Aims. This paper presents 2.5D numerical experiments of Alfvén wave phase mixing and aims to assess the effects of nonlinearities on wave behaviour and dissipation. In addition, this paper aims to quantify how effective the model presented in this work is at providing energy to the coronal volume. Methods. The model is presented and explored through the use of several numerical experiments which were carried out using the Lare2D code. The experiments study footpoint driven Alfvén waves in the neighbourhood of a two-dimensional x-type null point with initially uniform density and plasma pressure. A continuous sinusoidal driver with a constant frequency is used. Each experiment uses different driver amplitudes to compare weakly nonlinear experiments with linear experiments. Results. We find that the wave trains phase-mix owing to variations in the length of each field line and variations in the field strength. The nonlinearities reduce the amount of energy entering the domain, as they reduce the effectiveness of the driver, but they have relatively little effect on the damping rate (for the range of amplitudes studied). The nonlinearities produce density structures which change the natural frequencies of the field lines and hence cause the resonant locations to move. The shifting of the resonant location causes the Poynting flux associated with the driver to decrease. Reducing the magnetic diffusivity increases the energy build-up on the resonant field lines, however, it has little effect on the total amount of energy entering the system. From an order of magnitude estimate, we show that the Poynting flux in our experiments is comparable to the energy requirements of the quiet Sun corona. However a (possibly unphysically) large amount of magnetic diffusion was used however and it remains unclear if the model is able to provide enough energy under actual coronal conditions.

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 of γ to heat the corona. We also investigate the degree to which the γ of 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 of γ in response to leakage of waves out of a loop, the quantity of this reduction, and how increasing the number of excited harmonics affects γ. Methods. We calculated an upper bound for γ and compared this with the γ required to heat the corona. Analytic results were verified numerically.Results. We find that at observed frequencies γ 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. 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, 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 that are parallel to the field rather than perpendicular to it. Therefore, in this case, the system is not heated mainly by phase mixing.

This paper uses linear magnetohydrodynamics to model resonant absorption in coronal plasma with a Cartesian coordinate system. We impose line-tied boundary conditions and tilt the background magnetic field to be oblique to the transition region. Halberstadt & Goedbloed, Goedbloed & Halberstadt, and Arregui et al. show that line-tied boundary conditions cause their resonant absorption models to produce steep boundary layers/evanescent fast waves. We aim to study the importance of boundary layers and assess their significance in a solar context. We calculate solutions in a model where we impose line-tied boundary conditions and compare this with a model where we include the chromosphere instead. Results are calculated analytically and then verified numerically. We show that line-tied boundary conditions can cause the model to overestimate the boundary layers’ amplitude significantly. If the fast waves can propagate in the chromosphere, then the line-tied model accurately predicts the boundary layers’ amplitude. However, if the fast waves are evanescent, then the boundary layers’ size is reduced significantly, and the line-tied model overestimates their amplitude. This leads to the counterintuitive result that length scales tangential to the transition region can play an essential role in determining line-tied boundary conditions’ validity. The results suggest that line-tied boundary conditions can cause the model to generate unphysically large boundary layers. However, researchers may wish to continue to use them in their models for their simplicity and ability to significantly reduce computation time if they understand and are aware of their flaws.

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This paper uses linear magnetohydrodynamics to model resonant absorption in coronal plasma with a Cartesian coordinate system. We impose line-tied boundary conditions and tilt the background magnetic field to be oblique to the transition region. Halberstadt & Goedbloed (1993, 1995Goedbloed & Halberstadt (1994);Arregui et al. (2003) show that line-tied boundary conditions cause their resonant absorption models to produce steep boundary layers/evanescent fast waves. We aim to study the importance of the boundary layers and assess their significance in a solar context. We calculate the solutions in a model where we impose line-tied boundary conditions and compare this with a model where we include the chromosphere instead. Results are calculated analytically and then verified numerically. We show that line-tied boundary conditions can cause the model to overestimate the boundary layers' amplitude significantly. If the fast waves can propagate in the chromosphere, then the line-tied model accurately predicts the boundary layers' amplitude. However, if the fast waves are evanescent, then the boundary layers' size is reduced significantly, and the line-tied model overestimates their amplitude. This leads to the counter-intuitive result that the length scales tangential to the transition region can play an essential role in determining line-tied boundary conditions' validity. The results suggest that line-tied boundary conditions can cause the model to generate unphysically large boundary layers. However, researchers may wish to continue to use them in their models for their simplicity and ability to significantly reduce computation time if they understand and are aware of their flaws.