Universal Theory of Decaying Turbulence

Abstract

Using loop equations, we derive an exact solution for the statistical distribution of freely decaying incompressible turbulence in arbitrary spatial dimension $d>1$. By applying the Mandelstam identity to the loop dynamics, we prove that the nonlinear advection term reduces to a pure derivative and drops out of the momentum-loop equation. As a result, the momentum-loop equation becomes purely diffusive, admitting an exact geometric solution as a random walk on a circle. Despite this distinct local loop algebra, the dimension-independent Euler ensemble dictates macroscopic observables via the Mellin transform. This Mellin transform $M(p)$ for the energy scaling function $H(k\sqrt{νt})$ emerges as completely universal, independent of $d$. The applications for $d=3$ were studied previously; here we extend the theory to $d=2$. Our analytical solution extends the empirically observed $k^{-3.5}$ spectrum to a continuous effective index, providing an exact analytic alternative to classical Kraichnan--Batchelor--Leith phenomenology. We prove that previously reported ``multifractal'' transient exponents are merely local tangents of a single universal scaling function. We find an infinite cascade of finite-time transitions (a Stokes staircase associated with complex zeros $z = 1/2 + iρ_n$ of the Riemann zeta function), mimicking finite-time discontinuities with Berry smoothing by the error function. Thus, there are no true finite-time singularities; instead, as a consequence of the Riemann hypothesis, an essential singularity emerges at infinite time, manifesting as rapid transitions at $t_n \propto ρ_n^3$, sharpening as $1/\log t_n$. We compare the predicted energy spectrum with recent 3D DNS in two independent ways, each yielding a close match within statistical errors.