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
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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.
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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.
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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]).
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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).
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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.
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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)
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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).
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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)
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