Nonlinearity 5 (1992) 237-288. Printed in the UK
New formulations of the primitive equations of
atmosphere and applications
Jacques Louis Lionst, Roger TemamtB and Shouhong Wangf
t College de France
and Centre National dEtudes Spatiales, 3 Rue d'Ulm, 75005
Paris, France
$The Institute for Applied Mathematics and Scientific Computing, Indiana
University, Blwmingtan, IN 47405, USA
§ Laboratoire &Analyse NumBrique, Universite Paris-Sud, Batiment 425, 91405
Orsay. France
Received 18 August 1991
Accepted by J D Gibbon
Abshnct. The primitive equations are the fundamental equations of atmospheric
dynamics. With the purpose of understanding the mechanism of long-term weather
prediction and climate changes, we study in this paper as a first step towards this
long-range project what is widely considered as the basic equations of atmospheric
dynamics in meteorology, namely the primitive equations of atmosphere, The
mathematical formulation and attractors of the primitive equations. with or without
vertical viscosity, are studied. First of all, by integrating the diagnostic equations we
present a mathematical setting, and obtain the existence and time analyticity ofsolutions
to the equations. We then establish some physically relevant estimates for the Hausdorff
and fractal dimensions of the attractors of the problems.
AMS classification scheme numbers: 35MLO,35030,35Q35,86A10
Introduction
In order to understand the turbulent behaviour of long-term weather prediction and
climate changes we can use historical records and numerical computations to' detect
the future weather and the possible global changes. Alternatively, a complementary
method and source of information was advocated and developed by the pioneers of
meteorology such as V Bjerknes, L F Richardson and others. This consists of
studying the mathematical equations and models governing the motion of the
atmosphere, and to solve them numerically. What was a dream at the time of
Richardson becoming closer to reality due to the considerable increase in computer
capacity in recent years and that to come, and also to the real-time information
0951-7715/92/020237 + 52W.50 0 1992 IOP Publishing Ltd and LMS Publishing Ltd
237
238
I L Lions et ol
system allowing access to data. From the mathematical point of view, the problems
we address here are how well posed the initial value problem is for the equations of
meteorology, and the long-time behaviour of their solutions.
The general equations describing the motion and states of the atmosphere, that
is, a specific compressible fluid, are the hydrodynamic and thermodynamic equations
with Coriolis force. Because the resulting Row for the atmosphere is amazingly rich
in its organization and complexity, the full governing equations are extremely
complicated and seem to defy analysis, at least for the time being.
Fortunately, since the vertical scale of the global atmosphere is much smaller
than the horizontal one, scale analysis, meteorological observations and historical
data reveal that the large-scale atmosphere satisfies the quasi-static equilibrium
equation (1.7), which is also called the hydrostatic equation. This equation provides
a relationship between pressure and density. Based on this relationship, the original
equations of atmosphere are reduced to the so-called primitive equations, which we
will study in this paper. Because of its good accuracy, the quasistatic equilibrium
equation is well accepted as a fundamental equation of the atmosphere, and is
considered a s a starting point for studying the extremely complicated atmospheric
phenomena, and for predicting the weather and possible climate changes. These
equations were proposed and used by Richardson (1922) in his historic forecast [l].
However, at that time they were considered too complicated, and were replaced by
simplified models. Owing to the advances in meteorology and computing power
since then, there is now a tendency to turn from the simplified models used by
Charney, von Neumann and Philips in the early 1950s, and other simplified models,
back to the equations of Richardson. The primitive equations are also called the
predictive equations or the balance equations [2, ch 3, p 74; 31. Since the primitive
equations are the fundamental equations of the atmosphere, we intend to show in a
subsequent article how they can be derived from the general conservation equations
of physics and mechanics by an asymptotic procedure, using the fact that the height
of the atmosphere is very small compared to the radius of the earth. We will also
show in a subsequent paper that all other meteorologically interesting equations of
the atmosphere, such as the linear balanced model, the ZD model, and the
quasigeostropheric and the barotropic equations are derived from the primitive
ones.
As we have already briefly said, the objective of this paper is to establish the
mathematical foundations and to develop a rigorous mathematical framework for
these equations, which does not seem to have been done before. We will also study
the long-time behaviour of the solutions of these equations using some concepts
recently developed in the mathematical theory of (infinite-dimensional) dynamical
systems. Long-time aspects ought to be considered since the length of time for which
weather prediction is sought is very long compared to the natural time attached to
the equations and the physical parameters.
It is well known that if the forecast period is to be extended to 3 and more days,
the diabatic heating and the fridion of the atmosphere will have to he taken into
account. Therefore, we have to consider the primitive equations with viscosity. On
the other hand, the quasistatic equilibrium equations, as we have mentioned, is a
fundamental property of the atmosphere. So we consider two types of viscosity for
the primitive equations. First, we emphasize the importance of the quasistatic
equilibrium equations and neglect the viscosity in the vertical direction. More
The primitive equations of atmosphere
239
specifically, keeping the quasistatic equilibrium relations unchanged, we only attach
the viscosity to the horizontal velocity and the temperature. For simplicity, the
resulting equations will still be called the primitive equations (PES). The second type
of viscosity is introduced in such a way that in addition to the viscosity terms given
for the horizontal velocity and the temperature, we also keep the viscosity terms for
the vertical velocity U in the vertical direction. We call the resulting equations the
primitive equations with vertical viscosity (PEV’s).
Owing to the quasistatic equilibrium relation, the pressure p is a decreasing
function of the vertical coordinate z and we can make a coordinate transformation
from the usual spherical coordinate system (0. q ~ ,z ) to the p-coordinate one
(e, q ~ , p ) .We know that t h e atmosphere is made up of a compressible fluid. The
primitive equations in the p-coordinate system, however, possess a similar structure
to the equations of an incompressible fluid. More precisely, the continuity equation
takes the same form as that of an incompressible fluid. This is one of the advantages
of the quasistatic equilibrium relation, and therefore, of the p-coordinate system. Of
course, due also to this relation, the resulting equations are highly degenerate for
the vertical velocity w. So we have to overcome the difficulty caused by this
degeneracy. In conclusion, the PES using the p-coordinate system, are similar to hut
more complicated than t h e Navier-Stokes equations of incompressible fluids. They
are, however, much simpler and more tractable than the equations of completely
general compressible fluids from which they are derived.
For both the PES and the PEV’s, the prognostic feature for the vertical velocity w
is lost, and there are two diagnostic equations involving the geopotential @ and w.
In both cases the vertical velocity w changes with time, but only as specified by the
diagnostic equations. Moreover, the vertical momentum equation is replaced by the
quasistatic equilibrium relation, in which the vertical velocity w does not appear
explicitly. It is worth pointing out that using the quasistatic equilibrium equation
does not imply that w vanishes. It means, rather, that it must he found by diagnostic
techniques and that the resulting values are precisely those needed to maintain the
quasistatic equilibrium. From the mathematical point of view, both the PES and PE&
do not appear as evolution equations due to the diagnostic ones. Owing to the
specific form of the diagnostic relations they are integrated, and then yield two
evolution systems with a non-local constraint (1.52) for the PES and the PEV’S,
respectively. We cannot, however, use the usual techniques to handle these
problems because the constraints are not local. The specific considerations to
circumvent the difficulty caused by the non-local constraint are important aspects of
this paper.
To be more specific, w e consider the non-dimensional forms (1.41)-(1.44) of the
PES, and (4.1)-(4.4) of PEV’s. Obviously, the vertical velocity U and the geopotential
CP are given by the diagnostic equations (1.42) and (1.43). Since there are two
boundary value conditions for CO, integrating the continuity equation (1.42)
eliminates U from the system, but yields a non-local constraint (1.52). On the other
hand, if we consider the geopotential @ at the isobaric surfaces as a given function,
then integrating the quasistatic equilibrium equation provides an overdetermined
system. Surprisingly, if we introduce a two-variable unknown function, as,as the
function of the geopotential at the isobaric surface p = P , then we obtain two
reformulation, (ISO)-(l.52), of the PES, and (4.8)-(4.10) of the PEV’s. It is
remarkable from the mathematical point of view that
is exactly the Lagrange
240
J L Lions ef al
multiplier of the non-local constraint, and that the resulting systems are 3~ evolution
systems. Moreover, we can simply use the Galerkin method to solve the weak
formulations (problems 2.1 and 4.1) of the systems. In [ll],however, we did not find
the reformulations (1.50)-(1.52) of PESand (4.8)-(4.10) of the PEV’s and, therefore,
we had to construct some special kinds of function spaces for the vertical velocity of
w , in which the elements possess more regularity in the p-direction. At the same
time, we had to use the semi-discretization method to prove the existence of weak
solutions because we were not able to use the Galerkin procedure. It is also very
useful from the meteorological point of view that our reformulations and the
methods to solve them provide an effective means of determining the geopotential at
the surface p = P. Moreover, we would like to point out that ‘DSalso provides some
information about the topography of the earth.
In the functional frameworks of our reformulated problems, we have to establish
a regularity result concerning an elliptic system with the non-local constraint. The
result is obtained by combining the differencequotient method of Nirenberg and our
systematic study for the function spaces constructed in this paper. The result is not
only crucial for our problem, but also possesses its own independent interest.
Another difficulty arises from the high nonlinearity of our problems. The
nonlinear terms take the same form as that of the nonlinear terms in 3~
Navier-Stokes equations. However, they possess a quite different nature. In our
problems, due to the absence of the time derivative term
of w , all the
information provided by this term is lost here. Thus, the situation is more
complicated than that of 3~ Navier-Stokes equations. The information for the PES
from those viscosity terms of the vertical velocity is also lost.
This paper is divided into three parts. In the first part we consider the PES. For
completeness, in section 1, we present a brief derivation of the primitive equations.
Then, by integrating the diagnostic equations, we introduce another formulation of
the equations in the last part of this section. In section 2 we construct some special
function spaces for our problem, in which the non-local relation (1.52) appears as a
constraint. Then, after establishing some functionals and their associated operators,
we obtain the weak formulation of the PES. We also prove a regularity result on
solutions of an elliptic system with the non-local constraint, which, on one hand,
possesses its own meaning and, on the other, is used in the remaining part of the
paper. Finally, in section 3, we establish an existence theorem of global weak
solutions for PES.
Part 3 deals with the PEV’s. A s in section 2, we obtain in section 4 a mathematical
formulation of the problem. Then, the existence of global weak solutions and local
strong solutions, and time analyticity of the strong solutions, are established in
section 5.
The last part of the article is devoted to attractors and their dimensions for both
PES and PEV’s. Some physically relevant estimates for the Hausdorff and fractal
dimensions of the attractors for PEV2s are given in section 6. Moreover, the existence
and dimensions of global attractors for both the PES and PEV’s are also reported in
the last section of this paper. In forthcoming articles [5, 6) we will study the case
where the pressure variable is allowed to vary in the interval [0, PI; here we restrict
ourselves, more classically, to an interval [p,,, PI, p o > O (see section 1). Subsequently, we will investigate the PES of the ocean where the independent variables are
the original (natural) ones (0, p, z and f).
The principle notation used in this paper is listed in the appendix.
The primitive equatiom of atmosphere
241
CONTENTS
Introduction
Part 1. The primitive equations of atmosphere (PES)
1. The PES in meteorology
1.1. General equations of atmosphere
1.2. The PES
1.3. The Non-dimensional form of the PES
1.4. Reformulation of the PES
2. Mathematical setting of the PES
2.1.
2.2.
2.3.
2.4.
Some function spaces and their properties
Some functionals and operators
Functional setting of the PES
Regularity for the linear stationary problems
3. Existence of global weak solutions
Part U. The primitive equations with vertical vicosity (PEV*s)
4. Mathematical setting of the PEV’s
4.1. The equations and some physical considerations
4.2. Reformulation of the PEV’s
4.3. Functional setting of the PEV%
5. Existence of solutions and their properties
5.1. Existence of solutions
5.2. Time-analyticity of the solutions
Part UI. Attractors and their dimensions
6. Existence and dimensions of attractors for the PEV’s
6.1. Absorbing sets and attractors
6.2. Dimensions of the attractors
7. Global attractors for the PE%
Appendix. Table of principal notation
Part 1. The primitive equations of atmosphere (PES)
In this part, we study the mathematical formuation and the existence of solutions for
the PES. As we mentioned in the introduction, regarding long-term weather
I L Lions et a1
242
prediction and climate changes, we have to consider the equations together with
viscosity. The quasistatic equilibrium equation, however, provides a fundamental
property of the atmosphere. So in this part we emphasize the importance of this
relation, and consider the specific PES in which we keep the quasistatic equilibrium
relation unchanged and only attach the viscosity to the horizontal velocity and the
temperature.
