on the equations of the large-scale ocean*

Nonlinearity 5 (1992) 1007-1053. Printed in the UK

On the equations of the large-scale ocean*

Jacques-Louis Lionst, Roger Temam$§ and Shouhong Wan& T College de France and Centre National d ' h d e r Spatiales, 3 Rue d'Ulm, 75005 Paris, France Z Laboratoire <Analyse Numerique, Universiti Paris-Sud, Bitiment 425, 91405 Orsay, France 5 The Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405, USA

Received 22 May I992 Accepted by J D Gibbon

Abtract. As a preliminary step towards understanding the dynamics of the ocean and the impact of the ocean on the global climate system and weather prediction, we study in this article the mathematical formulations and a t t rac t~rs of three systems of equations of the ocean, i.e. the primitivc equations (the PES), the primitive equations with vertical viscosity (the PEV's), and the Boussinesq equations (the BEs), of the ocean.

These equations are fundamental equations of the ocean. The BEs are obtained from the general equations of a compressible fluid under the Boussinesq approximation. i.e. the density differences are neglected in the system except in the buoyancy term and in the equation of state. The PES are derived from the BEs under the hydrostatic approximation for the vertical momentum equation. The PEV's are the PES with the viscosity for the vertical velocity retained. This retention is partially based on the important role played by the viscosity in studying the long time behaviour of the ocean. and the earths climate.

From the mathematical point of view, by integrating the diagnostic equations we present two new formulations of the PES and the PEV's. Then we establish some mathematical settings far all the three systems, and obtain the existence and time-analyticity of solutions to the equations. Then we establish some physically relevant estimates for the Hausdorff and fractal dimensions of the a t t rac t~rs of the problems.

From the modelling paint of view, the PEV2s are introduced in this paper for the first time. Even though the PES have been studied for a period of time, the new formulation of the PES is introduced here for the first time. These new formulations of both the PES and the PEV's play a crucial role not only for the mathematical studies in this paper, but for the further numerical analysis, which will be developed elsewhere.

AMS classification scheme numbers: 35M10, 35420, 35476, 86A10

Introduction

The primary purpose of this paper is to establish a mathematical and physical foundation behind some three-dimensional circulation models of the ocean, and as a step towards understanding the impact of the ocean on the global climate system, in which the ocean is one of the principal components.

* This paper is part of a series of articles on the mathematical theory and numerical analysis of equations of climatology which includes [I, 3, 41.

095l-7715/92/05l007+47$04.50 0 1992 IOP Publishing Ltd and LMS Publishing Ltd 1007

1008 J - L Lions et ai

Generally speaking, it is well-accepted that the ocean is a slightly compressible fluid under Coriolis force. The basic quantities describing the motion and states of the ocean are the velocity field, the temperature, the salinity, the pressure and the density of the seawater. The full set of equations governing the behaviour of these quantities are the general equations of a compressible fluid under Coriolis force, consisting of the momentum equation, the continuity equation, the thermodynamical equation, the equation of state and the equation of diffusion of the salinity. These general equations of the ocean, however, are extremely complicated from both the mathematical and the computational points of view.

It is well-known that in any theoretical and practical investigation of a real physical system, we are always forced to make simplifying assumptions concerning the true nature of the system. Simplified systems were considered in [21, 24-25], etc. The approach here will be different.

For the ocean, one of the salient features is that the differences in density are significant only in the term of buoyancy and in the equation of state. Therefore, as a starting point, we can make the so-called Boussinesq approximation, i.e. the density differences are neglected in the system except in the buoyancy term and in the equation of state. The resulting system is called the Boussinesq equations of the ocean (BEs) . Due to the extremely high accuracy of the approximation, these equations are considered as the basic equations in oceanography. From the computational point of view, however, the BEs are still not suitable, because they are fully nonlinear and three-dimensional.

Fortunately, another important feature of the ocean is that the horizontal scale is much larger than the vertical one, i.e. the ocean composes a very thin layer of water. The scale analysis ensures that the dominant forces in the vertical momentum equation come from the vertical component of the pressure gradient and gravity. The related approximation is called the hydrostatic approximation. This amounts to replacing the vertical momentum equation by the hydrostatic equation, i.e. (1.24). The system obtained from the BEs with this assumption is called the primitiue equations of the ocean ( P E S ) . As we studied in Lions, Temam and Wang [I], the large-scale atmosphere is nearly always in hydrostatic balance. The hydrostatic approximation is an even better assumption in the ocean (see Washington and Parkinson [2]). Due to its high accuracy, the hydrostatic equation is well-accepted as a fundamental equation of the ocean. Even though they are still fully nonlinear and three-dimensional, the PES are suitable for practical computations. Since the primitive equations are the fundamental equations of the ocean, we intend to show in a subsequent article how they can be derived from the general conservation equations of physics and mechanics by an asymptotic procedure. This justification is based on the fact that the depth of the ocean is very small compared to the radius of the earth, and that a variable density is recognized only in the buoyancy term and in the equation of state.

It is known that the viscosity plays an active role in studying the dynamics of the ocean. Recalling that one of the objectives of this paper is to understand the impact of the ocean on the possible global changes, we will also study the long-time behaviour of the equations. Therefore we cannot neglect the viscosity effects, which is done, and appropriate, for short-term studies. Of course, this is certainly also important even if we consider the ocean as an independent system. Due to some physical considerations, the viscosity in the BEs and the PES is introduced in such a way that the vertical and horizontal eddy viscosity coefficients, and eddy diffusivity coefficients are different (see among others Washington and Parkinson [2]).

Moreover, emphasizing the importance of the viscosity in the vertical momentum

On the equations of the large-scale ocean 1009

equation, we replace the hydrostatic approximation by the so-called hydrostatic approximation with uertical viscosity, i.e. (1.31). The resulting equations are called the primitive equations with vertical viscosity (PE&). Similar equations for the large- scale atmosphere were also studied in [I] (see also [3, 41). As far as we know, the PEV’s for both the atmosphere and the ocean were first introduced in [ I , 3, 41 and the present paper.

For both the PES and the PEV’s, the prognostic feature for the’uertical uelocity w is lost; and there are two dingnostic equnrians incio!c,ing thc pres:~re and w. In both cases, the vertical velocity w changes with time, hut only as specified by the diagnostic equations. From the mathematical point of view, both the PES and the PEV’s do not appear as evolution equations due to the diagnostic ones. As in [l] for the PES and the PEV’s of the atmosphere, both diagnostic equations are integrated, and then yield two evolution systems with a non-local constraint, the fourth equation (2.4), for the PES and the PEV’s respectively. We cannot, howeverl use directly the usual techniques to handle these problems, because the constraints are not local. Therefore, we have to make some specific analysis similar to that in [I], hut however different in several respects.

To he more specific, we consider, for instance, the non-dimensional form (1.42) of the PES. Integrating the continuity equation in the vertical direction eliminates the vertical velocity w from the system, hut yields a non-local constraint, i.e. the fourth equation (2.4). On the other hand, if we consider the pressure p at the upper surface of the ocean as a given function, then integrating the hydrostatic equation provides an overdetermined system. If we introduce, however, a two-variable unknown function, ps , as the pressure function at the upper surface of the ocean, then we obtain a new formulation of the PES, namely (2.4).

From the mathematical point of view, it is very natural to introduce this two- variable function p, . For the usual Navier-Stokes equations, e.g. (I.l2)-(1.14), the term grad,p is, in a sense, the Lagrange multiplier associated to the constraint of incompressibility. This multiplier appears through the use of Hahn-Banach duality theorem and de Rham’s theorem (cf Lions [5], Temam [6] for the Navier-Stokes equations). These processes are linear.

If we modify the condition of incomprssibility by integration in one space variable j‘nere the vertical coordinatej, we obtain a new constraint, which is not iocai (see the fourth equation (2.4)). Because of the linearity observed above, the similar integration of grad, p leads to a new Lagrange multiplier, i.e. the two-variable function grad p, . It corresponds to the new (non-local) constraint. Of course, this has to be verified by a careful analysis. It is presented in the text to follow.

We can then apply the Galerkin method to the new PES expressed in a weak (variational) formulation. It is also very useful from the oceanographic point of view that our reformulation and the methods to solve it provide an effective way of determining the pressure at the upper surface of the ocean. Similar reformulation for the PEV2s is also established and used in this paper.

Again, from the mathematical point of view, the reformulated PES of the atmosphere studied in [I, 3, 41 possess a structure similar to that of the PES and the PEV’s here. The =ai:: diRe:e::ces betwee:: the p:ese-t :efcxx!ated e q - a r k s and those similar equations of the atmosphere in [I, 3, 41 are two-fold. Firstly, the domain here is much more complicated, so we have to make some more involved analysis in this paper. Secondly, the equation of state is quite different from that of the atmosphere. Therefore we have to overcome the difficulty caused here by the density terms to

1010 .I-L Lions et R I

obtain the R priori estimates. Moreover, the BEs are similar to the Benard equations studied in [7]. But in the BEs, we have one more equation for the salinity, a much more complicated domain, as we just mentioned, and the non-homogeneous boundary value conditions. So the problem here is much more complicated than the problem for the Benard equations. Due to the non-homogeneous boundary value conditions, for instance, we are not able here to use the maximum principle to obtain Lm-estimates for the temperature T as in [7].

This paper is divided into five parts. In the first part, we derive the BEs, the PES and the PEV’s from the general hydrodynamic and thermodynamic equations of a compressible fluid under Coriolis force. Some physical and mathematical explanations are given.

The second part deals with the PES. Firstly, by integrating the diagnostic equations, we introduce the reformulation of the PES. Then after establishing some functionals and their associated operators, we obtain the weak formulation of the PES. Finally, we prove an existence theorem of global weak solutions for the PES.

As in part 11, we obtain in section 4 a mathematical formulation of the problem. Then the existence of global weak solutions and local strong solutions, time-analyticity of the strong solutions are established in section 5.

