TWO–DIMENSIONAL INFINITE PRANDTL NUMBER
CONVECTION: STRUCTURE OF BIFURCATED SOLUTIONS
JUNGHO PARK
Abstract. This paper examines the bifurcation and structure of the bifurcated solutions of the two-dimensional infinite Prandtl number convection
problem. The existence of a bifurcation from the trivial solution to an attractor ΣR was proved by Park [14]. We prove in this paper that the bifurcated attractor ΣR consists of only one cycle of steady state solutions and
that it is homeomorphic to S 1 . By thoroughly investigating the structure and
transitions the solutions of the infinite Prandtl number convection problem
in physical space, we confirm that the bifurcated solutions are indeed structurally stable. In turn, this will corroborate and justify the suggested results
with the physical findings about the presence of the roll structure. This bifurcation analysis is based on a new notion of bifurcation, called attractor
bifurcation, and structural stability is derived using a new geometric theory
of incompressible flows. Both theories were developed by Ma and Wang; see
[11, 12].
1. Introduction
Rayleigh-Bénard convection, that is, a buoyancy-driven convection in a fluid
layer heated from below and cooled from above, is one of the prime examples
of bifurcating high-dimensional systems. It has long been a subject of intense
theoretical and experimental study and has been applied to many different areas
of study such as meteorology, geophysics, and astrophysics. In this paper, we
consider a two-dimensional (2D) convection problem. The governing equations are
the following Boussinesq equations:
·
¸
1 ∂u
(1.1)
+ (u · ∇)u + ∇p − ∆u − RT k = 0,
Pr ∂t
∂T
(1.2)
+ (u · ∇)T − ∆T = 0,
∂t
(1.3)
∇ · u = 0,
where u = (u1 , u2 ) is the velocity field, p is the pressure function, T is the temperature field and k = (0, 1) is the unit vector in x2 -direction.
In the Boussinesq equations, we have two important numbers: the Rayleigh
number,
gα(T2 − T1 )h3
,
R=
νκ
1991 Mathematics Subject Classification. 35Q, 35B, 37L, 76E.
Key words and phrases. Rayleigh-Bénard Convection, bifurcation, structure of solutions, structural stability, infinite Prandtl number.
The work was supported in part by the Office of Naval Research, and by the National Science
Foundation.
1
2
J. PARK
which measures the ratio of overall buoyancy force to the damping coefficients, and
the Prandtl number,
ν
Pr = ,
κ
which measures the relative importance of kinematic viscosity over thermal diffusivity. Here, ν and κ are the kinematic viscosity and thermal diffusive coefficients
respectively, α is the thermal expansion coefficient of the fluid, g is the gravitational
constant, h is the distance between two plates confining the fluid and T2 − T1 is the
temperature difference between the bottom and top plates.
Due to the fact that mathematical information is very limited, the complicated
equations have been simplified. For example, the infinite Prandtl number limit of
the Boussinesq equations has been used as the standard model for the convection
of earth’s mantle, where it is argued that Pr could be of the order 1024 , as well
as for many gasses under high pressure. A Broader rationale for investigating the
infinite Prandtl number convection is based on the observation of both the linear
and weakly nonlinear theories; that fluids with Pr > 1 convect in a similar fashion.
Moreover, the infinite Prandtl number model of convection can also be justified as
the limit of the Boussinesq approximation to the Rayleigh-Bénard convection as
the Prandtl number approaches infinity [21, 22].
In the limit of the infinite Prandtl number, the inertial terms in the momentum
equation can be dropped, thus we are left with a linear dependence of the velocity
field on temperature:
(1.4)
(1.5)
(1.6)
∇p − ∆u − RT k = 0,
∂T
+ (u · ∇)T − ∆T = 0,
∂t
∇ · u = 0.
Since the velocity field is linearly dependent on the temperature field in the infinite
Prandtl number convection, the velocity field has a behavior that is much more
regular than the finite Prandtl number model. Thus this is a distinct advantage of
studying this model. In particular, by investigating the structure of the attractor
generated only by T (much more convenient), it is possible to reconstruct the structure of the attractor in terms of (u, T ), which has the same topological structure
as is obtained in terms of only T .
Extensive mathematical studies have been conducted for the Rayleigh-Bénard
convection since Rayleigh’s work. In particular, for the Rayleigh-Bénard convection
with the finite Prandtl numbers, readers are referred to Chandrasekhar [1], to
Drazin and Reid [4] for linear theories; to Foias, Manley and Temam [6] for the
existence and physical bounds of attractors; to Rabinowitz [16] for the existence
of rectangular solutions; and to Ma and Wang [10, 11] for attractor bifurcation.
For the case regarding the infinite Prandtl number, the readers are referred to
Constantin and Doering [2, 3] for the upper bounds of the minimal conduction
value; to Schnaubelt and Busse [19] for two-dimensional convection rolls; to Keken
[8] and to Yanagisawa and Tamagishi [23] for mantle and spherical shell convection
studies, respectively; to X. Wang [21, 22] for the justification of the infinite Prandtl
number convection as the infinite Prandtl number limit of the Boussinesq equations;
and to Park [14] for the existence of attractor bifurcation.
T. Ma and S. Wang [9, 11] recently developed a new notion of bifurcation theory
