“Toy models” of turbulent convection and the hypothesis of the local isotropy restoration

A brief review is given of recent results devoted to the effects of large-scale anisotropy on the inertial-range statistics of the passive scalar quantity $\theta(t,{\bold x})$, advected by the synthetic turbulent velocity field with the covariance $\propto\delta(t-t')|{\bold x}-{\bold x'}|^{\varepsilon}$. Inertial-range anomalous scaling behavior is established, and explicit asymptotic expressions for the structure functions $S_n(\bold r)\equiv \langle[\theta(t,{\bold x}+\bold r)-\theta(t,{\bold x})]^{n}\rangle$ 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 $\varepsilon$ in any space dimension. The exponents are associated with tensor composite operators built of the scalar gradients, and exhibit a kind of hierarchy related to the degree of anisotropy: the less is the rank, the less is the dimension and, consequently, the more important is the contribution to the inertial-range behavior. The leading terms of the even (odd) structure functions are given by the scalar (vector) operators. The small-scale anisotropy reveals itself in odd correlation functions: for the incompressible velocity field, $S_{3}/S_{2}^{3/2}$ decreases going down towards to the depth of the inertial range, while the higher-order odd ratios increase; if the compressibility is strong enough, the skewness factor also becomes increasing.