viscosity has the dimension L2/T, where L(resp., T) is a typical length (resp., time) scale so that νxand νyhave the dimension L2 H/T, whereas νzhas the dimension L2 V/T, where LH(resp., LV) denotes a typical horizontal (resp., vertical) length scale. It follows that the ratio νz/νxand νz/νy= O(2).2 Now (2.4) becomes ν3∂3u i=τ i/, i =1,2. We see that in order to end up with an O(1)-wind force on the rescaled domain, we have to assume that τ i=·θi,i=1,2, where the θiare functions independent of . 2We do not delude ourselves with this sketchy argument. As far as we know, up to now there has been no rigorous derivation of any eddy viscosity model. Downloaded 05/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 850 PASCAL AZ´ ERAD AND FRANCISCO GUILL´ EN Remark. This last assumption can also be motivated by dimensional analysis, as follows. From τi=νz∂zvi, one derives that τihas the dimension of L2 V T·1 LV ·LH T=L2 H T2=O(). With the above considerations, problem (2.1)–(2.5) transforms into the following anisotropic Navier–Stokes equations: ∂tu 1+u·∇u 1−∆νu 1−αu  2+βu  3+∂1p=0 inΩ×(0,T),(2.7) ∂tu 2+u·∇u 2−∆νu 2+αu  1+∂2p=0 inΩ×(0,T),(2.8) 2{∂tu 3+u·∇u 3−∆νu 3}−βu  1+∂3p=0 inΩ×(0,T),(2.9) div u=0 inΩ×(0,T),(2.10) u=0 onΓ b×(0,T),(2.11) ν3∂3u 1=θ1,ν 3∂3u 2=θ2,u  3=0 onΓ s×(0,T),(2.12) u(·,t= 0) = u0in Ω.(2.13) Now ∇=(∂1,∂ 2,∂ 3), ∆ν=ν1∂2 11 +ν2∂2 22 +ν3∂2 33,Γ b=∂Ω\Γs,α=2fsin(l(x2)), and β=2fcos(l(x2)). If we assume that u=O(1), then neglecting the 2and terms in the first and third momentum equation, (2.7) and (2.9), we formally get the hydrostatic Navier– Stokes equations, also called the primitive equations: ∂tu1+u·∇u1−∆νu1−αu 2+∂1p=0 inΩ×(0,T),(2.14) ∂tu2+u·∇u2−∆νu2+αu 1+∂2p=0 inΩ×(0,T),(2.15) ∂3p=0 inΩ×(0,T),(2.16) div u=0 inΩ×(0,T),(2.17) u1=u2=u3n3=0 onΓ b×(0,T),(2.18) ν3∂u1=θ1,ν 3∂3u2=θ2,u 3=0 onΓ s×(0,T),(2.19) ui(·,t= 0) = u0iin Ω,i=1,2.(2.20) Remark. The boundary condition (2.18) differs from its counterpart (2.11) because u3is less regular than u1,u 2as we shall see below. Also, the initial condition (2.20) does not involve u3, the time derivative of which is missing in the hydrostatic model. The problem is not in the Cauchy–Kowalevska form.3 Remark. If u3were to be computed directly from (2.17), which is a first order equation, it is not obvious at all that it would fulfill the two boundary conditions on the bottom (2.18) and the surface (2.19). 3. Main theorem. Let Tbe a fixed positive duration. We make the natural assumption of a wind of finite energy: θ1,θ 2∈L2(0,T;H−1/2(Γs)).Our main result is the following theorem. Theorem 3.1. Let u0∈L2(Ω)3, with div u0=0,u0·n=0on ∂Ω, and θ1,θ 2∈ L2(0,T;H−1/2(Γs)); there exists a weak solution uof the hydrostatic Navier–Stokes equations (2.14)–(2.20), obtained as a limit of weak solutions uof the anisotropic Navier–Stokes equations (2.7)–(2.13), as the aspect ratio tends to zero. 3Meteorologists say that u3is no longer a prognostic variable (see [11, 12]). Downloaded 05/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php JUSTIFICATION OF THE HYDROSTATIC APPROXIMATION 851 The proof relies on a priori estimates in anisotropic spaces (Propositions 6.1 and 6.2), which are sufficient to take the limit in the linear terms (see [1]), whereas for the nonlinear terms, we establish a new time-compactness criterium (Theorem 5.1), which enables