Fundamentals of Plasma Physics

14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 401

so that ∫

f ̄ 0 (v,t)dv=1. (14.5)

This definition causesn 0 to show up explicitly so that terms such as

(

e^2 /mε 0

)

∂f 0 /∂vcan
be written asω^2 p∂f ̄ 0 /∂v.A possible initial condition forf 0 (v,t),namely a monotonically
decreasing velocity distribution, is shown in Fig.14.1(a).
Spatial averaging will be used in the mathematical procedure to filter out some types of
terms while retaining others. This averaging will be denoted by〈〉so that, for example, the
average off 1 is

〈f 1 (x,v,t)〉=

1

L

∫

dxf 1 (x,t) (14.6)

whereLis the length of the one-dimensional system and the integration is over this length.
The spatial average of a quantity is independent ofxand so, sincef 0 is assumed to be
independent of position,〈f 0 〉=f 0 .(In an alternate version of the theory, the averaging is
instead understood to be over a statistically large ensemble of systems containing turbulent
waves and in this version, the averaged quantity can be position-dependent.)
The 1-D electron Vlasov equation is

∂

∂t

(f 0 +f 1 +f 2 +...)+v

∂

∂x

(f 1 +f 2 +...)

−

e

m

(E 1 +E 2 +...)

∂

∂v

(f 0 +f 1 +f 2 +...) = 0; (14.7)

note that there is no∂f 0 /∂xterm becausef 0 is assumed to be spatially independent and
also there is noE 0 term because the system is assumed to be neutral in equilibrium. The
linear portion of this equation

∂f 1

∂t

+v

∂f 1

∂x

−

e

m

E 1

∂f 0

∂v

=0, (14.8)

forms the basis for the linear Landau theory of plasma waves discussed in Section 4.5.
Subtracting Eq.(14.8) from Eq.(14.7) leaves the remainder equation

∂

∂t

(f 0 +f 2 +...)+v

∂

∂x

(f 2 +...)−

e

m

(E 2 +...)

∂

∂v

(f 0 +f 1 +f 2 +...)

−

e

m

E 1

∂

∂v

(f 1 +f 2 +...) = 0.

(14.9)

We assume that the quantities with subscriptsn≥ 1 are waves and therefore have spatial
averages which vanish, i.e.,〈f 1 〉= 0,〈E 1 〉= 0,etc. Also, sincef 0 is independent of
position, it is seen that〈E 2 f 0 〉=f 0 〈E 2 〉= 0,etc. Spatial averaging of Eq.(14.9) thus
annihilates many terms, leaving

∂f 0

∂t

−

e

m

∂

∂v

[〈E 1 f 1 〉+〈E 2 f 1 〉+...] (14.10)

where∂/∂vhas been factored out of the spatial averaging becausevis an independent
variable in phase-space. The term〈E 1 f 1 〉is of orderǫ^2 whereas〈E 2 f 1 〉is of orderǫ^3 and
the terms represented by the+...are of still higher order. The essential postulate of quasi-
linear theory is that all terms of orderǫ^3 and higher may be neglected becauseǫis small.