Geoscience Reference
In-Depth Information
Figure 6.46. Radial (R) profile of azimuthal (tangential) wind (V) in a Rankine combined
vortex. R core is the core radius; V max is the maximum azimuthal wind speed (adapted from
Brown and Wood, 2012).
the work done is negative (i.e., work must be done on the system). Also, as r 0 ! 0,
the work required !1 : it becomes more and more dicult to bring in rings of
air the closer one gets to the center of the vortex.
The simplest model of a tornado is the ''Rankine combined vortex'', which is
a core of solid body rotation surrounded by (whence the adjective ''combined'' is
used) a region of potential flow, with no vertical motion in either region ( Figure
6.46 ) . This type of vortex wind profile satisfies the axisymmetric equations of
motion subject to the approximations of steady-state azimuthal flow ((6.9) and
(6.10)).
Outside
the
core
radius, where
angular momentum is
constant,
d 2
dr 2
2
¼ 0, so it takes no work to bring a ring of fluid radially inward, but
only down to the core radius—not within it. One may verify this claim simply by
rewriting (6.31) for the case when a cyclostrophic wind balance is imposed on a
potential flow vortex. The work needed to bring a ring of air radially inward from
the core radius or within must be related via the continuity equation to the inten-
sity of the updraft forced from above (in the parent storm or vortex chamber), as
the radial gradient of u is related to the vertical gradient in w (6.15) and w ¼ 0at
the surface.
Recent mobile Doppler radar measurements of the wind field in tornadoes
exhibit radial profiles of azimuthal wind that are similar to the Rankine combined
vortex, except that there is a smooth transition from solid body rotation to poten-
tial-like flow near the radius of maximum wind (RMW) ( Figure 6.47a ) instead of
an abrupt transition. The reader should note that in the Rankine combined vortex,
the RMW is identical to the core radius. However, for other radial profiles of
azimuthal wind, this may not necessarily be the case: we define the core radius as
that beyond which angular momentum is nearly constant (at least until we are
outside the parent vortex). The formal definition of the core radius is given by the
following:
=
½ r r Þ
r c ¼ G 1 =v c
ð 6
:
33 Þ
 
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