us to get strong convergence of the horizontal velocities; see Lemma 6.3. This theorem states essentially that a small perturbation of an Lp-equicontinuous family still possesses a strong convergent subsequence. Let us emphasize that this seemingly technical refinement is by no means superfluous. Indeed, the usual compactness estimate fails: as (u 1,u  2, 2u 3) is not divergence free, even if it is easy from (2.7)–(2.9) to control ∂t(u 1,u  2, 2u 3) in some dual space of divergence free velocities, it is not possible to apply the Aubin–Lions lemma to get compactness. Another major difficulty of the proof is the lack of regularity of the vertical velocity, which is determined only by the incompressibility equation (2.10). Remark. It is possible to handle a general force (f1,f 2,f 3) in problem (2.14)– (2.20), by simply adding f=(f1,f 2,f3 ) to (2.1), in order to end up with (f1,f 2,f 3) in (2.7)–(2.9). 4. Weak formulation and anisotropic spaces. We need the following Hilbert spaces: H1 b(Ω) = C∞ b(Ω)H1(Ω) =v∈H1(Ω); v=0onΓ b (where C∞ b(Ω) = ϕ∈C∞(¯ Ω); ϕ= 0 in some neighborhood of Γb), V=v∈H1 b(Ω) ×H1 b(Ω) ×H1 0(Ω); div v=0inΩ , H(∂3,Ω) = v∈L2(Ω); ∂3v∈L2(Ω) (endowed with the norm v2 H(∂3,Ω) =v2 L2(Ω) +∂3v2 L2(Ω) ), H0(∂3,Ω) = C∞ 0(Ω)H(∂3,Ω) ={v∈H(∂3,Ω); vn 3=0on∂Ω} (n3is the third component of the normal exterior vector on ∂Ω, and vn 3is understood in the H−1/2(∂Ω) sense (see [19] for these spaces)), W=u∈H1 b(Ω) ×H1 b(Ω) ×H0(∂3,Ω); div u=0inΩ . Let us denote that uH=(u1,u 2), θH=(θ1,θ 2), b(uH)=α(−u2,u 1), and ∇ν= (ν1/2 1∂1,ν1/2 2∂2,ν1/2 3∂3).The scalar product in L2(Ω)d, or the duality Lp(Ω),L p(Ω), is denoted by (·,·), and the duality H−1/2(Γs)H1/2(Γs), is denoted by ·,·Γs. The weak form of the hydrostatic Navier–Stokes equations (2.14)–(2.20) is then as follows. Find u=(uH,u 3)∈L2(0,T;W), with uH∈L∞(0,T;L2(Ω)2), such that T 0 −(uH,∂ tvH)−(uH,(u·∇)vH)+(b(uH),v H)+(∇νuH,∇νvH) =−(u0H,v H(0)) + T 0 θH,v HΓs (4.1) for all v=(vH,v 3)∈H1(0,T;W), with vH(T) = 0 and ∂3vH∈L∞(0,T;L3(Ω)2). Downloaded 05/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 852 PASCAL AZ´ ERAD AND FRANCISCO GUILL´ EN Remark. Notice that a weak solution of the Navier–Stokes equations verifies the following regularity: u∈L2(0,T;V)∩L∞(0,T;L2(Ω)3) (cf. [8, 10, 18]). Now the lack of regularity of u3makes it necessary to change Vto W. Moreover, in general, u3∈ L∞(0,T;L2(Ω)). Remark. The regularity L∞(0,T;L3(Ω)2) is required for ∂3vHto give a meaning to T 0(uH,u 3∂3vH) dt. The regularity L2(0,T;L∞(Ω)2) or any interpolated one L2/a(0,T;L3/(1−a)(Ω)2) with 0 ≤a≤1 can also be considered. 5. Compactness by perturbation. We give a compactness criterium, new to our knowledge, which generalizes the well-known translation criterium of Riesz– Fr´echet–Kolmogorov, extended to the vectorial case by Simon [17]. In the following, τhf(t) denotes f(t+h). Theorem 5.1. Let T>0, and let the Banach spaces Xcompact $→B$→Y.Let (f)>0be a family of functions of Lp(0,T;X),1≤p≤∞, with the extra condition (f)>0⊂C(0,T;Y)if p=∞, such that (H1) (f)>0is bounded in Lp(0,T;X), (H2) τhf−fLp(0,T −h;Y)≤ϕ(h)+ψ()with limh→0ϕ(h)=0, lim→0ψ()=0. Then the family (f)>0possesses a cluster point in Lp(0,T;B)and also in C(0,T;B) if p=∞,as→0. Proof. It is enough to prove that, for every sequence (n)nsuch as n>0 and n→0, the family (fn)nis relatively compact in