Ionospheric Alfvén Resonator (IAR) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 189

This equation holds true if

1
r @ ' ‰; and A ' D @ r ‰:

A r D (5.93)

Finally we arrive at the following representation

A D O r

r @ ' ‰ O'

@ r ‰ CO z A; (5.94)

where O r , O' , and O z stand for the unit vectors. Substituting Eq. ( 5.94 )for A into

Eq. ( 5.73 ) yields

1
r @ ' A z @ z A ' D

1
r @ ' A C @ r z ‰;

ıB r D (5.95)

1
r @ z ' ‰ @ r A;

ıB ' D @ z A r @ r A z D (5.96)

r @ r rA '

1 1

r @ ' A r D

1
r @ r .r@ r ‰/

1
r 2 @ '' ‰:

ıB z D (5.97)

Similarly, substituting Eq. ( 5.94 )for A into Eq. ( 5.74 ) yields

i!
r @ ' ‰;

E r D @ r dž C (5.98)

1
r @ ' dž i!@ r ‰;

E ' D (5.99)

E z D @ z dž C i!A: (5.100)

Here, as we have noted above, the terms depending on the potentials dž and A
describe the shear Alfvén mode, whereas the terms depending on the potential ‰
correspond to the compressional mode.

Appendix D: Solutions of the Boundary Problems

Solution of the Problem Associated with IAR

In the magnetosphere . z >L/ the solution of wave equations for the potentials dž
and ‰ is given by Eqs. ( 5.15 ) and ( 5.16 ). Inside the IAR region .0< z <L/ the
solution of Eq. ( 5.12 ) describing Alfvén waves can be written as

dž D dž.0/ cos ! z

V AI C C 3 sin ! z

; (5.101)

V AI