# The Stationary Concentrated Vortex Model

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## Abstract

**:**

## 1. Introduction

## 2. Vortex Models

#### 2.1. Rankine Vortex

#### 2.2. Burgers Vortex

#### 2.3. The Sullivan Vortex

## 3. Vortex Model for an Incompressible and Inviscid Fluid

- (a)
- in the vortex centre, for $r=0:$${v}_{r}={v}_{\varphi}=0$, ${v}_{z}$ and p are finite values;
- (b)
- at the bottom boundary, for $z=0:$${v}_{z}={v}_{\varphi}=0$, ${v}_{r}$ and p are finite values; and
- (c)
- at the vortex periphery, when $r/{r}_{0}\gg 1$ and $z/L\gg 1$, where ${r}_{0}$ and L are characteristic vortex scales in the radial and vertical directions, respectively,${v}_{r}={v}_{\varphi}={v}_{z}=0$ and $p=0$.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Maxworthy, T. A Vorticity Source for Large-Scale Dust Devils and Other Comments on Naturally Occurring Columnar Vortices. J. Atmos. Sci.
**1973**, 30, 1717–1722. [Google Scholar] [CrossRef] [Green Version] - Mullen, J.B.; Maxworthy, T. A laboratory model of dust devil vortices. Dyn. Atmos. Ocean.
**1977**, 1, 181–214. [Google Scholar] [CrossRef] - Trapp, R.J.; Fiedler, B.H. Tornado-like Vortexgenesis in a Simplified Numerical Model. J. Atmos. Sci.
**1995**, 52, 3757–3778. [Google Scholar] [CrossRef] [Green Version] - Kanak, K.M.; Lilly, D.K.; Snow, J.T. The formation of vertical Vortices in the convective boundary layer. Q. J. R. Meteorol. Soc.
**2000**, 126, 2789–2810. [Google Scholar] [CrossRef] - Kanak, K.M. Numerical simulation of dust devil-scale vortices. Q. J. R. Meteorol. Soc.
**2005**, 131, 1271–1292. [Google Scholar] [CrossRef] - Balme, M.; Greeley, R. Dust devils on Earth and Mars. Rev. Geophys.
**2006**, 44, RG3003. [Google Scholar] [CrossRef] - Gu, Z.; Qiu, J.; Zhao, Y.; Li, Y. Simulation of terrestrial dust devil patterns. Adv. Atmos. Sci.
**2008**, 25, 31–42. [Google Scholar] [CrossRef] - Zhao, Y.Z.; Gu, Z.L.; Yu, Y.Z.; Ge, Y.; Li, Y.; Feng, X. Mechanism and large eddy simulation of dust devils. Atmos. Ocean
**2010**, 61–84. [Google Scholar] [CrossRef] [Green Version] - Raasch, S.; Franke, T. Structure and formation of dust devil-like vortices in the atmospheric boundary layer: A high-resolution numerical study. J. Geophys. Res. Atmos.
**2011**, 116, D16120. [Google Scholar] [CrossRef] - Davies-Jones, R. A review of supercell and tornado dynamics. Atmos. Res.
**2015**, 158, 274–291. [Google Scholar] [CrossRef] - Horton, W.; Miura, H.; Onishchenko, O.; Couedel, L.; Arnas, C.; Escarguel, A.; Benkadda, S.; Fedun, V. Dust devil dynamics. J. Geophys. Res. Atmos.
**2016**, 121, 7197–7214. [Google Scholar] [CrossRef] [Green Version] - Neves, T.; Fisch, G.; Raasch, S. Local Convection and Turbulence in the Amazonia Using Large Eddy Simulation Model. Atmosphere
**2018**, 9, 399. [Google Scholar] [CrossRef] [Green Version] - Farrell, W.M.; Delory, G.T.; Cummer, S.A.; Marshall, J.R. A simple electrodynamic model of a dust devil. Geophys. Res. Lett.
**2003**, 30, 2050. [Google Scholar] [CrossRef] [Green Version] - Farrell, W.M.; Smith, P.H.; Delory, G.T.; Hillard, G.B.; Marshall, J.R.; Catling, D.; Hecht, M.; Tratt, D.M.; Renno, N.; Desch, M.D.; et al. Electric and magnetic signatures of dust devils from the 2000–2001 MATADOR desert tests. J. Geophys. Res. Planets
**2004**, 109, E03004. [Google Scholar] [CrossRef] - Farrell, W.M.; Renno, N.; Delory, G.T.; Cummer, S.A.; Marshall, J.R. Integration of electrostatic and fluid dynamics within a dust devil. J. Geophys. Res. Planets
**2006**, 111, E01006. [Google Scholar] [CrossRef] [Green Version] - Melnik, O.; Parrot, M. Electrostatic discharge in Martian dust storms. J. Geophys. Res.
**1998**, 103, 29107–29118. [Google Scholar] [CrossRef] - Zhou, Y.H.; Guo, X.; Zheng, X.J. Experimental measurement of wind-sand flux and sand transport for naturally mixed sands. Phys. Rev. E
**2002**, 66, 021305. [Google Scholar] [CrossRef] [PubMed] - Zheng, X.J.; Huang, N.; Zhou, Y.H. Laboratory measurement of electrification of wind-blown sands and simulation of its effect on sand saltation movement. J. Geophys. Res. Atmos.