called attractor bifurcation. The existence of bifurcation from the trivial solution
INFINITE PRANDTL NUMBER CONVECTION
3
to an attractor ΣR of the infinite Prandtl number convection was achieved by the
author [14] using the attractor bifurcation theories. In this paper, the following two
problems will be addressed:
(1) the classification of the structure of ΣR ,
(2) the transition of the bifurcated solutions in physical space.
For the first result, we use an approximation formula for the center manifold function given in [11]. Of course, the key ingredient of the analysis is to derive the
reduction equation for our problem. For the second result, it is essential to study
the topological structure of the divergence-free vector fields for 2D-incompressible
flows governed by the Navier-Stokes equations or Euler equations. Such a result
would involve a specific connection between solutions of the equations and flow
structure in physical space, i.e., this area of research links the kinematics to the
dynamics of fluid flows.
As a result, we have a bifurcation from the trivial state to an attractor ΣR
as R crosses the critical Rayleigh number Rc , the first eigenvalue of the eigenvalue
problem of linearized equations [14]. Next, we prove that ΣR is homeomorphic to S 1
and that ΣR consists of steady state solutions for Rc < R < Rc + ε for some ε > 0.
These results can be obtained by analyzing eigenvectors in terms of the temperature
field T and later on, the velocity field u(T ) associated with the temperature T can
be reconstructed by the eigenvectors of T . Thanks to the geometric theory of 2D
incompressible flows [12], the structure and its transitions between the convection
states in physical space is analyzed. Through this, we prove that the associated
velocity field is structurally stable and show what the asymptotic structure looks
like. This leads us, in particular, to a rigorous justification of the roll structure.
We take free–free boundary conditions on the top and bottom plates to study
the infinite Prandtl number convection. However, the free–free boundary condition
(which was first considered by Rayleigh) can be realized only under very artificial
conditions, such as by having the liquid layer floating on top of a somewhat heavier
liquid. From the point of view of realizability in the laboratory for precise quantitative experiments, the Dirichlet boundary condition (rigid-rigid) is, of course, of the
greatest interest. Nevertheless, the case is of theoretical interest because it allows
for an explicit solution.
The paper is organized as follows. After introducing the infinite Prandtl number convection and its mathematical settings in Section 2, we describe the center
manifold approximation and bifurcation theories for the infinite Prandtl number
convection in Section 3, followed by a topological structure of the bifurcated solutions. Then we study the structural stability and the asymptotic structure of the
bifurcated solutions in Section 4. Concluding remarks are given in Section 5.
2. Bénard Problem and its Mathematical Setting
2.1. Mathematical setting. In this paper we deal with the Boussinesq equations
on the non-dimensional domain Ω = R × (0, 1) ⊂ R2 . The basic linear profiles of
(1.4)–(1.6) are steady state solutions given by
u = 0,
T = 1 − x2 ,
p = p0 + R(x2 −
x22
)k.
2
4
J. PARK
Let q be the difference between p and the steady state solutions, and let θ be the
x2
difference between T and the steady states solutions, i.e, p = p0 + R(x2 − 22 )k + q
and T = (1 − x2 ) + θ. Then the equations for the perturbation of these trivial
solutions are derived as
∇q − ∆u − Rθk = 0,
∂θ
+ (u · ∇)θ − ∆θ − u2 = 0,
∂t
∇ · u = 0.
Let T =
(2.1)
(2.2)
(2.3)
√ √
R θ and p = q. Then we have
√
∇p − ∆u − RT k = 0,
√
∂T
+ (u · ∇)T − ∆T − Ru2 = 0,
∂t
∇ · u = 0.
We impose the periodic boundary condition with spatial periods L in the horizontal
direction x1 :
(2.4)
(u, T )(x, t) = (u, T )(x1 + kL, x2 , t), for any k ∈ Z.
The top and bottom plates are assumed to be free-free:
∂u1
= 0 at x2 = 0, 1.
∂x2
For an initial value problem we also provide an initial value as
(2.5)
(2.6)
T = 0,
u2 = 0,
(u, T )|t=0 = (u0 , T0 ).
Since (2.1) together with (2.3) is a Stokes equation, we have the solution u = u(T )
under the above boundary condition. Replacing u and its second component u2 in
(2.2) by the solution which depends on T , we obtain the following equation
√
∂T
(2.7)
+ (u(T ) · ∇)T − ∆T − Ru2 (T ) = 0,
∂t
where u(T ) and u2 (T ) satisfy (2.1).
Adjusting boundary and initial conditions yields
½
T =0
at x2 = 0, 1,
(2.8)
T (x, t) = T (x1 + kL, x2 , t), for x ∈ Ω,
(2.9)
T |t=0 = T0 .
We shall recall here the functional setting of the equation (2.7) with the boundary
and initial conditions (2.8) and (2.9). Let
H = L2 (Ω)
and
H1 = H01 (Ω) ∩ H 2 (Ω),
where H01 (Ω) is the space of functions in H 1 (Ω), which vanish at x2 = 0, 1 and are
periodic in x1 -direction.
Thanks to the existence result, we can define a semi-group
S(t) : T0 → T (t),
which enjoys the semi-group properties.
INFINITE PRANDTL NUMBER CONVECTION
5
In the bifurcation theory for the infinite Prandtl number convection, the critical
Rayleigh number, denoted by Rc , plays a key role. We consider the linearized
equations of (2.7) with the boundary conditions (2.8),
√
(2.10)
−∆T − Ru2 (T ) = 0,
where u2 (T ) satisfies (2.1).
Since this eigenvalue problem for the Rayleigh number R is symmetric, all eigenvalues Rk with multiplicities mk of (2.10) with (2.8) are real numbers, and
(2.11)
0 < R1 < · · · < Rk < Rk+1 < · · ·
in the sense that


