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