Supercells - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 164

So, the buoyancy term is

5 Þ
Suppose now that the vertical perturbation pressure gradient term is as large as
the effective buoyancy term. From the horizontal equation of motion (2.13) we
recall from (2.15) that

B 4CAPE = H ð 4 :

p 0 U 2

ð 4 : 6 Þ

where U is the scale of the horizontal wind component, so that

p 0 zU 2 2U 2 = 7 Þ

The ratio of the approximate magnitude of the acceleration due to the vertical
perturbation pressure gradient force to the acceleration due to buoyancy, which is
a measure of the relative importance of dynamic effects, is given by

j 1 @ =@ z j@=@ = H ð 4 :

p 0 z j=j B j 2 ð U 2

1

2 U 2

j 1 = @ =@ = H Þ=ð 4CAPE = H Þ = CAPE ð 4 : 8 Þ

This ratio is like a Froude number because it represents the ratio of the magnitude
of the inertial term to that of the term involving gravity (namely, CAPE). (Since
CAPE W 2 2, (4.8) represents the ratio of the kinetic energy of the horizontal

component of the wind to the kinetic energy of the vertical component of the
wind that is induced by buoyancy.) So, when 2 U 2 is comparable with the CAPE,
the effects of the dynamic perturbation pressure gradient are comparable with the
effects of effective buoyancy.
The empirical dimensionless parameter called the bulk Richardson number (R),
was first described in a seminal paper by Morris Weisman and Joe Klemp based
on numerical simulation experiments in homogeneous environments of varying
vertical shear and CAPE, and based on an analytical formulation of two-
dimensional steady convection in an environment of vertical shear by Mitch
Moncrieff and J. S. A. Green in 1972:

=

2 U 2

1

R ¼ CAPE =ð Þ ð 4 : 9 Þ

In this case, U is the approximate magnitude of the storm-relative inflow velocity
(generally, it can include both u and v components). In practice, U is computed
from the difference between the pressure-weighted mean wind vector in the moist
boundary layer and the pressure-weighted mean wind vector in the lowest 6 km.
The depth of the boundary layer is usually taken to be 500m, so U the speed of
the vector difference between mean wind in the lowest 6 km and the mean wind in
the lowest 500m. If the mean wind in the lowest 6 km represents storm motion, an
assumption that is supported by many observations, then U represents the storm-
relative wind speed of the inflow layer in the storm.
Since fundamental storm dynamics should not depend on our reference frame,
we should be able to analyze storm dynamics in the reference frame of the storm
and get the same ''answers'' to questions about dynamics as we would in the
ground-relative frame. From (4.8) and (4.9) we see that if U in (4.8) is interpreted
as the same as the storm-relative wind speed ( just transform the coordinate system
to one moving along with the storm, which is one moving along at a constant