1. The PES in meteorology
1.1. General equations of atmosphere
!f we use a non-inertia! coordinate system rotating with !he earth; then atmospheric
motion is described by the following general hydrodynamic and thermodynamic
equations of compressible fluids with Coriolis force:
Momentum equations
dV,
-=
dt
pressure gradient
=
+ gravity + Coriolis force +dissipative
force
_-I grad, p + G - 2R X V,+ D .
P
Continuiry equation
dp
- p div, V, = 0.
dt
+
The first law of thermodynamics
dT
dt
RTdp
dQ
c -- - -.
’
p dt
dt
The equation of state
p = RpT.
(1.4)
In the above equations, D denotes the viscosity terms, which will he specified later,
and dQ/dt is the heat flux per unit density in a unit time interval, which includes
molecular or turbulent, radiative and evaporative heating. In other words dQ/dl
represents the solar heating, the albedo of the earth, and the molecular heating.
With appropriate expressions of D and dQ/dt, (1.1)-(1.4) are the usual
equations of compressible fluids with the additions of Coriolis and gravitational
forces (e.g. see [7, vol. 11).
S i x e we wint to s ! ~ ! y!he atmosphere around the earth, the spherical
coordinate system obviously describes atmospheric motion most naturally. Let
0 (Os 0 G n) denote the colatitude of the earth, q ( 0 e q s 2n)the longitude of the
earth and r the radial distance; z = r - a is the height above sea level. Then we have
V, = ugeg+ u,e,
+ u,e, = roee + r sin 0@e, + iez,
243
The primitive equations of atmosphere
where e,, ,e and e, are unit vectors in 8-, q - and r-directions. Using the
geometrical notation, they are defined by
l
18
ea=-a 88
a
a
e =-.
= a2
e , =--a sin Oaq
Then (1.1)-(1.4) can be rewritten as
-+-(~,u,-uZ,CO~
dv, 1
dv, 1
-+-(u,v,
df
r
du,
dt
c
~ ) = - - -1+aP
~Qcos~u,+D,
pr ae
r
df
1
r
+ U ~ U Cot
,
e) =
1 aP
( v i + v i ) = ----g
P ar
d T RTdp
p dt
df
pr sin 8 aq
+ 2 Q sin Ov,
- 2Q cos
Bv, - 2Q sin Bv, + Dp
+ 0,
dQ
dt
p = RpT
where
a +--+-v,a
d =_
_
dt at
r a8
up,
a
r s m 039,
+U,-
a
ar
and D = (De, D,, D,), The underlined terms on the left-hand sides of (1.5) arise
from the curvature of the earth.
1.2. The PES
The scale analysis, meteorological observations and historical data show that the
atmosphere satisfies the following quasistatic equilibrium equation:
aP
-=
ar
-Pg
which provides a relation between pressure and density. As we mentioned in the
introduction, this relation is highly accurate for the large-scale atmosphere; thus, it
has become a fundamental equation in atmospheric science and serves as the
starting point of all theoretical investigations and practical weather predictions. As
indicated in the introduction, a rigorous mathematical justification of (1.7) will be
given elsewhere. This equation is derived from (1.5) by asymptotic expansions,
using the fact that the height of the atmosphere is small compared to the radius of
the earth.
Replacing the vertical velocity equation (the third equation) in (1.5) by the
quasistatic equilibrium equation (1.7), we can obtain the following PES of atmos-
244
J L Lions et a/
phere in the coordinate system ( t , 0, 9,z ) :
(1.8)
(1.9)
aP
(1.10)
-Pg
-=
ar
1
c
dv, sin 6
d T RTdp
p dt
dt
+--
(1.11)
dQ
dt
(1.12)
p =RpT
(1.13)
a v,a
d
dt
-=-+--+--
at
a
U?
38
a
a
sin839
a
+U,---.
ar
(1.14)
Since the vertical scale is much smaller than the horizontal one, we have replaced r
in (1.5) and (1.6) by the radius of the earth a and, moreover, we have omitted some
terms underlined in (1.5), which are thought to be small.? However, we still keep
the terms
_":cot
_ e
a
*cot
a
e
representing the force contributed by the curvature of the sphere S:.
1.21. Pressure vertical coordinate. It is easy to see from (1.7) that p is decreasing
with respect to the third independent variable z, and therefore the coordinate
transformation
(t,
e, 9,z)+(t*, o*, V * , P =p(t, e, 9,z))
can be made with
t*=t
e' = e
(1.15a)
ql* = 9.
The inverse transformation is
We have
(I,
e, 9,P)+
(t,
e, 9,z = z ( I * , e*,T*, P)).
(1.15b)
(1.16)
P =Po. e, 9,z ( I * , e*, v*, P)).
While z vanes from 0 (the surface of the earth or of the ocean) to infinity, the
pressure p vanes from p. (the (unknown) pressure on the sea or the earth level) to 0
(the pressure in the high atmosphere). However, as we will see later, we will restrict
p to an interval [po, P I , where po> 0 and P is a given approximate value of p.; the
case po= 0 leading to some special mathematical difficulties is studied in future
papers in this series [ 5 , 61.
TIf we keep all the terms, most of what follows remains true but the proofs become a little more
complicated.
The primitive equafions of atmosphere
245
I
We now derive the PESwritten using the new system of variables, which is called
the pressure coordinate system.
Differentiating (1.16) with respect t o p , e*, p*, respectively, we obtain
ap az
-1
az ap
az
ap ap
-+--=o
a0
azae*
ap
ap az
-+ - y = o .
--ap azap
(1.17)
Replacing apldz by -pg in (1.17) provides
P
aP
(1.18)
-agz
=-_
~
I
1 aP
pap]
Moreover, it is easy to see that
a ap
---_
a2 - a2
a
ap
1
_a= -
a +--a p a
ae ae* a e a p
-=-+--.J
a
a apa
aP
a'p*
1
(1.19)
avap
On the other hand, the material differentiation in the new coordinate system is
given by
a
(1.20)
These expressions may be used to form the gradient of a scalar function A and,
similarly, the 2~ divergence of a vector B with the results
grad, A = grad, A
dA ap
+ --grad,
ap az
z
aB ap
grad, . B = grad, .B + -- . grad,
ap az
2
where grad, and grad, denote the horizontal gradient operators in the p-coordinate
and z-coordinate systems, respectively. With these formulae the horizontal pressure
I L Lions et al
246
foices become
1
- - grad,
P
p = -grad, @
where @ = gz is the geopotential. SO the first two components of the momentum
equations (1.8) and (1.9) become
(1.21)
with d/dt* given by (1.20).
The first equation in (1.18) and the equation of state (1.13) yield
a@
RT
-+-=o.
(1.22)
P
Now we consider the continuity equation (1.11). By (1.7), the continuity
equation (1.11) can be rewritten as follows:
aP
+--a s m 0 aq
ar
(1.23)
Now we check each term of (1.23):
d
-
aP
( 2) = atd (dazp) + ve-' a a3e (3)
+ a s m O aaq (-)
az
dz
dt az
aP
V'Za (%)
7+
-
adp
avo ap
av,
=%(dt)
&a30
az a s i n O 3 p
1
ap
av,ap
3z 3r'
Owing to (1.19) we have
avo aP
1 hesin 0
+aP
az a d e azasin %
a0
1
3p av, sin O*
--
az a sin O* d e * .
Similarly,
_-av,
aP
az a sin e aP
ap
I Lav
ap
+
-?dz a sin e
a~
av,
dz a sin e* aq'* .
So (1.23) implies
(1.24)
The primitiue equations of atmosphere
241
Since the total derivative does not depend on the coordinate system, we can
define
(1.25)
Then, omitting the asterisk for the independent variables of the new coordinate
system, we can easily obtain the following PES of atmosphere in the coordinate
system (f, 8, q,p), which is called the p-coordinate system:.
dw,
dt
U',
a
cot O =
1 a@
---+
a a0
252 COS OW, + D e
(1.26)
(1.27)
(1.28)
Remark 1.1. It is well known that the atmosphere is made up of a compressible
fluid. Here in the p-coordinate system the continuity equation in (1.26), however,
takes the same form as that of an incompressible fluid. This is one of the advantages
of the p-coordinate system.
Remark 1.2. There are other alternative ways of treating the vertical coordinatz. By
taking the orography into consideration, for instance, we can use u = p I p 5 as the
vertical coordinate, where ps stands for the surface pressure on the earth. The
resulting coordinate system is called us-coordinate system. Even though it has some
drawbacks (see p 79 of [2]), the u-system is also widely accepted and practicely
used. We wiii study the priiiiiiive equaiions on this coordinate sjijieiii elsewhere.
We now want to make a further simplification of the term RTwlp in the last
equation of (1.26). The reason is two-fold. Firstly, the resulting equations are more
accessible from a mathematical point of view. Secondly, the 'energy' provided by
%a
I L Lions et a1
the simplified term will be balanced by the term RT/p in the third equation of
(1.26).
Let t = t ( p ) E C"([p,, P I ) be a given vertical distribution of the standard
temperature C" on the interval [ p o ,PI such that
3
C Z = R- - p (:pi'
=constant.
(1.29)
T can be considered as the climate average value of the temperature on isobaric
surfaces. Here P>po>O. Since we want to study the 3~ large-scale atmospheric
motion, as a first step, we can assume that the surface of the earth is nearly an
isobaric surface given by p = P where P denotes the approximate value of the
pressure on the surface of the earth. We also assume in this paper that the isobaric
r ythe atmosphere, ..here rn.u ."
i s 2 cmall
~ ~ r f z pc p= p o provides the y p e r b o ~ g d ~ of
positive number.
Moreover, we choose a standard distribution 6 = 6 ( p ) of @ defined by
-
a6 Rf
-+-=o.
aP
(1.30)
P
Then, by some straightforward computations, it is easy to see that the unknown
functions (U,o),CP' = @ - 6 and T' + T - t satisfy the following equations (for
simplicity we omit the primes):
I a@
cot e = - --+m
dv,
df
U',
dv
df
v v
a
2
a
a
ae
COS
eu, + o0
1 a@
0 = -__-2 9 cos Ov, + Dw
a sin 0 a~
aQ R
-+-T=O
aP P
dT
'dt
c
(1.31)
cpC2
dQ
U=-.
pR
dl
Here we have used the following meteorologically reasonable approximation (see [SI
for more details):
RT
P
-fJJ-(lJ
Rf
P
(1.32)
.
..
We wouid iike to point out that whiie the number jT - Ti may noi be smaii
compared to IT( li'i, it appears (see [SI) that IT - is small compared to p / R o :
+
( T - TI <<- P
RW
,
Thus it is legitimate to replace, as a first approximation, T by
i' in (1.32)
249
The primitive equotions of atmosphere
Since the main purpose of this paper, as we have mentioned, is to study the
mechanism of long-term weather prediction and climate changes, it is necessary to
add (or replace (De,
0,)and dQ/dt by) some viscosity terms in the equations
above. According to some physical considerations, we can introduce the viscosity in
such a way that the PES take the following form (the viscosity terms have been
underlined) (see [9, lo]):
av
-+
alJ
V,u+ o-+228cos(0k)v
at
aP
acp R
-+-T=O
aP P
GGV+-=G
ao
+ grad@-
[(")
ap RT
pl Au - v I -
a
-1
2au
ap
=O
'
(1.33)
with space domain S:x (pu. f),where E stands for the diabatic heating, including
mainly the radiative heating and the evaporative heating, such as the solar heating
and the albedo of the earth. One of the underlined terms concerning T in (1.33)
corresponds to the derivative
azT v
v2-=
az2
,d((E)???.
ap
RT
az
Flirthermore T i s replaced by T and this is legitimate as a first-order approximation,
since v2 is small.
Moreover, Vuu and V u T are covariant derivatives of U and T with respect to U
(see also (2.17)) given by
(1.34)
J
where e, and e? are the unit vectors in the 0- and cp-directions, respectively. In
(1.33), there are two horizontal Laplace-Beltrami operators for the scalar function
T and the vector field on Si, respectively. They are defined as follows:
(1.35)
(1.36)
where, in (1.36), AV@and Au, are given by (1.35).
1.2.2. Boundary value conditions. There are no conditions at the boundary of the
250
J L Lions er al
(0, q) domain except the obvious conditions resulting from periodicity. It remains
to specify the conditions at the boundary of the p-domain. As indicated before, p
helongs to the interval [p,. P] where po>O is a small number and P is an
approximate value of the pressure at the surface of the earth.
