This document discusses rotating Boussinesq equations, which model the motion of a viscous incompressible stratified fluid in a rotating system. It derives the rotating Boussinesq equations, considers them with and without Reynolds stress terms, and discusses existence and uniqueness of solutions. It focuses on Faedo-Galerkin approximations and LaSalle invariance principle to assess stability and attraction of solutions.

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On the qualitative theory of the rotating Boussinesq equations for
the atmosphere and ocean
Maleafisha Stephen Joseph Tladi ∗
¤Corresponding author address:
Maleafisha Stephen Joseph Tladi
Department of Mathematics and Applied Mathematics
Seminar for Dynamical Systems
University of Limpopo, South Africa
E-mail: Stephen.Tladi@ul.ac.za
1

2. ABSTRACT
An alternative title for this paper would perhaps be ”Well-posedness of the problem of
beta-plane ageostrophic flows for meteorology and oceanography.” The author studies the
rotating Boussinesq equations describing the motion of a viscous incompressible stratified
fluid in a rotating system which is relevant, e.g., for Lagrangian coherent structures. These
equations consist of the Navier-Stokes equations with buoyancy-term and Coriolis-term in
beta-plane approximation, the divergence-constraint, and a diffusion-type equation for the
density variation. They are considered in a plane layer with periodic boundary conditions
in the horizontal directions and stress-free conditions at the bottom and the top of the
layer. Additionally, the author considers this model with Reynolds stress, which adds hyper-diffusivity
terms of order 6 to the equations. This manuscript focuses primarily on deriving
the rotating Boussinesq equations for geophysical fluids, showing existence and uniqueness of
solutions, and outlining how Lyapunov functions can be used to assess stability. The main
emphasis of the paper is on Faedo-Galerkin approximations as well as LaSalle invariance
principle for asymptotic stability and attraction.
1. Models for rotating Boussinesq equations
a. Description of rotating Boussinesq equations
1) Rotating Boussinesq equations
The objective of this investigation is to develop prototype geophysical fluid dynamics models
as the first effort towards understanding the impact of rotating Boussinesq equations in the
ocean. First, we review the fundamental assumptions and techniques involved in the deriva-tion
of a system of partial differential equations governing geophysical fluid flows (5; 27; 26).
The evolution equations for the quantities of interest, which may be scalar or vector or ten-sor
valued are derived with the assistance of Reynolds’ transport theorem, conservation laws
and constitutive assumptions (4; 5; 19). The basic fields in the description of the motion
and states of geophysical phenomena are the velocity, the pressure, the density, the temper-ature
and the salinity. The equations governing these fields consists of the mass balance or
continuity equation, the momentum equation, the energy equation and the equation for salt
budget.
Consider a geophysical fluid occupying a region B ∈ IR3 as depicted in figure 1 for the
benchmark Lagrangian coherent structures utilizing the DsTool package program of Worfolk
et al (53).
The Boussinesq approximation gives the following system of primitive partial differential
1

3. equations
ρ0( @u
@t + u. ▽u + f × u) + gρk = − ▽p + μ△u,
▽.u = 0,
@½
@t + u.▽ρ = κ△ρ.
(1)
These equations govern the flow of a viscous incompressible stratified fluid with the Coriolis
force. Some preliminary remarks are offered on other geophysical fluid dynamics models that
are derived from the equations (1) where use is made of a perturbation analysis of the flow
fields at the Rossby number on the order of unity or less. In particular, at the zero-order we
obtain the geostrophic equations and in this limit the rotating Boussinesq equations reduces
to a balance between the Coriolis force and the pressure gradient. Subsequently, first-order
terms in the Rossby number yield the quasigeostrophic equations that govern the evolution
of geostrophic pressure.
Next, concerning the the boundary conditions we assume the fields to be periodic along the
horizontal coordinates x and y with periods L1 and L2, respectively. The flow domain is
assigned on an open rectangular periodic region B = {(x, y, z) ∈ (0,L1) × (0,L1) × (0, h)}
with boundary ¡ = {z = 0, h}. The domain B represents the flow region and the surfaces
z = 0 and z = h represent the lower and upper boundaries of the ocean, respectively. To
system (1) we prescribe boundary conditions
@u
@z = @v
@z = w = 0 at ¡,
ρ(z = 0) = ρl
ρ(z = h) = ρu
(2)
where ρl and ρu are constants. In order to eliminate rigid motions we consider the case
ZB
udx = ZB
vdx = 0.
We introduce the following conducting solutions
ρ = ¯ρ − ρl +
z
2
(ρu − ρl)
p = ¯p + zρlg −
g
4
z2(ρu − ρl)
for density and pressure, respectively. Substitution in (1) lead to the system of partial
2

4. differential equations
@u
@t + u. ▽u + f × u + g
½0
ρk = − 1
½0
▽p + ν △u,
▽.u = 0,
@½
@t + u. ▽ρ + (ρu − ρl)u.k = κ△ρ.
(3)
2) Rotating Boussinesq equations with Reynolds stress
We embark on the the following additive decomposition of flow fields for the rotating Boussi-nesq
equations (1):
u = < u > +´u,
p = < p > +´p,
ρ = < ρ > +´ρ.
(4)
Here the time-averaging operator < . > is defined by
< u >=
1
τ Z¿
u(x)χ(x)dx
with τ being a characteristic evolution time-scale and χ(x) a probability density function.
The terms in the splitting (4) represent coherent and incoherent flow fields, respectively.
Furthermore, we assume that the incoherent velocity field satisfies
< ´u >= 0.
The formal additive decomposition of the flow fields of the rotating Boussinesq equations
into coherent and incoherent terms is justified partly by the realization that solutions to
the primitive equations of geophysical fluid dynamics contain both fast and time scales with
frequencies proportional to the Rossby number.
Employing the decomposition (4) in the rotating Boussinesq equations (1) and invoking the
time-averaging operator, the system
ρ0( @u
@t + u. ▽u + f × u) + gρk = − ▽p + μ△u +▽.A
▽.u = 0
@½
@t + u.▽ρ = κ△ρ +▽.W.
(5)
3

5. for coherent flow fields in which A and W are Reynolds stress (26) fields. The presence of the
effective dissipative terms ▽.A and ▽.W implies that although the incoherent flow fields have
zero average, the momentum and density fluxes of the fluctions, which are quadratic in the
incoherent fluctions, need not vanish when the time-averaging operator is employed. As far
as the contributions of the incoherent flow fields are concerned, we observe that a prototypical
fluid parcel with fluctuation velocity ´u in the x-direction will transport z-momentum ρ0 ´ w
with a nonzero flux across the surface given by the quadratic relation −ρ0 < ´u ´ w >= Axz.
It is similarly convenient to introduce the following quadratic protocols for other Reynolds
stress fields
Axx = −ρ0 < ´u´u >,
Ayy = −ρ0 < ´v´v >,
Azz = −ρ0 < ´ w ´ w >,
Ayx = −ρ0 < ´u´v >= Axy,
Azx = −ρ0 < ´u ´ w >= Axz,
Azy = −ρ0 < ´v ´ w >= Ayz,
Wx = −ρ0 < ´u´ρ >,
Wy = −ρ0 < ´v ´ρ >,
Wz = −ρ0 < ´ w´ρ > .
(6)
It is useful from the oceanographic and meteorological perspectives to idealize the Reynolds
stress fields as a representation of wind stress quantities. To be more specific, we consider the
wind stress comprising primarily the trades that is communicated to ocean water through
absorption of a shear stress and is responsible for driving oceanic circulations in the form
of jets and eddying currents that impact the earth’s weather (27; 26). It is important to
realize that the additive decomposition of the ageostrophic flow quantities into coherent and
incoherent terms and consideration of the averaging operator to obtain the Reynolds stress
fields or wind stress fields have resulted in a set of evolution equations that are not closed.
4

6. We adopt the following Pedlosky closure protocols (26) for the Reynolds stress fields (6):
Axx = −2ερ0 △5 @u
@x ,
Ayy = −2ερ0 △5 @v
@y ,
Azz = −2ερ0 △5 @w
@z ,
Ayx = −ερ0 △5 ( @u
@x + @v
@y ) = Axy,
Azx = −ερ0 △5 ( @w
@z + @u
@x ) = Axz,
@z + @v
@y ) = Ayz,
Azy = −ερ0 △5 ( @w
(7)
and
Wx = −δ( @2u
@x2 +△5 @½
@x ),
Wy = −δ( @2v
@y2 +△5 @½
@y ),
Wz = −δ( @2w
@z2 +△5 @½
@z ),
(8)
where ε and δ are parameters. The geophysical motivation of the choice of the closure
protocols (7 − 8) will become clear in the sequel when we give a mathematical treatment
of the resulting evolution equations. As a validation from a dynamical systems approach,
the closure protocols lead to well-posedness in the sense of Hadamard and the existence
of attractors for the solution of the system of partial differential equations. The closure
protocols (7−8) are the simplest which allows existence of stability and attraction. Another
validation desirable from the craft (7; 13) of stability and attraction follows from the fact that
a siginificant quantity in analyzing whether the solution of an evolution equation is bounded
in some suitable norm is that of Lyapunov functional of the system with interpretations such
as fictitious energy, enstrophy and entropy production which are decreasing along solutions.
The equations governing the flow of ageostrophic equations with Reynolds stress (7 − 8)
induce damping mechanisms in the nature of viscosity, diffusion, stratification and rotation
effects that manifest themselves in the evolution equation with the existence of Lyapunov
functionals which are equivalent to some norm induced by the inner product of solution.
We now proceed with the derivation of our geophysical fluid dynamics model. Employing
the Pedlosky closure hypothesis into the equations (5) for coherent flow fields gives the
following primitive partial differential equations governing rotating Boussinesq equations
5

7. with Reynolds stress:
ρ0( @u
@t + u. ▽u + f × u + ε△6 u) + gρk = − ▽p + μ△u,
▽.u = 0,
@½
@t + u.▽ρ + δ △6 ρ = κ△ρ.
(9)
The presence of effective dissipative terms △6u and △6ρ is equivalent to stress due to the
impact of the incoherent terms on the coherent flow. Alternatively, the dissipative terms
account for the wind stress along the ocean surface. Next, concerning the boundary con-ditions,
we assume the fields to be periodic along the coordinates x, y and z with periods
L1,L2 and L2, respectively. The flow domain is assigned on an open rectangular periodic
region
B = {(x, y, z) ∈ (0,L1) × (0,L2) × (0,L3)}.
To system (9) we prescribe the following space-periodic boundary conditions for the flow
fields
u(x + Lei, t) = u(x, t)
ρ(x + Lei, t) = ρ(x, t)
p(x + Lei, t) = p(x, t)
(10)
where {e1, e2, e3} is the standard basis of IR3 and with the simplifying assumption that the
period L = L1 = L2 = L3. We further assume the derivatives of u, ρ and p are also space-periodic.
In order to eliminate rigid motions we consider the case where the average velocity
and pressure vanish
ZB
u(x, t)dx = 0,
Zp(x, t)dx = 0.
B
Substitution in (9) reduce to the following:
@u
@t + u. ▽u + f × u + g
½0
ρk + ε △6 u = − 1
½0
▽p + ν △u,
▽.u = 0,
@½
@t + u. ▽ρ + (ρu − ρl)u.k + δ △6 ρ = κ△ρ.
(11)
6

8. b. The initial-boundary value problems
1) Rotating Boussinesq equations
In this section we introduce the nondimensional form of the equations governing the flow
of a viscous incompressible stratified fluid under the Coriolis force without Reynolds stress.
In order to achieve this objective, the system of partial differential equations and boundary
conditions are made nondimensional with the following length and velocity scales of motions
in which rotation and stratification effects are essential:
x = L¯x, y = L¯y, z = H¯z, t =
L
U
¯t
u = U¯u, v = U¯v,w =
UH
L
¯ w,L1 = L¯L
1
L2 = L¯L
2, ρ =
1
gH
f0ρ0UL¯ρ, p = f0ρ0UL¯p
ρu − ρl =
1
g
N2ρ0, ¯ β =
L cos ϕ0
r0Ro sin ϕ0
where T = L
U represent the time scale, U, represent the horizontal velocity scale, W represent
the vertical velocity scale, L represent the length scale, H represent the vertical length scale,
etc. Distinguishing attributes of geophysical fluid dynamics are due to the effects of rotation,
density heterogeneity or stratification and scales of motion. To ensure the measure of the
effect of variations of density defined by stratification frequency
N2 = −
g
ρ0
dρ
dz
is real(which implies static stability of the stratified fluid), we assume d½
dz < 0 for 0 ≤ z ≤ h. It
is known that an unstable density stratification leads to rapid convective motions. Employing
the beta-plane approximation and substituting the above scaling into the system (3) and
omitting bars from nondimensional quantities, we obtain the following nondimensional form
of the equations governing the flow of a viscous incompressible stratified fluid with the
7

