Geoscience Reference
In-Depth Information
1. 5
a
3
|)
½
(
a
1
/|
0
0
0
0
10
50
Time
t
Time
t
Figure 1.7.
Solution to equation (1.11) and (1.12) showing the
approach to a steady equilibrium.
Figure 1.6.
A typical solution to equations (1.9) and (1.10)
showing sustained amplitude oscillations.
dA
dτ
=
a
1
A
A
3
,
−|
a
3
|
(1.11)
R
{
}
τ
=
A(τ )F(y
,
z)
exp
k(x
−
ct)
,
(1.7)
2
,
V
=
a
4
|
A
|
(1.12)
resulting typically in an asymptotic equilibration toward
a steady amplitude
A
=
(a
1
/
where
y
is the meridional coordinate and
z
is in the vertical
in the presence of a zonal flow of the form
)
1
/
2
(see Figure 1.7).
These models, of course, represent a considerable over-
simplification of the real equilibration processes in fully
developed baroclinic instability. Indeed,
Boville
[1981]
noted that
Pedlosky
's [1970] approach led to significant
inaccuracies in predicting the amplitude behavior of baro-
clinic waves and failed to observe the predicted amplitude
oscillations close to minimum critical shear. Subsequent
work [
Pedlosky
, 1982a, 1982b;
Warn and Gauthier
, 1989;
Esler and Willcocks
, 2012] suggests that the dynamical
equilibration is more typically dominated by the behav-
ior of a nonlinear critical layer that develops within the
flow, leading to the wrapping up and eventual homoge-
nization of potential vorticity. The equilibrated state may
then result in convergence toward a steady wave state via
a series of damped amplitude oscillations. The analyti-
cal solutions of
Warn and Gauthier
[1989] even produced
periodic amplitude oscillations under certain conditions,
resembling an amplitude vacillation, but in which poten-
tial vorticity is reversibly mixed and unmixed. This is
only strictly applicable under conditions of weak fric-
tion close to marginal instability. The recent work of
Willcocks and Esler
[2012] also suggests that the mode
of equilibration via a Landau equation, predicted in the
models of
Pedlosky
[1971] and
Romea
[1977], probably
applies mainly to the dissipatively destabilized instabil-
ity that occurs for shears less than the critical shear
in the inviscid problem in the presence of Ekman fric-
tion [
Holopainen
, 1961;
Boville
, 1981]. The full applica-
bility of any of these models, however, still remains to
be verified in detail in laboratory experiments or fully
nonlinear numerical simulations in continuously strati-
fied flows.
|
a
3
|
U
=
U(y
,
z)
+
V(τ )G(y
,
z)
.
(1.8)
Here
A
and
V
are respectively the slowly varying
amplitudes of the wave and the correction to the zonal
flow due to the nonlinear self-interaction of the wave,
whose spatial structure is represented by
Gy
,
z)
.The
resulting evolution equations for
A
and
V
depend upon
the relative magnitude of viscous dissipation:
(i) Weak Dissipation (
O(
1
/
2
)
)
Examples include the Eady or two-layer Phillips prob-
lems with no Ekman layers [e.g.,
Pedlosky
, 1987]. It
can be shown that the problem then reduces to coupled
ordinary differential equations (ODEs) of the form
1
/
2
/
Ro
E
d
2
A
dτ
2
=
a
1
A
A
3
,
−|
a
3
|
(1.9)
dV
dτ
dA
dτ
,
=
a
4
(1.10)
typically resulting in a sustained amplitude modula-
tion, or “vacillation,” associated with the exchange of
potential energy between wave and zonal flow (e.g., see
Figure 1.6).
(ii) Stronger Dissipation (
1
/
2
/
Ro =
O(
1
)
)
Examples include the Eady and Phillips problems with
Ekman damping. It can be shown that the amplitude
equations reduce to the well-known Landau equation
[e.g., see
Pedlosky
, 1971;
Romea
, 1977;
Drazin
, 1978;
Hocking
, 1978]
E
