Exact, Free-Surface Equatorial Flows with General Stratification in Spherical Coordinates
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This paper is concerned with the construction of a new exact solution to the geophysical fluid dynamics governing equations for inviscid and incompressible fluid in the equatorial region. This solution represents a steady purely-azimuthal flow with a free-surface. The novel aspect of the solution we derive is that the flow it prescribes accommodates a general fluid stratification: the density may vary both with depth, and with latitude. The solution is presented in the terms of spherical coordinates, hence at no stage do we invoke approximations by way of simplifying the geometry in the governing equations. Following the construction of our explicit solution, we employ functional analytic considerations to prove that the pressure at the free-surface defines implicitly the shape of the free-surface distortion in a unique way, exhibiting also the expected monotonicity properties. Finally, using a short-wavelength stability analysis we prove that certain flows defined by our exact solution are stable for a specific choice of the density distribution.
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We gratefully acknowledge the support of the Science Foundation Ireland (SFI) under the research grant 13/CDA/2117. The authors would like to thank the reviewer for constructive comments and suggestions.
- 1.Bayly, B.J.: Three-dimensional instabilities in quasi-two dimensional inviscid flows. In: Miksad, R.W., et al. (eds.) Nonlinear Wave Interactions in Fluids, pp. 71–77. ASME, New York (1987)Google Scholar
- 3.Chen, R.M., Walsh, S., Wheeler, M.: Existence and qualitative theory for stratified solitary water waves. Ann. Inst. H. Poincaré Anal. Non Linéaire 35(2), 517–576 2018Google Scholar
- 14.Fedorov, A.V., Brown, J.N.: Equatorial waves. In: Steele, J. (ed), Encyclopedia of Ocean Sciences, pp. 3679–3695. Academic Press, New York, 2009Google Scholar
- 17.Gill, A.: Atmosphere-Ocean Dynamics. Academic Press, New York (1982)Google Scholar
- 19.Henry, D., Martin, C.I.: Exact, purely azimuthal stratified equatorial flows in cylindrical coordinates. Dyn. Partial Differ. Equ. 15, 337–349 2018Google Scholar
- 20.Henry, D., Martin, C.-I.: Free-surface, purely azimuthal equatorial flows in spherical coordinates with stratification. J. Differential Equations, to appearGoogle Scholar
- 22.Henry, D., Matioc, B.-V.: On the existence of steady periodic capillary-gravity stratified water waves. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 12(4), 955–974 2013Google Scholar
- 35.Vallis, G.K.: Atmospheric and oceanic fluid dynamics. Cambridge University Press (2006)Google Scholar