Some Mathematical Problems in Geophysical Fluid Dynamics

Transcription

1 Some athematical Problems in Geophysical Fluid Dynamics adalina Petcu Laboratoire de athématiques et Applications, UR 686, Universîté de Poitiers, France Roger. Temam The Institute for Scientific Computing and Applied athematics, Indiana University, Bloomington, IN 4745, USA ohammed Ziane University of Southern California, athematics Department, 14 W. 36 Place, Los Angeles, CA 949, USA Abstract This chapter reviews the recently developed mathematical setting of the primitive equations (PEs) of the atmosphere, the ocean, and the coupled atmosphere and ocean. The mathematical issues that are considered here are the existence, uniqueness, and regularity of solutions for the time-dependent problems in space dimensions and 3, the PEs being supplemented by a variety of natural boundary conditions. The emphasis is on the case of the ocean that encompasses most of the mathematical difficulties. This chapter is devoted to the PEs in the presence of viscosity, while the PEs without viscosity are considered in the chapter by Rousseau, Temam, and Tribbia in the same volume. Whereas the theory of PEs without viscosity is just starting, the theory of PEs with viscosity has developed since the early 199s and has now reached a satisfactory level of completion. The theory of the PEs was initially developed by analogy with that of the incompressible Navier Stokes equations, but the most recent developments reported in this chapter have shown that unlike the incompressible Navier-Stokes equations and the celebrated illenium Clay problem, the PEs with viscosity are well-posed in space dimensions and 3, when supplemented with fairly Computational ethods for the Atmosphere and the Oceans Copyright 8 Elsevier B.V. Special Volume (Roger. Temam and Joseph J. Tribbia, Guest Editors) of All rights reserved HANDBOOK OF NUERICAL ANALYSIS, VOL. XIV ISSN P.G. Ciarlet (Editor) DOI 1.116/S (8) Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

2 568. Petcu et al. general boundary conditions. This chapter is essentially self-contained, and all the mathematical issues related to these problems are developed. A guide and summary of results for the physics-oriented reader is provided at the end of the Introduction (Section 1.4). 1. Introduction The aim of this chapter is to address some mathematical aspects of the equations of geophysical fluid dynamics, namely existence, uniqueness, and regularity of solutions. The equations of geophysical fluid dynamics are the equations governing the motion of the atmosphere and the ocean, and they are derived from the conservation equations from physics, namely conservation of mass, momentum, energy, and some other components such as salt for the ocean and humidity (or chemical pollutants) for the atmosphere. The basic equations of conservation of mass and momentum, i.e., the three-dimensional (3D) compressible Navier-Stokes equations, contain however too much information, and we cannot hope to numerically solve these equations with enough accuracy in a foreseeable future. Owing to the difference in sizes of the vertical and horizontal dimensions, both in the atmosphere and in the ocean (1 km versus several thousands of kilometers), the most natural simplification leads to the so-called primitive equations (PEs), which we study in this chapter. We continue this introduction by briefly describing the physical and mathematical backgrounds of the PEs Physical background The PEs are based on the so-called hydrostatic approximation, in which the conservation of momentum in the vertical direction is replaced by the simpler, hydrostatic equation (see Eq. (.5)). As far as we know, the PEs were essentially introduced by Richardson in 19; when it appeared that they were still too complicated, they were abandoned and, instead, attention was focused on simpler models, such as the barotropic and the geostrophic and quasigeostrophic models, considered in the late 194s by von Neumann and his collaborators, in particular Charney. With the increase of computing power, interest eventually returned to the PEs, which are now the core of many Global Circulation odels (GCs) or Ocean Global Circulation odels (OGCs) available at the National Center for Atmospheric Research and elsewhere. GCs and OGCs are very complex models that contain many physical components (for the atmosphere, the chemistry (equations of concentration of pollutants), the physics of the cloud (radiation of solar energy, concentration of vapor), the vegetation, the topography, the albedo, for the oceans, phenomena such as the sea ice or the topography of the bottom of the oceans). Nevertheless, the PEs that describe the dynamics of the air or the water and the balance of energy are the central components for the dynamics of the air or the water. For some phenomena, there is need to give up the hydrostatic hypothesis and then nonhydrostatic models are considered, such as 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

3 Some athematical Problems in Geophysical Fluid Dynamics 569 Table 1.1 Level of physical complexity (richness) Three-dimensional Navier-Stokes equations Nonhydrostatic models Primitive equations (hydrostatic equations) Shallow water equations Quasi-geostrophic models Two-dimensional barotropic equations in Laprise [199] or Smolarkiewicz, argolin and Wyszogrodzki [1]; these models stand at an intermediate level of physical complexity between the full Navier- Stokes equations and the PEs hydrostatic equations. Research on nonhydrostatic models is ongoing and, at this time, there is no agreement, in the physical community, for a specific model. In this hierarchy of models for geophysical fluid dynamics, let us add also the shallow water equation corresponding essentially to a vertically integrated form of the Navier- Stokes equations; from the physical point of view, they stand as an intermediate model between the primitive and the quasi-geostrophic equations. In summary, in terms of physical relevance and the level of complexity of the physical phenomena they can account for, the hierarchy of models in geophysical fluid dynamics is as in Table 1.1. We remark here also that much study is needed for the boundary conditions from both the physical and the mathematical point of views. As we said, our aim in this chapter is the study of mathematical properties of the PEs. In the above, and in all of this chapter, the PEs that we consider are the PEs with viscosity; the PEs without viscosity are studied in the chapter by Rousseau, Temam, and Tribbia [8] in this volume. The PEs without viscosity raise questions of a totally different nature. In particular, whereas the PEs with viscosity bear some similarity with the incompressible Navier-Stokes equations as we explain below, the PEs without viscosity are different from the Euler equations of incompressible inviscid flows in many respects (see the already quoted chapter of Rousseau, Temam, and Tribbia). 1.. athematical background The level of mathematical complexity of the equations in Table 1.1 is not the same as the level of physical complexity: at both ends, the quasi-geostrophic models and barotropic equations are mathematically well understood (at least in the presence of viscosity; see Wang [199a,b], and despite its well-known limitations, the mathematical theory of the incompressible Navier-Stokes equations is also relatively well understood. On the other hand, nonhydrostatic models are mathematically out of reach, and there are much less 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

