Geography Reference
In-Depth Information
4.5.2
The Barotropic Vorticity Equation
If the flow is purely horizontal (w
0), as is the case for barotropic flow in a
fluid of constant depth, the divergence term vanishes in (4.23) and we obtain the
barotropic vorticity equation
=
D
h
ζ
g
+
f
=
0
(4.27)
Dt
which states that absolute vorticity is conserved following the horizontal motion.
More generally, absolute vorticity is conserved for any fluid layer in which the
divergence of the horizontal wind vanishes, without the requirement that the flow
be geostrophic. For horizontal motion that is nondivergent (∂u/∂x
0),
the flow field can be represented by a
streamfunction
ψ (x, y) defined so that the
velocity components are given as u
+
∂v/∂y
=
=−
∂ψ/∂y, v
=+
∂ψ/∂x. The vorticity is
then given by
∂
2
ψ/∂x
2
∂
2
ψ/∂y
2
2
ψ
ζ
=
∂v/∂x
−
∂u/∂y
=
+
≡
∇
Thus, the velocity field and the vorticity can both be represented in terms of the
variation of the single scalar field ψ (x, y), and (4.27) can be written as a prognostic
equation for vorticity in the form:
f
∂
∂t
∇
2
ψ
2
ψ
=−
V
ψ
·∇
∇
+
(4.28)
where
V
ψ
ψ is a nondivergent horizontal wind. Equation (4.28) states
that the local tendency of relative vorticity is given by the advection of absolute
vorticity. This equation can be solved numerically to predict the evolution of the
streamfunction, and hence of the vorticity and wind fields (see Section 13.4).
Because the flow in the midtroposphere is often nearly nondivergent on the synoptic
scale, (4.28) provides a surprisingly good model for short-term forecasts of the
synoptic-scale 500-hPa flow field.
≡
k
×
∇
4.6
THE BAROCLINIC (ERTEL) POTENTIAL VORTICITY EQUATION
Section 4.3 used the circulation theorem and mass continuity to show that Ertel's
potential vorticity, P
g∂θ/∂p), is conserved following the motion in
adiabatic flow. If diabatic heating or frictional torques are present, P is no longer
conserved. An equation governing the rate of change of P following the motion
in such circumstances can, however, be derived fairly simply starting from the
equations of motion in their isentropic coordinate form.
≡
(ζ
θ
+
f)(
−
