  In this note we examine the dynamical role played by inertial forces on the
surface temperature (or buoyancy) variance in strongly rotating, stratified
flows with uniform potential vorticity fields. In particular, using a dynamic,
multi-scale averaging process, we identify a sufficient condition for the
existence of a direct temperature variance cascade across an inertial range.
While the result is consistent with the physical and numerical theories of SQG
turbulence, the condition which triggers the cascade is more exotic, a fact
reflecting the non-locality introduced by fractional dissipation.
