Geography Reference
In-Depth Information
cup there is an approximate balance between the radial pressure gradient and the
centrifugal force of the spinning fluid. However, near the bottom, viscosity slows
the motion and the centrifugal force is not sufficient to balance the radial pressure
gradient. (Note that the radial pressure gradient is independent of depth, as water
is an incompressible fluid.) Therefore, radial inflow takes place near the bottom
of the cup. Because of this inflow, the tea leaves always are observed to cluster
near the center at the bottom of the cup if the tea has been stirred. By continuity of
mass, the radial inflow in the bottom boundary layer requires upward motion and a
slow compensating outward radial flow throughout the remaining depth of the tea.
This slow outward radial flow approximately conserves angular momentum, and
by replacing high angular momentum fluid by low angular momentum fluid serves
to spin down the vorticity in the cup far more rapidly than could mere diffusion.
The characteristic time for the secondary circulation to spin down an atmo-
spheric vortex is illustrated most easily in the case of a barotropic atmosphere.
For synoptic-scale motions the barotropic vorticity equation (4.24) can be written
approximately as
f ∂u
g
Dt =−
∂v
∂y
f ∂w
∂z
∂x +
=
(5.39)
where we have neglected ζ g compared to f in the divergence term and have also
neglected the latitudinal variation of f . Recalling that the geostrophic vorticity is
independent of height in a barotropic atmosphere, (5.39) can be integrated easily
from the top of the Ekman layer (z
=
De) to the tropopause (z
=
H)to give
f w(H )
g
Dt =+
w(De)
(5.40)
(H
De)
Substituting for w(De) from (5.38), assuming that w(H )
=
0 and that H
De,
(5.40) may be written as
1/2
g
Dt =−
fK m
2H 2
ζ g
(5.41)
This equation may be integrated in time to give
=
ζ g (t)
ζ g (0) exp (
t/τ e )
(5.42)
=
where ζ g (0) is the value of the geostrophic vorticity at time t
0, and τ e
1/2
is the time that it takes the vorticity to decrease to e 1
H |
2/(f K m ) |
of its
original value.
This e-folding time scale is referred to as the barotropic spin-down time . Taking
typical values of the parameters as follows: H
10 4
s 1 , and
10 km, f
=
10 m 2
s 1 , we find that τ e
K m =
4 days. Thus, for midlatitude synoptic-scale
disturbances in a barotropic atmosphere, the characteristic spin-down time is a
few days. This decay time scale should be compared to the time scale for ordinary
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