Rigorous justification of the formal perturbation series for stationary powerful-tornado solutions of the reduced Navier–Stokes system, and the role of the Coriolis force in it

Gavrikov and Taiurskii (2020) reduced the axisymmetric incompressible Navier–Stokes equations on the half-space $z\ge0$ to a one-dimensional system for a complex velocity combination $u$ and an axial velocity $C$, using exact polynomial-in-$r$ special solutions, and classified stationary tornado-type versus antitornado-type flows. For 'powerful tornadoes', where the scaled boundary value $a=u_0\Gamma^{-1/2}$ is small, they constructed a formal double power series in the real and imaginary parts of $a$ whose coefficients solve a chain of linear boundary-value problems on the half-line, and used it to derive velocity and funnel-boundary formulas claimed to hold 'with any given accuracy'. The paper itself acknowledges that convergence of these series and the legitimacy of term-by-term differentiation are open, and that whether the governing equations should include the Coriolis force 'needs further discussion'. The problem posed here is to give this stationary powerful-tornado theory a rigorous analytical foundation: prove (or refute) the analytic dependence of the stationary solutions on $a$ with convergent expansions and controlled remainders, and settle the effect of incorporating the Coriolis force into the bulk reduced equations on the existence and classification of stationary tornado-type solutions.

Background

A tornado is a concentrated swirling upward air flow near the Earth's surface, believed to be generated by a rotating 'mother cloud' vortex ring in the upper atmosphere. Direct three-dimensional numerical simulation of tornadoes is difficult because it requires initial and boundary data that observations cannot supply, which motivated Gavrikov and Taiurskii to look for exact special solutions of the incompressible Navier–Stokes equations that capture the near-axis 'trunk' of the flow. They used the ansatz $U_r=rA(t,z)$, $U_\varphi=rB(t,z)$, $U_z=C(t,z)$, $p=\Phi(t,z)+r^2Q(t)$ in cylindrical coordinates, i.e. radial and azimuthal velocities linear in the radius $r$ and pressure quadratic in $r$; writing $u=A+\mathrm{i}B$, this ansatz satisfies the Navier–Stokes equations with constant density $\rho$ and viscosity $\mu$ exactly on the half-space $z\ge0$ if and only if $u$ and $C$ obey, in dimensionless form, the reduced system

$$\frac{\partial u}{\partial t}+C\frac{\partial u}{\partial z}-\frac{1}{R}\frac{\partial^2 u}{\partial z^2}+u^2+\Gamma=0,\qquad \frac{\partial C}{\partial z}=-2\operatorname{Re}u,$$

where $R=\rho U_0L_0/\mu$ is the Reynolds number built from the characteristic velocity and length scales, and $\Gamma=2Qt_0^2/\rho$ is the dimensionless version of the quadratic pressure-drop parameter $Q$ measuring the radial pressure growth from the tornado axis to the periphery. In the rotating frame of the Earth the Coriolis and centrifugal forces are neglected in the bulk equations (they are small for tornadoes), the Coriolis force being allowed to enter at most through the surface boundary value $u(0)=\mathrm{i}\omega_c$ with $\omega_c=-\Omega_E\sin\varphi$, where $\Omega_E$ is the Earth's angular velocity and $\varphi$ the latitude. Stationary solutions are studied through the boundary-value problem

$$u(0)=u_0,\qquad u(\infty)=\mathrm{i}\sqrt{\Gamma},\qquad C(0)=0,$$

and a stationary solution is called a tornado (antitornado) if $\lim_{z\to\infty}C(z)>0$ ($<0$). Numerically, using the stabilization method (time integration of the parabolic system until a steady state is reached), the authors found a critical value $Q_{\mathrm{cr}}=\rho\omega^2/2$ separating tornado-type from antitornado-type stationary solutions, where $\omega$ is the near-surface angular velocity entering the boundary condition $u(0)=\mathrm{i}\omega$ (in the model, the Coriolis-swirled value $\omega=\omega_c$), and the algebraic relation $\Omega^2=2Q/\rho$ as a necessary condition for a stationary tornado to exist, where $\Omega$ is the angular velocity of the air in the mother-cloud vortex ring, i.e. the far-field value $u(\infty)=\mathrm{i}\Omega$ in the underlying dimensional variables, which under the dimensionless normalization above becomes $u(\infty)=\mathrm{i}\sqrt{\Gamma}$ with $\sqrt{\Gamma}=\Omega t_0$; these statements rest on numerics, not on analytic proof. A 'powerful tornado' is the regime $|a|=|u_0|\Gamma^{-1/2}\ll1$, physically meaning that the mother-cloud angular velocity $\Omega$ is much larger than the near-surface angular velocity $\omega$. In this regime an exact rescaling (their Theorem 2) maps the stationary problem to the parameter-free canonical system

