Drag-controlled (f-plane) case

Accounting for baroclinicity improves the eddy scaling

In the classic eddy scalings, an assumption of complete eddy barotropization (i.e. negligible vertical structure --> neglecting baroclinicity) under an extensive inverse cascade is commonly invoked. However, this assumption is flawed, especially when the strength of bottom drag is moderate/large to limit the extent of inverse cascade. It was shown in Chen (2023) that incorporating the effect of variable eddy baroclinicity improves the eddy scaling theory. Variable baroclinicity also acts as a scale-selective damping: Larger eddies are more barotropic and thus more dissipative. This partly helps resolve a known paradox that the scale independent linear drag cannot arrest the inverse cascade. 

Ref:  Chen, S. N., 2023: Revisiting the baroclinic eddy scalings in two-layer, quasigestrophic turbulence: Effects of partial barotropization. Journal of Physical Oceanography, 53, 891-913.