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Function approximation via the subsampled Poincaré inequality
Stuart-type polar vortices on a rotating sphere
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 |
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.
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.
|









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