Tornadoes - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 390

Then

S ¼ðv 0 R Þ=ð 2hu 0 Þ ð 6 : 61 Þ

If the vortex chamber is constructed so that R 2h ¼ 1 (i.e., if the depth of the

inflow layer is twice the radius of the updraft hole), then

=

S ¼ v 0 = u 0 ð 6 : 62 Þ

The swirl ratio can be thought of as a measure of the relative amount of
azimuthal flow compared with the amount of radial flow into the bottom of the
vortex or, equivalently, the relative amount of vertical vorticity to (horizontal)
convergence.
The physical significance of the swirl ratio is illuminated by considering once
again the radial equation of motion for an axisymmetric vortex in a steady-state,
inviscid, constant density,

incompressible fluid in which w ¼ 0 at the bottom

surface (6.45)

2

p 0 u @ u =@ r v = r ¼ 0 @ =@ r ð 6 : 63 Þ

In this equation, the inertial acceleration term (u @ u =@ r) contains the effect of

2

radial r) contains the effect of swirl, the

azimuthal wind component. In a crude, qualitative manner, at a given radius,
when the swirl is large compared with the radial flow, the centripetal term
dominates; when the swirl is low compared with the radial flow, the inertial
acceleration dominates. Also, if the swirl is large, then there is a large drop in
perturbation pressure at the center of the vortex: If the pressure drop is large
enough, the resultant downward-directed perturbation pressure gradient force can
reverse the frictionally induced central updraft to induce a downdraft. The larger
the relative amount of aziumthal flow (circulation) to the updraft, the greater the
effect of the downward-directed perturbation pressure gradient force.
Rich Rotunno in the late 1970s and early 1980s, building on work done by
Davies-Jones, illuminated why the structure of a vortex changes as the swirl ratio
ranges from small to large. His work was motivated in part by the finding that the
size of the core of the vortex in vortex chambers and the nature of the secondary
circulation are determined by the swirl ratio ( Figure 6.55 ). For the approximate
wind field

inflow and the centripetal term ( v =

u ð r = R Þ u 0 ð 6 : 64 Þ

v ð R = r Þv 0 ð 6 : 65 Þ

which depicts potential flow expected outside the core and radial convergence
(

@ =@ <

0) as air decelerates as it enters the corner region. It follows then from
(6.63) that

u r

u 0 =ð R 2 r 3

4

S 2 p 0 Þ½ð r = R Þ ¼ 0 @ =@ r ð 6 : 66 Þ