A theory for the heating of coronal magnetic flux ropes is developed. The dissipated magnetic energy has two distinct contributions: (1) energy injected into the corona as a result of granule-scale, random footpoint motions, and (2) energy from the large-scale, nonpotential magnetic field of the flux rope. The second type of dissipation can be described in term of hyperdiffusion, a type of magnetic diffusion in which the helicity of the mean magnetic field is conserved. The associated heating rate depends on the gradient of the torsion parameter of the mean magnetic field. A simple model of an active region containing a coronal flux rope is constructed. We find that the temperature and density on the axis of the flux rope are lower than in the local surroundings, consistent with observations of coronal cavities. The model requires that the magnetic field in the flux rope is stochastic in nature, with a perpendicular length scale of the magnetic fluctuations of order 1000 km.

9 pages (emulateapj style), 4 figures, ApJ, in press (v. 679; June 1, 2008)