−∆u + ∇p − λT k = 0,
−∆T − λu2 = 0,


∇ · u = 0,
is equivalent to
Z h
Z h
i
i
∇u · ∇e
u + ∇T · ∇Te dx = λ
Tu
f2 + u2 Te dx,
√
Ω
Ω
where λ = R.
The first eigenvalue R1 depends on periods L. It is known that there is a period
Lc which minimizes R1 (L). We denote the critical Rayleigh number by Rc . Let
the multiplicity of Rc be m1 = m (m ≥ 1) and the first eigenvectors T1 , · · · , Tm of
(2.10) be orthonormal, such that
Z
hTi , Tj iH =
Ti · Tj dx = δij .
Ω
Then E0 , the first eigenspace of (2.10) with (2.8) is
(m
)
X
(2.12)
E0 =
αk Tk | αk ∈ R, 1 ≤ k ≤ m .
k=1
Let H and H1 be defined as above. Define Lλ = −A + Bλ : H1 → H and
G : H1 → H by


 A(T ) = −∆T,
Bλ (T ) = λu2 (T ),
(2.13)


G(T ) = −(u(T ) · ∇)T,
√
where λ = R. Then we have an operator equation, which is equivalent to the
Boussinesq equation (2.7):
(2.14)
dT
= Lλ T + G(T ).
dt
3. Center Manifold and Attractor Bifurcation Theory
3.1. Center manifold approximation. The purpose of this section is to recall
some results of the center manifold theory under general settings, i.e., general function spaces and general functional equations. It is a powerful tool for the reduction
method and for the dynamic bifurcation of abstract nonlinear evolution equations
developed in [11].
6
J. PARK
Let H and H1 be two arbitrary Hilbert spaces, and H1 ,→ H be a dense and
compact inclusion. Consider the following nonlinear evolution equations
dv
(3.1)
= Lλ v + G(v, λ),
dt
(3.2)
v(0) = v0 ,
where v : H × [0, ∞) → H is the unknown function, λ ∈ R is the system parameter, and Lλ : H1 → H are the parameterized linear completely continuous fields
continuously depending on λ ∈ R, which satisfy


