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=
1
2 , which is outside the domain of the vortex chamber for
this radius is R ð S Þ
S
R); thus, the radius at which the radial pressure gradient vanishes
increases monotonically with swirl ratio. Incoming air must therefore decelerate
and turn upward at a large distance from the axis of rotation when S is high:
surfaces of constant angular momentum are not allowed to converge to small
radius, so that azimuthal wind speeds are modest when S is large. The physical
interpretation of this result, which may be applied to the real atmosphere, is that
as an air parcel having a specified angular momentum (r v ¼ G
>
1 (i.e., r
>
) is forced radially
inward, v increases and r decreases, so that for (6.63) expressed as
2
p 0
v
=
r u
@
u
=@
r ¼ 0 @
=@
r
ð 6
:
68 Þ
2
the centrifugal acceleration ( v
r) increases on the LHS of (6.68). At high r in the
inertial region, however, v is small, so that v
=
2
=
r is very small and
p 0
u
@
u
=@
r 0 @
=@
r
ð 6
:
69 Þ
Since u
<
0 (radial inflow) and
@
u
=@
r
<
0 (radial inflow increases with increasing
p 0
radius) u
@
u
=@
r
<
0 at high r
;@
=@
r
<
0. The radially directed pressure gradient
p 0
(
0) at large radius is therefore adverse, but becomes less negative
(adverse) and eventually vanishes as r decreases because the centrifugal term in
(6.68)
@
=@
r
<
term ( u
@
=@
increases and opposes
the
inertial
u
r) and eventually
overwhelms it.
In addition, if the vortex is confined to low levels, then there is a downward-
directed dynamic perturbation pressure gradient force. At small radius there is
therefore sinking motion ( Figure 6.55 ), so that there is a two-cell vortex, like that
in the Sullivan vortex: upward motion at some radius near or beyond where the
inertial term and the centrifugal terms cancel each other out, and downward
motion at the center and at the far radius beyond where there is rising motion
( Figure 6.55 ). So, at high swirl ratio, air ascends at some radius and descends both
in the center and far from the center owing to (1) the adverse radial pressure
gradient resulting from the dominance of the inertial term at high radius; (2) the
downward-directed dynamic perturbation pressure gradient
resulting from the
decrease of vorticity with height; and (3) mass conservation.
If the updraft is very strong, then there is a relative minimum in pressure
associated with the updraft—consistent with (4.18) integrated from the surface up
to some level and neglecting B. In this case, not only might a condensation funnel
form near the center of the vortex where at a given altitude the pressure is a
minimum, but it might also form near the updraft, leading to a double-wall con-
densation funnel, which is sometimes observed ( Figure 6.56 ). Rotunno has noted
that double-pressure minima occur more often with relatively large swirl ratios.
This feature may, however, be an artifact, created as a consequence of the limited
model domain and the explanation may lie elsewhere.
When the swirl ratio is high, angular momentum does not converge beyond
some radius, so that a ring (annulus) of relatively large (cyclonic) shear vorticity
forms (inside this radius the air is stagnant and/or stagnant air from aloft is
advected downward); this ring of vorticity may be thought of as a curved vortex
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