The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 53

may now be expressed as

@v=@ t ¼ 1 = r @ P =@þ B cos ð 2 : 92 Þ

@ u =@ t ¼@ P =@ r þ B sin ð 2 : 93 Þ

r ð r 2 u cos @=@ð r v cos Þþ@=@ Þ¼ 0 ð 2 : 94 Þ

@ B =@ t ¼ 0 ð 2 : 95 Þ

The problem is now to solve for t (at t ¼ 0) in terms of the

independent variables (at t ¼ 0; we are not trying to determine the time-
dependent behavior of the wind field beyond t ¼ 0). To do so, we must eliminate
P and T. Actually, since (2.95) is not obviously coupled to the other three
equations, we eliminate only P.
First, we eliminate P from (2.92) and (2.93) by forming a vorticity equation
(in the longitudinal direction) by multiplying (2.92) by r and then differentiating
with respect to r and subtracting from this resulting equation (2.93) differentiated
with respect to

@ u =@ t and @v=@

. Making use of

@ B =@ ¼ dB = dz @ z =@ ð 2 : 96 Þ

and

@ B =@ r ¼ dB = dz @ z =@ r ð 2 : 97 Þ

we find that

1 = r ½@=@ @ u =@ t @=@ r ð r @v=@ t Þ¼ 0 ð 2 : 98 Þ

We now make use of the continuity equation (2.88) which is non-divergent in the
r-

-plane to define a streamfunction C such that

=ð r 2 cos

@ u =@ t ¼ 1 Þ@=@ð@C=@ t Þ ð 2 : 99 Þ

and

@v=@

t ¼ 1

=ð r cos Þ@=@

r ð@C=@ t Þ

ð 2 :

100 Þ

We now have eliminated two of the three variables and end up with just one
equation in terms of . Since the streamfunction appears differentiated with

respect to time, we define yet a new variable

C

¼ @C=@ t ð 2 : 101 Þ

which is reminiscent of how we define the geopotential height tendency variable

in quasi-geostrophic theory. The resulting second-order partial differential

equation for is

@=@Þþ r 2

2

r 2

@=@ð 1 = @ =@ ¼ 0 ð 2 :

102 Þ

cos cos

Once we know at t ¼ 0, we then can use (2.99) and (2.100) to find @ u =@ t and

@v=@

t. We need two boundary conditions to solve this equation. One is the
kinematic boundary condition and the other is the dynamic boundary condition,
both applied at the interface between the spherical bubble of buoyant fluid and its
non-buoyant environment.