4 57. Petcu et al. mathematical results available for the shallow water equations than for the Navier-Stokes equations, even in space dimension (see, however Orenga [1995]). The mathematical theory of the (viscous) PEs has developed in two stages. The first stage ranging from the article of Lions, Temam and Wang [1993a,b] to the review article by Temam and Ziane [4] concentrated on the analogy of the PEs with the 3D incompressible Navier-Stokes equations. Indeed, and as we show below, the PEs although physically poorer than the Navier-Stokes equations, in some sense, they structurally more complicated than the incompressible Navier-Stokes equations. Indeed, this is due to the fact that the nonlinear term in the Navier-Stokes equations, also called inertial term, is of the form velocity first-order derivatives of velocity, whereas the nonlinear term for the PEs is of the form first-order derivatives of horizontal velocity first-order derivatives of horizontal velocity. The mathematical study of the PEs was initiated by Lions, Temam and Wang [199a,b]. They produced a mathematical formulation of the PEs that resembles that of the Navier- Stokes due to Leray and obtained the existence for all time of weak solutions (see Section and the original articles by Lions, Temam and Wang [199a,b, 1995] in the list of references). Further works, conducted during the 199s and especially during the past few years, have improved and supplemented the early results of these authors by a set of results that, essentially, brings the mathematical theory of the PEs to that of the 3D incompressible Navier-Stokes equations, despite the added complexity mentioned above; this added complexity is overcome by a nonisotropic treatment of the equations (of certain nonlinear terms), in which the horizontal and vertical directions are treated differently. In summary, the following results have been obtained, which were presented in the review article by Temam and Ziane [4] and appear herein in Sections and 3: (i) Existence of weak solutions for all time (dimensions and 3) (see Sections ). (ii) In space dimension 3, existence of a strong solution for a limited time (local in time existence) (see Section 3.1). (iii) In space dimension, existence and uniqueness for all time of a strong solution (see Section 3.3). (iv) Uniqueness of a weak solution in space dimension (see Section 3.4). In the above, the terminology that is normally used for Navier-Stokes equations: the weak solutions are those with finite (fluid) kinematic energy (L (L ) and L (H 1 )), and the strong solutions are those with finite (fluid) enstrophy (L (H 1 ) and L (H )). Essential in the most recent developments (ii) (iv) above is the H -regularity result for a Stokes-type problem appearing in the PEs, the analog of the H -regularity in the Cattabriga Solonnikov results on the usual Stokes problem; the whole Section 4 is devoted to this problem. 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

5 Some athematical Problems in Geophysical Fluid Dynamics 571 The second stage of the mathematical theory of the (viscous) PEs is more recent. It is based on the observation that the pressure-like function (the surface pressure) is in fact a two-dimensional (D) function (a function of the horizontal variables and time) and because of that the 3D PEs are also close to a D system. Technically, by suitable estimates of the surface pressure, the difficulties related to the pressure are overcome. This approach was developed in the two independent articles (with different proofs) by Cao and Titi [7] and by Kobelkov [6], for the case of an ocean with a flat bottom and Neuman boundary condition. The case of a varying-bottom topography the Dirichlet boundary condition studied in the subsequent article of Kukavica and Ziane [7a,b]. These three articles combine the above mentioned results of local existence of a strong solution and the new a priori estimates to show that the strong solution is defined for all time. These newest results appear in Section Content of this chapter Because of space limitation, it was not possible to consider all relevant cases here. Relevant cases include, the ocean, the atmosphere, and the coupled ocean and atmosphere, on the one hand, and, on the other hand, the study of global phenomena on the sphere (involving the writing of the equations in spherical coordinates), and the study of midlatitude regional models in which the equations are projected on a space tangent to the sphere (the Earth), corresponding to the so-called β-plane approximation: here, x is the west east axis, y is the south north axis, and z is the ascending vertical. In this chapter, we have chosen to concentrate on the cases mathematically most significant. Hence for each case, after a brief description of the equations on the sphere (in spherical coordinates), we concentrate our efforts on the corresponding β-plane Cartesian coordinates). Indeed, in general, going from the β-plane Cartesian coordinates to the spherical case necessitates only the proper handling of terms involving lower order derivatives; full details concerning the spherical case can be found also in the original articles by Lions, Temam and Wang [199a,b, 1995]). In the Cartesian case of emphasis, generally, we first concentrate our attention on the ocean. Indeed, as we will see in Section, the domain occupied by the ocean contains corners (in dimension ) or wedges (in dimension 3); some regularity issues occur in this case, which must be handled using the theory of regularity of elliptic problems in nonsmooth domains (Grisvard [1985], Kozlov, azya and Rossmann [1997], azya and Rossmann [1994]). For the atmosphere or the coupled atmosphere ocean (CAO), the difficulties are similar or easier to handle hence, most of the mathematical efforts will be devoted to the ocean in Cartesian coordinates. In Section, we describe the governing equations and derive the result of existence of weak solutions with a method different from that of the original articles by Lions, Temam and Wang [199a,b, 1995], thus allowing more generality (for the ocean, the atmosphere, and the CAO). In Section 3, we study the existence of strong solutions in space dimensions 3 and and a wealth of other mathematical results, regularity in H m higher Sobolev spaces, 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

