the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 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.
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Status: open (until 19 Oct 2026)
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RC1: 'Comment on egusphere-2026-4368', Anonymous Referee #1, 18 Aug 2026
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AC2: 'Reply on RC1', Farrukh Chishtie, 08 Sep 2026
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I thank the referee for a careful reading and for the judgement that the work is fascinating but not yet well conveyed. I accept the diagnosis. I wrote the manuscript from inside the calculation, and I will rewrite it so that a geophysicist who has not met fractional calculus can follow every step. My specific commitments follow.
On motivation and the many-body origin (L35-53). I will replace the current paragraph with a self-contained account. The starting point is that the vertical momentum flux carried by coherent subvortices is not proportional to the local mean gradient, because a subvortex transports momentum across its full vertical extent in one event. A flux of that kind is represented by an integral kernel, and when the kernel is a power law the operator is a fractional derivative. Hierarchical statistics means that the flow is organised into levels, each containing $N_k$ similar coherent entities bound to one parent, and that the correlation functions of such a system are power laws whose exponent is fixed by $N_k$. The Riesz correspondence maps that exponent to the fractional order, which gives Eq. (1). I will state this in words, write the kernel form of the operator explicitly, and explain in two sentences how the continuum limit of Chishtie (2026b) carries it into an averaged momentum equation. I will also make clear, as Referee 4 requests, that the result is a nonlocal closure for the azimuthally averaged flow and not a replacement of the Navier-Stokes equations.
On the appendices. They will be cited at the point of use: Appendix A at Eq. (1), Appendix B in Section 5, Appendix C at the linear tail operator, Appendix D at the first mention of the Mittag-Leffler function.
On the list of terms. I will define each at first use with a reference. Anomaly exponent: the departure of the transport order from the classical value two, so that $2-\alpha$ measures the strength of the nonlocality. Riemann-Liouville and Caputo derivatives: I will write both as integral operators in the text, note that they differ only in whether differentiation acts before or after the convolution, and state the bridge relation between them. The Riemann-Liouville-Caputo choice then reads as a statement about which of the two integral forms is regular at a wall where the swirl is prescribed, and I will reword L124-125 accordingly. Reaction term: the term linear in the deviation that arises from expanding $u^2$ about the far-field state; I will show the expansion. Linear tail operator and Laplace symbol: I will write the operator and its transform explicitly and explain that the symbol is the expression in the transform variable that multiplies the transformed unknown. On Watson's lemma, the referee is right that no integral appears in the main text, and I should have been clearer: the integral is the branch-cut integral of Appendix C, and I will bring its form into Section 6 so that the lemma has an object to act on. Riemann-Liouville wall singularity: I will refer to Eq. (7) and explain that it is the operator's response to a constant, which has no counterpart at integer order. Mittag-Leffler relaxation: the fractional generalisation of exponential decay, with the function defined and its asymptotic behaviour stated.
L27-29, effective viscosity. I will expand this to explain that debris and water loading raise the effective momentum diffusivity of the two-phase flow well above the molecular value, cite the estimates in Gavrikov and Taiurskii and in Varaksin and Ryzhkov, and make clear that this motivates a transport coefficient far from molecular, while the coherent structure motivates the form of the operator.
Eq. (2). I will cite von Kármán (1921), show that the ansatz is a separation of variables consistent with the radial structure of the axisymmetric equations, and add the continuity and momentum steps that reduce the system to Eqs. (3)-(4). I accept the notation suggestion and will write the height-dependent factors as tilded versions of the original variables, and I will use complex velocity rather than complex swirl. The normalisation behind $u|_{z=0}=\mathrm{i}$ will be stated with the nondimensionalisation.
Remarks after Eqs. (3)-(4) and (5)-(6). I will add a paragraph after each pair describing the balance each term represents, the role of the far-field algebraic condition, and what changes when the second derivative becomes nonlocal.
Two-parameter phase space. Agreed. I will state at the outset that the system has two control parameters, $\Gamma$ and $\alpha$, and redraw Fig. 3 in the $(\Gamma,\alpha)$ plane. Because the threshold is $\Gamma=1$ for every $\alpha$, the boundary is a vertical line, which is the content of Proposition 1, and I will overlay $C(\infty)$ as a colour field to show how the jet amplitude varies with $\alpha$ within the tornado regime.
