Mesoscale cascades and the conundrum of energy transfer from large to dissipation scales in an adiabatic ocean

Abstract. A well-known conundrum in ocean dynamics has been expressed as follows: How does the energy of the general circulation cascade from the large climate scales, where most of it is generated, to the small scales, where all of it is dissipated? In particular, how is the dynamical transition made from an anisotropic, 2D-like, geostrophic cascade at large scales-with its strong inhibition of down-scale energy flux-to 3D-like, down-scale cascades at small scales. (Muller, McWilliams and Molemaker, 2002). To study this as yet unsolved problem, we introduce in the analysis a dynamical consideration based on the mesoscale model developed by Dubovikov (2003) and Canuto and Dubovikov (2005) within which in a quasi-adiabatic ocean interior the large scale baroclinic instability generates mesoscale eddy potential energy (EPE) at scales of the Rossby deformation radius ~ r_d. Since at those scales the mesoscale Rossby number is small, the generated EPE cannot convert into eddy kinetic energy (EKE) and cascades to smaller scales at which the spectral Rossby number Ro(k) increases until at some horizontal scales ~ ℓ it reaches Ro(1 / ℓ)~ 1. Under this condition, EPE converts into EKE and thus the cascade of the former terminates while the inverse EKE cascade begins. At scales ~ r_d the inverse EKE cascade terminates and reinforces the EPE cascade produced by the large scale baroclinic instability thus closing the mesoscale energy cycle. If the flow were exactly adiabatic, i.e. eddy energy were not dissipated, the latter would increase unlimitedly at the expense of the permanent production of the total eddy energy (TEE) by the mean flow. However, at the same scales ~ ℓ where the EPE cascade terminates and the inverse EKE cascade begins, the vertical eddy shear reaches the value of the buoyancy frequency N that gives rise to the Kelvin-Helmholtz instability. The latter generates the stratified turbulence which finally dissipates EKE. A steady state regime sets in when the dissipation balances the TEE production by the mean flow.