9. Coriolis force:
Ro(
∂u
∂t
+ u. ▽u) − (1 + yRoβ0)u × k + ρk = −▽p + Ek △u,
▽.u = 0,
Fr2
Ro
(
∂ρ
∂t
+ u. ▽ρ) + u.k =
Fr2
Ro(Ed)
△ρ,
where we set the aspect ratio H
L to unity. The evolution equations have been scaled so
that the relative order of each term is measured by the dimensionless parameter multiplying
it. Here the parameter Ro = U
Lf0
is the Rossby number which compares the inertial term
to the Coriolis force; Fr = U
NH is the Froude number which measures the importance of
stratification; Ek = º
H2f0
measures the relative importance of frictional dissipation; Ed =
º
2
is the nondimensional eddy diffusion coefficient. The ratio Frmeasures the significant
f0N2 Ro influence of both rotation and stratification to the dynamics of the flow. The corresponding
initial-boundary conditions are specified as follows: We make the vertical density boundary
conditions homogeneous using
ρ = ¯ρ + ρl +
1
g
zρ0N2
p = ¯p − zρl −
1
2g
z2ρ0N2
Omitting bars the conditions (2) become
∂u
∂z
=
∂v
∂z
= w = ρ = 0 at ¡,
Zudx = Zvdx = 0.
B
B
And the flow domain reduces to B = {(x, y, z) ∈ (0,L) × (0,L) × (0, 1)} with boundary
¡ = {z = 0, 1}. Furthermore, velocity and density are given at initial time t = 0
u(x, 0) = u0(x)
ρ(x, 0) = ρ0(x).
We conclude this section with recapitulation of the set of equations that will be analyzed in
this investigation. The evolutionary equation for the β-plane rotating Boussinesq equations
8

10. governing geophysical fluid flows is the following initial-boundary value problem:
Ro( @u
@t + u.▽u) − (1 + yRoβ0)u × k + ρk = −▽p + Ek △u,
▽.u = 0,
Fr2
Ro ( @½
@t + u.▽ρ) + u.k = Fr2
Ro(Ed) △ρ,
(12)
@u
= @v
= w = ρ = 0 at ¡, (13)
@z @z Zudx = Zvdx = 0.
B
B
u(x, 0) = u0(x)
ρ(x, 0) = ρ0(x).
(14)
2) Rotating Boussinesq equations with Reynolds stress
In this section we introduce the nondimensional form of the equations governing the flow
of a viscous incompressible stratified fluid under the Coriolis force with Reynolds stress. In
order to achieve this objective, the system of partial differential equations and boundary
conditions are made nondimensional with the following length and velocity scales of motions
in which rotation and stratification effects are essential:
x = L¯x, y = L¯y, z = H¯z, t =
L
U
¯t
u = U¯u, v = U¯v,w =
UH
L
¯ w,L1 = L¯L
1
L2 = L¯L2, ρ =
1
gH
f0ρ0UL¯ρ, p = f0ρ0UL¯p
ρu − ρl =
1
g
N2ρ0, ¯ β =
L cos ϕ0
r0Ro sin ϕ0
where T = L
U represent the time scale, U, represent the horizontal velocity scale, W represent
the vertical velocity scale, L represent the length scale, H represent the vertical length scale,
etc. Distinguishing attributes of geophysical fluid dynamics are due to the effects of rotation,
9

11. density heterogeneity or stratification and scales of motion. To ensure the measure of the
effect of variations of density defined by stratification frequency
N2 = −
g
ρ0
dρ
dz
is real(which implies static stability of the stratified fluid), we assume d½
dz < 0. It is known that
an unstable density stratification leads to rapid convective motions. Employing the beta-plane
approximation and substituting the above scaling into the system (3) and omitting
bars from nondimensional quantities, we obtain the following nondimensional form of the
equations governing the flow of a viscous incompressible stratified fluid under the Coriolis
force with Reynolds stress:
Ro(
∂u
∂t
+ u.▽u) − (1 + yRoβ0)u × k + ρk + Ek △6 u = −▽p + Ek △u,
▽.u = 0,
Fr2
Ro
(
∂ρ
∂t
+ u.▽ρ) + u.k +
Fr2
Ed(Ro)
△6 ρ =
Fr2
Ro(Ed)
△ρ,
where we set the aspect ratio H
L to unity. The evolution equations have been scaled so
that the relative order of each term is measured by the dimensionless parameter multiplying
it. Here the parameter Ro = U
Lf0
is the Rossby number which compares the inertial term
to the Coriolis force; Fr = U
NH is the Froude number which measures the importance of
stratification; Ek = º
H2f0
measures the relative importance of frictional dissipation; Ed =
º
2
is the nondimensional eddy diffusion coefficient. The ratio Frmeasures the significant
f0N2 Ro influence of both rotation and stratification to the dynamics of the flow. The corresponding
initial-boundary conditions are specified as follows: The space-periodicity of the fields and
the vanishing of the average velocity and pressure are represented by
u(x + Lei, t) = u(x, t),
ρ(x + Lei, t) = ρ(x, t),
p(x + Lei, t) = p(x, t),
ZB
u(x, t)dx = 0,
Zp(x, t)dx = 0,
B
10

12. where the flow domain B = {(x, y, z) ∈ (0,L) × (0,L) × (0,L)} is the periodic region.
Furthermore, velocity and density are given at initial time t = 0
u(x, 0) = u0(x)
ρ(x, 0) = ρ0(x).
We conclude this chapter with recapitulation of the set of equations that will be analyzed in
this investigation. The evolutionary equation for the β-plane rotating Boussinesq equations
with Reynolds stress is the following initial-boundary value problem:
Ro( @u
@t + u.▽u) − (1 + yRoβ0)u × k + ρk + Ek △6 u = − ▽p + Ek △u,
▽.u = 0,
Fr2
Ro ( @½
@t + u.▽ρ) + u.k + Fr2
Ed(Ro) △6 ρ = Fr2
Ro(Ed) △ρ,
(15)
u(x + Lei, t) = u(x, t),
ρ(x + Lei, t) = ρ(x, t),
p(x + Lei, t) = p(x, t),
(16)
ZB
u(x, t)dx = 0,
Zp(x, t)dx = 0,
B
u(x, 0) = u0(x)
ρ(x, 0) = ρ0(x).
(17)
2. Existence and uniqueness of solutions
a. Rotating Boussinesq equations
The purpose of this section is to develop results of existence, uniqueness and differentiability
of solution for the initial-boundary value problem of β-plane rotating Boussinesq equations
(12−14). We prove the significant aspects of well-posedness of solution utilizing the machin-ery
developed above and additional accessible techniques constructed in this section. The
11

13. functional formulation for the evolution equation of β-plane rotating Boussinesq equations
(12 − 14) now takes the form of the following initial value problem in the Hilbert space ¨:
dU
dt + LU + N(U) = 0
U(0) = U0
(18)
where U = (u, v,w, ρ). The linear operator LU and the nonlinear operator N(U) are given
by
N(U) = B(U,U) − FU
FU = −(0, 0,¦
ρ
Ro
,
Ro
Fr2w)
LU = TU + SU
TU = −µEk
Ro¦△u
1
Ed △ρ ¶ = µAsu
Alρ¶
u · ▽ρ ¶ = µbs(u, u)
B(U,U) = B(U) = µ¦u ·▽u
bl(u, ρ)¶
S = −¦(
1 + yβRo
Ro
)

0 −1 0 0
1 0 0 0
0 0 0 0
0 0 0 0
∀ U ∈ D(L) = D(T).
This problem as the the Navier-Stokes equations in the presence of stratification but without
Coriolis terms, this problem has been examined by Temam et al (36; 37; 14; 15) and cited
works therein. In order to develop basic results concerning whether there exists a unique
solution in the large for the general three-dimensional space variables for initial-value problem
(18), we derive a priori energy type estimates useful in proving that (18) generate a dynamical
system, denoted, U(t) = S(t)U0, where U(t) is the unique solution uniformly bounded in
finite-time and S(t) is a group of continuous nonlinear solution operators. The principal
result concerning the existence of such a unique solution will be proven by the utility of the
Faedo-Galerkin technique, a priori energy type and Leray-Schauder principle. As for the
linear operator LU of the initial-value problem (18), the following is valid
Ro k▽uk2 + 1
Edk ▽ρk2 (19)
(LU,U) = (TU,U) = Ek
12

14. It follows from (16; 17; 20; 21) that the operator T is self-adjoint and its spectrum σ(T)
satisfies σ(T) ⊆ [0,∞). The following result on the spectrum of L from (16; 17; 21; 20) will
be utilized:
Lemma 2.1 σ(L) ⊆ [0,∞) ∪ ∐, where ∐ is either empty or an at most denumerable set
consisting of isolated, positive eigenvalues λn = λn( 1
Ro + Ro
Fr2 ) with finite multiplicity such
that
∞ > λ1 ≥ λ2 ≥ · · · ≥ λn ≥ . . . ,
clustering at zero. Furthermore, whenever
1
Ro
+
Ro
Fr2 ≤ Ga
then ∐ = ∅ whereas if
1
Ro
+
Ro
Fr2 ≥ Ga
we have ∐6= ∅. Also, zero is not an eigenvalue whenever the following is satisified:
1
Ro
+
Ro
Fr2 ≤ Ga.
Before giving the proof of the lemma, we make remarks and investigate the steady linear
versions of the nonlinear initial-value problem. The examination demonstrate that the steady
linear system has a unique solution and provide useful properties which are employed in
the investigation of the nonlinear initial-value problem. We start by recalling that when
σ(L) ⊆ (0,∞) which hold if
1
Ro
+
Ro
Fr2 ≤ Ga
then we obtain
(LU,U) = (TU,U) ≥ λkUk2 > 0 (20)
which implies the bilinear form induced by L is coercive and L−1 is compact and self-adjoint.
This establishes existence and uniqueness of solution to the steady linear case by the Riesz-
Fr´echet representation theorem or the Lax-Milgram lemma. Through the utility a result from
(36; 16; 17; 20; 21), λ is the smallest eigenvalue of Laplace’s operator and Stokes operator.
Also, the symmetry of the bilinear form a(U,W), together with the coerciveness of a(U,U)
imply the existence of a sequence of positive eigenvalues and a corresponding sequence of
eigensolutions which yield an orthonormal basis of ¨ such that
LWi = λiWi
0 < λ1 ≤ λ2 ≤ · · · ≤ λi ≤ . . . ,
(21)
13

15. and λi → ∞ as i → ∞. Moreover,
kL
1
2Uk ≥ λ
1
2
1 kUk
kLUk ≥ λ
1
2
1 kL
1
2Uk
(22)
Next, we focus our attention to establishing the above lemma. The flavor of the proof is
similar to that given in Kato (21) for the existence of eigensolutions and eigenvalues of
perturbed elliptic operators
Proof: According to a perturbation theorem (16; 17; 21; 20), if TU is closed and SU
is relatively compact with respect to TU then σ(L = T + S) and σ(T) differ by at most
a denumerable number of isolated, positive eigenvalues of finite multiplicity clustering at
zero. Thus it suffices to show that S is relatively compact with respect to T since from
(16; 17; 21; 20) we have σ(T) ⊆ [0,∞). By definition, S is relatively compact with respect
to T if for any sequence {Un} ⊂ ¨,
kUnk + kTUnk ≤ α, uniformly in n, (23)
then there is a subsequence U´n ⊂ {Un} such that SU´n is strongly convergent. With strong
convergence, we can make strong statements. Indeed, the following inequality is valid
kSU´nk2 ≤ [5( 1+y¯Ro
Ro )2 + ( 1
Ro + Ro
Fr2 )2]kU´nk2
≤ [5( 1+y¯Ro
Ro )2 + ( 1
Ro + Ro
Fr2 )2]α2
(24)
and by the courtesy of Ascoli-Arzela theorem {SU´n} contains a uniformly convergent sub-sequence
which is a Cauchy sequence in a complete normed space ¨. Thus SU´n is strongly
convergent which illustrates that S is relatively compact with respect to T and the claim of
the lemma is achieved. We proceed put together a series of results that will be employed in
establishing properties of solution to the initial-value problem (18). The following mini-max
principle from (36; 14) will be utilized:
limt→∞ supkρ(t)k2 ≤ k­k (25)
where k­k is the volume of ­ which represents the flow region ­ = {(x, y, z) ∈ IR3}. Let
α = max{Ek
Ro , 1
Ek}. By the orthogonal property of the nonlinear operator, we have
(B(U),U) = R­[(u ·▽u · u) + (u ·▽ρ)ρ]dx = 0 (26)
Taking the inner product of (18) with U, utilizing Cauchy-Schwarz inequality and Young’s
inequality gives the following:
d
dtkUk2 + αλ1kUk2
≤ d
dtkUk2 + αkL
1
2Uk2
® ( Ro
Fr2 + 1
Ro )2kρk2.
≤ ¸1
(27)
14

