Canonical Transfer and Multiscale Energetics for Primitive and Quasigeostrophic Atmospheres

Abstract

The past years have seen the success of a novel multiscale energetics 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 study.

A Diagnosis of Some Dynamical Processes Underlying a Higher-Latitude Typhoon Using the Multiscale Window Transform

“…as proved in [20]. Physically this means that the transfer is a mere redistribution of energy among the scale windows, without generating or destroying energy as a whole.…”

“…[20]), and the MS-EVA-based theory of finite-amplitude baroclinic and barotropic instabilities. This is a systematic line of work; a comprehensive description is beyond the scope of this paper.…”

“…The following is just a very brief introduction; for details and comparison to the traditional local energetics formalism, refer to a recent comprehensive review [20].…”

“…One issue remains, that is, the distinguishment of transfer and transport processes from the intertwined nonlinear terms, which is very important for this study, since it is directly related to the cyclogenesis processes. For a scalar field u within a flow v, the transfer from the large-scale window to the cyclone window proves [20] to be,…”

“…With theory, the multiscale kinetic energy (K) and available potential energy (A) equations now can be obtained for a geophysical fluid system. The details are referred to [20]. Here, we just symbolically write them as,…”