Global Energetics of Solar Flares. IX. Refined Magnetic Modeling

A more accurate analytical solution of the {\sl vertical-current approximation nonlinear force-free field (VCA3-NLFFF)} model is presented that includes besides the radial $(B_r)$ and the azimuthal $(B_\varphi)$ magnetic field components a poloidal component $(B_{\theta} \neq 0)$ also. This new analytical solution is of second-order accuracy in the divergence-freeness condition, and of third-order accuracy in the force-freeness condition. We re-analyze the sample of 173 GOES M- and X-class flares observed with the {\sl Atmospheric Imaging Assembly (AIA)} and {\sl Helioseismic and Magnetic Imager (HMI)} onboard the {\sl Solar Dynamics Observatory (SDO)}. The new code reproduces helically twisted loops with a low winding number below the kink instability consistently, avoiding unstable, highly-twisted structures of the Gold-Hoyle flux rope type. The magnetic energies agree within $E_{VCA3}/E_W=0.99\pm0.21$ with the Wiegelmann (W-NLFFF) code. The time evolution of the magnetic field reveals multiple, intermittent energy build-up and releases in most flares, contradicting both the Rosner-Vaiana model (with gradual energy storage in the corona) and the principle of time scale separation ($\tau_{flare} \ll \tau_{storage}$) postulated in self-organized criticality models. The mean dissipated flare energy is found to amount to $7\%\pm3\%$ of the potential energy, or $60\%\pm26\%$ of the free energy, a result that can be used for predicting flare magnitudes based on the potential field of active regions.