Ionospheric Alfvén Resonator (IAR) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 200

and ( 5.164 ) with respect to z across the E layer from z D 0 to z D l and making
formally l ! 0, gives the boundary conditions at z D 0
0 ıB ' D † H E ' † P E r B 0 † H V r C † P V ' ;

(5.165)

0 ŒıB r D † P E ' C † H E r C B 0 † H V ' † P V r ;

(5.166)

where the square brackets denote the jump of magnetic field across the E-layer,
and the height-integrated Pedersen and Hall conductivities, † P and † H ,aregiven
by Eq. ( 5.25 ). As before the wind velocity V is assumed to be independent of the
z -coordinate.
Substituting

Eqs. ( 5.130 ) and ( 5.131 ) for the potentials into Eqs. ( 5.165 )

and ( 5.166 ) and applying a Bessel transform to these equations, we get

iǛ H x 0
L

Ǜ P
V AI dž.0/ F 1 ;

ŒA.0/ D ‰.0/ (5.167)

iǛ P x 0
L

Ǜ H
V AI dž.0/ C F 2 ;

Œ@ z ‰.0/ D ‰.0/ (5.168)

where the Bessel transform of the potentials A, dž and ‰ are given by Eq. ( 5.134 ).
Here as before Ǜ P D † P =† w and Ǜ H D † H =† w are the ratios of the height-
integrated Pedersen and Hall conductivities to the wave conductivity † w D
. 0 V AI / 1 , and the dimensionless frequency x 0 is again defined in Eq. ( 5.16 ).
Additionally we made use of the following abbreviations:

kV AI Ǜ H v r C Ǜ P v ' ;

B 0

F 1 D (5.169)

kV AI Ǜ P v r Ǜ H v ' ;

B 0

F 2 D (5.170)

where v r and v ' are the Bessel transform of the radial and azimuthal components of
the wind velocity

Z

v r;' .!;k/ D V r;' .!;r/rJ 1 .kr/dr: (5.171)

0

It is interesting to note that Eqs. ( 5.167 )-( 5.170 ) are identical to Eqs. ( 5.120 )-
( 5.123 ) if the parameter k ? is replaced by the factor k D ik O r where O r D r =r is a
unite vector directed along the vector r .