Geoscience Reference
In-Depth Information
consideration of lateral shears. This was considered by
Bell and White
[1988], who examined the stability of an
idealized internal zonal jet flow in a straight, rectangular
channel of the form
1.3.3. Steady Waves and Equilibration: Weakly
Nonlinear Theory
As waves grow in strength from an initial zonal flow,
they typically equilibrate to either a steady or periodically
varying amplitude (“amplitude vacillation”)or even to a
weakly chaotic flow. The linear models of baroclinic insta-
bility cannot account for equilibration and vacillation,
and so we must consider the effects of nonlinearity in the
interaction between the growing wave and the basic zonal
flow. Weakly nonlinear theory was developed in the late
1960s as a means of introducing nonlinearity into linear
instability problems while keeping the mathematics ana-
lytically tractable. The basic assumption of this approach
is that the flow in which the wave grows is only weakly
supercritical, and so only a small range of wave numbers
is unstable and grows relatively slowly. More detailed dis-
cussions can be found [e.g.,
Drazin
, 1978;
Hocking
, 1978;
Ghil and Childress
, 1987;
Pedlosky
, 1987].
Consider a zonal flow under conditions just inside the
stability threshold with weak supercriticality
. Because
the stability boundary is then asymptotically quadratic in
zonal wave number
k
about the wave number of the first
mode to go unstable
k
c
(see Figure 1.5), a small range of
k
U
=
1
2
(
1
−
a
s
+
a
s
−
sin
πy)
sin
πz
,
(1.5)
where
a
s
is a constant that determines the degree of
horizontal barotropic shear in the otherwise baroclinic
jet. If full account is taken of lateral shear in such an
internal jet (by varying
a
s
), however, the critical Burger
number for the onset of waves is found to vary by a factor
of
O
(10). The precisely applicable value is likely depen-
dent upon subtle details of the shape of the zonal flow
and the imposed lateral boundary conditions, since the
true boundary conditions at the sides of the geostrophic
interior ought really to take proper account of the com-
plex viscous boundary layer structures (e.g., Stewartson
layers), although impermeable, free-slip boundaries have
typically been employed (for mathematical convenience)
in most theoretical studies to date.
Recent exceptions to this include the two-layer studies
by
Mundt et al.
[1995a, 1995b] and the analysis of the
full thermal annulus problem by
Lewis and Nagata
[2004].
Mundt et al.
[1995a] examined the linear (and nonlinear)
stability of a quasi-geostrophic, two-layer jet in a recti-
linear channel in which internal viscosity was included in
deriving the zonally symmetric basic state. This led to the
formation of viscous (Stewartson) boundary layers adja-
cent to the sidewalls of the channel, within which strong
zonal shear developed as the flow adjusted to the nonslip
condition at each boundary. This was then shown to mod-
ify the critical Froude number for instability by a factor
O
(1) for the gravest modes. Similar results were obtained
by
Mundt et al.
[1995b] in cylindrical geometry, for which
improved agreement with experimental measurements
was shown compared with stability calculations assum-
ing a free-slip outer boundary. The most sophisticated
approach applied so far for the thermal annulus configu-
ration was by
Lewis and Nagata
[2004], who used numeri-
cal continuation techniques to solve for the linear stability
boundary (as a function of
and
1
/
2
is destabilized. In a periodic
x
domain where
k
is discretized, this may permit only one unstable wave
number. We introduce a “slow” time scale
τ
defined by
τ
=
1
/
2
t
,
∼
1,
(1.6)
(in the sense that it advances more slowly than “normal
time”
t
) and solve for normal modes of the following
wavelike form in the zonal (
x
) direction:
a
) of an axisymmetric
baroclinic zonal jet in cylindrical geometry using the full
Navier-Stokes equations for a viscous, Boussinesq fluid.
The results indicated good agreement with the location of
both the upper and lower symmetric transitions as found
in laboratory experiments. They also indicated the influ-
ence of centrifugal buoyancy in modifying the stability
boundary at the lower symmetric transition. These calcu-
lations all serve to demonstrate the quantitative success
of linear stability theory in accounting quantitatively for
the onset of the principal mode of baroclinic instability in
both two-layer and continuously stratified rotating tank
experiments as a supercritical global bifurcation.
T
½
Δ
Δ
a
c
0
k
c
k
Figure 1.5.
Schematic stability diagram showing the assumed
(quadratic) form of the critical curve
a(k)
as a function of wave
number
k
in the vicinity of the first unstable mode with wave
number
k
c
.
