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

Context. Coronal loops act as resonant cavities for low-frequency fluctuations that are transmitted from the deeper layers of the solar atmosphere. These 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, because some energy leaks down to the chromosphere on a long timescale, limiting the turbulent heating. Aims. 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. Methods. 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 that are equivalent, in the line-tied case, to forcing by the photospheric shear flows. While varying the turbulence strength, we systematically compare the average coronal energy level and dissipation in three models with increasing complexity: the classical closed model, the open corona, and the open corona including chromosphere (or three-layer model), with the last two models allowing energy leakage. Results. We find that (i) leakage always plays a role. Even for strong turbulence, the dissipation time never becomes much lower than the leakage time, at least in the three-layer model; therefore, both the energy and 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 an effective turbulent correlation time longer than the Alfvén coronal crossing time; (iii) the heating rate is close to the value 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 three-layer model, the two-component spectrum leads to a global decrease in damping equal to Kolmogorov damping reduced by a factor urms/Vac where Vac is the coronal Alfvén speed.