Adzhemyan L Ts, Antonov N V, Honkonen J
Department of Theoretical Physics, St. Petersburg University, Ulyanovskaya 1, St. Petersburg-Petrodvorez 198504, Russia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Sep;66(3 Pt 2B):036313. doi: 10.1103/PhysRevE.66.036313. Epub 2002 Sep 27.
The renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by the Gaussian self-similar velocity field with finite, and not small, correlation time. The inertial-range energy spectrum of the velocity is chosen in the form E(k) proportional, variant k(1-2 epsilon ), and the correlation time at the wave number k scales as k(-2+eta). Inertial-range anomalous scaling for the structure functions and other correlation functions emerges as a consequence of the existence in the model of composite operators with negative scaling dimensions, identified with anomalous exponents. For eta> epsilon, these exponents are the same as in the rapid-change limit of the model; for eta< epsilon, they are the same as in the limit of a time-independent (quenched) velocity field. For epsilon =eta (local turnover exponent), the anomalous exponents are nonuniversal through the dependence on a dimensionless parameter, the ratio of the velocity correlation time, and the scalar turnover time. The nonuniversality reveals itself, however, only in the second order of the epsilon expansion and the exponents are derived to order epsilon (2), including anisotropic contributions. It is shown that, for moderate order of the structure function n, and the space dimensionality d, finite correlation time enhances the intermittency in comparison with both the limits: the rapid-change and quenched ones. The situation changes when n and/or d become large enough: the correction to the rapid-change limit due to the finite correlation time is positive (that is, the anomalous scaling is suppressed), it is maximal for the quenched limit and monotonically decreases as the correlation time tends to zero.
重整化群和算符乘积展开被应用于由具有有限且并非很小的关联时间的高斯自相似速度场平流的被动标量场模型。速度的惯性范围能谱被选取为(E(k))与(k^{(1 - 2\epsilon)})成比例的形式，并且波数(k)处的关联时间按(k^{(-2 + \eta)})标度。结构函数和其他关联函数的惯性范围反常标度是由于模型中存在具有负标度维数的复合算符而产生的，这些复合算符与反常指数相关联。对于(\eta > \epsilon)，这些指数与模型的快速变化极限中的指数相同；对于(\eta < \epsilon)，它们与时间无关（淬火）速度场极限中的指数相同。对于(\epsilon = \eta)（局部周转指数），反常指数通过对一个无量纲参数（速度关联时间与标量周转时间之比）的依赖而具有非普适性。然而，这种非普适性仅在(\epsilon)展开的二阶中显现出来，并且指数被推导到(\epsilon^2)阶，包括各向异性贡献。结果表明，对于结构函数的中等阶数(n)和空间维数(d)，有限关联时间相比于快速变化极限和淬火极限这两种情况都增强了间歇性。当(n)和/或(d)变得足够大时情况会发生变化：由于有限关联时间对快速变化极限的修正为正（即反常标度被抑制），在淬火极限中修正最大，并且随着关联时间趋于零单调减小。
