Localized multiscale energy and vorticity analysis: I. Fundamentals

Abstract

A new methodology, multiscale energy and vorticity analysis (MS-EVA), is developed to investigate the inference of fundamental processes from oceanic or atmospheric data for complex dynamics which are nonlinear, time and space intermittent, and involve multiscale interactions. Based on a localized orthogonal complementary subspace decomposition through the multiscale window transform (MWT), MS-EVA is real problem-oriented and objective in nature. The development begins with an introduction of the concepts of scale and scale window and the decomposition of variables on scale windows. We then derive the evolution equations for multiscale kinetic and available potential energies and enstrophy. The phase oscillation reflected on the horizontal maps from Galilean transformation is removed with a 2D large-scale window synthesis. The resulting energetic terms are analyzed and interpreted. These terms, after being carefully classified, provide four types of processes: transport, transfer, conversion, and dissipation/diffusion. The key to this classification is the transfer–transport separation, which is made possible by looking for a special type of transfer, the so-called perfect transfer. The intricate energy source information involved in perfect transfers is differentiated through an interaction analysis. The transfer, transport, and conversion processes form the basis of dynamical interpretation for GFD problems. They redistribute energy in the phase space, physical space, and space of energy types. These processes are all referred to in a context local in space and time, and therefore can be easily applied to real ocean problems. When the dynamics of interest is on a global or duration scale, MS-EVA is reduced to a classical Reynolds-type energetics formalism.