A fractional von Kármán tornado model: anomalous vertical transport with a cascade-fixed order, and an advection-controlled far-field law

Abstract. We propose a reduced axisymmetric tornado model in which the vertical transport of momentum is nonlocal, governed by a fractional derivative whose order is not a free parameter but is fixed by the observed coherent substructure of the vortex through the hierarchical cascade relation α = 2−2/(N+1). Building on the classical von Kármán similarity reduction of Gavrikov and Taiurskii, we replace the ordinary vertical diffusion by a Riemann–Liouville operator of order α ∈ (1,2] acting on the deviation from the far-field swirl, and we retain the same complex swirl variable and the same coupling of the vertical velocity to the radial inflow. Three results follow. First, the bifurcation that separates the collapsing (vacuum-cleaner) regime from the tornado regime is an algebraic condition on the periphery-to-centre pressure drop and is therefore independent of α nonlocal transport of any admissible order preserves the threshold while reshaping the vertical structure. Second, the far-field swirl deviation decays as z^−(α+1), a law we confirm numerically across four orders to better than one percent. This exponent is two powers steeper than the naive linearised prediction z^−(α−1); we show that it is not a local dominant balance but is selected by the coupling of the advective jet to the radial continuity relation, with the fractional diffusion provably subdominant in the tail and the order α entering only through C_∞ – C(z) ∼ z^−α. Third, the wall layer is a saturating stretched exponential whose thickness collapses sharply as α → 1. The mapping from swirl ratio to coherent-subvortex count converts these results into a falsifiable prediction relating the anomaly exponent to observed vortex structure, illustrated for the multiple-vortex tornado of 31 May 2013.