Geography Reference
In-Depth Information
4.4.2
The Vorticity Equation in Isobaric Coordinates
A somewhat simpler form of the vorticity equation arises when the motion is
referred to the isobaric coordinate system. This equation can be derived in vec-
tor form by operating on the momentum equation (3.2) with the vector operator
k
now indicates the horizontal gradient on a surface of constant
pressure. However, to facilitate this process it is desirable to first use the vector
identity
·∇
×
, where
∇
V
·
V
(
V
·∇
)
V
=
∇
+
ζ
k
×
V
(4.18)
2
where ζ
=
k
·
(
∇
×
V
), to rewrite (3.2) as
V
∂
V
∂t
=−
∇
·
V
ω
∂
V
∂p
+
−
(ζ
+
f)
k
×
V
−
(4.19)
2
We now apply the operator
k
·∇
×
to (4.19). Using the facts that for any scalar A,
∇
×
∇
A
=
0 and for any vectors
a
,
b
,
∇
×
(
a
×
b
)
=
(
∇·
b
)
a
−
(
a
·∇
)
b
−
(
∇·
a
)
b
+
(
b
·∇
)
a
(4.20)
we can eliminate the first term on the right and simplify the second term so that
the resulting vorticity equation becomes
∂
V
∂p
×
∇
ω
∂ζ
∂t
=−
ω
∂ζ
V
·∇
(ζ
+
f )
−
∂p
−
(ζ
+
f )
∇·
V
+
k
·
(4.21)
Comparing (4.17) and (4.21), we see that in the isobaric system there is no vorticity
generation by pressure-density solenoids. This difference arises because in the
isobaric system, horizontal partial derivatives are computed with p held constant
so that the vertical component of vorticity is ζ
=
(∂v/∂x
−
∂u/∂y)
p
whereas
in height coordinates it is ζ
∂u/∂y)
z
. In practice the difference
is generally unimportant because as shown in the next section, the solenoidal
term is usually sufficiently small so that it can be neglected for synoptic-scale
motions.
=
(∂v/∂x
−
4.4.3
Scale Analysis of the Vorticity Equation
In Section 2.4 the equations of motion were simplified for synoptic-scale motions
by evaluating the order of magnitude of various terms. The same technique can
also be applied to the vorticity equation. Characteristic scales for the field variables
