Geography Reference
In-Depth Information
Using the vector identity
V
·
V
( V
·∇
) V
=
+
k
×
V ζ
2
we can rewrite (11.12) as
V ψ
∂t
V ψ ·
V ψ
=−
+
k
×
V ψ
+
f )
(11.13)
2
We next take k
·∇ ×
(11.13) to obtain the vorticity equation
∂t +
V ψ ·∇
+
f )
=
0
(11.14)
valid for nondivergent flow. This equation shows that in the absence of condensa-
tion heating, synoptic-scale circulations in the tropics in which the vertical scale
is comparable to the scale height of the atmosphere must be barotropic; absolute
vorticity is conserved following the nondivergent horizontal wind. Such distur-
bances cannot convert potential energy to kinetic energy. They must be driven by
barotropic conversion of mean-flow kinetic energy or by lateral coupling either to
midlatitude systems or to precipitating tropical disturbances.
Because both the nondivergent velocity and the vorticity can be expressed in
terms of the streamfunction, (11.14) requires only the field of ψ at any level in order
to make a prediction. The pressure distribution is neither required nor predicted.
Rather, it must be determined diagnostically. The relationship of the pressure and
streamfunction fields can be obtained by taking
(11.13). This yields a diagnostic
relationship between the geopotential and streamfunction fields, which is usually
referred to as the nonlinear balance equation :
∇·
2
ψ) 2
f
2 ψ
ψ
1
2 (
+
= ∇·
+
(11.15)
For the special case of stationary circularly symmetric flow, (11.15) is equivalent to
the gradient wind approximation. Unlike the gradient wind, however, the balance
in (11.15) does not require information on trajectory curvature, and thus can be
solved for from knowledge of the instantaneous distribution of ψ on an isobaric
surface. Alternatively, if the distribution is known, (11.15) can be solved for
ψ . In this case the equation is quadratic so that there are generally two possible
solutions, which correspond to the normal and anomalous gradient wind cases.
Such a balance condition is valid only when the above scaling arguments apply.
These, however, have been based on the assumptions that the depth scale is com-
parable to the scale height of the atmosphere and that the horizontal scale is of
 
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