Algebraic Energy Spectra in Stochastic Problems for the Incompressible Navier-Stokes Equation; Relation to Other Nonlinear Problems

Abstract

We use the Wiener-Hermite expansion based on the integral of the white noise process (Feynman path integral) to represent nonlinear, but near to Gaussian, processes. The energy transfer function is examined and, for the equilibrium 5/3-law energy spectrum, it is seen that the dependence on k ₀, the energy range wavenumber, drops out so that changes in the energy spectrum must be algebraic. This surprising result follows from the fact that the pressure term emphasizes the enormous region of phase space available for transfer, of order the large wavenumber k, compared with the small region in the vicinity of the energy spectrum peak. The discussion thus gives, to the author’s knowledge, the first explanation of the spectrum which goes beyond simple dimensional analysis (and assumptions concerning appropriate, influencing parameters). Applications to more general nonlinear stochastic problems are briefly discussed.