Investigations for the BEs similar to part 11, 111 are given in part IV. More specifically, after establishing the functional setting of the BEs, we prove the existence of global weak solutions and local strong solutions, time-analyticity of the strong solutions.

The last part of the article is devoted to attractors and their dimensions for both BEs and PEV’s. Some physically relevant estimates for the Hausdorff and fractal dimensions of the attractors or the functional invariant sets for these two systems are obtained.

In part 111, we consider the PEV’s.

CONTENTS Part I. 1. The equations of the large-scale ocean

1.1 . The Boussinesq approximation 1.2. The hydrostatic approximation 1.3. The boundary value conditions 1.4. Non-dimensional forms of the equations

Part 11. The primitive equations of the ocean 2. Mathematical setting of the PES

2.1. Reformulation of the PES 2.2. Some function spaces 2.3. Some functionals and operators 2.4. Weak formulation of the PES

The equations of the large-scale ocean

3. Existence of weak solutions Part 111. The primitive equations with vertical viscosity (PEV’s) 4. Weak formulation of the PEV2s

On the equations of the large-scale ocean

4.1. Reformulation of the PEV% 4.2. Variational formulation of the PEV2s

5. Existence of solutions and their properties Part IV. The equations with Boussinesq approximation

6. Existence and properties of solutions 6.1. Functional setting 6.2. Existence and properties of solutions

Part V. Attractors 7. Dimension estimates for the attractors of the PEV’s

7.1. Preliminaries 7.2. Dimensions of the attractors 7.3. Some further estimates

8. Dimensions of the attractors for the BEs 8.1. Dimension estimates 8.2. Some more estimates

Appendix. Table of principal notation

Part I. The equations of the large scale ocean

1. The equations of the large-scale ocean

1 .I. The Boussinesq approximation

1011

Generally speaking, it is considered that the ocean is ma slightly compressible fluid with Coriolis force. The full set equations of the large-scale ocean are the following: the momentum equation, the continuity equation, the thermodynamical equation (i.e. the equation for the temperature), the equation of state and the equation of diffusion for the salinity S :

up o

(1.1)

(1.2)

d “3 p x f 2pR x “3 + V3p+pg = D

9 + pdiv, V, = 0 di

d T -=e, dt

dS - = Q s dt

(1.3)

where as we have mentioned in the list of principal notation in the appendix, g = (O,O, g) is the gravity vector, D the molecular dissipation, Q, and Q, are the heat and salinity diffusions. The analytic expressions of D, QT and Q, will be given hereafter.

1012 J-L Lions et a1

From both the theoretical and the computational points of view, the above systems of equations of the ocean seem too complicated to study. So it is necessary to simplify them according to some physical and mathematical considerations. It is known that the differences of the density of the seawater are significant only in the term of buoyancy and in the equation of state. Therefore, as a starting point, we can make the following so-called Boussinesq approximation:

Boussinesq approximation: the density differences are neglected except in the buoyancy term and in the equation of state. (1.6)

This amounts to replacing ( l . l ) , (1.2) by

Consider the spherical coordinate system (G,*p,r), where Q (0 < Q < n) stands for the colatitude of the earth, rp (0 < rp < 27r) for the longitude of the earth, r for the radical distance, and z = r - a for the vertical coordinate with respect to the sea level, and let eo, e*, e, be the unit vectors in e-, rp- and z-directions, respectively. Then we write the velocity of the ocean in the form

V, = vueH + umeq + u,e, = U + w (1.9) where U = uueo + u,e, is the horizontal velocity field and w the vertical velocity.

Another common simplification is to replace, to first order, r by a the radius of the earth. This is based on the fact that the depth of the ocean is small compared with the radius of thc earth. In particular

a dt at r J e r s i n e aq ar

~ = - + --+ _- O!'" a +U,- d u s a

becomes

(1.10)

(1.11)

and, taking the viscosity into consideration, we obtain the equations of the large-scale ocean with Boussinesq approximation, which are simply called Boussinesq equations of the ocean (BEs) , i.e. equations (l . l2)-(l . l7) hereafter (for the equation of state ( l . l7 ) , see remark 1.1):

au au I a% at a Z Po az - + V,u + w- + - grad p + 2ncos Gk x U - p A u - Y = 0 (1.12)

aw aw I a p a Z w - + V"W + w- + -- + -g - p n w - v- = 0 at a Z p0 az a z Z

(1.13)

(1.14) aw az d ivu+-=O

aT JT a2T a t az a z - +V,T + w- - p T A T - v T T = O

as as azs at az - +V,S + w- - p s A S - v S p = 0

(1.15)

(1.16)


Some differential where U is the horizontal velocity field, w the vertical velocitv. operators are defined as follows. The (horizontal) gradient operator grad is

(1.18) gradp=--eR+-- 1 a p 1 ap a 8 0 asinOamev'

The (horizontal) divergence operator div is

(1.19)

The derivatives VUE and VuF of a vector function 5 and a scalar function covariant derivatives with respect to U ) are

(horizontal

(1.20)

(1.21)

Moreover, we have used the same notation A to denote the Laplace-Beltrami operators for both scalar functions and vector fields on S,". More precisely, we have

Au = A(v,e, + u,eJ

(1.22)

(1.23)

where in (1.23), Au,,Au, are defined by (1.22), and in (1.22), T is any given (smooth) function on Si.

Remark 1 .1 . Only empirical approximations of the function

are known (see Washington and Parkinson [2]). A key point is to make an explicit equation of state. This equation of state is generally derived on a phenomenological basis. It is natural to expect that p decreases if T increases and that p increases if S increases. A natural linear law is therefore (I.l7), where p,,, T,,S, are reference values of the density, the temperature and the salinity, PT and p, are expansion coefficients (constants).

More general polynomial equations in T and S have been considered (cf [ 2 ] , U

Remark 1.2. Since the BEs, equations (l.l2j-(l.l7), are fully nonlinear and three dimensional, they present considerable computational difficulties. However, their mathematical analysis is possible, together with the study of the large time behaviour of the BEs. Moreover, in the next subsection, we will make some further simplifications of the equations based on the hydrostatic assumption, which is considered to be extremely accurate. The resulting equations are called the primitive equations of the large-scale ocean (PES), which improve the situation from the computational point of view. 0

Generally speaking, the equation of state for the ocean is given by (1.5).

P = f(T,S,p)

pp 131-2). They lead to further mathematical difficulties.


1.2. The hydrostatic approximation

It is known that for the large scale ocean, the horizontal scale is much bigger than the vertical one. Therefore, the scale analysis shows that the large-scale ocean satisfies the following hydrostatic approximation :

( 1.24)

which provides a fundamental equation connecting the pressure and the density for the ocean. As we have mentioned in the introduction, this relation is highly accurate for the large-scale ocean, so it became a fundamental equation in oceanography. Due to the importance of this approximation, we will study elsewhere the justification of (1.24) using asymptotic expansions in E .

Using the hydrostatic approximation, we obtain the following equations called the primitive equations of the large-scale ocean ( P E S ) :

av av I azo at az az2 - + V , u + w- + -gradp+ 2QcosBk x U -pAv- v- = 0 (1.25)

a p az - = -pg

aT aT a2 T - + VuT + w- - pTAT - v T z = 0 at az az as as azs at az - +v,s +w- - p s A S - v s s = o

(1.26)

(1.27)

(1.28)

(1.29)

P=Po( l - -PT(T-To)+ I (s (S-So) ) . (1.30)

Remark 1.3. In the above PES of the ocean, we have replaced the vertical momentum equation (1.13) by the hydrostatic approximation (1.26), which does not contain the vertical velocity w explicitly. Therefore the prognosticfeature for w is lost in the PES. In other words, we have to find w through the other diagnostic equations, and, from the mathematical point of view, the initial value of w cannot be assigned: these equations are not of Cauchy-Kovalevsky type for w (and p ) , This is one of the major differences between the BEs and the PES. We have indicated in the introduction and will see later that one of our contributions in this paper is to overcome the difficulties caused by the

U

Remark 1.4. Due to the hydrostatic approximation (1.26) (or (l.24)), the PES are suitable for numerical computation. Based on this, the PES became the fundamental equations for studying the dynamics of the ocean, and serve as a starting point for oceanography. Other models of the ocean, such as the quasi-geostrophic model, are derived from the PES. U

absense of w in (1.26).

On the equations of the large-scale ocean IQ15

As we have mentioned before, if we study the long term behaviour of the ocean, the viscosity will play an important role in the dynamics of the ocean. So if we emphasize the importance of the viscosity, we can replace the hydrostatic approximation (1.24) by the following equations called (by us) the hydrostatic approximation with vertical viscosity :

Then the resulting equations of the large-scale ocean are the following primitive equations with vertical viscosity (PEV2s) :

av av I - + V,u + w- + -gradp+2QcosOk x U -pAu - at Jz Po

aw az

aT J T JZT - + VoT + w- - p T A T - v T m = Q at aZ

divv + - = Q

as dS a2s at az ar - + V,S + w- -psAS - vs7 = 0

(1.32)

Remark 1.5. The PEV’s for the atmosphere were considered in [l]. But here it is even more natural to retain (i.e. not to neglect) the vertical viscosity for w, because viscosity effects are more important in water than in air. Note also that it is easier to consider the vertical viscosity in z-coordinate than in the p-coordinate system. As far as we know, the PEV2s for both the atmosphere and the ocean were first introduced in [I , 3,

0 41 and in the present paper.

Remark 1.6. Another natural approximation for the vertical momentum equation is

i.e. we keep all the linear terms in the vertical momentum equation. From the mathematical point of view, the resulting equations are very similar to the BEs, but different from the PES and the PEV2s. Here the time derivative term a w l a t for w is retained, so that the equations are of Cauchy-Kovalevsky type for w (recall that the PES and the PEV2s are not of Cauchy-Kovalevsky type). The mathematical study of

U

Remark 1.7. From the mathematical point of view, the PEV2s are more tractable than the PES due to the viscosity for w. Some other motivations for introducing the vertical

0

these equations is similar to that of the BEs presented in this paper.

viscosity can be found in [4].


