Crossover Dimensions for Fully Developed Turbulence

Abstract

The analytic continuation of homogeneous, isotropic turbulence in nonintegral dimensions $d$ is not realizable for $d<\mathrm{}2\mathrm{}$ because the energy spectrum generally becomes negative. Recent arguments, in favor of an $\frac{8}{3}$ crossover dimension below which the Kolmogorov 1941 theory is exact, are found questionable. The existence of a crossover dimension $d_c (\approx 2.03)$ at which the direction of the energy cascade reverses is supported by a second-order closure calculation.