Coronal heating in coupled photosphere-chromosphere-coronal systems: turbulence and leakage

Coronal loops act as a resonant cavity for low frequency fluctuations that are transmitted from the deeper layers of the solar atmosphere. Such fluctuations are amplified in the corona and lead to the development of turbulence that in turn is able to dissipate the accumulated energy, thus heating the corona. However trapping is not perfect, some energy leaks down to the chromosphere on a long timescale, thus limiting the turbulent heating.

We consider the combined effects of turbulence and energy leakage from the corona to the photosphere in determining the turbulent energy level and associated heating rate in models of coronal loops which include the chromosphere and transition region. We use a piece-wise constant model for the Alfvén speed in loops and a Reduced MHD - Shell model to describe the interplay between turbulent dynamics in the direction perpendicular to the mean field and propagation along the field. Turbulence is sustained by incoming fluctuations which are equivalent, in the line-tied case, to forcing by the photospheric shear flows. We find that: (i) Leakage always plays a role, whatever the intensity of the photospheric forcing; the dissipation time never becomes much lower than the leakage time, at least in the three-layer model. Hence, the energy as well as the dissipation levels are systematically lower than in the line-tied model. (ii) In all models, the energy level is close to the resonant prediction, i.e., assuming effective turbulent correlation time longer than the Alfvén coronal crossing time. (iii) The heating rate is approximately given by the ratio of photospheric energy divided by the Alfvén crossing time. (iv) The coronal spectral range is divided in two, an inertial range with 5/3 spectral slope, and a large scale peak where nonlinear couplings are inhibited by trapped resonant modes. (v) In the realistic 3-layer model, the two-component spectrum leads to a global decrease of damping equal to Kolmogorov damping reduced by a factor u_rms/B₀.

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