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arXiv 2007.12468hep-th
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Turbulence中的Clebsch约束和瞬子

Clebsch Confinement and Instantons in Turbulence

Alexander Migdal

备注：
Invited review article for Int. J. of Mod. Phys. A. 67 pages, 10 figures, 1 table
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摘要：

我们引入了Clebsch禁闭的概念与不间断的规范不变性和研究Clebsch瞬子：奇异的涡度片与非平凡的螺旋。这是我们早在90年代就开始的“即时和快速”计划的实现\cite{FKLM}。这些奇异的解决方案是参与加强无穷小的随机力在远程边界导致临界现象。在欧拉方程中，涡量沿随机自回避表面集中，切向分量与法向距离的δ函数成比例。Navier-Stokes方程中的粘度将该δ函数涂抹到具有固定能量流的在$ν\ra 0$处的宽度为$h \propto ν^{\nicefrac{3}{5}}$的高斯。这些瞬子支配着耗散中的涡度拟能以及空间中固定环$C$周围速度环流$Γ_C$的PDF。在大循环，得到的对称指数分布完全符合数值模拟\cite{IBS20}，包括前指数因子$1/\sqrt{|Γ|}$。在小循环，我们主张产生的随机自回避表面理论与多重分形标度律在数值模拟中观察到的关系。这些法律被解释为波动的内部度规（刘维尔场）的结果。反常尺寸$ζ(n)$的曲线在小$n$处可以拟合到抛物线，该抛物线来自具有双参数$α, Q$的Liouville理论。在大的$n$，在我们的理论中的后续的时刻的比率与循环的大小线性增长，这对应于与DNS一致的$ζ(\infty)$的有限值。

英文摘要：

We introduce a concept of Clebsch confinement related to unbroken gauge invariance and study Clebsch instantons: singular vorticity sheets with nontrivial helicity. This is realization of the "Instantons and intermittency" program we started back in the 90ties\cite{FKLM}. These singular solutions are involved in enhancing infinitesimal random forces at remote boundary leading to critical phenomena. In the Euler equation vorticity is concentrated along the random self-avoiding surface, with tangent components proportional to the delta function of normal distance. Viscosity in Navier-Stokes equation smears this delta function to the Gaussian with width $h \propto ν^{\nicefrac{3}{5}}$ at $ν\ra 0$ with fixed energy flow. These instantons dominate the enstrophy in dissipation as well as the PDF for velocity circulation $Γ_C$ around fixed loop $C$ in space. At large loops, the resulting symmetric exponential distribution perfectly fits the numerical simulations\cite{IBS20} including pre-exponential factor $1/\sqrt{|Γ|}$. At small loops, we advocate relation of resulting random self-avoiding surface theory with multi-fractal scaling laws observed in numerical simulations. These laws are explained as a result of fluctuating internal metric (Liouville field). The curve of anomalous dimensions $ζ(n)$ can be fitted at small $n$ to the parabola, coming from the Liouville theory with two parameters $α, Q$. At large $n$ the ratios of the subsequent moments in our theory grow linearly with the size of the loop, which corresponds to finite value of $ζ(\infty)$ in agreement with DNS.

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