“Toy Models” of Turbulent Convection and the Hypothesis on Local Isotropy Restoration

Abstract

We give a brief review of recent results devoted to effects of large-scale anisotropy on the inertial-range statistics of the passive scalar quantity θ(t, x) advected by a synthetic turbulent velocity field with covariance ∝ δ(t – t')|x – x'|^ε. An inertial-range anomalous scaling behavior is established, and explicit asymptotic expressions for the structure functions S _n(r) ≡ 〈 [θ(t, x + r) – θ(t, x)]ⁿ〉 are obtained; they are represented by superpositions of power laws with nonuniversal (dependent on the anisotropy parameters) anomalous exponents, calculated to the first order in ε in any space dimension. The exponents are associated with tensor composite operators, built of scalar gradients, and exhibit a kind of hierarchy related to the degree of anisotropy: the less the rank, the less the dimension and, consequently, the more important the contribution to the inertial-range behavior. The leading terms of even (odd) structure functions are given by scalar (respectively, vector) operators. Small-scale anisotropy reveals itself in odd correlation functions: for an incompressible velocity field, S ₃/S ₂ ^3/2 decreases in the direction toward the depth of the inertial range, while higher-order odd ratios increase; if the compressibility is sufficiently strong, the skewness factor also becomes increasing. Bibliography: 33 titles.