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Multiscale interaction in geofluid flows

Multiscale windows transform

      Generally speaking, atmosphere-ocean problems such as atmospheric cyclogenesis, oceanic eddy shedding, turbulence production/relaminarization, climate variability, to name a few, are in nature highly nonlinear, multiscale interactive, and intermittent in space and time. Obviously they are beyond the capabilities of the currently available analysis tools. To overcome the difficulties, Liang and Anderson (2007) introduced the concept of “scale window”, which literally means a subspace with a range of scales, and henceforth developed a novel mathematical tool, called the multiscale window transform (MWT), to address the afore-mentioned issues. Loosely speaking, MWT decomposes a space of dynamical processes orthogonally* into the direct sum of several scale windows (usually two or three windows, which may be named, for example, large-scale, meso-scale, and sub-mesoscale windows), while keeping their localization, and allowing the local energy to be faithfully represented on the respective windows (in general this cannot be fulfilled with the available filters). Note that the classical (and the simplest) Reynolds decomposition is a very particular case of MWT; it has a neat energy representation because of its unique functional analysis property: In the framework of MWT, it is both a transform and a reconstruction.


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Liang and Anderson, 2007: Multiscale window transfrom. SIAM J. Multiscale Model. Simul., Vol.6 Issue 2, 437-467.