The turbulence in incompressible fluid is represented as a field theory in 3 dimensions. There is no time involved, so this is intended to describe stationary limit of the Hopf functional. The basic fields are Clebsch variables defined modulo gauge transformations (symplectomorphisms). Explicit formulas for gauge invariant Clebsch measure in space of generalized Beltrami flow compatible with steady energy flow are presented. 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 1990s.1 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 [Formula: see text] at [Formula: see text] with fixed energy flow. These instantons dominate the enstrophy in dissipation as well as the PDF for velocity circulation [Formula: see text] around fixed loop [Formula: see text] in space. At large loops, the resulting symmetric exponential distribution perfectly fits the numerical simulations2 including pre-exponential factor [Formula: see text]. 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 [Formula: see text] can be fitted at small [Formula: see text] to the parabola, coming from the Liouville theory with two parameters [Formula: see text], [Formula: see text]. At large [Formula: see text] the ratios of the subsequent moments in our theory grow linearly with the size of the loop, which corresponds to finite value of [Formula: see text] in agreement with DNS.

中文翻译：

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湍流中的 Clebsch 约束和瞬时子

不可压缩流体中的湍流被表示为 3 维的场论。不涉及时间，因此这是为了描述 Hopf 泛函的平稳极限。基本字段是定义模规范变换（辛同胚）的 Clebsch 变量。给出了与稳态能量流相容的广义Beltrami流空间中规范不变Clebsch测度的显式公式。我们引入了与不间断规范不变性相关的 Clebsch 约束概念，并研究了 Clebsch 瞬时子：具有非平凡螺旋度的奇异涡量片。这是我们在 1990 年代开始的“即时和间歇”计划的实现。1这些奇异解决方案涉及增强远程边界处的无穷小随机力，从而导致临界现象。在欧拉方程中，涡量集中在随机自回避表面上，切线分量与法向距离的增量函数成正比。Navier–Stokes 方程中的粘度将此 delta 函数涂抹在具有固定能量流的 [公式：参见文本] 处的具有宽度 [公式：参见文本] 的高斯函数上。这些瞬时子支配着耗散的熵以及空间中围绕固定环 [公式：见文本] 的速度循环 [公式：见文本] 的 PDF。在大循环中，得到的对称指数分布完全符合数值模拟2包括指前因子[公式：见正文]。在小循环中，我们提倡将所得随机自回避表面理论与数值模拟中观察到的多重分形比例定律联系起来。这些定律被解释为波动的内部度量（刘维尔场）的结果。异常维度的曲线[公式：见文本]可以在小的[公式：见文本]处拟合抛物线，来自刘维尔理论，具有两个参数[公式：见文本]，[公式：见文本]。在大[公式：见文本]中，我们理论中后续矩的比率随着循环的大小线性增长，这对应于与 DNS 一致的[公式：见文本]的有限值。