The boundary conditions at po and P are written as if po corresponds to the
upper atmosphere and P to the surface of the earth. However, it will appear that the
solution of the boundary value problem will provide the value of the geopotential
@ = gr at P and p o , i.e. we find afterwards the heights of the isobaric surfaces p = P
(close to the earth) a n d p =po (close to the upper atmosphere)
W O , w, P ‘Po.
t ) = gW0, T2t )
and @(e,q ~ p, = P, t) is a small function of t. The justification of these boundary
conditions and the (delicate) passage to the limit po+0 will he studied
subsequently.
The boundary value conditions for (1.33)-(1.36) are
where a,is a constant related to the turbulent transition on the surface of the earth
including the albedo of the earth, T, is the temperature on the surface of the earth
(more precisely, it is the temperature on the land and the surface of the ocean,
which we assume given). Without loss of generality, we assume that T, E C-(S:).
The boundary condition
comes from the natural boundary condition
aT
3.z
gpdT
RTap
We then replace T by
cu’(T,-T)
at p = P .
f (see after (1.32)) and we set that
We assume that CY,is a constant which leads to some slight simplifications, but the
case of a, non-constant can be treated in exactly the same way.
We would like to indicate that other boundary value conditions can also be
considered and treated by the methods of this paper. The following boundary value
conditions, for instance, are also widely used in meteorology (see [lo, 111):
The primitiue equariow of atmosphere
251
It is also worth pointing out that in the second part of this paper we will consider
equations with vertical viscosity, which are called the primitive equations with
vertical viscosity (PEV2s). Since (1.33) is obtained from the general equations of
atmosphere by making coordinate transformations, it is not obvious how to
introduce such vertical viscosity. We will discuss this in the first part of section 4.
Moreover, we should mention that in the case of ocean dynamics we can consider
the PES in the usual ( x ; y : z ) coordinates since water is an incompressible fluid:
incompressibility is written in the form div, V, = 0 in the original ( x , y, z ) system,
and the quasistatic equation (1.7) (or (1.22)) is a simplified form of the conservation
of momentum equation in the r-variable. Therefore, the viscosity in the z-direction
can be introduced very naturally. We will study elsewhere the equations of ocean
dynamics and the coupled system of ocean-atmosphere,
1.3. Nondimensionalform of the PES
It is usual, from the physical point of view, to introduce the non-dimensional form
of the PES. To this end, first of all, we introduce some non-dimensional unknown
functions and parameters. let
v=v'U
(1.38)
a
= - f'
U
U
RO =UP
1
(1.39)
(1.40)
The parameters R e , , Rez, R f , and RIz are Reynolds numbers. R o stands for the
Rossby number, which measures the significant influence of the earth's rotation on
252
I L Lions et al
the dynamic behaviour of the atmosphere. The iU, is a positive constant related to
the turbulent transition on the surface of the earth. Other notation is explained in
the appendix.
We can then obtain from (1.33) the following non-dimensional form of the PES
(for simplicity we omit the primes):
dU
-+V,u
at
au
f
+ w-+--k
ag Ro
div U
am
C-=
at
x
U
p ~
)
:
(
[
R e 2 a g PT
1
+grad@--Au
Re,
1
a
-a[1
2 ,a v
0
(1.41)
(1.42)
a@ b~
-+-T=O
a[ p
a l aT
(dt+~u~
=fi
1
1
a
Rt,ag
[(-I -1
pxl ’3T
PT a[
(1.43)
=A.
(1.44)
Here the covariant derivative operator V , , the gradient operator grad, the
divergence operator div and the Laplace-Beltrami operators A on S2 are defined by
(2.10)-(2.18). We have to point out that we use the same notation for these
operators as that for S: listed in (1.34)-(1.36). The only difference between the
operators given by (2.12)-(2.18) and those given by (1.34)-(1.36) is the factor U .
The space domain is now
M = Sz x (0, 1)
(1.45)
and the boundary vaiue conditions are given by
[=1:
C=O:
(U, w ) = 0
(U, w ) = 0
aT
-= iU,(T;
at
aT
-=o.
-T)
(1.46)
In the above equations, similar to (1.33), there are two horizontal Laplacian
operators on S2 for the scalar function T and the horizontal velocity field U ,
respectively. The explicit form of these two operators are given by (2.21) and (2.22)
in section 2. Henceforth, we use A to denote these two Laplacians on S2. Moreover,
V,u and V,T denote the covariant derivatives of U and T with respect to U, given by
(2,ii).
For the real physical situation, fi can be taken as zero. From the mathematical
point of view, however, we prefer to consider the general case. Moreover, for
simplicity, we still use ( p / P ) ’ in the viscosity terms instead of using ((P- p o ) [ +
Po/P12.
1.4. Reformulation of the PES
In (1.41)-(1.44) above, there are two diagnostic equations, (1.42) and (1.43). So, as
we mentioned before, the whole system is not an evolution one. Therefore, in this
section, we integrate the diagnostic equations, and then establish another formulation of the PES.
We would like to mention that it is significant to introduce. the two-variable
The primitiue equations of atmosphere
253
function Qs below. The reason is three-fold. Firstly, it is very nice from the
mathematical point of view that Qs is exactly the Lagrange multiplier of the
non-local constraint (1.52) (see section 2 below). Moreover, the whole system is a 3~
evolution one, but Qs is only a function of the first two independent variables.
Secondly, from a technical point of view we will see in section 3 that we can directly
use the Faedo-Galerkin method to solve the weak formulation of the reformulated
problem (1.50)-(1.54). In [ll], however, we did not find this reformulation and,
therefore, we had to construct some special kinds of function spaces for the vertical
velocity of w , in which the elements possess more regularity in the p-direction. In
the meantime we had to use the semi-discretization method to prove the existence of
weak solutions because we were not able to use the Galerkin procedure. Thirdly,
from the meteorological point of view, the isobaric surface p = P, in general, is not
just the surface of z = 0. In fact, the geopotential Q at the isobaric surface p = P
should be considered as an unknown function rather than as a known function
obtained from practical observations. This function also describes, at least partially,
the topography of the earth.
More precisely, integrating (1.42), and taking the boundary value conditions for
w into account, we obtain
Suppose that the geopotential Q = g r - 6 is equal to a certain function Qsat the
isobaric surface p = P, then we can integrate (1.43) as
Q ( t ; 8, 9,5) = Q& 8, p) +
which implies that
grad Q = grad Qs+
1
‘bP
-T ( t ; 8, 9,5’) d t ’
(1.48)
C P
I
:grad
T d5‘
(1.49)
Therefore, the PES (1.41)-(1.44) can be rewritten as
au
-+
at
V,u
au f
+ W ( u )-+--k
at RO
+grad Qs
aT
o I ( z + V,T
+W
xU
1
+
a
2au
[(-)pl;,
PT -ag
]=A
bP
1
1 a
W ( U )- - A T
[ pro ’ 3-1T
P
Rfi
R t Z a g PT ag
1
-Au - -Re2d[
Re,
a
at
( U ) q --
--
(1.50)
=f2
(
7
)
(1.51)
l dI i v u d t = O
(1.52)
where W ( u ) is defined by (1.47) and the physical parameters Ro, Re,, Re,, Rt, and
Rt2 are given by (1.40).
254
.
IL Lions er
al
The houndarv value conditions are
(1.53)
Henceforth, we use 3 to denote the pair
conditions can he given as
Tu).
-S I , = ~ = ~ ~ = ( U ~ ,
(U. T ) :
and then the initial value
In the remaining parts of this paper we will study the well-posedness and the
long-term behaviour of the initial boundary value problem (1.50)-(1.54), and its
modification (PEV's).
Intuitiveiy, iet (U, as,
T)be a smooth solution of the problem (1.50)-(1.54) for
= 0. For any test function SI= (U,, G)E G ( T M [ 7
's') x C ( M ) satisfying (1.52),
we can take the Lz inner product between (1.50) and (1.51), and the test function
&, where the function spaces are defined in section 2. The resulting equations can
be obtained by direct computation as follows:
d
-(=
;
I
+,lT
T \ I h l = s, q + d v 3 \ - ( l C
Fi
dt\-' .
'1,"
'I,
-\-,
\",, J1,,
(1.55)
"\I?
',
~
where the functionals a(., .), b(., ., .) and e ( . , .) are defined by (2.42), (2.48) and
(2.53), respectively and the inner product (., .)His given by (2.31). As we mentioned
before, the function Qsdisappears in the resulting equations (1.55). Therefore, the
basic idea of how to solve the problem is as follows. Firstly, we construct some
function spaces for the unknown function 3 = (U! T) satisfying the constraint
equation (1.52). Secondly, we prove that the function Qs is exactly the multiplier of
the constraint. Then, we obtain that (1.55) is the variational formulation of the
problem (1.50)-(1.54) for
Thus, we can solve the problem by the
Faedo-Galerkin method. Moreover, to study the long-time behaviour, we can use
the idea of [12] to estimate the H a u s d o s and fractal dimensions of the attractor of
the problem. Of course, in the case z # O , we have to find a function T* satisfying
the non-homogeneous boundary condition
dT* _
- Cs(T,- T * )
[=1:
z=O.
and inequality (2.58). This inequality is crucial to obtain some a priori estimates
and, therefore, for solving the problem.
2. Mathematical setting of the PES
2.1. Somefunction spaces and rheir properties
The space domain of our problem, M = Sz x (0, l ) , is a manifold with boundary
r = r, U r, given by
rt= Sz x
{i}
Vi = 0, 1.
(2.1)
Noticing that M is the product of Sz and the unit interval (0, l), we see that the
255
The primitive equalions of atmosphere
tangent space T(,,,,M of M at (4,
T,S2 and Tc(O, 1)= R as follows:
5) E M can
he decomposed into the product of
T,,,<,M = GS' x Tc(O,1) = T,S'X R .
(2.2)
Therefore, the Riemannian metric g , on M is given by
g,((q,
5); ( U .
U),( U I , 01))
=gsZ(q; U , V I )
t-
WO,
VU,U I E TJZ
where gs2 is the Riemannian metric on S2.
In particular, in the spherical coordinates ( x ' , x 2 , x ' ) = ( B .
given by the matrix
(g,)
=
(
W, WI
'p.
E R
<) on M , g , is
1 sin20 l)
where
(2.3)
(2.4)
a a
<);z,s).
g , j = g , ( ( B > 9,
So the unit vectors in the 0-, p- and <-directions are
a
e' --ae
l
e,=-- sin
a
3
'
B a9
et
'Z'
(2.6)
We used in (2.6) a classical notation in geometry but throughout the article we use
the notation e,, e, and ec. For simplicity, we denote the inner product and norm in
the tangent space T(,,m,t)Mby
X.Y=X'Y'+XZY2+X3Y'
1x1= ( X ' x)'n
for any
x = X ' e , t X2e, + X3ec
Y = Yle, + YZe, + Y3ec
1
(2.7)
E T < H . W , t,M.
After introducing this basic notation, we can consider functions on M. First, let
C-(h?) (resp. C-(TM)) he the function space for all C" functions from M into R
(resp. C" vector fields on &f). Then, by (2.2), C " ( T M ) possesses the following
decomposition:
cy7-M)= cy TLf I TSZ) x C"(M)
= { U : &f+ TS2 E C^ I ~ ( 0 q, , 5) E Tc,,,,S2}X
{ w : h?+ Iw
E
C").
(2.8)
Similarly, let Cj(M) (resp. C,(TM)) be the function space for all C" functions from
M into R (resp. C" vector fields on M) with compact supports. Then, we have the
following decomposition (similar to (2.8)) for C,(TM):
CF(7-M)= C,"(TM I TS') X C,(M)
= { v : M - TS'c C" with compact support Iu(0, p, <) t T~,,,,Sz)
x { o :M + R
C" with compact support).
(2.9)
Moreover, we consider C"(S') (resp. C"(TS2)) to be the space of all smooth
functions (resp. smooth vector fields) on Sz.
E
256
I L Lions et a1
We are now in a position to define some differential operators. First, the natural
generalization of the directional derivative on the Euclidean space to the covariant
derivative on Szis given as follows. Let T E Cm(Sz)and
+ u,e,
U = U&
u1= (U1),eH+ (ul),e,
cCD(TSz)
we then define the covariant derivative of ul and T with respect to
V,u,=
(
U
,
+
a(Ul),
ao
uv
a(U1)"
U , ( U ~ )Cot
~
s i n 0 aq
a(ul),+
.U,
d(UI),
aq
sine
+(U,?
