Problem 14 - In the simplest case E ’ obeys Ohm’s Law...

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Phys761/F10/Hassam/Problem 14 Alfven waves and resistivity 14.1 Consider a homogenous equilibrium with constant B 0 pointing in the z ^ direction. Derive the dispersion relation for ideal Alfven waves, with B ~ polarized along x ^ and the k vector lying in the y-z plane but not parallel to B 0 . Equilibrium density = n 0 . [Start with B x ~ and follow what it drives, to close the set.] Make a sketch of the total magnetic field lines, B = B 0 + x ^ B x ~ , as they would appear at any instant with respect to the y-z plane. 14.2 The condition E = - u x B /c is a statement of the fact that the electric field in the frame of the plasma, E ’, is zero, because the plasma is a very good conductor. In practice, however, E ’ is not exactly zero.
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Unformatted text preview: In the simplest case, E ’ obeys Ohm’s Law, ie, E ’ = & j , where j is the current density and & is the resistivity. Rederive the dispersion relation for the plane polarized Alfven waves of 14.1 including the resistivity. Comment on the physics of your result for the limiting case k.B = 0, ie, explain which purely k_perp mode (as discussed in class and in the notes) is being modified by resistivity and how so. Then, for general k , solve the dispersion relation for “small” resistivity, to first order. Describe the time evolution of the Alfven wave in this case. Define “small”. What would happen to the Alfven wave with resistivity if k y = 0....
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This document was uploaded on 12/29/2011.

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