A theoretical framework for energy and momentum consistency in subgrid-scale parameterization for climate models

A theoretical framework for the joint conservation of energy and momentum in the parameterization of subgrid-scale processes in climate models is presented. The framework couples a hydrostatic resolved (planetary scale) flow to a non-hydrostatic subgrid-scale (mesoscale) flow. The temporal and horizontal spatial scale separation between the planetary and mesoscales is imposed using multiple scale asymptotics. Energy and momentum are exchanged through subgrid-scale flux convergences of heat, pressure and momentum. The generation and dissipation of subgrid-scale energy and momentum is understood using wave-activity conservation laws which are derived by exploiting the (mesoscale) temporal and horizontal spatial homogeneities in the planetary scale flow. The relations between these conservation laws and the planetary-scale dynamics represent generalized non-acceleration theorems. A derived relationship between the wave-activity fluxes --- which represents the extension of the second Eliassen-Palm theorem to three dimensions, non WKB-type conditions and dissipative dynamics --- is key to ensuring consistency between energy and momentum conservation. The framework includes a consistent formulation of heating and entropy production due to kinetic energy dissipation.