ABSTRACT
Traditional formulations of Navier-Stokes turbulence do not provide a good match for many physically important concepts: multiscale dynamics, nonlocal effects, spectral transfer and spatial transport, similarity and scaling, intermittency, mixing, regularity, emergence of structures, and others. Defining flexion as the Laplacian of velocity (negative curl of vorticity for incompressible flow), it is shown that filtered flexion is related to most items on this list. The flexion equation is equivalent to the familiar Fourier version of the momentum equation. Gaussian filtering yields the evolution equation for the Mexican hat wavelet transform of velocity at the corresponding scale. The inverse wavelet transform is an alternative form of the Biot-Savart relation, and simple diffusion maps into a uniform translation of the wavelet coefficients toward smaller scale. Work in progress is outlined, including the formulation of nonlinear terms in the framework of complex systems dynamics and emergence of structures, Navier-Stokes regularity (see companion abstract), rapid-distortion theory, and recursive filtering and renormalization.
