American Institute of Mathematical Sciences

Advanced
January  2021, 41(1): 201-215. doi: 10.3934/dcds.2020263

Stuart-type polar vortices on a rotating sphere

Adrian Constantin 1,, , Darren G. Crowdy 2, , Vikas S. Krishnamurthy 1, and  Miles H. Wheeler 3,

1. 

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
2. 

Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom
3. 

Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom

* Corresponding author: Adrian Constantin

Received  December 2019 Published  January 2021 Early access  July 2020

Fund Project: This research was supported by the WWTF research grant MA16-009

Stuart vortices are among the few known smooth explicit solutions of the planar Euler equations with a nonlinear vorticity, and they can be adapted to model inviscid flow on the surface of a fixed sphere. By means of a perturbative approach we show that the method used to investigate Stuart vortices on a fixed sphere provides insight into the dynamics of the large-scale zonal flows on a rotating sphere that model the background flow of polar vortices. Our approach takes advantage of the fact that while a sphere is spinning around its polar axis, every point on the sphere has the same angular velocity but its tangential velocity is proportional to the distance from the polar axis of rotation, so that points move fastest at the Equator and slower as we go towards the poles, both of which remain fixed.

Keywords: Euler equation, rotating frame, stereographic projection, nonlinear elliptic equation, sub- and super-solutions.
Mathematics Subject Classification: Primary: 86A10; Secondary: 35Q3, 30C20, 35J15.
Citation: Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy, Miles H. Wheeler. Stuart-type polar vortices on a rotating sphere. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 201-215. doi: 10.3934/dcds.2020263
References:
[1]

A. C. B. Aguiar, P. L. Read, R. D. Wordsworth, T. Salter, R. H. Brown and Y. H. Yamazaki, A laboratory model of of Saturn's North Polar Hexagon, Icarus, 206 (2010), 755-763.  doi: 10.1016/j.icarus.2009.10.022.

[2] K. H. Baines, F. M. Flasar, N. Krupp and T. Stallard, Saturn in the 21st Century, Cambridge University Press, Cambridge, 2018.  doi: 10.1017/9781316227220.
[3]

K. H. Baines, L. A. Sromovsky, P. M. Fry and et al., The eye of Saturn's North Polar Vortex: Unexpected cloud structures observed at high spatial resolution by Cassini/VIMS, Geophys. Res. Lett., 45 (2018), 5867-5875.  doi: 10.1029/2018GL078168.

[4]

R. Brown, J. P. Lebreton and J. Waite, Titan from Cassini-Huygens., Springer Netherlands, 2010,535 pp. doi: 10.1007/978-1-4020-9215-2.

[5]

A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates., Proc. A., 473 (2017), 20170063, 17 pp. doi: 10.1098/rspa.2017.0063.

[6]

A. Constantin and V. S. Krishnamurthy, Stuart-type vortices on a rotating sphere, J. Fluid Mech., 865 (2019), 1072-1084.  doi: 10.1017/jfm.2019.109.

[7]

D. G. Crowdy, General solutions to the 2D Liouville equation, Internat. J. Engrg. Sci., 35 (1997), 141-149.  doi: 10.1016/S0020-7225(96)00080-8.

[8]

D. G. Crowdy, Stuart vortices on a sphere, J. Fluid Mech., 398 (2004), 381-402.  doi: 10.1017/S0022112003007043.

[9]

M. K. Dougherty, L. W. Esposito and S. M. Krimigis, Saturn from Cassini-Huygens, Springer Netherlnds, 2009,805 pp. doi: 10.1007/978-1-4020-9217-6.

[10] P. L. Duren, Theory of $H^p$ Spaces., Academic Press, New York-London, 1970. 
[11]

N. P. Fofonoff, Steady flow in a frictionless homogeneous ocean, J. Marine Res., 13 (1954), 254-262. 

[12] A. E. Gill, Atmosphere-Ocean Dynamics, Academic Press, 1982. 
[13]

B. Haurwitz, The motion of atmospheric disturbances on a spherical Earth, J. Marine Res., 3 (1940), 254-267. 

[14]

P. Henrici, Applied and Computational Complex Analysis, Vol. 3, John Wiley & Sons, Inc., New York, 1986.

[15]

M. S. Longuet-Higgins, Planetary waves on a rotating sphere Ⅱ, Proc. Roy. Soc. London Ser. A, 284 (1964), 40-68.  doi: 10.1098/rspa.1964.0116.