Asymptotics, Eqs. (8)-(10). I should clarify one point here, since the text evidently left the wrong impression: the asymptotic form in Section 6 is derived, and the numerics confirm it rather than suggest it. The derivation was compressed to the point of being invisible, which is my fault. I will add the explicit expansion in Section 6.2 with the term-by-term exponents, a log-log plot of $|w|$ against $z$ with fitted slopes for each order, and the corresponding plot of $C_\infty - C(z)$ verifying $z^{-\alpha}$. L205-208 and L222-240 will be expanded with the governing equations at each step, and the branch-cut inversion will be summarised in the text with the full calculation in Appendix C.
L247. I will write near the ground throughout.
L268, onset. The referee is right that a steady model cannot describe onset in time. I will replace onset with existence of the tornado regime and say explicitly that the classical result concerns the existence of a positive far-field jet, not its formation.
L360-361, El Reno. I will describe the event: near El Reno, Oklahoma, 31 May 2013, the widest tornado on record, observed by rapid-scan mobile Doppler radar, with multiple subvortices documented by Bluestein et al. (2018). I will also reposition it as an illustration of the mapping rather than the source of the reference order, in response to the community comment.
Citation: https://doi.org/10.5194/egusphere-2026-4368-AC2
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AC2: 'Reply on RC1', Farrukh Chishtie, 08 Sep 2026
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RC2: 'Comment on egusphere-2026-4368', Anonymous Referee #2, 22 Aug 2026
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Review of “A fractional von K.rm.n tornado model: anomalous vertical transport with a cascade-fixed order, and an advection-controlled far-field law” by Chishtie
Recommendation: Minor revisions
This manuscript is developing a fractional derivative tornado model. It does so by introducing a fractional derivative in the vertical component. This is a nice study and can be considered for publication after my comments have been addressed.
1) In my opinion it would strengthen the manuscript if the model would be evaluated against tornado observations. It is not clear to me how much the fractional derivative improves the model with respect to real tornadoes.
2) All variables should be defined at first use. E.g. abstract (alpha, N), Eq. (2): C, Q
3) The author should elaborate more on the physical meaning of the fractional derivate in the vertical component.
Citation: https://doi.org/10.5194/egusphere-2026-4368-RC2 -
AC3: 'Reply on RC2', Farrukh Chishtie, 08 Sep 2026
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I thank the referee for the positive assessment.
- Evaluation against observations. I agree this is the natural next step and I will be explicit about what the present paper can and cannot do. The similarity reduction yields a height profile of the azimuthal factor, whereas radar retrievals give radial profiles of azimuthal wind at a few heights, so a direct comparison requires an observational study with its own assumptions. What I can do in revision is twofold. I will compare the predicted compression of the near-surface layer with decreasing $\alpha$ against the heights of low-level wind maxima reported in Rotunno and Bluestein (2024), and I will state the far-field prediction $z^{-(\alpha+1)}$ as a quantitative target, with the El Reno value $z^{-5/2}$ and the observations that would test it. I will also add a sentence making clear that the contribution of the fractional operator is structural: it reshapes the jet and the wall layer at fixed threshold, and the classical model is recovered at $\alpha=2$.
- Definitions at first use. All symbols will be defined where they first appear, including $\alpha$ and $N$ in the abstract and $C$ and $Q$ at Eq. (2).
- Physical meaning of the fractional vertical derivative. I will add a dedicated paragraph in Section 3.1. The operator is a weighted integral of the velocity deviation over all heights below the point of evaluation, with a power-law weight. Physically it says that the momentum flux at height $z$ depends on the velocity profile over the whole column beneath, because coherent subvortices transport momentum across their full vertical extent rather than by local gradient diffusion. The order $\alpha$ controls how slowly the weight decays and hence how far the influence reaches; $\alpha=2$ is the local limit.
Citation: https://doi.org/10.5194/egusphere-2026-4368-AC3
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AC3: 'Reply on RC2', Farrukh Chishtie, 08 Sep 2026
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RC3: 'Comment on egusphere-2026-4368', Anonymous Referee #3, 27 Aug 2026
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There is a lot of mathematical exposition, but it's not clear that it sufficiently recognises physical reality. However, it seems par for the course and publishing it would be acceptable in view of the difficulty of making observations of tornadoes.
