Plasma oscillations and Landau damping

Mathematically, Landau damping is described by the Vlasov equation, which is a kinetic equation that describes the evolution of the distribution function of particles in phase space. The Vlasov equation, along with appropriate wave-particle interaction terms, can be used to analyze and predict the damping of waves through Landau damping in plasmas. Assume a uniform plasma with a distribution

for zeroth order.

For first order, we denote the perturbation in

Since v vector is an independent variable and is not to be linearized. The first order Vlasau equation for electron is,

We assume the ions are massive and fixed.

Considering the waves a plane waves in the x direction.

Then equation 2 becomes

We have poisson's equation

Putting f1 from 4 in 5, we get

where the factor no comes when replacing fo by its normalized function.

If fo is a Maxwellian or some other factorable distribution, the distribution over Vy and Vz can be carried out easily. Then dispersion relation is

Since we are dealing with a one dimensional problem, so we drop subscript Vx

This equation is the Dispersion relation and contains a singularity at v = w/k since velocity v is a real quantity, the denominator in 6 never vanishes.

Let us take the 1-D Maxwellian distribution as

Landau was the first to treat this equation properly. He found that even though the singularity lies off the path of integration, its presence introduces an important modification to the plasma wave dispersion relation.

The integral can be divided in two parts

a. a central region denoted by ∂fo/∂v

b. a region near V = V𝜙 = w/k denoted by singularity

Case I: V<< V𝜙

Now, the integral in equation 6 is integrated by parts we get

For a Maxwellian distribution assumed in equation 7

The integral over limits - ∞ to +∞ for the odd function vanish so only the even function are left i.e

Substituting in dispersion relation, we get

if the thermal correction is small, we may take w = wp

Although an exact analysis of the problem (eq 6) is complicated, we can obtain an approximate dispersion relation for the case of large velocity phase V𝜙 >> v and weak damping (i.e small imaginary term)

Case II: V = V𝜙

The integral 6 becomes

from above equation , we get

The imaginary term which has now appeared in dispersion relation implies that the wave is either damped [ for Im (w) < 0] or amplified [ for Im(w) > 0].

In evaluating this small imaginary term, it will be sufficiently accurate to neglect the thermal correction to the real part of w and w² ≈ wp² then the principal part is approximately k² / w²

If fo is 1-D Maxwellian then

Since Im(w) is negative there is a collision less damping of plasma walls, this is called 'Landau damping' This damping is extremely small for K𝜆D but becomes important for large K𝜆D as seen in equation 12.

Landau damping arises due to the resonant interaction between the wave and particles in the plasma. When a wave propagates through a plasma, it can interact with particles that have velocities near or equal to the phase velocity of the wave. These resonant particles experience a net energy exchange with the wave, leading to the damping of the wave.

The mechanism of Landau damping can be understood through the following steps:

1. Wave-Particle Interaction: The wave interacts with particles in the plasma as they move through space. The wave exerts forces on the particles, causing them to gain or lose energy.

2. Particle Energy Exchange: If a particle's velocity is close to the phase velocity of the wave, it can gain or lose energy from the wave. The particle absorbs energy from the wave if it is moving slower than the phase velocity, or it gives up energy to the wave if it is moving faster than the phase velocity.

3. Phase Mixing: As the wave-particle interactions occur, particles with different velocities undergo energy exchange with the wave. This leads to a process called phase mixing, where the initially coherent wave is gradually spread out in velocity space.

4. Energy Dissipation: The phase mixing process results in the transfer of energy from the wave to the particles. The resonant particles that gain energy from the wave become "trapped" in regions of velocity space and tend to have their velocities modified to match the phase velocity of the wave. This leads to a net energy transfer from the wave to the particles, causing the wave to lose energy and eventually damp out.

   Landau damping is most prominent for waves that have velocities comparable to the thermal velocities of the particles in the plasma. It is an important process in the study of plasma waves and their stability. Landau damping can impact phenomena such as the propagation of electromagnetic waves in plasmas, the stability of plasma oscillations, and the behaviour of waves in controlled fusion devices.

   This note is a part of the Physics Repository.