[16]

D. M. Mitchell, L. Montabone, S. Thomson and P. L. Read, Polar vortices on Earth and Mars: A comparative study of the climatology and variability from reanalyses, Quart. J. Roy. Meteorol. Soc., 141 (2015), 550-562.  doi: 10.1002/qj.2376.

[17]

K. Miyazaki and T. Iwasaki, On the analysis of mean downward velocities around the Antarctic Polar Vortex, J. Atmospheric Sci., 65 (2008), 3989-4003.  doi: 10.1175/2008JAS2749.1.

[18]

R. Narasimhan, Analysis on Real and Complex Manifolds, North-Holland Publishing Co., Amsterdam, 1985.

[19]

M. E. O'Neill, K. A. Emanuel and G. R. Flierl, Weak jets and strong cyclones: Shallow-water modeling of giant planet polar caps, J. Atmospheric Sci., 73 (2016), 1841-1855.  doi: 10.1175/JAS-D-15-0314.1.

[20]

George Polya and Gordon Latta, Complex Variables, John Wiley & Sons, Inc., New York-London-Sydney, 1974.

[21]

A. C. Ponce, Elliptic PDEs, Measures and Capacities. From the Poisson Equations to Nonlinear Thomas-Fermi Problems, EMS Tracts in Mathematics, Vol. 23, European Math. Soc., Zürich, 2016. doi: 10.4171/140.

[22]

R. K. Scott and D. G. Dritschel, Downward wave propagation on the polar vortex, J. Atmospheric Sci., 62 (2005), 3382-3395.  doi: 10.1175/JAS3526.1.

[23]

J. T. Stuart, On finite amplitude oscillations in laminar mixing layers, J. Fluid Mech., 29 (1967), 417-440.  doi: 10.1017/S0022112067000941.

[24] G. K. Vallis, Atmosphere and Ocean Fluid Dynamics, Cambridge University Press, Cambridge, 2006. 
[25]

W. T. M. Verkley, The construction of barotropic modons on a sphere, J. Atmospheric Sci., 41 (1984), 2492-2505.  doi: 10.1175/1520-0469(1984)041<2492:TCOBMO>2.0.CO;2.

[26]

T. von Larcher and P. D. Williams, Modeling Atmospheric and Oceanic Flows, Amer. Geophys. Union, 2015.

[27]

P. Wu and W. T. M Verkley, Nonlinear structures with multivalued ($q$, $\psi$) relationships–exact solutions of the barotropic vorticity equation on a sphere, Geophys. Astrophys. Fluid Dyn., 69 (1993), 77-94. 

show all references

References:
[1]

A. C. B. Aguiar, P. L. Read, R. D. Wordsworth, T. Salter, R. H. Brown and Y. H. Yamazaki, A laboratory model of of Saturn's North Polar Hexagon, Icarus, 206 (2010), 755-763.  doi: 10.1016/j.icarus.2009.10.022.

[2] K. H. Baines, F. M. Flasar, N. Krupp and T. Stallard, Saturn in the 21st Century, Cambridge University Press, Cambridge, 2018.  doi: 10.1017/9781316227220.
[3]

K. H. Baines, L. A. Sromovsky, P. M. Fry and et al., The eye of Saturn's North Polar Vortex: Unexpected cloud structures observed at high spatial resolution by Cassini/VIMS, Geophys. Res. Lett., 45 (2018), 5867-5875.  doi: 10.1029/2018GL078168.

[4]

R. Brown, J. P. Lebreton and J. Waite, Titan from Cassini-Huygens., Springer Netherlands, 2010,535 pp. doi: 10.1007/978-1-4020-9215-2.

[5]

A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates., Proc. A., 473 (2017), 20170063, 17 pp. doi: 10.1098/rspa.2017.0063.

[6]

A. Constantin and V. S. Krishnamurthy, Stuart-type vortices on a rotating sphere, J. Fluid Mech., 865 (2019), 1072-1084.  doi: 10.1017/jfm.2019.109.

[7]

D. G. Crowdy, General solutions to the 2D Liouville equation, Internat. J. Engrg. Sci., 35 (1997), 141-149.  doi: 10.1016/S0020-7225(96)00080-8.

[8]

D. G. Crowdy, Stuart vortices on a sphere, J. Fluid Mech., 398 (2004), 381-402.  doi: 10.1017/S0022112003007043.

[9]

M. K. Dougherty, L. W. Esposito and S. M. Krimigis, Saturn from Cassini-Huygens, Springer Netherlnds, 2009,805 pp. doi: 10.1007/978-1-4020-9217-6.

