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 Socie y o Indus ial and Applied Ma hema ics Vol. 33, No. 4, pp. 847–859 Abs ac . Geophysical fluids all exhibi a common ea u e: hei aspec a io (dep h o ho i- zon al wid h) is e y small. This leads o an asymp o ic model widely used in me eo ology, oceanog- aphy, and limnology, namely he hyd os a ic app oxima ion o he ime-dependen incomp essible Na ie –S okes equa ions. I elies on he hypo hesis ha p essu e inc eases linea ly in he e ical di ec ion. In he ollowing, we p o e a con e gence and exis ence heo em o his model by means o aniso opic es ima es and a new ime-compac ness c i e ium. Key wo ds. Na ie –S okes equa ions, shallow domains, geophysical fluid dynamics, hyd os a ic app oxima ion, singula pe u ba ion, compac ness c i e ium, asymp o ic analysis AMS subjec classifica ions. 35Q30, 35B40, 76D05, 34C35 PII. S0036141000375962 1. In oduc ion. A mosphe ic flow in me eo ology, wa e flow in oceanog aphy, and limnology a e all desc ibed by he Na ie –S okes equa ions. Due o he ac ha he aspec a io =cha ac e is ic dep h cha ac e is ic wid h is e y small in mos geophysical domains, asymp o ic models ha e been used; see, e.g., [9, 15, 22]. One such model is he p imi i e equa ions model; see, e.g., [11, 12], whe ein he unknown flow a iables a e eloci y, p essu e, empe a u e, and salini y (in he case o an ocean). Besides, mos geophysical fluids a e s a ified (i.e., densi y is a known unc ion o he empe a u e (and salini y, i any)) and ha e a ee su ace. We shall no in es iga e hese ea u es in his pape , lea ing i , a he , o o hcoming wo k. Ins ead we shall ocus on he assump ion ha he p essu e is hyd os a ic, i.e., inc eases linea ly wi h espec o he dep h, as in he s a ic case. This law ag ees well wi h expe imen (as fi s obse ed by Blaise Pascal a ound 1650; see [14])) and is equen ly aken as a hypo hesis in geophysical fluid dynamics. We jus i y his assump ion by means o asymp o ic analysis ( aking as he small pa ame e ). Ou de i a ion is made possible by he use o aniso opic eddy iscosi ies, namely ν= (νx,ν y,ν z), elying on he ac ha he a io be ween he ho izon al and e ical scales leads o e y diffe en sizes o he ho izon al and e ical eddies (see [9, 15]). Specifically, i we assume ha ν=(ν1,ν 2, 2ν3) wi h νi= O(1) o i=1,2,3, hen we will see ha weak solu ions o he Na ie –S okes equa ions con e ge o a weak solu ion o a limi p oblem wi h hyd os a ic p essu e. ∗Recei ed by he edi o s July 26, 2000; accep ed o publica ion (in e ised o m) Augus 22, 2001; published elec onically Decembe 18, 2001. This wo k was suppo ed by he onds F anco-Espagnol D.R.E.I.F. ( e . UC 815) and he p ojec MAR98-0486 C.I.C.Y.T. (Espa˜na). h p://www.siam.o g/jou nals/sima/33-4/37596.h ml †Labo a oi e de Mod´elisa ion, Analyse Non Lin´eai e e Op imisa ion, Uni e si ´e de Pe pignan, 52 a . de Villeneu e, F-66860 Pe pignan cedex, F ance (aze ad@uni -pe p. ). ‡Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico, c/ a fia s/n Uni e sidad de Se illa, 41012 Se illa, Spain (guillen@nume .us.es). 