论文标题
Clebsch限制和激动
Clebsch Confinement and Instantons in Turbulence
论文作者
论文摘要
我们介绍了与不间断的仪表不变性有关的Clebsch限制的概念，并研究了Clebsch Instantons：具有非平凡螺旋性的奇异涡度表。这是我们从90ties \ cite {fklm}开始启动的“ Instantons and Inthertilency”程序的实现。 这些奇异的溶液参与增强远程边界处的无限随机力，从而导致关键现象。在Euler方程中，涡度沿随机自我避免表面集中，切线成分与正常距离的三角洲功能成正比。 Navier-Stokes方程中的粘度将此三角洲功能涂抹在高斯，宽度$ h \ propto c n nicefrac {\ nicefrac {3} {5}}} $ at $ν\ ra 0 $带有固定能量流量。 这些Instantons在耗散方面占据了主导地位，以及用于固定回路$ c $的速度循环的PDF $γ_c$。 在大循环下，所得的对称指数分布完美地拟合了数值模拟\ cite {ibs20}，包括指数$ 1/\ sqrt {|γ|} $。 在小循环下，我们主张将随机自我避免表面理论与在数值模拟中观察到的多性缩放定律的关系。这些法律是由于内部度量的波动而解释的。异常尺寸的曲线$ζ（n）$可以将小$ n $安装给抛物线，来自liouville理论，带有两个参数$α，q $。在大致上，我们理论中后续时刻的比率与循环的大小线性增长，这对应于与DNS一致的有限值$ζ（\ infty）$。
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.
