Canonical transfer and multiscale energetics for primitive and quasi-geostrophic atmospheres

Abstract

The past years have seen the success of a novel multiscale energetic formalism in a variety of ocean and engineering fluid applications. In a self-contained way, this study introduces it to the atmospheric dynamical diagnostics, with important theoretical updates. Multiscale energy equations are derived using a new analysis apparatus, namely, multiscale window transform, with respect to both the primitive equation and quasi-geostrophic models. A reconstruction of the "atomic" energy fluxes on the multiple scale windows allows for a natural and unique separation of the in-scale transports and cross-scale transfers from the intertwined nonlinear processes. The resulting energy transfers bear a Lie bracket form, reminiscent of the Poisson bracket in Hamiltonian mechanics, we hence would call them "canonical". A canonical transfer process is a mere redistribution of energy among scale windows, without generating or destroying energy as a whole. By classification, a multiscale energetic cycle comprises of available potential energy (APE) transport, kinetic energy (KE) transport, pressure work, buoyancy conversion, work done by external forcing and friction, and the cross-scale canonical transfers of APE and KE which correspond respectively to the baroclinic and barotropic instabilities, among others, in geophysical fluid dynamics. A buoyancy conversion takes place in an individual window only, bridging the two types of energy namely KE and APE, it does not involve any processes among different scale windows, and is hence basically not related to instabilities. This formalism is exemplified with a preliminary application to the Madden-Julian Oscillation (MJO) study.