Geography Reference
In-Depth Information
viscous diffusion. For viscous diffusion the time scale can be estimated from scale
analysis of the diffusion equation
K
m
∂
2
u
∂z
2
∂u
∂t
=
(5.43)
If τ
d
is the diffusive time scale and H is a characteristic vertical scale for diffusion,
then from the diffusion equation
K
m
U
H
2
U
τ
d
∼
H
2
/K
m
. For the above values of H and K
m
, the diffusion time scale
is thus about 100 days. Hence, in the absence of convective clouds the spin-down
process is a far more effective mechanism for destroying vorticity in a rotating
atmosphere than eddy diffusion. Cumulonimbus convection can produce rapid
turbulent transports of heat and momentum through the entire depth of the tro-
posphere. These must be considered together with boundary layer pumping for
intense systems such as hurricanes.
Physically the spin-down process in the atmospheric case is similar to that
described for the teacup, except that in synoptic-scale systems it is primarily the
Coriolis force that balances the pressure gradient force away from the boundary,
not the centrifugal force. Again the role of the secondary circulation driven by
forces resulting from boundary layer drag is to provide a slow radial flow in the
interior that is superposed on the azimuthal circulation of the vortex above the
boundary layer. This secondary circulation is directed outward in a cyclone so that
the horizontal area enclosed by any chain of fluid particles gradually increases.
Since the circulation is conserved, the azimuthal velocity at any distance from the
vortex center must decrease in time or, from another point of view, the Coriolis
force for the outward-flowing fluid is directed clockwise, and this force thus exerts
a torque opposite to the direction of the circulation of the vortex. Fig. 5.7 shows a
qualitative sketch of the streamlines of this secondary flow.
It should now be obvious exactly what is meant by the term
secondary circula-
tion
. It is simply a circulation superposed on the primary circulation (in this case
the azimuthal circulation of the vortex) by the physical constraints of the system.
In the case of the boundary layer, viscosity is responsible for the presence of the
secondary circulation. However, other processes, such as temperature advection
and diabatic heating, may also lead to secondary circulations, as shown later.
The above discussion has concerned only the neutrally stratified barotropic
atmosphere. An analysis for the more realistic case of a stably stratified baroclinic
atmosphere is more complicated. However, qualitatively the effects of stratifica-
tion may be easily understood. The buoyancy force (see Section 2.7.3) will act
to suppress vertical motion, as air lifted vertically in a stable environment will be
so that τ
d
∼