V,T = U,-
aT
a0
)
U
as follows:
0 e,
+ u,(uI),
1
cot 0 e,
U, aT
+ -sinoaq'
(2.10)
(2.11)
We now define some natural operators on functions and vector fields on M .
Firstly, for any function h E C " ( M ) , we define the gradient grad, h E C"(TM) of h
by
ah
I ah
ah
grad, h = - e ,
-- e , + - - <
a0
smoaq
at
ah a
=grad h -(2.12)
+
+ a t ac
where grad is the gradient on
Szgiven by
ah
grad h =-e,
1 ah
+ -(2.13)
sin eaqe-'
Moreover, for any X = X'eR+ XZe, + X3es E C"(TM), we define the divergence of
ao
X , div, X E C"(M), by
div, X = div(X'e,
-
1
ax3
+ X'e,) + at
axlsino
-G( ao
axz + ax3
f-
avj
z
(2.14)
with div the divergence operator on Sz,which is given by
div(Xle,
1 (3X'sin 0 +-axz),
+ X2e,) = sin 0
80
av
(2.15)
The Laplace-Beltrami operator of a scalar function h E C"(M) is
A,h
= divM(grad, h )
= div,( grad h
=Ah+-
+et
ah
at
a2h
ac2
a sino.
- z z [ ~ H ! $1
-
I
1
J2h
+L'h]+z
sinoaqz
(2.16)
The primiliue equations of atmosphere
257
where A is the Laplace-Beltrami operator on S 2 defined by
Ah = div(grad h )
-
1
Sine
[$ (sin e g)
+G
a2h1
drpZ
1
(2.17)
Moreover, we need a horizontal Laplace-Beltrami operator for vector fields on
S2. For any U = ueeg u,ep E C"(TS2)we define the horizontal Laplace-Beltrami
operator, Au of U on S 2 as
+
i
2 c~~eau,
2 ~ 0 s e a u , up,
Au= A u ~ - T - - - e,+ A u , t 7 - - sin e aq sin20
ue
sin20 ap sin2e)eV
where Aue and A V , are given by (2.17).
By direct computation we can prove
)
(
(2.18)
L ( - A u ) . v , d M = l ( V , , u ~ V , , v , + V , ~ u ~ V , ~ u , + u ~ u , ) d(2.19)
M.
M
We are now in a position to introduce the notation of Sobolev-type function
spaces on M. First, let L 2 ( M ) , L 2 ( T M ) and L2(TM I TS') be the Hilbert spaces of
L2 functions, L2 vector fields and the first two components of L2 vector fields on M ,
respectively. We denote the inner products and norms by the same notation (., .)
and 1.1 given by
( h , , h ) = l h,.hZdM
Ihl= (h, h)'"
(2.20)
M
for any h , h,, h2 E L2(M),or L 2 ( T M ) , or L2(TM 1 TS'). Similarly, we can define the
spaces L2(S') and L2(TS2).
As usual, we can define the Sobolev spaces H"(SZ), H"(TS*), H"(M), H"(TM),
H'(TM I TSz), W m , p ( S 2 ) ,Wm,p(TS2),W"'P(M), W m . P ( T M )and W".p(TM I TS2)
for any 0 c m = integer < m. 1s p c m, s E R. More precisely, H'(S2) and H"(TS2)
are defined by (see [13])
H ~ ( S *=) D((-A).")
H"(TS2)
= D((-A)sn)
I
(2.21)
where the first Laplace-Beltrami operator is the scalar one defined by (2.17); the
second one is defined by (2.18) for vectors.
H"(TM) and Hs(TM 1 TS2) is as follows: when s is
The definition of Hs(M),
positive integer, these are the completed spaces of Cm(M), C"(Th?) and
C ( T M I TSz) for the respective norms
Ilhllw=
(c
jtk-
112
l ( - A ~ " ( v d ' h l ~ ~ ).
(2.22)
When s is zero, these spaces are the L2 ones. When s is not an integer or s is
negative, they are defined naturally by interpolation and duality (see [13]).
The norms in the W",p spaces are given by
Ilhllw-.P=
[I ( c
M
c
1 6 k S m 1,=1,2,3
j=l, ..., k
lVjL. . . Vi,hlP + Ihl") dM]"'
(2.23)
258
J L Lions el a[
for h in one of the spaces Wm.’(M), W”’,p(TM)and W’”,’(TM ITS’). For h in one
of the spaces Wm,”(S2)and Wm.’(TSz),we define
Ilhllw-n =
[I( 2
Lhkhm
c
1,=1.2
,=I.
.k
IV., . . . V,,hlP + Ihl’) dS’]’*.
(2.24)
Here we have used the notation
VI = VC@
a
v3=z.
v z = V<#
(2.25)
Before proceeding, we emphasize that in the following we use the standard
notation given below to denote the norms and inner products in the Sobolev-type
function spaces defined above:
11. II wm e
11. 11 H’
( ’ 9
(2.26)
‘)H’.
Now we turn to the definition of the function spaces for the unknown function
5 = (U, T). Motivated by the studies for Navier-Stokes equations (e.g. [14, 15]), we
construct the function spaces for U in such a way that (1.52), i.e.
I
Ldivudc=O
appears as a constraint for U in the spaces, which plays the same role as the
continuity equation div U = 0 in the mathematical theory of the Navier-Stokes
equations. More precisely, let
I
1
(2.27)
V l = { u ~ C ; ( T M I T S 2 )0I divudC=O
I
.
We then define
VI = the closure of V,for the H’ norm
V, = H 1 ( M )
HI =the closure of VI for the L2 norm
H, = L 2 ( M )
v = VI x V’
H = HI x H2.
1
(2.28)
Before studying some properties of these spaces, we should define their norms
and inner products. By definition, the inner product and norm in VI and V, are given
by
((U. U1))
=I,
@<#U
llull = ( ( U .
U))1iz
((T, TI))=
IM
. V&
+ v,u
v,Tu,
vu, u1 E VI
(grad, T . grad,
IlTll = ( ( T T ) Y n
’
au,
+-a~
. -+ U
ac a t
r, + TT,) dM
VT, TI E V,.
’
)dMl
u1
(2.29)
259
The primitive equarions of atmosphere
For simplicity, w e use the same notation 11.11 and ((., .)) as for VI and V, to
denote the norm and inner product of V given by
((321)) = ((U. U I ) ) + ( ( T , T ) )
/IB(I=((E, %))1’2
VZ=(u, T)
El = (U. T,) E
v.
I
(2.30)
Moreover, we use the same notation (., .) and 1.1 to denote the L2 inner products
and norms in the spaces H, (i = 1, 2) and H . However, since in (1.51) there is a
coefficient a, for the time derivative term d T / d t , we introduce the following
equivalent norm and inner product for the spaces H2 and H:
(~:,2l)H=(uvUl)+(alT
TI)
~
v E = (U. T )
lqH=(E, =)E2
2 , = ( U . T,) E H
1
(2.31)
( T , G ) H = ( 0 , T, TI)
V T, TI E H2.
lTlH=(Tr T ) E .
Henceforth, the norms and inner products in H, and H refer to those given in (2.31)
unless otherwise stated.
Now, by the Riesz representation theorem, we can identify the dual space H‘ of
H (resp. Hc’ of H,, I = 1, 2) with H (resp. H,), i.e. H’ = H (resp. H8’= H,, i = 1, 2).
We then have
V cH =H ’ c V ‘
(2.32)
where the two inclusions are compact continuous.
The following lemma characterizes the spaces H and V.
Lemma 2 1 . Let H: be the orthogonal complement of H I in L2(TM I TS2). Then
H: = {U
HI= {U
E
E
I
L2(TM TS’)
L*(TM I TS2)1
VI= { u E HA(TM I TS2)I
I U =grad I, I E H’(S*))
/
/
1
div U d 5 = 0
0
I
div u clc = O
0
I
i.
(2.33)
(2.34)
(2.35)
Proof.
(i) Proofof (2.22). It is easy to see that the right-hand side of (2.33) is included
in Hf.Conversely, we claim that the inverse inclusion is also true. Indeed, for any
U E Hf,we have
LuuldM=O
vu, E VI.
(2.36)
C,(TM I TS’)}.
(2.37)
Obviously,
I;:{
KI
- U E
So it follows from (2.36) and (2.37) that
(Mu$dM=O
vu E c ; ( T M
1 TS’).
260
I L Lions et a /
<
That is, &lag = 0 in the distribution sense. Therefore, U does not depend on by
theorem 3.1.4‘ of chapter I11 of [16], which we will quote in part (iii) of lemma 2.2
below.
Now, choosing a function h E CY(0. 1) such that
i,’h(S)d 5 f 0
we can infer from (2.36) that
JMuu dS’ = 0
Vu
E (U E
C‘(TS2) I divu = 0 ) .
Thus, lemma 2.2 below shows that there is a unique (up to a constant) function
I E H ’ ( S 2 ) such that
U = grad I
(2.38)
which proves (2.33).
(ii) Equations (2.34) and (2.35) can be proved similarly to the corresponding
results for the Navier-Stokes equations (see [14]), and w e omit the details. 0
Lemma 2.2.
(i) Let f E 9’TS’),
( then
U u)=O
Vu E
{U E
C”(TS2)Idiv U = 0)
(2.39)
if and only if there is an I E 9‘(S’) such that
f = grad 1.
(2.40)
(ii) I f f EL’(TS’) satisfies (2.39), then the I obtained in (2.40) possesses t h e
following regularity property:
I E H’(S’).
(2.41)
(iii) I f U E 9 ’ ( T M 1 TS’) such that aw/aC = 0 in the sense of distribution, then U
is a distribution on S’ independent of the third variable 6. Here 9 ’ ( T M I 73’)
represents the distribution space of the first two components of all the distribution
vector fields on M.
For the proof of parts (i) and (ii) of this lemma, we refer to 1141. The last part of
the lemma is the trivial generalization of theorem 3.1.4’ in chapter 111 of [161.
2.2. Some funcrionals and their associated operators
W e are now in a position to define some functionals and their associated operators,
which are related to different parts of the PES.
First, concerning the principal parts of the PES, w e define three bilinear forms
a : V X V + R , a i : VI x &-+ iw (i = 1, 2), and their corresponding linear operators
A : V + V ’ , Ai:&--tVI ( i = 1 , 2 ) by
a,) = (AB, E , ) H
= ( A , u , 2)’) + (AzT, K ) H
= d u , U , ) + a 2 ( T T,)
a(=,
(2.42)
The primitiue equations of atmosphere
ai(v, ui)=(A,u,
-
261
VI)
1-[Re,
1
(V,,v
‘ V#I
+ vevu ’ v,v, + U . U,)
1 pT; ,au av,
+
Re,PT
(
7
-.-]dM
at )
ac
az(T, ‘G) = (A27‘,
‘&)H
(-)
I,
1 p Z , ,aT
T . grad TI+ -.
RI, PT ac
1
-grad
[RI,
=
+
(2.43)
I,2
-ar,
1ac
dM
(2.44)
r G ) ’ T T , dS2
for any 5 = (U, T ) , El = (U,,TI)E V. Note that the appearance of the term U . U ,
ensures full coercivity of a , on (2.43), this term arising from (2.19). For T the same
is true but now the term involving TT,arises from the boundary value condition.
Lemma 2.3.
(i) a, a, (i = 1, 2) are coercive and continuous; therefore, A : V+ V’ and
A i : y.+ V ( (i = I, 2) are isomorphisms. Moreover, we have
{k,k,I
a ( 3 , Z l ) < c n m a x -,-
1
2-
ll~ll~ll’&ll
IIEII ’ II~III
&in
a(Z,Z)*chmin
“I
-,--{ R1t , Rt,
1 R12
+CO”
6-
llull.llu,ll
{k,
-,-
1
Rmax
;e2}
(2.45)
id, R:I,, :
Ilull‘+c;)min -
-
’
IlTll 2
J
1lEIl2
(2.46)
where cn, cb are absolute constants independent of the physically relevant constants
Cs,Re,, R f , , etct. The parameters R m i ,and R,, are defined
Rhl
Rt21
R,,,=--;max Rt,, Rt2, R e l , Re,,:
Cn
l
{
as
R,,,=-min
Cn
1
4
(2.47)
R t , , RI2, R e , , R e 2 , _ .
a,
(ii) The isomorphism A : V+ V’ (resp. A , : y + Vl (i = 1, 2)) can be extended to
a self-adjoint unbounded linear operator on H (resp. on Hi (i = 1, 2)), with compact
inverse A-l:H-.H (resp. A,:’:H,+Hj(i=1,2)),
and with domain D ( A ) = V n
H*(TM) (resp. &A,) = VI n HZ(TM 1 TS’), D(A2) = V, n H 2 ( M ) . )
t co, cb may depend on
E
= ( P -p,)RT,lagP
such that (4.19) holds true.
