A filtering and wavelet formulation for incompressible turbulence

Gaussian filtering and Hermitian wavelet, transforms lead to a new presentation of the Navier Stokes equations by adding an independent variable. The diffusive part takes the form of an invariant translation toward smaller scales. The filtered pressure term is spatially local and is a superposition of generalized stresses at all scales larger than the scale of observation. Dominant contributors to the stresses are identified in the wavelet domain. The wavelet representation of Navier Stokes is derived from the filtered version, and has a simple algebraic structure similar to its Fourier counterpart. All nonlinear terms are shown to involve spatial transport; within this scheme, spectral transfer involves triplets of scales, covering the entire spectrum with a concentration on nearby scales, consistently with cascade models. The physical content of the equations is interpreted anew from this perspective, and several lines of application are discussed.