Exact and approximate methods for Alfvén waves in dissipative atmospheres
Abstract The Alfvén wave equation in a dissipative atmosphere has been solved exactly (Campos [1–3]) or using the phase mixing approximation (Heyvaerts and Priest , Nocera, Leroy and Priest ). In the present paper, we compare the phase mixing approximation, as it appears the reference above (which we designate henceforth HP), with an exact solution of the same model problem (Section 1). It is shown that: (Section 2) the dissipative Alfvén wave equation in HP is correct for the magnetic field perturbation, but not for the velocity perturbation; (Section 3) HP makes assumes implicitly that both the static viscosity and resistive diffusivities are constant, and omits restrictions on the external magnetic field, so that we redefine the atmospheric model which is the background for the wave propagation and dissipation; (Section 4) the phase mixing ‘ansatz’, when combined with the principle of superposition, which must hold for linear waves, is incompatible with Fourier analysis; (Section 5) the exact solution for dissipative Alfvén waves in an atmosphere, demonstrates the existence of a critical level, separating regions of dominant viscous and resistive dissipation; (Section 6) none of these properties is apparent in the phase mixing approximation, which can be compared with the exact solution if three restrictions are made: (i) high- or low-altitude i.e. far from the critical layer; (ii/iii) high-frequency and weak damping. Even under these restrictions no satisfactory agreement is found.
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