**2003**, 108, 4322. [Google Scholar] [CrossRef] - Xie, L.; Ling, Y.; Zheng, X. Laboratory measurement of saltating sand particles’ angular velocities and simulation of its effect on saltation trajectory. J. Geophys. Res. Atmos.
**2007**, 112, D12116. [Google Scholar] [CrossRef] - Harrison, R.G.; Barth, E.; Esposito, F.; Merrison, J.; Montmessin, F.; Aplin, K.L.; Borlina, C.; Berthelier, J.J.; Déprez, G.; Farrell, W.M.; et al. Applications of Electrified Dust and Dust Devil Electrodynamics to Martian Atmospheric Electricity. Space Sci. Rev.
**2016**, 203, 299–345. [Google Scholar] [CrossRef] [Green Version] - Izvekova, Y.N.; Popel, S.I. Charged Dust Motion in Dust Devils on Earth and Mars. Contrib. Plasma Phys.
**2016**, 56, 263–269. [Google Scholar] [CrossRef] - Izvekova, Y.N.; Popel, S.I. Plasma Effects in Dust Devils near the Martian Surface. Plasma Phys. Rep.
**2017**, 43, 1172–1178. [Google Scholar] [CrossRef] - Izvekova, Y.N.; Popel, S.I.; Izvekov, O.Y. On the Possibility of Excitation of Oscillations in a Schumann Resonator on Mars. Plasma Phys. Rep.
**2020**, 46, 65–70. [Google Scholar] [CrossRef] - Izvekova, Y.N.; Reznichenko, Y.S.; Popel, S.I. On the Possibility of Dust Acoustic Perturbations in Martian Ionosphere. Plasma Phys. Rep.
**2020**, 46, 1205–1209. [Google Scholar] [CrossRef] - Stenflo, L. Nonlinear equations for acoustic gravity waves. Phys. Lett. A
**1996**, 222, 378–380. [Google Scholar] [CrossRef] - Shukla, P.K.; Stenflo, L. Acoustic gravity tornadoes in the atmosphere. Phys. Scr.
**2012**, 86, 065403. [Google Scholar] [CrossRef] - Sinclair, P.C. General Characteristics of Dust Devils. J. Appl. Meteorol.
**1969**, 8, 32–45. [Google Scholar] [CrossRef] [Green Version] - Onishchenko, O.; Pokhotelov, O.; Fedun, V. Convective cells of internal gravity waves in the earth’s atmosphere with finite temperature gradient. Ann. Geophys.
**2013**, 31, 459–462. [Google Scholar] [CrossRef] [Green Version] - Onishchenko, O.; Pokhotelov, O.; Horton, W.; Fedun, V. Dust devil vortex generation from convective cells. Ann. Geophys.
**2015**, 33, 1343–1347. [Google Scholar] [CrossRef] [Green Version] - Onishchenko, O.G.; Horton, W.; Pokhotelov, O.A.; Fedun, V. “Explosively growing” vortices of unstably stratified atmosphere. J. Geophys. Res. Atmos.
**2016**, 121, 11. [Google Scholar] [CrossRef] - Rafkin, S.; Jemmett-Smith, B.; Fenton, L.; Lorenz, R.; Takemi, T.; Ito, J.; Tyler, D. Dust Devil Formation. Space Sci. Rev.
**2016**, 203, 183–207. [Google Scholar] [CrossRef] - Rennó, N.O.; Burkett, M.L.; Larkin, M.P. A Simple Thermodynamical Theory for Dust Devils. J. Atmos. Sci.
**1998**, 55, 3244–3252. [Google Scholar] [CrossRef] - Smith, R.K.; Leslie, L.M. Thermally driven vortices: A numerical study with application to dust-devil dynamics. Q. J. R. Meteorol. Soc.
**1976**, 102, 791–804. [Google Scholar] [CrossRef] - Hicks, W.M. Researches in Vortex Motion. Part III: On Spiral or Gyrostatic Vortex Aggregates. Philos. Trans. R. Soc. Lond. Ser. A
**1899**, 192, 33–99. [Google Scholar] [CrossRef] [Green Version] - Moffatt, H.K. The degree of knottedness of tangled vortex lines. J. Fluid Mech.
**1969**, 35, 117–129. [Google Scholar] [CrossRef] - Wu, J.Z.; Ma, H.Y.; Zhou, M.D. Vorticity and Vortex Dynamics. 2006. Available online: https://www.springer.com/gp/book/9783540290278 (accessed on 5 January 2021).
- Onishchenko, O.G.; Pokhotelov, O.A.; Astafieva, N.M. A Novel Model of Quasi-Stationary Vortices in the Earth’s Atmosphere. Izv. Atmos. Ocean. Phys.
**2018**, 54, 906–910. [Google Scholar] [CrossRef] - Onishchenko, O.; Pokhotelov, O.; Horton, W.; Smolyakov, A.; Kaladze, T.; Fedun, V. Rolls of the internal gravity waves in the Earth’s atmosphere. Ann. Geophys.
**2014**, 32, 181–186. [Google Scholar] [CrossRef] [Green Version] - Onishchenko, O.G.; Fedun, V.; Horton, W.; Pokhotelov, O.; Verth, G. Dust devils: Structural features, dynamics and climate impact. Climate
**2019**, 7, 12. [Google Scholar] [CrossRef] [Green Version] - Onishchenko, O.G.; Pokhotelov, O.; Astafieva, N.M.; Horton, W.; Fedun, V. Structure and dynamics of concentrated mesoscale vortices in the atmospheres of planets. Phys. Usp.
**2020**. [Google Scholar] [CrossRef] - Rankine, W.J.M. A Manual of Applied Mechanics; Charles Griffin and Company Limited: London, UK, 1901. [Google Scholar]
- Battan, L.J. Energy of a Dust Devil. J. Atmos. Sci.
