Nonlinear cascade models for fully developed turbulence

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Abstract
The cascade model of Desnyansky and Novikov is extended to include energy flow inward as well as outward from a given k-space shell. The stationary solutions for the energy spectrum are of the 1941 Kolmogoroff form, ε2/3 k−5/3 f (k/kd), but the scaling function f (x) has a nonanalytic dependence on a parameter C of the model which represents the relative strength of inward and outward energy flow in k space. For C<1, f (0) is finite, and the energy dissipation rate ε is nonzero in the limit of zero viscosity. For C ≳1, f (x) behaves as x−ζ for small x with ζ=2 log C/log 2. This leads to an inertial range exponent for the energy spectrum of 5/3+ζ, and to an ε (ν) which goes slowly to zero as the viscosity, ν, goes to zero. The model may play the role of a mean field theory for strong turbulence in which fluctuations leading to intermittency are neglected. The parameter C would then be a smooth function of the spatial dimensionality. A comparison with other recent work indicates that the crossover phenomenon when C=1 occurs between two and three dimensions.