Geography Reference
In-Depth Information
The resulting vorticity equations are
2
ψ
1
∂
∂t
∇
β
∂ψ
1
f
0
δp
ω
2
2
ψ
1
+
V
1
·∇
∇
+
∂x
=
(8.5)
2
ψ
3
∂
∂t
∇
β
∂ψ
3
f
0
δp
ω
2
2
ψ
3
+
V
3
·∇
∇
+
∂x
=−
(8.6)
where we have used the fact that ω
0
=
0, and assumed that ω
4
=
0, which is
approximately true for a level-lower boundary surface.
We next write the thermodynamic energy equation (8.3) at level 2. Here we must
evaluate ∂ψ/∂p using the difference formula
(
∂ψ/∂p
)
2
≈
(
ψ
3
−
ψ
1
)
/δp
The result is
∂
∂t
(ψ
1
−
σδp
f
0
ψ
3
)
=−
V
2
·∇
(ψ
1
−
ψ
3
)
+
ω
2
(8.7)
The first term on the right-hand side in (8.7) is the advection of the 250- to 750-hPa
thickness by the wind at 500 hPa. However, ψ
2
, the 500-hPa streamfunction, is
not a predicted field in this model. Therefore, ψ
2
must be obtained by linearly
interpolating between the 250- and 750-hPa levels:
ψ
2
=
(ψ
1
+
ψ
3
) /2
If this interpolation formula is used, (8.5)-(8.7) become a closed set of prediction
equations in the variables ψ
1
, ψ
3
, and ω
2
.
8.2.1
Linear Perturbation Analysis
To keep the analysis as simple as possible, we assume that the streamfunctions ψ
1
and ψ
3
consist of basic state parts that depend linearly on y alone, plus perturbations
that depend only on x and t . Thus, we let
ψ
1
(x, t)
ψ
1
=−
U
1
y
+
ψ
3
(x, t)
ψ
3
=−
U
3
y
+
(8.8)
ω
2
(x, t)
ω
2
=
The zonal velocities at levels 1 and 3 are then constants with the values U
1
and
U
3
, respectively. Hence, the perturbation field has meridional and vertical velocity
components only.