**1958**, 15, 235–236. [Google Scholar] [CrossRef] [Green Version] - Sinclair, P.C. on the rotation of dust devils. Bull. Am. Meteorol. Soc.
**1965**, 46, 388–391. [Google Scholar] [CrossRef] - Sinclair, P.C. The Lower Structure of Dust Devils. J. Atmos. Sci.
**1973**, 30, 1599–1619. [Google Scholar] [CrossRef] - Williams, N.R. Development of Dust Whirls and Similar Small-Scale Vortices. Bull. Am. Meteorol. Soc.
**1948**, 29, 106–117. [Google Scholar] [CrossRef] [Green Version] - Bluestein, H.B.; Weiss, C.C.; Pazmany, A.L. Doppler Radar Observations of Dust Devils in Texas. Mon. Weather Rev.
**2004**, 132, 209. [Google Scholar] [CrossRef] - Toigo, A.D.; Richardson, M.I.; Ewald, S.P.; Gierasch, P.J. Numerical simulation of Martian dust devils. J. Geophys. Res. Planets
**2003**, 108, 5047. [Google Scholar] [CrossRef] [Green Version] - Kurgansky, M.V. A simple model of dry convective helical vortices (with applications to the atmospheric dust devil). Dyn. Atmos. Ocean.
**2005**, 40, 151–162. [Google Scholar] [CrossRef] - Kurgansky, M.V.; Lorenz, R.D.; Renno, N.O.; Takemi, T.; Gu, Z.; Wei, W. Dust Devil Steady-State Structure from a Fluid Dynamics Perspective. Space Sci. Rev.
**2016**, 203, 209–244. [Google Scholar] [CrossRef] - Burgers, J.M. A mathematical model illustrating the theory of turbulence. Adv. Appl. Mech.
**1948**, 1, 171–199. [Google Scholar] [CrossRef] - Rott, N. On the viscous core of a line vortex. Z. Angew. Math. Und Phys.
**1958**, 9, 543–553. [Google Scholar] [CrossRef] - Sullivan, R.D. A Two-Cell Vortex Solution of the Navier-Stokes Equations. J. Aerosp. Sci.
**1959**, 26, 767–768. [Google Scholar] [CrossRef] - Michaels, I.T.; Rafkin Scot, C.R. Large-eddy simulation of atmospheric convection on Mars. Q. J. R. Meteorol. Soc.
**2004**, 130, 1251–1274. [Google Scholar] [CrossRef] [Green Version] - Morton, B.R. The strength of vortex and swirling core flows. J. Fluid Mech.
**1969**, 38, 315–333. [Google Scholar] [CrossRef] - Velikhov, E.P. Stability of an ideally conducting liquid flowing between cylinders rotating in a magnetic field. JETP
**1959**, 36, 1398–1404. [Google Scholar] - Chandrasekhar, S. Hydrodynamic and Hydromagnetic Stability; Courier Corporation: Chelmsford, MA, USA, 1961. [Google Scholar]
- Clyne, J.; Mininni, P.; Norton, A.; Rast, M. Interactive desktop analysis of high resolution simulations: Application to turbulent plume dynamics and current sheet formation. New J. Phys.
**2007**, 9, 301. [Google Scholar] [CrossRef] [Green Version] - Clyne, J.; Rast, M. Visualization and Data Analysis 2005. In A Prototype Discovery Environment for Analyzing and Visualizing Terascale Turbulent Fluid Flow Simulations; Erbacher, R.F., Roberts, J.C., Gröhn, M.T., Börner, K., Eds.; International Society for Optics and Photonics: Bellingham, WA, USA, 2005; Volume 5669, pp. 284–294. [Google Scholar] [CrossRef]
- Li, S.; Jaroszynski, S.; Pearse, S.; Orf, L.; Clyne, J. VAPOR: A Visualization Package Tailored to Analyze Simulation Data in Earth System Science. Atmosphere
**2019**, 10, 488. [Google Scholar] [CrossRef] [Green Version] - Spiga, A.; Barth, E.; Gu, Z.; Hoffmann, F.; Ito, J.; Jemmett-Smith, B.; Klose, M.; Nishizawa, S.; Raasch, S.; Rafkin, S.; et al. Large-Eddy Simulations of Dust Devils and Convective Vortices. Space Sci. Rev.
**2016**, 203, 245–275. [Google Scholar] [CrossRef] [Green Version] - Izvekova, Y.N.; Popel’, S.I.; Izvekov, O.Y. On the Question of Calculating the Parameters of Vortices in the Near-Surface Atmosphere of Mars. Sol. Syst. Res.
**2020**, 53, 423–430. [Google Scholar] [CrossRef] - Smith, R.K.; Montgomery, M.T.; Persing, J. On steady-state tropical cyclones. Q. J. R. Meteorol. Soc.
**2014**, 140, 2638–2649. [Google Scholar] [CrossRef]

