Geography Reference
In-Depth Information
Strong positive meridional curvature can occur at the core of an easterly jet or on
the flanks of a westerly jet. Instability associated with such horizontal curvature is
referred to as
barotropic instability
. The norma
l
baroclinic inst
ab
ility in midlati-
tudes is associated with mean flows in which ∂q/∂y > 0 and ∂u/∂z
∗
> 0 at the
ground. Hence, a mean meridional temperature gradient at the ground is essential
for the existence of such instability. Baroclinic instability can also be excited at
the tropopause due to the rapid decrease of ε with height if there is a sufficiently
strong easterly mean wind shear to cause a local reversal in the mean potential
vorticity gradient.
8.4.3
The Eady Stability Problem
This section analyzes the structures (eigenfunctions) and growth rates (eigenval-
ues) for unstable modes in the simplest possible model that satisfies the necessary
conditions for instability in a continuous atmosphere given in the previous subsec-
tion. For simplicity we make the following assumptions:
(i) Basic state density constant (Boussinesq approximation).
(ii) f
-
plane geometry (β
=
0).
(iii) ∂u/∂z
∗
=
constant.
(iv) Rigid lids at z
∗
=
=
0 and H .
These conditions are only a crude model of the atmosphere, but provide a first
approximation for study of the dependence of vertical structure on horizontal scale
and stability. Despite the zero mean potential vorticity in the domain, the Eady
model satisfies the necessary conditions for instability discussed in the previous
subsection because vertical shear of the basic state mean flow at the upper bound-
ary provides an additional term in (8.61) that is equal and opposite to the lower
boundary integral.
Using the aforementioned approximations, the quasi-geostrophic potential vor-
ticity equation and thermodynamic energy equation are
∂
∂t
+
∇
ε
∂
2
ψ
∂z
∗
∂
∂x
2
ψ
+
u
=
0
(8.62)
2
and
∂
∂t
+
∂ψ
∂z
∗
−
∂ψ
∂x
N
2
f
0
=
∂
∂x
∂u
∂z
∗
+
w
∗
u
0
(8.63)
f
0
/N
2
.
respectively, where again ε
≡
Letting
ψ
x, y, z
∗
,t
=
z
∗
cos ly exp [ik (x
−
ct)]
u
z
∗
=
(8.64)
z
∗