6 57. Petcu et al. C regularity, Gevrey regularity, and backward uniqueness. We establish in dimension 3 the existence and uniqueness of strong solutions on a limited interval of time (Section 3.1) and then for all time (see Section 3.). In dimension, we prove the existence and uniqueness, for all time, of such strong solutions (see Section 3.3). Section 3.4 contains a technical result. In Section 3.5, we consider the D space-periodic case and prove the existence of solutions for all time, in all H m,m. In Section 3.6, we prove the Gevrey regularity of the solutions and in Section 3.7. the backward uniqueness result. Section 4 is technically very important, and many results of Sections and 3 rely on it: this section contains the proof of the H -regularity of elliptic problems, which arise in the PEs. This proof relies, as we said, on the theory of regularity of solutions of elliptic problems in nonsmooth domains. It is shown there that the solutions to certain elliptic problems enjoy certain regularity properties (H -regularity, i.e., the function and their first and second derivatives are square integrable); the problems corresponding to the (horizontal) velocity, the temperature, and the salinity are successively considered. The study in Section 4 contains many specific aspects that are explained in detail in the introduction to that section. ore explanations and references will be given in the introduction of or within each section. As mentioned earlier, the mathematical formulation of the equations of the atmosphere, ocean, and CAO was derived by Lions, Temam and Wang [199a,b, 1995]. For each of these problems, these articles also contain the proof of existence of weak solutions for all time (in dimension 3 with a proof that easily extends to dimension ). An alternative slightly more general proof of this result is given in Section. Concerning the strong solutions, the proof given here of the local existence in dimension 3 is based on the article by Hu, Temam and Ziane [3]. An alternate proof of this result is due to Guillén-González, asmoudi and Rodríguez-Bellido [1]. In dimension, the proof of existence and uniqueness of strong solutions, for all time, for the considered system of equations and boundary conditions is new and based on an unpublished manuscript by Ziane []. This result is also established, for a simpler system (without temperature and salinity), by Bresch, Kazhikhov and Lemoine [4]. ost of the results of Sections are due to. Petcu, alone or in collaboration with D. Wirosoetisno Summary of results for the physics-oriented reader The physics-oriented reader will recognize in Eqs. (.1) (.5) the basic conservation laws: conservations of momentum, mass, energy and salt for the ocean, equation of state. In Eqs. (.6) and (.7) appears the simplification due to the Boussinesq approximation, and in Eqs. (.11) (.16) the simplifications resulting from the hydrostatic balance assumption. Hence Eqs. (.11) (.16) are the PEs of the ocean. The PEs of the atmosphere appear in Eqs. (.116) (.11), and those of the CAO are described in Section.5. Concerning, to begin, the ocean, the first task is to write these equations, 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

7 Some athematical Problems in Geophysical Fluid Dynamics 573 supplemented by the initial and boundary conditions, as an initial value problem in a phase space H of the form du + AU + B(U, U) + E(U) = l, dt (1.1) U() = U, (1.) where U is the set of prognostic variables of the problem, i.e., the horizontal velocity v = (u, v), the temperature T, and the salinity S, U = (v,t,s); (see Eq. (.66)). The phase space H consists, for its fluid mechanics part, of (horizontal) vector fields with finite kinetic energy. We then study the stationary solutions of Eq. (1.1) in Section.., and in Theorem.; we prove the existence for all times of weak solutions of Eqs. (1.1) and (1.), which are solutions in L (,t 1 ; L ) and L (,t 1 ; H 1 ) (bounded kinetic energy and square integrable enstrophy for the fluid mechanics part). A parallel study is conducted for the atmosphere and the CAO in Sections.4 and.5. Section 4 is mathematically very important although technical. For the physics-oriented reader, the most important results are those of Sections and 3. Section contains the weak formulation of the PEs and shows the extensive use of the balance of energy principles to prove them. The tools of balance of energy are also those needed for the study of stability of numerical results, and they are therefore both physically and computationally revelant. The main results of Section 3 are the existence and uniqueness of strong solutions for all time, now both available in space dimensions and 3. Noteworthy also in this section are the results concerning the Gevrey regularity of the solutions, which implies in particular an exponential decay of the Fourier coefficients, results that have been used in the recent articles by Temam and Wirosoetisno [7, 8] to prove that the PEs can be approximated by a finite-dimensional model up to an exponentially small error. The results of existence and uniqueness for all time of strong solutions are also important for the study of the dynamical system generated by the PE (attractors, etc) (see the first developments of this theory in the article by Ju [7], and quoted therein some previous partial results). The study presented in this chapter is only a small part of the mathematical problems on geophysical flows, but we believe it is an important part. We did not try to produce here an exhaustive bibliography. Further mathematical references on geophysical flows will be given in the text, (see also the bibliography of the articles and books that we quote). There is also of course a very large literature in the physical context; we only mentioned some of the books that were very useful to us such as Haltiner and Williams [198], Pedlovsky [1987], Trenberth [199], Washington and Parkinson [1986], and Zeng [1979]. The mathematical theory presented in this chapter focuses on questions of existence, uniqueness, and regularity of solutions, the so-called issue of well-posedness. From the geophysics point of view, these issues relate, according to von Neumann [1963], to the short-term forecasting. The other issues as described in von Neumann [1963] relate to the long-term climate and intermediate climate dynamics. Pertaining to the long-term 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

8 574. Petcu et al. climates are the questions of attractors for the PEs, which have been addressed in e.g., Ju [7], Lions, Temam and Wang [199a,b, 1993a, 1995] (see also the references therein) For intermediate climate dynamics, the mathematical issues relate to successive bifurcations, transition, and instabilities (see, e.g., a and Wang [5a,b], and the chapter by Simonnet, Dijkstra and Ghil [8] in this volume). Besides the efforts of the authors, we mention in several places that this study is based on joint works with Lions, Wang, Hu, Wirosoetisno and others. Their help is gratefully acknowledged, and we pay tribute to the memory of Jacques-Louis Lions. The authors thank Denis Serre and Shouhong Wang for their careful reading of an earlier version of this manuscript and for their numerous comments that significantly improved the manuscript. They extend also their gratitude to Daniele Le eur and Teresa Bunge who typed significant parts of the manuscript. This chapter is an updated version of the article by Temam and Ziane [4]. It is included in this volume by invitation of PG Ciarlet, editor of the Handbook of Numerical Analysis. The authors thank PG Ciarlet for his invitation and the Elsevier Company for endorsing it.. The PEs: weak formulation, existence of weak solutions As explained in the introduction to this chapter, our aim in this section is first to present the derivation of the PEs from the basic physical conservation laws. We then describe the natural boundary conditions. Then, on the mathematical side, we introduce the function spaces and derive the mathematical formulation of the PEs. Finally, we derive the existence for all time of weak solutions. We successively consider the ocean, the atmosphere, and the CAO..1. The PEs of the ocean Our aim in this section is to describe the PEs of the ocean (see Section.1.1), and we then describe the corresponding boundary conditions and the associated initial and boundary value problems (Section.1.) The PEs It is considered that the ocean is made up of a slightly compressible fluid with Coriolis force. The full set of equations of the large-scale ocean are the following: the conservation of momentum equation, the continuity equation (conservation of mass), the thermodynamics equation (that is the conservation of energy equation), the equation of state, and the equation of diffusion for the salinity S: ρ dv 3 + ρ V p + ρg = D, dt (.1) dρ dt + ρ div 3 V 3 =, (.) 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

