Let us estimate the collisional heating rates in the upper chromosphere, where the problems already occur. Typical parameters may be for the density, , and barometric scale height, . The assumed perturbation values are: , , , . With these reasonable parameters the dissipation rates are (in cgs units) as follows: Through viscous shear, , through thermal conduction, , and through Ohmic resistance, . Here is the plasma current density, and the transport coefficients are viscosity, , heat conductivity, , and electrical conductivity, , for which values can be found in Braginskii (1965).
These numbers ought to be confronted with the losses due to radiative cooling, which amount to , with the radiative loss functions , for references see the book of Mariska (1992). is a factor of 106 or more larger than . Consequently a much smaller than the assumed scale, for instant , is required to match heating to cooling. Note, however, that then the assumption stated in the following Equation (27), which is implicit in the derivation of , and from the subsequent Equation (23), seriously breaks down, since is larger than this in the chromosphere.
The situation is no better under coronal conditions, where classical dissipation rates have to be grossly enhanced, by more than six orders of magnitude, to match the empirical damping of loop oscillations (Nakariakov et al., 1999), or dissipation of propagating waves (Ofman et al., 1999). This problem, however, cannot be healed by simply claiming anomalously high transport coefficients or correspondingly low Reynolds numbers, but only by revising the classical transport scheme and developing a new kinetic paradigm for coronal transport. This is even more so needed as the functional dependencies on local gradients of fluid parameters as employed in the subsequent Equations (23), (24) and (25) are not self-evident for a collisionless plasma, and may become meaningless in the corona, where global boundary effects superpose local processes.
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