Geometric Solution of Turbulent Mixing

TL;DR

The paper presents an analytic solution for the one-point distribution of a passive scalar in decaying, strong turbulence by leveraging a loop-space reformulation of Navier–Stokes. Velocity statistics are governed by the Euler ensemble, yielding a linear loop equation for the scalar that can be solved exactly at fixed Schmidt number, revealing a geometric shell structure in the scalar density. For a point source, this produces expanding concentric shells with quantized radii set by Euler totients and piecewise-parabolic interior profiles, smoothed by diffusion; volume-averaged quantities provide robust, observable signatures. The work connects turbulence intermittency to number-theoretic structures and provides parameter-free predictions testable by DNS, with potential relevance to ramp-cliff patterns and certain astrophysical or quantum-fluid regimes where dissipation is negligible.

Abstract

We derive an analytical solution for the one-point distribution of a passive scalar in decaying homogeneous turbulence, in the limit of strong turbulence (high Re, fixed Schmidt number). Velocity statistics are governed by the Euler ensemble, a spontaneously stochastic exact solution of the loop equation from the Navier-Stokes equations in the strongly turbulent regime. The scalar's advection-diffusion problem is also recast as a solvable linear loop equation. For a localized initial condition, the solution consists of expanding concentric shells: the radial scalar profile is quantized and piecewise parabolic, with gaps organized by Euler totients - an arithmetic structure distinct from conventional scaling. This shell pattern is the unique solution in the Euler ensemble, smoothed by any finite diffusivity. The result provides the underlying geometric structure for scalar transport in decaying strong turbulence, relevant in astrophysical or quantum-fluid regimes where dissipation is negligible. This may explain the "ramp-cliff" structures observed in turbulent mixing half a century ago. While this shell structure is hard to resolve in DNS, its statistical signature is robustly captured by the volume-averaged scalar density, a measurable quantity.

Figures (3)

• Figure 1: Log--log plot of the universal function $\xi^2\Phi\left(\lfloor*\rfloor{\frac{1}{ \xi}}\right)$ where $\xi = \frac{2 \pi r}{\left(\sqrt{2 \tilde{\nu} (t+ t_0)} -\sqrt{2 \tilde{\nu} t_0}\right)}$.
• Figure 2: The plot of universal function $U(\kappa)$ in \ref{['UkappaExp']} describing the amplitude of passive scalar plane wave with wavevector $\boldsymbol{q}$ at the time $t$ as a function of scaling variable $\kappa = |\boldsymbol{q}| \left(\sqrt{2\tilde{\nu} (t+ t_0)}-\sqrt{2\tilde{\nu} t_0}\right)$
• Figure 3: The plot of universal function $V(\xi)$ in \ref{['GlobalAv']} describing the average temperature as a function of the time $t$ as a function of scaling variable $\xi = \left(\sqrt{2\tilde{\nu} (t+ t_0)}-\sqrt{2\tilde{\nu} t_0}\right)/r_0$, where $r_0$ is the radius of the spherical averaging domain. Note the slight discontinuity of the derivative at $\xi=2, 3,...$, when there is one more term added to the sum. This universal function is almost constant, because of the small coefficient $\frac{1}{240 \pi ^2}$.