847 Downloaded 05/16/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php 848 PASCAL AZ´ ERAD AND FRANCISCO GUILL´ EN The s a iona y case has al eady been s udied (see [4] o he linea p oblem and [5] o he nonlinea one), whe eas he linea ime-dependen case was sol ed in [1]. The main ask o his pape is hen o sol e he nonlinea ime-dependen case. Ou esul was announced in [2], whe eas nume ical simula ions s emming om i we e discussed in [3]. Fluid flow in hin domains (fla , cu ed, and wi h a ious bounda y condi ions) has been ex ensi ely s udied; see [7, 13, 16, 20, 21]. In hese wo ks, an iso opic iscosi y is used, and he dep h is cons an . By a e aging along he e ical di ec ion, wo-dimensional (2D) limi models a e ob ained, oge he wi h exis ence and global egula i y esul s. Ou app oach is diffe en , because we nei he elimina e he e ical eloci y by a e aging no assume he dep h o he domain o be cons an . By making use o diffe - en ho izon al and e ical eddy iscosi ies, we a e able o de i e a h ee-dimensional (3D) limi nonlinea model. Le us emphasize ha he aniso opic iscosi y hypo h- esis is undamen al o he de i a ion o he p imi i e equa ions: in he s a iona y case, keeping an iso opic iscosi y, he asymp o ic model is linea , wi h anishing ho izon al diffusion; see [6]. The pape is o ganized as ollows. In sec ion 2, we p esen he physical model and he scaling leading o he p imi i e equa ions. We s a e he main heo em in sec ion 3. The unc ional se ing and weak o mula ion a e desc ibed in sec ion 4. In he nex sec ion, we s a e and p o e a ime-compac ness esul , which we shall use in he p oo o he main heo em in sec ion 6. Finally, in sec ion 7, we commen on he con e gence o he p essu e and he o de s o magni ude o he e ical eloci y wi h espec o he aspec a io. 2. Equa ions go e ning he flow and scaling. Le us conside an incom- p essible homogeneous fluid filling a hin domain defined by Ω=(x, y, z)∈R3;(x, y)∈ω,−h(x, y)<z<0, whe e ωis an open bounded Lipschi z domain in R2and h:ω→Ris a nonnega- i e lipschi zian applica ion, which is a bi a y p o ided ha Ωis lipschi zian. In pa icula , hmay anish, con a y o [12, 9], bu in o de ha he domain Ωhas no cusps, he slope mus no anish on he sho es.1We deno e by Γs=ω×{0} he fluid su ace and by Γ b=∂Ω Γs he basin bo om. The fluid flow in Ωis gene a ed by he wind ac ion on he su ace Γs, influenced by he Co iolis and cen i ugal o ces and go e ned by he Na ie –S okes equa ions, in which we ake diffe en eddy iscosi ies acco ding o he di ec ion; see [5, 9, 15]. Finally, we ake he densi y as iden ically equal o one. In a geophysical o a ing ame (zpoin ing upwa ds, xeas , and yno h), he ini ial-bounda y alue p oblem eads as ollows. Find =( 1, 2, 3) ( eloci y) and q(p essu e), such ha ∂ +( ·∇) −∆ν +∇q+2w× =gin Ω×(0,T),(2.1) di =0 inΩ ×(0,T),(2.2) =0 onΓ  b×(0,T),(2.3) νz∂z 1=τ1,ν z∂z 2=τ2, 3=0 onΓ s×(0,T),(2.4) (·, = 0) = 0in Ω.