1.3. The boundary value conditions In accordance with the approximation r N a, we approximate the domain filled by the sea by a region M , c S,‘ x R:

M, = U w d x (-~(~,rp),o). (O.vlEM,*

Here Si is the sphere of radius a, Ma, c S,’ is the 2D domain occupied by the ocean on the surface of the earth; finally H(B, rp) is the depth of the ocean at point of co-latitude 0 and longitude rp.

The region M,, is obtained from S,’ by removing some islands (or continents) I ; ( i = 1,2,. . . ,N), i.e.

M a , = s:\ (lj i;) \i=1 /

(1.33)

where the islands I b ( i = 1,. . . , N) are simply connected open sets in S,’ with smooth boundary such that r:nT:=$ i , j , = I ,,.., N i # j . (1.34)

Here the bar over 1: indicates the closure of 1: in S,‘. + R. We

assume that H is a smooth and positive function, i.e. H > 0 in &fah: so that in fact H is hounded from below on Mioh: H > Ho in & f a h .

Now we are in a position to give the boundary value conditions for the BEs, the PES and the PEV2s. They read:

At the upper surface of the ocean, z = 0:

Moreover, in the definition of Ma, we have a depth function H :

(1.35)

At the bottom of the ocean, z = -H :

I ( q w ) = o ( T : S ) = [ T , ; S , ) ,

At the lateral boundary, U(s,v, ,aM,{(e,rp)) x (H(Q, rp),O):

(1.36)

(1.37)

Here 7” is the (given) wind stress, which depends on the velocity of the atmosphere; xT is a positive constant related with the turbulent heating on the surface of the ocean, and TA is the (given) ‘apparent’ atmospheric equilibrium temperature, TH and S, are the (given) temperature and salinity of the sea at the bottom of the ocean, which are functions of the colatitude 8 and longitude rp. See Bryan [8, 91, Semtner [lo] and Washinton and Parkinson [2].

( v , w ) = O

a %(T, S) = 0


Remark 1.8. With the same method as in this article, we could treat the BEs, the PES and the PEV's with different boundary value conditions. For example, instead of (1.36), we can consider the boundary value condition at the bottom of the ocean to be

aU at - = 0 w = -u .gradH

a az - (T,S) = 0.

Of course the treatment of the boundary terms may be different because of new terms coming from the integration by parts.

1.4. Non-dimensional form of the equations

In this section we introduce the non-dimensional form of the PES, the PEV% and the BEs. To this end, let U , To, So, pa be the reference values respectively of the horizontal velocity, temperature, salinity, density. We also consider a to he the reference value of the horizontal length and Z to he the reference value of vertical length, E = Z/a being small. Then we set

(1.38)

(1.39)

(1.40)

Here Re,(i = 1,2) are the Reynolds numbers, Rti(i = 1,2) and Rsi(i = 1,2) are the non-dimensional eddy diffusion coefficients, Ro the Rossby number, which measures the significant influence of the earth's rotation to the dynamical behaviour of the ocean. The definition of these numbers appears in the list of principal notation.

Then by direct computation we obtain the following non-dimensional form of the BEs, the PES and the PEV's. For simplicity we drop all the super index prime in the equations.


The non-dimensional form of the BEs:

a u a v 1 1 1 a% -+VV,u+w- +gradp+-fk x u - -Au- -- = 0 at az Ro Re I Re, J z z

aw aw , a p . 1 1 a Z W - at +Vow + W - + E - (- + bp) - - A w - -- = 0 az dz Re I Re, Jz2

aw az divu+ - = 0

aT aT 1 1 JZT - +V,T + w - - - A T - -- = O at a z Rt, Rt, Jz2

as as 1 1 d2S - + VuS + w- - -AS - -- = 0 at J z Rs, Rs, J z z

P = I - & ( T - I )+Bs(S - 1).

The non-dimensional form of the P E S :

> (1.41)

a u a v 1 1 I a%

JP - az

- + V,u + w- + gradp + -fk x U - -Au - -- = 0 at az Ro Re I Re, Jz2

- + b p = O

divu + - = 0 aw dz

JT JT 1 1 J2T -+VV,T +w- --AT - -- = 0 at J z Rt, Rt, J z 2

as as 1 1 azs - +V"S + w- - -AS - -- = 0 at az RS, RS, a z 2

p = 1 - p,(T - 1) + Bs(S - I). The non-dimensional form of the PEV's:

a u a u 1 1 I a% at aZ Ro Re I Re, 822

-2 JP - 1 I a Z w

J Z Re, Re, dz2

- +V,u + w- + gradp+ -fk x u - -Au - -- = 0

E (- + b p ) - -Aw - -- = O

aw J Z

d i v u + - = O

aT 8T 1 1 a2T

as as 1 1 a2s - + V"S + w- - -AS - -- = 0

- + V,T + w- - -AT - -- = 0 at az Rt , Rt, Jz2

at J z Rs, Rs, dzZ

p = 1 -B,(T - 1) +$ , (S - I) .

(1.42)


In the above equations, the covariant derivative operator Vu, the gradient operator grad, the divergence operator div and the Laplace-Beltrami operators A on S' are defined as in (1.18)-(1.23) with a replaced by 1, the sphere S,' being replaced by the unit radius sphere S 2 (= S:).

Now the non-dimensional domain of our problems is

where M , is defined by (see (1.33))

(1.44)

(1.45)

where I' c S' (corresponding to 1;) ( i = 1,. , . , N ) are simply connected open sets with Cm boundary such that

finfj=$,i+j,i,j=l ,..., N. (1.46)

Obviously, M is a submanifold of S2 x IR (it carries the Riemannian metric of S' x IR). For simpiiciiy, we specify ihe bounciary ofthe domain ivi as fuihws (sce figures i

and 2).

r.

r, ,,j rr -

Figure 1. The domain Mh. Figure 2. A vertical section of M.

Let r, be the boundary of the ZD domaim M,, and r,, Tb and r, be the upper, the bottom and the lateral boundaries of the ocean, respectively. They are given by

N

I-, = J M , = U Jli i=l

(1.47)


Then obviously the boundary aM of M is

r = a~ = ru Ur, U rl. (1.48)

Then we can state the boundary conditions of the non-dimensional problem as follows :

(U, w) = 0 ) On ' b [ T , S ) = ( T b , S b ) ) (U, w ) = 0

a - (T,S) = 0 an

(1.49)

Part 11. The primitive equations of the large-scale Ocean

2. Mathematical setting of the PES

2.1. Reformulation of the PES

An important feature of the PES is that not all equations are prognostic equations. This terminology, classical in meteorology and oceanography, can be mathematically restated by saying that the PES are not of Cauchy-Kovalevsky type with respect to all variables. As we have mentioned earlier, there are two diagnostic equations in the PES, the vertical momentum equation, i.e. the second equation (1.42), and the continuity equation, i.e. the third equation (1.42). Therefore in this section, we integrate these two diagnostic equations, and then we establish another formulation of the primitive equations, which is an evolution system. The main idea here is to introduce an unknown function p,, indicating the pressure of the sea water on the surface of the ocean which depends on 0 and 'p. but not on the third independent variable z. This is significant because our whole system is three dimensional.

More precisely, integrating the continuity equation and taking the boundary conditions for w into account, we obtain

w(t;O,'p,z) = W(u)( t ;O, 'p , z ) =div u(t;B,rp,z')dz' l (2.1) div U dz' = 0.

- h ( W

We consider then the vertical momentum equation, i.e. the second equation (1.42). First of all, we define an unknown function on the surface of the ocean, say

p, : Mh - w (2.2)

On the equations o f the large-scale ocean 1021

which is the pressure of the sea water on the surface of the ocean. Then

p ( t ; Q , q , z ) = p , ( t ; B , q ) + &(t;Q,rp,z')dz'. (2.3) 1 Therefore the primitive equations (1.42) of the ocean can be rewritten as follows

Re, azz 1 + gradp, - -A0 -

Re 1

aT aT 1 1 a*T at az Rt, Rt, az, as as I I a% at aZ R ~ , RS, azz -

- + V , T + W(u) - - -AT - -- =

- +V,S + " ( U ) - - -AS - -- - div l u d z ' = 0

p = (1 + P T -Bs) -PrT + Pss

(2.4)

where W(u) is defined by (2.1) and the physical parameters Ro, Re,, Re,, RI,, Rt,, Rs, and Rs, are given by (1.40).

The corresponding initial-boundary conditions are

1 au a on ru : - = i, -(T,S) = (XT(TA - T),O) az az on r, : U = 0 ( T , S ~ = (Tb,Sb)

i (2.5)

From now on we use Z to denote the unknown function ( U , T,Sj.

Remark 2.1. We would like to point out that the reformulated system (2.4) of PES is three dimensional, but the unknown function p , depends only on ( t ; Q , q ) . Moreover, we also have a non-local constraint in the system, namely the fourth equation (2.4), whose Lagrange multiplier is just the unknown function p, . The reformulated PES of the atmosphere studied in [l , 3, 41 possess a structure similar to that of the equations here. 0

2.2. Some function spaces

The space domain M is an open submanifold of Sz x R. So the Riemaniann geometry of M is the same as that of Sz x R, or S2 x (0, I), which is also the domain for the primitive equations of the atmosphere in [l] . Therefore we quote briefly some notation, basic concepts , definitions of some differential operators and function spaces from that article.

