FLR in Plasma Configurations - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 164

Explicit Solutions

Dispersion Equation

In the general case, the boundary problem (6.94), (6.95) can be solved numer-
ically. In this section we shall consider a special case of the plasma density
distribution allowing us to solve explicitly the boundary problem. Consider
the power distribution of the plasma density (6.63) along a field-line. Then
the general solution of (6.94) at p = 6 is the simplest and is given by

e 2 = A cos( ωw/Ω )+ B sin( ωw/Ω ) .

If the Pedersen conductivities of the conjugate ionospheres are equal, that
is X + = X − = X , then the symmetric solution with respect to the equatorial
plane is

e 2 = A cos( ωw/Ω ) ,

(6.99)

and the anti-symmetric one is

e 2 = B sin( ωw/Ω ) . (6.100)

Substituting (6.99) and (6.100) into the boundary conditions (6.96), we
obtain the dispersion equations for the symmetrical and anti-symmetrical

modes:

cot( w 0 ω

Ω )= iq ( ν )

symmetrical mode ,

tan( w 0 ω

− iq ( ν )

anti-symmetrical mode ,

Ω )=

with

cν
Ω (1 + 3 w 0 ) 1 / 2

sin I 0
X ±

q ( ν )= − .

The FLR-frequencies are determined by

ω j ( ν )= jω A ( ν ) − iγ j ( ν ) , (6.101)

where

ω A ( ν )= π
2

Ω ( ν )
w 0 ( ν ) ,

γ j ( ν )= γ j ( ν )

2
jπ atanh q ( ν ) . ω A ( ν ) =

(6.102)

Symmetrical (anti-symmetrical) modes correspond to odd (even) j .
Let us estimate the FLR-frequencies ω A ( ν ) and decrements γ j ( ν ) for the
dipole field. The parameter ν = − 1 /LR E , the geomagnetic dipole moment

10 25 G cm 3 , the Earth's radius 10 8 cm, the proton M =8 × · R E =6 . 37 ×