Tornadoes - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 399

Depleted angular momentum is useful because its flux into the corner region from
the boundary layer is nearly equal to its flux out of the corner region into the core
(not shown here), whereas angular momentum flux itself is not because it varies
significantly within the corner region (also not shown here). So, instead of the
mass flux, we will use the total depleted angular momentum flux through the
corner region

r 1 ð z 1
0

U ¼ 2 u G d dz ð 6 : 73 Þ

where z 1 is the height of the top of the friction layer; and r 1 is just outside the
corner flow. To make sure that the mass flux M in the denominator of (6.71) has
units of (m s 1 )m 2 , we scale G 1 so that M ¼ U=G 1 . It follows that the

corner flow swirl ratio in (6.73) may be expressed with the aid of (6.33) as

U by

2
1 =U ðG 1 =

r c Þ=½U=ðG 1 r c Þ¼ v c =

S c ¼ r c G U ð 6 : 74 Þ

where U is a measure of the component of the wind flowing into the corner region
given by

U=ðG 1 r c Þ . It is nice that the dimensions of the hole and inflow depth in
the vortex chamber do not appear in (6.74), so that is more easily applied to the
real atmosphere. The reader is reminded that (6.74) is still an empirical parameter
and that it is not the only way to characterize flow.
Another way to interpret the corner flow swirl ratio is, using (6.33), to express
the numerator of (6.74) as

2

1 ¼ v c r c G 1 w c r c G 1

r c G ð 6 : 75 Þ

where vertical velocity upward from the corner region into the core region w c is
proportional to the azimuthal velocity in the core (the stronger the vertical jet in
an end-wall vortex, the higher the azimuthal velocity). We interpret (6.33) as
expressing the relationship between r c , the closest to the center of the axis of rota-
tion fluid may converge when the fluid is characterized by environmental angular
momentum G 1 and by v c , the azimuthal wind speed at the core of the vortex

imposed from above. The flux of angular momentum upward into the core is
given by (6.75), while the denominator of (6.74) is the depleted angular momen-
tum flux into the corner region: thus, the corner flow swirl ratio is a measure of the
ability of the converging boundary-layer flow to supply the core of the vortex with
upward-moving fluid depleted of angular momentum subject to the constraints of the
size of the core radius r c and the azimuthal wind speed in the core v c .
When the corner flow swirl ratio S c S c , the ''critical corner flow swirl
ratio'', there is an end-wall vortex with a strong upward jet that ends with vortex
breakdown just above the surface. This configuration results in the maximum
azimuthal wind speed possible as close to the ground as possible. Brian Fiedler
showed that the depleted angular momentum flux in the boundary layer of a
potential vortex characterized by a high Reynolds number is

2
1

¼ 2 : 6 G ð 6 : 76 Þ