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

Abstract

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 Ñow orthogonal to the magnetic Ðeld in weakly ionized chromospheric plasma. The currents are driven by a convection electric Ðeld generated by velocity oscillations of linear, slow, longitudinal magneto- acoustic waves with frequencies l[3.5 mHz in the lower chromosphere. The heating occurs in thin magnetic Ñux tubes and begins lower in the chromosphere in Ñux tubes with higher photospheric Ðeld strength. The lower chromosphere, which emits most of the net radiative loss in the network, is heated by Ñux tubes with photospheric Ðeld strengths D700È1500 G. A typical Ðeld strength and core diameter for a Ñux 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 D100 km in which the heating rate is an order of magnitude smaller. About ND102 of these Ñux tubes distributed over the boundary region of a granule with a diameter D103 km provide an average heating rate over the entire granule D107 ergs cm~2 s~1. If the core heating rate is changed by a factor f, then NDf ~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 satisÐed at a single height in a given Ñux tube. At this height, control of the proton dynamics begins to be dominated by the magnetic Ðeld 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 dissi- pation 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 magne- tization, 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 Ðeld aligned currents is insigniÐcant. Below the height in the atmosphere at which the onset condition is satis- Ðed, any current orthogonal to the magnetic Ðeld must be primarily a Hall current, which is nondis- sipative. 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 D107 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: D10 km, corre- sponding to Ñux tubes in which network heating occurs, and D104[105 km, perhaps corresponding to Ñux tubes in which active region heating might occur. The model has a singularity at the acoustic cuto† frequency, corresponding to periods near 3 minutes. Therefore, unless nonresistive damping mechanisms such as viscous dissipation and thermal conduction provide sufficiently strong damping, MHD oscil- lations with periods near 3 minutes in chromospheric magnetic Ñux tubes must be nonlinear.