Geography Reference
In-Depth Information
4.6.2
The Potential Vorticity Equation
If we take
k
·∇
θ
×
(4.31) and rearrange the resulting terms, we obtain the isentropic
vorticity equation:
F
r
−
θ
∂
V
∂θ
D
Dt
(ζ
θ
+
f )
+
(ζ
θ
+
f )
∇
θ
·
V
=
k
·∇
θ
×
(4.34)
where
D
Dt
=
∂
∂t
+
V
·∇
θ
is the total derivative following the horizontal motion on an isentropic surface.
Noting that σ
−
2
∂σ/∂t
∂σ
−
1
/∂t , we can rewrite (4.32) in the form
=−
D
Dt
σ
−
1
σ
−
1
∂θ
σ θ
∂
σ
−
2
−
∇
θ
·
V
=
(4.35)
Multiplying each term in (4.34) by σ
−
1
and in (4.35) by (ζ
θ
+
f)and adding, we
obtain the desired conservation law:
DP
Dt
=
F
r
−
θ
∂
V
∂θ
∂θ
σ θ
+
∂P
∂t
+
P
σ
∂
σ
−
1
k
V
·∇
θ
P
=
·∇
θ
×
(4.36)
where P
f )/σ is the Ertel potential vorticity defined in (4.12). If the
diabatic and frictional terms on the right-hand side of (4.36) can be evaluated, it
is possible to determine the evolution of P following the
horizontal
motion on
an isentropic surface. When the diabatic and frictional terms are small, potential
vorticity is approximately conserved following the motion on isentropic surfaces.
Weather disturbances that have sharp gradients in dynamical fields, such as jets
and fronts, are associated with large anomalies in the Ertel potential vorticity. In
the upper troposphere such anomalies tend to be advected rapidly under nearly
adiabatic conditions. Thus, the potential vorticity anomaly patterns are conserved
materially on isentropic surfaces. This material conservation property makes poten-
tial vorticity anomalies particularly useful in identifying and tracing the evolution
of meteorological disturbances.
≡
(ζ
θ
+
4.6.3
Integral Constraints on Isentropic Vorticity
The isentropic vorticity equation (4.34) can be written in the form
F
r
−
θ
∂
V
∂θ
∂t
=−
∇
θ
·
ζ
θ
+
f
V
+
∂ζ
θ
k
·∇
θ
×
(4.37)