262
J L Lions el a1
Proof. The proof of D ( A J = VI n H Z ( T M I TS') can be obtained from lemma 2.7
below, The other parts of the proof are obvious, so we omit the details. 0
-
Now, related to the nonlinear terms appearing in the equations, we define three
functionals b :H X H X H - Iw, b, : H IX H, X H, R (i = 1, 2) and their associated
operators B : H X H - H , & : H I x H,+ H, (i = 1, 2) by
b(8, E,, SZ)= ( E @ , El),8Z)H
= ( B I ( ~VI).
, uz)
+ (Bdu,
G), G)H
(2.48)
= b i ( u , u i r uz) + b 2 ( ~ ,T,, T2)
bi(u, ui3 VI)= (BI(u, V I ) , uz)
= L[V"u,
bdu, Ti,
T2)=
+ ((divu
d5)
$1
'
(2.49)
u,dM
( E A U , Ti), G)I<
=a,
jM[V,,T,+ (jcL
div
U
de)
%]&
(2.50)
dM
for any B = (U, T ) , E, = ( u L ,T ) E V or D ( A ) (i = 1,2). We then have the following
lemma.
Lemma 2.4.
(i) For any E E V , 8 , E D ( A ) ,
b(8,
81,
(b(8.8,, E,)[ s
(ii)
(2.51)
S 1 ) = b 1 ( u ,U,, ~ , ) = b 2 ( TI,
~ , T,)=O.
I821
{ cc Ilzllff~ll%llH-J~'
llW
'
11~11H='
(2.52)
' 11=211.
Proof.
(i) By definition, we have
v"~vll~=v"u,~uI+uI~v"u,=2v"u,.uI
'71
which yields
bdu,
U,, u l ) = j M I V u u l .211
=
jM[:Vu
=-I
=-
+'
(U,('
(Icldiv u de)
2
dM
.
a iuli27
dM ]
0,
[div(~u,l'u)-~u,~2divu+ I divudc
1
2
+ ?1( l d i v u d5) alull?
M
f I,[ 1u112(div u +-aw'u')]
35
1.__
";121
dM
dM
* ~ 2 1 ~ l l ' ( ~ d i v ~ d5=11.1
S ) dS'
l
= 0.
Similarly, we can prove bz(u,TI, TI) = 0 and, therefore, (2.51) follows.
261
The primitive equations of atmosphere
(ii) Consider the typical terms in b, as follows:
MI,
[([div
de)
U
4
3.
u.1 dMI s
{
c l u ~.l
J vI
. ldiv ulL*
C IuZIL*. -
. ldiv uIL?
LZ
c
l l ~ l I l ! P I=,/
c Il=llw.'
ll%ll.
'
I l = l l H 2 ~
1 ~ 1
I aUri
JC
11~111'
This gives (2.52).
0
As we mentioned in the introduction, the PES possess some highly nonlinear
terms like (J'kdivu dC) J u l J C , whose behaviour is the same as that of IV;vI'(i =
1 , 2 , 3 ) , where V j(i = 1, 2, 3) are the first-order differential operators given by
(2.25). So, as far as existence and uniqueness of solutions are concerned, the PES
are even more complicated than the usual 3~ Navier-Stokes equations, and we will
consider some regularized forms of these equations later.
Concerning the linear terms in the equations, we define a bilinear functional
e :H x H - R and the associated linear operator E : H - H by
e(=, 9,) = ( E ( = ) ,Z , ) H
bP
- ( k x U ) , U , + IbPgrad Td:) . u L-- W ( u ) ' T I ]dM
=
P
(2.53)
for any E, El E V , where W ( u ) is defined in (1.47). We then have the following
lemma.
IM KO
(LP
Lemma 2.5.
(i) There is a constant c independent of the physically relevant parameters Rej,
R I , (i = 1, 2) and R o , such that
C
I(W),=,)HI
e(E, 9) = 0
(ii)
- IUI '
=z
lull + c IEl 11=, 1 1
{:
'
(2.54)
- 1111. lull + c 11E11. I=&
v9 E
v.
(2.55)
Proof. We only have to prove (2.55). The other parts of the proof are obvious. To
this end, by definition we have
e(=,E)=/
"(-U
w-+-aJ e
U@ J ) . u d M
sinflarp
bP
\<IF
T d c ) div U
W ( u ) , T dM
IM [(IM [(-Ic':
=IM
M~~
+
=
-7
Td5I.G
0:
= 0,
which proves (2.55). 0
1
div U dC) - - W ( u )' 7'1 d M
P
( F W ( u ) ,T - - W
bPW ( u ) .
P
bP
T ) dM
264
J L Lions et al
2.3. Functional setting of the PES
First, we should homogenize the boundary value condition for T. To this end,
solving
a P_
- as(% - 7 ' )
a<
yields
f* = Q 1 -
exp(-a:.<)].
Then, we define a function T : by
T: = 7*0,(C)
VE E (0,
{:
f)
(2.56)
where 0, E C"([O, 11) is given by
1-&<<<1
1 - 2 E S < <1 o s 5 s 1- 2 E .
0,(C) = increasing
E
Obviously, then T : satisfies the boundary value conditions for T given in (1.53).
Moreover, we have the following lemma.
Lemma 2.6. For any 6 >O, there is an
M u , T : , T)l < 6
E (0, 1 ) such
E"
that
vz E v.
llull ' IlTll
(2.57)
Proof, By definition, we have
Ibz(~,C ,
T)I = lbz(u, T, T:)l
= l a l j [VuT+W(u)-]T:dM1
aT
ac
U
= la,
1
[Vu T
s~v[,-2Gl)
-ac1
+ W(u) d T
( U ( .(grad T I .
sa~i,x,l-2e,l,
.
T,( 1 - e-"35)0,( C) dS2 d <
If,\dSzd<
aT
1
lzl . lzl] dS2d<
+ a 1 ~ . x l l ~ z i . l l [ ( j l ~ z r ~d d4i v ~ ~
cal
x
I u l2 dS2dc)
I Z I . - ~ )S(~ ~x [ l - Z s . l [
o,,x,,-2,,l,
in
liZ
lgrad TI2dS2 dc)
1
+ ~ l l % l L = ~s'x[l-2i.ll
s ~ ) ~ (1-2cldivuldC')
I$
aT
dS2d5
j
265
The primitive equations of atmosphere
G a,
x
{(I
(Is2x[1-2.,II
{(I'
IlTll
l%-(s2)
dS2dl;)li2
S~x~l-2c.Il
+ a l I ~ I L - ~ . P )s=
/
]
112 112
l~14dS2dt)
1-2s
l d i ~ u l d C ) (I-2e
\ ~ ($(dC)}dS'
1 1 ~ 1 1. lS21"4 ( 2 ~ ) ' "' IuIL'
s a l ITJL-cs2)
111
1
+al l & - ( s z ~ ~
x
(I' 1$I2
SI
{(2&)(11-2r ldivu12dC)
dl;)'"} dS2
1-2e
~al(lSZI'"(2&)1'4+2E)I % I L Y S ~IlTll
~ '
llull
which proves the inequality we need.
Now we choose
E~ E
(0, 1/2) such that E* = (0, T') = (0, TZo)satisfies
1
1
\b(E, E*,=)I= Ib2(u, T', T)( 6Ilull . IITII s - - l13z. (2.58)
2Rmax
2Rmax
And we define another unknown function Y in terms of T as
S=T-T*.
(2.59)
Then we have the following equations describing the distribution of E = (U, s):
au
-+
at
V,u
au f
+ W(u) -+--k
at RO
+
[
?grad
x u +grad Qs
YdC'
1
Rel
a
1
- -Au --Re, a t
pZ,Za~
[
(z
d51)
=fl =fi - f g g r a d '7 dC
(2.60)
t P
:i
al -+VuY+w(u)-
ayl
at
i
+ a , V"T*+W(u)-
--bP W ( U )- - A1Y
P
RI,
1
I
div u dc = 0
rr3ras1
--~1t , aa g PT
= f , = f i +1- ~ ~ *+--I a
Rti
aT*l
at
pib , a T *
Rt2at[(2)
1;
TI
(2.62)
(2.62)
266
J L Liom et aI
I
with the initial and boundary value conditions as follows:
as
at
a s_
c=1
v=o
(=O
u=o
t=O
8 = (U, 9)
= (Uo.
-+b,S=O
ac
-0
s").
(2.63)
We are in a position to state the functional formulation of the problem
(2.60)-(2.63) or (1.50)-(1.54) as follows.
Problem 2.1. For F
that
=
(A.
(l/a&),
E,,= (vu, Yo)E H given, find 8 =(U, Y) such
8 E L*(o, t;v)n ~ " ( 0t;
, H)
Vt>O
(2.64)
d
-(8, 8,)H + ( A = , Z l ) H + ( B ( Z , E), E,),, + (B(Z,Z*), E l ) H
dt
+ ( E @ ) , E J H = (F, E,)+,
-
-so.
(2.65)
VEl E D ( A )
(2.66)
l/,=o=
Obviously, (2.65~)is equivalent to the following operator equation in H :
8,+ AE
+ B ( Z , Z) + B(Z,Z*) + E ( 8 ) = F.
(2.65b)
Moreover it is easy to see that for any solution (U. 9, @J of (2.60)-(2.63) or
(1.50)-(1.54), which is sufficiently smooth, 8 = (U. S)is a solution of problem 2.1.
Conversely, following the procedure of the proof of (2.33) (in lemma 2.1), w e can
obtain that, for any solution = = ( U , 5)of problem 2.1, there is a unique
distribution (up to a constant) @ $ E 9'(S*) such that (U. 9,
as)is a solution of
(2.60)-(2.63) or (1.50)-(1.54), at least in the sense of distributions. Thus, problem
2.1 is indeed the weak formulation of the PES, (2.60-(2.63) or (1.50)-(1.54).
2 4. Regularity of the linear stationary problem
In this subsection we want to prove some regularity result concerning the solutions
of the linear operator equation in VI:
I
A l u = f e L*(TM TS').
(2.67)
By definition, (2.67) is equivalent to the following weak formulation:
find
I
V, such that
U E
al(w VI)
=
U U,)
(2.68)
v u , E VI.
It follows from the Lax-Milgram theorem that there is a unique solution
for (2.68) or (2.67). Then, integrating by parts, we obtain
1
(-ZG
1 a
Au - -Re2dc
*au
[(-)pTo
-1
PT ay
-f, U,) = O
V u , E 'VI.
U E
VI
(2.69)
267
The primitive equations of atmosphere
Similar to the proof of (2.33) (in the proof of lemma 2.1), we can obtain that, as a
distribution.
_- 1
A.---[(-)1
a
PF
Rezag
Rel
-1-f
at
pi;, ’ a u
does not depend on the third variable
that
5.
and there is an Lz function I E L z ( S z )such
-1
[(-)
1 a pi;, z a u
AU - -+grad I =f
Rel
Re,dt
PF a t
_-
1
,
1
bdivudc=O
U E HA(TM
(2.70)
I TSz)
I E L’(S’)
L21dS2 = 0
where I is uniquely determined because of the restriction
constraint we have
I.5.1 dSZ= 0.
Under this
Ill~Clfl.
The following lemma provides the regularity of the solution
(2.70).
Lemma 2.7. Let f E H*(TM 1 TS’) (k > 0) and
(2.70). Then
U E
H*+’(TM
~
k
~
I TS’) n V,
IE
I~ l r l ~ l ~ *~ + ’ ~I~l fcl l* ~ A
where c is a constant independent of
U,
~
(U,
(U.
I ) obtained in
I ) E V, x L z ( S z ) be a solution
ffk+’(SZ)
+
I and f:
1
(2.71)
Proof.
(i) The case k = 0. We proceed by the difference quotient method of Nirenberg
[17]. For some local coordinates ( x , y ) E U c Sz,we consider a ball
I
BR = ( ( x , y ) E U c Sz 1x1’
Let
r)
be a C” function such that
{i
q=
Then
(VU,
V I ) satisfies
1
1
a
- -A ( ~ u ) --
Rel
+ IyI2 < R 2 ) .
