In the solar wind at the proton density is (electrons follow quasi-neutrally and…

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

Plasma Physics, Ch Cold magnetised plasma waves: the dielectric tensor, cutoffs, resonances and the CMA diagram

Problem

In the solar wind at

1\,\mathrm{AU}

the proton density is

n_i = 5.0\times10^{6}\,\mathrm{m^{-3}}

(electrons follow quasi-neutrally and their mass is negligible in the mass density) and the magnetic field is

B_0 = 5.0\times10^{-9}\,\mathrm{T}

. Treat the plasma as cold. ((a)) Starting from the single-particle polarisation drift

\mathbf{v}_{ps} = \dfrac{m_s}{q_sB_0^2}\dfrac{d\mathbf{E}_\perp}{dt}

of a species

s

(valid for

\omega\ll|\Omega_s|

), construct the polarisation current density and show that it gives a dielectric response

S = 1+\frac{\rho}{\epsilon_0B_0^2}, \qquad \rho=\sum_sn_sm_s.

((b)) Using the Alfven speed

v_A = B_0/(\mu_0\rho)^{1/2}

, show that

S = 1+c^2/v_A^2

. ((c)) For the stated solar-wind plasma calculate

v_A

S

and the refractive index

n=c/v_A

of the low-frequency mode, and check that the condition

\omega\ll|\Omega_i|

is satisfied for an ultra-low-frequency wave of

f=1.0\,\mathrm{mHz}