Tornadoes - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 385

at the center of the tornado is identical to the pressure drop at the surface from
the environment to the center

p 0 ð z ¼ 0 ÞD p 0 !1 ð 6 : 54 Þ

Integrating the cyclostrophic equation (6.46) from the perturbation pressure at the
surface at the center of the tornado to the perturbation pressure at the tropopause
above the center of the tornado and using (6.47), (6.48), and (6.53) we find that
ð p 0 ð z ¼ z trop Þ ð p 0 ð z ¼ z trop Þ ð r ½ p 0 ð z ¼ z trop Þ

0 dp 0 ¼ p 0

2

0 @ =@ rdr ¼ v = rdr

p 0 ð z ¼ 0 Þ p 0 ð z ¼ 0 Þ r ½ p 0 ð z ¼ 0 Þ

ð r ¼ r c
r ¼ 0 O

ð r ¼1
r ¼ r c ½ðv max Þ

2 rdr þ 2 2

r 3 ¼ =O ð 1 = Þ dr ¼ CAPE ð 6 : 55 Þ

It is seen that r ð p 0 ð z ¼ 0 ÞÞ corresponds to r ¼ 0 and r ð p 0 ð z ¼ z trop ÞÞ corresponds to
r ¼1 . It follows that

2 max ¼ CAPE

v ð 6 : 56 Þ

so that

1 = 2

v max ¼ð CAPE Þ

ð 6 :

57 Þ

which is similar to the parcel theory estimate of vertical velocity (3.7), save for the

absence of the p 2 factor. Other more accurate vortex solutions, which also include
radial and vertical wind components, may be used to find other expressions for
v max and for the radial pressure distribution.
Above the upper part of the corner region where the vortex is strongest, a
dynamic, downward-directed perturbation pressure gradient force develops (cf.
(4.48); as in Figure 4.58 , but for the tornado—not for the mesocyclone), which
acts to induce subsidence. Consider now what happens when the effects of
compressibility are taken into account. The dynamically induced subsidence
induces adiabatic warming at the center of the tornado and thereby produces a
warmer core that hydrostatically results in lower central pressure; this warmer
core is manifested as higher CAPE. If the air in the middle of the tornado is
unsaturated, then it will descend dry adiabatically rather than moist adiabatically,
and CAPE could be substantially increased. Thus, it seems that a dynamically
forced downdraft should increase the hydrostatic speed limit (cf. (6.57)). However,
it takes work to force a downdraft, which is warmer than its surroundings; if the
forcing for the downdraft were to disappear suddenly, the air would be positively
buoyant and thus accelerate back upward, a manifestation of static stability.
If a downdraft were to originate at the tropopause and the downdraft were
unsaturated, there would be the potential for very substantial warming near the
surface. However, the work needed to push an air parcel downward in a colder
environment beginning at the tropopause and ending at the surface is enormous.
Suppose that the air is saturated, so that it descended at the moist-adiabatic rate
(instead of at the greater dry-adiabatic rate): the initial downward vertical velocity
at the top of the storm (in the absence of mixing with the environment) would