 Lλ = −A + Bλ a sectorial operator,
A : H1 → H a linear homeomorphism,
(3.3)

 B : H → H the parameterized linear compact operators.
λ
1
We can see that Lλ generates an analytic semi-group {e−tLλ }t≥0 and then we
can define fractional power operators Lα
λ for any 0 ≤ α ≤ 1 with the domain
Hα = D(Lα
)
such
that
H
⊂
H
if
α
<
α1 , and H0 = H.
α1
α2
2
λ
We now assume that the nonlinear terms G(·, λ) : Hα → H for some 0 ≤ α < 1
are a family of parameterized C r bounded operators (r ≥ 1) continuously depending
on the parameter λ ∈ R, such that
(3.4)
G(v, λ) = o(kvkHα ),
∀ λ ∈ R.
For the linear operator A, we assume that there exists a real eigenvalue sequence
{ρk } ⊂ R and an eigenvector sequence {ek } ⊂ H1 such that


 Aek = ρk ek ,
0 < ρ 1 ≤ ρ2 ≤ · · · ,
(3.5)

 ρ → ∞ (k → ∞),
k
where {ek } is an orthogonal basis of H.
For the compact operator Bλ : H1 → H, we also assume that there is a constant
0 < θ < 1 such that
(3.6)
Bλ : Hθ −→ H bounded, ∀ λ ∈ R.
We know that the operator L = −A + Bλ satisfying (3.5) and (3.6) is a sectorial
operator. It generates an analytic semigroup {Sλ (t)}t≥0 . Then the solution of (3.1)
and (3.2) can be expressed as
v(t, v0 ) = Sλ (t)v0 ,
t ≥ 0.
We assume that the spaces H1 and H can be decomposed into
(
H1 = E1λ ⊕ E2λ , dim E1λ < ∞, near λ0 ∈ R1 ,
(3.7)
e1λ ⊕ E
e2λ , E
e1λ = E1λ , E
e2λ = closure of E2λ in H,
H=E
where E1λ and E2λ are two invariant subspaces of Lλ , i.e., Lλ can be decomposed
into Lλ = Lλ1 ⊕ Lλ2 such that for any λ near λ0 ,
( λ
e1λ ,
L1 = Lλ |E1λ : E1λ → E
(3.8)
e2λ ,
Lλ2 = Lλ |E2λ : E2λ → E
where all eigenvalues of Lλ2 possess negative real parts, and all eigenvalues of Lλ1
possess nonnegative real parts at λ = λ0 .
INFINITE PRANDTL NUMBER CONVECTION
7
Thus, for λ near λ0 , (3.1) can be rewritten as

dx


= Lλ1 x + G1 (x, y, λ),
dt
(3.9)

 dy = Lλ y + G (x, y, λ),
2
2
dt
ei
where u = x + y ∈ H1 , x ∈ E1λ , y ∈ E2λ , Gi (x, y, λ) = Pi G(u, λ), and Pi : H → E
are canonical projections. Furthermore, we let
E2λ (α) = E2λ ∩ Hα ,
with α given by (3.4).
Theorem 3.1. (Center Manifold Theorem) Assume (3.4)–(3.8). Then there exists
a neighborhood of λ0 given by |λ − λ0 | < δ for some δ > 0, a neighborhood Bλ ⊂ E1λ
of x = 0, and a C 1 function Φ(·, λ) : Bλ → E2λ (α), depending continuously on λ,
such that
(1) Φ(0, λ) = 0, Dx Φ(0, λ) = 0.
(2) The set
ª
©
Mλ = (x, y) ∈ H1 | x ∈ Bλ , y = Φ(x, λ) ∈ E2λ (α) ,
called the center manifold, is locally invariant for (3.1), i.e., for any v0 ∈
Mλ ,
vλ (t, v0 ) ∈ Mλ ,
∀ 0 ≤ t < tv0 ,
for some tv0 > 0, where vλ (t, v0 ) is the solution of (3.1).
(3) If (xλ (t), yλ (t)) is a solution of (3.9), then there are βλ > 0 and kλ > 0
with kλ depending on (xλ (0), yλ (0)), such that
kyλ (t) − Φ(xλ (t), λ)kH ≤ kλ e−βλ t .
By the help of the Center Manifold Theorem, we obtain the following bifurcation
equation reduced to the finite dimensional system
dx
= Lλ1 x + G1 (x + Φ(x), λ),
dt
for x ∈ Bλ ⊂ E1λ .
Now we recall an approximation of the center manifold function which will be
used in the proof of Theorem 3.8; see [11] for details. Let the nonlinear operator G
be given by
(3.10)
G(v, λ) = Gk (v, λ) + o(|v|k ),
where Gk (v, λ) is a k-multilinear operator(k ≥ 2).
Theorem 3.2. (T.Ma and S.Wang, [11]) Under the conditions in Theorem 3.1
and (3.10), we have the following center manifold function approximation:
Φ(v, λ) = (−Lλ2 )−1 P2 Gk (v, λ) + O(|Reβ(λ)| · ||v||k ) + o(||v||k ),
where β(λ) = (β1 (λ), · · · , βm (λ)) are the eigenvalues of Lλ1 .
8
J. PARK
3.2. Attractor bifurcation theory for the infinite Prandtl number convection. In this subsection, we shall recall the attractor bifurcation theory of the
infinite Prandtl number convection [14] and provide a sufficient condition which
implies that the bifurcated attractor of the system (3.1) is homeomorphic to S 1
[11].
Definition 3.3. A set Σ ⊂ H is called an invariant set of (3.1) if S(t)Σ = Σ
for any t ≥ 0. An invariant set Σ ⊂ H of (3.1) is said to be an attractor if Σ is
compact and there exists a neighborhood U ⊂ H of Σ such that for any ϕ ∈ U we
have
(3.11)
lim distH (v(t, ϕ), Σ) = 0.
t→∞
The largest open set U satisfying (3.11) is called the basin of attraction of Σ.
Definition 3.4.
(1) We say that the equation (3.1) bifurcates from (v, λ) =
(0, λ0 ) to an invariant set Ωλ , if there exists a sequence of invariant sets
{Ωλn } of (3.1), 0 ∈
/ Ωλn , such that
limn→∞ λn = λ0 ,
limn→∞ maxx∈Ωλn |x| = 0.
(2) If the invariant set Ωλ is an attractor of (3.1), then the bifurcation is called
attractor bifurcation.
(3) If Ωλ is an attractor and is homotopy equivalent to an m-dimensional sphere
S m , then the bifurcation is called S m -attractor bifurcation.
Let the eigenvalues (counting multiplicity) of Lλ be given by
βk (λ) ∈ C (k ≥ 1).
Suppose that


