A cylindrical laboratory plasma column of fully ionised hydrogen ( ) has radius and…

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

Plasma Physics, Ch Magnetohydrodynamics: Ohm's law for a moving plasma, frozen flux and magnetic pressure

Problem

A cylindrical laboratory plasma column of fully ionised hydrogen (

m_i = m_p

) has radius

a = 5\,\mathrm{cm}

and is confined by a uniform axial field

B = 0.2\,\mathrm{T}

. The plasma has

n_e = n_i = 10^{19}\,\mathrm{m^{-3}}

and

T_e = T_i = 20\,\mathrm{eV}

. Take the Coulomb logarithm as

\ln\Lambda = 10

and

Z = 1

. Use the quoted electron collision time and resistivity,

\tau_e = \frac{6\sqrt{2}\pi^{3/2}\epsilon_0^2m_e^{1/2}T_e^{3/2}}{n_iZ^2e^4\ln\Lambda},\qquad \eta_\parallel = \frac{0.51m_e}{n_ee^2\tau_e}.

((a)) Calculate

\tau_e

and the parallel resistivity

\eta_\parallel

. ((b)) The slowest resistive mode of a cylindrical column decays in

\tau_R = \mu_0a^2/(2.405^2\eta_\parallel)

. Calculate

\tau_R

. ((c)) Calculate the Alfven speed

v_A = B/(\mu_0n_im_i)^{1/2}

, the Alfven time

\tau_A = a/v_A

across the column, and hence the Lundquist number

S = \tau_R/\tau_A

. ((d)) Calculate the ion cyclotron frequency

\Omega_i = eB/m_i

and the ion inertial length

d_i = c/\omega_{pi}

, with

\omega_{pi} = (n_ie^2/\epsilon_0m_i)^{1/2}

. ((e)) A global Alfvenic disturbance crosses the column once per Alfven time, so

\omega \sim 1/\tau_A

. Using the results of (c) and (d), decide whether ideal MHD is a valid description of this disturbance, even though

S \gg 1

. Argue in terms of the two conditions

L \gg d_i

and

\omega \ll \Omega_i

, with

L=a

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