Investigating the damping rate of phase-mixed Alfvén waves

Abstract

Context.This paper investigates the effectiveness of phase mixing as a coronal heating mechanism. A key quantity is the wave damping rate,γ, defined as the ratio of the heating rate to the wave energy.Aims.We investigate whether or not laminar phase-mixed Alfvén waves can have a large enough value ofγto heat the corona. We also investigate the degree to which theγof standing Alfvén waves which have reached steady-state can be approximated with a relatively simple equation. Further foci of this study are the cause of the reduction ofγin response to leakage of waves out of a loop, the quantity of this reduction, and how increasing the number of excited harmonics affectsγ.Methods.We calculated an upper bound forγand compared this with theγrequired to heat the corona. Analytic results were verified numerically.Results.We find that at observed frequenciesγis too small to heat the corona by approximately three orders of magnitude. Therefore, we believe that laminar phase mixing is not a viable stand-alone heating mechanism for coronal loops. To arrive at this conclusion, several assumptions were made. The assumptions are discussed in Sect. 2. A key assumption is that we model the waves as strictly laminar. We show thatγis largest at resonance. Equation (37) provides a good estimate for the damping rate (within approximately 10% accuracy) for resonant field lines. However, away from resonance, the equation provides a poor estimate, predictingγto be orders of magnitude too large. We find that leakage acts to reduceγbut plays a negligible role ifγis of the order required to heat the corona. If the wave energy follows a power spectrum with slope −5/3 thenγgrows logarithmically with the number of excited harmonics. If the number of excited harmonics is increased by much more than 100, then the heating is mainly caused by gradients that are parallel to the field rather than perpendicular to it. Therefore, in this case, the system is not heated mainly by phase mixing.