To the theory of decaying turbulence

Abstract

We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This solution family is equivalent to a fractal curve in complex space $\mathbb C^d$ with random steps parametrized by $N$ Ising variables $σ_i=\pm 1$, in addition to a rational number $\frac{p}{q}$ and an integer winding number $r$, related by $\sum σ_i = q r$. This equivalence provides a dual theory describing a strong turbulent phase of the Navier-Stokes flow in $\mathbb R_d$ space as a random geometry in a different space, like ADS/CFT correspondence in gauge theory. From a mathematical point of view, this theory implements a stochastic solution of the unforced Navier-Stokes equations. For a theoretical physicist, this is a quantum statistical system with integer-valued parameters, satisfying some number theory constraints. Its long-range interaction leads to critical phenomena when its size $N \rightarrow \infty$ or its chemical potential $μ\rightarrow 0$. The system with fixed $N$ has different asymptotics at odd and even $N\rightarrow \infty$, but the limit $μ\rightarrow 0$ is well defined. The energy dissipation rate is analytically calculated as a function of $μ$ using methods of number theory. It grows as $ν/μ^2$ in the continuum limit $μ\rightarrow 0$, leading to anomalous dissipation at $μ\propto \sqrtν \to 0$. The same method is used to compute all the local vorticity distribution, which has no continuum limit but is renormalizable in the sense that infinities can be absorbed into the redefinition of the parameters. The small perturbation of the fixed manifold satisfies the linear equation we solved in a general form. This perturbation decays as $t^{-λ}$, with a continuous spectrum of indexes $λ$ in the local limit $μ\to 0$.