Rezag
in BRl4
in B,\B,,.
za(q~)
[ p%
-1a t + grad(r)l)
PT
(
7
)
1
= Izf---(A(~v)
1
Re,
I
[div(r)v) d f =
0
grad
r)
-
.U df.
v Au) +[grad
r)
1
(2.72)
I L Lions er al
268
We now consider the difference quotient D,,,f and the translation
function f defined by
Di,hf(x, Y.
5) =
f(x + h, Y ,
f"h
of a
5 ) -f@,Y , 5 )
h
5 ) =f(x + h, Y , 5).
P h ( x ,Y ,
Then, for any Ihl < R / 8 , let w = D,,,(qu). We want to cmpute al(w, w). To this
end, we observe by (2.72) that
a l ( D l , h ( v ) ,Dl.h(tlu)) =
lM{ [
Dl.h(tlf)
1
+ Di,h( - ~ e (A(11u)
,
- tl Au) + l grad tl
-I,
<c
)]
'
Di.h(V)) dM
grad(Di,h(tll)) ' Di.h(tlu) dM
lDi,-h(Di,h(W'))l ' (If1
f
f
l/MDl,h(tll)div(Di,h(tlu))
s
c If1
s
IuI + lhll + Ill)
dMl
'
IlDi.h(tlu)II + l/MDdlll)Di,h(Wd tl
c If1
'
IlDl,h(W)II + ~ ~ D l , h ( ~ l ) D l . h (V~.rua)d
s c If1
'
IIDi.h(Ou)II + ~/M$Dl,-hDl,h(@ad tl ' u )
'
IlDi.h(tlu)ll f (/Mi{Di,-&i,h[Wd 11 ' (Vu)]
< c If1
'
U
f
tl divu) dMI
dMl
dMi
- (Di,-hDi.htl) ' (grad 11 ' U ) - (Di.htl)i'h' Di.-h(grad
tl ' U )
- (&-h11"*) ' Di,h(grad tl ' U ) } dM1
c If1
s c If1
s
'
IIDi,h(tlU)II +
c 111
. IlDi.h(tlu)II+ c Iflz.
'
(lDl,-hDw(tlu)\
+ llull + IDi.hUI + IDl.-hUo
(2.73)
It follows that
1
-~ ~ ~ ~ , h ( c
t lIf1 ~ ') llDi,h(tlu)ll
~ ~ z s + c Ifl*.
Hence
R.W
c
~ ~ ~ i , h (s1 1I f l~
. )~~
i
Passing to the limit h-+O in the above inequality, we find
?@9
E HI( TM I 7-9)
ax
I
l
m
x
ul
)
HI
s
c If I.
(2.74)
The primifiue equations of atmosphere
269
Similarly, we can prove that (2.74) is still true if we replace a(qv)/ax by
a ( t p ) / a y . So we obtain
c Ifl.
v(Vu)llH's
I
ITS2)
V(qu) E H'(TM
(2.75)
Since Sz is a compact manifold without boundary, the above argument yields
Vu E H'(TM I 73')
lIvuIIH'
ZZ
c If\.
I
(2.76)
Then, by applying the horizontal divergence operator to the first equation of (2.70).
we have
so
A1 E H-'(S2)
I E H'(S')
IllllHl e c Ifl.
I
(2.77)
Then, returning to (2.70), we have f - grad I E L2(TM I TS'). Therefore, by the
usual regularity of elliptic equations, we obtain
UE
H2( TA4 I TSZ)
1
(2.78)
IIuIIHZsC Iff.
The case. k = 0 has been proved.
(ii) The case k > 0. This case can be shown more easily by using the results of
the case k = O . As an example, we only prove the case k = l . To this end,
integrating the first equation in (2.70) in 5 E (0, l), we have
1
A ( l u d t ) +grad I
Rel
=(by (2.78))
=b+k[(m)
zI+,-(m)2IJ
To
I
2du
p,,&,
2 a ~
TF).
E HI"(
(2.79)
Sa the usual regularity result for the 2D Navier-Stokes equations on S2 implies
6'.
It follows that f
have
Then
U E
d c E Hz"n(TSz)
1 E H'+'O(S2).
(2.80)
- grad 1 E HIn(TM I TS'). Thus, by the first equation of (2.70), we
H*+'O(TM
I TS*).
t
av
+&[(m)
zlC='
- (iqz))
zIc=ol
=au
I
bfdt
(2.81)
pu~o
2
E
H'(TS*).
(2.82)
Thus, we infer from (2.79) that I E Hz(S2) and then, by the first equation of (2.70)
again, U E H ~ ( T M n2).
0
I
I L Lions
270
el a1
3. Existence of global weak solutions
The main result in this section is the following existence theorem of global weak
solutions to problem 2.1.
Theorem 3.1. For any time interval [0, a], there is at least one solution I = (U, T)
to problem 2.1 such that
Zt E L*(O, t; H-'( T M ) )
2 E C([O, a]; H,)
I
I
(3.1)
6
1
'
13(f)IL+ - \\I(t)Il' dt G IE& + (F, 8(t)),, dt
a.e. t E [0, a]
Rmss U
where H,, is the space H endowed with the weak topology.
(3.2)
Proof. We proceed by the Faedo-Galerkin procedure. Since this method is now
standard, we only need to present an outline of the proof. Let {q,I i = 1, 2, . . . } be
the eigenfunctions of A , then we look for an approximate solution Z,(t) =
E;"=, gJ,(t)qJ satisfying
d
-("
dt
qJ)Hf (AZm+ B(B,, Z m ) + B(Z,, 2*)+ E(Em), q,)"
I
"
-
=mJ,=U
-
=
V I $,I"
j = 1,.
.. ,m
= Pm=*
I
(3.3)
where P, is the projector from H onto the space spanned by { q l , .. . , q,). We
then have from (3.3) that
Id
~~I""(')l;+-
1
1
ll%(~)lIZ~ll~m(t)llz+ ( E %"H
Rmax
2Rm.,
(3.4)
which implies (3.2) (by passing to the limit) and
8,
E
a bounded set of L"(0, a; H ) n L2(0, t;V ) .
Since
(3.5)
llB(Z, E1)ll,,-'<C llEll ' lZ,l"
we have
(H,),
a bounded set of Lz(O, t;H-'(TM)).
(3.6)
Equations (3.5) and (3.6) show that we can pass to the limit m + m to obtain a
solution Z = (U, F ) satisfying (3.1); (3.2) follows from (3.4) by passing to the
limit. 0
E
Remark 3.1.
(i) A similar result was obtained in theorem 2.1 in chapter 2 of [ I l l . Since in
[ l l ] we did not find the reformulation (1.50)-(1.52) of the PES we had to solve the
original PES (1.41)-(1,44). Therefore, we had to construct some special function
spaces for the vertical velocity w, and use different functional formulation.
Moreover, we were not able to prove the result by the Galerkin method due to the
degeneracy of the equations with respect to the vertical velocity w. So we proved
the result there by the semi-discretization method. Here we reformulate the PES by
The primitive equations of atmosphere
27 1
integrating the vertical momentum and the continuity equations, so that the variable
o disappears. Therefore, we can simply use the Faedo-Galerkin method here to
prove the existence of global weak solutions.
(ii) By directly considering (2.60)-(2.63) or (1.50)-(1.54), using the LeraySchauder principle we can prove the existence and uniqueness of local strong
solutions (in L2(0, t ;H4 n V ) n C([O, t];H’ n V) for t small) to the problem. Since
the method and the result are very similar to those in Ill], we omit them.
Part
n. Primitive equations with vertical viscosity (PEV’s)
4. Mathematical setting of the PE%
4.1. The equations and some physical considerations
In this part, we consider the PEV2s as
aU
-+
at
f
au
V,v + U-+
85
-k
Ro
x
U
a@ bP
E~
-+-T--Ao--X P
Re,
€2 3
Re,35
[(-)pi;,
PT
2aw
-]=o
af
aw
+=0
div U
aT
a,(x+v,~
[(T) 2-3C1a =~A
1
1 a
pi;,
+grad ‘4 - -Au - - ~
Re I
R e 2 a 5 PT
(4.2)
(4.3)
35
ZT) y
1
---a - - A T
+ 0-
(4.1)
Rf,
~ [ ,aa [r3)2”]=h
t PT a t
--1
(4.4)
with the same initial and boundary value conditions as those in part I. In (4.2)
E = ( P -p,,)Ri;,/agP, indicating the ratio between the height of the atmosphere and
a.
Comparing these equations with the PES in section 1, we can see that the only
difference is the underlined terms in (4.2), representing the viscosity in the
p-direction. It is reasonable from the meteorological point of view to have these
viscosity terms for w . Recalling the procedure of deriving the PES in section 1, we
use the quasistatic equilibrium assumption d p / d z = -pg to approximate the vertical
momentum equation. So, in ( l . l O ) , we have omitted the term D,, denoting the
viscosity in the vertical direction. If we keep this term D,, then (1.10) is replaced by
_
a’ -- - p g
az
+ pD,.
Since 0, is smaller than the term - p g , we can still make the coordinate
transformation from (0, 9 , z ; t ) to (0, 9 , p ; t ) . Then the first equation of (1.18)
becomes
1
_
agz ap - p + p&ig
0, 1
= _ -1_ -D,
P Pg
+o( M P
_ -1- -0,
-
P
Pg.
~
J L Lions et ai
272
Keeping in mind that 0,is small, we can omit this term in the second and third
equations of (1.18). Therefore, (1.28) is replaced by
a@
-+
R
1
-T f-
D, = 0
aP P
Pg
and (1.26) and (1.27), (1.29) and (1.32) remain unchanged. Finally, we can add, in
the non-dimensional forms of the PE, those viscosity terms underlined in (4.2).
It is worth pointing out that for large-scale oceanic dynamics, owing to the
incompressibility of sea water, the PES are written in the usual (e, 9,r ; t ) coordinate
system rather than transforming the equations into the p-coordinate system.
Therefore, we can consider more easily the PEV2s large-scale oceanic circulation, and
we will report this in a forthcoming paper.
4.2. Reformulation of the PEV's
I
Similar to section 1.4, integrating (4.2) and (4.3), and taking the boundary value
conditions for w into account, we have
1)
I
div U d 5 = 0
1'
+ it'F +
i:
w = W ( u )=
div U dg
@ = Q5
T dc
(4.5)
L W ( u )d 5
(4.6)
where Qs is a function on S2, and L is a linear differential operator given by
&2
&*
a
L W ( u ) = - -A W ( u )- Rezag
Re,
' m q
[ pT,
PT
ag
(4.7)
(
7
)
'
Recalling the non-homogeneous boundary value conditions for T, we define
T=T-T*
where T' = T : given by (2.56) with
reformulation of the PEV2s as
aU
such that (2.58) holds. We then obtain the
du + f
+ V,,u + W ( u )ag RO k x U + grad @, + grad
-
at
E
+ IIhPgrad
5 P
a,(%
+V,Y+ W ( u ) -
(
- bP
_ W ( U ) - -1A Y - - -
1
L W ( u )d 5
1
1
T d 5 - -Au - Re2df
Rel
aT*
+ a l V , , T * +W ( U ) - )
a
acy )
P
IC'
a
Rti
2 a
[(-)pTo
PT -ag
1
~
=fl
(44
X
1 a pT,, 2 a y
Rt, ag [ PT)
=f2
5
1
(4.9)
I
divudg=O
(4.10)
The primitive equations of atmosphere
273
with the same initial and boundary value conditions (2.63) or (1.53) and (1.54) as
those in the PES in part I, since we have not changed the maximum order of the
C-derivatives.
4.3. Functional setting of the PEV's
Now we begin with the mathematical formulation of the PEV2s. To this end, recall
that in (4.8) there is one term corresponding to the viscosity of w as follows:
grad l L W ( u ) dc.
If we take the inner product in Lz between this term and
IC .
=-lM[(i
U,
then
1
JM
[(grad
LW(u) de)
U]
dM
I
LW(u)dt)divu]dM
I
= kLW(u)(\
t
div U dC) dM
= k L W ( u ) . W ( v )dM
E2
-grad
W(u) ' grad W ( v )
=L[Re,
E'
pT, 2 a w ( u ) a w ( u )
_
_
~
Re, PT
a6
a[ dM.
+-(+
]
Thus, it is natural that the space for
regularity:
U
(4.11)
in this case should have the following
W ( v )E H ' ( M ) .
Therefore, we can define the function spaces for U as follows.
