Heating and activity of the solar corona: 1. Boundary shearing of an initially homogeneous magnetic field

To contribute to the understanding of heating and dynamic activity in boundary-driven, low-beta plasmas such as the solar corona, we investigate how an initially homogeneous magnetic field responds to random large-scale shearing motions on two boundaries, by numerically solving the dissipative MHD equations, with resolutions ranging from 24 3 to 136 3 . We find that even a single application of large-scale shear, in the form of orthogonal sinusoidal shear on two boundaries, leads to the formation of tangential discontinuities (current sheets). The formation time scales logarithmically with the resistivity and is of the order of a few times the inverse shearing rate for any reasonable resistivity, even though no mathematical discontinuity would form in a finite time in the limit of vanishing resistivity. The reason for the formation of the current sheets is the interlocking of two magnetic flux systems. Reconnection in the current sheets is necessary for the field lines to straighten out. The formation of current sheets causes a transition to a very dynamic plasma state, where reconnection drives supersonic and super-Alfvenic jet flows and where these, in turn, cause the formation of smaller-scale current sheets. A statistically steady state level for the average Poynting flux and the average Joule dissipation is reached after a few correlation times, but both boundary work and Joule dissipation are highly fluctuating in time and space and are only weakly correlated. Strong and bursty Joule dissipation events are favored when the volume has a large length/diameter ratio and is systematically driven for periods longer than the Alfven crossing time. The understanding of the reason for the current sheet formation allows a simple scaling law to be constructed for the average boundary work. Numerical experiments over a range of parameter values, covering over 3 orders of magnitude in average dissipation, obey the scaling law to within a factor of 2. The heating rate depends on the boundary velocity amplitude and correlation time, the Alfven speed, and the initial magnetic field strength but appears to be independent of the resistivity because of the formation of a hierarchy of current sheets. Estimates of the photospheric boundary work on the solar coronal magnetic field using the scaling law are consistent with estimates of the required coronal heating rates. We therefore conclude that the work supplied to the solar corona as a consequence of the motion of the magnetic foot points in the solar photosphere and the emergence of new flux is a significant contributor to coronal heating and flaring and that it quite plausibly is the dominant one.