These three parts check warm results against cold-plasma tools from earlier chapters.

PHYSICS · Multi-part · Fermi University Physics · Plasma Physics · Waves in plasmas · Waves in plasmas

Plasma Physics, Ch Waves that need pressure or tension: Langmuir, ion-acoustic, Alfven and magnetosonic

Problem

These three parts check warm results against cold-plasma tools from earlier chapters. ((a)) The cold unmagnetised dielectric function

P(\omega)=1-\omega_{pe}^2/\omega^2

gives the parallel electrostatic resonance

P=0

\omega=\omega_{pe}

. Using the one-dimensional adiabatic closure (

\gamma_e=3

, valid because the wave moves faster than a typical electron), state how this frequency is modified at finite

k

, and evaluate

\omega/\omega_{pe}

for

k\lambda_{De}=0.25

. ((b)) The cold magnetised function

S(\omega)

has a resonance at

S=0

; in the limit

\omega_{pe}\gg|\Omega_e|

this becomes the lower hybrid frequency

\omega_{LH}^2\approx\Omega_i|\Omega_e|

. For deuterium ions (

m_i=2.014\,\mathrm{u}

) in

B_0=0.5\,\mathrm{T}

with

n_e=1.0\times10^{20}\,\mathrm{m^{-3}}

, check that

\omega_{pe}\gg|\Omega_e|

holds and evaluate

f_{LH}=\omega_{LH}/2\pi

. ((c)) The cold magnetised compressional (fast) wave travels at

v_A

for

\mathbf{k}\perp\mathbf{B}_0

. Show algebraically that the warm fast magnetosonic speed at

\theta=90^\circ

v_{fast}(90^\circ)=(v_A^2+c_s^2)^{1/2}

, reduces to

v_A

\beta\to0

, and evaluate

v_A

for the same deuterium plasma as in part (b).