(2.5) 1This is a echnical hypo hesis. One could p obably dispense wi h i due o he specific shape o he domain. Downloaded 05/16/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php JUSTIFICATION OF THE HYDROSTATIC APPROXIMATION 849 In (2.1), ∇=(∂x,∂ y,∂ z) deno es he g adien ec o , and ∆νdeno es he aniso- opic Laplacian defined by ∆ν=νx∂2 xx +νy∂2 yy +νz∂2 zz wi h ν=(νx,ν y,ν z) he eddy kinema ic iscosi y ec o . Mo eo e , w= (0,cos(l(y)),sin(l(y))) ep esen s he ea h o a ion angula speed ( he module and l(y) he la i ude), 2w× ep esen s he Co iolis accele a ion (×deno es he c oss-p oduc in R3), and g ep esen s he o ce due o g a i y (which also includes he cen i ugal effec ). I is well known (c . [15, p. 18]) ha gis a po en ial, i.e., g=∇ϕ. I is cus oma y o inco po a e he g a i y po en ial in he p essu e e m; hus we se p=q−ϕ. Equa ion (2.2) ep esen s he incomp essibili y condi ion, and (2.3) ep esen s he no-slip condi ion on he bo om. In (2.4), τi,i=1,2, s and o he ho izon al ac ions exe ed by he wind on he (fixed) su ace Γso he fluid, and w=0onΓ scomes om he igid lid hypo hesis. In (2.5), 0=( 01, 02, 03) designa es he ini ial eloci y. Rema k. We ha e neglec ed he ea h’s cu a u e, and hence ou analysis is alid only locally, e.g., o lakes; o seas o oceans, sphe ical coo dina es should be used [12], al hough his can be somewha cumbe some. As usual in asymp o ic analysis, we pe o m a e ical scaling o make he domain independen o , ha is, x=x1,y=x2,z=x 3, so ha Ω = (x1,x 2,x 3)∈R3;(x1,x 2)∈ω, −h(x1,x 2)<x 3<0is he new fixed domain. The co esponding kinema ic scaling is 1=u 1, 2=u 2, 3=u  3,p=p,(2.6) so ha u=(u 1,u  2,u  3) is he new unknown eloci y and pis he new p essu e. I is necessa y o scale he mechanical quan i ies acco dingly. Fi s , i is only na u al o assume 01 =u01, 02 =u02, and 03 =u03, whe e u0idoes no depend on ,i=1,2,3. Nex we assume νx=ν1,νy=ν2, and νz=2·ν3, whe e ν1,ν 2,ν 3 a e cons an s. As men ioned in he in oduc ion, in oceanog aphy he e ical eddy iscosi y is usually e y small compa ed o he ho izon al one. We e e o [5] o a ma hema ical discussion o his assump ion, and he e we con en ou sel es wi h one heu is ic commen . Basically, a kinema ic iscosi y has he dimension L2/T, whe e L( esp., T) is a ypical leng h ( esp., ime) scale so ha νxand νyha e he dimension L2 H/T, whe eas νzhas he dimension L2 V/T, whe e LH( esp., LV) deno es a ypical ho izon al ( esp., e ical) leng h scale. I ollows ha he a io νz/νxand νz/νy= O(2).2 Now (2.4) becomes ν3∂3u i=τ i/, i =1,2. We see ha in o de o end up wi h an O(1)-wind o ce on he escaled domain, we ha e o assume ha τ i=·θi,i=1,2, whe e he θia e unc ions independen o . 