Abstract
In astrophysics and geophysics, one often has to deal with rotating flows. This rotation can be in a large-scale streaming pattern or in the form of vortices: small swirls, or both. Also, in geophysics and planetary physics one usually describes an atmospheric flow in a rotating reference frame , for instance a frame that is fixed to the Earth’s surface. This choice has an effect on the description of the flow as such a frame is not an inertial frame: centrifugal and Coriolis terms appear in the equation of motion. Fluid dynamics in a rotating frame will be dealt with in the next chapter. Here we will look at vorticity.
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See also the Mathematical Appendix.
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Strictly speaking one has \(\varvec{{\varvec{\nabla }}} P = (\rho \mathcal{R} / \mu ) \varvec{{\varvec{\nabla }}} T + (T \mathcal{R}/\mu ) \varvec{{\varvec{\nabla }}} \rho \), but since \(\varvec{{\varvec{\nabla }}} \rho {{\, \times \,}}\varvec{{\varvec{\nabla }}} \rho = 0\) the term involving the density gradient does not contribute to the vorticity-generation term \(\propto \varvec{{\varvec{\nabla }}} \rho {{\, \times \,}}\varvec{{\varvec{\nabla }}} P\).
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This ‘magnetic analogy’ goes further. For instance, the magnetic field \(\varvec{B}\) can be defined in terms of a so-called vector potential \(\varvec{A}\) as \(\varvec{B} = {\varvec{\nabla }}{{\, \times \,}}\varvec{A}\) so that \({\varvec{\nabla }}{{\, \cdot \,}}\varvec{B} = 0\). For more details see for instance [12, 15], Sect. 7.
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Achterberg, A. (2016). Vorticity. In: Gas Dynamics. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-195-6_11
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DOI: https://doi.org/10.2991/978-94-6239-195-6_11
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