On probability density functions of locally averaged dissipation and 1D surrogate in isotropic turbulence

In the Kolmogorov theory in 1962 the energy dissipation rate locally averaged over a sphere with scale r is introduced and is assumed to obey the lognormal distribution. The variance is a power law function of r with the exponent mu, the intermittency exponent. In many efforts to find the exponent, the 1D dissipation surrogate instead of full dissipation has been used, but which raised arguments whether the two exponents are the same. We explored the relation between two exponents by using the transform formula for the PDFs of the full dissipation and 1D surrogate. It was found that the difference between two exponents decreases

as (log R_\lambda)^{-1}, which was confirmed by the DNS data.