Fractional diffusion framework for atmospheric turbulent transport: Self-similar solutions and universal scaling laws

Atmospheric turbulent transport exhibits anomalous scaling with variance growth $\langle r^2 \rangle \propto t^{2.5}$ observed in stratospheric balloon experiments, fundamentally deviating from classical Fickian diffusion. We present a rigorous framework based on space-time fractional diffusion equations with Riemann-Liouville time derivatives ($\mu > 0$) and Riesz spatial derivatives ($\alpha \in (1,2]$). Self-similarity analysis proves the fractional equation is invariant under $(\mathbf{r}, t, f) \mapsto (\lambda \mathbf{r}, \lambda^{\alpha/\mu} t, \lambda^{-n} f)$, establishing the universal scaling law $\langle r^2 \rangle \propto t^{2\mu/\alpha}$ directly from dimensional invariance. Monte Carlo simulations with $N = 10^5$ particles using the Chambers-Mallows-Stuck algorithm yield $\beta_{\text{sim}} = 2.499 \pm 0.002$ versus $\beta_{\text{theory}} = 2.500$ (0.03\% error), with radial distributions exhibiting heavy tails $P(r) \sim r^{-2.2}$ confirming L\'evy stable behavior. Parameters $\alpha = 1.2$, $\mu = 1.5$ reproduce EOLE stratospheric observations within 0--4\% error, providing physics-based foundations for atmospheric transport parameterizations.