First, for the 3 0 Navier-Stokes equations, we can define a function space
I
I
V : = { ( u , w)cHA(TM)ldivu+-=O
aw
ac
which is the completion in H' norm of
pY= {(U, w ) E G ( T M ) 1 div u +-=
0
aco
ag:
Then, it is easy to see that
VY = [ U
V;=(
E
U E
I,
=I,
G(TM I TS') I W(u) =
HA(TM I TS2)I W(u)
1
= {U E H&(TM TS')
(4.12)
,
I
div U dC E CF(M)}
1
div U d6 E H A ( M ) }
I W ( u )E H'(M),
I
1
0
I
div U dC = 0 .
1
(4.13)
(4.14)
274
J L Lions et (11
Here V ; is equipped with the following inner product and norm:
(u2
Ul)"
llUllr
= ((U. U,)) + ( W ( u ) , W ( u , ) ) ,
vu, U,E
=(U. U):"
v;.
We let
1
(4.15)
V" = V ; X H ' ( M ) = V ; X V,
(4.16)
and we use the same notation (., .)" and I l . I I U as for V ; to denote the norm and
inner product in V".
As in section 2 we define two bilinear functionals a" :V" x V"+ R, a ; : V ; x
V;+ R, and their corresponding linear operators A'":V"+(V")', A;: V ; + (V;)'
hY
a"(=, E,) = ( A T : ,E,)H
+ (AzT, Ti)"
=a;(u, U I ) + 4 T , TI)
= (AYu, U I )
a;(u,
UI)
= (ATV, U,)
= al(u, vi)
+
1
[?grad
Rel
E'
pT, , a w ( u )
~
.
Re, PT
ag
(4.17)
W ( u ) .grad W ( u , )
M
+-(-)
aw(uI)
_
_
ac
Id,
(4.18)
for any E = (U, T ) , 3, = ( u l , TI)E V". Then, as in lemma 2.3, we have the following
lemma.
Lemma 4.1.
(i) a" and a; are coercive and continuous; therefore, A " : V"+ (V")' and
A;: V;+ (V;)' are isomorphisms. Moreover, we have
a"(%
U
21) ~
-
1
-
Rmi,
~
~11Elll,"
2 ~
~
"
~
1
(=,=)a- ll2lli
U -
R,,
1
(4.19)
a ; ( q V I ) < -II~II". llulll"
Rmi,
1
a;(u, U ) 3 -Ilulli.
Rm,
J
(ii) The isomorphism A " : V"+ (V")' (or A;: V;+ (V;)') can be extended as a
self-adjoint unbounded linear operator on H (or HI), with compact inverse
(A")-':H-*H (or ( A ; ) - ' : H l + H l ) , and with domain D(A") = V n (D(A;) x
H Z ( M ) )(or D(A;) = V ; n { U E Hz(TM I 2
's) I W ( u )E H Z ( M ) ) . )
Proof. We only have to prove that
D ( A ; ) = V ; I ~ { U E H ' ( T MTS')I
I
W(U)EH'(M)).
The primitiue equalions of atmosphere
275
To this end, we consider
I
A;u = / E Lz(TM TS')
then following the same procedure as in section 2.4 we obtain the existence of a
function a$E H - ' ( S Z ) such that
1
1 a
Au--Rel
Re,af
--
1
[(-)pz,
-]igradO,+grad
PT af
2 a ~
I
i:
LW(u)df=f
div U d f = 0.
We define
+
@ = as L
We can then obtain that
I:
W ( u )df.
1
I a
-_
Au - -Re,
Re,af
z-1a +grad
~
[(-)pT;
3
' =f
PT af
_
a@- _
1 A W ( u ) - -R-1e[,(a-af)
a f Re,
PZ12%]
af
PT
=O
divu+-- a w ( u ) - 0
ac
which is essentially the same problem as the 3~ Stokes problem. By the method of
the Cattabriga regularity theorem (see [14, 181) the result follows. 0
We now state the functional formulation of the PEV's as follows.
Problem 4.1. For F = (fl,&/al), E, = (U",
=
d
dt
-(3,
=,)H
E
s,)E H given, find E = ( U . F) such that
L*(o, T ; v")n ~ " ( 0 t;
, H)
+(AT:,
%1)H
+ ( E ( = , =), E J H
+ ( E ( = ) , S I ) H= (F, q H
-"I,=" = a-,,.
VT>O
(4.20)
+ ( E ( = , E*), El)"
VI,
E
D(A'")
(4.21)
(4.22)
Remark 4. I.
( i ) Problem 4.1 is indeed a weak formulation to the initial boundary problem of
the PEV's. That is, for any solution ( U , .T, QS) of the PEV's which is sufficiently
smooth, Z = ( U . F) is a solution of Problem 4.1. Conversely, following the
procedure of the proof of (2.33) (in lemma 2.1), we can prove that for any solution
= = ( U , Y) of problem 4.1, there is a unique distribution (up to a constant)
asE '3'(S2) such that (U, F, @J satisfies the PEV's, at least in the sense of
distribution.
( i i ) Equation (4.21) is just the same as (2.65) in problem 2.1 with the operator A
replaced by A'". Formally, both possess the same nonlinear terms E @ , E). But, in
(4.21). the principal term A'"= provides some a priori estimates for W ( v ) =
J L Lions er a1
276
I;div U d t , the absence of which causes some serious difficulty for solving problem
2.1. In comparison to the 3 0 Navier-Stokes equations, we do not have the time
derivative term of W ( u ) here and, therefore, we cannot obtain some a priori
estimates from d W ( u ) / d t as in the case of 3~ Navier-Stokes equations. Due to the
absence of this term, for instance, we cannot use the Sobolev-Lieb-Thirring
inequality when estimating the dimensions of the attractors of the equations (see
section 6).
Before ending this subsection, we would like to write explicitly some estimates of
the nonlinear term E ( = , 8 , ) in terms of I l . I I W and A'".
Lemma 4.2.
llw" . 1 1 ~ l 1 1' IW ' l182111n
Ib(8, SI,&)I
G
C
. llE1ll'" IAEIJiii IZ2IH
llw" . I%l ' ll%IlH'..
~~~~~w
(4.23)
Proof. The proof of (4.23) is the same as that for the corresponding result for
Navier-Stokes equations. 0
30
Remark 4.2. We would also like to mention that in the case of 3 0 Navier-Stokes
equations we can obtain some better estimates for the nonlinear terms in terms of
the norms 11.11 and 1.1, which cannot be proved in our case here. The reason is that
the norm I.IH in the space H does not provide any information about the Lz norm of
W ( u ) , owing to the absence of the term d W ( u ) / d tin the equations.
5. Existence of solutions and their properties
5.1. Existence ofsolutionr
As for theorem 3.1, we can prove the following existence result of solutions for
problem 4.1.
Theorem 5.1. For any time interval [0, a] there is at least one solution E = (U. T)to
problem 4.1 such that
2, E ~ ( 0T ;,H P ( T M ) ) I
E E C([0, a ] ;H")
(5.1)
where H , is the space H endowed with weak topology.
Moreover, we have the following theorem
Theorem 5.2. If the initial value 8, E V" then there is a time al = rI(llEollw)
> 0,
such that there is a unique solution Z = (U, T)to problem 4.1 satisfying
I
8, E L2(0, a l ; H )
3 E L2(0, a,; D(A'"))flC([O, zJ;V").
(5.3)
The primitive equations of atmosphere
211
Here t, = T , ( ~ ~ can
Z ~be~ given
~ ~ ) by
I
Proof of theorem 5.1. We proceed by the Faedo-Galerkin procedure. Let { V i i =
1, 2, . . .) be eigenfunctions of A". Let P, be the orthogonal projector of H onto the
space spanned by {$,, . . . , qm;. Theii, ii is eajy io find
appimiiiiate sG!i;:iGn
E,,,([)= E$l gj,(t)vj satisfying
d
-("
df
-"
Vj)H
+ ( A T , + B(Z,, E,) + B(B,,
-
=
a,,,
P,aw
j = 1,.
= ( F , VI)"
,
5')
+ E(=,),
$,)H
. ,m
(5.5)
(5.6)
Then, by (5.5) and (5.6), we have
which implies that
E, E a bounded set of L"(0, t;H) f?L2(0,5 ; V w ) .
(5.8)
On the other hand, by (4.23), we have
IIB(%!
%J!!H--~
C
!lZ"lw
IZ~IH
which yields
(Z,),
E
a bounded set of L2(0, t ; H-"*(TM)).
The combination of (5.7)-(5.9)
proves theorem 5.1.
(5.9)
0
Proof oftheorem 5.2. From (5.5) we have
-- (AwZ,,,,Em)H + IA"S,(f)lL= ( F - B(Z,,,, 3,)
Id
2 dt
- B(B,,
E * ) - E@,), A T , ) ,
s IA"E,IH. [IFIH+ lB(%, %)IH
+ IB(%, B*)IH+ IE(S,)IH]
G [by (4.23))
G+IA'"Z,(&+C((FI2+llDmllt+ //Z,lIt).
It follows that
d
-(A"=,,
dt
Let
E,),+IA"E,(1)12,sC(1+(AWZ,,
B,,,),)3.
y ( t ) = 1+ ( A T , , E,)H
then
Y (0)
for 0 s t <
1
2Y2(0)C
(5.10)
278
I L Lions el a1
So we can choose
(5.11)
such that whenever 0 s f < t,,
(AT,,,, E,,&,SZ(l+
( A T : , , 2")")
(5.12)
which proves that the solution S = (U, 5 )satisfies
E € L"(0. t; V").
(5.13)
By (5.10) and (5.13) we have
2 E L2(0, t,; D(A")).
(5.14)
and then, by (4.21). we obtain
2, E LZ(0, s,; H ) .
(5.15)
Remark 5.1. Here we have only shown the existence of local strong solutions. In a
forthcoming paper we intend to prove the existence of global strong solutions to the
problem using the smallness of the vertical scale in comparison to the horizontal
scale of the global atmosphere.
5.2. Time-analylicify of the solufions
The main objective in this subsection is to prove that the strong solutions of problem
4.1 given by theorem 5.2 are analytic in time D(A")-valued functons. To this end,
be the complex spaces of H , D(A") and V". We use the
let HC,D(A'")' and
same notation as for A", E , E, etc., to denote the naturally extended complex
operators on the corresponding complex spaces. Moreover, the complex equation of
(4.21) become (here z is the complex time variable)
d
dz
- (E(z), E,)*
+ (A"E(z) + B ( % ( z ) ,E(z)) + E ( ~ ( z )E*),
, E,)"c
+ ( E ( E ( z ) ) ,E,)"" = (F, El)@,
VE,
E
D(A")~.
(5.16)
Then, our main result in this subsection is as follows.
Theorem 5.3. Let Eo€V". Then there exists a
to=
to(llSoll~))given by
c
(5.17)
such that the complex problem (5.16) and (4.22) possesses a unique solution
2 = (U, F),which is analytic from A(llZollw)into D(A")", and which coincides with
the unique solution of problem 4.1 when restricted in the positive real interval
IO, al(llZollw)l. Here A(llEollw) is given by
The primitive equations of atmosphere
279
Proof. We proceed by applying the Galerkin method, following the ideas of Foias
and Temam (see [19,20]). Since the procedure is now standard, we only have to
present some formal a priori estimates, which will become rigorous if the procedure
works out in detail.
First, we take E, in (5.16) to be A T : , and we multiply the resulting equation by
e" (when z = reie), and then we consider the real part. This yields (101 < n/2)
I d
(A"'E(re'B), E(re")),, t cos 8 IA"E(rei8)1L
2 dr
= -Re[eje(B(Z(r),:E(z)) + B ( E ( z ) , E*) t ,?(E@)) - F, A'"Z(z)),,]
C
S $ cos 0 IA"E(re")IL +(1 + llE(reie)ll~)3.
cos e
--
It follows that
d
(A"E(reis), E(reiB)),, + cos 8 IA'"E(re")I$
dr
C
s[l t (AWZ(reiB),E(rei"))H]3.
cos 8
-
On integrating (5.19) we have
(A"E(reis), Z(reis)),,S C ( l
+ eo^^^)
(5.19)
(5.20)
as long as
(5.21)
Following the method developed by Foias and Temam [19] we conclude that the
solution E of (5.16) and (4.22) is analytic from a region in C,which comprises
A(llZollw) given by (5.18), into D(A")". The other parts of the proof follow the
standard procedure we mentioned before. 0
Part 111. Attractors and their dimensions
6. Existence and dimensions of attractors for the PEV's
The main purpose of this section is to estimate the Hausdorff and fractal dimensions
of the attractors and/or functional invariant sets for the problem in terms of some
physically relevant parameters, using the CFI theory developed by Constantin, Foias
and Temam (121. Here we follow Temam [4].