**Figure 1.**The normalised three-dimensional stream lines of the vortex velocity field. On the left panel only four stream lines are shown. The light blue and green colours of the stream lines correspond to the internal structure of the vortex. The red and amber colours correspond to the external part of the vortex. On the right panel more than forty velocity stream lines are shown. The red, green and blue orthogonal vectors colours indicate the x, y and z axis, correspondingly. The visualisation was computed for ${r}_{0}=1$ and $L=10$.

**Figure 2.**The behaviour of the normalised pressure $\left({P}^{*}=\frac{p-{p}_{at}}{\rho {v}_{0}^{2}}\right)$ across the vortex at different heights $Z=z/L$ for the values of ${v}_{\varphi 0}/{v}_{0}$ equal to 9.34, 14.0 and 18.7 (solid, dash and dash-dot plots, correspondingly).

**Figure 3.**The radius–height plot of dimensionless isobars for ${v}_{\varphi 0}/{v}_{0}=9.34$. Axis labels given by $Z=\frac{z}{L}$ and $X=\frac{x}{{r}_{0}}$, where $x\in [-r,r]$.

**Figure 4.**The radius–height contour plots of dimensionless toroidal velocity, ${\overline{v}}_{\varphi}={v}_{\varphi}/{v}_{\varphi 0}$. Axis labels given by $Z=\frac{z}{L}$ and $R=\frac{r}{{r}_{0}}$.

**Figure 5.**The dimensionless stream functions (or angular momentum surfaces) $\overline{\psi}=\psi /{v}_{0}{r}_{0}^{2}$. Axis labels given by $Z=\frac{z}{L}$ and $X=\frac{x}{{r}_{0}}$, where $x\in [-r,r]$.

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**MDPI and ACS Style**

Onishchenko, O.; Fedun, V.; Horton, W.; Pokhotelov, O.; Astafieva, N.; Skirvin, S.J.; Verth, G.
The Stationary Concentrated Vortex Model. *Climate* **2021**, *9*, 39.
https://doi.org/10.3390/cli9030039

**AMA Style**

Onishchenko O, Fedun V, Horton W, Pokhotelov O, Astafieva N, Skirvin SJ, Verth G.
The Stationary Concentrated Vortex Model. *Climate*. 2021; 9(3):39.
https://doi.org/10.3390/cli9030039

**Chicago/Turabian Style**

Onishchenko, Oleg, Viktor Fedun, Wendell Horton, Oleg Pokhotelov, Natalia Astafieva, Samuel J. Skirvin, and Gary Verth.
2021. "The Stationary Concentrated Vortex Model" *Climate* 9, no. 3: 39.
https://doi.org/10.3390/cli9030039