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Variational formulations of steady rotational equatorial waves. (English) Zbl 1464.76203

Summary: When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the \(f\)-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional \(\mathcal{H}\) in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

MSC:

76U60 Geophysical flows
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M30 Variational methods applied to problems in fluid mechanics
76E20 Stability and instability of geophysical and astrophysical flows
35Q35 PDEs in connection with fluid mechanics
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