## 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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