2We do no delude ou sel es wi h his ske chy a gumen . As a as we know, up o now he e has been no igo ous de i a ion o any eddy iscosi y model. Downloaded 05/16/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php 850 PASCAL AZ´ ERAD AND FRANCISCO GUILL´ EN Rema k. This las assump ion can also be mo i a ed by dimensional analysis, as ollows. F om τi=νz∂z i, one de i es ha τihas he dimension o L2 V T·1 LV ·LH T=L2 H T2=O(). Wi h he abo e conside a ions, p oblem (2.1)–(2.5) ans o ms in o he ollowing aniso opic Na ie –S okes equa ions: ∂ u 1+u·∇u 1−∆νu 1−αu  2+βu  3+∂1p=0 inΩ×(0,T),(2.7) ∂ u 2+u·∇u 2−∆νu 2+αu  1+∂2p=0 inΩ×(0,T),(2.8) 2{∂ u 3+u·∇u 3−∆νu 3}−βu  1+∂3p=0 inΩ×(0,T),(2.9) di 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(·, = 0) = u0in Ω.(2.13) Now ∇=(∂1,∂ 2,∂ 3), ∆ν=ν1∂2 11 +ν2∂2 22 +ν3∂2 33,Γ b=∂Ω Γs,α=2 sin(l(x2)), and β=2 cos(l(x2)). I we assume ha u=O(1), hen neglec ing he 2and  e ms in he fi s and hi d momen um equa ion, (2.7) and (2.9), we o mally ge he hyd os a ic Na ie – S okes equa ions, also called he p imi i e equa ions: ∂ u1+u·∇u1−∆νu1−αu 2+∂1p=0 inΩ×(0,T),(2.14) ∂ u2+u·∇u2−∆νu2+αu 1+∂2p=0 inΩ×(0,T),(2.15) ∂3p=0 inΩ×(0,T),(2.16) di 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(·, = 0) = u0iin Ω,i=1,2.(2.20) Rema k. The bounda y condi ion (2.18) diffe s om i s coun e pa (2.11) be- cause u3is less egula han u1,u 2as we shall see below. Also, he ini ial condi ion (2.20) does no in ol e u3, he ime de i a i e o which is missing in he hyd os a ic model. The p oblem is no in he Cauchy–Kowale ska o m.3 Rema k. I u3we e o be compu ed di ec ly om (2.17), which is a fi s o de equa ion, i is no ob ious a all ha i would ulfill he wo bounda y condi ions on he bo om (2.18) and he su ace (2.19). 3. Main heo em. Le Tbe a fixed posi i e du a ion. We make he na u al assump ion o a wind o fini e ene gy: θ1,θ 2∈L2(0,T;H−1/2(Γs)).Ou main esul is he ollowing heo em. Theo em 3.1. Le u0∈L2(Ω)3, wi h di u0=0,u0·n=0on ∂Ω, and θ1,θ 2∈ L2(0,T;H−1/2(Γs)); he e exis s a weak solu ion uo he hyd os a ic Na ie –S okes equa ions (2.14)–(2.20), ob ained as a limi o weak solu ions uo he aniso opic Na ie –S okes equa ions (2.7)–(2.13), as he aspec a io  ends o ze o. 3Me eo ologis s say ha u3is no longe a p ognos ic a iable (see [11, 12]). Downloaded 05/16/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php JUSTIFICATION OF THE HYDROSTATIC APPROXIMATION 851 The p oo elies on a p io i es ima es in aniso opic spaces (P oposi ions 6.1 and 6.2), which a e sufficien o ake he limi in he linea e ms (see [1]), whe eas o he nonlinea e ms, we es ablish a new ime-compac ness c i e ium (Theo em 5.1), which enables us o ge s ong con e gence o he ho izon al eloci ies; see Lemma 6.3. This heo em s a es essen ially ha a small pe u ba ion o an Lp-equicon inuous amily s ill possesses a s ong con e gen subsequence. Le us emphasize ha his seemingly echnical efinemen is by no means supe fluous. Indeed, he usual compac ness es i- ma e ails: as (u 1,u  2, 2u 3) is no di e gence ee, e en i i is easy om (2.7)–(2.9) o con ol ∂ (u 1,u  2, 2u 3) in some dual space o di e gence ee eloci ies, i is no possible o apply he Aubin–Lions lemma o ge compac ness. Ano he majo difficul y o he p oo is he lack o egula i y o he e ical e- loci y, which is de e mined only by he incomp essibili y equa ion (2.10). Rema k. I is possible o handle a gene al o ce ( 1, 2, 3) in p oblem (2.14)– (2.20), by simply adding =( 1, 2, 3 ) o (2.1), in o de o end up wi h ( 1, 2, 3) in (2.7)–(2.9). 