Tornadoes - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 371

where is a constant, which is the rotation rate of a solid body. It is noteworthy

that radial or vertical turbulent diffusion may be responsible for bringing about a
linear azimuthal wind profile close to the axis of the tornado, which itself is asso-
ciated without any radial turbulent diffusion (i.e., diffusion produces a profile that
reduces diffusion to zero; numerical model simulations show nearly solid body
rotation close to the center of the axis of rotation and the friction term is
negligible). The vortex model in (6.28) is attributed to William John Macquorn
Rankine in the late 19th century. In effect, the Rankine vortex is a solution to the
inviscid equations of motion for which w ¼ 0, u ¼ 0, and v is a function of r only.
We will now address the second question posed at the end of the previous
paragraph.
The Rayleigh criterion for stability in an axisymmetric,

O

inviscid vortex of

azimuthal flow only, which is independent of height, is that
d 2 dr 2

2

= ½ r vð r Þ > 0 ð 6 : 29 Þ

that is, that the square of the angular momentum increases with increasing radius
(squared). In the core of the tornado vortex, the square of the angular momentum
is r 2 , whose square increases with increasing r 2 ; the more rapid the rotation, the

more stable the vortex. If the vortex is stable with respect to lateral displacements,
then it takes work to bring air in closer to the axis of rotation.
Howard and Gupta in 1962 showed that an axisymmetric, inviscid vortex
with both azimuthal and vertical motions is stable with respect to axisymmetric
perturbations if the ''Richardson number'' defined as

O

2 r 2

=½ r 3

2

1
4

Ri ð r Þ¼@ðv Þ=@ r ð@ w =@ r Þ > ð 6 : 30 Þ

In the corner region, where there are prominent vertical motions, the flow is even
more resistant to axisymmetric radial perturbations (i.e., is more stable) than the
core region because there is large radial shear in vertical velocity, which appears in
the denominator of (6.30)—compare (6.30) with (6.29).
Let us compute the work needed to bring a ring of air radially inward from
the core radius in the inertial layer by forcing it against the restoring force on the
ring. In effect, we are finding out how much work is needed to bring the outer
edge of the core radially inward. Neglecting turbulent diffusion and vertical
motion (or simply at z ¼ 0 where w ¼ 0), we find from (6.9) that for a steady state
a vortex in solid body rotation that is in cyclostrophic balance (and for which
angular momentum is conserved) follows the following equation of motion in the
radial direction:

2 r þ v c r c =

r 3 u @ u =@ r ¼O ð 6 : 31 Þ

To compute the work needed to bring a ring of air radially inward from the core
radius r c to an arbitrary radius r 0 , we integrate (6.31) from r c to r 0 and find that
1

2 u r 0 2 u r c ¼ðO

2

2 Þð r 0 2

r c Þþðv

c

r 0 Þ

2

1

= =

2 Þ½ 1 ð r c =

ð 6 :

32 Þ

where v c is the azimuthal velocity at the core radius, the radius at which solid
body flow changes to potential flow. First, note from the RHS that since r 0 < r c