232 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma
Since the kinetic energymv^2 / 2 of the unperturbed orbits is a constant of the motion, the
quantityexp
[
−v^2 /vTσ^2]
has been factored out of the time integral.This orbit integral may
be simplified by noting
d
dt′exp[ik·x(t′)]=ik·v(t′)exp[ik·x(t′)] (8.15)so that Eq.(8.14) becomes
fσ 1 (x,v,t) = −
qσ
κTσφ ̃ 1 fσ 0∫t−∞dt′{
exp[−iωt′]
d
dt′exp[ik·x(t′)]}
= −
qσ
κTσ
φ ̃ 1 fσ 0{
[exp(ik·x(t′)−iωt′)]t−∞+iωIphase(x,t)}
(8.16)
where the phase-history integral is defined as
Iphase(x,t)=∫t−∞dt′exp(ik·x(t′)−iωt′).Evaluation ofIphaserequires knowledge of the unperturbed orbit trajectoryx(t′). This
trajectory, determined by solving Eqs.(8.9) with boundary conditions specified by Eq.(8.7),
has the velocity time-dependence
v(t′)=v‖Bˆ+v⊥cos[ωcσ(t′−t)]−Bˆ×v⊥sin[ωcσ(t′−t)]. (8.17)Equation (8.17) satisfies both the dynamics and the boundary conditionv(t) =vand so
gives the correct helical ‘unwinding’ into the past for a particle at itspresent position in
phase space. The position trajectory, found by integrating Eq.(8.17), is
x(t′) = x+v‖(t′−t)Bˆ
+1
ωcσ{
v⊥sin[ωcσ(t′−t)]+Bˆ×v⊥(cos[ωcσ(t′−t)]−1)} (8.18)
which satisfies the related boundary conditionx(t)=x.
To proceed further, we defineφto be the velocity-space angle between the fixed quan-
tityk⊥and the dummy variablev⊥so thatk⊥·v⊥=k⊥v⊥cosφ andk⊥·Bˆ×v⊥=
k⊥v⊥sinφ.Using this definition, the time history of the spatial part of the phase can be
written as
k·x(t′)=k·x+k‖v‖(t′−t)+k⊥v⊥
ωcσ{sin[ωcσ(t′−t)+φ]−sinφ}. (8.19)The phase integral can now be expanded
Iphase(x,t) = e−iωt∫t−∞dt′eik·x(t′)−iω(t′−t)= eik·x−iωt∫ 0
−∞dτexp
i
(
k‖v‖−ω)
τ+
k⊥v⊥
ωcσ{sin[ωcστ+φ]−sinφ}