16. The consideration of Gronwall’s inequality into (27) and invoking the mini-max principle
(25) provides a priori estimate
Fr2 + 1
Ro )4 ¸2
kU(t)k2 ≤ kU0k2exp(−αt) + ( Ro
1
®2 k­k(1 − exp(−αt)) (28)
which is asymptotic stability with flow energy and entropy production provided by
E(t) =
1
2
kU(t)k2 =
1
2 Z­
(u2 + ρ2)dx
Ro , 1
Ek}. Furthemore, in the limit as time goes
and exponential dissipation rate α = max{Ek
to infinity we obtain
Fr2 + 1
Ro )4 ¸2
limt→∞ supkU(t)k2 ≤ ( Ro
1
®2 k­k = ρ21
(29)
which shows U(t) is uniformly bounded for all time in ¨. Next, by the mini-max principle
(25) and uniform bound (29), it follows that for any U0 ∈ ¨ and ε there exists T1 = T1(U0, ε)
such that for t ≥ T1 then
kρ(t)k ≤ k­k
1
2 + ε (30)
Integration of the second inequality in (27) and the courtesy (30) provide a priori energy
type estimate
Z T1
0
kL
1
2U(t)k2dt ≤
1
α
{kU0k2 + (
Ro
Fr2 +
1
Ro
)2λ1
α
(k­k
1
2 + ε)2} = ρ2 11.
Taking the inner product of (18) with LU and making use of Cauchy-Schwarz inequality,
Young’s inequality and Sobolev inequality yield the estimates
d
dtkL
1
2Uk2 + λ1αkL
1
2Uk2
≤ d
dtkL
1
2Uk2 + αkLUk2
Fr2 + 1
Ro )2kρk2 + c2
≤ 2α( Ro
0
4®kL
1
2Uk6.
(31)
By setting c = max{ c2
Fr2 + 1
Ro )2(k­k
0
4®, 2α( Ro
1
2 + ε)2} then (31) and (30) imply that
d
dt (1 + kL
1
2Uk2) ≤ c(1 + kL
1
2Uk2)3. (32)
Invoking Gronwall’s inequality in the differential inequality (32) gives
kL
1
2Uk2 ≤ 1 + kL
1
2U0k2 (33)
for t ≤ T2(kL
1
2U0k) = 3
8c(1+kL
1
2 U0k2)
. Consequently putting the pieces together we obtain
Supt∈[0,T2]kL
1
2Uk2 ≤ 1 + kL
1
2U0k2 = ρ2 20 (34)
15

17. which shows that U(t) is bounded for finite-time T2 in D(L). Integrating the differential
inequality (32) provide a priori energy type estimate
Z T2
0
kLU(t)k2dt ≤
1
α
{kL
1
2U0k2 + 2α(
Ro
Fr2 +
1
Ro
)2(k­k
1
2 + ε)2 +
c20
4α
ρ6 20} = ρ2 21.
We now state existence, uniqueness, continuity and differentiability with respect to initial
conditions of solution to the initial-value problem (18) and utilize the above a priori estimates
to prove the results.
Proposition 2.2 Under the above hypothesis for U0 ∈ ¨ given, (18) generate a dynamical
system U(t) = S(t)U0 satisfying
U ∈ C([0, T]; ¨) ∩ L2(0, T;D(L
1
2 ))
for all T > 0 and if U0 ∈ D(L
1
2 ) then
U ∈ C([0, T];D(L
1
2 )) ∩ L2(0, T;D(L))
for all T > 0 where T = min{T1, T2} is finite-time.
Proof: For fixed m, the Faedo-Galerkin approximation
Um =
m
Xi=1
gim(t)Wi
of the solution U of (18) is defined by the finite-dimensional system of nonlinear ordinary
differential equations for gim(t) given by
dUm
dt + LUm + PmN(Um) = 0
Um(0) = PmU0
(35)
where Pm is the orthogonal projection in ¨,D(L
1
2 ),D(L−1
2 ) and D(L) onto the space spanned
by the m eigenfunctions of L given in (21). Let X be the finite-dimensional Hilbert space
spanned by the m eigenfunctions of L given in (21) with inner product ((., .)) induced by
D(L). Consider a closed ball of radius ρ2 21 contained in D(L). Integration of (35) and the
above a priori energy type estimates provide a continuous mapping defined by the integral
of (35) from the closed ball of radius ρ2 21 into itself. Thus, by the Leray-Shauder principle
(11; 20) the finite-dimensional system of nonlinear ordinary differential equations (35) has
at least one solution Um inside the ball of radius ρ2 21. The passage to the limit
m → ∞ and Tm → T
16

18. follows from the following a priori estimates: Utilizing the differential inequality (27) yields
d
dtkUmk2 + αλ1kUmk2
® ( Ro
Fr2 + 1
Ro )2kρmk2
≤ ¸1
(36)
which together with a priori estimate (28) and the subsequent one imply Um remains bounded
in
L∞((0, T); ¨) ∩ L2(0, T;D(L
1
2 )).
This useful result combined with the weak compactness implies there is a subsequence also
denoted by Um and
U ∈ L∞(0, T; ¨) ∩ L2(0, T;D(L
1
2 ))
such that
Um → ½ U ∈ L2(0, T;D(L
1
2 )) weakly
U ∈ L∞(0, T; ¨) weak-star.
Invoking the same inequalities that let to the differential inequality (31), give the boundeness
1
of N(Um) and PmN(Um) in L2(0, T;D(L−2 )). Furthermore, from (35), it follows that dUm
dt
is also bounded in L2(0, T;D(L−1
2 )). Utility of this result and weak compactness yield
dUm
dU
1
→
∈ L2(−0, T;D(L
2 ))
dt
dt
weakly. The assertion of Lebesgue dominated convergence theorem as well as the above
convergence results yield Um → U ∈ L2(0, T; ¨) strongly. With strong convergence we can
declare strong assertions. We pass the limit in (35) and obtain the required result (18). We
obtain
U ∈ C([0, T]; ¨) ∩ L2(0, T;D(L
1
2 )).
The result
U ∈ C([0, T];D(L
1
2 )) ∩ L2(0, T;D(L))
We conclude that if U0 ∈ D(L
1
2 ) and by the Faedo-Galerkin technique
U ∈ C([0, T];D(L
1
2 )) ∩ L2(0, T;D(L))
for all T > 0. According to Gronwall’s inequality, uniqueness and continuity with respect to
initial conditions of solution U(t) follows from considering the difference between two solu-tions
of the initial-value problem (18) and employing a priori estimates (28−34). Uniqueness
and continuity with respect to initial conditions of solution U(t) generate a dynamical system
which is described by continuous solution operators S(t), t ∈ IR defined by
S(t)U0 = U(t) ≡ S(U0, t)
17

19. satisfying the group property
S(t)U(s) = S(s)U(t)
S(0)U = U ∀ t, s ∈ IR
(37)
such that t+s ≤ T. That the solution operators S(t) are continuous is a consequence of the
continuity of U(t) in time and in initial conditions. The group property is a consequence of
the injectivity of the solution operators which is equivalent to the backward uniqueness of so-lution
for the initial-value problem (18). Next, we turn our attention to establish uniqueness,
continuity and Fr´echet differentiability with respect to initial conditions of solution U(t) for
the initial-value problem (18) with the work of Temam (36; 37; 38) as an exploration upon
which we build. The technique of the proof uses the linearization of the evolution equation
(18) about the difference of two given solutions. Let u and v be solutions of the initial value
problem (18) corresponding to the initial conditions u(0) = u0 and v(0) = v0, respectively.
Then the difference w = v − u satisfies
dw
dt + Lw + N(v) − N(u) = 0
w(0) = v0 − u0
(38)
where
N(v) − N(u) = B(v, v) − B(u, u)
= B(v,w) + B(w, u)
= B(u,w) + B(w, u) + B(w,w).
(39)
The linearization of the nonlinear equation (38) about a given solution reduces to the vari-ational
equation
dª
dt + Lª + l0(t)ª = 0
ª(0) = v0 − u0 = ξ
(40)
where l0(t)w = B(u(t),w) + B(w, u(t)) and l0(t) is a linear bounded operator from D(L
1
2 )
to D(L−1
2 ). Consequently, the equation of variation is a linear nonautonomous ordinary
differential equation. The difference ϕ = w − ª satisfies
d'
dt + Lϕ + l0(t)ϕ + l1(t;w(t)) = 0
ϕ(0) = 0
(41)
where l1(t;w(t)) = B(w,w). From the last equality of (39), we have N(v)−N(u) = l0(t)w+
l1(t;w).
18

20. Proposition 2.3 If u0 ∈ ¨ and u is the unique solution of the nonlinear initial-value prob-lem
(18) then the corresponding variational equation (40) has a unique solution ª satisfying
ª ∈ C([0, T]; ¨) ∩ L2(0, T;D(L
1
2 ))
for all T > 0 where T = min{T1, T2} is finite-time. Moreover, the dynamical system u0 →
S(t)u0 is Fr´echet differentiable in ¨ with differential L(t, u0) : ξ → ª(t) given by the solution
of the variational equation.
Proof: Existence and uniqueness of solution to the variational equation (40) is proved by
the courtesy of the splendid Faedo-Galerkin technique and the above a priori energy type
estimates. Taking the inner product of the equation (38) with w and application of Cauchy-
Schwarz inequality, Young inequality and Sobolev inequalities yield
d
dtkwk2 + kL
1
2wk2 ≤ k2kwk2 (42)
where k = c1ρ21 and ρ21 is given in the estimate (34). By the virtue of the Gronwall’s
inequality we obtain
kv(t) − u(t)k2 ≤ exp(k2T)kv0 − u0k2 (43)
∀t ∈ (0, T). The sharp a priori estimate (43) illustrates forward uniqueness and continuity
with respect to initial conditions of solution. Furthermore, invoking the estimates (42) yield
Z t
0
kL
1
2w(s)k2ds ≤ exp(k2T)kv0 − u0k2.
Additionally, taking the inner product of equation (41) with ϕ we obtain
d
dtkϕk2 + 1
2kL
1
2ϕk2 ≤ k2kϕk2 + c2
1
¸1
kL
1
2w(t)k3 (44)
By the utility of Gronwall’s inequality we obtain the estimate
kϕ(t)k2 ≤
c21
λ1 Z T
0
kL
1
2w(s)k3ds
for all t ∈ [0, T]. Consequently it follows that
kϕ(t)k2 ≤ c2
1
¸1
exp(3k2T/2)kv0 − u0k (45)
which gives the required result
kS(t)v0 − S(t)u0 − L(t, u0).(v0 − u0)k2
kv0 − u0k2 = o(kv0 − u0k)
19

21. as v0 → u0. Thus, the dynamical system S(t) : u0 → u(t) is Fr´echet differentiable at u0 in
the Hilbert space ¨.
Next, we investigate the problem of proving backward uniqueness of solution for the initial-value
equation (18) that establish the injectivity properties and group properties of the
solution operators S(t). The backward uniqueness of solution is proved in approximately
the same way as Fr´echet differentiability result using the standard log-convexity method that
has been employed in (3; 37). In order to motivate this objective of the injectivity of the
solution operators, consider two solutions u and v of the initial-value problem elaborated
above such that at time t = t∗ both u and v satisfy the equation (18) for t ∈ (t∗ − ǫ, t∗)
and u(t∗) = v(t∗). Given that this hypothesis is valid, backward uniqueness of the solution
operators is accomplished if u(t) = v(t) for all time t < t∗ whenever the solutions are
well-defined. Alternatively, suppose the solution operators satisfy
S(t) = u(t + s)∀ t > 0 s ∈ IR
then for τ ∈ (0, ǫ), the problem of backward uniqueness is whether the following hold:
S(τ )u(t∗ − τ ) = S(τ )v(t∗ − τ ) ⇒ u(t∗ − τ ) = v(t∗ − τ ) (46)
which is the injectivity of the solution operators S(τ ).
Proposition 2.4 Suppose the following hold
u, v ∈ L∞(0, T; V = D(L
1
2 ) ∩ L2(0, T;H = D(L))
then the difference w = v − u satisfies the differential equation (38) for t ∈ (0, T) and the
solution operators S(t) are injective.
We give a sketch of the proof due to Temam (37).
Proof: We will use the notation ϕ = ϕ(t) for the quotient
ϕ =
kϕ(t)k2
V
kϕ(t)k2
H
=
(Lϕ(t), ϕ(t))
(ϕ(t), ϕ(t))
.
Invoking Cauchy-Schwarz inequality, Young’s inequality, Sobolev inequalities, continuity
properties of the nonlinear operator B(w,w) established above and the fact that (Lw −
20

22. ϕw,w) = 0 we obtain
1
2
d
dtϕ = (( dw
dt ,w))
kwk2 − kwk2
V
( dw
kwk4 dt ,w)
= 1
kwk2 ( dw
dt ,Lw − ϕw)
= 1
kwk2
V
(N(v) − N(u) − Lw,Lw − ϕw)
= −kLw−'wk2
kwk2 + 1
V ertwk2 (Lw,N(v) − N(u))
≤ −kLw−'wk2
2kwk2 + kN(v)−N(u)k2
2kwk2
≤ −kLw−'wk2
2kwk2 + k2ϕ
(47)
where N(v) − N(u) is given in (39) and the function k(t) = k1(t) + B(w,w) is defined by
the mapping
1
2
V kLu(t)k
k1(t) : t → c(ku(t)k
1
2
V kLv(t)k
1
2 + kv(t)k
1
2 ) ∈ L2(0, T).
Utility of Gronwall’s inequality into (47) gives
ϕ(t) ≤ ϕ(0)exp(2 Z t
0
k2(s)ds) t ∈ (0, T).
Thus, if the difference w = v − u satisfies the above conditions and
w(τ ) = 0 ⇒ w(t) = 0, t ∈ [0, T].
We proceed by contradiction. Hence, suppose that kw(t0)6= 0k for t0 ∈ [0, T). As a result by
continuity we have kw(t)6= 0k on some open interval (t0, t0 +ǫ) and we employ the notation
t∗ ≤ T for the largest time for which
w(t)6= 0 ∀t ∈ [t0, t∗).
It follows that w(t∗) = 0. In the interval [t0, t∗) the mapping given by t → logkw(t)k is
well-defined and the fact that w(t) satisfies the differential equation (38) we obtain
d
dt
log
1
kwk
≤ 2ϕ + k2.
Consequently for the time t ∈ [t0, t∗) the following is valid:
log
1
kwk
≤ log
1
kw(t0)k
+ Z T
t0
2ϕ(s) + k2(s)ds
which illustrates the boundedness of 1
w(t) from above as the time t → t∗ from below and this
is a contradiction.
21