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AC4: 'Reply on RC3', Farrukh Chishtie, 08 Sep 2026
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I thank the referee for a generous and thoughtful report, and for the references, which I will use. I take the general remarks first and then the commentary in order.
On rapid intensification and dissipation. The referee is right that a steady model cannot represent them, and I will say so plainly in the Introduction rather than only in the limitations. The stationary solutions computed here are the long-time states of an unsteady problem, and the time-fractional extension I outline in Section 9.3 is the route to intensification dynamics, in which Mittag-Leffler relaxation toward the steady profile replaces exponential stabilisation. I will make clear that the present paper establishes the steady balances against which that extension is to be checked.
On energetics. I agree that the generation of tornadic wind speeds is a question about the small fraction of available energy that can be converted into kinetic energy by nearly entropy-free work, and that the supply in a supercell comes from the phase changes of water. The model takes that supply as given, through the upper-atmosphere vortex strength and the periphery-to-centre pressure drop that together define $\Gamma$, and asks how the vertical structure of the vortex follows from them. I will add a paragraph in the Discussion that places the model in this energetic context, citing Gill (1982), Dutton (1986), and Tuck (2025), and that identifies the generalised viscosity $\nu_\alpha$ as the parameter through which dissipation enters, so that the referee's point at lines 315-325 is addressed directly: a quantitative account of $\nu_\alpha$ requires the system's energetics, and I will state this as the condition for turning the structural predictions into quantitative ones.
Line 17, turbulence. The referee is right that the word does not appear, and it should. The nonlocal, non-Gaussian character of the vortex is a turbulence statement, and the fractional operator is a closure for turbulent momentum flux in the azimuthally averaged flow, a framing that Referee 4 also asked for and that the revision will adopt throughout. I will also cite Tuck (2025), because the picture of atmospheric structure as vortices interacting on all scales is close to the hierarchical picture underlying the cascade relation, as I explain under lines 281-299 below.
Line 23, control. Agreed. I will write that the swirl ratio orders the transition between vortex types, which is the laboratory result, rather than that it controls it.
Lines 43-48, many-body dynamics. The bodies are not molecules. They are coherent fluid entities at each level of a hierarchy: subvortices within a vortex, and at finer levels the eddies within them. Each level consists of $N_k$ similar entities bound to one parent, and the power-law correlation statistics of such a system fix the fractional order at that level through Eq. (1). I will state this at first use and contrast it with the atomistic approach of Kadau et al. (2010), which operates at the molecular scale where the local Navier-Stokes description is recovered.
Lines 57-60, rotation sense, pressure gradient, and Coriolis. The rotation is cyclonic in the case I have in mind, but the model itself is indifferent to the sense, because the equations are invariant under reversal of the azimuthal direction. The Coriolis force is neglected on the standard scale argument: for a tornado of radius of order 100 m and azimuthal speed of order 100 m/s the Rossby number is of order $10^4$, so the Coriolis acceleration is negligible within the vortex. The cyclonic preference in observed tornadoes is inherited from the parent mesocyclone, which in the model enters through the upper-atmosphere vortex that sets the far-field boundary condition, not through Coriolis acting within the vortex itself. The pressure drop is parameterised by $\Gamma$, the squared angular velocity of the upper vortex in nondimensional form; I will give the dimensional correspondence and note that core pressure deficits of order 10 to 100 hPa are consistent with the tornado regime $\Gamma>1$. All of this will be stated explicitly in Section 2.
Line 63, jet. The vertical velocity $C(z)$, which for $\Gamma>1$ is positive at large height. I will define it at first use.
Lines 281-299, vortices on all scales and the value of selecting integers. I think this is the point where the hierarchical picture and the referee's own picture meet, and I am grateful for the question because it shows what the text failed to say. A sequence of vortices on all scales is exactly a hierarchy of levels, and the cascade relation assigns an order to each level from the number of coherent entities it contains. A level with many entities, such as the fine-scale eddy population, has $\alpha_k\to 2$ and contributes classical local diffusion; a level with few entities, such as three subvortices in a parent vortex, has an order well below two and contributes the anomalous transport. The integers are therefore not selected in preference to the continuum of scales; they identify which level in that continuum departs from local behaviour, and the full operator is a sum over levels. The present paper keeps only the anomalous level, and I will state this truncation explicitly, with the distributed-order extension identified as the next step. The tornado is itself a vortex, as the referee says, and in the hierarchy it is the parent at the level whose inner entities are the subvortices.