< 0
Reβi (λ) = 0


>0
(3.12)
(3.13)
if λ < λ0 ,
if λ = λ0
if λ > λ0 ,
Reβj (λ0 ) < 0
(1 ≤ i ≤ m),
(m + 1 ≤ j).
Let the eigenspace of Lλ at λ0 be
E0 =
m
[
©
ª
v ∈ H1 | (Lλ0 − βi (λ0 ))k v = 0, k = 1, 2, · · · .
i=1
The following dynamic bifurcation theorem for (2.7) was proved based on the principle of exchange of stabilities and asymptotic stability of trivial solution [14].
Theorem 3.5. [14] For the problem (2.7) with (2.8), we have the following assertions:
(1) If R ≤ Rc , the steady state T = 0 is a globally asymptotically stable equilibrium point of the equation.
(2) The equation (2.7) bifurcates from (T, R) = (0, Rc ) to an attractor ΣR for
R > Rc , with m − 1 ≤ dim ΣR ≤ m, which is connected when m > 0.
INFINITE PRANDTL NUMBER CONVECTION
9
(3) For any T ∈ ΣR , the associated velocity field u = u(T ), which is achieved
from a given T , can be expressed as
!
Ãm
m
X
X
αk ek ,
αk ek + o
(3.14)
u=
k=1
k=1
where ek are eigenvectors of (2.7) corresponding to each Tk .
(4) The attractor ΣR has the homotopy type of an (m − 1)-dimensional sphere
S m−1 provided ΣR is a finite simplicial complex.
(5) For any bounded open set U ⊂ L2 (Ω) with 0 ∈ U there is an ε > 0 such
that as Rc < R < Rc + ε, the attractor ΣR attracts U/Γ in L2 (Ω), where Γ
is the stable manifold of T = 0 with co-dimension m.
Remark 3.6. We can easily see that Theorem 3.5 is true for the 2D-Bénard problem as well as for the 3D-Bénard problem.
So far, we have introduced some crucial definitions and theorems which have
established the existence of bifurcation. We now introduce another theorem related
to the structure of the bifurcated solutions.
Let v be a two-dimensional C r (r ≥ 1) vector field given by
(3.15)
vλ (x) = λx − G(x, λ),
for x ∈ R2 . Here, G(x, λ) is defined as in (3.10) and satisfies an inequality
(3.16)
C1 |x|k+1 ≤< Gk (x, λ), x >H ≤ C2 |x|k+1 ,
for some constants 0 < C1 < C2 and k = 2m + 1, m ≥ 1.
Theorem 3.7. (T. Ma and S. Wang, [11]) Under the condition (3.16), the vector
field (3.15) bifurcates from (x, λ) = (0, λ0 ) to an attractor Σλ for λ > λ0 , which is
homeomorphic to S 1 . Moreover, one and only one of the following is true.
(1) Σλ is a periodic orbit,
(2) Σλ consists of infinitely many singular points, or
(3) Σλ contains at most 2(k + 1) = 4(m + 1) singular points, and has 4N +
n(N + n ≥ 1) singular points, 2N of which are saddle points, 2N of which
are stable node points (possibly degenerate), and n of which have index zero.
3.3. Structure of Bifurcated Solutions for the infinite Prandtl number
convection. In this subsection, we shall state and prove the first results which are
related to the classification of the structure of ΣR .
Theorem 3.8. For the problem (2.7) with (2.8), we have the following assertions:
the equation bifurcates from the trivial solution (T, R) = (0, Rc ) to an attractor ΣR
for R > Rc such that
(1) ΣR consists of exactly one cycle of steady state solutions.
(2) ΣR is homeomorphic to S 1 .
Proof. Step 1.We divide the proof into several steps. In the first step, we shall
consider the eigenvalue problem of the linearized equation of (2.7) and shall find
the eigenvectors and the critical Rayleigh number Rc .
Let’s consider the eigenvalue problem of the linear equation,
(3.17)
Lλ T = β(λ)T.
10
J. PARK
It is equivalent to
where λ =
√
−∆T − λu2 (T ) + β(λ)T = 0,
R. Because u(T ) satisfies
−∆u(T ) + ∇p − λT k = 0,
we can put two equations together as follows:


 − ∆u(T ) + ∇p − λT k = 0,
− ∆T − λu2 (T ) + β(λ)T = 0,
(3.18)

