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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.

A magnetohydrodynamic (MHD) model of an electron-ion, collision-dominated plasma that includes the electrical conductivity and thermoelectric tensors in Ohm's law is used to generate current sheet solutions in parameter ranges that correspond to those of the solar transition region and lower corona. The model contains a prescribed sheared magnetic field with a characteristic length scale L. The characteristic sheet width is 2L, but it is found that the temperature has transition region or coronal values only within a diffusion region (DR) with a width several orders of magnitude smaller than 2L. The heating rate per unit mass and flow speed in the DR are orders of magnitude larger, and the density is orders of magnitude smaller than in the surrounding plasma. The heating rate per unit volume is a maximum in the DR and falls off steadily outside the DR. The Joule heating rate and current density each consist of a conduction component driven by the center-of-mass electric field and a thermoelectric component driven by the temperature gradient. It is found that these components largely cancel, leading to a total heating rate and current density orders of magnitude smaller than either of their components. This suggests that thermoelectric current drive is important in determining current sheet structure. The center-of-mass electric field that provides the energy to maintain the plasma in a steady state is almost entirely the convection electric field. The electron magnetization Me is the product of the electron cyclotron frequency and the electron-ion collision time. Nonzero values of Me cause the conductivity and thermoelectric tensors to be anisotropic. It is found that the large values of Me that occur in the DR increase the heating rates per unit volume and mass by several orders of magnitude and can change the sign of the heating rate per unit mass from negative to positive, corresponding to a change from a cooling process to a heating process. This suggests that electron magnetization, and hence anisotropic transport, is a major factor in current sheet heating.

Three points and research directions are discussed: The outstanding problem of identifying the mechanisms of solar atmospheric heating and wind acceleration can be solved only by combining quantitative models that include realistic descriptions of relevant transport processes with observational constraints on the inputs and outputs of these models. Most solar atmospheric heating, with the possible exception of flares, takes place in the chromosphere, emphasizing the importance of identifying the mechanisms of chromospheric heating, which may be important for understanding coronal heating and wind acceleration. Recent modeling leads to the conclusion that the onset of proton magnetization with increasing height in thin magnetic flux tubes triggers the onset of chromospheric network heating by resistive dissipation of Pedersen currents driven by the convection electric field of slow, longitudinal magnetoacoustic waves.

A mechanism for chromospheric network heating and a necessary and sufficient condition for its onset are presented. The heating mechanism consists of resistive dissipation of proton Pedersen currents, which flow orthogonal to the magnetic field in weakly ionized chromospheric plasma. The currents are driven by a convection electric field generated by velocity oscillations of linear, slow, longitudinal magnetoacoustic waves with frequencies ? 3.5 mHz in the lower chromosphere. The heating occurs in thin magnetic flux tubes and begins lower in the chromosphere in flux tubes with higher photospheric field strength. The lower chromosphere, which emits most of the net radiative loss in the network, is heated by flux tubes with photospheric field strengths ~700-1500 G. A typical field strength and core diameter for a flux tube in the lower chromosphere with a core heating rate of 107 ergs cm-2 s-1 are 170 G and 10 km. This core region is contained in a region with a diameter ~100 km in which the heating rate is an order of magnitude smaller. About N ~ 102 of these flux tubes distributed over the boundary region of a granule with a diameter ~103 km provide an average heating rate over the entire granule ~107 ergs cm-2 s-1. If the core heating rate is changed by a factor f, then N ~ f-1/2102. The condition for the onset of heating is that the ratio of the proton cyclotron frequency to the proton-hydrogen collision frequency equal unity. This ratio increases with height, and the condition is satisfied at a single height in a given flux tube. At this height, control of the proton dynamics begins to be dominated by the magnetic field rather than by collisions with hydrogen, and the anisotropic nature of the electrical conductivity begins to play a critical role in resistive dissipation. The protons become magnetized. Heating by dissipation of heavy ion and, to a lesser extent, proton Pedersen currents causes the temperature to start increasing. The heating increases hydrogen ionization. With increasing height, and hence proton magnetization, the Pedersen current density rapidly increases with hydrogen ionization via positive feedback, and the proton number density rapidly reaches and exceeds the heavy ion number density, resulting in an increase in heating rate by an order of magnitude over only 1 pressure scale height. During this process the protons rapidly dominate the Pedersen current. Heating by dissipation of magnetic field aligned currents is insignificant. Below the height in the atmosphere at which the onset condition is satisfied, any current orthogonal to the magnetic field must be primarily a Hall current, which is nondissipative. Heating by this mechanism must occur to some degree in the chromospheric network of all solar-type stars. It is proposed to be the dominant mechanism of chromospheric network heating, although viscous dissipation may also be important if the core heating rate is much larger than ~107 ergs cm-2 s-1 or if the linear MHD waves studied here evolve into shock waves with increasing height. Flux tubes in the quiet chromosphere are predicted to have two possible core diameters: ~10 km, corresponding to flux tubes in which network heating occurs, and ~104-105 km, perhaps corresponding to flux tubes in which active region heating might occur. The model has a singularity at the acoustic cutoff frequency, corresponding to periods near 3 minutes. Therefore, unless nonresistive damping mechanisms such as viscous dissipation and thermal conduction provide sufficiently strong damping, MHD oscillations with periods near 3 minutes in chromospheric magnetic flux tubes must be nonlinear.

