Geography Reference
In-Depth Information
As in the two-layer model, boundary conditions are required at lower and upper
boundary pressure surfaces. Assuming that the flow is adiabatic and that the vertical
motion w
∗
vanishes at these boundaries, the linearized form of the thermodynamic
energy equation (8.45) valid at horizontal boundary surfaces is simply
∂
∂t
+
∂ψ
∂z
∗
∂ψ
∂x
∂
∂x
∂u
∂z
∗
u
−
=
0
(8.50)
The sidewall boundary conditions are
∂ψ
/∂x
hence,ψ
=
=
0,
0
at
y
=±
L
(8.51)
We now assume that the perturbation consists of a single zonal Fourier compo-
nent propagating in the
x
direction:
Re
y, z
exp
ik
x
ct
ψ
(x, y, z, t)
=
−
(8.52)
=
r
+
where (y, z)
i
i
is a complex amplitude, k is the zonal wave number,
=
c
r
+
and c
ic
i
is a complex phase speed. Note that (8.52) can alternatively be
expressed as
ψ
x, y, z, t
=
e
kc
i
t
r
cos k
x
c
r
t
−
i
sin k
x
c
r
t
−
−
Thus, the relative magnitudes of
r
and
i
determine the phase of the wave for
any y, z
∗
.
Substituting from (8.52) into (8.47) and (8.50) yields
c)
∂
2
∂y
2
ερ
0
1
ρ
0
∂
∂z
∗
∂
∂z
∗
∂q
∂y
k
2
(u
−
−
+
+
=
0
(8.53)
and
c)
∂
∂u
∂z
∗
0 tz
∗
=
(u
−
∂z
∗
−
=
0
(8.54)
If the upper boundary is taken to be a rigid lid at a finite height, as is sometimes
done in theoretical studies, the condition (8.54) is appropriate at that boundary as
well. Alternatively the upper boundary condition can be specified by requiring
to remain finite as z
∗
→∞
.
Equation (8.53), together with its boundary conditions, constitutes a linear
boundary value problem for (y, z
∗
). It is generally not simple to obtain solu-
tions to (8.53) for realistic mean zonal wind profiles. Nevertheless, we can obtain
some useful information on stability properties simply by analyzing the energetics
of the system.
