Fundamentals of Plasma Physics

246 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma

where it is assumed thatǫthis approximately constant. To prove this is the correct form,
consider the following two statements: (i) ifǫth→ 0 , Eq.(8.75) reverts to Eq.(6.108)
(Fourier-analyzed in thezdirection, but not thexdirection), (ii) ifǫthis finite andSand
Pare assumed uniform, then Fourier analysis inxrestores Eq.(8.49).
We now suppose that the plasma is non-uniform such thatS(x) = 0at some particular
value ofx.Thexorigin is defined to be at this location and so corresponds to the location
of the hybrid resonance. In order to be specific, we assume that we are dealing with a mode
whereS> 0 andP< 0 , sinceSandPmust have opposite signs for the cold plasma wave
to propagate. Then, in the vicinity ofx= 0Taylor expansion ofSgivesS=S′xso that
Eq.(8.75) becomes

ǫth

d^4 φ

dx^4

+S′

dφ

dx

+S′x

d^2 φ

dx^2

−k‖^2 Pφ= 0 (8.76)

which has a coefficient which vanishes atx= 0just like the Airy equation. Although
Eq.(8.76) could be analyzed as it stands, it is better to clean it up somewhat bychanging to
a suitably chosen dimensionless coordinate. This is done by first defining

ξ=

x

λ

(8.77)

whereλis a yet-undetermined characteristic length which will be chosen to provide maxi-
mum simplification of the coefficients. Replacingxbyξin Eq.(8.76) gives

ǫth

λ^4

d^4 φ

dξ^4

+

S′

λ

dφ

dξ

+

S′

λ

ξ

d^2 φ

dξ^2

−k^2 ‖Pφ= 0. (8.78)

This becomes
d^4 φ
dξ^4

+

dφ

dξ

+ξ

d^2 φ

dξ^2

+μφ= 0 (8.79)

ifλis chosen to haveǫth/S′λ^3 = 1and we setμ=−k^2 ‖λP/S′.These choices imply

λ=

(ǫ

th

S′

) 1 / 3

, μ=−

k‖^2 Pǫ^1 th/^3

(S′)^4 /^3

. (8.80)

Equation (8.79) describes a boundary layer problem having complicated behavior in the
inner|ξ|< 1 region in the vicinity of the hybrid resonance and, presumably, simple WKB-
like behavior in the outer|ξ|>> 1 region away from the hybrid resonance. To solve this
problem the technique discussed for the Airy equation will now be applied and generalized:

1. Laplace Transform: While Eq.(8.79) cannot be Fourier analyzed in theξdirection, it
   can be Laplace transformed giving

p^4 ̃φ(p) +p ̃φ(p)−

d

dp

[

p^2 φ ̃(p)

]

+μ ̃φ(p) = 0 (8.81)

or equivalently

1

[

p^2 ̃φ(p)

]

d

dp

[

p^2 ̃φ(p)

]

=p^2 +

1

p

+

μ

p^2

. (8.82)

This has the solution

̃φ(p) =Aexp(p^3 / 3 −μ/p−lnp) (8.83)