Lp(0,T;B). We apply Theorem 5 of Simon [17, p. 84] to the sequence (fn)n, while observing that hypothesis (H2) implies that τhfn−fnLp(0,T −h;Y)→0ash→0 uniformly with respect to n. Indeed, (H2) implies that ∀n, τhfn−fnLp(0,T −h;Y)≤ϕ(h)+ψ(n). Let >0 and then ∃N, such that for all n≥N,ψ(n)≤/2. On the other hand, ∃δ>0, such that for all h:0≤h<δ,ϕ(h)≤/2. Therefore, we get the estimate ∀n≥Nand ∀h:0≤h<δ, τhfn−fnLp(0,T −h;Y)≤. In addition, for each k≤N,∃δk>0, such that for all h:0≤h<δ k τhfk−fkLp(0,T −h;Y)≤. This follows from the Lp-continuity by translation of an Lpfunction for p<∞and for p=∞; this is precisely a hypothesis. Defining η= min{δ, δ1,...,δ N}, we obtain the desired uniform estimate ∀h:0≤h<η, τhfn−fnLp(0,T −h;Y)≤∀n. The family (fn)nfulfills the hypotheses of Simon’s theorem. Downloaded 05/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php JUSTIFICATION OF THE HYDROSTATIC APPROXIMATION 853 6. Proof of the main theorem. For simplicity in the notation, from now on, unless we specify otherwise, we will denote u=uas a weak solution of the anisotropic Navier–Stokes equations (2.7)–(2.13). 6.1. Energy estimates. The usual energy inequality (cf. [10]) for the Navier– Stokes equations gives, for a.e. t∈[0,T], uH(t)2 L2+2u3(t)2 L2+t 0 {∇νuH(τ)2 L2+2∇νu3(τ)2 L2}dτ ≤u0H2 L2+2u032 L2+t 0 θH,u HΓs. Hence we obtain as in the isotropic Navier–Stokes system (cf. [1]) the following proposition. Proposition 6.1. The sequences u1,u 2,u 3are bounded in L∞(0,T;L2(Ω)) ∩ L2(0,T;H1(Ω)). For the vertical velocities, we prove the following. Proposition 6.2. The sequences u3and ∂3u3are bounded in L2(0,T;L2(Ω)); i.e., u3is bounded in L2(0,T;H0(∂3,Ω)). Proof. As div u= 0, ∂3u3=−∂1u1−∂2u2is bounded in L2(0,T;L2(Ω)). Moreover, the Poincar´e inequality in the vertical direction, owing to u3=0onΓ s, yields u3L2≤hmax ∂3u3L2,where hmax = max ωh. Therefore, we have proved the proposition. 6.2. Fractional time derivatives in horizontal spaces. First, we define the auxiliary Hilbert spaces BH=PHU(L2)2 ,W H=PHU(H1)2 ,and YH=PHU(H2)2 , where U=ϕ∈C∞ b(Ω)2×C∞ 0(Ω); div ϕ=0  and PHis the projection PH:(x1,x 2,x 3)∈R3→ (x1,x 2)∈R2. Then, from the Sobolev–Rellich embeddings, one deduces easily that YH$→WH$→BH≡B H$→W H$→Y H,(6.1) where all are dense and compact embeddings. Here and henceforth, Xdenotes the dual space of X. Now, we have the following lemma. Lemma 6.3. The estimate τhuH−uHL∞(0,T −h;Y H)≤C(h1/4+)holds. Proof. The spatial weak form of the Navier–Stokes equation (2.7)–(2.13) is d dt(uH,v H)−(uH,(u·∇)vH)+(b(uH),v H)+(∇νuH,∇νvH) +2d dt(u3,v 3)+(u·∇u3,v 3)+(∇νu3,∇νv3) +(βu3,v 1)−(βu1,v 3)=θH,v HΓsin D(0,T) ∀v=(vH,v 3)∈V. (6.2) Downloaded 05/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 854 PASCAL AZ´ ERAD AND FRANCISCO GUILL´ EN Letting vH∈YH, there is a null divergence lifting v=(vH,v 3)∈H2 b(Ω)2×H1 0(∂3,Ω) such that v3H1+∂3v3H1≤CvHYH.(6.3) Here, the spaces H2 b(Ω) and H1 0(∂3,Ω) are the natural extensions of the spaces H1 b(Ω) and H0(∂3,Ω): H2 b(Ω) = C∞ b(Ω)H2(Ω) =v∈H2(Ω); v=∂v ∂n =0onΓ b, H1(∂3,Ω) = v∈H1(Ω); ∂3v∈H1Ω), H1 0(∂3,Ω) = C∞ 0(Ω)H1(∂3,Ω) =v∈H1(∂3,Ω); v=∂3v=0on∂Ω. Indeed, as vH∈YH, there exists a sequence (ϕn H,ϕ n 3)∈Usuch that ϕn H→vHin H2(Ω)2.Then ∂3ϕn 3=−∂1ϕn 1−∂2ϕn 2is a Cauchy sequence in H1(Ω), and by vertical Poincar´e inequality, ϕn 3is also a Cauchy sequence in H1(Ω).Therefore, ϕn 3, being a Cauchy sequence in H1(∂3,Ω),converges to a function v3, which provides the desired lifting function. The continuous dependence (6.3) results from the above construction. Now we take this v=(vH,v 3) as a test function in (6.2) and integrate over (t, t +h); i.e., (τhuH(t)−uH(t),v H)+2(τhu3(t)−u3(t),v 3)=t+h t g(s)ds,(6.4) where g(s)=(uH,(u·∇)vH)−2(u·∇u3,v 3)−(b(uH),v H)−(∇νuH,∇νvH) −(∇ν(u3),∇νv3)−{(βu3,v 1)−(βu1,v 3)}+θH,v HΓs. Now we prove that gL4/3(0,T )≤CvHYH.