 ∇ · u = 0.
For the free-free boundary conditions, the following separation of variables is
appropriate: for T ∈ H1 ,
(u, T ) = (u1 , u2 , T )
µ
¶
2kπ
2kπ
2kπ
2kπ
= − sin
x1 H 0 (x2 ),
cos
x1 H(x2 ), cos
x1 Θ(x2 ) ,
L
L
L
L
or,
(u, T ) = (u1 , u2 , T )
µ
¶
2kπ
2kπ
2kπ
2kπ
= cos
x1 H 0 (x2 ),
sin
x1 H(x2 ), sin
x1 Θ(x2 ) .
L
L
L
L
Then, from (3.18) we can derive a system of ODEs
( 2
(D − a2k )2 H = λak Θ,
(3.19)
(D2 − a2k )Θ = −λak H + β(λ)Θ,
supplemented with the boundary conditions
H = D2 H = Θ = 0
(3.20)
at x2 = 0, 1,
2
d
2kπ
where D2 = dx
2 and ak =
L .
2
It is clear that for each k ≥ 0, the solutions of (3.19) with (3.20) are given by
Hj (x2 ) = sin jπx2 ,
[(jπ)2 + a2k ]3/2
,
a
qk
Θkj (x2 ) = (jπ)2 + a2k sin jπx2 ,
λkj =
and
λak
− [(jπ)2 + a2k ]
(jπ)2 + a2k
βk,2j−1 (λ) = βk,2j (λ) = p
for j = 1, 2, · · · .
From the above arguments, it can be seen that the following sequences are the
set of eigenvectors of (3.18):
(3.21)
(uk,2j−1 , Tk,2j−1 )
¶
µ
2kπ
2kπ
2kπ
2kπ
2
x1 cos jπx2 ,
cos
x1 sin jπx2 , ckj cos
x1 sin jπx2 ,
= −(jπ) sin
L
L
L
L
INFINITE PRANDTL NUMBER CONVECTION
11
(3.22)
(uk,2j , Tk,2j )
µ
¶
2kπ
2kπ
2kπ
2kπ
2
= (jπ) cos
x1 cos jπx2 ,
sin
x1 sin jπx2 , ckj sin
x1 sin jπx2 ,
L
L
L
L
q
2
where k ≥ 0, j ≥ 1 and ckj = (jπ)2 + ( 2kπ
L ) . Here, L = Lc leads λ11 (L) to be
the critical Rayleigh number.
Therefore, we get


< 0 if λ < λ11 ,
β11 (λ) = β12 (λ) = 0 if λ = λ11 ,


> 0 if λ > λ11 ,
βkj (λ11 ) < 0
if
(k, j) 6= (1, 1),
and the first eigenvectors of (3.18) corresponding to β11 are
(u11 , T11 ) and
(u12 , T12 ).
Step 2.We shall show that the bifurcated attractor of (2.7) and (2.8) contains
a singularity cycle. In order to show that, we note that each steady state solution
of (2.7) generates one S 1 in H. We can easily see that the above claim is true from
the following arguments: since the solutions of (2.7) are translation invariant,
T (x, t) → T (x1 + θ, x2 , t),
for any θ ∈ R,
and if T0 is a steady state solution of (2.7), the set
Γ = {T0 (x1 + θ, x2 )|θ ∈ R}
1
represents S in H1 .
Applying the classical Krasnoselskii theorem, we can see that the rest of Step 2
holds true. To this end, let’s consider the odd function spaces
H odd = {T ∈ H | T is an odd function in the x1 direction},
H1odd = H1 ∩ H odd .
Then H odd and H1odd are invariant for Lλ + G : H1odd → H odd and a set
{Tk,2j−1 |k ≥ 0, j ≥ 1}
forms a basis of H1odd .
Since the first eigenvalue β11 (λ) is simple in H odd , Lλ + G bifurcates from the
trivial solution to a steady state solution in H1odd , i.e., (2.7) bifurcates from (T, R) =
(0, Rc ) to a steady state solution. From the first argument in the Step 2, we can
conclude that the bifurcated attractor contains a singularity cycle of (2.7).
Step 3. In the third step, we shall investigate the topological structure of ΣR .
Let E1λ = E0 = span{T11 , T12 } and E2λ = E0⊥ . For T ∈ H, we have the Furier
expansion
X
T =
ykj Tkj .
k≥0,j≥1
12
J. PARK
Then the reduction equations of (2.14) are as follows:
(
dy11
1
dt = β11 (λ)y11 + <T11 ,T11 >H < G2 (T, T ), T11 >H ,
(3.23)
dy12
1
dt = β11 (λ)y12 + <T12 ,T12 >H < G2 (T, T ), T12 >H ,
where G2 is the bilinear operator of G such that
G2 (T, T ) = −(u(T ) · ∇)T,
Z
< G2 (T1 , T2 ), T3 >H = − (u(T1 ) · ∇T2 )T3 dx.
Ω
Let Φ be the center manifold function. Then
T = y11 T11 + y12 T12 + Φ,
and (3.23) becomes
(3.24)
 dy11

dt = β11 (λ)y11 +



dy12

= β11 (λ)y12 +


 dt
1
<T11 ,T11 >H
1
<T12 ,T12 >H
< G2 ( y11 T11 + y12 T12 + Φ(y),
y11 T11 + y12 T12 + Φ(y)), T11 >H ,
< G2 ( y11 T11 + y12 T12 + Φ(y),
y11 T11 + y12 T12 + Φ(y)), T12 >H ,
where y = y11 T11 + y12 T12 .
Note that for T11 and T12 ,
< G2 (T1` , T1m ), T1n >H = 0
for
`, m, n = 1 or 2,
< G2 (Φ, T1i ), T1i >H = 0
for i = 1, 2
< G2 (T1` , T1m ), T1n >H = − < G2 (T1` , T1n ), T1m >H .
Therefore,
< G2 (y11 T11 + y12 T12 + Φ(y), y11 T11 + y12 T12 + Φ(y)), T11 >H
(3.25)
= − < G2 (T11 , T11 ), Φ(y) >H y11 + < G2 (T12 , Φ(y)), T11 >H y12
+ < G2 (Φ(y), T12 ), T11 >H y12 + < G2 (Φ(y), Φ(y)), T11 >H ,
and
< G2 (y11 T11 + y12 T12 + Φ(y), y11 T11 + y12 T12 + Φ(y)), T12 >H
(3.26)
= − < G2 (T12 , T12 ), Φ(y) >H y12 + < G2 (T11 , Φ(y)), T12 >H y11
+ < G2 (Φ(y), T11 ), T12 >H y11 + < G2 (Φ(y), Φ(y)), T12 >H .
It is easy to check the following inner products:
½
6= 0
if k = 0, 2, and j = 3,
< G2 (T11 , T11 ), Tkj >H
=0
otherwise,
½
< G2 (T12 , T12 ), Tkj >H
½
< G2 (T12 , Tkj ), T11 >H
½
< G2 (T11 , Tkj ), T12 >H
6= 0
if
k = 0, 2, and j = 3,
=0
otherwise,
6= 0
if
=0
otherwise,
6= 0
if
=0
otherwise,
k = 0, 2, and j = 4,
k = 0, 2, and j = 4,
INFINITE PRANDTL NUMBER CONVECTION
½
< G2 (Tkj , T12 ), T11 >H
6= 0
if
=0
otherwise.
13
k = 0, 2, and j = 4,
By Theorem 3.2, it is seen that the center manifold function Φ, near λ = λ11 ,
can be expressed as
X
(3.27)
Φ(y) =
Φkj (y)Tkj ,
(k,j)6=(1,1),(1,2)
where
Φkj (y) = −
akj
pq
1
βkj (λ)
X
akj
pq y1p y1q ,
1≤p,q≤2
=< G2 (T1p , T1q ), Tkj >H .
Due to the above calculations, (3.25) and (3.26) can now be written as
< G2 (y11 T11 + y12 T12 + Φ(y), y11 T11 + y12 T12 + Φ(y)), T11 >H
= − [< G2 (T11 , T11 ), T03 >H Φ03 + < G2 (T11 , T11 ), T23 >H Φ23 ] y11
(3.28)
+ [< G2 (T12 , T04 ), T11 >H Φ04 + < G2 (T12 , T24 ), T11 >H Φ24 ] y12
+ [< G2 (T04 , T12 ), T11 >H Φ04 + < G2 (T24 , T12 ), T11 >H Φ24 ] y12
+ < G2 (Φ(y), Φ(y)), T11 >H ,
and
< G2 (y11 T11 + y12 T12 + Φ(y), y11 T11 + y12 T12 + Φ(y)), T12 >H
= − [< G2 (T11 , T04 ), T12 >H Φ04 + < G2 (T11 , T24 ), T12 >H Φ24 ] y11
(3.29)
+ [< G2 (T04 , T11 ), T12 >H Φ04 + < G2 (T24 , T11 ), T12 >H Φ24 ] y11
− [< G2 (T12 , T12 ), T03 >H Φ03 + < G2 (T12 , T12 ), T23 >H Φ23 ] y12
+ < G2 (Φ(y), Φ(y)), T12 >H .
By (3.27), near λ = λ11 we have
(3.30)
Φ03 = −