The transition region (TR) is assumed to be a collision-dominated plasma. The dissipation and transport of energy in such a plasma is accurately described by the classical transport coefficients, which include the electrical and thermal conductivity, viscosity, and thermoelectric tensors. These tensors are anisotropic and are functions of local values of temperature, density, and magnetic field strength. The transport coefficients are valid for all magnetic field strengths and so may be used to study the physics of weakly as well as strongly magnetized regions of the TR. They may be used in an MHD model to obtain a self-consistent, realistic description of the TR. The physics of kinetic processes is included in the MHD model through the transport coefficients. As a first step in studying heating and cooling processes in the TR in a realistic, quantitative manner, a 1.5 dimensional, steady state MHD model with a specified temperature profile is developed. The momentum equation includes the inertial, pressure, magnetic, and gravitational forces. Ohm's law includes the exact expressions for the electrical conductivity and thermoelectric tensors. It is found that the contribution of the dissipation of large-scale electric currents to in situ heating of the TR is negligible, but that thermal energy flowing into the TR from the corona can provide the energy required to heat the TR. The possibility that significant in situ heating of the TR takes place through viscous dissipation or small-scale electric current dissipation such as may occur in current sheets or filaments is discussed, although these processes are not described by the model. The importance of thermoelectric and electron pressure gradient effects in Ohm's law, and in determining the electron heat flux, is demonstrated. Results of the model suggest that the force-free approximation is not valid over most of the TR. Justification for assuming that the TR is collision dominated is presented. In particular, a self-consistent calculation of the ratio of the electric field parallel to the magnetic field to the Dreicer electric field yields a value 10-3, which suggests that anomalous transport processes are not important. The necessity of using a realistic description of transport processes in modeling heating mechanisms in the solar atmosphere is stressed.

A magnetohydrodynamic model that includes a complete electrical conductivity tensor is used to estimate conditions for photospherically driven, linear, non-plane Alfvenic oscillations extending from the photosphere to the lower corona to drive a chromospheric heating rate due to Pedersen current dissipation that is comparable to the net chromospheric net radiative loss of $\sim 10^7$ ergs-cm$^{-2}$-sec$^{-1}$. The heating rates due to electron current dissipation in the photosphere and corona are also computed. The wave amplitudes are computed self-consistently as functions of an inhomogeneous background (BG) atmosphere. The effects of the conductivity tensor are resolved numerically using a resolution of 3.33 m. The oscillations drive a chromospheric heating flux $F_{Ch} \sim 10^7 - 10^8$ ergs-cm$^{-2}$-sec$^{-1}$ at frequencies $\nu \sim 10^2 - 10^3$ mHz for BG magnetic field strengths $B \gtrsim 700$ G and magnetic field perturbation amplitudes $\sim 0.01 - 0.1$ $B$. The total resistive heating flux increases with $\nu$. Most heating occurs in the photosphere. Thermalization of Poynting flux in the photosphere due to electron current dissipation regulates the Poynting flux into the chromosphere, limiting $F_{Ch}$. $F_{Ch}$ initially increases with $\nu$, reaches a maximum, and then decreases with increasing $\nu$ due to increasing electron current dissipation in the photosphere. The resolution needed to resolve the oscillations increases from $\sim 10$ m in the photosphere to $\sim 10$ km in the upper chromosphere, and is proportional to $\nu^{-1/2}$. Estimates suggest that these oscillations are normal modes of photospheric flux tubes with diameters $\sim 10-20$ km, excited by magnetic reconnection in current sheets with thicknesses $\sim 0.1$ km., Comment: 55 pages, 19 figures

