Landau Damping

Our starting point is the Vlasov equation for an unmagnetized, collisionless
plasma:

where

Here, is the number density of ions (which is the same as the number density of electrons).

Since we are dealing with small amplitude waves, it is appropriate to
*linearize* the Vlasov equation. Suppose that the electron
distribution function is written

(988) |

and

respectively.

Let us now follow the standard procedure for analyzing small amplitude
waves, by assuming that all perturbed quantities vary with
and like
.
Equations (989) and (990) reduce to

(991) |

(992) |

We can interpret Eq. (993) as the dispersion relation for electrostatic plasma
waves, relating the wave-vector, , to the frequency, .
However, in doing so, we run up against a serious problem, since the integral has
a *singularity* in velocity space, where
,
and is, therefore, not properly defined.

The way around this problem was first pointed out by Landau^{} in a very
influential paper which laid the basis of much subsequent
research on plasma oscillations and instabilities. Landau showed that,
instead of simply assuming that varies in time as
,
the problem must be regarded as an initial value problem in which
is given at and found at later times.
We may still Fourier analyze with respect to , so we write

(994) |

and

In order to solve Eqs. (995) and (996) as an initial value problem, we
introduce the Laplace transform of with respect to :

(997) |

Noting that the Laplace transform of
is
(as is easily shown by integration by parts), we can Laplace transform Eqs. (995)
and (996) to obtain

(998) |

(999) |

(1000) |

where

(1002) |

The Laplace transform of the distribution function is written

(1003) |

Having found the Laplace transforms of the electric field and the perturbed
distribution function, we must now invert them to obtain
and as functions of time. The inverse Laplace transform
of the distribution function is given by

Rather than trying to obtain a general expression for , from
Eqs. (1004) and (1005), we shall concentrate on the behaviour of the
perturbed distribution function at *large* times. Looking at
Fig. 31, we note that if
has only a finite
number of simple poles in the region
, then
we may deform the contour as shown in Fig. 32, with a loop around
each of the singularities. A pole at gives a contribution
going as
, whilst the vertical part of the
contour goes as
. For sufficiently long times
this latter contribution is negligible, and the behaviour is
dominated by contributions from the poles furthest to the right.

Equations (1001)-(1004) all involve integrals of the form

If is sufficiently well-behaved that it can be continued off the real axis as an analytic function of a complex variable then the continuation of (1006) as the singularity crosses the real axis in the complex -plane, from the upper to the lower half-plane, is obtained by letting the singularity take the contour with it, as shown in Fig. 33.

Note that the ability to deform the contour into that of Fig. 32, and find
a dominant contribution to and
from a few poles, depends on and
having *smooth* enough
velocity dependences that the integrals appearing in
Eqs. (1001)-(1004) can be continued sufficiently far into the left-hand
half of the complex -plane.

If we consider the electric field given by the inversion of Eq. (1001), we see that its behaviour at large times is dominated by the zero of which lies furthest to the right in the complex -plane. According to Eqs. (1004) and (1005), has a similar contribution, as well as a contribution going as . Thus, for sufficiently long times after the initiation of the wave, the electric field depends only on the positions of the roots of in the complex -plane. The distribution function has a corresponding contribution from the poles, as well as a component going as . For large times, the latter component of the distribution function is a rapidly oscillating function of velocity, and its contribution to the charge density, obtained by integrating over , is negligible.

As we have already noted, the function
is equivalent to the
left-hand side of Eq. (993), provided that is replaced by
.
Thus, the dispersion relation, (993), obtained via Fourier transformation of the
Vlasov equation,
gives the correct behaviour at large times as long as the singular integral
is treated correctly. Adapting the procedure which we found using the
variable , we see that the integral is defined as it is written for
, and analytically continued, by deforming the
contour of integration in the -plane (as shown in Fig. 33), into the region
. The simplest way to remember how to do the
analytic continuation is to note that the integral is
continued from the part of the -plane corresponding to *growing*
perturbations, to that corresponding to *damped* perturbations. Once we
know this rule, we can obtain kinetic dispersion relations in a fairly direct manner
via Fourier
transformation of the Vlasov
equation, and there is no need to attempt the more complicated Laplace transform
solution.

In Sect. 4, where we investigated the cold-plasma dispersion relation, we found that
for any given there were a finite number of values of , say ,
, , and a general solution was a linear superposition of
functions varying in time as
,
, *etc*. This set of values of is called the
*spectrum*, and the cold-plasma equations yield a *discrete* spectrum.
On the other hand, in the kinetic problem we obtain contributions
to the distribution function going as
,
with taking any real value.
All of the mathematical difficulties of the kinetic
problem arise from the existence of this *continuous spectrum*. At
short times, the behaviour is very complicated, and depends on the details
of the initial perturbation. It is only asymptotically that a mode
varying as
is obtained, with determined
by a dispersion relation which is solely a function of the unperturbed state.
As we have seen, the emergence of such a mode depends on the initial velocity
disturbance being sufficiently smooth.

Suppose, for the sake of simplicity, that the background plasma state is a
Maxwellian distribution. Working in terms of , rather than , the kinetic dispersion
relation for electrostatic waves takes the form

(1008) |

(1009) |

Suppose that is sufficiently small that
over the
range of where
is non-negligible. It follows
that we can expand the denominator of the principal part integral in a
Taylor series:

(1010) |

(1011) |

where is the Debye length, and it is assumed that . We can regard the imaginary part of as a small perturbation, and write , where is the root of Eq. (1012). It follows that

(1013) |

giving

If we compare the above results with those for a cold-plasma, where
the dispersion relation for an electrostatic plasma wave was found to
be simply
, we see, firstly, that now depends on ,
according to Eq. (1012), so that in a warm plasma the electrostatic plasma
wave is a *propagating* mode, with a non-zero group velocity. Secondly, we
now have
an imaginary part to , given by Eq. (1015), corresponding, since
it is negative, to the *damping* of the wave in time. This damping is generally
known as *Landau damping*. If
(*i.e.*,
if the
wave-length is much larger than the Debye length) then the imaginary part
of is small compared to the real part, and the wave is only
lightly damped. However, as the wave-length becomes comparable to the
Debye length, the imaginary part of becomes comparable to the
real part, and the damping becomes strong.
Admittedly, the approximate solution given above
is not very accurate in the short wave-length case, but it is sufficient to indicate
the existence of very strong damping.

There are *no* dissipative effects included in the collisionless Vlasov equation.
Thus, it can easily be verified that if the particle velocities are
reversed at any time then the solution up to that point is simply reversed in
time. At first sight, this reversible behaviour does not seem to be
consistent with the fact that an initial perturbation dies out. However,
we should note that it is only the electric field which decays. The
distribution function contains an undamped term going as
. Furthermore, the decay of the electric field depends on there being a
sufficiently smooth initial perturbation in velocity space. The presence
of the
term means that as time advances the velocity space dependence of the
perturbation becomes more and more convoluted. It follows that if we
reverse the velocities after some time then we are not starting
with a smooth distribution. Under these circumstances, there is
no contradiction in the fact that under time reversal the electric field will
grow initially, until the smooth initial state is recreated, and subsequently
decay away.