Geography Reference
In-Depth Information
This problem can be overcome by defining a vertical coordinate that is propor-
tional to pressure normalized by the surface pressure. The most common form of
this coordinate is the
sigma coordinate,
defined as σ
p/p
s
, where p
s
(x,y,t)
is the pressure at the surface. Thus, σ is a nondimensional-independent vertical
coordinate that decreases upward from a value of σ
≡
0at
the top of the atmosphere. In sigma coordinates the lower boundary condition will
always apply exactly at σ
=
1 at the ground to σ
=
=
1. Furthermore, the vertical σ velocity defined by
Dσ
Dt
σ
˙
≡
will always be zero at the ground even in the presence of sloping terrain. Thus, the
lower boundary condition in the σ system is merely
σ
˙
=
0atσ
=
1
To transform the dynamical equations from the isobaric system to the σ system
we first transform the pressure gradient force in a manner analogous to that shown
in Section 1.6.3. Applying (1.28) with p replaced by , s replaced by σ , and z
replaced by p, we find that
∂
∂x
σ
=
∂
∂x
p
+
σ
∂ ln p
s
∂x
∂
∂σ
(10.28)
Because any other variable will transform in an analogous way, we can write
the general transformation as
ln p
s
∂ ()
∂σ
∇
p
()
=
∇
σ
()
−
σ
∇
(10.29)
Applying the transformation (10.29) to the momentum equation (3.2), we get
D
V
Dt
+
∂
∂σ
f
k
×
V
=−
∇
+
σ
∇
ln p
s
(10.30)
where
∇
is now applied holding σ constant, and the total differential is
D
Dt
=
∂
∂t
+
∂
∂σ
V
·∇
+˙
σ
(10.31)
The equation of continuity can be transformed to the σ system by first using
(10.29) to express the divergence of the horizontal wind as
∂
V
∂σ
∇
p
·
V
=
∇
σ
·
V
−
σ (
∇
ln p
s
)
·
(10.32)
To transform the term ∂ω/∂p we first note that since p
s
does not depend on σ
∂
∂p
=
∂
∂ (σp
s
)
=
1
p
s
∂
∂σ
