Some Mathematical Problems in Geophysical Fluid Dynamics

136 Pages·2004·0.628 MB·English

by  Temam R., Ziane M.

Handbook of Mathematical Fluid Dynamics, Vol 3
S. Friedlander and D. Serre Editors, Elsevier. To appear
SOME MATHEMATICAL PROBLEMS IN GEOPHYSICAL
FLUID DYNAMICS
Roger Temam*# and Mohammed Ziane%
∗The Institute for Scientific Computing and Applied Mathematics
Indiana University, Bloomington, IN 47405
#Laboratoire d’Analyse Num´erique, Universit´e Paris-Sud
Bˆatiment 425, 91405 Orsay, France
%University of Southern California
DRB 155; 1042 W. 36 Place; Los Angeles, CA 90049
Abstract. Thisarticleaddressessomemathematicalaspectsoftheequations
of geophysical fluid dynamics namely, existence, uniqueness, and regularity of
solutions of the Primitive Equations (PEs) of the ocean, the atmosphere and
thecoupledatmosphere-ocean. Theemphasisisonthecaseoftheoceanwhich
encompasses most of the mathematical difficulties.
A guide and summary of results for the physics oriented reader is provided
at the end of the Introduction (Section 1).
Contents
1. Introduction 2
2. The Primitive Equations. Weak Formulation. Existence of Weak
Solutions. 9
2.1. The Primitive Equations of the Ocean 9
2.2. Weak formulation of the PEs of the ocean.
The stationary PEs 18
Date: January 14, 2004.
1
2 SOME MATHEMATICAL PROBLEMS IN GFD 14-1-04
2.3. Existence of weak solutions for the PEs of the ocean 27
2.4. The Primitive equations of the atmosphere 32
2.5. The coupled atmosphere and ocean 40
3. Strong Solutions of the Primitive Equations in
Dimension 2 and 3. 45
3.1. Strong solutions in space dimension 3 45
3.2. Strong solutions of the two dimensional primitive equations: Physical
Boundary Conditions 56
3.3. The space periodic case in dimension 2: Higher regularities 71
4. Regularity for the elliptic linear problems in GFD. 87
4.1. Regularity of solutions of elliptic boundary value problems in
cylinder type domains 89
4.2. Regularity of Solutions of a Dirichlet-Robin Mixed Boundary Value
Problem 97
4.3. Regularity of Solutions of a Neumann-Robin Boundary Value
Problem 104
4.4. Regularity of the Velocity 116
Proof of Theorem 4.4 125
4.5. Regularity of the coupled system 130
References 134
1. Introduction
Theaimofthisarticleistoaddresssomemathematicalaspectsoftheequations
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 are derived from the conservation
equations from physics, namely conservation of mass, momentum, energy, and
some other components such as salt for the ocean, humidity (or chemical pol-
lutants) for the atmosphere. The basic equations of conservation of mass and
momentum, that is the three-dimensional compressible Navier–Stokes equations
contain however too much information and we can not hope to numerically solve
SOME MATHEMATICAL PROBLEMS IN GFD 14-1-04 3
these equations with enough accuracy in a foreseeable future. Owing to the dif-
ference of sizes of the vertical and horizontal dimensions, both in the atmosphere
and in the ocean (10 to 20 kms versus several thousands of kilometers), the most
natural simplification leads to the so-called Primitive Equations (PEs) which we
study in this article.
We continue this Introduction by briefly describing the physical and mathe-
matical backgrounds of the PEs.
Physical Background
The primitive equations 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 e.g. equation (2.25)).
As far as we know, the primitive equations were essentially introduced by
L. F. Richardson (1922); when it appeared that they were still too complicated,
they were left out and, instead, attention was focused on even simpler mod-
els, the geostrophic and quasi-geostrophic models, considered in the late 1940’s
by J. von Neumann and his collaborators, in particular J. G. Charney. With
the increase of computing power, interest eventually returned to the PEs, which
are now the core of many Global Circulation Models (GCM) or Ocean Global
Circulation Models (OGCM), available at the National Center for Atmospheric
Research (NCAR) and elsewhere. GCMs and OGCMs are very complex models
which contain many components, but still, the PEs are the central component for
the dynamics of the air or the water. For some phenomena there is need to give
up the hydrostatic hypothesis and then non-hydrostatic models are considered,
such as in Laprise [18] or Smolarkiewicz, Margolin and Wyszogrodzki [34]; these
models stand at an intermediate level of physical complexity between the full
Navier–Stokes equations and the PEs-hydrostatic equations. Research on non-
hydrostatic 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.
4 SOME MATHEMATICAL PROBLEMS IN GFD 14-1-04
In summary, in term 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 follows:
3D Navier–Stokes Equations
⇓
Nonhydrostatic Models
⇓
Primitive Equations
(hydrostatic equations)
⇓
Shallow Water Equations
⇓
Quasi-Geostrophic Models
⇓
2D Barotropic Equations
We remark here also that much study is needed for the boundary conditions
from both the physical and mathematical points of views. As we said, our aim
in this article is the study of mathematical properties of the primitive equations.
