Geography Reference
In-Depth Information
Thus, in log-pressure coordinates the continuity equation becomes simply
∂ (ρ
0
w
∗
)
∂z
∗
∂u
∂x
+
∂v
∂y
+
1
ρ
0
=
0
(8.44)
It is left as a problem for the reader to show that the first law of thermodynamics
(3.6) can be expressed in log-pressure form as
∂
∂t
+
∂
∂z
∗
+
κJ
H
w
∗
N
2
V
·∇
=
(8.45)
where
(R/H)
∂T /∂z
∗
+
κT/H
N
2
≡
is the buoyancy frequency squared (see Section 2.7.3) and κ
R/c
p
. Unlike the
static stability parameter, S
p
, in the isobaric form of the thermodynamic equation
(3.6), the parameter N
2
varies only weakly with height in the troposphere; it can
be assumed to be constant without serious error. This is a major advantage of the
log-pressure formulation.
The quasi-geostrophic potential vorticity equation (6.24) has the same form as
in the isobaric system, but with q defined as
≡
ερ
0
1
ρ
0
∂
∂z
∗
∂ψ
∂z
∗
2
ψ
q
≡∇
+
f
+
(8.46)
f
0
/N
2
where ε
≡
.
8.4.2
Baroclinic Instability: The Rayleigh Theorem
We now examine the stability problem for a continuously stratified atmosphere
on the midlatitude β plane. The linearized form of the quasi-geostrophic potential
vorticity equation (6.24) can be expressed in log-pressure coordinates as
∂
∂t
+
q
+
∂ψ
∂x
=
∂
∂x
∂q
∂y
u
0
(8.47)
where
ερ
0
∂ψ
∂z
∗
1
ρ
0
∂
∂z
∗
q
≡∇
2
ψ
+
(8.48)
and
ερ
0
∂
2
u
∂y
2
∂q
∂y
=
1
ρ
0
∂
∂z
∗
∂u
∂z
∗
β
−
−
(8.49)
