Multiscale energetics analysis begins with a faithful conceptualization of multiscale energy, which however has been mostly overlooked in generalizing the two classical global formalisms, namely, Fourier transform and Reynolds averaging, to local processes. The two formalisms prove to be equivalent with the deviation part in the Reynolds formalism equal to the summation of the harmonics on all nonzero frequencies, which form a subspace which we will henceforth call a scale window. The resulting eddy energy in the Reynolds formalism bears an averaging operator. This is a must, as by the Parseval equality, only in this way can the eddy energy be equal to the summation of the Fourier energies on all nonzero frequencies. (The removal of the averaging operator to achieve localization, a common practice in the literature, is therefore groundless in physics and mathematics.)

In order to achieve a decomposition by scale with localness preserved, we developed a functional analysis apparatus called multiscale window transform (MWT), on the basis of the aforementioned scale window. In a function space, energy with respect to a vector field is the inner product of it with itself. It is shown that the above conservation of energy (and any quadratic quantities) through Parseval equality is equivalent to that the scale windows must be orthogonal, a property that MWT possesses. 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. That is to say, the common practice with buoyancy conversion for instability indication may not work.


 

Multiscale window transform,Parseval equality, orthogonality, canonical transfer, Poisson bracket, barotropic/baroclinic instability, subgrid process parameterization