Editorial Type:
Article
Article Type:
Research Article

A Theoretical Framework for Energy and Momentum Consistency in Subgrid-Scale Parameterization for Climate Models

Tiffany A. Shaw Department of Physics, University of Toronto, Toronto, Ontario, Canada

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Theodore G. Shepherd Department of Physics, University of Toronto, Toronto, Ontario, Canada

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Print Publication:
01 Oct 2009
DOI:
https://doi.org/10.1175/2009JAS3051.1
Page(s):
3095–3114
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Abstract

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 nonhydrostatic subgrid-scale (mesoscale) flow. The temporal and horizontal spatial scale separation between the planetary scale and mesoscale 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 that 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 nonacceleration theorems. A derived relationship between the wave-activity fluxes—which represents a generalization of the second Eliassen–Palm theorem—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.

Corresponding author address: Dr. Theodore G. Shepherd, Department of Physics, University of Toronto, 60 St. George St., Toronto, Ontario M5S 1A7, Canada. Email: tgs@atmosp.physics.utoronto.ca

Abstract

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 nonhydrostatic subgrid-scale (mesoscale) flow. The temporal and horizontal spatial scale separation between the planetary scale and mesoscale 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 that 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 nonacceleration theorems. A derived relationship between the wave-activity fluxes—which represents a generalization of the second Eliassen–Palm theorem—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.

Corresponding author address: Dr. Theodore G. Shepherd, Department of Physics, University of Toronto, 60 St. George St., Toronto, Ontario M5S 1A7, Canada. Email: tgs@atmosp.physics.utoronto.ca

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