1022 J-L Lions et al

First of all, the tangent space T(,,z)M of M at (q , z ) E M can be decomposed into the product of T,M, = TqS2 and T,(-k(q),O) = R:

T(n,z) M = Tq Mh X R. (2.6) Therefore, the Riemannian metric g, on M is given by

gM ( ( 4 , Z ) ; ( U , W ) , ( U i , W , ) ) =g,+,,(4;U,UO + W W i VU,UI E T q M h , W , W l E R (2.7)

where gM, IS the Riemannian metric on M, t S'.

by the following matrix In particular, in the spherical coordinates (x',x2,x3) = (O,q,r) on M, g, is given

( g i j ) = ( sin2 8 , ) where

So the unit vectors in e-, rp- and r-directions are the following

(2.9)

(2.10)

For simplicity, we denote the inner product and norm in the tangent space T(,,,,z)M by

(2.11)

for any pair X , Y ,

x = x'e, + X 2 e , + X3e ,

Y = Y l e , + Y 2 e , + Y 3 e , E T(B,p,l)M.

Now we introduce the notation for some standard function spaces on M. We let C m ( a ) (respectively Cm(TM)) be the function space of all Cm functions from M into IR (respectively Cm vector fields on M ) . Then Cm(TM) possesses the following decomposition:

C m ( T M ) = CyTMIJTM,) x C " ( A ) = { U : M --+ T M , E cmlu(e,rp,z) E T , ~ , , , M , } x { W : lir --+ IRE cm}

(2.12)

Similarly, let C F ( M ) (respectively C r ( T M ) ) be the function space of all C m functions from M into R (respectively C m vector fields on M ) with compact supports. Then we have the following decomposition of C ? ( T M ) :

C,m(TM) = C,"(TMITM,) x C,m(M) = { U : M -+ T M , E C" with compact support i u ( Q , q , z ) E T(,,&4,}

x {w : M -+ R E C m with compact support} (2.13)


Moreover, the standard notation for the Soblolev spaces on Mh as Cm(Mh), C?(Mh),

Moreover we define Cm(Mh), Cm(TMh) , C,"(TM,) and C m ( T M h ) are also used in this paper.

Cgb,o(TMITMh) = {U E Cm(TM1TMh)lu is wro near rl U r,} C G ( M ) = { h E Cm(M)lh is zero near rb}.

(2.14)

(2.15)

We also need some notation and formulae for some basic differential operators on

(1) Let T E Cm(Mh) and M and M,. They are the same as those in [l], so we only list them.

U = use, + uoep uI = ('l)8e@ + h ) q e , E Cm(TMh)

then we define the (horizontal) covariant derivatives of U, and T , with respect to U, by formula (1.20), (l.21), where a is replaced by I ;

(2) For any function T E Cm(M), we define the (horizontal) gradient grad and the gradient on M, grad,, by

(2.16)

(2.17)

(3) For any X = X'es + X'e, + X3e, E Cm(TM), we define the divergence of X, div, X E Cm(M), and the tangential divergence, div(X'e, + X'e,), of Xle, + X 2 e , by

aT J Z

grad, T = grad T + -ez.

div (X'e, + X2e,) = - 1 ( J x g i I l e + g ) sin 8 aq

(2.18)

(2.19)

(4) The (horizontal) Laplace-Beltrami operator AT of a scalar function T E Cm(M), and the (horizontal) Laplace-Beltrami operator Au of a vector field U = use, t u,e, E Cm(TS2), are defined by formula (1.22), (1.23), where a is replaced by 1.

Now we are in a position to introduce some notation of Sobolev type function spaces on M. First of all, let L2(M) , L 2 ( T M ) and Lz(TMITMh) be the Hilbert spaces of L2-functions, Lz-vector fields and the first two components of Lz-vector fields on M , respectively. We denote the inner products and norms in these spaces by the same notation c, .) and I 1 given by

ax3 az div, X = div (X'e, + X'e,) + -.

(h , ,h2) = 1 h, . hzdM Ihl = (h,h)''2 (2.20) M

for any h,h , ,h2 E L 2 ( M ) , or L2(TM), or L z ( T M I T M h ) . Similarly we can define the spaces L2(Mh) and L2(TMh) .


The definition of the Sobolev spaces H ' ( M ) , H S ( T M ) and H " ( T M / T M , ) for any s E IR is the following. When s is a positive integer, these are the completed spaces of Cm(M), Cm(TM) and Cm(TMITMh) for the respective norms

for h in one of the spaces. When s is zero, these spaces are the L'-spaces. When s is not an integer or s is negative, they are defined naturally by interpolation and duality (see Lions and Magenes [ll]). In (2.21), we have used the notation

Moreover, for 0 Q m = integer < m,1 Q p Q 00, Wm+'(M), Wm'(TM) and W",P(TMITMh) are the Sobolev spaces with norm

for h in one of Wm,p(M), WmJ'(TM) and Wm,P(TMITMh). We can also define similar Sobolev spaces on Mh, hut we omit the details.

Nevertheless, we always use the following standard notation to denote the norms -4 inn-- n m A n r t . in thece fnnrtinn m n r ~ e . Y1.V 11.1111 y'Lv....-." I.. ... ..".. ..... ".._.. -y...,.,-.

I/ . /I W".P 11 ' IlH' ( ' , ' ) H x . (2.24)

Finally we also need the following notation. For s > 1/2, we define

Hi,o(M) = the closure of C,qb(M) in H'

H;+,,o(TMITM,) = the closure of CP;,,,(TMITM,,) in H S . (2.25)

The function spaces V and H Now we turn to define the function spaces for the unknown function = (u ,T ,S) . Motivated by the studies for Navier-Stokes equations (see among others Lions [51 and Temam [6]) , we construct the function spaces for U in such a way that the fourth equation (2.4), i.e.

div J_", udz' = 0

appears as a constraint for v in the spaces, which plays the same role as the continuity equation div U = 0 in the mathematical theory of the Navier-Stokes equations. More precisely, let

0 U E C,m+b,o(TMITMh)/div/ -h udz'=O}. (2.26)


Then we define

VI = the closure of Y , for the H' - norm

V2 = V, = H&,(M)

H I = the closure of Y , for the L2 - norm

H2 = H, = L2(M)

v = v, x v2 x v, H = H , x H2 X H,.

(2.27)

The norms and inner products for the spaces H and Hi are the L'-ones, denoted by (;) and I . I. By definition, the inner products and norms in VI, V2 and V, are given by

(2.28) I I

llhll = ( (h ,h ) ) ' l 2 Vh,h, E V2 or V,. J For simplicity, we use the same notation 11 . 11, ((.;)) to denote the norm and inner

product of V given by

((=,=I)) = ( ( U . U I ) ) +U, T , ) ) + ( ( S ? S , ) )

11EIl = ((Z,Z))I'2 vz = (u ,T ,S ) , E, = (U,,TI,SI) E v. (2.29)

Now by the Riesz representation theorem, we can identify the dual space H' of H (respectively H: of Hi, i = 1,2,3) with H (respectively Hi) , i.e. H' = H (respectively H ; = Q, I = i, & 3 j . Tien we have .-, .. .

V c H =HI c V' (2.30)

where the two inclusions are compact continuous. The following lemma characterizes the spaces H and V ,

Lemma 2.2. Let H: be the orthogonal complement of H I in L2(TMlTMh) . Then

H f = {U E L'(TMJTM,)\U = gradI,l E H ' ( M h ) ) (2.31)

H I = u ~ L ~ ( T M l T M , ) l d i v u d z ' = O (2.32) { L > { L } VI = U E H:+,,o(TMITM,)I div udz' = 0 . (2.33)

Proof: The proof of this lemma relies in part on a similar result proved in [l].

1026 J-L Lions e t a1

We start by observing that for any I E H'(Mh) and uI E Y,, we have

grad 1 . U, dM = k, (l,,,, grad I . U, dz dM, ) = - Lh I div (1' v1 dz ) dM, + l, I div (so U, dz) . n dTh

= o - h ( W - h ( W

which proves that the right-hand side of (2.31) is included in H:.

U E H:, we have Conversely, we want to show that the inverse inclusion is also true. Indeed for any

/, vuI dM = 0 Vu, E Y,. (2.34)

As in the proof of lemma 2.1 in [l], we see that

therefore (2.34) implies that U does not depend on z. Then we write (2.34) with u1 = ku, such that

UI E C,"(TMh) div u1 = 0

k E CF(-minh,O) s_", k ( z ' ) dz' # 0. Mh

As in the proof of lemma 2.1 in [l], we can reduce the problem into a 2D Stokes problem, then we can obtain the function e, which depends only on the first two space variables. We omit the details. 0

2.3. Some functionals and operators

We will need the following lemma of integration by parts, which is a direct consequence of the Stokes formula on the manifold M .

Lemma 2.3. For any pair v,u , E Cm(TMIMh), k , k , E C"(M),


where n is the unit outward normal of M, and the (covariant) derivative operator D is defined by linear extension of the following formulae

3 ax3 as D,(Xle, + X2e , + X e,) = V,(X'es +X2evj + -e,

/ (--uAk - /12) a r 2 , k, dM = k (or grad k.grad k, + M

(2.36)

Based on the above formulae, we can define four bilinear forms R : V x V - R, ai : Vi x Vi -+ W (i = 1,2,3), and the corresponding linear operators A : V -+ V ' ,

1028 J-L Lions et a/

A, : Vi + V: (i = 1,2,3) by setting

@,E,) = (AH,8 , ) = (A,u,u,j +(A,T,T,j + ( A , S , S , ) (2.37)

al(u,uI) = (A,u,uJ

(2.38)

(2.39)

grads .g rads , + RS, az az (2.40)

for any B = ( U , T,S),ZI = (uI,TI,SI) E V . Note that the appearance of the term u ' u I in (2.38) ensures the full coercivity of a, ; this term appears in the integration by parts formula (2.35). For T the same is true but now the term involving T T , is due to the boundary value condition. For S, we make use of the Poincare inequality. The following lemma follows readily.

