Turbulent cascades

Turbulent cascades at high Reynolds numbers are explained briefly in terms of multipliers and multiplier distributions. Two properties of the multipliers ensure the existence of power laws for locally averaged energy dissipation rate: (a) the existence of a multiplier probability density function that is independent of the level of the cascade, and (b) the statistical independence of multipliers at one level on those at previous levels. Under certain conditions described in the paper, the same two properties of multipliers guarantee that velocity increments over inertial-range separation distances also possess power laws. This work is specifically motivated by the need to understand the influence on scaling of the experimental observations that property (a) is true for turbulence, but property (b) is not; and additional motivation is the need to relate cascade models to intermittent vortex stretching (and folding). This effect has been modeled by allowing the multiplier distribution to depend on the magnitude of the local strain rate, and it is demonstrated that this rate-dependent model accounts for the statistical dependence observed in experiments. It is also shown that this model is consistent with the uncorrelated cascade models except for very weak singularity strengths (or for negative moments below a certain order), leading to the conclusion that, for all practical purposes, the uncorrelated level-independent multipliers abstract the essence of the breakdown process in turbulence.