9 Some athematical Problems in Geophysical Fluid Dynamics 575 dt dt = Q T, (.3) ds = Q S, dt (.4) ρ = f(t, S, p). (.5) Here, V 3 is the 3D velocity vector, V 3 = (u,v,w), ρ, p, T are the density, pressure, and temperature, S is the concentration of salinity, g = (,,g)is the gravity vector, D is the molecular dissipation, and Q T and Q S are the heat and salinity diffusions; is the rotation vector. The analytic expressions of D, Q T, and Q S will be given below. We denote by 3, 3, div 3, the 3D Laplacian, gradient, and divergence, leaving,, div to their D versions more frequently used. The Boussinesq approximation. From both the theoretical and the computational point of views, the above systems of equations of the ocean seem to be too complicated to study. So it is necessary to simplify them according to some physical and mathematical considerations. The ach number for the flow in the ocean is not large and therefore, as a starting point, we can make the so-called Boussinesq approximation in which the density is assumed constant, ρ = ρ, except in the buoyancy term and in the equation of state. This amounts to replacing Eqs. (.1) and (.) by dv 3 ρ + ρ V p + ρg = D, dt (.6) div 3 V 3 =. (.7) Consider the spherical coordinate system (θ,φ,r), where θ ( π/ < θ < π/) stands for the latitude of the earth, φ ( φ π) on the longitude of the earth, r for the radial distance, and z = r a for the vertical coordinate with respect to the sea level, and let e θ, e φ, and e r be the unit vectors in the θ-, φ-, and z-directions, respectively. Then we write the velocity of the ocean in the form V 3 = v θ e θ + v φ e φ + v r e r = v + w, (.8) where v = v θ e θ + v φ e φ is the horizontal velocity field and w is the vertical velocity. Another common simplification is to replace, to first order, r by the radius a of the earth. This is based on the fact that the depth of the ocean is small compared with the radius of the earth. In particular, d dt = t + v θ r θ + v φ r cos θ φ + v r r (.9) becomes d dt = t + v θ a θ + v φ a cos θ φ + v z z, (.1) 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

10 576. Petcu et al. 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, i.e., Eqs. (.11) (.16) hereafter (for the equation of state (Eq. (.16)), see Remark.1): v t + vv + w v z + 1 v p + sin θ v μ v v ν v =, ρ z (.11) w + v w + w w t z + 1 p ρ z + ρ w g μ v w ν v =, ρ z (.1) div v + w =, z (.13) T + v T + w T t z μ T T T ν T z =, (.14) S t + vs + w S z μ S S S ν S =, (.15) z ( ρ = ρ 1 βt (T T r ) + β S (S S r ) ), (.16) where v is the horizontal velocity of the water, w is the vertical velocity, and T r and S r are averaged (or reference) values of T and S. The diffusion coefficients μ v,μ T,μ S and ν v,ν T,ν S are different in the horizontal and vertical directions, accounting for some eddy diffusions in the sense of Smagorinsky [1963]. The differential operators are defined as follows. The (horizontal) gradient operator grad = is defined by grad p = p = 1 p a θ e θ + 1 p a cos θ φ e φ. (.17) The (horizontal) divergence operator div = is defined by div(v θ e θ + v φ e φ ) = v = 1 ( (vθ cos θ) + v ) φ. (.18) a cos θ θ φ The derivatives v ṽ and v T of a vector function ṽ and a scalar function T (covariant derivatives with respect to v) are v ṽ = { vθ ṽ θ θ + v φ ṽ θ a cos θ φ v } φṽ φ cot θ e θ a a { vθ ṽ φ + a θ + v φ ṽ φ a cos θ φ ṽθv φ a tan θ } e φ, (.19) v T = v θ T a θ + v φ T a cos θ φ. (.) 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

11 Some athematical Problems in Geophysical Fluid Dynamics 577 oreover, we have used the same notation to denote the Laplace Beltrami operators for both scalar functions and vector fields on Sa, the D sphere of radius a centered at. ore precisely, we have T = { ( 1 a cos θ T ) + 1 } T cos θ θ θ cos θ φ, (.1) v = (v θ e θ + v φ e φ ) { = v θ { + v φ sin θ v φ a cos θ φ sin θ v θ a cos θ φ } v θ a cos e θ θ v φ a cos θ } e φ, (.) where in Eq. (.), v θ, and v φ are defined by Eq. (.1), and in Eq. (.1), T is any given (smooth) function on Sa, the D sphere of radius a. Remark.1. The equation of state for the ocean is given by Eq. (.5). Only empirical forms of the function ρ = f(t, S, p) are known (seewashington and Parkinson [1986, pp ]). This equation of state is generally derived on a phenomenological basis. It is natural to expect that ρ decreases if T increases and that ρ increases if S increases. The simplest law is Eq. (.16) corresponding to a linearization around average (or reference) values ρ,t r,s r of the density, the temperature, and the salinity, and β T and β S are positive constant expansion coefficients. uch of what follows extends to more general nonlinear equations of state. Remark.. The replacement of r by Eq. (.1) in the differential operators implies a change of metric in R 3, where the usual metric is replaced by that of S a R, S a, the D sphere of radius a centered at O. Remark.3. In a classical manner, the Coriolis force ρ V 3 produces the term sin θk v and a horizontal gradient term that is combined with the pressure so that p in Eq. (.11) is the so-called augmented pressure. The hydrostatic approximation. It is known that for large-scale ocean, the horizontal scale is much bigger than the vertical one (5 1 km versus a few thousands kilometers). Therefore, the scale analysis (see Pedlovsky [1987]) shows that p/ z and ρg are the dominant terms in Eq. (.1), leading to the hydrostatic approximation p = ρg, (.3) z which then replaces Eq. (.1). The approximate relation is highly accurate for the large-scale ocean, and it is considered as a fundamental equation in oceanography. From the mathematical point of view, its justification relies on tools similar to those used in Section Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