Mathematical Background
The level of mathematical complexity of the equations above 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 S. Wang [40, 41]), and we know the level of complexity of the
Navier–Stokes equations to which this handbook is devoted. On the other hand,
non-hydrostatic 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 two (see however Orenga [31]).
As we indicate hereafter, the primitive equations although physically simpler
are, in fact, slightly more complicated than the incompressible Navier–Stokes
equations.
SOME MATHEMATICAL PROBLEMS IN GFD 14-1-04 5
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 primitive equations, is of the form
first order derivatives of horizontal velocity
× first order derivatives of horizontal velocity.
The mathematical study of the primitive equations was initiated by J. L. Lions,
R. Temam, and S. Wang in the early 1990s. They produced a mathematical for-
mulation of the PEs which resembles that of the Navier–Stokes due to J. Leray,
and obtained the existence for all time of weak solutions; see Section 2, and
the original articles [21], [22], [24], in the list of references. Further works, con-
ducted during the 1990s and more especially during the past few years, have
improved and supplemented the early results of [21], [22], [24] by a set of re-
sults which, essentially, brings the mathematical theory of the PEs to that of the
3D incompressible Navier–Stokes equations, despite the added complexity men-
tioned above; this added complexity is overcome by a non-isotropic treatment of
the equations (of certain nonlinear terms), in which the horizontal and vertical
directions are treated differently.
In summary the following results are now available which will be presented in
details in this article:
(i) Existence of weak solutions for all time (dimension two and three).
(ii) In space dimension three, existence of a strong solution for a limited time
(local in time existence).
(iii) In space dimension two, existence and uniqueness for all time of a strong
solution.
(iv) Uniqueness of a weak solution in space dimension two.
In the above, the terminology is that normally used for Navier–Stokes equa-
tions: 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
6 SOME MATHEMATICAL PROBLEMS IN GFD 14-1-04
analog of the H2 regularity in the Cattabriga–Solonnikov results on the usual
Stokes problem; the whole Section 4 is devoted to this problem.
Content of this article
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 the south-north axis, and 0z the
ascending vertical.
For this more mathematically oriented article 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 concen-
trate our efforts on the corresponding β-plane case (in Cartesian coordinates).
Indeed, in general, going from the β-plane case in Cartesian coordinates to the
spherical case necessitates only the proper handling of terms involving lower or-
der derivatives; full details concerning the spherical case can be found also in the
original articles [21], [22], [24],...
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 2D) or wedges (in 3D); some regularity issues occur in
this case which must be handled using the theory of regularity of elliptic problems
innonsmoothdomains(Grisvard[11],Kozlov,Maz’yaandRossmann[17],Maz’ya
and Rossmann [25]). For the atmosphere or the coupled atmosphere–ocean, 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 exis-
tence of weak solutions with a different method than in the original articles [21],
[22], [24], thus allowing more generality (for the ocean, the atmosphere, and the
coupled atmosphere–ocean).
SOME MATHEMATICAL PROBLEMS IN GFD 14-1-04 7
In Section 3 we study the existence of strong solutions in space dimension three
and two (solutions local in time in dimension three, and for all time in dimension
two). We establish in dimension three the existence and uniqueness of strong
solutions on a limited interval of time(Section 3.2); in dimension two we prove
the existence and uniqueness, for all time, of such strong solutions. Finally in
Section 3.3, we consider the two-dimensional space-periodic case and prove the
existence of solutions for all time, in all Hm,m ≥ 2.
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 primitive equations. This proof relies, as we said, on the theory
of regularity of solutions of elliptic problems in nonsmooth domains.
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, oftheoceanandofthecoupledatmosphereoceanwerederivedinthe
articles by Lions, Temam and Wang, [21], [22], [24]. For each of these problems,
these articles also contain the proof of existence of weak solutions for all time
(in dimension three with a proof which easily extends to dimension two). An
alternative proof of result, slightly more general is given in Section 2. Concerning
the strong solutions, the proof given here of the local existence in dimension three
is based on the article by Hu, Temam and Ziane [16]. An alternate proof of this
result is due to [12]. In dimension two, the proof of existence and uniqueness of
strongsolutions, foralltime, fortheconsideredsystemofequationsandboundary
conditionsisnew, andbasedonanunpublishedmanuscriptofM.Ziane[45]. This
result is also established, for a simpler system (without temperature and salinity)
by Bresch, Kazhikov and Lemoine, [6]. In the space periodic case, existence and
uniqueness of solutions in all the spaces Hm is proved in Petcu, Temam and
Wirosoetisno [29].
Summary of results for the physics oriented reader.
The physics oriented reader will recognize in (2.1)- (2.5) the basic conservation
laws: conservations of momentum, mass, energy and salt for the ocean, equa-
tion of state. In (2.6) and (2.7) appears the simplification due to the Boussinesq
approximation, and in (2.11)-(2.16) the simplifications resulting from the hydro-
static assumptions. Hence (2.11)-(2.16) are the Primitive Equations of the ocean.