s
1  3
π
β03 (λ)
µ
π2 +
2π
L
¶2

2
2 
(y11
+ y12
) ,
Φ04 = Φ23 = Φ24 = 0.
Replacing (3.28) and (3.29) by (3.30), we obtain
< G2 (y11 T11 + y12 T12 + Φ(y), y11 T11 + y12 T12 + Φ(y)), T11 >H
"
µ ¶2 #
£ 2
¤
1
2π
2
=
π6 π2 +
y11 y11
+ y12
+ < G2 (Φ, Φ), T11 >H ,
β03 (λ)
L
and
< G2 (y11 T11 + y12 T12 + Φ(y), y11 T11 + y12 T12 + Φ(y)), T12 >H
"
µ ¶2 #
£ 2
¤
1
2π
6
2
2
=
π π +
y12 y11
+ y12
+ < G2 (Φ, Φ), T12 >H .
β03 (λ)
L
Therefore, the reduction equations (3.24) are transformed into
(
dy11
α
2
2
dt = β11 (λ)y11 + <T11 ,T11 >H y11 (y11 + y12 )+ < G2 (Φ, Φ), T11 >H ,
(3.31)
dy12
α
2
2
dt = β11 (λ)y12 + <T12 ,T12 >H y12 (y11 + y12 )+ < G2 (Φ, Φ), T12 >H ,
14
where
J. PARK
"
µ ¶2 #
1
2π
α=
π6 π2 +
< 0 for
β03 (λ)
L
λ11 < λ < λ11 + ε.
We know that the center manifold function involves only higher order terms
Φ(y) = O(|y11 |2 , |y12 |2 ),
which implies
< G2 (Φ, Φ), T1i >H = o(|y11 |3 , |y12 |3 ),
Since the following equations,
(
dy11
dt = β11 (λ)y11 +
(3.32)
dy12
dt = β11 (λ)y12 +
α
2
<T11 ,T11 >H y11 (y11
α
2
<T12 ,T12 >H y12 (y11
(i = 1, 2).
2
+ y12
) + o(|y11 |3 , |y12 |3 ),
2
+ y12
) + o(|y11 |3 , |y12 |3 ),
satisfy conditions of Theorem 3.7, we can conclude that ΣR is homeomorphic toS 1 .
This completes the proof.
¤
Remark 3.9. It is known that under free-free boundary condition, the critical
4
Rayleigh number can be calculated analytically as 27π
4 = 657.5, which is independent of the Prandtl number of the fluid; see [1, 4] for details.
4. Structural Stability for the Infinite Prandtl Number Convection
In this section, we shall study the structural stability and asymptotic structure
of the bifurcated solutions of the infinite Prandtl number convection in physical
space. To do so, it is crucial to study the structure and transitions of velocity
fields for two dimensional incompressible fluid flows governed by the Navier-Stokes
equations or the Euler equations. First, we shall recall some basic results about the
structural stability; see [12] for details.
Let C r (Ω, R2 ) be the space of all C r (r ≥ 1) vector fields on Ω, which are periodic
in x1 direction with periods L, and let
Dr (Ω, R2 ) = {v ∈ C r (Ω, R2 ) | ∇ · v = 0, v2 = 0 at x2 = 0, 1},
∂v1
B1r (Ω, R2 ) = {v ∈ Dr (Ω, R2 ) | v2 =
= 0 at x2 = 0, 1},
∂x2
Z
B2r (Ω, R2 ) = {v ∈ B1r (Ω, R2 ) |
udx = 0}.
Ω
Definition 4.1. Two vector fields u, v ∈ C r (Ω, R2 ) are called topologically equivalent if there exists a homeomorphism of ϕ : Ω → Ω, which takes the orbits of u to
that of v and preserves their orientations.
Definition 4.2. Let X = Dr (Ω, R2 ). A vector field v ∈ X is called structurally
stable in X if there exists a neighborhood U ⊂ X of v such that for any u ∈ U , u
and v are topologically equivalent.
Lemma 4.3. (Connection Lemma, [12]) Let u, v ∈ Dr (Ω, R2 ) with v sufficiently
small, and L ⊂ Ω be an extended orbit of u starting at p. Then an extended orbit
γ of u + v starting at p passes through a point q ∈ L if and only if
Z
v × d` = 0,
L[p,q]
INFINITE PRANDTL NUMBER CONVECTION
15
where L[p, q] is the curve segment on L from p and q.
Theorem 4.4. [12] Let u ∈ B1r (Ω, R2 ) (r ≥ 2)(resp. u ∈ B2r (Ω, R2 )). Then u is
structurally stable in B1r (Ω, R2 ) (resp. B2r (Ω, R2 )) if and only if
(1) u is regular.
(2) All interior saddle points of u are self-connected, i.e., each interior saddle
point is connected only to itself.
(3) Each boundary saddle point of u is connected to boundary saddle points on
the same connected component of ∂Ω (resp. each boundary saddle point
of u is connected to boundary saddle points not necessarily on the same
connected component).
Since ((α, 0), 0) is a solution of (2.1)–(2.3) for any constant α under the free-free
boundary condition, we have to take velocity field u from the following function
spaces:
Z
e = {u ∈ L2 (Ω)2 | ∇ · u = 0,
H
udx = 0, u1 is periodic in x1 direction
Ω
and u2 = 0 at x2 = 0, 1},
e 1 = {u ∈ H
e ∩ H 2 (Ω)2 | u satisfies (2.4) and (2.5)}.
H
Now, we introduce one simple but strongly necessary lemma, which links the structure of the solutions in terms of (u, T ) and that of the solutions in terms of only T .
This occurs only in the infinite Prandtl number systems, not in the finite Prandtl
number system.
Lemma 4.5. Let T ∈ ΣR , T = Σk≥0,j≥1 ykj Tkj and u = u(T ) be the associated
velocity field which is obtained from T . Then u can be expressed as
u=
2
X
α1k u1k + o(
k=1
2
X
α1k u1k ),
k=1
where u1k are eigenvectors of (2.7) corresponding to T1k . Moreover,
α1k = τ y1k , for some τ ∈ R.
Proof. We infer from Theorem 3.5 that for any T ∈ ΣR , the associated velocity
field u = (u1 , u2 ) can be expressed as
u=
2
X
α1k u1k + o(
k=1
2
X
α1k u1k ) for some α1k ,
k=1
where u1k are eigenvectors of (2.7) corresponding to T1k . Thus it suffices to prove
that α1k are proportional to y1k .
From the first equation in (3.18), we have
∆(
∂u1
∂u2
∂T
−
)−λ
= 0.
∂x2
∂x1
∂x1
16
J. PARK
With the sequences of the eigenvectors (3.21) and (3.22), we can rewrite the above
equality as follows:
·µ
¶
¸
2π
2π
2π
2π
2π
π 3 ( )2 + ( )4 − π 5 − ( )2 π 2 α11 − λy11 c11 ( ) sin
x1 sin πx2 = 0,
L
L
L
L
L
·µ
¶
¸
2π
2π
2π
2π
2π
π 3 ( )2 + ( )4 − π 5 − ( )2 π 2 α12 − λy12 c11 ( ) cos
x1 sin πx2 = 0,
L
L
L
L
L
where c11 is as defined in (3.21) and (3.22). Therefore, we arrive at
α1k = τ y1k for some τ ∈ R (k = 1, 2).
¤
With the help of Lemma 4.5, we can finally state and prove structural stability
theorem.
Theorem 4.6. For any T0 ∈ H\Γ, there exists a time t0 ≥ 0 such that for any
t ≥ t0 , the associated vector field u(t, T ) = u(t, T, Ψ0 ) is structurally stable and
is topologically equivalent to the structure as shown in Figure 4.1, where Ψ =
e × H)\Γ
e
(u(t, Ψ0 ), T (t, Ψ0 )) is the solution of (2.1)–(2.5), Ψ0 = (u0 (T0 ), T0 ) ∈ (H
e
and Γ(resp. Γ) is the stable manifold of T = 0(resp. of (u, T ) = 0) with coe × H).
dimension 2 in H(resp. in H
Figure 4.1. The roll structure of solutions of Benard convection
with infinite Prandtl number.
Proof. By [6], we know that the solution (u(t, ψ0 ), T (t, Ψ0 )) is analytic in time
and all the H m norms of u(t, ψ0 ), T (t, Ψ0 ) remain uniformly bounded in time for
t ≥ δ > 0, m ≥ 0.
INFINITE PRANDTL NUMBER CONVECTION
17
From Lemma 4.5 and equation (3.32) at the end of the proof of Theorem 3.7, we
have
(
2π
2
u1 = α11 (−π 2 sin 2π
L x1 cos πx2 ) + α12 (π cos L x1 cos πx2 ) + v1 (α11 , α12 , β11 ),
2π
2π
2π
u2 = α11 ( 2π
L cos L x1 sin πx2 ) + α12 ( L sin L x1 sin πx2 ) + v2 (α11 , α12 , β11 ).
where α11 = τ y11 , α12 = τ y12 and τ is the constant obtained in the proof of
Lemma 4.5.
Now, note that the above expansions were done in terms of the eigenvectors.
Thus, they must hold not only on H, but also on higher dimensional Sobolev
spaces H m . By simple modifications, the following velocity fields are obtained:
(
u1 = π 2 r sin 2π
L (θ − x1 ) cos πx2 + v1 (α11 , α12 , β11 ),
(4.1)
2π
u2 = L r cos 2π
L (θ + x1 ) sin πx2 + v2 (y11 , y12 , β11 ),
for some 0 ≤ θ ≤ 2π. Here,
β11
< T11 , T11 >H for λ > λ11 ,
α
vi (α11 , α12 , β11 ) = o(r) (i = 1, 2).
2
2
r2 = α11
+ α12
= τ2
Since
R
Ω
udx = 0, we have
Z
(4.2)
vi dx = 0 (i = 1, 2)
Ω
for the velocity field u given above. By the connection Lemma, it follows that
2π
2π
2π
x1 cos πx2 ,
r cos
x1 sin πx2 )
L
L
L
is topologically equivalent to u = (u1 , u2 ). Now it suffices to prove that u0 is
structurally stable.
For any singular point
(4.3)
u0 = (−π 2 r sin
(x1 , x2 ) = (
kL 1
(2k + 1)L 1