[10] P. L. Duren, Theory of $H^p$ Spaces., Academic Press, New York-London, 1970. 
[11]

N. P. Fofonoff, Steady flow in a frictionless homogeneous ocean, J. Marine Res., 13 (1954), 254-262. 

[12] A. E. Gill, Atmosphere-Ocean Dynamics, Academic Press, 1982. 
[13]

B. Haurwitz, The motion of atmospheric disturbances on a spherical Earth, J. Marine Res., 3 (1940), 254-267. 

[14]

P. Henrici, Applied and Computational Complex Analysis, Vol. 3, John Wiley & Sons, Inc., New York, 1986.

[15]

M. S. Longuet-Higgins, Planetary waves on a rotating sphere Ⅱ, Proc. Roy. Soc. London Ser. A, 284 (1964), 40-68.  doi: 10.1098/rspa.1964.0116.

[16]

D. M. Mitchell, L. Montabone, S. Thomson and P. L. Read, Polar vortices on Earth and Mars: A comparative study of the climatology and variability from reanalyses, Quart. J. Roy. Meteorol. Soc., 141 (2015), 550-562.  doi: 10.1002/qj.2376.

[17]

K. Miyazaki and T. Iwasaki, On the analysis of mean downward velocities around the Antarctic Polar Vortex, J. Atmospheric Sci., 65 (2008), 3989-4003.  doi: 10.1175/2008JAS2749.1.

[18]

R. Narasimhan, Analysis on Real and Complex Manifolds, North-Holland Publishing Co., Amsterdam, 1985.

[19]

M. E. O'Neill, K. A. Emanuel and G. R. Flierl, Weak jets and strong cyclones: Shallow-water modeling of giant planet polar caps, J. Atmospheric Sci., 73 (2016), 1841-1855.  doi: 10.1175/JAS-D-15-0314.1.

[20]

George Polya and Gordon Latta, Complex Variables, John Wiley & Sons, Inc., New York-London-Sydney, 1974.

[21]

A. C. Ponce, Elliptic PDEs, Measures and Capacities. From the Poisson Equations to Nonlinear Thomas-Fermi Problems, EMS Tracts in Mathematics, Vol. 23, European Math. Soc., Zürich, 2016. doi: 10.4171/140.

[22]

R. K. Scott and D. G. Dritschel, Downward wave propagation on the polar vortex, J. Atmospheric Sci., 62 (2005), 3382-3395.  doi: 10.1175/JAS3526.1.

[23]

J. T. Stuart, On finite amplitude oscillations in laminar mixing layers, J. Fluid Mech., 29 (1967), 417-440.  doi: 10.1017/S0022112067000941.

[24] G. K. Vallis, Atmosphere and Ocean Fluid Dynamics, Cambridge University Press, Cambridge, 2006. 
[25]

W. T. M. Verkley, The construction of barotropic modons on a sphere, J. Atmospheric Sci., 41 (1984), 2492-2505.  doi: 10.1175/1520-0469(1984)041<2492:TCOBMO>2.0.CO;2.

[26]

T. von Larcher and P. D. Williams, Modeling Atmospheric and Oceanic Flows, Amer. Geophys. Union, 2015.

[27]

P. Wu and W. T. M Verkley, Nonlinear structures with multivalued ($q$, $\psi$) relationships–exact solutions of the barotropic vorticity equation on a sphere, Geophys. Astrophys. Fluid Dyn., 69 (1993), 77-94. 

