Refined Similarity Hypothesis for a Randomly Advected Passive Scalar

Kolmogorov's refined similarity hypothesis (RSH) is extended to study the inertial-range scaling of a passive scalar advected by a rapidly changing incompressible velocity field in d dimensions. For ζ₂>d, the non-negativity of the scalar dissipation rate constrains the 2nth order scaling exponents, ζ_2n, to be linear in n asymptotically. With the RSH formulated in terms of a stochastic variable θ, the molecular-diffusion terms are evaluated in general d dimensions. For d≥2, the exponents are found to be ζ_2n  =  1 / 2 sqrt[[d-ζ₂-g(n)ζ₂]²+4ng(n)ζ₂(d-ζ₂)]-1 / 2 [d-ζ₂-g(n)ζ₂], where g(n)  =  (2n-1) 〈θ^2n-2〉 〈θ²〉/〈θ²ⁿ〉.