23. b. Rotating Boussinesq equations with Reynolds stress
The goal of this section is to establish results of existence, uniqueness and differentiability of
the solutions for the initial-boundary value problem of β-plane rotating Boussinesq equations
with Reynolds stress (15−17). We prove the significant aspects of well-posedness of solution
utilizing the machinery developed above and additional accessible techniques constructed
in this section. The functional formulation for the evolution equation of β-plane rotating
Boussinesq equations with Reynolds stress (15 − 17) takes the following form of an initial
value problem in the Hilbert space ¨:
dU
dt + LU + N(U) = 0
U(0) = U0
(48)
where U = (u, v,w, ρ). The boundary conditions have been taken to be space-periodic in
order to simplify the analysis. Concerning the linear operator LU and the nonlinear operator
N(U) we have
N(U) = B(U,U) + CU − FU
FU = −(0, 0,
ρ
Ro
,
Ro
Fr2w)
CU = −µEk
Ro △u
1
Ed △ρ¶ = µAlu
Alρ¶
B(U,U) = B(U) = µu ·▽u
u ·▽ρ¶ = µbl(u, u)
bl(u, ρ)¶
LU = TU + SU
TU = −µEk
Ro △6 u
1
Ed △6 ρ¶ = µA6l
u
A6l
ρ¶
S = −(
1 + yβRo
Ro
)

0 −1 0 0
1 0 0 0
0 0 0 0
0 0 0 0

∀ U ∈ D(L) = D(T).
22

24. In order to develop basic results concerning whether there exists a unique solution in the
large for the general three-dimensional space variables for initial-value problem (48), we
derive a priori energy type estimates useful in proving that (48) generate a dynamical system,
denoted, U(t) = S(t)U0, where U(t) is the unique solution uniformly bounded in time and
S(t) is a group of continuous nonlinear solution operators. The principal result concerning
the existence of such a unique solution will be proven by the utility of the Faedo-Galerkin
technique, a priori energy type estimates and Leray-Schauder principle. The linear operator
LU of the initial-value problem (48) satisfies
(LU,U) = (TU,U) ≡ a(U,U) (49)
where a(., .) is symmetric bilinear form. It follows from (36; 16; 20; 21) that this implies L
is self-adjoint. Additionally,
(LU,U) = (TU,U) ≥ λkUk2 > 0 (50)
which implies the bilinear form induced by L is coercive and L−1 is compact and self-adjoint.
By the courtesy of an excellent result from (36; 16; 20; 21), λ is the smallest eigenvalue of
Laplace’s operator and Stokes operator. Also, the symmetry of the bilinear form a(U,W),
together with the coerciveness of a(U,U) imply the existence of a sequence of positive eigen-values
and a corresponding sequence of eigensolutions which yield an orthonormal basis of
¨ such that
LWi = λiWi
0 < λ1 ≤ λ2 ≤ · · · ≤ λi ≤ . . . ,
(51)
and λi → ∞ as i → ∞. Moreover,
kL
1
2Uk ≥ λ
1
2
1 kUk
kLUk ≥ λ
1
2
1 kL
1
2Uk
(52)
The following useful inequalities are utilized in the derivation of a priori energy type esti-mates:
(B(U),U) = R­[(u ·▽u · u) + (u ·▽ρρ]dx = 0 (53)
kCUk2 ≤ λ1kL
1
2Uk2 ≤ λ
1
2
1 kLUkkL
1
2Uk (54)
Since LC = CL, the inequality
kL
1
2CUk2 ≤ λ1kLUk2 (55)
holds. By the courtesy of the Sobolev inequality and continuity properties of the nonlinearity
B(U) the following bounds are satisfied
kB(U)k2 ≤ c20
(CU,U)[kCUk2 + kUk2]
≤ c20
(λ1 + 1/λ1)kUkkL
1
2Uk2kLUk
(56)
23

25. By the virtue of Cauchy-Schwarz inequality, Young’s inequality and (56) the following is
satisfied
|(B(U) + CU,LU)| ≤ 54(c41
kUk2kL
1
2Uk2 + λ1)kL
1
2Uk2 + 1
4kLUk2 (57)
where c1 = c20
(λ1 + 1/λ1). Also, from (15) the following a priori estimates is valid
kL
1
2B(U)k2 ≤ ckLUk2. (58)
Concerning the solution to the steady linear case we have
(LU + CU,U) = (TU,U) + (CU,U)
≥ (λ + λ
1
6
1 )kUk2 > 0,
(59)
which implies L + C is positive and hence the bilinear form it induces is coercive. This
establishes existence and uniqueness of solution to the steady linear case by the Riesz-Fr´echet
representation theorem or the Lax-Milgram lemma. The following mini-max principle from
(36; 14) will be utilized:
Fr2 + 1
Ro )2k­k2 (60)
limt→∞ supkFk2 ≤ ( Ro
where k­k is the measure of ­. Taking the inner product of (48) with U, and invoking the
inequalites (59), (52) and Young’s inequality yield the following differential inequality:
d
dtkUk2 + αλ1kUk2
≤ d
dtkUk2 + αkL
1
2Uk2
®( Ro
Fr2 + 1
Ro )2k­k2
≤ 1
(61)
where α = λ + λ
1
6
1 . The consideration of Gronwall’s inequality in (61) provides a priori
estimate
kU(t)k2 ≤ kU0k2exp(−αt) + 1
®2 ( Ro
Fr2 + 1
Ro )2k­k2(1 − exp(−αt)) (62)
which is asymptotic stability with flow energy and entropy production provided by
E(t) =
1
2
kU(t)k2 =
1
2 Z­
(u2 + ρ2)dx
and exponential dissipation rate α = λ + λ
1
6
1 . Thus, we are able to assert that the following
asymptotic estimate is valid
limt→∞ supkU(t)k2 ≤ 1
®2 ( Ro
Fr2 + 1
Ro )2k­k2 (63)
24

26. which shows U(t) is uniformly bounded in time in ¨. Taking the inner product of (18) with
LU and using inequalities (52) and (57), we find that
d
dtkL
1
2Uk2 + λ1kL
1
2Uk2
≤ d
dtkL
1
2Uk2 + kLUk2
Fr2 + 1
Ro )2k­k2 + 108(c41
≤ 2( Ro
kUk2kL
1
2Uk4 + λ1kL
1
2Uk2)
(64)
We estimates (61) using the inequality
lim
t→∞
sup Z t+1
t
kL
1
2U(s)k2ds ≤ (1/α2 + 1/α3)(
Ro
Fr2 +
1
Ro
)2k­k2
and by the virtue of Gronwall’s inequality in (64) we obtain a priori estimate
kL
1
2U(t)k2 ≤ ρ21
(65)
where ρ21
= ν1ν2, and the parameters ν1 and ν2 are given by the following involved and
tedious expressions, respectively,
ν1 = (2 + 107λ1(1/α2 + 1/α3) + 1/α2 + 1/α3)(
Ro
Fr2 +
1
Ro
)2k­k2
and
ν2 = exp(108c20
(1/α2 + 1/α3)(
Ro
Fr2 +
1
Ro
)2k­k2).
Consequently from the last inequality the following is valid
limt→∞ supkL
1
2U(t)k2 ≤ ρ21
(66)
which shows U(t) is uniformly bounded in time in D(L
1
2 ). Similarly, taking the inner product
of (48) with L2U and making use of the inequality (52) and (58) yield
d
dtkLUk2 + λ1kLUk2
≤ d
dtkLUk2 + kL
3
2Uk2
Ro2 + Ro2
Fr4 )ρ21
≤ 3c2kLUk4 + 3λ1kLUk2 + 3( 1
(67)
Using the differential inequality (64) and invoking H¨older inequality we obtain a priori energy
type estimate
Z t+1
t
kLU(s)k2ds ≤ ν2
3
25

27. where ν2
3 = a1 + a2 denotes the sum of the involved expressions
a1 = λ1(1/α2 + 1/α3)(
Ro
Fr2 +
1
Ro
)2k­k2 + ρ21
and
a2 = 108c20
(1/α2 + 1/α3)2(
Ro
Fr2 +
1
Ro
)4k­k4.
Substitution of this a priori energy type estimate into (67) yields
kLU(t)k2 ≤ ρ22
(68)
with ρ22
denoting
ρ22
3 + 2λ1ν2
= 3c2ν4
3 + 3(
1
Ro2 +
Ro2
Fr4 )ρ21
.
Consequently, putting the estimates together we obtain
limt→∞ supkLU(t)k2 ≤ ρ22
(69)
which shows U(t) is uniformly bounded in time in D(L). We now state existence, uniqueness,
continuity and differentiability with respect to initial conditions of solution to the initial-value
problem (48) and invoke the above a priori estimates to prove the result.
Proposition 2.5 Under the above hypothesis for F ∈ D(L
1
2 ) and U0 ∈ ¨ given, (48) gen-erate
a dynamical system U(t) = S(t)U0 satisfying
U ∈ C([0, T]; ¨) ∩ L2(0, T;D(L
1
2 ))
for all T > 0. Moreover, if U0 ∈ D(L
1
2 ) then
U ∈ C([0, T];D(L
1
2 )) ∩ L2(0, T;D(L))
for all T > 0.
Proof: For fixed m, the Faedo-Galerkin approximation
Um =
m
Xi=1
gim(t)Wi
of the solution U of (48) is defined by the finite-dimensional system of nonlinear ordinary
differential equations for gim(t) given by
dUm
dt + LUm + PmB(Um) + CUm = PmF
Um(0) = PmU0
(70)
26

28. where Pm is the orthogonal projection in ¨,D(L
1
2 ),D(L−1
2 ) and D(L) onto the space spanned
by the m eigenfunctions of L given in (51). Let X be the finite-dimensional Hilbert space
spanned by the m eigenfunctions of L given in (51) with inner product ((., .)) induced by
D(L). Consider a closed ball of radius ρ22
given by (69) contained in D(L). Integration of
(70) and the above a priori energy type estimates provide a continuous mapping defined
by the integral of (70) from the closed ball of radius ρ22
given by (69) into itself. Thus,
22
by the Leray-Shauder principle (11; 20) the finite-dimensional system of nonlinear ordinary
differential equations (70) has at least one solution Um inside the ball of radius ρ. The
passage to the limit
m → ∞ and Tm → T = ∞
follows from the following a priori estimates: Utilizing the differential inequality (61) provides
d
dtkUmk2 + αλ1kUmk2
≤ 1
®( Ro
Fr2 + 1
Ro )2
(71)
which implies Um remains bounded in
L∞((0, T); ¨) ∩ L2(0, T;D(L
1
2 )).
This useful result combined with the weak compactness implies there is a subsequence also
denoted by Um and
U ∈ L∞(0, T; ¨) ∩ L2(0, T;D(L
1
2 ))
such that
Um → ½ U ∈ L2(0, T;D(L
1
2 )) weakly
U ∈ L∞(0, T; ¨) weak-star.
Using (53) and (56), we obtain the boundeness of B(Um) and PmB(Um) in L2(0, T;D(L−1
2 )).
dt is also bounded in L2(0, T;D(L−1
Furthermore, from (70), dUm
2 )). Utility of this result and
weak compactness yield
dUm
dt
→
dU
dt
∈ L2(0, T;D(L
−1
2 ))
weakly. The assertion of Lebesgue dominated convergence theorem as well as the above
convergence results yield Um → U ∈ L2(0, T; ¨) strongly. We pass the limit in (70) and
obtain the required result (48). We obtain
U ∈ C([0, T]; ¨) ∩ L2(0, T;D(L
1
2 )).
Similarly, from (64) and (67) we conclude that if U0 ∈ D(L
1
2 ) then by the Faedo-Galerkin
technique
U ∈ C([0, T];D(L
1
2 )) ∩ L2(0, T;D(L))
27

