Scale-Separation and Quasi-Equilibrium Principles in Arakawa and Schubert’s Cumulus Parameterization

Abstract

Arakawa and Schubert?s cumulus parameterization is formulated under the two asymptotic limits: σ ? 0 and τc/τL ? 0, where σ is the fractional area occupied by cumulus convection, and τc and τL are the characteristic timescales for convection and large scales, respectively. The present note shows that the two asymptotic limits tend to contradict each other, so that the smallness of these two quantities is established only under a compromise. This contradiction is formally established by referring to the heat engine theories by Rennó and Ingersoll, and by Emanuel and Bister. The climatological estimate for σ indicates that the timescale separation is only marginally satisfied with respect to the tropical planetary-scale circulation.