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depends on the parameters, but we do not indicate this
dependence explicitly. The steady axisymmetric solution
can be found directly using a stream function approach.
See
Lewis and Nagata
[2003] for details. Analytical solu-
tions cannot be found, and therefore numerical approx-
imations are used. We discuss the numerical methods
briefly below. The solution corresponds to a toric convec-
tion cell in which the fluid rises at the outer cylinder and
falls at the inner while being deflected to the right as it
passes from the outer to the inner cylinder in the top of
the annulus and from the inner to the outer cylinder in
the bottom.
It is convenient to compute the linearized stability of
the steady axisymmetric solution from the perturbation
equations, which can be obtained by substituting
may be written in the form of a generalized linear eigen-
value problem
λ
A
m
U
m
=
L
m
U
m
,
(2.12)
where
⎛
⎞
u
m
˜
˜
U
m
=
⎝
⎠
w
m
T
m
×
and
A
m
and
L
m
are 3
3 matrices of linear opera-
tors. If
m
= 0, a stream function method can be used
in exactly the same manner as in the calculation of the
axisymmetric solution. Again numerical approximations
are implemented.
If, for a given set of parameter values, all eigenvalues
λ
have negative real parts, then all (small) perturbations
from the axisymmetric solution will decay, see (2.11), and
thus, the steady axisymmetric solution is asymptotically
stable. If any eigenvalue
λ
has a positive real part, then a
perturbation in the direction of the corresponding eigen-
function will grow, and the steady axisymmetric solution
is unstable. The steady solution is called neutrally stable
if there are a finite number of eigenvalues with zero real
part while all other eigenvalues have negative real part. In
a two-dimensional parameter space, there is a curve along
which the steady axisymmetric solution is neutrally stable.
On one side of this curve the steady solution is linearly sta-
ble, while on the other side the solution is linearly unstable.
Thus, we expect that a transition will occur as parameters
are varied from the stable side to the unstable side.
Procedurally, because the eigenvalue problem takes the
form (2.12), a neutral stability curve is constructed for
each azimuthal wave number
m
, where a neutral stability
curve is a curve in the space of parameters along which at
least one eigenvalue corresponding to a given wave num-
ber
m
has zero real part while all other eigenvalues for
that wave number have negative real part. Thus, to one
side of the neutral stability curve for wave number
m
,
the axisymmetric solution is stable to all perturbations of
azimuthal wave number
m
, while to the other side it is
unstable to some perturbation with that wave number and
thus is unstable. Consequently, the portions of the neutral
stability curves, for which the eigenvalues corresponding
to all other wave numbers have negative real part, form a
segment of the transition curve.
In Figure 2.1 is a transition curve for the case where
water is the working fluid; specifically, the experimental
results of
Fein
[1973] are compared to the linear stability
analysis of
Lewis and Nagata
[2003]. It can be seen that
the linear stability analysis is successful in predicting the
location of the primary transition. The approximations
are not as accurate at large Rossby number; this may be a
consequence of discretization error, which becomes larger
at higher differential heating.
u
=
u
(
0
)
+
u
,
p
=
p
(
0
)
+
p
,
T
=
T
(
0
)
+
T
ˆ
ˆ
(2.9)
into equations (2.4)-(2.6) and forming equations for the
hatted variables, i.e., the perturbations. The trivial solu-
tion
p
=0,
T
= 0 then satisfies the perturbation
equations and corresponds to the steady axisymmetric
solution. In what follows, we drop the hats.
It is also convenient to represent the perturbation equa-
tions in the following (abstract) form:
ˆ
ˆ
u
=
0
,
U
=
L
U
+
N
(U)
,
(2.10)
where
U
is a vector of the functions
u
and
T
,
L
is a (par-
tial differential) linear operator such that
L
U
is the linear
part of the equation,
N
(U)
is the nonlinear part, and
the velocity components of
U
are divergence free. Writ-
ten as such, some important theoretical properties have
been established that enable the analysis described below
[
Henry
, 1981].
Assuming that the perturbations are sufficiently small
that the nonlinear part may be neglected, the linear stabil-
ity of the steady axisymmetric solution can be determined
from the spectrum of the linearization
L
. In particular, we
assume that the perturbations may be written as
u
=
u
(r
,
ϕ
,
z
,
t)
=
e
λt
u
m
(r
,
z)e
imϕ
,
˜
(2.11)
with
m
an integer and likewise for
T
and
p
, where the
azimuthal dependence of these functions can be assumed
due to periodicity in the azimuthal variable. Consequently,
we obtain a linear eigenvalue problem for the eigenval-
ues
λ
and the eigenfunctions
u
m
(r
,
z)
,
T
m
(r
,
z)
e
imϕ
for
each azimuthal wave number
m
. The eigenfunctions will
determine the form of the bifurcating solutions, and thus
we anticipate that they will be azimuthal waves of some
integer azimuthal wave number
m
.
If
m
˜
= 0, it is possible to eliminate the pressure and
azimuthal velocity. The resulting three equations in the
three remaining unknowns
w
m
(r
,
z)
,and
T
m
(r
,
z)
˜
˜
u
m
(r
,
z)
,
