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Parker's model is one of the most discussed mechanisms for coronal heatingand has generated much debate. We have recently obtained new scaling results ina two-dimensional (2D) version of this problem suggesting that the heating ratebecomes independent of resistivity in a statistical steady state [Ng andBhattacharjee, Astrophys. J., 675, 899 (2008)]. Our numerical work has now beenextended to 3D by means of large-scale numerical simulations. Randomphotospheric footpoint motion is applied for a time much longer than thecorrelation time of the motion to obtain converged average coronal heatingrates. Simulations are done for different values of the Lundquist number todetermine scaling. In the high-Lundquist number limit, the coronal heating rateobtained so far is consistent with a trend that is independent of the Lundquistnumber, as predicted by previous analysis as well as 2D simulations. In thesame limit the average magnetic energy built up by the random footpoint motiontends to have a much weaker dependence on the Lundquist number than that in the2D simulations, due to the formation of strong current layers and subsequentdisruption when the equilibrium becomes unstable. We will present scalinganalysis showing that when the dissipation time is comparable or larger thanthe correlation time of the random footpoint motion, the heating rate tends tobecome independent of Lundquist number, and that the magnetic energy productionis also reduced significantly.

High-Lundquist Number Scaling in Three-Dimensional Simulations of Parker's Model of Coronal Heating