Lemma 2.4. (i) The forms a,ai ( i = 1,2,3) are coercive, continuous, and therefore, the operators A : V --+ V' and A, : Vi - V: ( i = 1,2,3) are isomorphisms. Moreover, we have

where the parameters and R,,,,, are defined by

where c,,, cb are two absolute constants (independent of the physically relevant constants Re,, Rt,, Rsi, etc.. . j ;

(ii) The isomorphism A : V - V' (respectively A, : Vi -+ V: ( i = 1,2,3)) can be extended to a self-adjoint unbounded linear operator on H (respectively on Hi (i = 1,2,3)), with compact inverse A-' : H --t H (respectively A;' : Hi + Hi ( i = 1,2,3)).

On the equations oJ the large-scale ocean 1029

Now we define three functionals b : H x H x H A IR, b, : HI x H, x H, -.+ IR ( i = 1,2) and the associated operators B : H x H -t H , B, : H , x H, -t H, ( i = 1,2) by setting

b ( E , Z l , E J = (B(8,El),8J = b l ( ~ , ~ i , ~ * ) + b 2 ( ~ . T ~ . T ~ ) +b,(u,Si,SJ (2.43)

for any 8 = (U, T , S), ai = (ui, Ti, Si) E V or D(A) and h, E V2 or V, (i = 1,2). Then we have

Lemma 2.5. (i) For any E V , SI E V n H 2 ,

(ii)

Prooj (i) By the formula,

2 V"lI+l = V"U, . U] + U , ' v,u, = 2V,U, ' U]

we see that

= 0.

""C La', p o u r ; 111 2 JllllllaL . . 1x1.. --- :- r.-.~-_ ....... t k n r h I.. T T i - A

"a). una.1 q,", 1 I , '1, - ". (ii) The proof of (2.46) is the same as that of (2.52) in [l].

(2.46)


We have to define another functional e : H x H -+ R and the associated linear operator E : H - H by

@,El) = (E(B),8,)

x U) ' uI + 6 grad (-&T + &S) dz' ) , v I ] dM. (2.47)

The!! we have

Lemma 2.6. (i)

(2.48) 1 0 - - e(=,=) = - 6 L [l (-&T +&S)dz' divudM, V ~ E V.

(jj) Tilere an coneiant (iadepeiideai of the phyPica:py ie:evaiii parameters Rei, Rti, Rsi ( i = 1,2) and Ro), such that

(2.49)

Proof We only have to establish that

(2.50)

and this can be obtained from the integration by parts formula

grad k(B,q, z) . uI d M = - k(0 , q , z ) div u, dM (2.51) J, valid for any HL-function k. -

I O prove ihis formuia, we consider ihe expiejsioii

The Stokes formula gives

On the other hand,

(2.52)

Hence (2.51) follows.


2.4. Weak formulation of the PES

For the sake of simplicity we assume that

(T,S)Irb =O. (2.53)

The more general case, where Fb, s, may not be zero, can he treated similarly a t the price of some technical complications. Of course, in the genera! case, we also have to deal with the non-homogeneous boundary value condition for the salinity.

Now we want to 'homogenize' the boundary value conditions (2.5) as follows. Suppose i, E C2(fih) is such that

div i, = 0. (2.54)

From the physical point of view, i, is the wind stress, which depends on the velocity of the atmosphere at the interface of the ocean and the atmosphere. From the mathematical point of view, the condition (2.54) for the wind stress is a sort of compatibility of the boundary value conditions; therefore, it is reasonable.

Then we construct a scalar function T; and a vector function U; in the form

T,' = p'O,(z) U* = zO,(z)?,, Vq E (0,1/2minh)

Mh 'I

(2.55)

where 0, E Cm([-minMh h,0]) is given by

- q < z < o

-2q < z < -q -minh < z < -217

O,(O = { [ncreasing

Mh

and F * is defined by

T * = TA(l -eexp(-a,z)).

For any S > 0, there exists qo E (0,1/2minM, h) such that for any Lemma 2.7. o < q < ' l o

(2.56)

The proof of this lemma is similar to that of lemma 2.6 in [I]. We omit the details. U

We infer easily from lemma 2.7 that we can choose qo such that the H2-function

(2.57) -. a = . (u*,T*,O) --I (U;~ ,T~~ ,O)


satisfies

We define the new unknown functions U and 9 by

u = u - v U . .T = T - T'.

The primitive equations (2.4) can then be rewritten as follows:

+- -kxu+gradpS+6grad f ( -&T+&S)dz ' Ro

as as 1 1 a2s - + vus + W ( u ) - + V". s - -as - -- = 0 a t aZ Rs I RS, az2

divJ_0hudz = O

and the initial-boundary conditions (2.5) can be rewritten:

au a 82 az o n r , : - = 0 - ( 9 - , S ) = (-+?j-,O)

( F , S ) = 0 a

an

on rh : U = 0

o n r , : u = O -(Y,S) = 0

at t = 0 : Z = S o = (uo,.To,So).

In (2.60), f l and f, are defined by

1 * 1 a2T' f 2 = - A T +----V".T*. Rt, Rt, az

(2.58)

(2.59)

(2.60)

(2.61)

(2.62)

We are in position to state the functional formulation of the PES. It reads:


For 9, = (uo, F,,, So) E H given, find E = (U, F, S) such that

1033

Problem 2.8.

E L2(0,T; v) n Lm(O,T;H) V T > 0 (2.63) d -@,E1) + (AS, Z,) + @(=,E), El) + ( B ( 2 , E*), El) dt

+ (B(E*,Z),EJ + (E(Z),Zl) = (F,ZJ V q E V n H 2 (2.64)

E'lt=o = q. (2.65) - -

In problem 2.8, &' is defined by (2.57), and F by

F = (fi,f2,0). (2.66)

Remark 2.9. There are several ways to see that problem 2.8 is exactly the weak formulation of the PES with the initial-boundary conditions (2.5) using lemma 2.2. See, for instance, the interpretations of the weak formulation of the Navier-Stokes equations in [5] and [6], and the interpretation of the PES of the atmosphere in [l , 3, 41. Here we only want to point out that for any smooth solution ( u , 9 , S , p s ) of (2.60)-(2.61), Z = ( u , F , S ) is a solution of problem 2.8. Conversely, for any solution Z = (u ,F,S) of problem 2.8, lemma 2.2 says that there is a unique distribution p , E g'(.nt,) such that ( u , F , S , p s ) is a solution of (2.60)-(2.61) at least in the sense of distributions. In

U other words, problem 2.8 is indeed the weak formulation of the PES.

3. Existence of weak solutions

The main result in this section is the following existence theorem of global weak solutions to problem 2.8.

Theorem 3.1. For any 7 z 0, there exists at least one solution Z = ( u , Y , S ) for problem 2.8 defined on ( 0 , ~ ) such that

where H , is the space H endowed with the weak topology.

Proof: This theorem can be proved by the FaedoCalerkin method. Here we only present some formal R priori estimates. The remaining parts of the proof can be accomplished by following the procedure in [5] and [6].

Replacing Z, = (u l ,F l ,S l ) by Z in (2.64), we obtain

It follows that


By Gronwall's inequality, (3.2) yields

Iz(t)lz < 1s012eC' + C(eC' - 1)IFI2 (3.3)

The Galerkin procedure and (3.2)-(3.3) imply that there is at least one solution Z

Moreover it is easy to see that satisfies (2.63) for our problem.

l lB(E,E~)ll~-~ d CllSl/ ' IEJ

So (3.1) follows. U

Part 111. Primitive equations With vertical velocity (PE%)

In this part, we want to establish the mathematical setting of the PEVZs, and to show the existence of global weak solutions and local strong solutions. Moreover, the time-analyticity of the local strong solutions is also obtained.

4. Weak formulation of the PEV2s

4.1. Reformulation of the PEV*s

As in part 11, by integrating the continuity equation and the hydrostatic equation with vertical viscosity in (1.46), we obtain

(4.2)

where L is a linear differential operator given by

Moreover, let

u = u - - u 7 - T - T ' (4.4)

with T' and U * given by (2.57), satisfying (2.58). reformulation of the PE&, equations (1.43):

Then we obtain the following

au au av* - + v v , u + W ( u ) & + v , u ' + at W(u)-+V".U az

+ - k f x u + grad p , + 6grad (-&T + & S ) dz' Ro


1 1 J2u Re, Re, Jz2

+ grad 1' LW(u) dz' - - Au - - - = f , (4.5)

J 9 - a r aT' 1 1 J 2 F at az JZ Rtl Rt, az2

- + V o , F + W ( ~ ) - + V V , T ' + W ( u ) - + V V , . F - - A F - - - =f2(4.6)

divJ_Uhudi = O

with the same initial-boundary conditions (2.61) as those of the PES. As we have mentioned before, the above system is three-dimensional, but the

unknown function p , is only a function of 0, 'p and time t, and does not depend on the third space variable z, Moreover, p , is just the Lagrange multiplier of the non-local constraint (4.8).

4.2. Variational formulation of the PEV's First of all, for any u,ul E C&,o(Tf i lTMh) and if w = W(u) , w, = W ( u , ) E C F ( M ) , we nave

1 [(gradl 'LW(u)dz') .U]] dM

To see this, by (2.51), we write the left-hand side of (4.9) as

= / M LW(u)W(u,)dM

=the right-hand side of (4.9)

and (4.9) follows.

Due to (4.9) it is natural to define the function spaces for U as 0

y w - - {U E C&,,(TMITMh)lW(u) = 1 divudz' E C;(M)> (4.10)

(4.11)


Moreover, we claim that Y; is dense in V;. To see this, we only have to observe that Y; and V; are obtained respectively from the following two function spaces:

(U, w ) E C&,n(TMITMh) x CF(M)Jdivu + ( u , w ) E H:+,,n(TMITM,) x Hd(M)Idivu+

az

by integrating the constraint equation. It is well-known that the first space above is dense in the second one. So our claim follows.

