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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 & 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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

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

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

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

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

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

[26]

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

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

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

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

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

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

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

[26]

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

[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.   Google Scholar

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
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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 $
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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
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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 $
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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]
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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 $
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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]
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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
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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
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