### 3.4 The failure to heat chromosphere or corona by collisions

The coronal heating problem as formulated in the literature encompasses three main topics: The
generation and release of energy in the photosphere, its transport and propagation into the corona,
and its conversion and dissipation at different heights in various coronal magnetic structures.
The ultimate energy source is magnetoconvection and flux emergence that render the coronal
magnetic field dynamic and energetic. Coronal MHD waves and oscillations are assumed to
be the main carrier of the energy (Nakariakov and Verwichte, 2005). Conventionally, ohmic,
conductive, and viscous heating is supposed to provide the heating in the transition region and
corona. However, the problems already start at low heights, since even in the chromosphere the
collisional heating rates are much too small. Therefore, shock heating is presently favoured there
(see, e.g., Ulmschneider and Kalkofen, 2003). Can fluid modes and kinetic plasma waves then
provide the heating of the transition region and the corona? What is the microphysics of their
dissipation? Ideas and heating scenarios abound, but these basic questions until today remain
unanswered.
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 10^{6} 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.