On the remark that the approach circumvents dissipation and closure: I would put it the other way. The fractional operator is the closure, and its order is fixed by the coherent structure rather than fitted; dissipation enters through $\nu_\alpha$, which, as noted above, is where the energetics must eventually be supplied.
Lines 326-341. I thank the referee for this.
Citation: https://doi.org/10.5194/egusphere-2026-4368-AC4
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AC4: 'Reply on RC3', Farrukh Chishtie, 08 Sep 2026
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RC4: 'Comment on egusphere-2026-4368', Anonymous Referee #4, 07 Sep 2026
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The manuscript is written in a rather terse manner, particularly in the Introduction, and it is difficult to understand clearly the assumptions underlying the proposed approach. For example, it seems clear that tornadoes, as fluid-mechanical phenomena, are governed by the Navier–Stokes equations. However, on page 3, line 36, the author refers to “fractional generalizations of the Navier–Stokes equations.” It is not clear why such generalizations are required in the present context, and this point is not justified in the manuscript.
The Navier–Stokes equations already give rise to intrinsically nonlocal and multiscale properties. In homogeneous and isotropic turbulence, this is reflected, for example, in the non-trivial Kolmogorov −5/3 scaling associated with the energy cascade. More generally, intermittency and the Richardson energy-cascade picture involve long-range properties in the dissipation field. Therefore, the introduction of fractional operators cannot simply be motivated by the need to account for “nonlocality”; the precise physical and mathematical motivation should be stated.
My understanding is that the author may instead be attempting to formulate an averaged description of the Navier–Stokes equations. If this is the case, then nonlocal closure models can indeed arise. Several recent studies have explored such approaches, including formulations involving fractional Laplacian operators (see, for example, the references suggested below). From this perspective, the objective would be to construct a nonlocal closure for Reynolds-averaged Navier–Stokes equations rather than to introduce a fractional generalization of the fundamental Navier–Stokes equations themselves. If this is indeed the approach proposed here, the Introduction and the first part of the manuscript should be substantially rewritten to make this distinction and the underlying assumptions much clearer.
Section 2 introduces the model of Gavrikov and Taiurskii (2019), which appears to rely on earlier work by von Kármán. However, the relevant original work is not cited, and, more importantly, the connection between the Navier–Stokes equations and Eq. (3) is not explained. How is Eq. (3) derived, and what assumptions or approximations are required to obtain it? This derivation, or at least a sufficiently detailed explanation of its physical and mathematical basis, is essential for the reader to assess the validity of the subsequent developments. Similarly, in Section 3, Eq. (3) is replaced by the nonlocal expression given in Eq. (5), again without sufficient justification. The manuscript should explicitly explain the logical and mathematical connection between these two equations. In particular, it should clarify why a nonlocal formulation is introduced at this stage, what physical mechanism it is intended to represent, and under which assumptions Eq. (5) can be regarded as an appropriate replacement or generalization of Eq. (3).
The remainder of the manuscript develops the consequences of these equations. However, unless their origin, underlying assumptions, and mutual relationships are clearly established at the beginning, the subsequent developments do not rest on a sufficiently solid theoretical foundation. I therefore recommend that the authors substantially revise the first sections of the manuscript before the later mathematical developments can be properly assessed.
Additional comment on the bibliography
The manuscript contains only nine references, which seems clearly insufficient given the topic addressed. Such a limited bibliography might perhaps be understandable for a study opening an entirely new field of research, but this is not the case here. There is an existing literature on nonlocal and fractional approaches to turbulence modelling and closure problems that should be discussed and properly positioned with respect to the present work. The authors should considerably broaden the literature review.
Relevant recent examples include:
Samiee et al., Physics of Fluids, 32, 055102 (2020).
Hadi Seyedi and Zayernouri, Physics of Fluids, 34, 035104 (2022).
