Figure 1.
A cross section of Saturn depicting the composition of its interior
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July
2022, 21(7): 2327-2336.
doi: 10.3934/cpaa.2022035
On the spherical geopotential approximation for Saturn
Department of Mathematics, Brown University, Box 1917, Providence, RI 02912, USA |
In this paper, we show by means of a diffeomorphism that when approximating the planet Saturn by a sphere, the errors associated with the spherical geopotential approximation are so significant that this approach is rendered unsuitable for any rigorous mathematical analysis.
Keywords:
Atmospheric flow,
spherical geopotential approximation.
Mathematics Subject Classification:
Primary: 86-10.
Citation:
Susanna V. Haziot. On the spherical geopotential approximation for Saturn. Communications on Pure and Applied Analysis,
2022, 21
(7)
: 2327-2336.
doi: 10.3934/cpaa.2022035
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References:
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A. Affholder, F. Guyot, B. Sauterey, R. Ferrière and S. Mazevet,
Bayesian analysis of Enceladus's plume data to assess methanogenesis, Nat. Astron., 8 (2021), 805-814.
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P. Benard,
An oblate-spheroid geopotential approximation for global meteorology, Quart. J. Roy. Meterol. Soc., 140 (2014), 170-184.
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H. Busemann and W. Feller,
Krümmungseigenschaften konvexer Flächen, Acta Math., 66 (1936), 1-47.
doi: 10.1007/BF02546515. |
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A. Constantin and R. S. Johnson,
On the modelling of large-scale atmospheric flow, J. Differ. Equ., 285 (2021), 751-798.
doi: 10.1016/j.jde.2021.03.019. |
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A. Constantin and R. S. Johnson, On the propagation of waves in the atmosphere, Proc. A, 477 (2021), 25 pp. |
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A. Constantin, D. G. Crowdy, V. S. Krishnamurthy and M. H. Wheeler,
Stuart-type polar vortices on a rotating sphere, Discrete Cont. Dyn. Syst., 41 (2021), 201-215.
doi: 10.3934/dcds.2020263. |
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G. Faure and T. M. Mensing, Introduction to Planetary Science: The Geological Perspective, Springer, Dordrecht, The Netherlands, 2007. |
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R. A. Freedman and W. J. Kaufmann III, Universe: The Solar System, Freeman, New York, NY, 2002. |
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B. Galperin and P.L. Read, Zonal Jets: Phenomenology, Genesis and Physics, Cambridge University Press, Cambridge, 2019.
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E. Gregersen, The Outer Solar System, Britannica Educational Publishing, New York, NY, 2009.
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P. Gruber, Convex and Discrete Geometry, Springer-Verlag, Berlin-Heidelberg, 2007.
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W. K. Hartmann, Moons and Planets, Brooks/Cole, Belmont, CA, 2005. |
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A. P. Ingersoll, et al., Atmospheres of the giant planets in The New Solar System, Sky Publishing, Cambridge, MA, 1999. |
| [14] |
J. E. Klepeis,
Hydrogen-helium mixtures at megabars pressure: Implications for Jupiter and Saturn, Science, 254 (1991), 986-989.
|
| [15] |
J. Lunine,
Saturn's Titan: A Strict Test for Life's Cosmic Ubiquity, Proceed. Amer. Phil. Soc., 153 (2009), 403-418.
|
| [16] |
NASA/Jet Propulsion Laboratory, Life on Titan? New clues to what's consuming hydrogen, acetylene on Saturn's moon, Sci. Daily, 2010. |
| [17] |
A. Staniforth,
Spherical and spheroidal geopotential approximations, Quart. J. Roy. Meterol. Soc., 140 (2014), 2685-2692.
|
show all references
References:
| [1] |
A. Affholder, F. Guyot, B. Sauterey, R. Ferrière and S. Mazevet,
Bayesian analysis of Enceladus's plume data to assess methanogenesis, Nat. Astron., 8 (2021), 805-814.
