  We investigate physical scaling laws for magnetic energy dissipation in solar
flares, in the framework of the Sweet-Parker model and the Petschek model. We
find that the total dissipated magnetic energy $E_{diss}$ in a flare depends on
the mean magnetic field component $B_f$ associated with the free energy $E_f$,
the length scale $L$ of the magnetic area, the hydrostatic density scale height
$\lambda$ of the solar corona, the Alfv\'en Mach number $M_A=v_1/v_A$ (the
ratio of the inflow speed $v_1$ to the Alfv\'enic outflow speed $v_A$), and the
flare duration $\tau_f$, i.e., $E_{diss} = (1/4\pi) B_f^2\ L\ \lambda\ v_A\
M_A\ \tau_f$, where the Alfv\'en speed depends on the nonpotential field
strength $B_{np}$ and the mean electron density $n_e$ in the reconnection
outflow. Using MDI/SDO and AIA/SDO observations and 3-D magnetic field
solutions obtained with the vertical-current approximation nonlinear force-free
field code (VCA-NLFFF) we measure all physical parameters necessary to test
scaling laws, which represents a new method to measure Alfv\'en Mach numbers
$M_A$, the reconnection rate, and the total free energy dissipated in solar
flares.
