Ed. note: The Bass-cited paper by Schluter and Lust is required reading in magnetohydrodynamic courses, such as the one at the University of Montana (PHY! 515: Plasma Physics & MHD). Sample homework assignment is below: (http ://soiar. physics, montana .edu/martens/plasma/calendar, html)
1. Working with the MHD equations. Consider an inviscid plasma of uniform resistivity with azimuthal symmetry, that is in a stale of steady (but possibly non-uniform) rotation about the z-axis and permeated with a magnetic field that has no azimuthal component.
where j, v, and B are vectors with the usual meaning, eta is the resistivity, and c as usual the velocity of light.
b) Prove that the current is wholly azimuthal.
c) Use div.B=0 to show that the plasma has constant angular velocity, (omega=v_phi/r)
2. Force-free fields. Consider the Lust and Schlueter expression for force-free fields with cylindrical symmetry (i.e. axial plus azimuthal symmetry).
a) For the constant twist field, that we covered in class, derive an expression for alpha, the ratio between current and magnetic field vector.
b) The magnetic field vector is tangent to the magnetic field line at any point. Hence the tangent vector satisfies rdphi dz dr
B_phi B_z B_r which is the field line equation. From the solution for a constant twist field derive the number of turns in the fieldlines on an axial segment of length L.
c)Now assume alpha is constant and use that to simplify the Lust and Schluter differential equation to derive Bessel's equation. Find and sketch the solutions for B_z and B_phi.
Robert W. Bass is a physicist working for an aerospace firm and can be reached at 45960 Indian Way #612, Lexington Park, MD 20653, [email protected]
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