8 SOME MATHEMATICAL PROBLEMS IN GFD 14-1-04
The Primitive Equations of the atmosphere appear in (2.116)-(2.121), and those
of the coupled atmosphere and ocean 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
dU
(1.1) +AU +B(U,U)+E(U) = (cid:96),
dt
(1.2) U(0) = U ,
0
where U is the set of prognostic variables of the problem, that is the horizontal
velocity v = (u,v), the temperature T and the salinity S,U = (v,T,S); see
(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 (1.1) in Section 2.2.2 and, in Theorem 2.2, we prove the existence for all
times of weak solutions of (1.1) - (1.2), which are solutions in L∞(0,t ;L2) and
1
L2(0,t ;H1) (bounded kinetic energy and square integrable enstrophy for the
1
fluid mechanics part). A parallel study is conducted for the atmosphere and the
coupled atmosphere-ocean in Sections 2.4 and 2.5.
In Section 3 we consider in dimension 3 and 2 the strong solutions, which
are solutions bounded for all times in the Sobolev space H1 (“finite enstrophy”
space). The main results are Theorems 3.1 and 3.2. Section 4 mathematically
very important although technical. It is shown there, that the solutions to certain
elliptic problems enjoy certain regularity properties (H2 regularity, that is the
functionandtheirfirstandsecondderivativesaresquareintegrable); theproblems
corresponding to the (horizontal) velocity, the temperature and the salinity are
successively considered. The study in Section 4 contains many specific aspects
which are explained in details in the long introduction to that section.
The study presented in this article 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 which were very
useful to us such as [14, 28, 39, 42, 46].
SOME MATHEMATICAL PROBLEMS IN GFD 14-1-04 9
Beside the efforts of the authors, we mention in several places that this study
is based on joint works with J.L Lions, S. Wang, C. Hu and others. Their help is
gratefully acknowledged and we pay tribute to the memory of Jacques-Louis Li-
ons. The authors wish to thank Denis Serre and Shouhong Wang for their careful
reading of the whole manuscript and for their numerous comments which signifi-
cantly improved the manuscript. They extend also their gratitude to Daniele Le
Meur and Teresa Bunge who typed significant parts of the manuscript.
2. The Primitive Equations. Weak Formulation. Existence of
Weak Solutions.
As explained in the Introduction to this article, our aim in this section is first
to present the derivation of the Primitive Equations from the basic physical con-
servation laws. We then describe the natural boundary conditions. Then, on the
mathematical side, we introduce the function spaces and derive the mathemati-
cal formulation of the PEs. Finally we derive the existence for all time of weak
solutions.
Wesuccessivelyconsidertheocean,theatmosphereandthecoupledatmosphere-
ocean.
2.1. The Primitive Equations of the Ocean. Our aim in this section is to
describethePrimitiveEquationsoftheocean(seeSection2.1.1), wethendescribe
the corresponding boundary conditions and the associated initial and boundary
value problems (Section 2.1.2).
2.1.1. The Primitive Equations. Generally speaking, 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 ther-
modynamics equation (that is the conservation of energy equation), the equation
10 SOME MATHEMATICAL PROBLEMS IN GFD 14-1-04
of state and the equation of diffusion for the salinity S:
dV
3
(2.1) ρ +2ρ Ω×V +∇ p+ρ g = D,
3 3
dt
dρ
(2.2) +ρdiv V = 0,
3 3
dt
dT
(2.3) = Q ,
T
dt
dS
(2.4) = Q ,
S
dt
(2.5) ρ = f(T,S,p).
Here V is the three-dimensional velocity vector, V = (u,v,w), ρ, p, T are
3 3
the density, pressure and temperature and S is the concentration of salinity;
g = (0,0,g) is the gravity vector, D the molecular dissipation, Q and Q are
T S
the heat and salinity diffusions. The analytic expressions of D, Q and Q will
T S
be given below.
The Boussinesq Approximation
From both the theoretical and the computational points of view, 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 Mach number for the flow in the ocean is not large and
therefore,asastartingpoint,wecanmaketheso-calledBoussinesq approximation
in which the density is assumed constant, ρ = ρ , except in the buoyancy term
0
and in the equation of state.
This amounts to replacing (2.1), (2.2) by
dV
3
(2.6) ρ +2ρ Ω×V +∇ p+ρg = D,
0 0 3 3
dt
(2.7) div V = 0.
3 3
Consider the spherical coordinate system (θ,φ,r), where θ (−π/2 < θ < π/2)
stands for the latitude on the earth, φ (0 ≤ φ ≤ 2π) 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 ,e ,e be the unit vectors in the θ-, φ- and z-directions,
θ φ r
respectively. Then we write the velocity of the ocean in the form
(2.8) V = v e +v e +v e = v +w,
3 θ θ φ φ r r
where v = v e +v e is the horizontal velocity field and w is the vertical velocity.
θ θ φ φ