| 17 Aug 2026
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.
It looks like that the author has performed a very fascinating theoretical work on the tornado dynamics. However, unfortunately, this fascination is not conveyed in the manuscript in any effective manner. As it stands for now, it is written like a "memoir" for the author himself, with various details not well explained. Personally, I would like to read it through with a better written version before I can provide my judgment on this manuscript. For this reason, I return the present manuscript with the "major revision".
Although I do not insist that the full details should be presented, the author should add few words of elaboration whenever he introduces a particular technical procedure, rather than just naming it. The manuscript must be written in a self-contained manner in the minimum sense. The author should also realize that NPG is directed to the geophysicists. Thus, every slightest unusual mathematical method must be well explained, at least, by few words. Otherwise, I suggest to re-send the present manuscript to a applied mathematical journal.
The most innovative aspect of the work is to introduce a fractional derivative to represent a diffusion term. However, the motivation is not well explained in the text. It appears to be presented over the paragraph of L35-53. However, the main difficulty for me to follow this part is the fact that the original formulation is introduced in a many-body problem. Although the author argues (by citing a paper in press) that a continuous version for the Navier-Stokes equations is also available, the text does not provide any elaboration how such a generalization is possible.
It appears to me that the key ingredient necessary for introducing fractional differential is an existence of "hierarchical statistics" (what is this to start with?). However, few words would be required to follow, how the "hierarchical statistics" (and for the hierarchy of what?) leads to a fractional differential.
As it stands for now, the appendices A-D appear to elaborate some of the technical procedures. However, they do not help much, because those appendices are never referred in the main text.
Though I did not pick all those systematically, the following terms must be better elaborated by few words with appropriate references:
anomaly exponent (L54)
Riemann-Liouville derivative
Caputo derivative
Riemann-Liouville-Caputo choice
reaction term (L203)
linear tail operator (L223)à
Laplace symbol
Watson's lemma: this must concern an asymptotic approximation for an integral. however, I do not see any integrals
Riemann-Liouville wall singularity
Mittag-Leffler relaxation (L321)
*
Some specifics:
L27-29, "effective viscosity" enhanced by presence of "large loading of water and sand or soil dust" : full elaboration is required to explain what this argument is about.
Eq. (2): just calling it "Karman similarity ansatz" is handy helpful, and does not explain anything with even no citation of Karman's original paper. This is a simple consequence of assuming a separation of variables. Then it can easily be shown that the radial dependeces are given in the form of Eq. (2): say it.
The notations, A, B, C, in the above: please find more reader-friendly notations. Those are not any mathematically abstract variables, but representing the height-dependence of particular variables after factoring out the radial dependence. Add simply tilde to the notations of the original variables, for example.
L74, u=A+iB : we more often call it "complex velocity" I believe. this term would be more accessible for more readers than "complex swirl"
L78, u|_z=i : this boundary condition must involve a certain normalization: say it
the pair equations (3) (4) and (5) (6): make some remarks on the basic behaviors of those two pair equation immediately after introducing them, that must be helpful to follow the subsequent reductions as well as discussions better.
L124-125, "the framework is therefore Riemann-Liouville in the interior....": is this a proper English?
The phase space analysis and bifurcation diagram: the system has two free parameters, Gamma and alpha. This very basic fact must be explicitly stated in the text. Fig. 3 is not quite the bifurcation diagram in this respect: the bifurcation diagram is a summary of the behavior in the phase space of the system parameters. Thus, it is better drawn in the phase space of (Gamma, alpha).
Asymptotic behaviors as z to infinity, e.g., Eqs. (8), (9), (10): those asymptotics appear to be inferred from the numerical results. If that is the case, the author must show the plots that backup those asymptotic formulas. Of course, it is much desirable to derive them more explicitly with asymptotic expansions, that should not be too difficult.
L205-208: a reduction leading to those conclusions must more explicitly be shown with some key equations
L247, "at the ground"?: it means the conditions at z=0. What the author really wants to mean must be the behavior of the solution "close to the ground"
L222-240: the discussion of this part must be fully elaborated with some details of reductions with equations
L268, "onset"?: how we can discuss about it with a steady model?
L360-361, "multiple-vortex tornado of 31 May 2013": if this event is so important, some details must be described to prove the importance of this event. Where it happened, to start with?