Investigating the damping rate of phase-mixed Alfvén waves

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. This paper is primarily concerned with answering the question: Can laminar phase-mixed Alfvén waves have a large enough value of γ to heat the corona? Other questions this paper aims to answer are: How well can the γ of standing Alfvén waves which have reached steady-state be approximated with a relatively simple equation, namely, equation (3.5)? Why does leakage of waves out of a loop reduce γ and by how much? How does increasing the number of excited harmonics affect γ?
Methods. We calculate an upper bound for γ and compare this with the γ required to heat the corona. Analytic results are verified numerically.
Results. We find that γ is too small at observed frequencies by approximately 3 orders of magnitude to heat the corona. Therefore, we believe that laminar phase mixing is not a viable standalone heating mechanism for coronal loops. To arrive at this conclusion, several assumptions were made. The assumptions are discussed in Section 2.1. A key assumption is that we model the waves as strictly laminar. We show that γ is largest at resonance. Equation (3.5) 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, with it predicting γ to be orders of magnitude too large. We find that leakage acts to reduce γ but plays a negligible role if γ is of the order required to heat the corona. If the wave energy follows a power spectrum with slope -5/3 then γ grows 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 parallel to the field rather than perpendicular. Therefore, in this case, the system is not heated mainly by phase mixing.