Geography Reference
In-Depth Information
(hPa)
ω
0
= 0
0
p = 0
ψ
1
1
p = 250
ω
2
2
p = 500
ψ
3
3
p = 750
ω
4
4
p = 1000
Fig. 8.2
Arrangement of variables in the vertical for the two-level baroclinic model.
vorticity equation for the midlatitude β plane is applied at the 250- and 750-hPa
levels, designated by 1 and 3 in Fig. 8.2, whereas the thermodynamic energy equa-
tion is applied at the 500-hPa level, designated by 2 in Fig. 8.2.
Before writing out the specific equations of the two-layer model, it is convenient
to define a
geostrophic streamfunction
, ψ
/f
0
. Then the geostrophic wind (6.7)
and the geostrophic vorticity (6.15) can be expressed respectively as
≡
2
ψ
V
ψ
=
k
×
∇
ψ,
ζ
g
=∇
(8.1)
The quasi-geostrophic vorticity equation (6.19) and the hydrostatic thermodynamic
energy equation (6.13) can then be written in terms of ψ and ω as
2
ψ
∂
∂t
∇
β
∂ψ
f
0
∂ω
∂p
2
ψ
+
V
ψ
·∇
∇
+
∂x
=
(8.2)
∂ψ
∂p
∂ψ
∂p
∂
∂t
σ
f
0
=−
V
ψ
·∇
−
ω
(8.3)
We now apply the vorticity equation (8.2) at the two levels designated as 1 and
3, which are at the middle of the two layers. To do this we must estimate the
divergence term ∂ω/∂p at these levels using finite difference approximations to
the vertical derivatives:
∂ω
∂p
1
≈
,
∂ω
∂p
ω
2
−
ω
0
ω
4
−
ω
2
3
≈
(8.4)
δp
δp
where δp
500 hPa is the pressure interval between levels 0-2 and 2-4, and
subscript notation is used to designate the vertical level for each dependent variable.
=