4. Weak o mula ion and aniso opic spaces. We need he ollowing Hilbe spaces: H1 b(Ω) = C∞ b(Ω)H1(Ω) = ∈H1(Ω); =0onΓ b (whe e C∞ b(Ω) = ϕ∈C∞(¯ Ω); ϕ= 0 in some neighbo hood o Γb), V= ∈H1 b(Ω) ×H1 b(Ω) ×H1 0(Ω); di =0inΩ , H(∂3,Ω) =  ∈L2(Ω); ∂3 ∈L2(Ω) (endowed wi h he no m  2 H(∂3,Ω) = 2 L2(Ω) +∂3 2 L2(Ω) ), H0(∂3,Ω) = C∞ 0(Ω)H(∂3,Ω) ={ ∈H(∂3,Ω); n 3=0on∂Ω} (n3is he hi d componen o he no mal ex e io ec o on ∂Ω, and n 3is unde s ood in he H−1/2(∂Ω) sense (see [19] o hese spaces)), W=u∈H1 b(Ω) ×H1 b(Ω) ×H0(∂3,Ω); di u=0inΩ . Le us deno e ha 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 scala p oduc in L2(Ω)d, o he duali y Lp(Ω),L p(Ω), is deno ed by (·,·), and he duali y H−1/2(Γs)H1/2(Γs), is deno ed by ·,·Γs. The weak o m o he hyd os a ic Na ie –S okes equa ions (2.14)–(2.20) is hen as ollows. Find u=(uH,u 3)∈L2(0,T;W), wi h uH∈L∞(0,T;L2(Ω)2), such ha T 0 −(uH,∂ H)−(uH,(u·∇) H)+(b(uH), H)+(∇νuH,∇ν H) =−(u0H, H(0)) + T 0 θH, HΓs (4.1) o all =( H, 3)∈H1(0,T;W), wi h H(T) = 0 and ∂3 H∈L∞(0,T;L3(Ω)2). Downloaded 05/16/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php 852 PASCAL AZ´ ERAD AND FRANCISCO GUILL´ EN Rema k. No ice ha a weak solu ion o he Na ie –S okes equa ions e ifies he ollowing egula i y: u∈L2(0,T;V)∩L∞(0,T;L2(Ω)3) (c . [8, 10, 18]). Now he lack o egula i y o u3makes i necessa y o change V o W. Mo eo e , in gene al, u3∈ L∞(0,T;L2(Ω)). Rema k. The egula i y L∞(0,T;L3(Ω)2) is equi ed o ∂3 H o gi e a mean- ing o T 0(uH,u 3∂3 H) d . The egula i y L2(0,T;L∞(Ω)2) o any in e pola ed one L2/a(0,T;L3/(1−a)(Ω)2) wi h 0 ≤a≤1 can also be conside ed. 5. Compac ness by pe u ba ion. We gi e a compac ness c i e ium, new o ou knowledge, which gene alizes he well-known ansla ion c i e ium o Riesz– F ´eche –Kolmogo o , ex ended o he ec o ial case by Simon [17]. In he ollowing, τh ( ) deno es ( +h). Theo em 5.1. Le T>0, and le he Banach spaces Xcompac $→B$→Y.Le ( )>0be a amily o unc ions o Lp(0,T;X),1≤p≤∞, wi h he ex a condi ion ( )>0⊂C(0,T;Y)i p=∞, such ha (H1) ( )>0is bounded in Lp(0,T;X), (H2) τh − Lp(0,T −h;Y)≤ϕ(h)+ψ()wi h limh→0ϕ(h)=0, lim→0ψ()=0. Then he amily ( )>0possesses a clus e poin in Lp(0,T;B)and also in C(0,T;B) i p=∞,as→0. P oo . I is enough o p o e ha , o e e y sequence (n)nsuch as n>0 and n→0, he amily ( n)nis ela i ely compac in Lp(0,T;B). We apply Theo em 5 o Simon [17, p. 84] o he sequence ( n)n, while obse ing ha hypo hesis (H2) implies ha τh n− nLp(0,T −h;Y)→0ash→0 uni o mly wi h espec o n. Indeed, (H2) implies ha ∀n, τh n− nLp(0,T −h;Y)≤ϕ(h)+ψ(n). Le >0 and hen ∃N, such ha o all n≥N,ψ(n)≤/2. On he o he hand, ∃δ>0, such ha o all h:0≤h<δ,ϕ(h)≤/2. The e o e, we ge he es ima e ∀n≥Nand ∀h:0≤h<δ, τh n− nLp(0,T −h;Y)≤. In addi ion, o each