Shell-models of RMHD turbulence and the heating of solar coronal loops

A simplified non-linear numerical model for the development of incompressible magnetohydrodynamics (MHD) in the presence of a strong magnetic field B0 and stratification, nicknamed Shell-Atm, is presented. In planes orthogonal to the mean field, the non-linear incompressible dynamics is replaced by 2D shell-models for the complex variables u and b, allowing one to reach large Reynolds numbers while at the same time carrying out sufficiently long time integrations to obtain a good statistics at moderate computational cost. The shell-models of different planes are coupled by Alfven waves propagating along B0. The model may be applied to open or closed magnetic field configurations where the axial field dominates and the plasma pressure is low; here we apply it to the specific case of a magnetic loop of the solar corona heated via turbulence driven by photospheric motions, and we use statistics for its analysis. The Alfven waves interact non-linearly and form turbulent spectra in the directions perpendicular and, via propagation, also parallel to the mean field. A heating function is obtained, and is shown to be intermittent; the average heating is consistent with values required for sustaining a hot corona, and is proportional to the aspect ratio of the loop to the power -1.5; characteristic properties of heating events are distributed as power-laws. Cross-correlations show a delay of dissipation compared to energy content.