Supercells - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 228

motion that is in part responsible for the downscale cascade of energy through
nonlinear scale interactions (the Lamb vector) is absent. Recall also the results
from classical Rayleigh-Be´ nard theory in which it is shown that rotation is a
stabilizing effect (cf. (2.266) and (2.267)).
To understand the downscale cascade of energy, consider the Eulerian rate of
change of the u-component of the wind as a result of the nonlinear advection term
in the x-component of the equation of motion
@

52 Þ
If we represent the variation of u in space as a wave in the x-direction
characterized by a wavenumber k, then
u e ikx

=@

t ¼ u

@ =@ ð 4 :

u u x

ð 4 : 53 Þ

It follows from (4.52) that

x ð ik Þ e i2kx @ u =@ t u @ u =@ ð 4 : 54 Þ

Since

2k ¼ 2 ð 2 = L Þ¼ 2 =ð L = 2 Þ ð 4 : 55 Þ

where L is the horizontal wavelength, or scale; then fluctuations on the scale of
L 2 are introduced. Further action (at later time steps) of the nonlinear advection

term produces fluctuations again on even shorter space scales, and ''so on to
viscosity'' as L. F. Richardson once put it.
Helicity (H), which is equivalent to streamwise vorticity,

=

is simply the dot

product between velocity and vorticity
H ¼ v

E !

ð 4 : 56 Þ

When helicity is maximized (i.e., when v and ! point in the same direction),

! T V ¼ 0, so that the term in the equation of motion in part responsible for the

downscale cascade of energy disappears.
Helicity has been hypothesized to be important also in understanding the
amplification of rotation in supercells. Consider a case in the Northern Hemi-
sphere when there is westerly vertical shear, so that the vorticity vector points to
the north. When air enters an updraft at low levels from its western side (i.e.,
when the storm-relative wind vector is oriented in the direction normal to the
vorticity vector), there is purely ''crosswise'' vorticity ( Figure 4.44, top). When air
enters the updraft at low levels from the southern side (i.e., when the storm-
relative wind vector is oriented in the same direction as that of the vorticity
vector), there is purely ''streamwise'' vorticity ( Figure 4.44, bottom panel). The
relative amounts of streamwise and crosswise vorticity are determined by storm-
relative motion. Storm motion is determined by the mean wind, propagation due
to vertical perturbation pressure gradients, propagation due to density current
behavior, among other things, and is not easily determined precisely from a
hodograph because cloud microphysics, temperature, and moisture stratification
also play a role, among other factors. When there is streamwise vorticity, vertical
vorticity is stretched by convergence as air parcels are tilted and enter the base of
an updraft. Bob Davies-Jones at NSSL, in a classic paper in 1984, demonstrated,