29. for all T > 0. According to Gronwall’s inequality, uniqueness and continuity with respect to
initial conditions of solution U(t) follows from considering the difference between two solu-tions
of the initial-value problem (48) and employing a priori estimates (63−69). Uniqueness
and continuity with respect to initial conditions of solution U(t) generate a dynamical system
which is prescribed by continuous solution operators S(t), t ∈ IR defined by
S(t)U0 = U(t) ≡ S(U0, t)
satisfying the group property
S(t)U(s) = S(s)U(t)
S(0)U = U ∀ t, s ∈ IR.
(72)
That the solution operators S(t) are continuous is a consequence of the continuity of U(t) in
time and in initial conditions. The group property is a consequence of the injectivity of the
solution operators which is equivalent to the backward uniqueness of solution for the initial-value
problem (48). Next, we turn our attention to establish uniqueness, continuity and
Fr´echet differentiability with respect to initial conditions of solution U(t) for the initial-value
problem (48). As in the previous section, the technique of the proof uses the linearization
of the evolution equation (48) about the difference of two given solutions. For the sake
of completeness, we repeat the analysis here. Consider u and v satisfying the evolution
equation (48) corresponding to the initial conditions u(0) = u0 and v(0) = v0, respectively.
The precise analysis of the solution difference w = v − w yields
dw
dt + Lw + N(v) − N(u) = 0
w(0) = v0 − u0
(73)
where
N(v) − N(u) = B(v, v) − B(u, u) + Cw
= B(v,w) + B(w, u) + Cw
= B(u,w) + B(w, u) + B(w,w) + Cw.
(74)
The linearization of (73) along the dynamical system trajectory satisfies the variational
equation
dª
dt + Lª + l0(t)ª = 0
ª(0) = v0 − u0 = ξ
(75)
where l0(t)w = B(u(t),w)+B(w, u(t))+Cw and l0(t) is linear bounded operator from D(L
1
2 )
to D(L−1
2 ). Consequently, the equation of variation is a linear nonautonomous ordinary
28

30. differential system. We shall be concerned with the difference ϕ = w − ª and the following
is valid
d'
dt + Lϕ + l0(t)ϕ + l1(t;w(t)) = 0
ϕ(0) = 0
(76)
where l1(t;w(t)) = B(w,w). The remarkable feature of the last equality is that from (74),
we have N(v) − N(u) = l0(t)w + l1(t;w). We are now in a position to state the keystone
result:
Proposition 2.6 If u0 ∈ ¨ and u is the unique solution of (48) then the variational equation
(75) has a unique solution ª satisfying
ª ∈ C([0, T]; ¨) ∩ L2(0, T;D(L
1
2 ))
for all T > 0. Moreover, the dynamical system u0 → S(t)u0 is Fr´echet differentiable in ¨
with differential L(t, u0) : ξ → ª(t) given by the solution of the variational equation.
Proof: Existence and uniqueness of solution to (75) is a consequence the Faedo-Galerkin
technique and the use of a priori estimates together with the Gronwall’s inequality. Next,
we exhibit a series of a priori estimates which will be required in establishing the above
proposition. Taking the inner product of (73) with w and application of Cauchy-Schwarz
inequality, Young’s inequality and Sobolev inequalities yield
d
dtkwk2 + kL
1
2wk2 ≤ k2kwk2, (77)
where k = c1ρ2 and ρ2 is given in the estimate (68). According to the Gronwall’s inequality
the following is valid
kv(t) − u(t)k2 ≤ exp(k2T)kv0 − u0k2 (78)
∀t ∈ (0, T). The sharp a priori estimate (78) yield forward uniqueness and continuity with
respect to initial conditions of solution. Furthermore, from (77) we obtain a priori energy
type estimate
Z t
0
kL
1
2w(s)k2ds ≤ exp(k2T)kv0 − u0k2.
By the utility of the inner product of (76) with ϕ we obtain the differential inequality
d
dtkϕk2 + 1
2kL
1
2ϕk2 ≤ k2kϕk2 + c2
1
¸1
kL
1
2w(t)k3 (79)
We repeat the construction using Gronwall’s inequality and find a priori estimate
kϕ(t)k2 ≤ Z T
0
kL
1
2w(s)k3ds
29

31. ∀t ∈ (0,∞). Consequently we obtain the bound
kϕ(t)k2 ≤ c2
1
¸1
exp(3k2T/2)kv0 − u0k (80)
which implies the needed result
kS(t)v0 − S(t)u0 − L(t, u0).(v0 − u0)k2
kv0 − u0k2 = o(kv0 − u0k)
as v0 → u0. Hence, the dynamical system S(t) : u0 → u(t) is Fr´echet differentiable at
u0 ∈ ¨.
Next, we examine the problem of proving backward uniqueness of solution for the initial-value
equation (48) that establish the injectivity properties and group properties of the
solution operators S(t). The backward uniqueness of solution is proved in approximately
the same way as Fr´echet differentiability result using the standard log-convexity method as
in the previous section. For the sake of completeness of presentation we cover the proof
of the injectivity of solution operators here. In order to motivate this objective, consider
two solutions u and v of the initial-value problem such that at time t = t∗ both u and
v satisfy (48) for t ∈ (t∗ − ǫ, t∗) and u(t∗) = v(t∗). Given that this hypothesis is valid,
backward uniqueness of the solution operators is accomplished if u(t) = v(t) for all time
t < t∗ whenever the solutions are well-defined. Alternatively, suppose the solution operators
satisfy
S(t) = u(t + s)∀ t > 0 s ∈ IR
then for τ ∈ (0, ǫ), the problem of backward uniqueness is whether the following hold:
S(τ )u(t∗ − τ ) = S(τ )v(t∗ − τ ) ⇒ u(t∗ − τ ) = v(t∗ − τ ) (81)
which is the injectivity of the solution operators S(τ ). The following significant result on
the injectivity of the solution operators is valid:
Proposition 2.7 Suppose the following hold
u, v ∈ L∞(0, T; V = D(L
1
2 ) ∩ L2(0, T;H = D(L))
then the difference w = v − u satisfies the differential equation (73) for t ∈ (0, T) and the
solution operators S(t) are injective.
We give a sketch of an elegant proof due to Temam (37).
Proof: We will use the notation ϕ = ϕ(t) for the quotient
ϕ =
kϕ(t)k2
V
kϕ(t)k2
H
=
(Lϕ(t), ϕ(t))
(ϕ(t), ϕ(t))
.
30

32. Invoking Cauchy-Schwarz inequality, Young’s inequality, Sobolev inequalities, continuity
properties of the nonlinear operator B(w,w) + Cw developed above and the fact that
(Lw − ϕw,w) = 0 we obtain
1
2
d
dtϕ = (( dw
dt ,w))
kwk2 − kwk2
V
( dw
kwk4 dt ,w)
= 1
kwk2 ( dw
dt ,Lw − ϕw)
= 1
kwk2
V
(N(v) − N(u) − Lw,Lw − ϕw)
= −kLw−'wk2
kwk2 + 1
V ertwk2 (Lw,N(v) − N(u))
≤ −kLw−'wk2
2kwk2 + kN(v)−N(u)k2
2kwk2
≤ −kLw−'wk2
2kwk2 + k2ϕ
(82)
where N(v)−N(u) is given in (74) and the function k(t) = k1(t)+B(w,w)+Cw is defined
by the mapping
1
2
V kLu(t)k
k1(t) : t → c(ku(t)k
1
2
V kLv(t)k
1
2 + kv(t)k
1
2 ) ∈ L2(0, T).
Utility of Gronwall’s inequality into (82) gives
ϕ(t) ≤ ϕ(0)exp(2 Z t
0
k2(s)ds) t ∈ (0, T).
Thus, if the difference w = v − u satisfies the above conditions and
w(τ ) = 0 ⇒ w(t) = 0, t ∈ [0, T].
We proceed by contradiction. Hence, suppose that kw(t0)6= 0k for t0 ∈ [0, T). As a result by
continuity we have kw(t)6= 0k on some open interval (t0, t0 +ǫ) and we employ the notation
t∗ ≤ T for the largest time for which
w(t)6= 0 ∀t ∈ [t0, t∗).
It follows that w(t∗) = 0. In the interval [t0, t∗) the mapping given by t → logkw(t)k is
well-defined and the fact that w(t) satisfies the differential equation (38) we obtain
d
dt
log
1
kwk
≤ 2ϕ + k2.
31

33. Consequently for the time t ∈ [t0, t∗) the following is valid:
log
1
kwk
≤ log
1
kw(t0)k
+ Z T
t0
2ϕ(s) + k2(s)ds
which illustrates the boundedness of 1
w(t) from above as the time t → t∗ from below and this
is a contradiction.
3. Aspects of stability and attraction
In this section we consider nonlinear stability and attractors of solutions of rotating Boussi-nesq
equations by adapting a priori estimates employed in establishing well-posedness and
differentiability of solutions. One of the advantages of the results presented in this chapter
is that they may be considered as more refined generalization of results such as (28) and
(62) for the dissipation of flow energy and the entropy production provided by
E(t) =
1
2
kU(t)k2 =
1
2 Z­
(u2 + ρ2)dx,
which is specification of energy and the entropy production in the L2-norm. Hence, the
results of this section on nonlinear stability differ in form rather than in essence from their
counterparts utilized in establishing well-posedness and differentiability of solutions. The
problem of showing that the rest state of system (18) or (48) is nonlinearly stable is inves-tigated
using ideas developed in Galdi et al (16; 17). An important quantity in analyzing
whether the solution of an evolution equation is bounded in some suitable norm is that of
Lyapunov functional or a priori energy type estimates of the system as we have already noted
in establishing well-posedness. We will illustrate in the sequel a technique for constructing
Lyapunov functionals with interpretations such as energy, enstrophy and entropy which are
decreasing along solutions. The equations governing the flow of an incompressible stratified
fluid under the Coriolis force induce damping mechanisms in the nature of viscosity, diffusion,
stratification and rotation effects that manifest themselves in the evolution equation with
the existence of Lyapunov functionals which are equivalent to some norm induced by the
inner product of solution. Here we state a technique for constructing generalized Lyapunov
functionals that furnish necessary and sufficient conditions for nonlinear stability. Specif-ically,
the tools used are spectral properties of the linear operator and a priori estimates
for the nonlinear operator. Now we recall definitions which are required in the formulation
of the nonlinear stability criteria to be customized for use in the solutions of the rotating
Boussinesq equations. Consider the following prototype initial-value problem
dU
dt = LU + N(U), U(0) = U0. (83)
32

34. where LU is linear operator and N(U) is nonlinear operator in the sense that its Fr´echet
derivative at zero vanishes. A self-adjoint linear operator L for the initial-value problem (83)
is called essentially dissipative if
(LU,U) ≤ 0 ∀ U ∈ D(L)
(LU,U) = 0 ⇒ U = 0
(84)
The complement of essentially dissipative is called essentially non-dissipative. By the spectral
theorem (16; 21), essential dissipativity is equivalent to the spectrum of L being nonnegative
and zero not an eigenvalue. For every essentially dissipative operator L, the bilinear form
(U,W)L = −(LU,W) ∀ U,W ∈ D(L) (85)
defines a scalar product in D(L). We denote by HL the completion of D(L) in the norm
k.kL given by
kUk2
L = −(LU,U).
According to the condition (84) on essential dissipativity of linear operators, we are in a
position to state necessary and sufficient criteria for nonlinear stability using the Lyapunov
functional
J(U) = −(LU,U) (86)
and generalized energy flow energy and entropy production
E(t) = 1
2{kUk2 + kUk2
L} (87)
for the paradigmatic example of initial-value problem (83).
Proposition 3.1 Suppose L is essentially dissipative and the nonlinear operator satisfies
kNUk2 ≤ ckUk2
L(kLUk2 + kUk2
L) (88)
where c > 0. Then the rest state of system (83) is asymptotically stable, that is,
J(U) ≤ 1
´{2E(0)(1 + 1
°μ0
)exp[ 4(1−°)
μ0° ]}/(t + 1), (89)
whenever
γ = (1 − cE(0)) > 0, (90)
where η > 0 and μ0 = 1
2 . Next assume L is essentially non-dissipative and the nonlinear
operator satisfies
kNUk2 ≤ cJ(U)(kLUk2 + kUk2) (91)
c > 0. Then the rest state is unstable in the sense of Lyapunov.
33

35. This proposition is proved in approximately the same way as the analogous fact for the
finite-dimensional linear problem with the assistance of a priori energy type estimates that
dominates the nonlinearity by linear elliptic operators.
Proof: Taking the inner product of the equation (83) with U yields
1
2
d
dt
(U,U) = (LU,U) + (NU,U).
Similarly, taking the inner product of the equation (83) with LU gives
1
2
d
dt
(LU,U) = (LU,LU) + (LU,NU).
Putting the estimates together provide the following equation for the evolution the general-ized
energy functional (87)
dE
dt = (LU,U) + (NU,U) − (LU,LU) − (LU,NU). (92)
Applying Cauchy-Schwarz inequality and Young’s inequality in (92) yields
dE
dt ≤ (LU,U) + 3
2kLUk2 + kUkkNUk + 1
2kNUk2
L) + kUkkNUk + 1
≤ −μ0(kLUk2 + |U|2
2kNUk2.
(93)
Substitution of a priori estimate (88) in (93) and the courtesy of the generalized energy
functional (87) provides the differential inequality
dE
dt ≤ −μ0(kLUk2 + kUk2
L)[1 − c
L)−1
1
2 kUkLkUk(kLUk2 + kUk2
2 − c
2kUk2
L]
L)[1 − c
≤ −μ0(kLUk2 + kUk2
2kUk2
L]
≤ −μ0(kLUk2 + kUk2
L)[1 − cE].
(94)
According to the inequality (94) and assumption (90) we obtain a priori energy estimate
μ0(kLUk2 + kUk2
L)(1 − cE(0)) ≤ −
dE
dt
which implies
Z ∞
0
μ0(kLUk2 + kUk2
L)(1 −
c
μ1
E(0))dt ≤ 2E(0).
In this respect it should be noticed that the following a priori energy type estimate holds
Z ∞
0
(kLUk2 + kUk2
L)dt ≤
2E(0)
μ0γ
.
34