Figure 1.  Image showing the hot spot centered on Saturn's South Pole (the core of the southern polar vortex, at the bottom of the image), taken from the observatory in Hawaii with infrared radiation sensitive to temperatures in Saturn's upper troposphere [Image credit: NASA/JPL-CalTech/Space Science Institute]. A similar hot spot is found at Saturn's North Pole
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The spherical coordinate system describing flow on a rotating planet. The coordinate system is fixed with respect to the planet rotating with an angular speed $ \Omega' $ about the $ z' $ axis of the Cartesian coordinate system $ (x',y',z') $. The spherical coordinates are $ (r',\theta,\phi) $ where $ r' = |\bf{r'}| $ is the distance from the origin at the planet's center, $ \theta $ is the polar angle (co-latitude) and $ \phi $ is the angle of longitude. The North Pole of the planet is at $ \theta = 0 $ and the South Pole is at $ \theta = \pi $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The stereographic projection maps the point $ (x,y,z) $ on the unit sphere with the North Pole $ N $ excised to the intersection point $ (X,Y) $ of the equatorial plane with the ray from $ N $ to $ (x,y,z) $. The point $ N $ itself is mapped to the point at infinity on the equatorial plane
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Stereographic projection of a spherical cap onto the equatorial plane. The cap near the South pole ($ S $) encloses the vortex region and has boundary co-latitude $ \theta_s $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  The eye of the stationary vortex centered on Saturn's South Pole and extending to 88.5$ ^\circ $S, captured in 2008 by NASA's Cassini spacecraft using thermal radiation, is about 2000 km across and features peak prograde winds of 170 m$ \, $s$ ^{-1} $ [Image credit: NASA/JPL-CalTech/Space Science Institute]
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The projected vortex region $ \mathcal{V} $ in the projected plane, with maximum radius $ \delta $ and minimum radius $ \mu $. The boundary of $ \mathcal{V} $ is denoted by $ \mathcal{V}_B $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  The eye of the stationary vortex at Saturn's North Pole (with a surrounding hexagonal jet stream), captured in 2017 by NASA's Cassini spacecraft, is more than 2000 km wide and features prograde wind speeds of 200 m$ \, $s$ ^{-1} $ on its outer edge at 88$ ^\circ $N (decreasing within the eye to zero at the pole) [Image credit: NASA/JPL-CalTech/Space Science Institute]
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Depiction of the streamline pattern (40) inside the polar vortex at the South Pole (the black dot). The velocity and vorticity fields are smooth inside the polar vortex region
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Depiction of the streamline pattern inside the polar vortex at the South Pole (the black dot) for the choice (41) with $ A = 1 $ and $ B^6 = .001 $. The velocity and vorticity fields are smooth inside the polar vortex region
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure and Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761
[2]

Yanqin Fang, De Tang. Method of sub-super solutions for fractional elliptic equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3153-3165. doi: 10.3934/dcdsb.2017212
[3]

Shu-Yu Hsu. Super fast vanishing solutions of the fast diffusion equation. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5383-5414. doi: 10.3934/dcds.2020232
[4]

A. M. Micheletti, Monica Musso, A. Pistoia. Super-position of spikes for a slightly super-critical elliptic equation in $R^N$. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 747-760. doi: 10.3934/dcds.2005.12.747
[5]

Xudong Shang, Jihui Zhang. Multi-peak positive solutions for a fractional nonlinear elliptic equation. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3183-3201. doi: 10.3934/dcds.2015.35.3183
[6]

A. El Hamidi. Multiple solutions with changing sign energy to a nonlinear elliptic equation. Communications on Pure and Applied Analysis, 2004, 3 (2) : 253-265. doi: 10.3934/cpaa.2004.3.253
[7]

Amine Laghrib, Abdelkrim Chakib, Aissam Hadri, Abdelilah Hakim. A nonlinear fourth-order PDE for multi-frame image super-resolution enhancement. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 415-442. doi: 10.3934/dcdsb.2019188
[8]

Chunqing Lu. Asymptotic solutions of a nonlinear equation. Conference Publications, 2003, 2003 (Special) : 590-595. doi: 10.3934/proc.2003.2003.590
[9]

Lipeng Duan, Shuying Tian. Concentrated solutions for a critical elliptic equation. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 4061-4094. doi: 10.3934/dcds.2022046
[10]

Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056
[11]

Miao Chen, Youyan Wan, Chang-Lin Xiang. Local uniqueness problem for a nonlinear elliptic equation. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1037-1055. doi: 10.3934/cpaa.2020048
[12]

Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675
[13]

David Mumford, Peter W. Michor. On Euler's equation and 'EPDiff'. Journal of Geometric Mechanics, 2013, 5 (3) : 319-344. doi: 10.3934/jgm.2013.5.319
[14]

Berat Karaagac. New exact solutions for some fractional order differential equations via improved sub-equation method. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 447-454. doi: 10.3934/dcdss.2019029
[15]

Vikas S. Krishnamurthy. The vorticity equation on a rotating sphere and the shallow fluid approximation. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6261-6276. doi: 10.3934/dcds.2019273
[16]

Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036
[17]

Stephen C. Preston, Alejandro Sarria. One-parameter solutions of the Euler-Arnold equation on the contactomorphism group. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2123-2130. doi: 10.3934/dcds.2015.35.2123
[18]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[19]

Henri Berestycki, Juncheng Wei. On least energy solutions to a semilinear elliptic equation in a strip. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1083-1099. doi: 10.3934/dcds.2010.28.1083
[20]

Goong Chen, Zhonghai Ding, Shujie Li. On positive solutions of the elliptic sine-Gordon equation. Communications on Pure and Applied Analysis, 2005, 4 (2) : 283-294. doi: 10.3934/cpaa.2005.4.283

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (391)
  • HTML views (266)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy