We are investigating the inviscid limit of the Navier–Stokes equation, and we find previously unknown anomalous terms in Hamiltonian, Dissipation, and Helicity, which survive this limit and define the turbulent statistics.

We find various topologically nontrivial configurations of the confined Clebsch field responsible for vortex sheets and lines. In particular, a stable vortex sheet family is discovered, but its anomalous dissipation vanishes as $\sqrt{\nu }$.

Topologically stable stationary singular flows, which we call Kelvinons, are introduced. They have a conserved velocity circulation ${\Gamma }_{\alpha }$ around the loop $C$ and another one ${\Gamma }_{\beta }$ for an infinitesimal closed loop $\tilde{C}$ encircling $C$, leading to a finite helicity. The anomalous dissipation has a finite limit, which we computed analytically.

The Kelvinon is responsible for asymptotic PDF tails of velocity circulation, perfectly matching numerical simulations.

The loop equation for circulation PDF as functional of the loop shape is derived and studied. This equation is exactly equivalent to the Schrödinger equation in loop space, with viscosity $\nu$ playing the role of Planck’s constant.

Kelvinons are fixed points of the loop equation at WKB limit $\nu \to 0$. The anomalous Hamiltonian for the Kelvinons contains a large parameter $log\frac{\left|{\Gamma }_{\beta }\right|}{\nu }$. The leading powers of this parameter can be summed up, leading to familiar asymptotic freedom, like in QCD. In particular, the so-called multifractal scaling laws are, as in QCD, modified by the powers of the logarithm.

我们正在研究 Navier-Stokes 方程的无粘性极限，我们在哈密顿量、耗散和螺旋度中发现了以前未知的反常项，它们在这个极限下仍然存在并定义了湍流统计。

我们发现受限 Clebsch 场的各种拓扑非平凡配置负责涡流片和线。特别是，发现了一个稳定的涡流片族，但其异常耗散消失了$\sqrt{\nu }$.

引入了拓扑稳定的平稳奇异流，我们称之为 Kelvinons。它们具有守恒的速度循环${\Gamma }_{\alpha }$循环$C$和另一个${\Gamma }_{\beta }$对于无限小闭环$\tilde{C}$包围$C$，导致有限的螺旋性。异常耗散有一个有限的限制，我们通过分析计算得出。

Kelvinon 负责速度循环的渐近 PDF 尾部，与数值模拟完美匹配。

推导和研究了作为环路形状函数的循环 PDF 的环路方程。该方程完全等价于循环空间中的薛定谔方程，具有粘性$\nu$扮演普朗克常数的角色。

Kelvinons 是循环方程在 WKB 极限处的不动点$\nu \to 0$. Kelvinons 的反常哈密顿量包含一个大参数$日志\frac{\left|{\Gamma }_{\beta }\right|}{\nu }$. 可以总结这个参数的主要力量，导致熟悉的渐近自由，就像在 QCD 中一样。特别是，所谓的多重分形比例定律，如在 QCD 中一样，由对数的幂修改。