Near a shear Alfven resonance layer at the local perpendicular wavenumber follows

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

Plasma Physics, Ch Waves in a non-uniform plasma: rays, reflection, mode conversion and radiofrequency heating

Problem

Near a shear Alfven resonance layer at x=xrx = x_r the local perpendicular wavenumber follows n2(x)≈a/(x−xr)n^2(x) \approx a/(x-x_r), where a=0.12 ma = 0.12\,\mathrm{m} is fixed by the density and field gradients at the layer. Following the standard treatment of resonance absorption, a small amount of dissipation, whatever its physical origin, is represented by shifting the resonance off the real axis, xr→xr−iδx_r \to x_r - i\delta with δ>0\delta > 0 small. ((a)) Show that Im n2(x)\mathrm{Im}\,n^2(x) has the Lorentzian form −aδ/[(x−xr)2+δ2]-a\delta/\left[(x-x_r)^2+\delta^2\right], and that ∫−∞∞Im n2(x) dx\int_{-\infty}^{\infty}\mathrm{Im}\,n^2(x)\,\mathrm{d}x does not depend on δ\delta. ((b)) Evaluate ∣∫−∞∞Im n2(x) dx∣\left|\int_{-\infty}^{\infty}\mathrm{Im}\,n^2(x)\,\mathrm{d}x\right| for the given aa, and state in one sentence why this means the fraction of the incident power absorbed at the layer does not depend on how the dissipation actually happens.

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