Clark Di Leoni et al., Journal of Fluid Mechanics, 914, A6 (2021).
Hamba, Journal of Fluid Mechanics, 950, A38 (2022).
These and many related studies could help the authors better clarify the relationship between fractional/nonlocal formulations, Reynolds averaging, and turbulence closure.
Citation: https://doi.org/10.5194/egusphere-2026-4368-RC4 -
AC5: 'Reply on RC4', Farrukh Chishtie, 08 Sep 2026
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I thank the referee for a comment that identifies the framing the manuscript should have had from the start, and for the references, which I will use.
The referee's reading is correct, and I should have made it explicit. The fractional operator does not modify the Navier-Stokes equations, which govern the flow at every scale. It is a nonlocal closure for the azimuthally averaged flow, in which the momentum flux carried by the unresolved coherent structures is represented by a fractional operator rather than by a local eddy viscosity. I will rewrite the Introduction and Section 3 to state this, to distinguish it from the phenomenological fractional Navier-Stokes proposals I previously cited, and to position it within the nonlocal closure literature, including Samiee et al. (2020), Seyedi and Zayernouri (2022), Clark Di Leoni et al. (2021), Hamba (2022), and Epps and Cushman-Roisin (2018). What my approach adds to that literature is that the order is fixed by the count of coherent structures rather than fitted, and I will make that the stated contribution.
On Kolmogorov scaling and nonlocality. I agree that the Navier-Stokes equations are already nonlocal and that the cascade is the source of that nonlocality. The closure literature the referee cites exists because an averaged description still needs a model for the unresolved flux, and that is exactly where a nonlocal operator legitimately enters. In the hierarchical framework the fine-scale eddy population is a level with large $N$, for which Eq. (1) gives $\alpha\to 2$, the classical viscous limit. The anomaly in the present paper comes from the level with small $N$, the subvortices. I will state this and note that a two-level formulation is distributed-order.
One consequence of the closure framing needs to be addressed and I will do so. The references above use the symmetric fractional Laplacian, whereas I use a one-sided Riemann-Liouville operator in $z$. The choice is deliberate for a half-space with a rigid ground at $z=0$, where the nonlocal influence is bounded below by the surface and carried upward by the jet, and I will justify it in Section 3 and note the symmetric alternative.
On the derivation of Eq. (3) and its replacement by Eq. (5). I will cite von Kármán (1921), give the reduction from the axisymmetric Navier-Stokes equations to Eqs. (3)-(4) with the assumptions stated, and then show that Eq. (5) follows from Eq. (3) by replacing the local vertical flux with the nonlocal closure flux, acting on the deviation from the far-field state so that the algebraic far-field balance is preserved exactly.
On the bibliography. It will be broadened to cover the nonlocal closure literature, the von Kármán and Bödewadt boundary-layer literature, and the relevant tornado observational and simulation literature.
Citation: https://doi.org/10.5194/egusphere-2026-4368-AC5
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AC5: 'Reply on RC4', Farrukh Chishtie, 08 Sep 2026
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CC1: 'Comment on egusphere-2026-4368', Johannes Dahl, 08 Sep 2026
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While I found the application of fractional derivatives to describe the mixing terms interesting, there are several issues with the treatment.
1. I wonder how the choice of N = 3 can be justified. This number is selected from a random, documented high-end tornado, which at the time of observation may temporarily have had three subvortices. But many tornadoes have a different number of subvortices, and even the El Reno tornado probably exhibited a varying number of subvortices during its lifetime.
2. Importantly, aside from this random choice, an m = 3 asymmetry violates the axisymmetric assumption. The modified Karman similarity equations assume axisymmetry and your model hinges on an m = 3 asymmetry, which is inconsistent.
3. The subvortices only form once a 2-celled vortex is present, so even ignoring the above inconsistency for now, the model could not be applied to the low-swirl single-celled case (w > 0). The treatment of the mixing terms thus does not match the structure of the modeled vortices.4. Further, assuming N = 3 implies that there are literally only three turbulent eddies available to achieve the mixing. While the subvortices do contribute to radial transport, there are still the large number of smaller-scale turbulent eddies that facilitate mixing, which are all ignored in your treatment.
While this paper is a nice mathematical exercise, it seems to exhibit major logical and conceptual discrepancies.
