FLR in Plasma Configurations - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 140

The obtained general solution of the system (6.15) fully proves the main
results of the theory of FLR for a 2D inhomogeneous plasma: a wave field has
a singularity near a resonance field-line.
The scheme of calculations of coecient matrices in expansion (6.17) is the
following. Inserting a solution in the form of (6.5) into (6.15) and equating
coecients at the same powers of x − x 0 , we obtain a system of recurrent

equations for coecients P n :

n− 1

n P n =[ T 0 P n ]+ T n−ν P ν , =0 , 1 , 2 ,.... (6.18)

ν =0

Here the equality

[ AB ]= AB − BA .

is the commutator of matrices A and B . When n = 0 the relationship (6.18)
becomes

[ T 0 P 0 ]= 0 .
It can be put here that P 0 = 1 ; this choice also ensures the invertibility of the
matrix U . Next, the coecients P 1 , P 2 ,... are determined from (6.18). Now
we take into consideration the block structure of matrices T 0 , T 1 ,... and a
specific form of the block Q 0 . There is only one non-zero row in the latter
block. The following identities hold:

S 0 Q 0 = 0 , Q 0 S 0 = 0 .

The first identity is evident: the s -th column of matrix S 0 is zero while in Q 0
all the elements, besides probably some elements of the s -th line, are zeros.
Note that

Q k Q m
c 4 A

= Q s Q m
c 2 A

Q k = Q s Q m
c 2 A

= δ sm ,

Q k
c 2 A

Q s Q k

then the other identity is

ω 2 m ( x 0 ) k y d

d x ω s ( x 0 ) − 1

Q k Q m
c 4 A

− ω 2 ( Q 0 S 0 ) nm = − δ ns Q s Q k

ω m ( x 0 ) k y d

d x ω s ( x 0 ) − 1

− ω 2

− δ ns δ sm =0 .

=

Again, here we make summation over repeated indexes. Thus
T 0 T 1 = 00

,

T 1 T 0 = 00

,

R 0 Q 0 0 R 0 Q 0 0

( T 0 T 1 ) 2 =( T 1 T 0 ) 2 = 0 ,

T 0 T 1 T 0 = 0 .

T 0 T 1 T 0 = 0 ,