An earlier heating pulse has left the electrons in a one-dimensional reduced…

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Kinetic theory, collisions and transport · Kinetic theory, collisions and transport

Plasma Physics, Ch The Vlasov equation and Landau damping

Problem

An earlier heating pulse has left the electrons in a one-dimensional reduced distribution

\hat f_{0e}(v)

that is Maxwellian everywhere except for a flat plateau,

\hat f_{0e}(v)=\text{constant}

for

3.5v_{te}\le v\le 4.5v_{te}

, matched continuously onto the Maxwellian tails outside that range. A Langmuir wave in this plasma has wavenumber

k

such that its real frequency satisfies

\omega_r/k = 4.0v_{te}

, in the middle of the plateau. ((a)) Using the weak-damping formula

\gamma=(\pi/2)(\omega_{pe}^3/k^2)\hat f_{0e}'(\omega_r/k)

, predict the value of

\gamma

to leading order and justify it in one sentence. ((b)) A colleague argues that the wave must still damp weakly because the distribution is 'almost Maxwellian' apart from a small region. Explain, referring to the formula rather than to the whole distribution, why this argument is wrong.

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