k≤N,∃δk>0, such ha o all h:0≤h<δ k τh k− kLp(0,T −h;Y)≤. This ollows om he Lp-con inui y by ansla ion o an Lp unc ion o p<∞and o p=∞; his is p ecisely a hypo hesis. Defining η= min{δ, δ1,...,δ N}, we ob ain he desi ed uni o m es ima e ∀h:0≤h<η, τh n− nLp(0,T −h;Y)≤∀n. The amily ( n)n ulfills he hypo heses o Simon’s heo em. Downloaded 05/16/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php JUSTIFICATION OF THE HYDROSTATIC APPROXIMATION 853 6. P oo o he main heo em. Fo simplici y in he no a ion, om now on, unless we speci y o he wise, we will deno e u=uas a weak solu ion o he aniso opic Na ie –S okes equa ions (2.7)–(2.13). 6.1. Ene gy es ima es. The usual ene gy inequali y (c . [10]) o he Na ie – S okes equa ions gi es, o a.e. ∈[0,T], uH( )2 L2+2u3( )2 L2+ 0 {∇νuH(τ)2 L2+2∇νu3(τ)2 L2}dτ ≤u0H2 L2+2u032 L2+ 0 θH,u HΓs. Hence we ob ain as in he iso opic Na ie –S okes sys em (c . [1]) he ollowing p opo- si ion. P oposi ion 6.1. The sequences u1,u 2,u 3a e bounded in L∞(0,T;L2(Ω)) ∩ L2(0,T;H1(Ω)). Fo he e ical eloci ies, we p o e he ollowing. P oposi ion 6.2. The sequences u3and ∂3u3a e bounded in L2(0,T;L2(Ω)); i.e., u3is bounded in L2(0,T;H0(∂3,Ω)). P oo . As di u= 0, ∂3u3=−∂1u1−∂2u2is bounded in L2(0,T;L2(Ω)). Mo e- o e , he Poinca ´e inequali y in he e ical di ec ion, owing o u3=0onΓ s, yields u3L2≤hmax ∂3u3L2,whe e hmax = max ωh. The e o e, we ha e p o ed he p oposi ion. 6.2. F ac ional ime de i a i es in ho izon al spaces. Fi s , we define he auxilia y Hilbe spaces BH=PHU(L2)2 ,W H=PHU(H1)2 ,and YH=PHU(H2)2 , whe e U=ϕ∈C∞ b(Ω)2×C∞ 0(Ω); di ϕ=0  and PHis he p ojec ion PH:(x1,x 2,x 3)∈R3→ (x1,x 2)∈R2. Then, om he Sobole –Rellich embeddings, one deduces easily ha YH$→WH$→BH≡B H$→W H$→Y H,(6.1) whe e all a e dense and compac embeddings. He e and hence o h, Xdeno es he dual space o X. Now, we ha e he ollowing lemma. Lemma 6.3. The es ima e τhuH−uHL∞(0,T −h;Y H)≤C(h1/4+)holds. P oo . The spa ial weak o m o he Na ie –S okes equa ion (2.7)–(2.13) is d d (uH, H)−(uH,(u·∇) H)+(b(uH), H)+(∇νuH,∇ν H) +2d d (u3, 3)+(u·∇u3, 3)+(∇νu3,∇ν 3) +(βu3, 1)−(βu1, 3)=θH, HΓsin D(0,T) ∀ =( H, 3)∈V. (6.2) Downloaded 05/16/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php 854 PASCAL AZ´ ERAD AND FRANCISCO GUILL´ EN Le ing H∈YH, he e is a null di e gence li ing =( H, 3)∈H2 b(Ω)2×H1 0(∂3,Ω) such ha  3H1+∂3 3H1≤C HYH.(6.3) He e, he spaces H2 b(Ω) and H1 0(∂3,Ω) a e he na u al ex ensions o he spaces H1 b(Ω) and H0(∂3,Ω): H2 b(Ω) = C∞ b(Ω)H2(Ω) = ∈H2(Ω); =∂ ∂n =0onΓ b, H1(∂3,Ω) =  ∈H1(Ω); ∂3 ∈H1Ω), H1 0(∂3,Ω) = C∞ 0(Ω)H1(∂3,Ω) = ∈H1(∂3,Ω); =∂3 =0on∂Ω. Indeed, as H∈YH, he e exis s a sequence (ϕn H,ϕ n 3)∈Usuch ha ϕn H→ Hin H2(Ω)2.Then ∂3ϕn 3=−∂1ϕn 1−∂2ϕn 2is a Cauchy sequence in H1(Ω), and by e ical Poinca ´e inequali y, ϕn 3is also a Cauchy sequence in H1(Ω).The e o e, ϕn 3, being a Cauchy sequence in H1(∂3,Ω),con e ges o a unc ion 3, which p o ides he desi ed li ing unc ion. The con inuous dependence (6.3) esul s om he abo e cons uc ion. Now we ake his =( H, 3) as a es unc ion in (6.2) and in eg a e o e ( , +h); i.e., (τhuH( )−uH( ), H)+2(τhu3( )−u3( ), 3)= +h g(s)ds,(6.4) whe e g(s)=(uH,(u·∇) H)−2(u·∇u3, 3)−(b(uH), H)−(∇νuH,∇ν H) −(∇ν(u3),∇ν 3)−{(βu3, 1)−(βu1, 3)}+θH, HΓs. Now we p o e ha gL4/3(0,T )≤C HYH.