Geoscience Reference
In-Depth Information
Ekman suction into the boundary layer from the free atmosphere (see textbooks
on synoptic meteorology: e.g., Bluestein, 1992). The depth of the Ekman layer is
approximately
ð
2
2
, where f is the Coriolis parameter; this is the height at
which the ageostrophic component of the wind falls to a factor of 1
1
=
=
f
Þ
e of the
ageostrophic wind component at the anemometer level. For a flat Earth, f
¼
2
=
O
,
2
. In the case of an Ekman layer,
we have a vortex having much less vorticity (
10
5
s
1
) than that of the rotating
surface below (10
4
s
1
). We remind the reader, for future reference, that the wind
actually overshoots its geostrophic value at and just above the ''gradient wind
level'', the height at which the wind direction first becomes identical to that of the
geostrophic wind.
The behavior of the secondary circulation in a tornado boundary layer
depends on the radial profile of the azimuthal wind. Two extremes are represented
by solid body rotation for which vorticity is constant, and for a potential vortex
for which there is no vorticity at all. So far, we have considered what happens
when there is a potential vortex. In general
v
r
1
=
so that the depth of the Ekman layer is
ð=OÞ
ð
6
:
23
Þ
where for solid body rotation
¼
1.
Von Bo¨ dewadt in 1940 found analytically that the secondary circulation for
solid body rotation is similar to that described by Ekman theory. To see why,
consider the equations of motion ((6.9) and (6.10)) for steady-state flow in which
v ¼ O
¼
1 and for potential flow
r and for which vertical eddy transports of momentum are much greater
than horizontal eddy transports
p
0
2
u
z
2
u
@
u
=@
r
þ
w
@
u
=@
z
¼
0
@
=@
r
þOvþ@
=@
ð
6
:
24
Þ
2
z
2
u
@v=@
r
þ
w
@v=@
z
¼O
u
þ@
v=@
ð
6
:
25
Þ
Note that the pressure gradient term does not appear in (6.25) as a result of
axisymmetry. When the advective terms are neglected on the LHS, equations
(6.24) and (6.25) are similar in form to the Ekman-layer equations for synoptic-
scale flow, except that the Coriolis parameter is replaced by
.
There is convergence at low levels and divergence just above, which is
accompanied by rising motion in the layer of convergence and just above it. The
depth of the friction layer in a tornado vortex characterized by solid body rotation
is
ð=OÞ
O
10
3
m
2
s
1
and
O
0.5 s
1
(i.e., vorticity
1s
1
), the depth of the friction layer is
40m. The
reader is reminded of the similarity of the formula for the depth of the tornado
friction layer to that of the Ekman layer, in which 2
1
=
2
, where 2
O
is the vorticity of the vortex. For
is the vorticity of the rotat-
ing surface. Just as in an Ekman boundary layer the wind speed overshoots the
geostrophic value in the layer aloft, the azimuthal wind speed overshoots the cyclos-
trophic value in the layer aloft, in this case by as much as
20%. The reason for
overshooting is that as air parcels converge radially inward, they spin up, but are
also spun down through frictional dissipation; the effect of the former is greater
than that of the latter, however, and enhanced vorticity is advected upward. When
air parcels get higher, they encounter divergence, and therefore spin down. The
O
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