# American Institute of Mathematical Sciences

November  2019, 39(11): 6261-6276. doi: 10.3934/dcds.2019273

## The vorticity equation on a rotating sphere and the shallow fluid approximation

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

* Corresponding author

The paper is for the special theme: Mathematical Aspects of Physical Oceanography, organized by Adrian Constantin

Received  October 2018 Revised  January 2019 Published  August 2019

The material conservation of vorticity in fluid flows confined to a thin layer on the surface of a large rotating sphere, is a central result of geophysical fluid dynamics. In this paper we revisit the conservation of vorticity in the context of global scale flows on a rotating sphere. Starting from the vorticity equation instead of the Euler equation, we examine the kinematical and dynamical assumptions that are necessary to arrive at this result. We argue that, in contrast to the planar case, a two-dimensional velocity field does not lead to a single component vorticity equation on the sphere. The shallow fluid approximation is then used to argue that only one component of the vorticity equation is significant for global scale flows. Spherical coordinates are employed throughout, and no planar approximation is used.

Citation: Vikas S. Krishnamurthy. The vorticity equation on a rotating sphere and the shallow fluid approximation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6261-6276. doi: 10.3934/dcds.2019273
##### References:
 [1] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9780511800955.  Google Scholar [2] V. A. Bogomolov, Dynamics of vorticity at a sphere, Fluid Dynamics, 12 (1977), 863-870.  doi: 10.1007/BF01090320.  Google Scholar [3] A. V. Borisov, I. S. Mamaev and S. M. Ramodanov, Coupled motion of a rigid body and point vortices on a two-dimensional spherical surface, Regular and Chaotic Dynamics, 15 (2010), 440-461.  doi: 10.1134/S1560354710040040.  Google Scholar [4] A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates, Proc. R. Soc. Lond. A, 473 (2017), 20170063, 17 pp. doi: 10.1098/rspa.2017.0063.  Google Scholar [5] A. Constantin and R. S. Johnson, Steady large-scale ocean flows in spherical coordinates, Oceanography, 31 (2018), 42-50.  doi: 10.5670/oceanog.2018.308.  Google Scholar [6] A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal flow as a model for the antarctic circumpolar current, Journal of Physical Oceanography, 46 (2016), 3585-3594.  doi: 10.1175/JPO-D-16-0121.1.  Google Scholar [7] B. Cushman-Roisin and J.-M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, With a foreword by John Marshall. International Geophysics Series, 101. Elsevier/Academic Press, Amsterdam, 2011. doi: 10.1016/c2009-0-00052-x.  Google Scholar [8] T. Gerkema, J. T. F. Zimmerman, L. R. M. Maas and H. van Haren, Geophysical and astrophysical fluid dynamics beyond the traditional approximation, Reviews of Geophysics, 46 (2008), RG2004.  doi: 10.1029/2006RG000220.  Google Scholar [9] A. E. Gill, Atmosphere-Ocean Dynamics, International Geophysics, vol. 30. Academic Press, Elsevier Science, 1982. doi: 10.1016/s0074-6142(08)x6002-4.  Google Scholar [10] R. S. Johnson, Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Phil. Trans. R. Soc. A, 376 (2018), 20170092, 19 pp. doi: 10.1098/rsta.2017.0092.  Google Scholar [11] Y. Kimura and H. Okamoto, Vortex motion on a sphere, Journal of the Physical Society of Japan, 56 (1987), 4203-4206.  doi: 10.1143/JPSJ.56.4203.  Google Scholar [12] M. S. Longuet-Higgins, Planetary waves on a rotating sphere, Proc. R. Soc. Lond. A, 279 (1964), 446-473.  doi: 10.1098/rspa.1964.0116.  Google Scholar [13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511613203.  Google Scholar [14] J. E. Marsden and A. J. Tromba, Vector Calculus, 6$^{th}$ edition, W. H. Freeman & Company, New York, 2012. Google Scholar [15] C. I. Martin, On the vorticity of mesoscale ocean currents, Oceanography, 31 (2018), 28-35.  doi: 10.5670/oceanog.2018.306.  Google Scholar [16] N. R. McDonald, The motion of geophysical vortices, Phil. Trans. R. Soc. A, 357 (1999), 3427-3444.  doi: 10.1098/rsta.1999.0501.  Google Scholar [17] P. Müller, Ertel's potential vorticity theorem in physical oceanography, Reviews of Geophysics, 33 (1995), 67-97.  doi: 10.1029/94RG03215.  Google Scholar [18] W. F. Newns, Functional dependence, The American Mathematical Monthly, 74 (1967), 911-920.  doi: 10.1080/00029890.1967.12000050.  Google Scholar [19] P. K. Newton, The N-Vortex Problem: Analytical Techniques, Applied Mathematical Sciences, 145. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4684-9290-3.  Google Scholar [20] L. D. Talley, G. L. Pickard, W. J. Emery and J. H. Swift, Descriptive Physical Oceanography: An Introduction, 6th Edition, Academic Press, Elsevier Science, 2011. Google Scholar [21] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation, Cambridge University Press, Cambridge, 2017.  doi: 10.1017/9781107588417.  Google Scholar [22] E. Zermelo [Translated by Enzo de Pellegrin], Hydrodynamical investigations of vortex motions in the surface of a sphere, Ernst Zermelo - Collected Works/Gesammelte Werke II. Schriften der Mathematisch-naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften(eds. H. D. Ebbinghaus, A. Kanamori), Springer, Berlin-Heidelberg, 23 (2013), 300–483. Google Scholar

