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

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

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  • 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.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  • Figure 1.  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 $

    Figure 2.  Decomposition of the orthonormal unit vectors in the spherical coordinate system into the Cartesian unit vectors

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