  We investigate the properties of an abstract family of advection diffusion
equations in the context of the fractional Laplacian. Two independent diffusion
parameters enter the system, one via the constitutive law for the drift
velocity and one as the prefactor of the fractional Laplacian. We obtain
existence and convergence results in certain parameter regimes and limits. We
study the long time behaviour of solutions to the general problem and prove the
existence of a unique global attractor. We apply results to two particular
active scalar equations arising in geophysical fluid dynamics, namely the
surface quasigeostrophic equation and the magnetogeostrophic equation.
