Geometric solution of turbulent mixing

We derive an analytic solution for the density of a passive scalar in decaying homogeneous turbulence, in the limit of high Reynolds number and fixed Schmidt number. The velocity statistics are described by the Euler ensemble, previously obtained as a spontaneously stochastic solution of the loop equation associated with the Navier-Stokes equations. The scalar advection-diffusion problem is formulated as a closed linear loop equation and solved within this framework. For a localized initial condition, the solution consists of a sequence of expanding concentric shells. The radial scalar profile is piecewise parabolic and supported at discrete radii, with amplitudes determined by Euler totients. This structure differs from conventional scaling descriptions of scalar turbulence. Finite diffusivity or sustained forcing smooths the discontinuities while preserving the leading-order geometry. The results provide a geometric description of scalar transport in decaying turbulence and may be relevant in regimes where dissipation is weak, such as astrophysical or quantum fluids. The predicted shell structure is difficult to resolve directly in numerical simulations; however, its statistical signature is captured by the volume-averaged scalar density, which provides a practical observable. This article is part of the theme issue 'Frontiers of turbulence and statistical physics'.