Some Mathematical Problems in Geophysical Fluid Dynamics

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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 2 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 1990s 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 Millenium Clay problem, the PEs with viscosity are well-posed in space dimensions 2 and 3, when supplemented with fairly 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).

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 (10-20 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.

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. (2.25)).

As far as we know, the PEs were essentially introduced by Richardson in 1922; 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 1940s 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 Models (GCMs) or Ocean Global Circulation Models (OGCMs) available at the National Center for Atmospheric Research and elsewhere. GCMs and OGCMs 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 in Laprise [1992] or Smolarkiewicz, Margolin and Wyszogrodzki [2001]; 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 [2008] 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).

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 [1992a,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 mathematical results available for the shallow water equations than for the Navier-Stokes equations, even in space dimension 2 (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 [2004] 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 [1992a,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 2 and the original articles by Lions, Temam and Wang [1992a,b, 1995] in the list of references). Further works, conducted during the 1990s 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 [2004] and appear herein in Sections 2 and 3:

  • (i)

    Existence of weak solutions for all time (dimensions 2 and 3) (see Sections 2).

  • (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 2, existence and uniqueness for all time of a strong solution (see Section 3.3).

  • (iv)

    Uniqueness of a weak solution in space dimension 2 (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(L2) and L2(H1)), and the strong solutions are those with finite (fluid) enstrophy (L∞(H1) and L2(H2)). Essential in the most recent developments (ii)-(iv) above is the H2-regularity result for a Stokes-type problem appearing in the PEs, the analog of the H2-regularity in the Cattabriga-Solonnikov results on the usual Stokes problem; the whole Section 4 is devoted to this problem.

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 (2D) function (a function of the horizontal variables and time) and because of that the 3D PEs are also close to a 2D 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 [2007] and by Kobelkov [2006], 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 [2007a,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 3.2.

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 mid-latitude 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, 0x is the west-east axis, 0y is the south-north axis, and 0z 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 [1992a,b, 1995]).

In the Cartesian case of emphasis, generally, we first concentrate our attention on the ocean. Indeed, as we will see in Section 2, the domain occupied by the ocean contains corners (in dimension 2) 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, Mazya and Rossmann [1997], Mazya 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 2, 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 [1992a,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 2 and a wealth of other mathematical results, regularity in H m-higher Sobolev spaces, 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 3.2). In dimension 2, 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 2D space-periodic case and prove the existence of solutions for all time, in all H m, m≥ 2. In Section 3.6, we prove the Gevrey regularity of the solutions and in Section 3.7.2 the backward uniqueness result.

Section 4 is technically very important, and many results of Sections 2 and 3 rely on it: this section contains the proof of the H2-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 (H2-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.

More 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 [1992a,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 2). An alternative slightly more general proof of this result is given in Section 2. 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 [2003]. An alternate proof of this result is due to Guillén -Gonz Ález, Masmoudi and Rodr Íguez -Bellido [2001]. In dimension 2, 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 [2000]. This result is also established, for a simpler system (without temperature and salinity), by Bresch, Kazhikhov and Lemoine [2004]. Most of the results of Sections 3.4 –3.7.2 are due to M. Petcu, alone or in collaboration with D. Wirosoetisno.

The physics-oriented reader will recognize in (2.1), (2.2), (2.3), (2.4), (2.5) the basic conservation laws: conservations of momentum, mass, energy and salt for the ocean, equation of state. In Eqs. (2.6) and (2.7) appears the simplification due to the Boussinesq approximation, and in (2.11), (2.12), (2.13), (2.14), (2.15), (2.16) the simplifications resulting from the hydrostatic balance assumption. Hence (2.11), (2.12), (2.13), (2.14), (2.15), (2.16)are the PEs of the ocean. The PEs of the atmosphere appear i n (2.116), (2.117), (2.118), (2.119), (2.120), (2.121), and those of the CAO are described in Section 2.5. Concerning, to begin, the ocean, the first task is to write these equations,supplemented by the initial and boundary conditions, as an initial value problem in a phase space H of the form dUdt+AU+B(U,U)+E(U)=, U(0)=U0,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. (2.66)). 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 2.2.2, and in Theorem 2.2; we prove the existence for all times of weak solutions of Eqs. (1.1) and (1.2), which are solutions in L(0,t1;L2) and L2(0,t1;H1) (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 2.4 and 2.5.Section 4 is mathematically very important although technical.

For the physics-oriented reader, the most important results are those of Sections 2 and 3.Section 2 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 2 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 [2007, 2008] 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 [2007], 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 [1980], Pedlovsky [1987], Trenberth [1992], 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 climates are the questions of attractors for the PEs, which have been addressed in e.g., Ju [2007], Lions, Temam and Wang [1992a,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., Ma and Wang [2005a,b], and the chapter by Simonnet, Dijkstra and Ghil [2008] 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 Meur and Teresa Bunge who typed significant parts of the manuscript.

This chapter is an updated version of the article by Temam and Ziane [2004]. 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.

Section snippets

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.

Strong solutions of the PEs in dimensions 2 and 3

In this section, we first show, in Section 3.1, the existence, local in time, of strong solutions to the PEs in space dimension 3, i.e., solutions whose norm in H1 remains bounded for a limited time. Then, in Section 3.2, we show the existence and uniqueness, global in time, of strong solutions to the PEs in space dimension 3.

In Section 3.3, we consider the PEs in space dimension 2 in view of adapting to this case the results of Sections 2 and 3.1. The 2D PEs are presented in Section 3.3.1, as

Regularity for the elliptic linear problems in geophysical fluid dynamics

We have used many times, in particular, inSection 3 the result ofH2-regularity of the solutions to certain linear elliptic problems. Following the general results from Agmon, Douglis and Nirenberg [1959, 1964], we know that the solutions to second-order elliptic problems are inHm+2if the right-hand sides of the equations are inHm,m≥ 0, and the other data are in suitable spaces (see also Lions and Magenes [1972] form< 0). Results of this type are proved in this section.

There are several specific

Acknowledgments

This work was partially supported by the National Science Foundation under the grants NSF-DMS-0074334, NSF-DMS-0204863, NSF-DMS-0505974, and NSF-DMS-0604235 and by the Research Fund of Indiana University.

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