Geometric Solution of Turbulent Mixing

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.