36. We proceed by recalling the useful property
−
1
2
d
dt
(LU,U) = −(LU,LU) + (LU,NU)
and invoking the Cauchy-Schwarz inequality, Young’s inequality and a priori estimate (88)
which yield the following differential inequality
d
dtkUk2
L ≤ 2ckUk2
L(kLUk2 + kUk2
L). (95)
In order to establish the hypotheses of Gronwall’s inequality we note that (95) may be
restated equivalently as follows:
d©
dt ≤ h1(t) + 2c
μ1
h2(t)© (96)
where
©(t) = kUk2
L(t + 1),
h1(t) = kUk2
L,
and
h2(t) = kLUk2 + kUk2
L.
Thus, the following holds
©(t) ≤ exp[2c Z t
0
h2(s)ds]{©(0) + Z t
0
exp[−2c Z s
0
h2(α)dα]h1(s)ds}.
which is the assertion of Gronwall’s inequality. Moreover, through the use of estimates
from above we deduce the following a priori estimate that guarantee the hypotheses of the
Gronwall’s lemma are valid
Z ∞
0
h1(t)dt ≤ Z ∞
0
h2(t)dt ≤
2E(0)
μ0γ
.
Another application of relation (90) yields a priori estimate
2c Z ∞
0
h2(t)dt ≤
4cE(0)
μ0γ
=
4(1 − γ)
μ0γ
.
Consequently, we have
exp[
2c
μ1 Z ∞
0
h2(t)dt] ≤ exp[
4(1 − γ)
μ0γ
].
35

37. Similarly, putting the above estimates together we obtain the following finite energy solution
©(t) ≤ 2E(0)(1 + 1
°μ0
)exp[ 4(1−°)
μ0° ]. (97)
Substitution of ©(t) = kUk21
(t + 1) into (97) gives the desired nonlinear stability result (89)
since J(U) ≤ 1
´kUk21
whenever L is essentially dissipative. Hence, as a bonus we obtain
the proof of Gronwall’s lemma. The instability result holds by following the proof given in
(16; 17).
For the rest states, asymptotic stability and attraction coincide. However, for more compli-cated
invariant sets than rest states, an attractor may be considered as the stronger notion
of stability than the one given above. The following proposition (9; 10; 37; 38; 14; 15) on
the characterization of attractors and determining of the compactness of an ω-limit set is
required:
Proposition 3.2 Suppose that for Q ⊂ H, Q6= ∅, and that for some s > 0 the set
∪t≥sS(t)Q is relatively compact in H. Then ω-limit set of Q denoted by ω(Q) is nonempty,
compact and invariant.
The proposition is of central importance in the construction of attractors. In order to prove
the major hypothesis that ∪t≥sS(t)Q is relatively compact in H, it is adequate to illustrate
that this set is bounded in a space V compactly imbedded in H by employing the existence
of the following injections
D(L
1
2 ) = V ⊂ D(L) ⊂ ¨ = H ⊂ D(L
−1
2 ).
The density and compactness of the injections will find applications in the sequel when we
prove existence of attractors for the initial-value problems.
a. Rotating Boussinesq equations
The goal of this section is to develop nonlinear stability results for β-plane rotating Boussi-nesq
equations (12 − 14). In order to give nonlinear stability bounds that agree approxi-mately
with established paradigmatic example of initial-value problem (83), the functional
formulation for the evolution equation of β-plane rotating Boussinesq equations is restated
equivalently as follows:
dU
dt + LU + N(U) = 0
U(0) = U0
(98)
where U = (u, v,w, ρ). Concerning the linear operator LU and the nonlinear operator N(U)
we have the following appropriate setup:
N(U) = B(U,U)
36

38. LU = TU + SU − FU
FU = −(0, 0,¦
ρ
Ro
,
Ro
Fr2w)
TU = −µEk
Ro¦△u
1
Ed △ρ ¶ = µAsu
Alρ¶
u · ▽ρ ¶ = µbs(u, u)
B(U,U) = B(U) = µ¦u ·▽u
bl(u, ρ)¶
S = −¦(
1 + yβRo
Ro
)

0 −1 0 0
1 0 0 0
0 0 0 0
0 0 0 0

∀ U ∈ D(L) = D(T).
According to the above setup of the linear operator LU, we obtain the following useful result
Ro k ▽uk2 + 1
Edk▽ρk2 + ( 1
Ro + Ro
Fr2 ) R­ ρwdx (99)
(LU,U) = Ek
which is a slight modification of the Lyapunov functional (19) that was employed in estab-lishing
well-posedness of solution for the problem of β-plane rotating Boussinesq equations
(12 − 14). With the attendant relation (99) we are able to demonstrate a necessary and
sufficient nonlinear stability criterion and hence the elegant power of energy methods is fully
realized. The first effort is then to let R and
I(u, ρ) (100)
be defined as follows
1/2R = sup{Z­
ρwdx}/J(U) = supI(u, ρ)
where the supremum is taken for U ∈ D(L). We shall compile various inequalities about the
equation (98). Substituting
| Z­
ρwdx| ≤ (kuk2 + kρk2)/2
into (100) and utilizing the Poincar´e-Friedrichs inequality yields R ≥ Ga = 80. As it
is known, the nonlinear stability bounds often give conditions that agree with numerical
37

39. approximations. Moreover, the adequacy cited above may be interpreted by noting that the
size of the critical value Ga depends on the conditions of the numerical approximations and
can be considerably improved by implementing more refined and accurate numerical schemes.
In the asymptotic stability analysis we take the number Ga = 80 and we note that we have
reformulated the initial-boundary value problem for β-plane rotating Boussinesq equations in
order to reflect the influence of Galdi et al (16; 17) investigations on energy stability theory.
The above stability criterion number provides a great deal of light on the potentialities of
the energy stability theory and validates this exposition from a benchmark point of view
and offers the opportunity of invoking established results as a guide for nonlinear stability
criteria. It is worth mentioning that the stability criterion number serve as an addition to
available paradigmatic examples of stability criteria such as the Richardson number Ri = 1
4
for stratified shear instability, the Rayleigh number Ra = 1015 for penetrative convection
and the Serrin stability criterion number Se = 33 (27; 26; 30).
The attendant relation (99) may be restated as follows
(LU,U) = (1 + ( 1
Ro + Ro
Fr2 )I(u, ρ))J(U), (101)
with the Lyapunov functional J(U) defined in (86) given by
Ro k▽uk2 + 1
Edk▽ρk2. (102)
J(U) = Ek
It is appropriate at this point to note that if ( 1
Ro + Ro
Fr2 ) < 2R then the following is valid
−(LU,U) ≤ (−1 + ( 1
2RRo + Ro
2RFr2 ))J(U) < 0, (103)
and in this case −L is essentially dissipative. Additionally, the following inequalities hold
(1 − ( 1
L ≤ (1 + ( 1
2RRo + Ro
2RFr2 ))J(U) ≤ kUk2
¼2Ro + Ro
¼2Fr2 ))J(U) (104)
which shows that kUk2
L is topologically equivalent to J(U). The next step is to note that if
( 1
Ro + Ro
Fr2 ) > 2R then there is a sequence {Un} ⊂ ¨ such that
lim
n→∞
I(un, ρn) = 1/2R.
Hence given ǫ < ( 1
2R − ( 1
Ro + Ro
Fr2 )), there exists U such that
−(LU,U) ≥ [−1 + ( 1
Ro + Ro
Fr2 )( 1
2R − ǫ]J(U) > 0, (105)
and in this case −L is essentially non-dissipative. The following result on the spectrum of
−L from (16; 21; 20) will be utilized:
Lemma 3.3 σ(−L) ⊆ (−∞, 0] ∪ ∐, where ∐ is either empty or an at most denumerable
set consisting of isolated, positive eigenvalues λn = λn( 1
Ro + Ro
Fr2 ) with finite multiplicity such
that
∞ > λ1 ≥ λ2 ≥ · · · ≥ λn ≥ . . . ,
clustering only at zero.
38

40. Proof: The validation of the dependence of the positive eigenvalues λn on 1
Ro + Ro
Fr2 is due
to the relation (105). According to a perturbation theorem (16; 21; 20), if TU is closed and
MU = SU−FU is relatively compact with respect to T then σ(−L = −(T +S−F)) and σ(T)
differ by at most a denumerable number of isolated, positive eigenvalues of finite multiplicity
clustering only at zero. Thus it suffices to show that M = S − F is relatively compact with
respect to T since from (36; 16; 21; 20) we have σ(−T) ⊆ (−∞, 0]. By definition, M = S−F
is relatively compact with respect to T if for any sequence {Un} ⊂ ¨,
kUnk + kTUnk ≤ α, uniformly in n, (106)
then there is a subsequence U´n ⊂ {Un} such that MU´n is strongly convergent. Indeed, the
following inequality is valid
kMU´nk2 ≤ [5( 1+y¯Ro
Ro )2 + ( 1
Ro + Ro
Fr2 )2]kU´nk2
≤ [5( 1+y¯Ro
Ro )2 + ( 1
Ro + Ro
Fr2 )2]α2
(107)
and by the courtesy of Ascoli-Arzela theorem {MU´n} contains a uniformly convergent sub-sequence
which is a Cauchy sequence in a complete normed space ¨. Thus MU´n is strongly
convergent which illustrates that M = S − F is relatively compact with respect to T.
It remains to estimate the nonlinearity N(U). Next, we exhibit a series of a priori estimates
on the nonlinearity which will be needed in establishing necessary and sufficient nonlinear
stability of the rotating Boussinesq equations (98). Consideration of the Sobolev inequality
and continuity properties of the nonlinearity B(U) yield the following uniform bounds
kNUk2 ≤ (TU,U)c20
(kTUk2 + kUk2)
≤ c20
kT
1
2Uk(λ1kL
1
2Uk2 + kUk2)
≤ c20
λ−1
2 kTUk(λ1kL
1
2Uk2 + kUk2)
≤ c20λ−1
2 λ
1
2
1 kL
1
2Uk(λ
1
2
1 kLUk2 + kUk2)
≤ c20
cλ−1
2 λ
1
2
1 kL
1
2Uk(kLUk2 + kUk2)
= c20
cλ−1
2 λ
1
2
1 J(U)(kLUk2 + kUk2)
(108)
for the nonlinearity N(U) = B(U) where c = max{λ
1
2
1 , 1}. Consequently, a priori estimates
(108) on the nonlinearity coincide with (91) which means that the nonlinear operator is
suitably dominated by the linear operator LU. Thus, the purpose of proving Lyapunov
39

41. instability aspect has been achieved. Moreover, when the following ( 1
Ro + Ro
Fr2 ) < 2R holds,
we observe the topological equivalence of the Lyapunov functional J(U) with the norm kUk2
L
given in inequality (104). In this case, putting the estimates (52) and(108) together and
invoking the equivalence relation (104) gives the needed a priori estimate for the nonlinearity
kNUk2 ≤ c20
cλ−1
2 λ
1
2
1 J(U)(kLUk2 + kUk2)
≤ c1c20
cλ−1
2 λ
1
2
1 kUk2
L(kLUk2 + λ−1
1 kUk2
L)
(109)
that is necessary for stability with the parameter c1 denoting c1 = (1 − ( 1
2RRo + Ro
2RFr2 ))−1.
According to the sharp a priori estimate (109), aspect of nonlinear stability provided by the
inequality (88) is satisfied. Hence, energetic stability criteria for necessary and sufficient
nonlinear stability of the rest state f β-plane rotating Boussinesq equations (12 − 14) or
equivalently (98) hold with the energy and entropy production provided by the following:
E(t) =
1
2
{kUk2 +
Ek
Ro
k▽uk2 +
1
Ed
k▽ρk2 + 2RZ­
ρwdx}
which is specification of energy and the entropy production in the Sobolev H1-norm if the
criterion Ga ≤ R < ( 1
Ro + Ro
Fr2 ) is valid. In the asymptotic stability analysis we take the num-ber
Ga = 80 and we note that we have reformulated the initial-boundary value problem for
β-plane rotating Boussinesq equations in order to reflect the influence of Galdi et al (16; 17)
investigations on energy stability theory. The above stability criterion number provides a
great deal of light on the potentialities of the energy stability theory and validates this ex-position
from a benchmark point of view and offers the opportunity of invoking established
results as a guide for nonlinear stability criteria. The result serve as a paradigm for con-structing
generalized Lyapunov functionals with interpretations such as energy and entropy
production which are decreasing along solution of the initial-value problem. Additionally,
we observe that in the investigation of nonlinear stability, the equations governing the flow
of an incompressible stratified fluid under the Coriolis force induce damping mechanisms in
the nature of viscosity, diffusion, combined stratification and rotation effects that manifest
themselves in the evolution equation with the existence of Lyapunov functionals which are
equivalent to some fictitious energy or enstrophy or entropy production. Thus, it is reason-able
in an attempt to give geophysical meaning to the above established nonlinear stability
criteria to recall the roles of the nondimensional parameters employed in the construction of
the above generalized Lyapunov functional: Ro is the Rossby number which compares the
inertial term to the Coriolis force; Fr is the Froude number which measures the importance
of stratification; Ek is the Ekman number which measures the relative importance of fric-tional
dissipation; Ed is the nondimensional eddy diffusion coefficient. And the geophysically
relevant ratio Fr2
Ro measures the significant influence of combined rotation and stratification
to the dynamics of the flow. It is crucial to note the techniques employed in developing
the nonlinear stability criteria have been completed without carrying out closed form wave
40