|
| [2] |
P. Benard,
An oblate-spheroid geopotential approximation for global meteorology, Quart. J. Roy. Meterol. Soc., 140 (2014), 170-184.
|
| [3] |
H. Busemann and W. Feller,
Krümmungseigenschaften konvexer Flächen, Acta Math., 66 (1936), 1-47.
doi: 10.1007/BF02546515. |
| [4] |
A. Constantin and R. S. Johnson,
On the modelling of large-scale atmospheric flow, J. Differ. Equ., 285 (2021), 751-798.
doi: 10.1016/j.jde.2021.03.019. |
| [5] |
A. Constantin and R. S. Johnson, On the propagation of waves in the atmosphere, Proc. A, 477 (2021), 25 pp. |
| [6] |
A. Constantin, D. G. Crowdy, V. S. Krishnamurthy and M. H. Wheeler,
Stuart-type polar vortices on a rotating sphere, Discrete Cont. Dyn. Syst., 41 (2021), 201-215.
doi: 10.3934/dcds.2020263. |
| [7] |
G. Faure and T. M. Mensing, Introduction to Planetary Science: The Geological Perspective, Springer, Dordrecht, The Netherlands, 2007. |
| [8] |
R. A. Freedman and W. J. Kaufmann III, Universe: The Solar System, Freeman, New York, NY, 2002. |
| [9] |
B. Galperin and P.L. Read, Zonal Jets: Phenomenology, Genesis and Physics, Cambridge University Press, Cambridge, 2019.
|
| [10] |
E. Gregersen, The Outer Solar System, Britannica Educational Publishing, New York, NY, 2009.
|
| [11] |
P. Gruber, Convex and Discrete Geometry, Springer-Verlag, Berlin-Heidelberg, 2007.
|
| [12] |
W. K. Hartmann, Moons and Planets, Brooks/Cole, Belmont, CA, 2005. |
| [13] |
A. P. Ingersoll, et al., Atmospheres of the giant planets in The New Solar System, Sky Publishing, Cambridge, MA, 1999. |
| [14] |
J. E. Klepeis,
Hydrogen-helium mixtures at megabars pressure: Implications for Jupiter and Saturn, Science, 254 (1991), 986-989.
|
| [15] |
J. Lunine,
Saturn's Titan: A Strict Test for Life's Cosmic Ubiquity, Proceed. Amer. Phil. Soc., 153 (2009), 403-418.
|
| [16] |
NASA/Jet Propulsion Laboratory, Life on Titan? New clues to what's consuming hydrogen, acetylene on Saturn's moon, Sci. Daily, 2010. |
| [17] |
A. Staniforth,
Spherical and spheroidal geopotential approximations, Quart. J. Roy. Meterol. Soc., 140 (2014), 2685-2692.
|
Figure 3.
Saturn is approximated by an ellispoid $ \mathcal{E} $ , with polar radius $ d_P' $ and equatorial radius $ d_E' $ . The polar radius is equal to only about $ 90.2\% $ of the equatorial one. In comparison, for Earth the polar radius is $ 99.67\% $ of the equatorial one
Figure 4.
The spherical and geopotential coordinate systems for fixed longitude $ \varphi $ . Here $ \theta $ denotes the geocentric latitude of the point $ Q $ . The normal vector of the ellipsoid $ \mathcal{E} $ at $ Q $ intersects the equatorial plane at the focus point $ A $ , at an angle $ \beta $ , called the geodetic latitude angle of $ P $
Figure 5.
Saturn viewed as an ellipsoid with a point $ P $ on its surface expressed in terms of the geocentric latitude $ \alpha $ . The angle $ \alpha $ is related to the geodetic latitude $ \beta $ by (2.1)
Figure 6.
The spherical geopotential approximation for fixed longitude $ \varphi $ . The point $ P $ on the ellipsoid is mapped into $ \hat{P} $ on the sphere of radius $ R' $ . This is obtained by setting the geocentric latitude angle of $ \hat{P} $ to be equal to the geodetic latitude of $ P $
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