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, n = 1010 cm −3, and barometric scale height, hG = 400 km. The assumed perturbation values are: L = 200 km, ΔB = 1 G, −1 ΔV = 1 km s, ΔT = 1000 K. With these reasonable parameters the dissipation rates are (in cgs units) as follows: Through viscous shear, QV = η (ΔV ∕ΔL )2 = 2 × 10− 8, through thermal conduction, Qc = κ (ΔT ∕ΔL )2 = 3 × 10− 7, and through Ohmic resistance, QJ = j2∕σ = (c∕4π )2(ΔB ∕ΔL )2∕σ = 7 × 10−7. Here j is the plasma current density, and the transport coefficients are viscosity, η, heat conductivity, κ, and electrical conductivity, σ, for which values can be found in Braginskii (1965Jump To The Next Citation Point).

These numbers ought to be confronted with the losses due to radiative cooling, which amount to QR = n2 Λ(T ) = 10 −1 erg cm − 3 s−1, with the radiative loss functions Λ, for references see the book of Mariska (1992Jump To The Next Citation Point). QR is a factor of 106 or more larger than QV,C,J. Consequently a much smaller than the assumed scale, for instant L = 200 m, is required to match heating to cooling. Note, however, that then the assumption stated in the following Equation (27View Equation), which is implicit in the derivation of η, κ and σ from the subsequent Equation (23View Equation), seriously breaks down, since λc = 1 –10 km is larger than this L 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 (23View Equation), (24View Equation) and (25View Equation) 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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