14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 409and the principle part integral in Eq.(14.58) can similarly be approximatedas
P
∫
dv∂f ̄ 0 /∂v
(v−ωr/k)^2= −
∫
dvf ̄ 0∂
∂v1
(v−ωr/k)^2= 2∫
dvf ̄ 0
(v−ωr/k)^3≃− 2
k^3
ω^3 r. (14.60)
Using Eq.(14.59), Eq. (14.56) becomes∂WP
∂t=−ω^2 p∫
dk 2 E(k,t)[
−
ωi(k)
ω^2 r+
ωr
k^2π(
∂f ̄ 0
∂v)
v=ωr/k]
(14.61)
and using Eq.(14.60), Eq. (14.58) becomes
− 2 ωik^2
ω^3 r+π(
∂f ̄ 0
∂v)
v=ωr/k=0. (14.62)
Comparison of the above two expressions shows that the second term in the square brackets
of Eq.(14.61) has twice the magnitude of the first term and is of the opposite sign. Since
ω^2 r≃ω^2 p,this means that Eq.(14.61) is of the form
∂WP
∂t=
∂
∂tWP,non−resonant+∂
∂tWP,resonant=∫
d^3 kE(k,t)[2ωi− 4 ωi] (14.63)where the 2 ωiterm prescribes the rate of change of the kinetic energy of the non-resonant
particles, and the− 4 ωiterm prescribes the rate of change of the kinetic energy of the
resonant particles. On the other hand, Eq.(14.52) showed that the rate of change ofthe total
particle kinetic energy was equal and opposite to the total rate of change of the electric field
energy. These two statements can be reconciled by asserting that the wave energy consists
of equal parts of non-resonant particle kinetic energy and electric field energy and that the
resonant particles act as a source or sink for this wave energy. This energy budgeting is
shown schematically as
∫
d^3 k 2 ωi(k)E(k,t)
︸ ︷︷ ︸
kineticenergy
of non-resonant particles
+
∫
d^3 k 2 ωi(k)E(k,t)
︸ ︷︷ ︸
energy stored in
electric field
︸ ︷︷ ︸
wave
energy⇐⇒
∫
d^3 k 4 ωi(k)E(k,t)
︸ ︷︷ ︸
kineticenergy
of resonant particles.
(14.64)
Behavior of the resonant particles
We definefresas the velocity distribution of the resonant particles, i.e., the particles
with velocitiesv≃ω/kfor whichE(k,t)is finite. Since the velocity rangev=ω(k)/k
of the resonant particles maps to the spectrumE(k,t),the upper and lower bounds of the