42. solutions for mesoscale and synoptic eddies. The presence of stratification and Reynolds
stress introduces some new considerations that need special attention since in this more
general setting, exact close form solutions need not be easily accessible. Thus, most cases
of oceanographic or meteorological interest such as vortex spin-down, the decay of energy
in the nonlinear stability criteria need not account for the dissipation of energy and enstro-phy
for mesoscale and synoptic wave motions. However, we consider the energetic stability
criteria as a first effort towards elucidating such geophysical phenomena. Furthermore, we
observe that the nonlinear stability criteria may give the possibility of a sharper examination
of the numerical approximation of the solution in the sense of the Lax equivalence theorem:
a finite-difference scheme to a well-posed system of partial differential equation converges to
the solution with the rate of convergence specified by the order of accuracy of the scheme.
Next, we focus our attention to the development of results for the attractor of the β-plane
rotating Boussinesq equations with Reynolds stress (15−17) for both the three-dimensional
and two-dimensional space variables. We show existence of the attractor for the dynamical
system and proceed by customizing the splendid techniques employed in (9; 10; 36; 37; 14; 15)
which have been summarized in the above proposition.
Next, we turn our attention to develop results on the attractor for β-plane rotating Boussi-nesq
equations (12 − 14) in the case of two-dimensional space variables. We show existence
of the attractor for the dynamical system and proceed essentially using the techniques em-ployed
in (9; 10; 37; 38; 14; 15) which have been summarized in the above proposition, but
the calculations that we present in this examination are tedious and involved. Putting the
estimate (28) and (27) together gives the bound specified by
lim
t→∞
sup Z t+1
t
kL
1
2U(s)k2ds ≤
k­k
α
[
λ21
α2 (
Ro
Fr2 +
1
Ro
)4 +
λ1
α
(
Ro
Fr2 +
1
Ro
)2]
Taking the inner product of the in the initial-value problem (18) with LU yields the following
differential inequalities
d
dtkL
1
2Uk2 + λ1kL
1
2Uk2
≤ d
dtkL
1
2Uk2 + kLUk2 ≤ ρ2 22
(110)
such that ρ2 22 = ν0 + ν1ν2 follows successive application of Gronwall’s inequality. And the
41

43. parameters ν0, ν1 and ν2 are given by
ν0 =
3k­k
ν
(
Ro
Fr2 +
1
Ro
)2,
ν1 =
27c
4ν3 +
2
κ3 (a23 + a22)exp(a21),
ν2 =
k­k
ν2 (
Ro
Fr2 +
1
Ro
)4(a13 + a12)2exp(2a11),
a11 =
27c
4ν3 ((
1
ν
)2(
Ro
Fr2 +
1
Ro
)2 +
1
ν3 )(
Ro
Fr2 +
1
Ro
)2 k­k2
ν2 (
Ro
Fr2 +
1
Ro
)4,
a12 = 3k­k2(
Ro
Fr2 +
1
Ro
)2,
a13 = ((
1
ν
)2(
Ro
Fr2 +
1
Ro
)2 +
1
ν3 )(
Ro
Fr2 +
1
Ro
)4k­k,
a21 =
9ck­k3
2κ5 (
1
κ
+ (
Ro
Fr2 +
1
Ro
)7κ2ν2)(
1
ν2 (
Ro
Fr2 +
1
Ro
)2 +
1
ν3 )2(
Ro
Fr2 +
1
Ro
)4,
a22 =
k­k3
2ν4κ
(
Ro
Fr2 +
1
Ro
)4(
1
ν2 (
Ro
Fr2 +
1
Ro
)2 +
1
ν3 )2(
Ro
Fr2 +
1
Ro
)4,
a23 = (
1
κ
+
1
κ2ν2 )(
Ro
Fr2 +
1
Ro
)6k­k.
Consideration of the Gronwall’s inequality in the differential inequality (27) provides a priori
estimate
limt→∞ supkL
1
2U(t)k2 ≤ ½2
22
® = ρ22
(111)
which shows U(t) is uniformly bounded in time in D(L
1
2 ). Putting the estimate (111) and
(110) together we obtain
limt→∞ sup Rt+1
t kLU(s)k2ds ≤ (1 + 1
®)ρ22
. (112)
Concerning the results for the attractor of the two-dimensional space variables initial-value
problem for viscous β-plane ageostrophic equations without Reynolds stress we infer from
the more refined a priori estimates (111) that the attractor for (18) is provided by the ω-limit
set of Q = B2½2 ,
A = ω(Q) = ∩s≥0Cl(∪t≥sS(t)Q).
where B2½2 denotes an open ball in phase space of radius 2ρ2, which depends on the geo-physically
relevant parameters. The closure is taken in the Hilbert space ¨. Utility of the
42

44. group property and continuity of of the solution operators S(t) defined for all time t ∈ IR,
gives the following invariance property of the above established attractor:
S(t)A = A, ∀t ∈ IR.
Consequently, examining the expression for the invariance of the attractor we observe that
the attractor for the two-dimensional space variables initial-value problem for viscous β-
plane rotating Boussinesq equations without Reynolds stress is both positively and negatively
invariant and consists of orbits or trajectories that are defined for all t ∈ IR.
According to a priori estimates (28) and (111), we have the following result which holds for
all time.
Proposition 3.4 If u0 ∈ ¨ and u is the unique solution of (18) for the case of two-dimensional
space variables then the corresponding variational equation (40) has a unique
solution ª satisfying
ª ∈ C([0, T]; ¨) ∩ L2(0, T;D(L
1
2 )) ∀T > 0.
And if u0 ∈ D(L
1
2 ) then
ª ∈ C([0, T];D(L
1
2 )) ∩ L2(0, T;D(L)) ∀T > 0.
The proof of the proposition follows from the stronger a priori energy type estimate (111)
and by repeating the arguments of the previous chapter.
b. Rotating Boussinesq equations with Reynolds stress
The aim of this section is to establish nonlinear stability results for the rest state of ro-tating
Boussinesq equations with Reynolds stress (15 − 17). In order to give nonlinear
stability bounds that agree approximately with established paradigmatic example of initial-value
problem (98), the functional formulation for the evolution equation of β-plane rotating
Boussinesq equations with Reynolds stress is restated equivalently as follows:
dU
dt + LU + N(U) = 0
U(0) = U0
(113)
where U = (u, v,w, ρ). The boundary conditions have been taken to be space-periodic as in
the investigation of well-posedness of solution. Concerning the linear operator LU and the
nonlinear operator N(U) we have the following appropriate setup:
N(U) = B(U,U)
43

45. LU = TU + CU + SU − FU
FU = −(0, 0,
ρ
Ro
,
Ro
Fr2w)
CU = −µEk
Ro △u
1
Ed △ρ¶ = µAlu
Alρ¶
B(U,U) = B(U) = µu ·▽u
u ·▽ρ¶ = µbl(u, u)
bl(u, ρ)¶
TU = −µEk
Ro △6 u
1
Ed △6 ρ¶ = µA6l
u
A6l
ρ¶
S = −(
1 + yβRo
Ro
)

0 −1 0 0
1 0 0 0
0 0 0 0
0 0 0 0

∀ U ∈ D(L) = D(T).
From the above definition of the linear operator LU, we have
(LU,U) = (TU,U) + (CU,U) + ( 1
Ro + Ro
Fr2 ) R­ ρwdx. (114)
which is a modification of the bilinear form (49) that was employed in establishing well-posedness
of solution for the problem of rotating Boussinesq equations with Reynolds stress.
As in the previous section, with the attendant relation (114) we are able to demonstrate a
necessary and sufficient nonlinear stability criterion and hence the elegant power of energy
methods is fully realized. For the sake of completeness, we repeat the analysis in this section
for rotating Boussinesq equations with Reynolds stress. The first effort is then to let R and
I(u, ρ) (115)
be defined as follows
1/2R = sup{Z­
ρwdx}/J(U) = supI(u, ρ)
where the supremum is taken for U ∈ D(L). We shall compile various inequalities about the
equation (113). Substituting
| Z­
ρwdx| ≤ (kuk2 + kρk2)/2
44

46. into (115) and utilizing the Poincar´e-Friedrichs inequality yields R ≥ Ga. The attendant
relation (114) may be restated as follows
(LU,U) = (1 + ( 1
Ro + Ro
Fr2 )I(u, ρ))J(U), (116)
with the generalized energy functional J(U) defined in (86) given by the Lyapunov functional
Ro k▽uk2 + 1
Edk ▽ρk2. (117)
J(U) = (TU,U) + Ek
It is appropriate at this point to note that if ( 1
Ro + Ro
Fr2 ) < 2R then the following is valid
−(LU,U) ≤ (−1 + ( 1
2RRo + Ro
2RFr2 ))J(U) < 0, (118)
and in this case −L is essentially dissipative. Additionally, the following inequalities hold
(1 − ( 1
L ≤ (1 + ( 1
2RRo + Ro
2RFr2 ))J(U) ≤ kUk2
¼2Ro + Ro
¼2Fr2 ))J(U) (119)
which shows that kUk2
L is topologically equivalent to J(U).
The next step is to note that if ( 1
Ro + Ro
Fr2 ) > 2R then there is a sequence {Un} ⊂ ¨ such
that
lim
n→∞
I(un, ρn) = 1/2R.
Hence given ǫ < ( 1
2R − ( 1
Ro + Ro
Fr2 )), there exists U such that
−(LU,U) ≥ [−1 + ( 1
Ro + Ro
Fr2 )( 1
2R − ǫ]J(U) > 0, (120)
and in this case −L is essentially non-dissipative. We require the following lemma on the
spectrum of −L whose validity may be proved by using the technique of proof similar to
those given above:
Lemma 3.5 σ(−L) ⊆ (−∞, 0] ∪ ∐, where ∐ is either empty or an at most denumerable
set consisting of isolated, positive eigenvalues λn = λn( 1
Ro + Ro
Fr2 ) with finite multiplicity such
that
∞ > λ1 ≥ λ2 ≥ · · · ≥ λn ≥ . . . ,
clustering only at zero.
Next, we proceed to illustrate that suitable a priori estimates for the nonlinear operator
N(U)) are available. Putting estimates (52) and (54) together into the inequality (56) yield
45

47. the desired bounds
kNUk2 ≤ (CU,U)c20
(kCUk2 + kUk2)
≤ c20
kC
1
2Uk(λ1kL
1
2Uk2 + kUk2)
≤ c20
λ−1
2 kCUk(λ1kL
1
2Uk2 + kUk2)
≤ c20
λ−1
2 λ
1
2
1 kL
1
2Uk(λ
1
2
1 kLUk2 + kUk2)
≤ c20
cλ−1
2 λ
1
2
1 kL
1
2Uk(kLUk2 + kUk2)
= c20
cλ−1
2 λ
1
2
1 J(U)(kLUk2 + kUk2)
(121)
for the nonlinearity N(U) = B(U) where c = max{λ
1
2
1 , 1}. Consequently, a priori estimates
(108) on the nonlinearity coincide with (91) which means that the nonlinear operator is
suitably dominated by the linear operator LU. Thus, the purpose of proving Lyapunov
instability aspect has been achieved. Moreover, when the following ( 1
Ro + Ro
Fr2 ) < 2R holds,
we observe the topological equivalence of the Lyapunov functional J(U) with the norm kUk2
L
given by the inequality (119). In this case, putting the estimates (52) and(122) together and
invoking the equivalence relation (119) gives the needed a priori estimate for the nonlinearity
kNUk2 ≤ c20
cλ−1
2 λ
1
2
1 J(U)(kLUk2 + kUk2)
≤ c1c20
cλ−1
2 λ
1
2
1 kUk2
L(kLUk2 + λ−1
1 kUk2
L)
(122)
where c1 = (1 − ( 1
2RRo + Ro
2RFr2 ))−1. According to the sharp a priori estimate (122), aspect
of nonlinear stability provided by the inequality (88) is satisfied. Hence, energetic stability
criteria for necessary and sufficient for nonlinear stability of the rest state of β-plane rotating
Boussinesq equations with Reynolds stress (113) or equivalently (15 − 17) hold with the
energy and entropy production provided by the following:
E(t) =
1
2
{kUk2 + J(U) + 2RZ­
ρwdx}
which is specification of energy and the entropy production in the Sobolev H1-norm if the
criterion Ga ≤ R < ( 1
Ro + Ro
Fr2 ) is valid. The result serve as the archetype for constructing
generalized Lyapunov functionals with interpretations such as energy and entropy which
are decreasing along solution of the system. Furthermore, we observe that in the craft of
nonlinear stability, the equations governing the flow of an incompressible stratified fluid under
the Coriolis force induce damping mechanisms in the nature of viscosity, diffusion, combined
46

