Mesoscale convective systems - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 287

At z ¼ 0, w ¼ 0 (the kinematic lower-boundary condition for a level surface), so
the integral of (5.4) from 1 to þ1 yields the following:

2 U 0 ¼ 0 p s þ 2 U c

1 1

ð 5 : 5 Þ

where p 0 ð x ¼1;

z ¼ 0 Þ¼ 0 (i.e., the atmosphere at the surface is hydrostatic far
upstream from the squall line leading edge); and p 0 ð x ¼1; z ¼ 0 Þ¼ p s (the

pressure far downstream at the surface). From (5.3) evaluated at z ¼ 0 and from
(5.5) we find that upon eliminating p s we get the relation
1

2 U 0 ¼ 2 U c þ 0 p t B u ð H 2h c Þ B d h c

1

ð 5 : 6 Þ

which is valid at z ¼ 0. We will use this equation shortly to solve for p t . Now
consider the continuity equation
@ u =@ x þ@ w =@ z ¼ 0 ð 5 : 7 Þ

The first equation in our model is derived from the two-dimensional, horizontal
vorticity equation for the y-component of vorticity (a technique used previously
by Mitch Moncrieff). The steady-state, frictionless form of the y-component of
the vorticity (

¼ @ =@

z @

=@

x) equation in flux form is based on (2.51) (the
divergence term is zero because the atmosphere is Boussinesq and the tilting term
is zero because the flow is two dimensional in the x-z-plane) and the continuity
equation (5.7):

u w

@=@ x ð u Þþ@=@ z ð w Þ¼@ B =@ x ð 5 : 8 Þ

This equation is integrated over the domain (x ¼1 to x ¼þ1 , and z ¼ 0to
z ¼ H þ , where H þ is just above the tropopause, in the lower stratosphere). By
intruding into the stratosphere, we are able to assume that u is zero at z ¼ H þ .
The result is the following:
1

2 U 0 ¼ 2 U c B u ð H 2h c Þ B d h c

1

ð 5 : 9 Þ

So, we now have a relation among the inflow velocity U 0 , the geometry of the
problem in terms of H, h 0 , and h c , rear inflow U c , and updraft and downdraft
buoyancies B u and B d : Since there are three dependent variables (h c , U 0 ,andU c ),
we need three independent equations to find solutions. Equation (5.9) is one such
equation. The quantities H, h 0 , B u , B d are specified constants. H is the depth of
the troposphere, B u and h 0 are determined from an environmental sounding (lapse
rate of temperature and depth of the boundary layer), and B d is determined from
both an environmental sounding and an estimate of liquid water content in the
storm.
We now compare (5.9) with (5.6) and see that p t must be zero. From (5.4)
and (5.7) we derive the flux form of the x-component of the equation of motion
as

u 2 p 0 10 Þ

Integrating (5.10) over the domain in the x-z-plane from 0 to H in z and 1 to
þ1 in x, we find that

@ =@ x þ@ð uw Þ=@ z ¼ 0 @ =@ x ð 5 :

0 0 p 0

ð H

x ¼þ1

U 0 h 0 ¼ 2U c h c þð H 2h c Þ U b þ

ð 5 :

11 Þ dz

x ¼1