随机纳维-斯托克斯方程的非微扰重整化群研究
A non-perturbative renormalization group study of the stochastic Navier--Stokes equation
摘要：
研究了具有幂律强迫的随机纳维-斯托克斯方程的平均作用量的重整化群流。使用伽利略不变性，我们引入了一个非微扰近似适应于零频率扇区的理论在参数范围内的霍德尔指数$4-2\,\varepsilon$的强制实空间的局部相互作用是相关的。在任何空间维度$d$，我们观察到的收敛所产生的重整化群流到一个独特的固定点，产生一个动能谱标度与正则维数分析。因此，对于$\varepsilon=2$，Kolmogorov的-5/3定律是恢复的，正如微扰重整化所预测的那样。与微扰预测不同的是，当根据微扰重整化，速度场变得与红外相关时，在$\varepsilon=3/2$处涡扩散率的标度维数的$\varepsilon$依赖性中存在\emph{saturation}时，出现-5/3定律。
英文摘要：
We study the renormalization group flow of the average action of the stochastic Navier--Stokes equation with power-law forcing. Using Galilean invariance we introduce a non-perturbative approximation adapted to the zero frequency sector of the theory in the parametric range of the Hölder exponent $4-2\,\varepsilon$ of the forcing where real-space local interactions are relevant. In any spatial dimension $d$, we observe the convergence of the resulting renormalization group flow to a unique fixed point which yields a kinetic energy spectrum scaling in agreement with canonical dimension analysis. Kolmogorov's -5/3 law is, thus, recovered for $\varepsilon=2$ as also predicted by perturbative renormalization. At variance with the perturbative prediction, the -5/3 law emerges in the presence of a \emph{saturation} in the $\varepsilon$-dependence of the scaling dimension of the eddy diffusivity at $\varepsilon=3/2$ when, according to perturbative renormalization, the velocity field becomes infra-red relevant.
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