(4.12)

In order to obtain the weak formulation of the PEV2s, we have to modify the definitions of the bilinear functionals and their associated linear operators, relating with the principal parts of the equations. To this end, we define two bilinear functionals a'" : V'" x V w A R, a; : V;' x V;' - R, and their corresponding linear operators A" : V w - (V")', A; : V; - (V;)' by setting aw(Z,El) = (A%,EI)

(4.13)

(4.14)

for any Z = (u ,T ,S) , E, = (u,,TI,SI) E V". Then we have

Lemma 4.1. (i) The forms aW and a; are bilinear, coercive, continuous, on V w and V;. Therefore, A" : V" - ( V w y and A; : V; - (V;)' are isomorphisms. Moreover, we have

(4.15) I 1 aw(E,El) < -llSllw IIElllw

aw(E,E) 2 - l l i l l w k i n

1 - 2

k a x

(ii) The isomorphism AW : V w -+ (V")' (respectively A; : V; - (V;) ' ) can be extended as a self-adjoint unbounded linear operator on H (respectively on HI), with compact inverse (AW)-' : H - H (respectively (Ay)-' : HI -+ HI). Moreover, the domain D ( A ; ) of A; is defined by

(4.16) U

D ( A ; ) = v; n {U E H * ( T M I T M ~ ) I W ( U ) E H~(M)}. The proof is similar to that of lemma 4.1 in [I].


Now we can state the functional formulation of the PEV2s:

1037

Problem 4.2. E E L ~ ( o , T ; v ~ ) ~ L ~ ( o , T ; H ) , V T > O (4.17) d -(E,=,) + (AWE,HI) + (B(E,E),E,) + (B(E,9'),8,)

For 8, = (uo,9,,S0) E H given, find H = ( u , F , S ) such that

dt + (B(E',a),E!) + (E(E),8!) = ( F , E ! ) , VH! E D(AW) (4.18) - - i l l = , = 1 0

where 8' is given by (2.57), and F is given by (2.66). (4.19)

Remark 4.3. Problem 4.2 is indeed a weak formulation of the initial-boundary problem of the PEV2s. We can show as for remark 2.9 that for any smooth solution ( u , F , S , p J of the PEC"s, E = ( u , F , S ) is a soiution of probiem 4.2; converseiy, for any soiution 9 = ( u , F , S ) of problem 4.2, there is a unique distribution p , E B'(Mh) such that ( u , 9 , S , p s ) is a solution of the PEv2s at least in the sense of distributions. 0

Remark 4.4. Since the norm / / ' \ I w is stronger than 1) .I), we have the following improved estimates for the functional b :

Ilall, ' l l q I/ ' l8,1"2 ' llH211"2

1 1 ~ 1 1 , ' 13,l. 1 1 ~ ~ 1 1 ~ 3 / 2 .

/b(=,~,,H,)I < c ~ ~ s ~ ~ w ' I ~ q p 2 ~ ~ 9 , ~ ~ ~ ; '1E21 (4.20)

0 The proof of (4.20) is the same as that of the corresponding result for the 3D Navier- Stokes equations. See also [l].

{ 5. Existence of solutions and their properties

As for the existence of weak solutions of the PES, we have

Theorem 5.1 . For any 7 > 0, there exists at least one solution S = (u,F,S) for the prnhlem 4.2 defined on (0; r) and such that

E E C ( [ O , r l ; H , ) (5.1)

E,EL2(0 ,r ; (VW n H 3 ' 2 ) ' ) . Here H , is the space H endowed with the weak topology.

We also have

Theorem 5.2 . If the initial value E, belongs to Vu, then there exists a time T , = r l ( ~ ~ E o ~ I w ) > 0, given by

such that there is a unique solution E = ( u , T , S ) for problem 4.2 satisfying H, E L2(0, 7I ; H )

E E L 2 ( 0 , r , ; D ( A " ) ) n C ( [ O , ~ , ] ; Vw). (5.3)

Moreover, H is time-analytic from ( 0 , ~ ~ ) into D ( A w ) .


The proof of theorem 5.1 is similar to that of theorem 3.1, and the proof of theorem 5.2 is similar to that of theorems 5.2 and 5.3 in [l]. The basic method of proving these theorems is the Galerkin method. Here we only give a sketch of the proof.

Proof of theorem 5.1. Let {ipili = 1,2, ...} be eigenfunctions of A". Let P, be the orthogonal projector in H onto the space spanned by {ip,, , , , , ip,}. Then it is easy to find an approximate solution 8,(t) ... = Cy=! gjm(t) ipj satisfying

d -(" dt -,' Vi) + (AW=:, + B(R,,B,) + E ( S J , V j )

+ (B(S,,S') + B(Z,Sm) , ip j ) = (F , ip j ) j = 1 , _ _ _ , m (5.4)

(5.5) - - aml t=O = P m q .

We infer from (5.4) that

d 1 &(t)l2 + -llE,(t)ll; Q C(IF12 + lq?"). (5.6)

&ax

By Gronwall's inequality, (5.6) yields

8, E a hounded set of Lm(O,r;H)nL2(0,7; V") . (5.7)

By (4.20), we have

(8m)c E a bounded set of L 2 ( 0 , ~ ; ( Y w nH3I2)'). (5.8)

U The theorem follows from (5.7), (5.8).

Proof of theorem 5.2.

I d 2 dt

(i) From (5.4), we also obtain

--(AWB,,H,j + I A " q t ) 1 2

= (F-B(Bm,8,)-B(B,,B') -B(Z* ,S , ) -E(B,),A"E,)

< CIAWB,(. (IF1 + ll~ml13w/21AWSmI~'2 + l18mllw) 1 < ZIA"B,12+C{IF12+ IlS,Il;+ llS,ll:}.

It follows that

d dt - ( A W S , , S , ) + I A W 8 , ( t ) l 2 ~ C { l +(AWE,,E,)}'.

Let

_I " l t ) \., = 1 + (a-s,; a,! then


So we can choose

such that whenever 0 < t < zl,

(AWE:,,&,) < 2(1 + ( A " z o , q )

Thus the solution E = (U, 9, S) satisfies

(5.9)

(5.10)

E E LyO, 7; V") (5.1 1)

z E L~(o,T~;D(A~)) (5.12)

and finally,

&[ E L 2 ( 0 , T l ; H ) . (5.13)

U (ii) A5 for the time analyticity of the solutions, let H", D(A")" and (V")" be

the complexified spaces of H, D(AW) and V". We use the same notation as AW, E , E, etc, to denote the naturally extended complexified operators on the corresponding complexified spaces. Then the complexified equation of (4.18) becomes(here we use the same notation t to denote the complexified time variable)

d - (E ( t ) ,2 , ) + (AwE(t) + E(E(t),E(t)) + B(&(t),E*),E,)

The existence of local in time strong solutions follows.

dt + (B(&*,&(t)) + E(Z(z)),E,) = (F,E,), El E D(Aw)c. (5.14)

Then formally, we can take E, in (5.14) to be A"&, and we multiply the resulting equation by eiS (when t = re"), and then we consider the real part of the resulting equation. This yields (see [I]) (IS1 < n/2)

I d --(A"E(reiu),E(reiB)) +cos e IAwZ(reiB)l* 2 dr

It follows that

as long a5

(5.16)

Following the method developed by Foias and Temam [12], we conclude that the solution E of (5.14) is analytic in a region of C, which comprises a neighborhood of ( O , T ~ ) , into D(AW)". Of course, the solution of (5.14) coincides with the local strong solution of problem 4.2 on ( 0 , ~ ~ ) .

The proof is complete. I3


Part IV. The equations with Boussinesq approximation (BEs)

As we have mentioned before, the equations df the ocean with only Boussinseq approximation, the BEs, are similar to the B h a r d equations studied in [7]. But in the BEs, we have one more equation for the salinity, a much more complicated domain and the non-homogeneous boundary value conditions. So the problem here is much more complicated than the problem for the B h a r d equations. Due to the non-homogeneous boundary value conditions, for instance, here we are not able to use the maximum principle to obtain Lm-estimates for the temperature T as in 171. Therefore, from both mathematical and physical points of view, it is necessary for us to present some mathematical foundations of the BEs of the ocean. In part IV, we establish the mathematical setting of the BEs, then we prove the existence and the time-analyticity of the solutions of the BEs.

6. Existence and properties of solutions

6.1. Functional setting

Define

-Vy =({U,.} E Czb,o(TM\TMh) x C:(M)Jdivu+ aw - = O } (6.1) az

and

(6.2) I VI” = the closure of Yf for the H ’ -norm

HIS = the closure of -V: for the L2 - norm

V B = VI” x v, x v, H B = HIS X H , x H, .

Here V / is equipped with the following inner product and norm

where ((.;)) is defined by (2.28). For simplicity, we use the same notation to denote the product norm and inner product in V B . Moreover, the norm and inner product in HIS are given as follows:

[ ~ U ~ W ~ , { ~ l . W I ~ 1 S = (u>uI)Lz + E 2 ( W , W I ) L I (6.4)

and we use the same notation for the norm and inner product of H B as those for H f .

I{u,w}lB= [ {u ,W} , {U,W}1~’2 v{u,w}3{ul,wl} E

Now let

I I = u - - V r = T - T T ’


with T' and U' given by (2.57), satisfying (2.58). Then the BEs can be rewritten as follows :

au au au* f - +V,U+ W - + VuV* + W- +V, .u+ - k x u at az aZ Ro

1 1 + gradp - - Au - -- - f a Re I Re, az2 - I

aw 2 (e + V , W + W Y + v , . w at dZ

E2

Re1 +b(-&F+BsS)- -Aw-- - Re, a 9 = f B

aw az d ivu+-=O

a r a 9 JT' 1 1 a 2 F

a t az az Rtl Rt, a z 2

as as 1 1 a2s at az Rs1 RS, az2

- + V , F + W- +V,T'+ W- + V v . Y - -AY - -- = f,"

- + V,,S + W - + V,.S - -AS - -- = 0

where

1 1 a%* . f --AV'+ -- - V . u - - k x U " - Re, Re, azz " Ro ' 1

The initial-boundary value conditions are

a o n r ' - = 0

U ' aZ az o n r , : { u , w ) = O ( F , S ) = O

o n r , : { u , w } = O - ( F , S ) = O

w = o - ( F , S ) = (-aTY,o) au

a a n

at t = 0 : E = (U, w , Y , S ) = So = (uo, wo,7,,,S0).