An MHD model of a Hydrogen plasma with flow, an energy equation, NLTE ionization and radiative cooling, and an Ohm's law with anisotropic electrical conduction and thermoelectric effects is used to self-consistently generate atmospheric layers over a $50$ km height range. A subset of these solutions contain current sheets, and have properties similar to those of the lower and middle chromosphere. The magnetic field profiles are found to be close to Harris sheet profiles, with maximum field strengths $\sim 25-150$ G. The radiative flux $F_R$ emitted by individual sheets is $\sim 4.9 \times 10^5 - 4.5 \times 10^6$ ergs-cm$^{-2}$-s$^{-1}$, to be compared with the observed chromospheric emission rate of $\sim 10^7$ ergs-cm$^{-2}$-s$^{-1}$. Essentially all emission is from regions with thicknesses $\sim 0.5 - 13$ km containing the neutral sheet. About half of $F_R$ comes from sub-regions with thicknesses 10 times smaller. A resolution $\lesssim 5-130$ m is needed to resolve the properties of the sheets. The sheets have total H densities $\sim 10^{13}-10^{15}$ cm$^{-3}$. The ionization fraction in the sheets is $\sim 2-20$ times larger, and the temperature is $\sim 2000-3000$ K higher than in the surrounding plasma. The Joule heating flux $F_J$ exceeds $F_R$ by $\sim 4-34 \%$, the difference being balanced in the energy equation mainly by a negative compressive heating flux. Proton Pedersen current dissipation generates $\sim 62-77\%$ of the positive contribution to $F_J$. The remainder of this contribution is due to electron current dissipation near the neutral sheet where the plasma is weakly magnetized., Comment: 29 pages, 11 figures

The mathematical formulation of an iterative procedure for the numerical implementation of an ionosphere-magnetosphere (IM) anisotropic Ohm's law boundary condition is presented. The procedure may be used in global magnetohydrodynamic (MHD) simulations of the magnetosphere. The basic form of the boundary condition is well known, but a well-defined, simple, explicit method for implementing it in an MHD code has not been presented previously. The boundary condition relates the ionospheric electric field to the magnetic field-aligned current density driven through the ionosphere by the magnetospheric convection electric field, which is orthogonal to the magnetic field B, and maps down into the ionosphere along equipotential magnetic field lines. The source of this electric field is the flow of the solar wind orthogonal to B. The electric field and current density in the ionosphere are connected through an anisotropic conductivity tensor which involves the Hall, Pedersen, and parallel conductivities. Only the height-integrated Hall and Pedersen conductivities (conductances) appear in the final form of the boundary condition, and are assumed to be known functions of position on the spherical surface R=R1 representing the boundary between the ionosphere and magnetosphere. The implementation presented consists of an iterative mapping of the electrostatic potential ψ the gradient of which gives the electric field, and the field-aligned current density between the IM boundary at R=R1 and the inner boundary of an MHD code which is taken to be at R2>R1. Given the field-aligned current density on R=R2, as computed by the MHD simulation, it is mapped down to R=R1 where it is used to compute ψ by solving the equation that is the IM Ohm's law boundary condition. Then ψ is mapped out to R=R2, where it is used to update the electric field and the component of velocity perpendicular to B. The updated electric field and perpendicular velocity serve as new boundary conditions for the MHD simulation which is then used to compute a new field-aligned current density. This process is iterated at each time step. The required Hall and Pedersen conductances may be determined by any method of choice, and may be specified anew at each time step. In this sense the coupling between the ionosphere and magnetosphere may be taken into account in a self-consistent manner.

