The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 55

will determine after applying the boundary conditions. It follows from (2.112), the
boundary conditions (2.107) and (2.111), and since R ð r ¼ a Þ¼ R ð r ¼ a þ Þ that
ð d F= d Þ a ½ð dR = dr Þ a ð dR = dr Þ a þ ¼ 2aB a sin cos ð 2 : 115 Þ

Since ½ð dR = dr Þ a ð dR = dr Þ a þ is a constant, we let

1 =½ð dR = dr Þ a ð dR = dr Þ a þ A ð 2 : 116 Þ

so that

ð d F= d Þ a ¼ A2aB a sin cos ð 2 : 117 Þ

Integrating (2.117) with respect to we find that

FðÞ a ¼ AaB a sin 2 þ C ð 2 : 118 Þ

where C is an arbitrary constant of integration. It follows from substituting
(2.118) into (2.113) that C ¼ 0 and S ¼ 2. So

¼ A 0 R ð r Þ cos 2

ð 2 : 119 Þ

where

A 0 ¼ AaB a

ð 2 : 120 Þ

We now find R ð r Þ from the equation
d 2 R dr 2 r 2 R ¼ 0 = 2 = ð 2 : 121 Þ

which is an ''equi-dimensional'' equation, for which solutions have the form
R ¼ r

ð 2 : 122 Þ

It follows from substituting (2.122) into (2.121) that can be 2 or 1, so that

R ð r Þ¼ D 1 r 1 þ D 2 r 2 ð 2 : 123 Þ

We expect R not to tend to infinity as r ! 0, so that D 1 ¼ 0 for r < a. Also, as

r !1 , should ! 0 (there should be no accelerations at infinity in the far

environment), so that D 2 ¼ 0 for r a. Then

¼ A 0 int r 2 cos 2

>

for r < a ð 2 : 124 Þ

¼ A 0 ext ð 1 r Þ cos 2 = for r > a ð 2 : 125 Þ

where

A 0 int ¼ A 0 D 2 ð 2 : 126 Þ

and

A 0 ext ¼ A 0 D 1

ð 2 :

127 Þ

Applying the boundary conditions (2.107) and (2.111) to (2.124) and (2.125) at
r ¼ a, we find that A 0 int and A 0 ext are determined such that
¼ð B a = 3 Þ r 2 cos 2 for r < a ð 2 : 128 Þ

3 Þð a 3 r Þ cos 2 and ¼ð B a = = for r > a ð 2 : 129 Þ