In order to establish the mathematical setting of the BEs, we have to modify the definitions of the functionals and their associated linear operators used in parts 11, 111. To this end, we define

a B (=,El) = [ A 8 Z . , ~ 1 1 B = a~({u,w},{u,,wI})8 +a2(T,TI) +a3(S,Sl) (6.12)

w } ? { u l , w l } ) w l } l B

= al(u,ul) +E,L [& gradw , gradw, + (6.13) Re, az az

Now we state the definition of the weak solutions of the BEs as follows.


6.2. Exisfence of solutions and their properties

The main results in this part are as in theorem 5.2

Theorem 6.2. For any T > 0, there exists a t least one solution E = (U, w, F, S) for the problem 6.1 such that

(6.23)

where H; is the space H B endowed with the weak topology.

Theorem 6.3. T~ =

If the initial value So belongs to V B , then there exists a time > 0, given by

c

such that there exists a unique solution 3 = (U, w , Y , S ) for problem 6.1 satisfying

(6.23)

Moreover, E is time-analytic from ( 0 , ~ ~ ) into D(AB) . The proof of these theorems is very similar to that of theorems 5.1 and 5.2. So we

omit the details. 0

Part V. Attractors and their dimensions

7. Dimension estimates for the attractors of the PEV‘s

7.1, Preliminaries

The main purpose of this part is to study the attractors and/or functional invariant sets for our problems and estimate their Hausdorff and fractal dimensions in terms of some physically relevant parameters. We have proved in the previous parts the existence of global in time weak solutions and local strong solutions. As we have pointed out earlier, it is not known in general whether there exist global strong solutions for our problems. Therefore we restrict ourselves to the study of attractors and functional invariant sets associated to global strong solutions, when such solutions exist.

For this purpose, we recall now the general concepts about functional invariant sets, attractors, and some general results about them.

Consider problem 4.2. According to theorem 5.2, we can define the solution operator S(t) (t > 0 given), whenever it makes sense, that maps the initial value Z,, to the solution 3( t ) of problem 4.2. For any t > 0, S(t) is an operator from V w into V w .


Since generally speaking, S ( t ) ( t > 0) is only defined and continuous on some part of V w into V" , we define

Obviously we have the following semigroup properties of S(t):

S(0) = I D ( S ( 0 ) ) = V w

S(t + s) = S( t ) 'S(s) (7.2)

on D ( S ( t + s)) Vt,s > 0.

Moreover, for the sake of convenience,. we recall the definitions of functional invariant sets and attractors from Temam [13].

Definition. if

7.1 (i) A set X c V w is a functional invariant set for the semigroup { S ( t ) ) ,

S(t)Eo exists VZo E x VI > 0

S(t)X = x V I > 0 (7.3)

(ii) An attractor set is a set d c V w such that (a) d is an invariant set; (b) d is the w-limit sett of one of its open neighborhoods U As we mentioned at the beginning of this section, our main objective here is to

estimate the Hausdorf and fractal dimensions of the functional invariant sets and attractors for the PEV2s. To this end, we assume the existence of a particular solution E = ( u , F , S ) of problem 4.2 satisfying the following houndedness property

(7.4)

Then we have

Theorem 7.2.t Let Z = ( u , F , S ) be a solution of problem 4.2 with initial value Eo = (uo, Yo, So), satisfying (7.4). Then there exists a functional invariant set X = X(Z,,) for the semigroup S ( t ) , which is bounded in D ( A w ) , and Z(t) converges to X as t + CO.

0 This is the direct application of theorem 1.2 of Temam [13, p 3821.

t That is, .d = & * o m , the closures being taken in V'". $ If it is true that

S(r)% exists VZn E V w Vc > 0 (7.4')

then we have

Theorem 7.2'. attractor .d c D(A"), which is bounded in D(A").

Assume (7.4'); then the semigroup (S(t)) , ,n possesses an absorbing set in V w and a maximal


In order to estimate the dimensions of the functional invariant set X in theorem 7.2 or the attractor d given by theorem 7.2', we should study the linearized form of problem 4.2. So consider a solution E = (u , .F ,S) of problem 4.2 satisfying

E E L"(0,m; V") . (7.5) The linearized equation of (4.18) is

U'+A"U + B ( E , U ) + B ( U , Z ) + B ( U , Z * ) +B(E',U) + € ( U ) = 0

U(0) = 5 E H . (7.6)

We quote as follows a direct application of proposition 2.1 of Temam [13, p 3871 without proof

Theorem 7.3. (i) There exists a unique solution U of (6.6) satisfying U E L2(o,7; v") n Lm(O,T;H) VT > 0 (7.7)

(ii) For any t l > 0, p > 0, D,(S(t)) is open in V", and S ( t , ) is differentiable in D ( S ( t l ) ) equipped with the norm of H . The differential of S ( q ) at a point Eo of D , ( S ( t , ) ) is the mapping

where U is the solution of (7.6).

7.2. Dimensions of the attractors Now we are in position to estimate the Hausdorff and fractal dimensions of the functional invariant set X given by theorem 7.2, or the attractor s2 given by theorem 7.2', of the semigroup S(t) , or problem 4.2. The method used here is the theory developed by Constantin, Foias and Temam [14] (see also [13]).

We follow the notation and procedures of [13]. Let Ui, Uz,. . . , U , be m solutions of (7.6) with initial values tl, rz,. . . ,[, respectively. Then we have

5 E H 4 L(t,;E0)5 = U(t , ) (7.8)

[U, ( f ) A ,. . A Um(t)lAmH = Itl A ' ' . A 5,lAmH exp

Qm(W = Span{ U , (r), . . . , U,,,Cr)).

Tr F'(S(T)SO) 0 Q,@) dr (7.9) L1 where Q,(r) is the orthogonal projection in H onto

Choosing {vj(7)}7=, to be an orthonormal basis of Q,(z)H, we have m

Tr(F'(E(7)) 0 Qm(7)) = I ( F ' ( W ) V ~ ( ~ , ~ ~ ( ~ 1 1 (7.10) j = l

where F'(E(r) ) is defined by F'(Z(7))U - A W U - B ( S ( r ) , U ) - B I ( U , E ( ~ ) ) - B ( U , E * ) - B(Z' ,U) - € ( U ) . (7.11)

Moreover, we introduce

(7.12)

i=l,.lli

qm = lim supq,(t). (7.13)

Then we quote the main result of the Constantin, Foias and Temam theory from [13], which will be used in the remaining part of this section.

I+"


Theorem 7.4. If

q j < -njs + p Q j > 1 (7.14)

then the Hausdorff dimension of X and the fractal dimension of X satisfy

dH(X) < m dp(X) < 2m (7.15)

To obtain the explicit dimension estimates of the functional invariant set X or the attractor d, we should estimate the right-hand side of (7.10). To this end, write

l y j = ( u j , Fi, Si) j = I , . . . , m

and omit temporarily T, then

Tr(F'(8) o Q,) m

= - C ( A W l y j , l y j ) j=!

m

- C ( B ( 2 , w j ) + B ( ~ ~ , 8 ) + B ( ~ ~ , 3 ' ) + 8 ( 8 * , l y ~ ) + E ( l y . ) I ' ly.) J j= I

m m

= - ~ ( A W ~ j . ~ j ) - ~ ( B ( ~ i . 8 ) + B ( ~ , , 2 * ) + E ( l y j ) . l y j ) . (7.17) i-l j=! 1-1

(7.18)

(7.19)

(7.20)

(7.21)

Now we define two quantities y and K, which are related to the energy dissipation, as follows:

Y = e,, S U P 1 1 ~ 1 1 * (7.22) * E X

(7.23)


It follows from (7.21), from the definition of q, and lemma 7.7 below that

(7.24) C 2 512 . < - - m 5 ~ 3 + C m c a x ( Y K + ( ( ~ + E , ) ) &ax

So by theorem 7.4, we have the following theorem, in which we obtain explicit estimates of the dimensions of the functional invariant set X or the attractor a2 in terms of the physically relevant parameter kaX and the enstropy related quantities y and K defined by (7.22), (7.23).

Theorem 7.5. Consider the dynamical system, problem 4.2 in H, and define m by

m - 1 < tea, ( Y K + ( 1 + E , ) * ) ~ ’ * < m (7.25)

where C is an absolute constant independent of the physically relevant parameters. Then

dH(X) < m dF(X) < 2m. (7.26)

Remark 7.6. In the next section, we are going to estimate K in terms of the external forcing, the Reynolds numbers I$,,,, and &ax etc, with a condition on the expansion coefficient & of the density with respect to the temperature. We are, however, not able

0 to give explicit estimates for y in terms of the physically relevant parameters.

Lemma 7.7. We have m

C I I V ~ I I ~ 2 c m s i 3 . j= l

(7.27)

ProoJ that the eigenvalues of AW satisfies

First of all, as in the proof of corollary of theorem 13.6 in [15] , we can show

A j ( A W ) 2 Cj213

Then lemma 2.1 in p 302 of [13] implies (7.27). U

7.3. Some further estimates In this subsection, we intend to estimate the enstropy related quantity K, defined by (7.23), in terms of the physically relevant parameters. To do this, we assume that

where CO a positive constant such that

(7.28)

(7.29)


Now we consider a solution E = ( u , T , S ) of problem 4.2, salisfying (7.4). Set E, = ( O , O , S ) in (4.18); we obtain

so that

(7.30)

Now we set E, = E in (4.18); hence

It follows that

Then by definition (7.23), we have

K < CQFI3.