(6.5) To this end, we estimate every piece of g. For the nonlinear terms, we have (uH,(u·∇)vH)≤uHL3uL2∇vHL6≤CuHL3uL2vHYH and 2(u·∇u3,v 3)≤uL3∇(u3)L2v3L6≤CuL3u3H1vHYH. By interpolation between L∞(0,T;L2) and L2(0,T;L6), uHis bounded in L4(0,T;L3); i.e., uHL3is bounded in L4(0,T). As uL2is bounded in L2(0,T), we have (uH,(u·∇)vH) bounded in L4/3(0,T). Similarly, as uL3is bounded in L4(0,T) and u3H1is bounded in L2(0,T), we have 2(u·∇u3,v 3) bounded in L4/3(0,T). The linear terms of gare handled easily by the Cauchy–Schwarz inequality: (b(uH),v H)≤uHL2vHL2bounded in L∞(0,T), (∇νuH,∇νvH)≤uHH1vHH1bounded in L2(0,T), (∇ν(u3),∇νv3)≤u3H1v3H1bounded in L2(0,T), β {(u3,v 1)−(u1,v 3)}≤2fuL2vL2bounded in L∞(0,T), θH,v HΓs≤CθHH−1/2(Γs)vHH1bounded in L2(0,T). Downloaded 05/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php JUSTIFICATION OF THE HYDROSTATIC APPROXIMATION 855 Therefore, taking into account (6.3), according to all previous bounds, (6.5) holds. Next, applying the H¨older inequality to (6.5), we see that t+h t |g(s)|ds ≤Ch 1/4vHYH. On the other hand, |2(τhu3(t)−u3(t),v 3)|≤{τh(u3)(t)L2+u3(t)L2}v3L2≤CvHYH by virtue of Proposition 6.1. These last two estimates together with (6.4) yield the required result. 6.3. Convergence. Here we come back to the notation u. The space-time weak form of the anisotropic Navier–Stokes equations (2.7)–(2.13) is as follows. Find u=(u H,u  3)∈L2(0,T;V)∩L∞(0,T;L2(Ω)3) such that T 0 −(u H,∂ tvH)−(u H,(u·∇)vH)+(b(u H),v H)+(∇νu H,∇νvH) +2T 0 −(u 3,∂ tv3)+(u·∇u 3,v 3)+(∇νu 3,∇νv3) +βT 0 (u 3,v 1)−(u 1,v 3)=−(u0H,v H(0)) −2(u03,v 3(0)) + T 0 θH,v HΓs ∀v=(vH,v 3)∈H1(0,T;V),with v(T)=0. (6.6) The purpose of the following is to take the limit as →0 in (6.6) to come to (4.1). By Propositions 6.1 and 6.2, it follows that uis bounded in L2(0,T;W) and u His bounded in L∞(0,T;BH), allowing us to extract a subsequence, still denoted by u, such that u=(u H,u  3)2u=(uH,u 3)inL2(0,T;W) weak, u H  2u Hin L∞(0,T;BH) weak −3. These weak convergences are enough to take the limit in the linear terms of (6.6) (cf. [1]). In particular, the terms of O() associated with the Coriolis acceleration vanish as tends to zero. Indeed, β T 0 (u 3,v 1)−(u 1,v 3)≤2fT 0 uL2vL2 ≤2fuL2(0,T ;L2)vL2(0,T ;L2)≤CvL2(0,T ;L2)≤C . On the other hand, combining (6.1), Proposition 6.1, and Lemma 6.3, we can apply Theorem 5.1 for p=∞and the spaces BH compact $→W H$→Y H. Therefore, there exists a subsequence u H→uHin C(0,T;W H) strong. Thus we get the weak time-continuity uH∈C(0,T;W H), so that the initial condition (2.20) makes sense for the horizontal velocities. On the other hand, the term of 0(2) related to the initial condition for the vertical velocity vanishes as tends to zero. Indeed, −2(u03,v 3(0)) ≤2u03L2v3(0)L2≤2u0L2v3C(0,T ;L2)≤C 2.(6.7) Downloaded 05/16/16 to 150.214.182.169. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php