The MHD coupling between coronal dynamics and photospheric motions

Context. Whether it be the heating problem or the destabilization of coronal structures, use is often made of the so-called “line-tying” boundary conditions, which amounts to imposing the photospheric velocity at the photosphere as a boundary condition for coronal dynamics. Directly coupling the low beta coronal evolution to prescribed photospheric motions of the magnetic footpoints allows strong magnetic energy accumulation in the corona. But this amounts to ignoring possible feedback from the coronal loops on photospheric motions, a neglect that is commonly justified by the strong density contrast between the photosphere and the corona. On the other hand, the energy injected into the corona comes from the photosphere, so in principle the coronal loop might act as a conduit communicating photospheric dynamics from one region to another. Aims. Our objective is to test the degree of validity of this line-tying approximation by considering the role of the dense photosphere explicitly. Methods. We consider here a 1.5D MHD model of a magnetic loop including a strongly stratified solar-like atmosphere and consider free (instead of prescribed/line-tied) boundary conditions applied deep in the photosphere, so as to quantify the coupling between the photosphere and corona as determined by stratification. We give an initial kick to one of the footpoints in the form of an upwardly propagating Alfvénic perturbation rising from the lower boundary, and then allow waves to freely escape the numerical domain from the boundaries, seated deep in the photosphere. Results. We find that the response of the loop differs in many aspects from what is predicted by the line-tied condition. a) The magnetic energy density available in the corona is limited to a value equal to the kinetic energy density in the photospheric motion. b) The initial velocity shear between the opposite loop footpoints vanishes after a time proportional to the loop length. The shear between the coronal boundaries on opposite sides of the loop is quasi-uniform and is relaxed slowly by Alfvén waves propagating downwards through the high-β photospheric layers. This process is insensitive to details of the thermal structure. c) Coronal loops are thus shown to exert a strong feedback on the photospheric dynamics, intermediate between friction and diffusion, instead of no reaction at all.