Geography Reference
In-Depth Information
Fig. 4.11
Dependence of depth on radius in
a rotating cylindrical vessel.
case of an easterly wind, the disturbance in the streamlines damps out away from
the barrier.
The situation described above, which assumes that the mountain barrier is
infinitely long in the meridional direction, is highly idealized. In reality, because
static stability tends to suppress vertical motions, large-scale flows in statically
stable environments are blocked by topographic barriers and are forced to flow
around mountains rather than over them. However, whether fluid columns go over
or around topographic barriers, the potential vorticity conservation constraint still
must be satisfied.
The Rossby potential vorticity conservation law, (4.13), indicates that in a
barotropic fluid, a change in the depth is dynamically analogous to a change in the
Coriolis parameter. This can be demonstrated easily in a rotating cylindrical vessel
filled with water. For solid-body rotation the equilibrium shape of the free sur-
face, determined by a balance between the radial pressure gradient and centrifugal
forces, is parabolic. Thus, as shown in Fig. 4.11, if a column of fluid moves radially
outward, it must stretch vertically. According to (4.13), the relative vorticity must
then increase to keep the ratio (ζ
f)/hconstant. The same result would apply if a
column of fluid on a rotating sphere were moved equatorward without a change in
depth. In this case, ζ would have to increase to offset the decrease of f . Therefore,
in a barotropic fluid, a decrease of depth with increasing latitude has the same
effect on the relative vorticity as the increase of the Coriolis force with latitude.
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4.4
THE VORTICITY EQUATION
The previous section discussed the time evolution of the vertical component of
vorticity for the special case of adiabatic frictionless flow. This section uses the
equations of motion to derive an equation for the time rate of change of vorticity
without limiting the validity to adiabatic motion.
