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The present paper concerns Alfven waves, in a resistive and viscous atmosphere, under a steep temperature gradient (Section 1). The dissipative Alfven wave equation is deduced assuming uniform vertical background magnetic field, and allowing for arbitrary profiles of Alfven speed, and viscous and resistive diffusivities as functions of altitude (Section 2). A three-parameter family of temperature profiles, allowing for independent choice of initial and asymptotic temperature, and of initial temperature gradient, is used to re-write the wave equation, with the temperature as the independent variable, instead of altitude (Section 3). It is shown that, for the conditions prevailing in the solar transition region between the chromosphere and corona, two approximations of the dissipative wave equations may be considered, the simplest leading to solution in terms of Gaussian hypergeometric functions (Section 4). The exact analytical solution allows calculation of the (i) velocity and (ii) magnetic field perturbations, (iii) kinetic, (iv) magnetic and (v) total energy density, (vi) energy flux, (vii) rate-of-strain and (viii) electric current, and (ix) viscous, (x) resistive and (xi) total rate of dissipation (Section 5). These are plotted versus temperature, across the transition region from the chromosphere to the corona, for the quiet and active Sun (Section 6). The feasibility of heating of the transition region by dissipation of Alfven waves is discussed (Section 7), by comparing empirical heating rates, with theoretical values for a range of physical conditions, including initial velocity perturbations 5 to 15 km s −1, background magnetic field 12 to 120 G, wave periods 60 to 300 s, thickness of the transition region 100 to 300 km, resistive and anomalous diffusivities χ to 100 χ and viscous and turbulent diffusivities ν to 100 ν. The conclusion is that dissipation of Alfven waves is not an effective heating mechanism for the transition region and corona, although it may be for the chromosphere (see Campos and Mendes, 1995, and references therein). |
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