One-exponent scaling for very high-Reynolds-number turbulence

We show that two strong but physically plausible assumptions allow all the measurable scaling exponents for very high-Reynolds-number incompressible turbulence to be expressed simply in terms of the exponent $\mu$ characterizing the dissipation fluctuations. The first assumption, introduced by Obukhov in 1962, relates the locally fluctuating dissipation to the locally fluctuating nonlinear energy transfer. The second assumption is that the same dissipation-length scale $\eta$ defines the crossover from the dissipation range to the inertial range for all of the velocity structure functions. The resulting exponent relations are the same as recently obtained from a simple geometrical model by Frisch, Sulem, and Nelkin, but the results appear less model dependent in the present context. In addition we introduce a locally defined energy transfer variable $T\left(\overrightarrow{\mathrm{x}}\right)={\eta }^2{\psi }^3\left(\overrightarrow{\mathrm{x}}\right)$, where $\psi \left(\overrightarrow{\mathrm{x}}\right)$ is any component of the velocity derivative tensor. We suggest that $T\left(\overrightarrow{\mathrm{x}}\right)$ has the same statistical properties as the locally defined viscous dissipation $\tilde{\epsilon }\left(\overrightarrow{\mathrm{x}}\right)=\nu {\psi }^2\left(\overrightarrow{\mathrm{x}}\right)$, where $\nu$ is the kinematic viscosity. This suggestion is compatible with our other results, and is capable of experimental test.