A magnetohydrodynamic model is used to determine conditions under which the Lorentz force accelerates plasma to type 2 spicule speeds in the chromosphere. The model generalizes a previous model to include a more realistic pre-spicule state, and the vertical viscous force. Two cases of acceleration under upper chromospheric conditions are considered. The magnetic field strength for these cases is ?12.5 and 25?G. Plasma is accelerated to terminal vertical speeds of 66 and 78?km?s?1 in 100?s, compared with 124 and 397?km?s?1 for the case of zero viscosity. The flows are localized within horizontal diameters ~80 and 50?km. The total thermal energy generated by viscous dissipation is ~10?times larger than that due to Joule dissipation, but the magnitude of the total cooling due to rarefaction is this energy. Compressive heating dominates during the early phase of acceleration. The maximum energy injected into the corona by type 2 spicules, defined as the energy flux in the upper chromosphere, may largely balance total coronal energy losses in quiet regions, possibly also in coronal holes, but not in active regions. It is proposed that magnetic flux emergence in intergranular regions drives type 2 spicules.

A 2.5D, time-dependent magnetohydrodynamic model is used to test the proposition that observed type II spicule velocities can be generated by a Lorentz force under chromospheric conditions. It is found that current densities localized on observed space and time scales of type II spicules and that generate maximum magnetic field strengths ?50 G can generate a Lorentz force that accelerates plasma to terminal velocities similar to those of type II spicules. Maximum vertical flow speeds are ~150-460?km?s?1, horizontally localized within ~2.5-10?km from the vertical axis of the spicule, and comparable to slow solar wind speeds, suggesting that significant solar wind acceleration occurs in type II spicules. Horizontal speeds are ~20 times smaller than vertical speeds. Terminal velocity is reached ~100 s after acceleration begins. The increase in the mechanical and thermal energy of the plasma during acceleration is (2-3) ? 1022?ergs. The radial component of the Lorentz force compresses the plasma during the acceleration process by factors as large as ~100. The Joule heating flux generated during this process is essentially due to proton Pedersen current dissipation and can be ~0.1-3.7 times the heating flux of ~106?ergs?cm?2?s?1 associated with middle-upper chromospheric emission. About 84%-94% of the magnetic energy that accelerates and heats the spicules is converted into bulk flow kinetic energy.

A Harris sheet magnetic field with maximum magnitude B 0 and length scale L is combined with the anisotropic electrical conductivity, viscosity, and thermoelectric tensors for an electron-proton plasma to define a magnetohydrodynamic model that determines the steady state of the plasma. The transport tensors are functions of temperature, density, and magnetic field strength, and are computed self-consistently as functions of position x normal to the current sheet. The flow velocity, magnetic field, and gravitational force lie along the z-axis. The plasma is supported against gravity by the viscous force. Analytic solutions are obtained for temperature, density, and velocity. They are valid over a broad range of temperature, density, and magnetic field strength, and so may be generally useful in astrophysical applications. Numerical examples of solutions in the parameter range of the solar atmosphere are presented. The objective is to compare Joule and viscous heating rates, determine the velocity shear that generates viscous forces that support the plasma and are self-consistent with a mean outward mass flux comparable to the solar wind mass flux, and compare the thermoelectric and conduction current contributions to the Joule heating rate. The ratio of the viscous to Joule heating rates per unit mass can exceed unity by orders of magnitude, and increases rapidly with L. The viscous heating rate can be concentrated outside the region where the current density is localized, corresponding to a resistively heated layer of plasma bounded by viscously heated plasma. The temperature gradient drives a thermoelectric current density that can have a magnitude greater than that of the electric-field-driven conduction current density, so thermoelectric effects are important in determining the Joule heating rate.