The Anomalous Diffusion of the Self-Interacting Passive Scalar in the Turbulent Environment

Two variants of the statistical model of diffusing self-interacting passive scalar θ(x, t) driven by the incompressible Navier–Stokes turbulence were studied by means of the field-theoretical renormalization group technique and ∊-expansion scheme, where ∊ denotes the parameter of the forcing spectrum. Dual ∫ ddxdt[θ(x, t)]2 and triple ∫ ddxdt[θ(x, t)]3 interaction terms of the action represent two different mechanisms of the self-interaction matching two alternative values of the critical dimension: d c =4 and d c =6. The major part of the calculations was carried out in the one loop order, nevertheless, the inclusion of the specific two loop contributions represents the important step of the analysis of some renormalization group functions. In the basic model variant the effective action is renormalizable for the supercritical dimensions d > d c . This theory exhibits the presence of the asymptotical regime, which is stable for the inertial-conductive range of wave numbers. It was also shown that stability of this regime remains preserved for a variety of the parametric paths connecting domain ∊>0, d>d c with ∊<2, d=3. In the second model variant, the effective action is constructed to be renormalizable at dimensions d≥d c and to justify the realizability of the continuation from ∊>0, d>d c to ∊< 2, d=3. This variant of the model was analyzed using "double expansion" method with the expansion parameters (d-d c )/2 and ∊. The negative correction ζ(ζ≃0.039 for d=3) to the universal Richardson exponent 4/3 is the physical consequence stemming from the calculations.