湍流环境对自组织临界行为的影响：各向同性与各向异性
Effects of turbulent environment on self-organized critical behavior: Isotropy vs Anisotropy
摘要：
研究了在环境湍流运动影响下的自组织临界系统。系统由Hwa和Kardar [{ \it Phys. Rev. Lett.}]提出的各向异性连续随机方程描述。{ \bf 62}：1813（1989）]。环境的运动由各向同性的Kazantsev-Kraichnan“快速变化”系综为不可压缩流体建模：它是高斯的，相关时间为零，对相关函数的形式为$\proptoδ(t-t') / k^{d+ξ}$，其中$k$是波数，$ξ$是一个任意指数，最真实的值为$ξ= 4/3$（柯尔莫哥洛夫湍流）和$ξ\to 2$（巴彻勒极限）。利用场论重整化群，我们找到了与普适类相关的重整化群方程的红外吸引不动点，即，批评行为的制度。最现实的值的空间维度$d=2$和指数$ξ=4/3$对应的普适类的纯湍流平流的Hwa-Kardar（HK）方程的非线性是无关的。然而，普适类的（各向异性）非线性的HK方程和（各向同性）平流速度场是相关的也存在的参数$\varepsilon=4-d$和$ξ$的某些值。根据哪些条款（各向异性，各向同性，或两者）是相关的特定的普适性类，不同类型的标度行为（普通的或广义的）建立。
英文摘要：
We study a self-organized critical system under influence of turbulent motion of the environment. The system is described by the anisotropic continuous stochastic equation proposed by Hwa and Kardar [{\it Phys. Rev. Lett.} {\bf 62}: 1813 (1989)]. The motion of the environment is modelled by the isotropic Kazantsev--Kraichnan "rapid-change" ensemble for an incompressible fluid: it is Gaussian with vanishing correlation time and the pair correlation function of the form $\proptoδ(t-t') / k^{d+ξ}$, where $k$ is the wave number and $ξ$ is an arbitrary exponent with the most realistic values $ξ= 4/3$ (Kolmogorov turbulence) and $ξ\to 2$ (Batchelor's limit). Using the field-theoretic renormalization group, we find infrared attractive fixed points of the renormalization group equation associated with universality classes, i.e., with regimes of critical behavior. The most realistic values of the spatial dimension $d=2$ and the exponent $ξ=4/3$ correspond to the universality class of pure turbulent advection where the nonlinearity of the Hwa--Kardar (HK) equation is irrelevant. Nevertheless, the universality class where both the (anisotropic) nonlinearity of the HK equation and the (isotropic) advecting velocity field are relevant also exists for some values of the parameters $\varepsilon=4-d$ and $ξ$. Depending on what terms (anisotropic, isotropic, or both) are relevant in specific universality class, different types of scaling behavior (ordinary one or generalized) are established.
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