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 = 1/2upsilon2, and dissipation rate epsilon=nu0(partial derivativeupsilon(i)/partial derivativex(j))2 are treated as independent dynamical variables, each obeying its own equation of motion. The relations DELTAu(DELTAK)2BAR is-proportional-to r, DELTAU(DELTAepsilon)2BAR is-proportional-to r0, and (DELTAu)5BAR almost-equal-to rDELTAepsilonDELTAK are derived. If velocity scales as (DELTAupsilon)rms is-proportional-to r(gamma/3)-1, then simple power counting gives (DELTAK)rms is-proportional-to r1-(gamma/6) and (DELTAepsilon)rms is-proportional-to 1/square-root (DELTAupsilon)rms is-proportional-to r(1/2)-(gamma/6). In the Kolmogorov turbulence (gamma=4) the intermittency exponent mu = (gamma/3)-1 = 1/3 and (DELTAepsilon)2=O(Re1/4). The scaling relation for the epsilon fluctuations is a consequence of cancellation of ultraviolet divergences in the equation of motion for the dissipation rate.