12 578. Petcu et al. The rigorous mathematical justification of the hydrostatic approximation is given by Azérad and Guillén [1]. In this chapter, the authors studied the asymptotic behavior of the incompressible Navier-Stokes equations when the depth goes to zero, and they proved that the solutions of the Navier-Stokes equations converge to a weak solution of the PEs. The mathematical details will not be discussed in this chapter, (see however Azérad and Guillén [1], as well as Remark 4.1 in Section 4.1). Using the hydrostatic approximation, we obtain the following equations called the PEs of the large-scale ocean: v t + vv + w v z + 1 v p + sin θk v μ v v ν v ρ z = F v, (.4) p = ρg, z (.5) div v + w =, z (.6) T + v T + w T t z μ T T T ν T z = F T, (.7) S t + vs + w S z μ S S S ν S z = F S, (.8) ( ρ = ρ 1 βt (T T r ) + β S (S S r ) ). (.9) Note that F v, F T, and F S corresponding to volumic sources (of horizontal momentum, heat, and salt) vanish in reality; they are introduced here for mathematical generality. We also set = k, where k is the unit vector in the direction of the poles (from south to north). Remark.4. At this stage, the unknown functions can be divided into two sets. The first one is called the prognostic variables, v,t,s(four scalar functions); we aim to write the PEs as an initial-boundary value problem for these unknowns, and we set U = (v,t,s). The second set of variables comprises p, ρ, w; they are called the diagnostic variables. In Section.1., we will see how, using the boundary condition, one can, at each instant of time, express the diagnostic variables in terms of the prognostic variables (a fact that is already transparent for ρ in Eq. (.9). Remark.5. We integrate Eq. (.8) over the domain occupied by the fluid that is described in Section.1.. Using the Stokes formula and taking into account Eq. (.6) and the boundary conditions (also described in Section.1.), we arrive at d S d = F S d; (.3) dt hence, t S d = S d + F S d dt. t 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

13 Some athematical Problems in Geophysical Fluid Dynamics 579 In practical applications, F S = as we said, and the total amount of salt S d is conserved. In all cases, we write S = S 1 S d, F S = F S 1 F S d, (.31) where is the volume of, and we see that S satisfies the same Eq. (.15), with F S replaced by F S. From now on, dropping the primes, we consider Eq. (.15) as the equation for S and we thus have S d =, F S d =. (.3).1.. The initial and boundary value problems We assume that the ocean fills a domain of R 3, which we describe as follows (see Figure.1): The top of the ocean is a domain Ɣ i included in the surface of the earth S a (sphere centered at of radius a). The bottom Ɣ b of the ocean is defined by (z = x 3 = r a) z = h(θ, ϕ), where h is a function of class C at least on Ɣ i ; it is assumed also that h is bounded from below, <h h(θ, ϕ) h, (θ, ϕ) Ɣ i. (.33) The lateral surface Ɣ l consists of the part of cylinder (θ, ϕ) Ɣ i, h(θ, ϕ) z. (.34) Remark.6. Let us make two remarks concerning the geometry of the ocean; the first one is that for mathematical reasons, the depth is not allowed to be (h h > ) and thus beaches are excluded. The second one is that the top of the ocean is flat (spherical), not allowing waves; this corresponds to the so-called rigid lid assumption in oceanography. The assumption h> can be relaxed for some of the following results, but this will not be discussed here. The rigid lid assumption can be also relaxed by the introduction of an additional equation for the free surface, but this also will not be considered. G i G l G l G b Fig..1 The ocean. 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

14 58. Petcu et al. Boundary conditions. There are several sets of natural boundary conditions can be associated with the PEs, for instance, the following: On the top of the ocean Ɣ i (z = ): v ν v z + α ( v v v a ) = τ v, w =, T ν T z + α ( T T T a ) =, (.35) S z =. At the bottom of the ocean Ɣ b (z = h(θ, ϕ)): v =, w =, (.36) T S =, =. n T n S On the lateral boundary Ɣ l ( h(θ,ϕ)<z<,(θ,ϕ) Ɣ i ): T S v =, w =, =, =. (.37) n T n S Here, n = (n H,n z ) is the unit outward normal on decomposed into its horizontal and vertical components; the co-normal derivatives / n T and / n S are those associated with the linear (temperature and salinity) operators, i.e., ( ) μ T = n T 1 + h μ T h h + (ν T μ T ) z, ( ) (.38) μ S = n S 1 + h μ S h h + (ν S μ S ) z, where h is the (D) covariant derivative in the direction of h (see, e.g., in Lions, Temam and Wang [1993a] after Eq. (1.1) and after Eq. (3.7)). Remark.7. (i) The boundary conditions (which are the same) on Ɣ b and Ɣ l express the no-slip boundary conditions for the water and the absence of fluxes of heat or salt. For Ɣ i,w= is the geometrical (kinematical) boundary condition required by the rigid lid assumption; the Neumann boundary condition on S expresses the absence of salt flux. (ii) In general, the boundary conditions on v and T on Ɣ i are not fully settled from the physical point of view. These correspond to some resolution of the viscous boundary layers on the top of the ocean. Here, α v and α T are given, v a and T a correspond to the values in the atmosphere, and τ v corresponds to the shear of the wind. (iii) The first boundary condition Eq. (.35) could be replaced by v = v a expressing a no-slip condition between air and sea. However, such a boundary condition 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

