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(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.
=
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