$$\tilde C\frac{\mathrm{d}\tilde u}{\mathrm{d}x}-\frac{\mathrm{d}^2\tilde u}{\mathrm{d}x^2}+\tilde u^2+1=0,\qquad \frac{\mathrm{d}\tilde C}{\mathrm{d}x}=-2\operatorname{Re}\tilde u,$$

on the half-line $x\ge0$ with boundary conditions $\tilde u(0)=a$, $\tilde u(\infty)=\mathrm{i}$, $\tilde C(0)=0$. Assuming — without proof — that stationary solutions of this canonical problem depend analytically on the boundary value $a=a_1+\mathrm{i}a_2$, the authors write the double power series $u_a(x)=\sum_\alpha u_\alpha(x)a^\alpha$, $C_a(x)=\sum_\alpha C_\alpha(x)a^\alpha$ over multi-indices $\alpha=(\alpha_1,\alpha_2)$ (pairs of nonnegative integers, with $a^\alpha=a_1^{\alpha_1}a_2^{\alpha_2}$ and $|\alpha|=\alpha_1+\alpha_2$), whose coefficients with $|\alpha|>0$ solve a chain of linear boundary-value problems sharing one linear differential operator, and they compute the first- and second-order coefficients numerically (the leading canonical profile satisfies $C_0(\infty)\approx1.35$). These expansions underlie the paper's practical output: asymptotic formulas for the vertical trunk velocity, the boundary-layer thickness, and an explicit funnel-shaped tornado boundary $r(z)$ obtained by matching the trunk-side pressure $Qr^2+\Phi(z)$ to an ambient adiabatic atmosphere. The paper states verbatim that 'above we have dealt with formal power series' and that 'the convergence of the series and the possibility of their term-by-term differentiation ... are much more difficult questions and require special consideration', and concludes that 'the question whether the air dynamics equation should take into account the Coriolis force needs further discussion', since within the present model tornado solutions exist even with the Coriolis force absent from the bulk dynamics while viscosity is essential.

Problem Statement

The common objective is to place the stationary powerful-tornado theory of Gavrikov and Taiurskii on a rigorous analytical footing, in two linked components concerning the same stationary boundary-value problem. (A) For the canonical stationary problem on the half-line $x\ge0$,

$$C\frac{\mathrm{d}u}{\mathrm{d}x}-\frac{\mathrm{d}^2u}{\mathrm{d}x^2}+u^2+1=0,\qquad \frac{\mathrm{d}C}{\mathrm{d}x}=-2\operatorname{Re}u,$$

with boundary conditions $u(0)=a$, $u(\infty)=\mathrm{i}$, $C(0)=0$ and small complex $a=a_1+\mathrm{i}a_2$: prove that stationary solutions depend analytically on $a$ in a neighborhood of $a=0$ — so that the formal double power series $u_a=\sum_\alpha u_\alpha a^\alpha$, $C_a=\sum_\alpha C_\alpha a^\alpha$ actually converge (uniformly on the half-line, or in an explicitly stated function space compatible with the far-field condition), term-by-term differentiation used to derive the coefficient problems is justified, and the $O(|a|^p)$ remainder estimates hold — or refute this by proving divergence of the series, non-analyticity of the solution map, or nonexistence of stationary solutions for arbitrarily small $|a|\neq0$, thereby delimiting the validity of the approximation scheme from which the velocity and funnel-boundary formulas are derived. (B) Within the same polynomial-in-$r$ ansatz class, incorporate the Coriolis force into the bulk reduced equations (rather than only into the boundary condition at $z=0$) in a fully consistent stationary boundary-value formulation, and determine whether stationary tornado-type solutions persist and how the classification criteria — the critical value $Q_{\mathrm{cr}}=\rho\omega^2/2$ and the necessary condition $\Omega^2=2Q/\rho$ — are quantitatively modified, thereby settling the paper's question of whether the governing equations for tornadoes should include Coriolis effects. An answer must address component (A) fully. Component (B) must declare the Coriolis-modified stationary boundary-value formulation (surface and far-field data), derive the corresponding reduced system, and determine whether the tornado/antitornado classification persists and whether the analytic dependence on $a$ established or refuted in (A) carries over to it, quantifying any change in the critical value $Q_{\mathrm{cr}}$ and in the necessary condition $\Omega^2=2Q/\rho$ within that declared formulation; a purely qualitative robustness statement does not resolve it.

