The Cluster's Landau Damping

A page about Landau damping — collisionless damping of plasma waves due to resonance with particles near the phase velocity.

The Landau damping mechanism

Landau damping is the damping of plasma waves (typically Langmuir waves or ion acoustic waves) due to energy exchange with particles whose velocity is near the wave's phase velocity v_phi = omega / k. Particles slightly slower than the wave are accelerated (gain energy from the wave); particles slightly faster are decelerated (lose energy to the wave). The net effect is determined by the slope of the velocity distribution at v_phi: damping rate gamma ~ - (pi omega_p^2 / k^2) (df_0/dv) |_{v=omega/k} If df_0/dv < 0 (more particles slower than v_phi than faster), the wave loses energy — damping. If df_0/dv > 0 (inverted population), the wave gains energy — instability. In the cluster, the edit Landau damping mechanism gives an edit damping rate.

The dispersion relation

For a cold plasma: omega^2 = omega_p^2 + 3 k^2 v_th^2 (Bohm-Gross dispersion). For a warm plasma, the kinetic theory gives: 1 = (omega_p^2 / k^2) integral (df_0/dv) / (v - omega/k) dv where the integral is taken as a Cauchy principal value. Landau solved this by deforming the contour into the complex plane, picking up the pole at v = omega/k. The real part of omega gives the frequency shift; the imaginary part gives the damping rate.

In the cluster, the edit dispersion relation is solved by an edit contour deformation.

The plasma dispersion function

The plasma dispersion function is defined as: Z(zeta) = (1 / sqrt(pi)) integral_{-infinity}^{infinity} exp(-t^2) / (t - zeta) dt where zeta = omega / (k v_th sqrt(2)). For zeta >> 1 (small damping): Z(zeta) ~ -1/zeta - 1/(2 zeta^3) - ... - i sqrt(pi) exp(-zeta^2) The real part gives the frequency shift; the imaginary part (from the pole) gives the Landau damping.

In the cluster, the edit plasma dispersion function encodes an edit frequency shift and an edit damping.

The applications

• Fusion plasmas: Landau damping of ion acoustic waves, electron Bernstein waves
• Space plasmas: Wave-particle interactions in the magnetosphere and solar wind
• Plasma heating: Landau damping is used for heating fusion plasmas (ECRH, ICRH, LHCD)
• Particle accelerators: Collective effects in charged particle beams
• Laser-plasma interactions: Stimulated Raman scattering, Brillouin scattering

In the cluster, edit applications include:

• edit Fusion plasmas
• edit Space plasmas
• edit Plasma heating
• edit Particle accelerators
• edit Laser-plasma interactions

The Vlasov equation

Landau damping is derived from the Vlasov-Poisson system: df/dt + v . nabla f + (E/m) . grad_v f = 0 nabla . E = rho / epsilon_0 where f(x,v,t) is the distribution function. The Vlasov equation describes the collisionless evolution of the plasma. In the cluster, the edit Vlasov equation describes an edit collisionless evolution.

This damping

This page is about Landau damping. gamma ~ - (pi omega_p^2/k^2) (df_0/dv)|_{v=omega/k}. df_0/dv < 0 -> damping. Z(zeta) from plasma dispersion. Vlasov equation. The damping is real.