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##### References:
 [1] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9780511800955.  Google Scholar [2] V. A. Bogomolov, Dynamics of vorticity at a sphere, Fluid Dynamics, 12 (1977), 863-870.  doi: 10.1007/BF01090320.  Google Scholar [3] A. V. Borisov, I. S. Mamaev and S. M. Ramodanov, Coupled motion of a rigid body and point vortices on a two-dimensional spherical surface, Regular and Chaotic Dynamics, 15 (2010), 440-461.  doi: 10.1134/S1560354710040040.  Google Scholar [4] A. Constantin and R. S. Johnson, Large gyres as a shallow-water asymptotic solution of Euler's equation in spherical coordinates, Proc. R. Soc. Lond. A, 473 (2017), 20170063, 17 pp. doi: 10.1098/rspa.2017.0063.  Google Scholar [5] A. Constantin and R. S. Johnson, Steady large-scale ocean flows in spherical coordinates, Oceanography, 31 (2018), 42-50.  doi: 10.5670/oceanog.2018.308.  Google Scholar [6] A. Constantin and R. S. Johnson, An exact, steady, purely azimuthal flow as a model for the antarctic circumpolar current, Journal of Physical Oceanography, 46 (2016), 3585-3594.  doi: 10.1175/JPO-D-16-0121.1.  Google Scholar [7] B. Cushman-Roisin and J.-M. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, With a foreword by John Marshall. International Geophysics Series, 101. Elsevier/Academic Press, Amsterdam, 2011. doi: 10.1016/c2009-0-00052-x.  Google Scholar [8] T. Gerkema, J. T. F. Zimmerman, L. R. M. Maas and H. van Haren, Geophysical and astrophysical fluid dynamics beyond the traditional approximation, Reviews of Geophysics, 46 (2008), RG2004.  doi: 10.1029/2006RG000220.  Google Scholar [9] A. E. Gill, Atmosphere-Ocean Dynamics, International Geophysics, vol. 30. Academic Press, Elsevier Science, 1982. doi: 10.1016/s0074-6142(08)x6002-4.  Google Scholar [10] R. S. Johnson, Application of the ideas and techniques of classical fluid mechanics to some problems in physical oceanography, Phil. Trans. R. Soc. A, 376 (2018), 20170092, 19 pp. doi: 10.1098/rsta.2017.0092.  Google Scholar [11] Y. Kimura and H. Okamoto, Vortex motion on a sphere, Journal of the Physical Society of Japan, 56 (1987), 4203-4206.  doi: 10.1143/JPSJ.56.4203.  Google Scholar [12] M. S. Longuet-Higgins, Planetary waves on a rotating sphere, Proc. R. Soc. Lond. A, 279 (1964), 446-473.  doi: 10.1098/rspa.1964.0116.  Google Scholar [13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, 27. Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511613203.  Google Scholar [14] J. E. Marsden and A. J. Tromba, Vector Calculus, 6$^{th}$ edition, W. H. Freeman & Company, New York, 2012. Google Scholar [15] C. I. Martin, On the vorticity of mesoscale ocean currents, Oceanography, 31 (2018), 28-35.  doi: 10.5670/oceanog.2018.306.  Google Scholar [16] N. R. McDonald, The motion of geophysical vortices, Phil. Trans. R. Soc. A, 357 (1999), 3427-3444.  doi: 10.1098/rsta.1999.0501.  Google Scholar [17] P. Müller, Ertel's potential vorticity theorem in physical oceanography, Reviews of Geophysics, 33 (1995), 67-97.  doi: 10.1029/94RG03215.  Google Scholar [18] W. F. Newns, Functional dependence, The American Mathematical Monthly, 74 (1967), 911-920.  doi: 10.1080/00029890.1967.12000050.  Google Scholar [19] P. K. Newton, The N-Vortex Problem: Analytical Techniques, Applied Mathematical Sciences, 145. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4684-9290-3.  Google Scholar [20] L. D. Talley, G. L. Pickard, W. J. Emery and J. H. Swift, Descriptive Physical Oceanography: An Introduction, 6th Edition, Academic Press, Elsevier Science, 2011. Google Scholar [21] G. K. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation, Cambridge University Press, Cambridge, 2017.  doi: 10.1017/9781107588417.  Google Scholar [22] E. Zermelo [Translated by Enzo de Pellegrin], Hydrodynamical investigations of vortex motions in the surface of a sphere, Ernst Zermelo - Collected Works/Gesammelte Werke II. Schriften der Mathematisch-naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften(eds. H. D. Ebbinghaus, A. Kanamori), Springer, Berlin-Heidelberg, 23 (2013), 300–483. Google Scholar
A spherical co-ordinate system $(r,\theta,\phi)$, with $\theta$ being the polar angle (or colatitude) and $\phi$ (azimuth) defined with respect to the $x$-axis of the corresponding Cartesian system $(x,y,z)$. In this paper, we consider a stationary sphere, as well as a rotating sphere with angular velocity $\boldsymbol{\varOmega} = \mathit\Omega\boldsymbol{e}_z$
Decomposition of the orthonormal unit vectors in the spherical coordinate system into the Cartesian unit vectors
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