Spectra of fluctuations of velocity, kinetic energy, and the dissipation rate in strong turbulence

Following the ideas of operator product expansion, the velocity v, kinetic energy K=1/2${\mathit{v}}^2$, and dissipation rate ε=${\nu }_0$(∂${\mathit{v}}_{\mathit{i}}$/∂${\mathit{x}}_{\mathit{j}}$ ${)}^2$ are treated as independent dynamical variables, each obeying its own equation of motion. The relations Δu(ΔK${)}^2$ ∝ r, Δu(Δε${)}^2$ ∝ ${\mathit{r}}^0$, and (Δu${)}^5$≊rΔεΔK are derived. If velocity scales as (Δv${)}_{\mathrm{rms}}$∝ ${\mathit{r}}^{(\gamma /3)\mathrm{-}1}$, then simple power counting gives (ΔK${)}_{\mathrm{rms}}$ ∝ ${\mathit{r}}^{1\mathrm{-}(\gamma /6)}$ and (Δε${)}_{\mathrm{rms}}$ ∝ 1/√(Δv${)}_{\mathrm{rms}}$ ∝ ${\mathit{r}}^{\mathrm{}(1/2)\mathrm{}\mathrm{-}(\gamma /6)}$. In the Kolmogorov turbulence (γ=4) the intermittency exponent μ=(γ/3)-1=1/3 and (Δε${)}^2$=O(${\mathrm{Re}}^{1/4}$). The scaling relation for the ε fluctuations is a consequence of cancellation of ultraviolet divergences in the equation of motion for the dissipation rate.