The basic equations - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 54

So, at r ¼ a, the component of the wind normal to the sphere must be
continuous across the interface (kinematic boundary condition) at t ¼ 0.
u ð r ¼ a Þ¼ u ð r ¼ a þ Þ ð 2 : 103 Þ

at t ¼ 0 and also at t ¼ t þD t. Note that u ð t ¼ 0 Þ¼ 0.

Expanding both the left and right-hand sides of (2.103) into Taylor series
expansions in time, it is seen that
u ð r ¼ a ; t þD t Þ¼ u ð r ¼ a ; t ¼ 0 Þþð@ u =@ t Þ t ¼ 0 ; a D t þ higher order terms ð 2 : 104 Þ

and
u ð r ¼ a þ ; t þD t Þ¼ u ð r ¼ a þ ; t ¼ 0 Þþð@ u =@ t Þ t ¼ 0 ; a þ D t þ higher order terms ð 2 : 105 Þ

Then, in the limit as D t goes to zero

ð@ u =@ t Þ t ¼ 0 ; a ¼ð@ u =@ t Þ t ¼ 0 ; a þ ð 2 : 106 Þ

and so from (2.99) and (2.101) it follows that
ð@=@Þ a ¼ð@=@Þ a þ

ð 2 : 107 Þ

Pressure must be continuous across the spherical bubble surface (the dynamic
boundary condition) so that

P ð r ¼ a Þ¼ P ð r ¼ a þ Þ ð 2 : 108 Þ

Then the latitudinal gradient of pressure is also continuous; that is
ð 1

= r @ P =@Þ a ¼ð 1 = r @ P =@Þ a þ ð 2 : 109 Þ

It follows from the latitudinal component of the equation of motion (2.92) that

= t Þ a ¼ðg T 0 ð@v=@ t Þ a þ ð@v=@ T Þ cos ð 2 : 110 Þ

Substituting the left-hand side of (2.110) using the definition of —(2.100) and

(2.101)—we find that

r Þ a þ ¼ aB a cos 2

ð@=@

r Þ a ð@=@

ð 2 :

111 Þ

= where B a ¼ g T 0

r Þ a and
one on ð@=@Þ a . The solutions to (2.102) subject to the boundary conditions
(2.107) and (2.111) are separable and take the form
¼ R ð r ÞFðÞ

T. We now have two boundary conditions: one on ð@=@

ð 2 : 112 Þ

Substituting (2.112) into (2.102), we find that
ð cos =FÞ d = d ð d F= d = cos Þ¼ S ð 2 : 113 Þ

and

d 2 R dr 2 r 2 = ¼ð R = Þ S ð 2 : 114 Þ

where S is the ''separation constant''. We now have two equations in two
unknowns (R and F ), and they are linked by the separation constant, which we