Geomagnetic Perturbations (GMPs) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 304

It is convenient to introduce new dimensionless functions g 1 D .B 1 C 2B 2 /=B 0
and g 2 D .B 1 B 2 /=B 0 . Summing and subtracting Eq. ( 7.85 ) and ( 7.86 ), and
rearranging Eq. ( 7.87 ), leads to
@ t g 1 D m @ r g 1 C 2 @ r V C 2 V

r

;

2@ r g 1
r

(7.89)

@ t g 2 D m @ r g 2 C

r 2

2@ r g 2
r

6g 2 V

r ;

C @ r V (7.90)

6g 2
r D 0:

@ r .g 1 C 2g 2 / C (7.91)

Such a form of the equations are preferable because either of Eqs. ( 7.89 )or( 7.90 )
includes only a single unknown function. Note that Eq. ( 7.89 ) can be transformed
to a classical 1D diffusion equation via changing the unknown function g 1 .r;t/ by
the function g 0 1 .r;t/ D g 1 .r;t/=r. The solution of this equation at zero initial and
boundary conditions are known (Surkov 1989a )

Z

Z t

dt 0
r 0

g 3 r;r 0 ;r 0 r 0 @ r 0 V C 2V dr 0 ;

2
1=2 r

g 1 .r;t/ D (7.92)

0 0

where V D V .r 0 ;t 0 / is the mass velocity of the medium. Here we made use of the
following abbreviations:
g 3 r;r 0 ;r 0 D exp

!

exp

! ;

.r r 0 / 2
r 0

.r C r 0 / 2
r 0

(7.93)

and

r 0 D 2 m t t 0 1=2 : (7.94)

Substituting Eq. ( 7.92 )forg 1 into Eq. ( 7.91 ) we come to a differential equation
of the first order with respect to the function g 2 . The solution of this equation takes
the form

Z

r

r 03 @ r 0 g 1 r 0 ;t dr 0 :

1
2r 3

g 2 .r;t/ D (7.95)

0

So the functions g 1 and g 2 given by Eq. ( 7.92 ) and Eq. ( 7.95 ) are the solutions of
Eq. ( 7.89 ) and ( 7.91 ).