(6.5) To his end, we es ima e e e y piece o g. Fo he nonlinea e ms, we ha e (uH,(u·∇) H)≤uHL3uL2∇ HL6≤CuHL3uL2 HYH and 2(u·∇u3, 3)≤uL3∇(u3)L2 3L6≤CuL3u3H1 HYH. By in e pola ion be ween 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 ha e (uH,(u·∇) H) bounded in L4/3(0,T). Simila ly, as uL3is bounded in L4(0,T) and u3H1is bounded in L2(0,T), we ha e 2(u·∇u3, 3) bounded in L4/3(0,T). The linea e ms o ga e handled easily by he Cauchy–Schwa z inequali y: (b(uH), H)≤uHL2 HL2bounded in L∞(0,T), (∇νuH,∇ν H)≤uHH1 HH1bounded in L2(0,T), (∇ν(u3),∇ν 3)≤u3H1 3H1bounded in L2(0,T), β {(u3, 1)−(u1, 3)}≤2 uL2 L2bounded in L∞(0,T), θH, HΓs≤CθHH−1/2(Γs) HH1bounded in L2(0,T). Downloaded 05/16/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php JUSTIFICATION OF THE HYDROSTATIC APPROXIMATION 855 The e o e, aking in o accoun (6.3), acco ding o all p e ious bounds, (6.5) holds. Nex , applying he H¨olde inequali y o (6.5), we see ha  +h |g(s)|ds ≤Ch 1/4 HYH. On he o he hand, |2(τhu3( )−u3( ), 3)|≤{τh(u3)( )L2+u3( )L2} 3L2≤C HYH by i ue o P oposi ion 6.1. These las wo es ima es oge he wi h (6.4) yield he equi ed esul . 6.3. Con e gence. He e we come back o he no a ion u. The space- ime weak o m o he aniso opic Na ie –S okes equa ions (2.7)–(2.13) is as ollows. Find u=(u H,u  3)∈L2(0,T;V)∩L∞(0,T;L2(Ω)3) such ha T 0 −(u H,∂ H)−(u H,(u·∇) H)+(b(u H), H)+(∇νu H,∇ν H) +2T 0 −(u 3,∂ 3)+(u·∇u 3, 3)+(∇νu 3,∇ν 3) +βT 0 (u 3, 1)−(u 1, 3)=−(u0H, H(0)) −2(u03, 3(0)) + T 0 θH, HΓs ∀ =( H, 3)∈H1(0,T;V),wi h (T)=0. (6.6) The pu pose o he ollowing is o ake he limi as →0 in (6.6) o come o (4.1). By P oposi ions 6.1 and 6.2, i ollows ha uis bounded in L2(0,T;W) and u His bounded in L∞(0,T;BH), allowing us o ex ac a subsequence, s ill deno ed by u, such ha 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 con e gences a e enough o ake he limi in he linea e ms o (6.6) (c . [1]). In pa icula , he e ms o O() associa ed wi h he Co iolis accele a ion anish as  ends o ze o. Indeed, β T 0 (u 3, 1)−(u 1, 3)≤2 T 0 uL2 L2 ≤2 uL2(0,T ;L2) L2(0,T ;L2)≤C L2(0,T ;L2)≤C . On he o he hand, combining (6.1), P oposi ion 6.1, and Lemma 6.3, we can apply Theo em 5.1 o p=∞and he spaces BH compac $→W H$→Y H. The e o e, he e exis s a subsequence u H→uHin C(0,T;W H) s ong. Thus we ge he weak ime-con inui y uH∈C(0,T;W H), so ha he ini ial condi ion (2.20) makes sense o he ho izon al eloci ies. On he o he hand, he e m o 0(2) ela ed o he ini ial condi ion o he e ical eloci y anishes as  ends o ze o. Indeed, −2(u03, 3(0)) ≤2u03L2 3(0)L2≤2u0L2 3C(0,T ;L2)≤C 2.(6.7) Downloaded 05/16/16 o 150.214.182.169. Redis ibu ion subjec o SIAM license o copy igh ; see h p://www.siam.o g/jou nals/ojsa.php