Investigating the damping rate of phase-mixed Alfven waves

Context: This paper investigates the effectiveness of phase mixing as a coronal heating mechanism. A key quantity is the wave damping rate, $\gamma$, defined as the ratio of the heating rate to the wave energy. Aims: We investigate whether or not laminar phase-mixed Alfv\'en waves can have a large enough value of $\gamma$ to heat the corona. We also investigate the degree to which the $\gamma$ of standing Alfv\'en 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 of $\gamma$ in response to leakage of waves out of a loop, the quantity of this reduction, and how increasing the number of excited harmonics affects $\gamma$. Results: We find that at observed frequencies $\gamma$ is 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. We show that $\gamma$ is largest at resonance. We find our simple equation 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, predicting $\gamma$ to be orders of magnitude too large. We find that leakage acts to reduce $\gamma$ but plays a negligible role if $\gamma$ is of the order required to heat the corona. If the wave energy follows a power spectrum with slope -5/3 then $\gamma$ 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 that are parallel to the field rather than perpendicular to it. Therefore, in this case, the system is not heated mainly by phase mixing.