Geography Reference
In-Depth Information
Thus, the continuity equation (3.5) can be written as
p
s
∇
p
·
V
+
∂ω
∂σ
=
0
(10.33)
Now the sigma vertical velocity can be written as
∂σ
∂t
+
σ
∂p
s
∂t
+
p
s
ω
p
s
Differentiating the above with respect to σ , eliminating ∂ω
∂σ with (10.33), and
rearranging yields the transformed continuity equation
ω
∂σ
σ
p
s
σ
˙
=
V
·∇
p
+
∂p
=−
V
·∇
+
∂p
s
∂t
+
∇·
p
s
∂
σ
∂σ
=
˙
(p
s
V
)
+
0
(10.34)
With the aid of the equation of state and Poisson's equation (2.44) the hydrostatic
approximation can be written in the sigma system as
p
p
0
κ
∂
∂σ
=−
RT
σ
Rθ
σ
=−
(10.35)
where p
0
=
1000 hPa.
Expanding the total derivative in (2.46), we may write the thermodynamic energy
equation for sigma coordinates as
∂θ
∂t
+
σ
∂θ
J
c
p
θ
T
V
·∇
θ
+˙
∂σ
=
(10.36)
10.3.2
The Zonal-Mean Angular Momentum
We now transform the angular momentum equation (10.27) into sigma coordinates
with the aid of (10.28) and (10.35) to yield
∂
∂t
+
M
a cos φ
∂
∂τ
E
∂σ
∂
∂σ
RT
p
s
∂p
s
∂x
+
g
p
s
V
·∇
+˙
σ
=−
∂x
+
(10.37)
Multiplying the continuity equation (10.34) by M and adding the result to (10.37)
multiplied by p
s
we obtain the flux form of the angular momentum equation:
1
∂ (p
s
M)
∂t
∂ (p
s
M
σ )
˙
=−
∇·
(p
s
M
V
)
−
∂σ
a cos φ
p
s
(10.38)
ga cos φ
∂τ
E
∂σ
∂
∂x
+
RT
∂p
s
∂x
−
−
1
It may be shown (Problem 10.2) that in sigma coordinates the mass element ρ
0
dxdydz takes the
form
−
g
−
1
p
s
dxdydσ. Thus, p
s
in sigma space plays a role similar to density in physical space.
