On the Mechanism of Chromospheric Network Heating and the Condition for Its Onset in the Sun and Other Solar-Type Stars

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.