15 Some athematical Problems in Geophysical Fluid Dynamics 581 necessitating an exact resolution of the boundary layer would not be practically (computationally) realistic, and as indicated in (ii), we use instead some classical resolution of the boundary layer (Schlichting [1979]). (iv) As we said the boundary condition of Ɣ i is standard unless more involved interactions are taken into consideration. However, for Ɣ b and Ɣ l, different combinations of the Dirichlet and Neumann boundary conditions can be (have been) considered. (Lions, Temam and Wang [199b]). Beta-plane approximation. For midlatitude regional studies, it is usual to consider the beta-plane approximation of the equations in which is a domain in the space R 3 with Cartesian coordinates denoted as x, y, z or x 1,x,x 3. In the beta-plane approximation, = f k, f= f + βy, k the unit vector along the south to north poles, = / x + / y, is the usual nabla vector ( / x, / y), and v = u / x + v / y (v = (u, v)). With these notations, the Eqs. (.4) (.9) and the boundary conditions (Eqs. (.35) (.38)) keep the same form; here the depth h = h(x, y) satisfies, like Eq. (.33), <h h(x, y) h, (.39) and the boundary of consists of Ɣ i,ɣ b,ɣ l defined as before. As indicated in the Introduction, we will emphasize in this chapter the regional model, which is slightly simpler, in particular because of the use of Cartesian coordinates. Usually, the general model in spherical coordinates simply requires the treatment of lower order terms. From now on, we consider the regional (Cartesian coordinate) case. The diagnostic variables. The first step in the mathematical formulation of the PEs consists in showing how to express the diagnostic variables in terms of the prognostic variables, thanks to the equations and boundary conditions. Since w = onɣ i and Ɣ b, integration of Eq. (.6) in z gives and w = w(v) = h Note that div z div v dz (.4) div v dz =. (.41) h v dz = h div v dz + h v, z= h and since v vanishes on Ɣ b, condition (Eq. (.41)), is the same as div v dz =. (.4) h 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

16 58. Petcu et al. Similarly, integration of Eq. (.5) in z gives p = p s + P, P = P(T, S) = g z ρ dz. (.43) Here, ρ is expressed in terms of T and S through Eq. (.9), and p s = p s (x,y,t)= p(x, y,,t)is the pressure at the surface of the ocean. Hence, Eqs. (.4) and (.43) provide an expression of the diagnostic variables in terms of the prognostic variables (and the surface pressure), and Eq. (.4) is an additional equation that, we will see, is mathematically related to the surface pressure. Remark.8. The introduction of the nonlocal constraint (Eq. (.41)) and of the surface pressure p s was first carried out by Lions, Temam and Wang [199a,b]. This new formulation has played a crucial role in much of the mathematical analysis of the PEs in various cases... Weak formulation of the PEs of the ocean: the stationary PEs We denote by U the triplet (v,t,s)(four scalar functions). In summary, the equations that we consider for the subsequent mathematical theory (the PEs) are Eqs. (.4), (.7), and (.8), with w = w(v) given by Eq. (.4) and p given by Eq. (.43) (ρ given by Eq. (.9)); furthermore, v satisfies Eq. (.41); hence, v t + vv + w v z + 1 v p + f k v μ v v ν v ρ z = F v, (.44) T + v T + w T t z μ T T T ν T z = F T, (.45) S t + vs + w S z μ S S S ν S z = F S, (.46) w = w(v) = z div v dz, (.47) div v dz =, (.48) h p = p s + P, P = P(T, S) = g z ρ dz, (.49) ( ρ = ρ 1 βt (T T r ) + β S (S S r ) ), (.5) S d =. (.51) The boundary conditions are Eqs. (.35) (.37). 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

17 Some athematical Problems in Geophysical Fluid Dynamics Weak formulation and functional setting For the weak formulation of this problem, we introduce the following function spaces V and H: V = V 1 V V 3, H = H 1 H H 3, { V 1 = v H 1 (), div v dz =, v = onɣ b Ɣ l }, V = H 1 (), V 3 = Ḣ 1 () = H 1 = h { S H 1 (), } S d =, { v L (), div v dz =,n H h h } v dz = on Ɣ i (i.e., on Ɣ l ), H = L (), { } H 3 = L () = S L (), S d =. These spaces are endowed with the following scalar products and norms: (( )) U, Ũ = (( v, ṽ )) 1 + K (( T T, T )) + K (( )) S S, S 3, ( (( )) v, ṽ 1 = v μ v v ṽ + ν v ) ṽ d, z z (( T, T )) ( = T T μ T T T + ν T z z (( )) ( ) S, S 3 = S S μ S S S + ν S d, z z ( ) U, Ũ H = ( v ṽ + KT T T + K S S S ) d, ) d + Ɣ i α T T T dɣ i, U = ( (U, U) ) 1/, U H = (U, U) 1/ H. Here, K T and K S are suitable positive constants chosen below. The norm on H is of course equivalent to the L -norm, and because of the Poincaré inequality, v vanishes on Ɣ b Ɣ l, and Eq. (.51), i = ((, )) 1/ i is a Hilbert norm on V i, and is a Hilbert norm on V ; more precisely we have, with c > a suitable constant depending on : U H c U U V. (.5) Let V 1 be the space of C (D) vector functions v, which vanish in a neighborhood of Ɣ b Ɣ l and such that div h v dz =. 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