, ) or (
, ), k = 1, 2, · · · ,
2 2
4
2
of u0 in Ω,
det Du0 (x1 , x2 ) 6= 0.
Therefore, u0 is structurally stable and we can easily show that the topological
structure is as shown in Figure 4.1. It provides us the structural stability of u and
its structure as well.
This completes the proof.
¤
5. Concluding Remarks
Throughout this paper, we studied the bifurcation and structure of the bifurcated
solutions of the two-dimensional infinite Prandtl number convection problem and
have obtained two main results, which are summarized as follows:
(i) The problem bifurcates from the basic steady state to an attractor ΣR as
the system parameter crosses the critical Rayleigh number Rc , and ΣR is
homeomorphic to S 1 . Moreover, ΣR consists of one cycle of steady state
solutions.
18
J. PARK
(ii) The structure and the transitions of the solutions are investigated and it
is proved that the associated velocity fields are structurally stable. It is
worthwhile to mention that structural stability is different from dynamical
stability such as the asymptotic stability. A solution of partial differential
equations is called asymptotically stable if any solution with perturbed initial data converges to the given solution. Structural stability, used in this
paper, refers to the stability of vector fields in physical space, and the idea
of structural stability can be extended to vector fields on higher or infinite dimensional manifolds. A vector field is said to be structurally stable
if, for any small perturbation, the perturbed vector field is topologically
equivalent to the original vector field. More precisely, we are led to study
the set of certain vector fields at once by defining a topology over it and
hence an equivalence relation between arbitrarily close vector fields (connected for instance to the existence of some homeomorphism identifying
different orbits). At this point, structural stability is obviously a global requirement, implying that the perturbation of the whole vector field leaves
its qualitative properties preserved no matter what the initial conditions
are. There are examples of steady states which are structurally stable but
dynamically unstable and vice versa. The above structural stability result
leads us to the justification of the roll structure for the problem as physical
experiments have suggested.
From a physical point of view, the two-dimensional convection problem can be
considered an idealized model for many physical phenomena, including the Walker
circulation over the tropics [17], which has the same topological structure as the cells
given in Figure 4.1 in Theorem 4.6. Simulations of the Rayleigh-Bénard convection
with the infinite Prandtl number and high Rayleigh numbers in spherical shell
geometry have often been carried out to understand the thermal structure of the
mantle and the evolution of the Earth [23]. They also tell us that the convection cells
have almost uniform size for the lower Rayleigh number, but become irregular for
the higher Rayleigh number. Near the critical Rayleigh number Rc , the convection
patterns are the same as described in this paper. Moreover, the growth rate of
the instability rate vanishes with Pr = ∞; while the two-dimensional rolls grow at
a finite rate from arbitrary initial conditions, where the rolls are effectively stable
[19].
In this paper we studied the infinite Prandtl number convection under the
f ree − f ree boundary condition on the top and bottom plates. We might be able
to combine different boundary conditions for further studies, such as rigid − rigid,
rigid−f ree and f ree−rigid, which are normally used in different physical settings.
However that requires more advanced techniques since we do not have explicit forms
of the eigenfunctions. Thus we need to develop new approaches supplemented with
numerical studies. Although we can also study the three-dimensional convection
problem, which would also be more technically involved and requires more complicated calculations, we may expect similar results to the two-dimensional case.
References
[1] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publications, Inc.
1981.
[2] C. Doering and P. Constantin, Infinite prandtl number convection, J. Statistical Physics,
vol.94, (1999), pp. 159–172.
INFINITE PRANDTL NUMBER CONVECTION
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
19
, On upper bounds for infinite Prandtl number convection with or without rotation, J.
Mathematical Physics, vol.42, No.2, (2001), pp. 784–795.
P. Drazin and W. Reid, Hydrodynamic Stability, Cambridge University Press, second ed.,
2004.
R. Ecke, F. Zhong, and E. Knobloch, Hopf-Bifurcation with broken reflection symmetry
in rotating Rayleigh-Bénard convection, Europhys. Lett. vo.19 (1992), pp. 177–182.
C. Foias, O. Manley, and R. Temam, Attractors for the Bénard problem: existence and
physical bounds on their fractal dimension, Nonlinear Anal., 11 (1987), pp. 939–967.
E. Kaminski and C. Jaupart, Laminar starting plumes in high-Prandtl number fluids, J.
Fluid Mech. vol.478 pp. 287–298.
P. E. Van Keken, Evolution of starting mantle plumes: a comparison between numerical
and laboratory models, Earth. Planet. Sci. Lett., vol.148, (1997), pp. 1–11.
T. Ma and S. Wang, Dynamic bifurcation of nonlinear evolution equations, Chinese Annals
of Mathematics-Ser. B, vol.26, No.2, (2005), pp. 185–206.
, Dynamic bifurcation and stability in the Rayleigh-Bénard convection, Comm. math.
sci. vol.2, No.2, (2004), pp. 159–183.
, Bifurcation Theory and Applications, World Scientific, 2005.
, Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics, vol.
119 of Mathematical Surveys and Monographs, Americal Mathematical Society, Providence,
RI, 2005.
, Structure of bifurcated soultions of 2D Rayleigh-Benard convection, preprint, (2005).
J. Park, Dynamic bifurcation theory of Rayleigh-Bénard convection with infinite prandtl
number, Disc. Cont. Dyn. Sys.B, Vol.6, No.3, (2005), pp 591–604.
A. Pazy, Semigroups of linear operators and applications to partial differential equations,
vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.
P.H. Rabinowitz, Existence and nonuniqueness of rectangular solutions of the Bénard problem, Arch. Rational Mech. Anal., 29 (1968), pp. 32-57.
M. Salby, Fundamentals of atmospheric physics, Academic Press, 1996.
S. Scheel and N. Seehafer, Bifurcation to oscillations in three-dimensional RayleighBénard convection, Phys. Rev. E, vol.56, No.5 (1997), pp. 5511–5516.
M. Schnaubelt and F. Busse, On the stability of two-dimensional convection rolls in an
infinite Prandtl number fluid with stress-free boundaries, J. Appl. Math. Phys., vol.40, (1989),
pp. 153–162.
R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, vo. 68 of
Applied Mathematical Sciences, Springer Verlag, New York, second ed., 1997.
X. Wang, Infinite Prandtl number limit of Rayleigh-Bénard convection, Comm. Pure Appl.
Math., vol.57, (2004), pp. 1265-1282.
, A note on long time behavior of solutions to the Boussinesq system at large Prandtl
number, Contemporary Mathematics (2005), pp. 315-323.
T. Yanagisawa and Y. Yamagishi, Rayleigh-Bénard convection in spherical shell with infinite Prandtl number at high Rayleigh number, J. Earth Simulator, vol.4, (2005), pp. 11–17.
Department of Mathematics & Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN 47405
E-mail address: [email protected]