  We consider the forced surface quasi-geostrophic equation with supercritical
dissipation. We show that linear instability for steady state solutions leads
to their nonlinear instability. When the dissipation is given by a fractional
Laplacian, the nonlinear instability is expressed in terms of the scaling
invariant norm, while we establish stronger instability claims in the setting
of logarithmically supercritical dissipation. A key tool in treating the
logarithmically supercritical setting is a global well-posedness result for the
forced equation, which we prove by adapting and extending recent work related
to nonlinear maximum principles. We believe that our proof of global
well-posedness is of independent interest, to our knowledge giving the first
large-data supercritical result with sharp regularity assumptions on the
forcing term.