18 584. Petcu et al. Then, V 1 V 1, and it has been shown by Lions, Temam and Wang [199a] that V 1 is dense in V 1. (.53) We also denote by V V the set of C functions on and by V 3 V 3 the set of C functions on with zero average; V = V 1 V V 3 is dense in V. To derive the weak formulation of this problem, we consider a sufficiently regular test function Ũ = (ṽ, T, S) in V. We multiply Eq. (.44) by ṽ, Eq. (.45) by K T T, Eq. (.46) by K S S, integrate over and add the resulting equations; K T,K S > are two constants to be chosen later on. The term involving grad p S vanishes; indeed, by the Stokes formula, p S ṽ d = p S n H ṽ d( ) p S ṽd, where n = (n H,n z ) is the unit outward normal on, and n H its horizontal component. The integral on vanishes because n H ṽ vanishes on ; the remaining integral on vanishes too since by Fubini s theorem, Eq. (.48), and ṽ = onɣ b, ( ) p S ṽd = p S ṽdzdɣ i = p S ṽ dz dɣ i =. Ɣ i h Ɣ i h Using Stokes formula and the boundary conditions (Eqs. (.35) (.38)) we arrive after some easy calculations at ( ) d dt U, Ũ + a ( U, Ũ ) + b ( U, U, Ũ ) + e ( U, Ũ ) = l ( Ũ ). (.54) H The notations are as follows: ( ) U, Ũ H = ( v ṽ + KT T T + K S S S ) d, a ( U, Ũ ) ( = a ) ( 1 U, Ũ + K T a ) ( U, Ũ + K S a ) 3 U, Ũ, ( a ) ( ) v ṽ 1 U, Ũ = μ v v ṽ + ν v d z z P(T,S) ṽd + α v vṽ dɣ i, Ɣ i P(T,S) = g ( β T T + β S S)dz ( ) see Eqs. (.49) and (.5), z ( a ) ( ) T T U, Ũ = μ T T T + ν T d + α T T T dɣ i, z z Ɣ i 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

19 Some athematical Problems in Geophysical Fluid Dynamics 585 ( a ) ( S 3 U, Ũ = μ S S S + ν S z b = b 1 + K T b + K S b 3, ( b 1 U, Ũ,U ) ( = v ṽ + w(v) ṽ z ( b U, Ũ,U ) ( = v T + w(v) T z ( b 3 U, Ũ,U ) ( = v S + w(v) S z e ( U, Ũ ) = (f k v) ṽ d ) S d, z ) v d, ) T d, ) S d, and l ( Ũ ) ( = Fv ṽ + K T F T T + K S F ) S S d [ ] + g (1 + β T T r β S S r )dz ṽd z [ + (gv ) ṽ + g T T ] dɣ i, (.55) Ɣ i where (see Eq. (.35)) g v = τ v + α v v a, g T = α T T a. For l, we observe that if T r and S r are constant, then [ ] g (1 + β T T r β S S r )dz vd z [ ] = g (1 + β T T r β S S r )dz n H v d( ) z =. (.56) It is clear that each a i, and thus a, is a bilinear continuous form on V ; furthermore, if K T and K S are sufficiently large, a is coercive (a,a 3 are automatically coercive on V,V 3 ): a(u, U) c 1 U U V(c 1 > ). (.57) Similarly, e is bilinear continuous on V 1 and even H 1, and e(u, U) = U H. (.58) 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

20 586. Petcu et al. Before studying the properties of the form b, we introduce the space V () : V () is the closure of V in ( H () ) 4. (.59) Then, we have the following Lemma.1. The form b is trilinear continuous on V V V () and V V () V, 1 b ( U, Ũ,U ) c U Ũ U V(), U, Ũ V, U V (), c U Ũ U (.6) V(), U, U V, Ũ V (), or ( b U, Ũ,U ) c U Ũ 1/ Ũ 1/ U V(), U, Ũ V, U V (). (.61) H Furthermore, and b ( U, Ũ,Ũ ) = for U V, Ũ V (), (.6) b ( U, Ũ,U ) = b ( U, U, Ũ ) (.63) for U, Ũ,U # V, and Ũ or U in V (). Proof. To show first that b is defined on V V V (), let us consider the typical and most problematic term w(v) T z T d. (.64) We have w(v) T z T d w(v) T L () z T L L (). () The first two terms in the right-hand side of this inequality are bounded by const v 1 (using Eq. (.4)) and T. In dimension 3, H () L () so that the third term 1 For Eqs. (.6) and (.61), the specific form of V and V () is not important: b is as well trilinear continuous on H 1 () 4 H () 4 H 1 () 4 and H 1 () 4 H 1 () 4 H (), and the estimates are similar, the H 1 and H norms replacing the V and V () norms. 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

21 Some athematical Problems in Geophysical Fluid Dynamics 587 is bounded by const T V(), and hence the right-hand side of the last inequality is bounded by c U Ũ U V(). With similar (and easier) inequalities for the other integral, we conclude that b is defined and trilinear continuous on V V V (). For the continuity on V V () V, the typical term above is bounded by w(v) T L () z L 4 () T L 4 (), which is bounded by c v 1 T H T H 1 c U Ũ U V() since H 1 () L 6 (), hence the second bound (Eq. (.6)). We easily prove Eqs. (.6) and (.63) by integration by parts for U, Ũ,U V; the relations are then extended by continuity to the other cases, using Eq. (.6). To establish the improvement (Eq. (.61)) of the first inequality (Eq. (.6)), we observe that b(u, Ũ,U ) = b(u, U, Ũ) and consider again the most typical term w(v)( U / z) Ũ d that we bound by w(v) L U z Ũ L 3. L 6 Remembering that H 1 L 6 and H 1/ L 3, in space dimension 3, we bound this term by c v U V() Ũ 1/ Ũ 1/, and Eq. (.61) follows. The operator form of the equation. We can write Eq. (.54) in the form of an evolution equation in the Hilbert space V (). For that purpose, we observe that we can associate to the forms a, b, e above, the following operators: A linear, continuous from V into V defined by AU, Ũ = a ( U, Ũ ) U, Ũ V, B bilinear, continuous from V V into V () defined by ( ) B U, Ũ,U = b ( U, Ũ,U ) U, Ũ V, U V (), E linear, continuous from H into itself defined by E(U), Ũ = e ( U, Ũ ) U, Ũ H. 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