References

1. M B Gavrikov and A A Taiurskii, 'Mathematical Theory of Powerful Tornadoes in the Atmosphere', Journal of Physics: Conference Series 1640, 012002 (2020). DOI: 10.1088/1742-6596/1640/1/012002. URL: https://doi.org/10.1088/1742-6596/1640/1/012002
2. M B Gavrikov and A A Taiurskii, 'A Mathematical Model of Tornado', Journal of Physics: Conference Series 1336, 012001 (2019). DOI: 10.1088/1742-6596/1336/1/012001. URL: https://doi.org/10.1088/1742-6596/1336/1/012001
3. M B Gavrikov and A A Tayurskiy, 'Mathematical model of powerful tornado in the atmosphere' (Математическая модель мощного торнадо в атмосфере), Keldysh Institute Preprints, 2020, No. 44 (2020). DOI: 10.20948/prepr-2020-44. URL: https://doi.org/10.20948/prepr-2020-44
4. M B Gavrikov and A A Taiurskii, 'Mathematical Model of a Powerful Tornado in the Atmosphere', Mathematical Models and Computer Simulations (2021). DOI: 10.1134/S2070048221060065. URL: https://doi.org/10.1134/S2070048221060065
5. M B Gavrikov and A A Taiurskii, 'Mathematical Model of Magnetic Tornado in Solar Plasma', Journal of Physics: Conference Series 2055, 012015 (2021). DOI: 10.1088/1742-6596/2055/1/012015. URL: https://doi.org/10.1088/1742-6596/2055/1/012015
6. Afifah Maya Iknaningrum, Pen-Yuan Hsu, Tsuyoshi Yoneda and Hirofumi Notsu, 'Energy Transfer Dynamics Generated by Non-Axisymmetric Tornado-Type Flows', arXiv:2509.05546 (2025). URL: https://arxiv.org/abs/2509.05546
7. Mustamina Maulani and Valentinus Galih Vidia Putra, 'Tensor analysis of tornadoes: a new analytical and numerical model', Mathematical Models in Engineering (2024). DOI: 10.21595/mme.2024.24041. URL: https://doi.org/10.21595/mme.2024.24041
8. S N Artekha, 'An analytical model of tornado generation', Physics of Fluids (2024). DOI: 10.1063/5.0213431; arXiv:2410.16087. URL: https://arxiv.org/abs/2410.16087
9. S N Artekha, 'Some generalizations of the tornado model', Atmospheric Research (2026). DOI: 10.1016/j.atmosres.2025.108541. URL: https://doi.org/10.1016/j.atmosres.2025.108541
10. A V Baev, 'Steady-State Solutions to the Navier–Stokes Equation', Moscow University Computational Mathematics and Cybernetics (2024). DOI: 10.3103/S0278641924700109. URL: https://doi.org/10.3103/S0278641924700109
11. Zachary Bradshaw and Dakota Palmer, 'Singular stationary Navier-Stokes flows: examples and stability', arXiv:2606.22291 (2026). URL: https://arxiv.org/abs/2606.22291
12. M D Arnold and A V Khokhlov, 'Implementation for blow up of tornado-type solutions for complex version of 3D Navier-Stokes system', arXiv:0806.4286 (2008). URL: https://arxiv.org/abs/0806.4286
13. S K Pandey and Shruti, 'A viscous model of wind fields in single-cell tornado-like vortices', International Journal of Numerical Methods for Heat & Fluid Flow (2025). DOI: 10.1108/HFF-09-2024-0668. URL: https://doi.org/10.1108/HFF-09-2024-0668