Atmospheric Energetics in the Wavelet Domain. Part I: Governing Equations and Interpretation for Idealized Flows

Abstract

Orthonormal wavelet analysis of the primitive momentum equations enables a new formulation of atmospheric energetics, providing a new description of transfers and fluxes of kinetic energy (KE) between structures that are simultaneously localized in both scale (zonal-wavenumber octave) and location spaces. Unpublished modified formulas for global Fourier energetics (FE) are reviewed that conserve KE for the case of a single latitude-circle and pressure level. The new wavelet energetics (WE) is extended to arbitrary orthogonal analyses of compressible, hydrostatic winds, and to formulating triadic interactions between components. In general, each triadic interaction satisfies a detailed conservation rule. Component ?self-interaction? is examined in detail, and found to occur (if other components catalyze) in common analyses except complex Fourier. Wavelet flux functions are new spatially localized measures of flux across scale, or wavenumber cascade. They are constructed by appropriately constrained partial sums over the scales of wavelet transfer functions. The sum constraints prevent KE ?double counting.? Application to Burgers-shock and Stuart-vortex 1D flow models illustrates appropriate physical interpretations of the new energy budget, compared to purely spatial or wavenumber energetics, and demonstrates methods that deal with asymmetry and lack of translation invariance. Such methods include incorporating all possible periodic translations into the analysis, known as the shift-equivariant wavelet transform. The Burgers shock exhibits in FE a global downscale cascade, whose spatial localization and upscale backscatter near the shock is revealed by WE. The Stuart vortex has zero FE, but its pure translation generates a WE picture that reflects the purely spatial energetics picture.