Fractional diffusion framework for stratospheric turbulent transport: Mechanistic origin from hierarchical many-body dynamics, self-similar solutions, and Richardson’s law

Stratospheric turbulent transport exhibits strongly anomalous behaviour, with relative dispersion growing as 〈 r 2 〉 ∝ t 2.5 in the EOLE balloon experiment at 200 mb, a scaling that classical Fickian diffusion and standard eddy-diffusivity parameterisations cannot reproduce. Existing stochastic models either fit this exponent empirically or require separate mechanisms for spatial and temporal non-locality. Here we show that a space–time fractional diffusion equation for stratospheric transport is not a phenomenological postulate but the continuum limit of an underlying hierarchical many-body process. Building on the hierarchical fractional dynamics of Chishtie (2026b) , in which power-law correlation functions generate Riesz fractional operators with a spatial order fixed by the scaling law α k = 2 − 2 / ( N k + 1 ) , we derive the governing equation from the correlation structure of the stratospheric field, with the spatial order α set by the statistics of coherent ejection structures and the temporal order μ set by the decay of Lagrangian velocity correlations. The anomalous scaling law 〈 r 2 〉 ∝ t 2 μ / α then follows from the dimensional self-similarity of the equation without empirical fitting. Parameters α = 1.2 and μ = 1.5 , with α realised as the ensemble average of the hierarchical law over a distribution of subsystem sizes that brackets Richardson’s classical value α = 4 / 3 , simultaneously predict the Richardson spatial exponent γ = 1.20 , within the observationally constrained range γ ≈ 1.2 –1.25. We confirm by direct simulation that the spatial order and the super-ballistic dispersion exponent emerge from the hierarchical cascade rather than being imposed, address normalization and positivity of the superdiffusive Green’s function explicitly, and validate against synthetic ground truth a protocol that infers the fractional orders from relative-displacement tails, relative dispersion, and a finite-scale Lyapunov exponent diagnostic. The framework provides a mechanistic, physics-based explanation for Richardson’s century-old empirical law and suggests implementable parameterisations for non-local and memory-dependent transport in atmospheric models.