On dissipative Alfvén waves in atmospheres with steep temperature gradients

Abstract

The present paper concerns (Section 1) dissipative Alfvén waves in an atmosphere with a steep temperature gradient, such that the background temperature varies on a scale shorter than the wavelength; an application is the transition region between the solar chromosphere and corona (Section 7). for which the temperature rises from 104 K to 1.8 × 106 K over a distance of a few hundred kilometers. The starting point is the Alfvén wave equation, in an atmosphere under an uniform vertical magnetic field, allowing (Section 2) the Alfvén speed and viscous and resistive diffusivities to be arbitrary functions of altitude. A three-parameter family of temperature temperature profiles is introduced, allowing independent choice (Section 3) of initial and asymptotic temperature, and initial temperature gradient. The wave equation for the magnetic field perturbation is written with the temperature as independent variable, instead of altitude. This leads to a second-order wave equation, whose coefficients are polynomials of degree ten, and which is solved (Section 4) in terms of power series, allowing the application of boundary, asymptotic, radiation and dissipation conditions (Section 5). The wave fields are plotted, versus altitude (Figs. 4, 5, 7) or temperature (Figs. 3,6,8), for several values of dimensionless frequency and viscous and resistive damping, of the order of magnitude of those relevant (Section 6) to dissipative Alfvén waves in the solar transition region, between the chromosphere and corona.