Citation: https://doi.org/10.5194/egusphere-2026-4368-CC1 -
AC6: 'Reply on CC1', Farrukh Chishtie, 08 Sep 2026
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I thank Professor Dahl for a comment that goes directly to the conceptual core of the model, and I welcome the chance to clarify it, because the four points share a single premise that the manuscript did not adequately explain. The premise is that $N$ counts the turbulent eddies responsible for mixing and that the model is therefore tied to a specific azimuthal wavenumber. That is not what the model asserts, and I take responsibility for not saying so clearly. $N$ counts the coherent entities at the hierarchical level that sets the anomalous order, the equations describe the azimuthally averaged flow, and the fractional operator is the closure for the momentum flux carried by the unresolved coherent structures. With that clarified, each point resolves as follows.
- The choice $N=3$. Professor Dahl is right that the subvortex count of a real tornado varies in time and across events, and I did not intend El Reno at one instant to be the model's value. The contribution is the mapping $N\mapsto\alpha$: single-cell, two-cell, and multiple-vortex regimes correspond to distinct orders, and a tornado whose subvortex count changes in time has an order that changes with it. That is the time-varying-order problem I identify as the unsteady extension, and I will say so. In revision $N=3$ will be one illustrative point on the ladder, with the full ladder shown.
- Axisymmetry and $m=3$. Axisymmetry is a property of the mean flow, not of the fluctuations about it. The subvortices are coherent unresolved structures whose azimuthal average is axisymmetric, and the operator represents the flux they carry, exactly as Reynolds-averaged equations for an axisymmetric mean remain consistent with non-axisymmetric fluctuations. There is no inconsistency once the averaged-flow interpretation is stated, and Referee 4 independently asked for the same statement, so the revised Introduction and Section 3 will make it explicit.
- The single-celled case. The anomalous order arises from the level with a small number of coherent entities. A single-celled vortex has no such level, and the operator reduces to the classical local form, so the model recovers Gavrikov and Taiurskii for the low-swirl case rather than failing to apply to it. This is the correct limit and I will show it on the $(\Gamma,\alpha)$ phase diagram requested by Referee 1, with the single-cell regime at $\alpha=2$ and the multiple-vortex regimes at lower orders.
- Small-scale eddies. The model does not assume three eddies do the mixing. The fine-scale eddy population is a further hierarchical level with large $N$, for which Eq. (1) gives $\alpha\to 2$ and hence the classical viscous term. The single-order model in this paper retains only the anomalous level; the full operator is a sum over levels, which is distributed-order. I will state this truncation, state that the classical viscous term is its large-$N$ limit, and identify the distributed-order extension as the natural next step.
I am grateful for the comment, because it has clarified what the model claims. The revision will present it as a nonlocal closure for the azimuthally averaged vortex whose order is set by coherent structure, with the classical model as its local limit.
Citation: https://doi.org/10.5194/egusphere-2026-4368-AC6
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AC6: 'Reply on CC1', Farrukh Chishtie, 08 Sep 2026
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AC1: 'Reply on RC1', Farrukh Chishtie, 08 Sep 2026
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I thank the referee for a careful reading and for the judgement that the work is fascinating but not yet well conveyed. I accept the diagnosis. I wrote the manuscript from inside the calculation, and I will rewrite it so that a geophysicist who has not met fractional calculus can follow every step. My specific commitments follow.
On motivation and the many-body origin (L35-53). I will replace the current paragraph with a self-contained account. The starting point is that the vertical momentum flux carried by coherent subvortices is not proportional to the local mean gradient, because a subvortex transports momentum across its full vertical extent in one event. A flux of that kind is represented by an integral kernel, and when the kernel is a power law the operator is a fractional derivative. Hierarchical statistics means that the flow is organised into levels, each containing $N_k$ similar coherent entities bound to one parent, and that the correlation functions of such a system are power laws whose exponent is fixed by $N_k$. The Riesz correspondence maps that exponent to the fractional order, which gives Eq. (1). I will state this in words, write the kernel form of the operator explicitly, and explain in two sentences how the continuum limit of Chishtie (2026b) carries it into an averaged momentum equation. I will also make clear, as Referee 4 requests, that the result is a nonlocal closure for the azimuthally averaged flow and not a replacement of the Navier-Stokes equations.
