A coronal loop carries a magnetic field of magnitude

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Plasma Physics · Fluid and magnetohydrodynamic description

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

Problem

B = 0.020\,\mathrm{T}

, taken uniform in magnitude both along the loop and across it over the region considered, whose field lines are curved with radius of curvature

R_c = 3.0\times10^{7}\,\mathrm{m}

. The loop plasma is hydrogen (

m_i = m_p

) with

n_i = n_e = 1.0\times10^{15}\,\mathrm{m^{-3}}

and

T_i = T_e = 100\,\mathrm{eV}

, and its resistivity is

\eta = 9.0\times10^{-7}\,\Omega\,\mathrm{m}

. Take the loop length scale for the Alfven time and Lundquist number to be

L = R_c

. ((a)) Using the ion Larmor radius

r_{Li} = m_iv_{\perp i}/eB

with

v_{\perp i} = (2T_i/m_i)^{1/2}

, check that

r_{Li} \ll R_c

, so that treating the field as a smooth curved line (rather than resolving individual gyro-orbits) is justified. ((b)) Because

B

is uniform in magnitude along and across the loop,

\nabla_\perp(B^2/2\mu_0) = 0

, so the perpendicular-gradient (pressure) part of

\mathbf{j}\times\mathbf{B}

vanishes and the force is pure tension,

B^2\boldsymbol{\kappa}/\mu_0

with

|\boldsymbol{\kappa}| = 1/R_c

. Find its magnitude, and also find the ratio of plasma to magnetic pressure,

\beta = n(T_i+T_e)/(B^2/2\mu_0)

, treating

p = n_i T_i + n_e T_e

as the sum of the two fluid pressures. ((c)) Find the Alfven speed

v_A

, the Alfven time

\tau_A = L/v_A

, and the Lundquist number

S = \mu_0 L v_A/\eta