Moreover, by direct computation and by the definition of F , we obtain

(7.31)

(7.32)

(7.33)

Therefore

Theorem 7.8. dimensions of the functional invariant set X or the attractor d satisfy

Assume (7.28) and the condition in theorem 7.5 are satisfied. Then the

dH(X) < m dF(X) < 2m (7.34)

with m given by

Here C is an absolute constant independent of the physically relevant parameters, and y is defined by (7.22).


8. Dimension estimates for the attractors of the BEs

8.1. Dimension estimates

As in section 7, we study in this section the dimension estimates of the attractors or the functional invariant sets of the BEs.

Thanks to theorem 6.3, we can define the solution operator S E @ ) (t > 0 given), whenever it makes sense, that maps the initial value Zo to the solution E(t) = (U, w, Y, S ) of problem 6.1. For any t > 0, S E @ ) is an operator from V B into V B . Then similar results, and definitions from definition 7.1 to theorem 7.4 can be applied to this case. In particular, if it is true that there exists a solution E = Z(t) = (U, w,.T,S) of problem 6.1 satisfying

then there exists a functional invariant set X B = X B ( Z ) for the semigroup S B or the problem 6.1 or the BEs, which is bounded in D(AB) , and E(t) converges to X B as t + CD. Moreover, if it is true that

SB(t)s,, exists vz0 E vB vt a o (8.2)

then the semigroup {SB(t)},,, possesses an absorbing set in V B and a maximal attractor dB c D(AB) , which is bounded in D(AB) .

To estimate the dimensions of the functional invariant set X B or the attractor dB of the BEs, as in the previous section, we consider a solution E = (U, w, T, S ) of nrnhlem 6 1 rit irfrrino 1% 1) 1 et r i 11. I 1 hp m mllntinn. nf Y.VV.I... ".&, "11.1 .,U. ,-..,. 1 - 1 -,,- ,...,- "-."..--." -. U' + A B U + B E @ , U) + B B ( U , - E) + B E ( U , sB) -' + BB(E.;I, U) + E B ( U ) = 0 in H B (8.3)

with initial values tl, t2,. . . , t, respectively. Then we define

(8.4)

q, = limsupq,(t) (8.5) 1-C2

where &(rj is the orihogonai projection in ii" onto

Q,(TW = span{u, (~) ,..., ~ , ( r ) )

and

FL(S(T))U = -ABU -EB(=(?), U) -BB(U,Z(r ) ) - BB(U,S.;I) -BB(E.;I, U) - E B ( U ) .

(8.6)

Choosing

u , . = ( u j , w j , . T p S j ) J j = 1 ,..., m


to be an orthonormal basis of Q,(r)HB, and omitting temporarily T, we obtain

m

Tr(FL(E) 0 em) = - z [ A B W j , Wjls j=l

m

< (as in (7.18)-(7.20))

By Mttivier’s result on the eigenvalues of the Navier-Stokes equations and lemma 2.1 in p 302 of 1131, we have

So it suffices to see that

where y B and K’ are defined by

(8.10)

(8.11)

(8.12)

Therefore

Theorem 8.1. Consider the dynamical system, problem 6.1 in H B ,and define m by

m - 1 i c e a X (y’KB + (1 + 8T)2)3’2 < m (8.13)

where C is an absolute constant independent of the physically relevant parameters. Then

d H ( X B ) 4 m d p ( X B ) < 2m. (8.14)

U Remark 7.6. Observe that (8.13) has the same form as (7.25) in theorem 7.5.


8.2. Some more estimates

Now we want to estimate the enstropy related quantity K ~ , defined by (7.23), in terms of the physically relevant parameters. As in section 7, we assume that

with CO a positive constant such that

Then with computations similar to that of section 7.2, we can establish

(8.15)

(8.16)

(8.17)

Finally, we have proved the following theorem

Theorem 8.3. As in theorem 7.8, assume that (8.15) and the condition in theorem 8.1 are true. Then the dimensions of the functional invariant set XB or the attractor dE satisfy

d H ( X E ) < m d F ( X E ) < 2m (8.18)

with m given by

where C is an absolute constant independent of the physically relevant parameters, and y E is defined by (8.1 I).

Acknowledgments

This work was partially supported by the National Science Foundation under Grant NSF-DMS-9024769, by the Department of Energy under grant DOE-DE-F602- 92ER25120 and by the Research Fund of Indiana University.

Appendix. Table of principal notation

Functions f = ~ C O S B

"3 U

W

T

Coriolis parameter the 3D velocity horizontal velocity vertical velocity temperature

1052 J - L Lions et a/

S P P Tb

'b

Tu, i,

Constants and parameters

h, ff

a Z

u > o To > 0 so > 0 Po > 0 R > O P> " Ps3 "s pT3 "T

E = z/a

Bs, Bs f l T , B T

Others k S2

A grad

s.'

grad, grad,

Old,

salinity pressure density temperature on the bottom of the ocean salinity on the bottom of the ocean depth functions of the ocean wind stress at the interface

radius of the earth reference value of the depth of the ocean acceleration due to gravity reference value of the horizontal velocity reference value of the temperature reference value of the salinity reference value of the density angular velocity of the earth eddy viscosity coefficients eddy diffusivity coefficients for T eddy diffusivity coefficients for S

expansion coefficients of p with respect to T expansion coefficients of p with respect to S

vertical unit vector 2D unit sphere 2D sphere of radius a Laplacian on s2 or S.' gradient on S 2 or S,' gradient in the 3D ocean gradient in M

of magnitude of the a N 6.4 x lo6 m

leuant scales ( c f Bryan 181)

7.9 10-4 N lo-' m s-I z E 5 x io3 m

g 1 9 . 8 m s - ~ R 10-4 S-I po N IO3 kg m-3 RO 1 2.1 x 10-4

- 3.8 1 0 - ~ Re2 - 3.8 x IO-^

p N lo4 m2 s-' p, E ps E 2.5 x 10' m2 s-'

- N 1.5 x lo-'

v cz vT N vs N 1.5 x m2 s-I

1

1

1

I Re I - N 3.9 x IO-' Rt. Rt,

l * Rs2 - N 3.8 x


References

Lions J L, Temam R and Wang S 1992 New formulations of the primitive equations of the atmosphere

Washington W M and Parkinson C L 1986 An Introduction to Three-Dimenaionol Climate Modeling

Lions 1 L, Temam Rand Wang S 1991 Some mathematical remarks concerning the primitive equations

Linns I Li Temam R. and Wang S !?RI %me ma!hema!ica! rernarks coEcerning !he primi!iw equations

Lions J L 1969 Quelques Mkthodes de R4solution des Problemzs aux Lirnitrs iVon L i h i r e s (Paris:

Temam R 1984 Nnuier-Stokes Equations, Raised Edition (Studies in Mathematics and its Applications

Foias C, Manley 0 and T a t " R 1987 Attractors far the Benard problem: existence and physical

Bryan K 1969 Climate and the ocean circulation: 111. The ocean model Mon. Weather Reu. 97 80627 Bryan K 1969 A numerical method for the study of the circulation of the world ocean J . Comput.

Semtner A J Jr 1974 An oceanic general circulation model with hattam topography Numerical

Lions J L and Magenes E 1972 Nonhomogeneous Boundary Value Problems and Applications (New

Foias C and Temam R 1979 Some analytic and geometric properties of the solutions 0 1 the Navier

Temam R 1988 Infinite Dimensional Dynamical Systems in Mechanics ond Physics (Applied Mothemoticol

Constantin P, Fabas C and Temam R 1985 Attractors Representing Turbulent Flows Memoirs of AMS

Agmon S I965 Lectures on clliptic boundary V ~ U C problems Mathematical Studies (New York: Van

Dautray R and Lions J L 1990 Mathematical Analysis and Numericol Methods for Science and

Temam R 1983 Navier-Stokes Equations and Nonlinear Functionol Analysis (CBMS-NSF Regionol

Wang S 1988 On solvability for the equations of the large-scale atmospheric motion ntesis Lanrhou

Wang S 1992 On the 20 model o f large scale atmospheric motion: well-posedness and attractors

Wang S 1990 Attractors for the 2D model of large scale atmosphere to appear Wang S 1992 Attractors for !he 30 haroclinic quasi-geostrophic equations of large scale atmosphere

Wang S, Huang 1 and Chou J 1989 Some properties of the equations of large-scale atmaspheric

Charney J G 1962 Integration of the primitive and balance equations Proc. In t . Symp. Numerical

Charney I G, Fjnrtaft R and von Neumann J 1950 Numerical integration of the barotropic vorticity

Gill A E 1982 Atmosphere-Ocean Dynamics (New York: Academic) Lorenz E 1963 Deterministic non-periodic flow J . Atm. Sei. 20 130-41 MCtivier G 1978 Valeurs propres d'ap&teurs definis par la restriction de systemes variaationnels a

Richardson L F 14ti w2other Prediction by Numerical Process (Cambridge: Cambridge University

and applications Nonlinearity 5 237-88

(Oxford: Oxford University Press)

of the atmosphere, to appear

of the atmosphere with vertical velocity, to appear

Dunod)

2) (Amsterdam: North-Holland)

hounds on their fractal dimension Nonlinear Anal. 11 939-67

Phys. 4 347-76

Simulation of Weather and Climate, Tech. Report No. 9, UCLA

York: Springer)

Stokes equations J . Moth. P u r a Appl. 58 339-68

Sciences 68) (Berlin: Springer)

53 no 314

Nostrand)

Technology, 6 uolumes (New York: Springer)

Conference Series in Applied Mathematics (Philadelphia, PA: SIAM)

University

Nonlinear Anal-TMA I8:l 17-60

3. Moth. Anal. Appl. 165 266-83

motion Science in Chino B 33 328-36

Weather Piediction Tokyo

equation Tellus 2 237-54

des sous-espaces J. Math. P u r a Appl. 133-56

Press (reprinted 1965 New York: Dover))

on the equations of the large-scale ocean*

Documents