6.1. Absorbing sets and attractors
As we have mentioned before, it is not known in general that there exist global
strong solutions of problem 4.1, even though we have proved the local existence of
280
I L Lions a al
strong solutions in theorem 5.2. Therefore, in this article we restrict ourselves to the
study of attractors and functional invariant sets associated with global strong
solutions for the PEV2s, when such solutions exist.
According to theorem 5.2, we can define the solution operator S ( f ) (t > 0 given),
whenever it makes sense, that maps the initial value S,, to the solution E(t) of
problem 4.1. For any f > 0, S ( f ) is an operator from V" into V". Since, generally
speaking, S ( f )( t > 0) is only defined and continuous on some part of V" into V", we
define
W ( t ) )=
I
U qm0)
P>O
(6.1)
D P ( S ( f ) ) = { s o EV" I iiS(t)soliwSp. O ~ ~ e t t ) .
Obviously, we have the following semigroup properties of S ( I ) :
S(0) = I
D ( S ( 0 ) )= V"
(6.2)
S(t + s) = S ( f ) S ( s )on D(S(t + s))
vt,s 0.
Moreover, for the sake of convenience, we quote the definitions of functional
invariant sets and attractors from [4] as follows.
Definition 6.1.
(i) A set X
1
I
c V I Is a functionai invariant set for the semigroup ( S ( t ) } , If
S(f)Z"
exists E,
E,
X
Vf>O
S ( f ) X= X
vt 2 0.
(ii) An attractor set is a set SP c V" such that:
(a) SP is an invariant set;
(b) .d is the w-limit set of one of its open neighbourhoods
(6.3)
(I.?
As we mentioned at the beginning of this section, the main objective in this
section is to estimate Hausdorff and fractal dimensions of the functional invariant
sets and attractors for the PEV2s. To this end, we assume the existence of a particular
solution Z = (U, T) of problem 4.1 satisfying the following boundedness property:
(6.4)
S U ~
iiq<m,
A
We have then the following theorem
Theorem 6.1.$ Let E=@, T) be a solution of problem 4.1 with initial value
Zo = (q,,Tn),
satisfying (6.4). Then there exists a functional invariant set X =
X ( Z o ) for the semigroup S ( f ) , which is bounded in D(A"), and E(f) converges to X
as r + @ .
tThat is, .d =nZoU,,,S(r)U, the closure being taken in V " .
$If it is true that
S(r)& exists VEo E V"
Vr 3 0
(6.4n)
then we have the following theorem.
Theorem 6.1'. Assuming (6.Q), then the semigroup ( S ( t ) ) , = " possesses an absorbing
maximal attractor c D(A'"), which is bounded in D(Aw).
set
in V" and a
281
The primitive equations of atmosphere
This is the direct application of theorem 1.2 of [4, p 3821.
In order to estimate the dimensions of the attractor &4 given by theorem 6.1, we
should study the linearized problem of problem 4.1. Thus, consider a solution
Z = (U, F ) of problem 4.1 satisfying
E E L"(0, m; V").
(6.5)
The linearized equation of (4.21) is
U ' + A " U + B ( S , U ) + B ( U ,E ) + B ( U , % * ) + E ( U ) = O
U(0) = E E H .
(6.6)
We quote a direct application of property 2.1 of [4, p 3871 without proof as follows.
.Theorem 6.2.
(i) There exists a unique solution U of (6.6) satisfying
vr > 0.
U E ~ ' ( 0r.; v")n ~ " ( 0t;, H)
(6.7)
(ii) For any t , > 0, p > 0, D,(S(t)) is open in V", and S ( t , ) is differentiable in
D ( S ( t , ) ) equipped with the norm of H. The differential of S ( t , ) at a point Eo of
D,,(S(t,)) is the mapping
E E H + L ( ~ , ; & ) E =U ( t l )
(6.8)
where U is the solution of (6.6).
6.2. Dimemiom of the attractors
We are now in a position to estimate the Hausdorff and fractal dimensions of the
functional invariant set X given by theorem 6.1, or the attractor d given by theorem
6.1', of the semigroup S ( t ) , or problem 4.1. Before we proceed, we would like to
mention that, in the case where ( 6 . h ) holds, we can obtain the exponential decay of
the volume elements (see the footnote to theorem 6.3). For the general case,
however, we are not able to establish the exponential decay of the volume elements.
We follow the notation and procedures of [4]. Let U,, U', . . . , U, be m
solutions of (6.6) with initial values Cl, E2,. . . , Em, respectively. Then, we have
IU,(f)h... ~ L l , ( t ) l ~ ~ ~ =. I. .EhE,lnmnexp
~h
rlTrF'(S(r)g,).Q,(r)dr
JO
(6.9)
where Q,(r) is the orthogonal projection in H onto
Q,(r)H=
Span{U,(r),
.. . , U,(r)}.
Choosing {tfji(r)}G1to be an orthonormal basis of Q,(r)H,
m
Tr[F'(E(r))oQ,(r)l
=
j-1
we have
V ' F ( r ) ) V j ( r ) ,tfjj(4h
(6.10)
where F ' ( 3 ( r ) )is defined by
F'(S(T))U=-A"U-B(S(r), U)-B,(LI, E(r))-B(U, E * ) - E ( U ) .
(6.11)
I L Lions et al
282
Moreover, we introduce
(6.12)
a,-
(6.13)
= lim sup q,=(t).
C"
Then, w e quote the main result of [12] from [4], which will be used in the remaining
part of this section.
Theorem 6.3.1.If
qj s - aj"
+g
Vial
(6.14)
then the Hausdorff dimension of X and the fractal dimension of X satisfy
d H ( X )< m
&(X)s 2m
(6.15)
(6.16)
In order to apply this theory to estimate dimensions of X,we should estimate the
right-hand side of (6.10). To this end, for simplicity, we omit temporarily r, then
Tr(F'(S)oQ,)
=
-2(A"qj,qj)H
j=1
Before we check each term on the right-hand side of (6.17), we define a number,
which is related to the energy dissipation, as follows:
Y = Lax
SUP
11~112.
Sex
(6.18)
Now, for simplicity, let
qj = ( U j ,
t In the case where (6.Q)
q,(1)
q
holds, we can obtain further that, if for some m and I o > 0.
-6
<a
VI 3I,
then the volume element lUl(t) A
E,. . . . . L E H ,
lU,(f)A
E = (U, 9)
... A
. . . A U,,,(f)lAmH
decays exponentially as I-m,
U,(r)l,,sCexp(-Gt).
uniformly for q e d ,
The primitive equations of atmosphere
283
(6.20)
s (by lemma 6.1 below)
,-
(6.22)
(6.23)
Integrating (5.2), we can easily obtain that
K S
CR;, IF[;
(6.24)
s (by definition of F )
<-- C mSJ3+ CR;,[YR;,
IFI;
+ (1 + c?~)']~~.
Finally, by theorem 6.3, we have proved the following theorem.
(6.25)
284
I L Lions ef al
Theorem 6.4.t We consider the dynamical system, problem 4.1 in H, and define m
by
~-~<cR,,.[YR~,,IFI:,+(~+~;)']"~~
(6.26~)
01
(6.266)
where C is an absolute constant independent of the physically relevant parameters.
Then
d H ( X )c m
d F ( X ) 2m.
(6.27)
Remark 6.1.
(i) In (6.26a) or (6.266), R,,., Rmi. are given by (2.35), which represent the
Reynolds numbers.
(ii) y is given by (6.18). We do not know how to estimate y explicitly in this
paper. Basically, estimates of y will depend on the external forcing cfi.f2), the
Reynolds numbers Rmax,Rmi. and the Rossby number Ro (see (1.40) for the
definition of Ro). The Rossby number represents a measure of the significant
influence of the rotation of the earth on the dynamics (long-term behaviour and
climate changes) of the atmosphere.
(iii) Since, in the equations, there is no time derivative term aw(v)/ar, we are
not able to obtain any estimate for Iw(v)lLz from the norm of H. Therefore, in
(6.19), we cannot use the Sobolev-Lieb-Thirring-type inequality to improve the
estimates of the dimensions of the attractors.
Lemma 6.1. We have
(6.28)
Proof. Applying the generalized Sobolev-Lieb-Thirrir
[4, p 466]), we obtain
in pality (theorem 4.1 of
(6.29)
On the other hand,
f
Equation (6.28) then follows. 0
t In the case where (6.Q) holds, we also have that the volume element lU,(l) A . .
exponentially as 1-m in the phase space.
. A Um(i)lAm,,
decays
The primitive equations of onnosphere
285
7. Global anmetors for the PEV*s
In this section we consider the PES and the PEVZsby adding to the equations some
enhanced viscosity terms. First, corresponding to problem 2.1 (the weak formulation
of PES), we study briefly the following Cauchy problem obtained from (2,656) with
the enhanced viscosity term E K E :
+
- + &Am=
Zo H
S,
AE
=
+ B ( E , Z) + B ( Z , Z * ) + E ( E ) = F
E
(7.1)
m > 1, & > 0 .
As &+O, (7.1) reduces to problem 2.1. The main result in this section is the
following theorem.
Theorem 7.1. Assume m 3 4, then:
(i) There is a unique global solution Z of (7.1) satisfying
sE~
~ (T 0;v, n H"(TM)) n
c([o,t]; H )
Wr>O.
(7.2)
Moreover, for any I 0, the mapping S m ( t ): H - + H, defined by S"(t)S, = Z(t), the
solution of (7.1) with initial value Zo, is continuous.
(ii) If &,E D(Am")c V n H m ( T M ) , then the unique solution of (7.1) obtained
in part (i) satisfies
sE~
~ (t;
0 v, n H ~ ( T M )n) c([o,r ] ;v
nH ~ T M ) )
wr>0.
(7.3)
(iii) There is a ball B,, with radius po in H such that for any ball E , c H with
radius R there is a time to = t,(R) > 0 satisfying
S"(f)BR c
wt 3 to.
(7.4)
B,, is called an absorbing ball in H. Similarly, there is an absorbing ball B,, in
D(A"/2). Moreover, the o-limit set of B,,,
is a compact, connected global attractor of the semigroup S"'(.)(t
(iv) si possesses finite Hausdorff and fractal dimensions.
3 0)
in H
Sketch ofthe proof. First, we claim that if m satisfies (7.2) then
E ( Z , S ) E L2(0, r ; H)
wr > 0.
Indeed, by (2.40), we have
lB(E, =)IH
S
C llSIl~*11S:ll~x
s c IlSllL?
-
(4m-7)""
13IH
which implies that
[ p ( ~E, ) ~ $ sC 11ZIlLP IZ:lg"-7)/2mdt
< c [ ( l ~ ~ l l & + IEI~$4m-7)'(2m-7)
)dr<m.
Thus, (7.6) follows.
(7.6)
286
J L Lions er a1
The theorem can then be proved by the same procedure as that used for the
similar results of the ZD Navier-Stokes equations (see [4]), and we omit the
details. 0
Remark 7.1. Consider the following Cauchy problem obtained from problem 4.1
(for the PEV's) by adding the same enhanced viscosity term E A T to (4.21):
- + &A'"= +H
Z,
A'%
=
E
+ E(%, 3 ) + B ( Z , E*) + E ( Z ) = F
m>l
&>O.
(7.7)
Then theorem 7.1 is true for this problem. Moreover, problem 7.7 will reduce to
problem 4.1 as E-O.
Acknowledgment
This work was partially supported by the National Science Foundation under Grants
NSF-DMS-8802596, NSF-DMS-9024769 and by the Research Fund of Indiana
University.
Appendix. Principal notation
Functions
~ = Z C O S ~
v,
U
w = dpldt
T
@=gr
P
P
T =T(p)
r,
E
Coriolis parameter
the 3~ velocity
horizontal velocity
vertical velocity in p-coordinate system
temperature
geopotential
pressure
density
average vertical distribution of the temperature
temperature on the surface of the earth
diabatic heating
Constants and parameters
a
R
CP
cu = in - R
radius of the earth
gas constants for dry air
specific heat of dry air at constant pressure
acceleration due to gravity
standard wind velocity
P>O
standard atmosphere pressure
T,
reference value of the temperature of T
Q>O
angular velocity of the earth
pi, vj(i= 1, 2) viscosity coefficients
g
u>o
The primitive equations of atmosphere
287
Others
k
S2
3:
A
grad
grad3
vertical unit vector
ZD unit sphere
ZD sphere of radius U
Laplacian on S2 or
gradient on S2 or S:
gradient in the 3~ atmosphere
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