48. stratification and rotation effects that manifest themselves in the evolution equation with the
existence of Lyapunov functionals which are equivalent to some fictitious energy or enstrophy
or entropy. In the final analysis, it is reasonable in an attempt to give geophysical meaning
to the above established nonlinear stability criteria to recall the roles of the nondimensional
parameters employed in the construction of the above generalized Lyapunov functional: Ro
is the Rossby number which compares the inertial term to the Coriolis force; Fr is the
Froude number which measures the importance of stratification; Ek is the Ekman number
which measures the relative importance of frictional dissipation; Ed is the nondimensional
eddy diffusion coefficient. And the geophysically relevant ratio Fr2
Ro measures the significant
influence of combined rotation and stratification to the dynamics of the flow. It is crucial
to note that the techniques employed in developing the nonlinear stability criteria have
been completed without carrying out closed form wave solutions for mesoscale and synoptic
eddies. The presence of stratification and Reynolds stress introduces some new considerations
that need special attention since in this more general setting, exact close form solutions
need not be easily accessible. Thus, most cases of oceanographic or meteorological interest
such as vortex spin-down, the decay of energy in the nonlinear stability criteria need not
account for the dissipation of energy and enstrophy for mesoscale and synoptic wave motions.
However, we consider the energetic stability criteria as a first effort towards elucidating such
geophysical phenomena. Furthermore, we observe that the nonlinear stability criteria may
give the possibility of a sharper examination of the numerical approximation of the solution
in the sense of the Lax equivalence theorem: a designed finite-difference scheme to an initial-value
problem converges to the solution with the rate of convergence specified by the order
of accuracy of the scheme.
Next, we focus our attention to the development of results for the attractor of the β-plane
rotating Boussinesq equations with Reynolds stress (15−17) for both the three-dimensional
and two-dimensional space variables. We show existence of the attractor for the dynamical
system and proceed by customizing the splendid techniques employed in (9; 10; 37; 38; 14;
15) which have been summarized in the above proposition. Concerning the results for the
attractor of the general three-dimensional space variables initial-value problem for viscous
β-plane rotating Boussinesq equations with Reynolds stress we infer from the more refined
a priori estimates (69) that the attractor for (48) is provided by the ω-limit set of Q = B2½2 ,
A = ω(Q) = ∩s≥0Cl(∪t≥sS(t)Q).
where B2½2 denotes an open ball in phase space of radius 2ρ2, which depends on the geophys-ically
relevant parameters. The closure is taken in the Hilbert space ¨. Utility of the group
property and continuity of the solution operators S(t) defined for all time t ∈ IR, gives the
following invariance property of the above established attractor:
S(t)A = A, ∀t ∈ IR.
Consequently, examining the expression for the invariance of the attractor we observe that
47

49. the attractor for the three-dimensional space variables initial-value problem for viscous β-
plane ageostrophic equations with Reynolds stress is both positively and negatively invariant
and consists of orbits or trajectories that are defined for all t ∈ IR.
This investigation represents the first effort in the theoretical and applied aspects of β-
plane rotating Boussinesq equations. The results are striking and encouraging, and we
point out that based on the manifestation of well-posedness and stability, it is possible to
design numerical algorithms such as finite-difference schemes. In the derivation of of β-plane
rotating Boussinesq equations with Reynolds stress, it was essential to note that the additive
decomposition of the rotating Boussinesq equations quantities into coherent and incoherent
terms and consideration of the averaging operator to obtain the Reynolds stress fields or
wind stress fields resulted in a set of evolution equations that were not closed. In order to
close the set of equations, we adopted the Pedlosky closure protocols.
Advances in understanding the mesoscale and Lagrangian coherent structures including dy-namics
of oceanic eddies, atmospheric vortex blocking, rings of the Agulhas Current, and
the Gulf Stream ring systems has been based on modons which are nonlinear Rossby solitary
wave solutions of quasigeostrophic equations. This research and innovation focuses primarily
on geometric singular perturbation theory and Melnikov techniques to address the problem of
adiabatic chaos and transport for modons to the quasigeostrophic potential vorticity system:
∂q
∂t
+ J(ψ, q) =
Ek
Ro
△q,
∂ρ
∂t
+ J(ψ, ρ) =
1
Ed
△ρ,
q = △ψ −
∂2ψ
∂z2 + βy,
ρ = −
∂ψ
∂z
,
u = −
∂ψ
∂y
,
v =
∂ψ
∂x
.
Singularly perturbed equations have singular solutions, which are unions of solutions to the
limiting quasigeostrophic potential vorticity system. A very general and central question
is what hypotheses on the equations and singular solutions guarantee that the solutions
approximate some solutions for the perturbed quasigeostrophic potential vorticity system.
We present a geometric approach to the problem which gives more refined a priori energy type
estimates on the position of the invariant manifold and its tangent planes as the manifold
48

50. passes close to a normally hyperbolic piece of a slow manifold. The main object of this result
gives geometric detection of adiabatic chaos and transport in singularly perturbed dynamical
systems as depicted in figure 2 using the DsTool package program of Worfolk et al (53).
Acknowledgments.
Dedicated with deep respect to Professor Daya Reddy on the occasion of his 60th Birthday.
It was given to me to speak at UCT’s CERECAM in Cape Town, or simply to attend: it
was always a humbling and unforgettable experience for a young researcher. In grateful
reminiscence of the kind support and attention that Professor Daya Reddy devoted to the
young researcher that I was when I wrote this progressive work, I dedicate this article, with
deep respect, to him. This paper is also dedicated to Professor Hlengani Siweya, who taught
me calculus. I am deeply grateful for his expressions of encouragement and support when
I began my profession as a senior lecturer. His generosity of spirit and fundamental good
nature have inspired me.
REFERENCES
[1] Acheson D.J., Elementary fluid dynamics, Clarendon Press, Oxford, 1990.
[2] Adams R. A., Sobolev spaces, Academic Press, New York, 1975.
[3] Agmon S., Nirenberg L., Lower bounds and uniqueness theorems for solutions of differ-ential
equations in a Hilbert space, Comm. Pure Appl. Math. 20(1967)207-229.
[4] Baehr H., Stephan K., Heat and mass transfer, transl. by Janepark N., Springer-
Verlag,Inc., New York, 1998.
[5] Batchelor G.K., Introduction to fluid dynamics, Cambridge University Press, Cam-bridge,
1967.
[6] Bourgeois A.J., Beale J.T., Validity of the quasigeostrophic model for large-scale flow
in the atmosphere and ocean, SIAM J. Math. Anal.25(4)(1994)1023-1068.
[7] Chow S.-N., Hale J.K., Methods of bifurcation theory, Spriger-Verlag,Inc., New York,
1982.
[8] Constantin A., Nonlinear water waves with applications to wave-current interactions
and tsunamis CBMS regional conference, SIAM, 2011.
49

51. [9] Dafermos C.M., Contraction semigroups and trend to equilibrium in continuum me-chanics,
Springer Lecture Notes in Math. 503(1976).
[10] Dafermos C.M., Slemrod M., Asymptotic behavior of nonlinear semigroups J. Funct.
Anal.13(1973)97-106
[11] Hale J. K., Ordinary differential equations, Wiley-Interscience, New York, 1969.
[12] John F., Partial differential equations, Spriger-Verlag,Inc., New York, 1982.
[13] Guckenheimer J., Holmes P., Nonlinear oscillations, dynamical systems and bifurcations
of vector fields , Spriger-Verlag,Inc., New York, 1983.
[14] Foias C., Manley O., Temam R., Attractors for the Bernard problem: Existence and
physical bounds on their fractal dimension, Nonlinear Anal. 11(1987)939-967.
[15] Foias C., Sell G., Temam R., Inertial manifolds for nonlinear evolutionary equations, J.
Differential Equations 73(1988)309-353.
[16] Galdi G.P., Padula M., A new approach to energy theory in the stability of fluid motion,
Arch. Rational. Mech. Anal. 110(1990)187-286.
[17] Galdi G.P., Rionero S., Weighted energy methods in fluid dynamics and elasticity,
Spriger-Verlag, New York, 1985.
[18] Galdi G.P., An Introduction to the Mathematical Theory of the Navier-Stokes Equa-tions,
Springer-Verlag, New York, 1994.
[19] Gurtin M.E., An introducation to continuum mechanics, Academic Press Inc., San
Diego, 1981.
[20] Ladyzhenskaya O.A., The mathematical theory of viscous incompressible flow, transl.
by Silverman R.A., Chu J., Gordon and Breach Science Publishers, New York, 1969.
[21] Kato T., Perturbation theory for linear operators, Spinger-Verlag Inc., New York, 1966.
[22] Kato T., Ponce G.,Well-posedness of the Euler and Navier-Stokes equations in Lebesgue
spaces Lps
(IR2), Revista Mathematica Iberoamericana2(1986)73-88.
[23] Klainerman S., Majda A., Singular limits of quasilinear hyperbolic systems with large
parameters and the incompressible limit of compressible fluids, Comm. Pure Appl.
Math.43(1981)481-524.
[24] Majda A., Vorticity and the mathematical theory of incompressible flow, Comm. Pure
Appl. Math.39(1986)S187-S220.
50

52. [25] Majda A., Introduction to PDEs and waves for the atmosphere and ocean, Courant
Lecture Notes, Vol. 9 New York University, 2003.
[26] Pedlosky J., Geophysical fluid dynamics, Springer-Verlag, Inc., 2nd Ed. New York,
1987.
[27] Cushman-Roisin B., Introduction to geophysical fluid dynamics, Prentice-Hall, Inc.,
Englewood Cliffs, New Jersey, 1994
[28] Reed M., Simon B., Functional analysis, Academic Press Inc., San Diego, 1980
[29] Royden H.L., Real Analysis, Macmillan Publishing Co., New York, 1988
[30] Serrin J., On the stability of viscous fluid motions, Arch. Rational. Mech. Anal.
110(1959)1-13.
[31] Serrin J., The initial value problem for the Navier-Stokes equations, Univ. of Wisconsin
Press, 110(1963)69-98.
[32] Spiegel E. A., Veronis G., On the Boussinesq approximation for a compressible fluid,
Astrophys. J. 131(1960)442-447.
[33] Titi E.S., On approximate inertial manifolds to the Navier-Stokes equations, J. Math.
Anal. and Appl. 149(1990)540-557.
[34] Titi E.S., C. Cao, Global well-posedness of the three-dimensional stratified primitive
equations with partial vertical mixing turbulence diffusion, Communications in Mathe-matical
Physics 310 (2012), 537-568.
[35] Titi E.S., C. Cao, Global well-posedness of the three-dimensional viscous primitive
equations of large scale ocean and atmosphere dynamics, Annals of Mathematics 166
(2007), 245-267.
[36] Temam R., Navier-Stokes equations:Theory and Numerical Analysis, AMS Chelsea Ed.,
Providence, 2001.
[37] Temam R., Infinite-dimensional dynamical systems in mechanics and physics, Spriger-
Verlag,Inc., New York, 1988.
[38] Temam R., Navier-Stokes equations and nonlinear functional analysis, CBMS regional
conference, SIAM, 1983.
[39] Friedlander S.J.,Introduction to the mathematical theory of geophysical fluid dynamics,
North Holland, New York, 1980
51

53. [40] Friedlander S.J., Lectures on stability and instability of an ideal fluid, IAS/Park City
Math. Ser. 5(1999)229-304.
[41] Jones C.K.R.T., Stability of the travelling wave solution of the Fitzhugh-Nagumo sys-tem,
Trans. Amer. Math. 286(2)(1984)431-469.
[42] Sternberg N.C., Blow up near higher modes of nonlinear wave equations, Trans. Amer.
Math. 296(1)(1986)315-325.
[43] Swaters G.E., Stability conditions and a priori estimates for equivalent-barotropic mod-ons.
Phys. Fluids 29 (1986) 1419-1422.
[44] Lumley J.L.,Berkooz G., Holmes P., Turbulence, coherent structures, dynamical systems
and symmetry, Cambridge University Press, Cambridge 1996.
[45] Lumley J.L., ed., Turbulence at the crossroads, Spinger-Verlag Inc., New York, 1990
[46] Tladi M.S., Adiabatic chaos and transport in mesoscale vortex rings, SIAM Conference,
(DS17) 2017.
[47] Tladi M.S., Well-posedness and long-time dynamics of beta-plane ageostrophic flows,
Ph.D. Thesis, Department of Math. and Applied Math., University of Cape Town, 2004.
[48] Tsinober A., Moffatt H.K., eds., Topological Fluid Mechanics, Cambridge University
Press, Cambridge, 1990.
[49] Wang S., Lions J.L., Temam R., On the equations of the large-scale ocean,
Nonlinearity5(1992)1007-1053.
[50] Wang S., Lions J.L., Temam R., New formulations of the primitive equations of the
atmosphere and applications, Nonlinearity5(1992)237-288.
[51] Wang S., Attractors for the 3D baroclinic quasi-geostrophic equations of large scale
atmosphere, J. Math. Anal. and Appl. 165(1992)266-283.
[52] Wiggins S., Introduction to applied nonlinear dynamical systems and chaos, Spinger-
Verlag Inc., New York, 1990
[53] Worfolk P., Guckenheimer J., Myers M., Wicklin F., Bak A., DsTool: Computer assisted
exploration of dynamical systems, Notices Amer. Math. Soc. 39(1992)303-309.
52

54. List of Figures
1 Lorenz chaotic attractor 54
2 Lagrangian coherent structures 55
53

55. Fig. 1. Lorenz chaotic attractor
54

56. Fig. 2. Lagrangian coherent structures
55