On the appendices. They will be cited at the point of use: Appendix A at Eq. (1), Appendix B in Section 5, Appendix C at the linear tail operator, Appendix D at the first mention of the Mittag-Leffler function.
On the list of terms. I will define each at first use with a reference. Anomaly exponent: the departure of the transport order from the classical value two, so that $2-\alpha$ measures the strength of the nonlocality. Riemann-Liouville and Caputo derivatives: I will write both as integral operators in the text, note that they differ only in whether differentiation acts before or after the convolution, and state the bridge relation between them. The Riemann-Liouville-Caputo choice then reads as a statement about which of the two integral forms is regular at a wall where the swirl is prescribed, and I will reword L124-125 accordingly. Reaction term: the term linear in the deviation that arises from expanding $u^2$ about the far-field state; I will show the expansion. Linear tail operator and Laplace symbol: I will write the operator and its transform explicitly and explain that the symbol is the expression in the transform variable that multiplies the transformed unknown. On Watson's lemma, the referee is right that no integral appears in the main text, and I should have been clearer: the integral is the branch-cut integral of Appendix C, and I will bring its form into Section 6 so that the lemma has an object to act on. Riemann-Liouville wall singularity: I will refer to Eq. (7) and explain that it is the operator's response to a constant, which has no counterpart at integer order. Mittag-Leffler relaxation: the fractional generalisation of exponential decay, with the function defined and its asymptotic behaviour stated.
L27-29, effective viscosity. I will expand this to explain that debris and water loading raise the effective momentum diffusivity of the two-phase flow well above the molecular value, cite the estimates in Gavrikov and Taiurskii and in Varaksin and Ryzhkov, and make clear that this motivates a transport coefficient far from molecular, while the coherent structure motivates the form of the operator.
Eq. (2). I will cite von Kármán (1921), show that the ansatz is a separation of variables consistent with the radial structure of the axisymmetric equations, and add the continuity and momentum steps that reduce the system to Eqs. (3)-(4). I accept the notation suggestion and will write the height-dependent factors as tilded versions of the original variables, and I will use complex velocity rather than complex swirl. The normalisation behind $u|_{z=0}=\mathrm{i}$ will be stated with the nondimensionalisation.
Remarks after Eqs. (3)-(4) and (5)-(6). I will add a paragraph after each pair describing the balance each term represents, the role of the far-field algebraic condition, and what changes when the second derivative becomes nonlocal.
Two-parameter phase space. Agreed. I will state at the outset that the system has two control parameters, $\Gamma$ and $\alpha$, and redraw Fig. 3 in the $(\Gamma,\alpha)$ plane. Because the threshold is $\Gamma=1$ for every $\alpha$, the boundary is a vertical line, which is the content of Proposition 1, and I will overlay $C(\infty)$ as a colour field to show how the jet amplitude varies with $\alpha$ within the tornado regime.
Asymptotics, Eqs. (8)-(10). I should clarify one point here, since the text evidently left the wrong impression: the asymptotic form in Section 6 is derived, and the numerics confirm it rather than suggest it. The derivation was compressed to the point of being invisible, which is my fault. I will add the explicit expansion in Section 6.2 with the term-by-term exponents, a log-log plot of $|w|$ against $z$ with fitted slopes for each order, and the corresponding plot of $C_\infty - C(z)$ verifying $z^{-\alpha}$. L205-208 and L222-240 will be expanded with the governing equations at each step, and the branch-cut inversion will be summarised in the text with the full calculation in Appendix C.
L247. I will write near the ground throughout.
L268, onset. The referee is right that a steady model cannot describe onset in time. I will replace onset with existence of the tornado regime and say explicitly that the classical result concerns the existence of a positive far-field jet, not its formation.
L360-361, El Reno. I will describe the event: near El Reno, Oklahoma, 31 May 2013, the widest tornado on record, observed by rapid-scan mobile Doppler radar, with multiple subvortices documented by Bluestein et al. (2018). I will also reposition it as an illustration of the mapping rather than the source of the reference order, in response to the community comment.
Citation: https://doi.org/10.5194/egusphere-2026-4368-AC1
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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?