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Canonical Transfer and Multiscale Energetics for Primitive and Quasigeostrophic Atmospheres

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  • 1 School of Marine Sciences, and School of Atmospheric Sciences, Nanjing Institute of Meteorology, Nanjing, China
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Abstract

The past years have seen the success of a novel and rigorous localized 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 and clarifications of some common misconceptions about multiscale energy. Multiscale equations are derived using a new analysis apparatus—namely, multiscale window transform—with respect to both the primitive equation and quasigeostrophic 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; hence, we 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 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 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 study of the Madden–Julian oscillation.

Denotes Open Access content.

Corresponding author address: X. San Liang, Center for Ocean-Atmosphere Dynamical Studies, S-308 Wende Building, Nanjing Institute of Meteorology, 219 Ningliu Blvd., Nanjing 210044, China. E-mail: san@pacific.harvard.edu

Abstract

The past years have seen the success of a novel and rigorous localized 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 and clarifications of some common misconceptions about multiscale energy. Multiscale equations are derived using a new analysis apparatus—namely, multiscale window transform—with respect to both the primitive equation and quasigeostrophic 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; hence, we 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 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 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 study of the Madden–Julian oscillation.

Denotes Open Access content.

Corresponding author address: X. San Liang, Center for Ocean-Atmosphere Dynamical Studies, S-308 Wende Building, Nanjing Institute of Meteorology, 219 Ningliu Blvd., Nanjing 210044, China. E-mail: san@pacific.harvard.edu
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