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An electrical conductivity tensor is used in a 1.5D magnetohydrodynamic (MHD) simulation to describe how MHD shock waves may form, propagate, and heat the photosphere and chromosphere by compression and resistive dissipation. The spatial resolution is 1 km. A train of six shock waves is generated by a sinusoidal magnetic field driver in the photosphere with a period T = 30 s, mean of 500 G, and variation of 250 G. The duration of the simulation is 200 s. Waves generated in the photosphere evolve into shock waves at a height z ~ 375 km above the photosphere. The transition of the atmosphere from weakly to strongly magnetized with increasing height causes the Pedersen resistivity η P to increase to ~2000 times the Spitzer resistivity. This transition occurs over a height range of a few hundred kilometers near the temperature minimum of the initial state at z ~ 500 km. The initial state is a model atmosphere derived by Fontenla et al., plus a background magnetic field. The increase in η P is associated with an increase in the resistive heating rate Q. Shock layer thicknesses are ~10-20 km. They are nonzero due to the presence of resistive dissipation, so magnetization-induced resistivity plays a role in determining shock structure, and hence the compressive heating rate Qc . At t = 200 s the solution has the following properties. Within shock layers, Q maximum ~ 1.4-7 erg cm–3 s–1, and Q c,maximum ~ 10-103 Q maximum. Between shock waves, and at some points within shock layers, Qc < 0, indicating cooling by rarefaction. The integrals of Q and Qc over the shock wave train are F ~ 4.6 × 106 erg cm–2 s–1 and Fc ~ 1.24 × 109 erg cm–2 s–1. A method based on the thermal, mechanical, and electromagnetic energy conservation equations is presented for checking the accuracy of the numerical solution, and gaining insight into energy flow and transformation. The method can be applied to higher dimensional simulations. It is suggested that observations be performed to map out the transition region across which the transition from weakly ionized, weakly magnetized plasma to weakly ionized, strongly magnetized plasma occurs, and to correlate it with net radiative loss.

A 1.5-dimensional, time-dependent magnetohydrodynamic (MHD) model that includes an energy equation and anisotropic electrical conductivity tensor for a variably ionized, multispecies plasma is presented. The model includes an algorithm that reduces the numerical dissipation rate far below the dissipation rate determined by the conductivity tensor. This is necessary for accurate calculation of resistive heating rates. The model is used to simulate the propagation of Alfven waves launched near the base of the middle chromosphere. The background state is the FAL CM equilibrium with a vertical magnetic field. The initial magnetic energy of a wave is almost completely damped out in the chromosphere by the time the disturbance propagates a distance of one wavelength. The energy is converted mainly into thermal energy. The remainder is converted into bulk flow kinetic energy and a Poynting flux with nonzero divergence. The thermal energy is generated almost entirely by Pedersen current dissipation. The corresponding heating rates are close to the FAL CM values near the base of the middle chromosphere. Dynamo action is observed. The damping of a continuously driven Alfven wave train is also simulated, yielding results similar to those of the single wave cases. It is the strong magnetization and weak ionization of the chromosphere that allows for strong heating by Pedersen current dissipation. This distinguishes the chromosphere from the weakly magnetized and weakly ionized photosphere, and the strongly magnetized and strongly ionized corona where Pedersen current dissipation is not significant on the length and timescales simulated.