230 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma
Consider an electrostatic wave with potential
φ 1 (x,t)=φ ̃ 1 exp(ik·x−iωt). (8.3)As before the convention will be used that a tilde refers to the amplitude of aperturbed
quantity;if there is no tilde, then the exponential phase factor is understood to be included.
Because the wave is electrostatic, Poisson’s equation is the relevant Maxwell’s equation
relating particle motion to fields, i.e.,
k^2 φ 1 =1
ε 0∑
σnσ 1 qσ (8.4)wherenσ 1 is the density perturbation for each speciesσ. Since the density perturbation is
just the zeroth moment of the perturbed distribution function,
nσ 1 =∫
fσ 1 d^3 v, (8.5)the problem reduces to determining the perturbed distribution functionfσ 1.
In the presence of a uniform magnetic field the linearized Vlasov equation is
∂fσ 1
∂t
+v·∂fσ 1
∂x+
qσ
mσ(v×B)·∂fσ 1
∂v=
qσ
mσ∇φ 1 ·∂fσ 0
∂v(8.6)
where the subscript 0 refers to equilibrium quantities and the subscript 1 tofirst-order per-
turbations (no 0 has been used for the magnetic field, because the wave has been assumed
to be electrostatic and so does not have any perturbed magnetic field, thusBis the equilib-
rium magnetic field).
Consider an arbitrary pointx,vin phase space at timet.All particles at this pointx,v
at timethave identical phase-space trajectories in both the future and the pastbecause the
particles are subject to the same forces and have the same temporal initial condition. By
integrating the equation of motion starting from this point in phase space,the phase-space
trajectoryx(t′),v(t′)can be determined.The boundary conditions on such a phase-space
trajectory are simply
x(t)=x, v(t)=v. (8.7)
Instead of treatingx,vas independent variables denoting a point in phase space, let us
think of these quantities as temporal boundary conditions for particles with phase-space
trajectoriesx(t),v(t)that happen to be at locationx,vat timet.Thus, the velocity distri-
bution function for all particles that happen to be at phase-space locationx,vat timetis
fσ 1 =fσ 1 (x(t),v(t),t)and sincexandvwere arbitrary, this expression is valid for all
particles. The time derivative of this function is
d
dtfσ 1 (x(t),v(t),t)=∂fσ 1
∂t+
∂fσ 1
∂x·
dx
dt+
∂fσ 1
∂v·
dv
dt. (8.8)
In principle, one ought to take into account the wave force on the particles when calculating
their trajectories. However, if the wave amplitude is small enough, the particle trajectory
will not be significantly affected by the wave and so will be essentially the same as the
unperturbed trajectory, namely the trajectory the particle would have had if there were no
wave. Since the unperturbed particle trajectory equations are
dx
dt=v,dv
dt=
qσ
mσ(v×B) (8.9)