Ordinary-cell convective storms - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 139

Figure 3.38. Illustration showing how gust front relative flow (black vectors) is decelerated as
it encounters an adverse, dynamic (non-hydrostatic) pressure gradient force (green vector
labeled PGF). The gust front relative wind comes to a grinding halt at the edge of the cold
pool (solid red line) because we have assumed that there is no mixing between the cold pool air
and the ambient air and that we are in the reference frame of the density current.

traveling at infinite speed—a real fast-talking gust front!). The wedge of an immis-
cible cold air mass advancing against a resting air mass is like an airfoil advancing
against a resting air mass: air slows down and flows up and over the air foil
( Figure 3.39 ) and lift is created as an upward-directed perturbation pressure
gradient.
Another way to diagnose the perturbation pressure is to use the inviscid form
of the divergence equation (2.62) applied to this simple model

2 p 0 ¼ð@ 2 2

=r =@

x Þ

ð@ =@

z Þ

2 ð@ =@ @ =@

z Þþ@

=@ ð 3 :

19 Þ 1 u w w x u B z

Consider first what happens just ahead of the density current, where B ¼ 0. Since
@ z, are both nonzero just ahead of the leading edge

of the density current, it follows that the first two terms on the right-hand side of
(3.19) are negative and that the last term is zero. Also,

u =@ x and, by continuity, @ w =@

@ w =@ x < 0, but @ u =@ z ¼ 0.

So, p 0

> 0. Far ahead of the density current @ u =@ x and @ w =@ z are both zero, so

p 0 ¼ 0. Just above the density current, 0, so there is a negative contribu-

tion to p 0 ; this is also the case when the leading edge of the density current is
wedge shaped, so that

@ B =@ z >

0 everywhere at its interface.
The perturbation pressure high ahead of the leading edge of the density
current, which is dynamically induced, falls off with distance ahead of the density
current. If it did not exist, then there would be a discontinuity in pressure across
the leading edge of the density current, which would violate the dynamic boundary
condition, and the horizontal pressure gradient at the leading edge, which is
hydrostatic, would be infinite. The non-hydrostatic high pressure allows the
pressure to be continuous across the leading edge.
We now integrate (3.18) from the far left, well behind the leading edge of the
gust front (i.e., the leading edge of the dense air mass), where u ð1Þ¼ 0, to the
far right, well ahead of the gust front, where u ð1Þ¼ c and p 0 ¼ 0: We find that

@ B =@ z >

2 c 2

1

¼ 0 p t þg h ½ð 1 0 Þ= 0 ð 3 : 20 Þ