Convective kinetic energy equation under the mass-flux subgrid-scale parameterization

Abstract

The present paper originally derives the convective kinetic energy equation under mass-flux subgrid-scale parameterization in a formal manner based on the segmentally-constant approximation (SCA). Though this equation is long since presented by Arakawa and Schubert (1974), a formal derivation is not known in the literature. The derivation of this formulation is of increasing interests in recent years due to the fact that it can explain basic aspects of the convective dynamics such as discharge–recharge and transition from shallow to deep convection.The derivation is presented in two manners: (i) for the case that only the vertical component of the velocity is considered and (ii) the case that both the horizontal and vertical components are considered. The equation reduces to the same form as originally presented by Arakwa and Schubert in both cases, but with the energy dissipation term defined differently. In both cases, nevertheless, the energy “dissipation” (loss) term consists of the three principal contributions: (i) entrainment–detrainment, (ii) outflow from top of convection, and (iii) pressure effects. Additionally, inflow from the bottom of convection contributing to a growth of convection is also formally counted as a part of the dissipation term. The eddy dissipation is also included for a completeness.The order-of-magnitude analysis shows that the convective kinetic energy “dissipation” is dominated by the pressure effects, and it may be approximately described by Rayleigh damping with a constant time scale of the order of 102–103s. The conclusion is also supported by a supplementary analysis of a cloud-resolving model (CRM) simulation. The Appendix discusses how the loss term (“dissipation”) of the convective kinetic energy is qualitatively different from the conventional eddy-dissipation process found in turbulent flows.