Exact solution for the passive scalar in decaying turbulence

Abstract: We present the exact analytic solution for the one-point distribution of a passive scalar in decaying homogeneous turbulence, in the limit of vanishing viscosity and diffusivity at fixed Schmidt number. The velocity statistics are governed by the Euler ensemble, obtained previously as the spontaneously stochastic solution to the loop equation derived from the Navier-Stokes equations in the extreme turbulent limit. The resulting advection-diffusion problem is solved explicitly via loop calculus. For an initially localized scalar distribution, the solution develops a sequence of concentric spherical shells with a quantized, piecewise-parabolic radial profile of temperature with a finite limit at the center -- an outcome not predicted by conventional theories. This shell structure is a unique solution of the transport equation within the Euler ensemble, but may be smeared by finite diffusivity or forcing. It represents the geometric skeleton of scalar transport in ideal turbulence, and may be relevant in astrophysical or quantum-fluid regimes where dissipation is negligible. This work redefines expectations for scalar mixing in the high-Reynolds-number limit.