22 588. Petcu et al. Since V () V H, with continuous injections, each space being dense in the next one, we also have the Gelfand Lions inclusions V () V H V V (). (.65) With this, we see that Eq. (.54) is equivalent to the following operator evolution equation du + AU + B(U, U) + E(U) = l, (.66) dt understood in V () and with l defined in Eq. (.55). To this equation, we will naturally add an initial condition: U() = U. (.67)... The stationary PEs We now establish the existence of solutions of the stationary PEs. Besides its intrinsic interest, this result will be needed in the next section for the study of the time-dependent case. The equations to be considered are the same as Eq. (.54), with the only difference that the derivatives v/ t, T/ t, and S/ t are removed, and that the source terms F v,f T,F S are given independent of time t. The weak formulation proceeds as before: Given F = (F v,f T,F S ) in H ( or L () 4), and g = (g v,g T ) in L (Ɣ i ) 3, find U = (v,t,s) V, such that a ( U, Ũ ) + b ( U, U, Ũ ) + e ( U, Ũ ) = l ( Ũ ), for every Ũ V () ; (.68) a, b, e, and l are the same as above. We have the following result. Theorem.1. We are given F = (F v,f T,F S ) in L () 4 (or in H) and g = (g v,g T ) in L (Ɣ i ) 3 ; then problem (Eq. (.68)) possesses at least one solution U V such that U 1 c 1 l V. (.69) Proof. The proof of existence is done by Galerkin method, a priori estimates and passage to the limit. The proof is essentially standard, but we give the details because of some specificities in this case. We consider a family of elements { j } j of V (), which is free and total in V (V () is dense in V ); for each m N, we look for an approximate solution of Eq. (.68), U m = m j=1 ξ jm j, such that a(u m, k ) + b(u m,u m, k ) + e(u m, k ) = l( k ), k = 1,...,m. (.7) 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

23 Some athematical Problems in Geophysical Fluid Dynamics 589 The existence of U m is shown below. An a priori estimate on U m is obtained by multiplying each equation (Eq. (.7)) by ξ km and summing for k = 1,...,m. This amounts to replacing k by U m in Eq. (.7); since b(u m,u m,u m ) = by Lemma.1, we obtain a(u m,u m ) = l(u m ) and, with Eq. (.57), c 1 U m l V U m, U m 1 c 1 l V. (.71) From Eq. (.71), we see that there exists U V and a subsequence U m such that U m converges weakly to U as m. Since we cannot replace Ũ by U in Eq. (.68), it is useful to notice that U lim inf m Um 1 l V, c 1 so that Eq. (.69) is satisfied. Then, we pass to the limit in Eq. (.7) written with m, and k fixed less than or equal to m. We observe below that b(u m,u m, k ) b(u, U, k ) (.7) so that at the limit, U satisfies Eq. (.68) for Ũ = k,kfixed arbitrary; hence Eq. (.68) is valid for any Ũ linear combination of k and, by continuity (Lemma.1), for Ũ V (). The proof is complete after we prove the results used above. Convergence of the b term. To prove Eq. (.7), we first observe, with Eq. (.63), that b(u m,u m, k ) = b(u m, k,u m ). We also observe that each component of U m converges weakly in H 1 () to the corresponding component of k. Therefore, by compactness, the convergence takes place in H 3/4 () strongly; by Sobolev embedding, H 3/4 () L 4 () in dimension 3, and the convergence holds in L 4 () strongly. Writing k = = (v,t,s ), the typical most problematic term is w(v m ) v z v m d. (.73) Since div v m converges weakly to div v in L (), w(v m ) converges weakly in L () to w(v); v m converges strongly to v in L 4 () as observed before, and since v / z belongs to L 4 (), the term above converges to the corresponding term where v m is replaced by v(u = (v,t,s)), hence Eq. (.69). 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page:

24 59. Petcu et al. Existence of U m. Equation (.7) amounts to a system of m nonlinear equations for the components of the vector ξ = (ξ 1,...,ξ m ), where we have written ξ jm = ξ j for simplicity. Existence follows from the following consequence of the Brouwer fixed point theorem. (see Lions [1969]). Lemma.. Let F be a continuous mapping of R m into itself such that [ ] F(ξ), ξ > for [ξ] =k, for some k>, (.74) where [, ] and [ ] are the scalar product and norm in R m. Then there exists ξ R m with [ξ] <ksuch that F(ξ) =. Proof. If F never vanishes, then G = kf(ξ)/[f(ξ)] is continuous on R m, and we can apply the Brouwer fixed point theorem to G, which maps the ball C centered at of radius k into itself. Then G has a fixed point ξ in C and we have [ G(ξ ) ] =[ξ ]=k, [ G(ξ ), ξ ] = k [F(ξ ), ξ ] [F(ξ )] =[ξ ]. This contradicts the hypothesis (Eq. (.74)) on F; the lemma is proven. We apply this lemma to Eq. (.7) as follows: F = (F 1,...,F m ), with F k (ξ) = [F(U m ), k ] = a(u m, k ) + b(u m,u m, k ) + e(u m, k ) l( k ). (.75) The space R m is equipped with the usual Euclidean scalar product so that [ ] m F(ξ), ξ = F k (ξ)ξ k k=1 = a(u m,u m ) + b(u m,u m,u m ) l(u m ) ( with Eqs. (.57), (.58), (.6) and Schwarz inequality ) c 1 U m l V U m. (.76) Since the last expression converges to + as U m [ξ] converges to +, there exists k> such that Eq. (.74) holds. The existence of U m follows. Remark.9. A perusal of the proof of Theorem.1 shows that we proved the following more general result. Lemma.3. Let V and W be two Hilbert spaces with W V, the injection being continuous. Assume that ā is bilinear continuous coercive on V, and that b is trilinear 13-Ch1-N51893 [11: 8/9/15] Temam & Tribbia: Computational ethods for the Atmosphere and the Oceans Page: