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Estimating Flare-related Photospheric Lorentz Force Vector Changes within Active Regions

It is shown that expressions for the global Lorentz force associated with a flaring active region derived by Fisher {\it et al.} (2012) can be used to estimate the Lorentz force changes for strong fields in large structures over photospheric subdomains within active regions. Gary's~(2001) model for the stratified solar atmosphere is used to demonstrate that in large-scale structures with typical horizontal magnetic length scale $\gg 300$~km and with strong magnetic fields ($\ge 1$~kG at the $τ=1$ opacity layer at 5000~Å), the Lorentz force acting on the photosphere may be approximated by a surface integral based on photospheric boundary data alone. These conditions cover many of the sunspot fields and major neutral lines that have been studied using Fisher {\it et al.}'s (2012) expressions over the past few years. The method gives a reasonable estimate of flare-related Lorentz force changes based on photospheric magnetogram observations provided that the Lorentz force changes associated with the flare have a lasting effect on the observed fields, and are not immediately erased by post-flare equilibration processes.