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

N. V. Antonov

Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 16, Tome 269 (2000), pp. 79-91

Résumé

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.