Geography Reference
In-Depth Information
Using the fact that any vector
A
satisfies the relationship
k
·
(
∇
θ
×
A
)
=
∇
θ
·
(
A
×
k
)
we can rewrite (4.37) in the form
ζ
θ
+
F
r
−
θ
∂
V
∂θ
k
f
V
∂ζ
θ
∂t
=−
∇
θ
·
−
×
(4.38)
Equation (4.38) expresses the remarkable fact that isentropic vorticity can only be
changed by the divergence or convergence of the horizontal flux vector in brackets
on the right-hand side. The vorticity cannot be changed by vertical transfer across
the isentropes. Furthermore, integration of (4.38) over the area of an isentropic
surface and application of the divergence theorem (Appendix C.2) show that for
an isentrope that does not intersect the surface of the earth the global average
of ζ
θ
is constant. Furthermore, integration of ζ
θ
over the sphere shows that the
global average ζ
θ
is exactly zero. Vorticity on such an isentrope is neither created
nor destroyed; it is merely concentrated or diluted by horizontal fluxes along the
isentropes.
PROBLEMS
4.1.
What is the circulation about a square of 1000 km on a side for an easterly
(i.e., westward flowing) wind that decreases in magnitude toward the north
at a rate of 10 m s
−
1
per 500 km? What is the mean relative vorticity in the
square?
4.2.
A cylindrical column of air at 30˚N with radius 100 km expands to twice
its original radius. If the air is initially at rest, what is the mean tangential
velocity at the perimeter after expansion?
4.3.
An air parcel at 30˚N moves northward conserving absolute vorticity. If its
initial relative vorticity is 5
10
−
5
s
−
1
, what is its relative vorticity upon
×
reaching 90˚N?
4.4.
An air column at 60˚N with ζ
0 initially stretches from the surface to a
fixed tropopause at 10 km height. If the air column moves until it is over
a mountain barrier 2.5 km high at 45˚N, what is its absolute vorticity and
relative vorticity as it passes the mountain top assuming that the flow satisfies
the barotropic potential vorticity equation?
4.5.
Find the average vorticity within a cylindrical annulus of inner radius 200 km
and outer radius 400 km if the tangential velocity distribution is given by
V
=
10
6
m
2
s
−
1
and r is in meters. What is the average
vorticity within the inner circle of radius 200 km?
=
A/r, where A
=
