Geometrical phases, such as the Berry phase, have proven to be powerful concepts to understand numerous physical phenomena, from the precession of the Foucault pendulum to the quantum Hall effect and the existence of topological insulators. The Berry phase is generated by a quantity named Berry curvature, describing the local geometry of wave polarization relations and known to appear in the equations of motion of multi-component wave packets. Such a geometrical contribution in ray propagation of vectorial fields has been observed in condensed matter, optics and cold atoms physics. Here, we use a variational method with a vectorial Wentzel-Kramers-Brillouin (WKB) ansatz to derive ray tracing equations in geophysical waves and reveal the contribution of Berry curvature. We detail the case of shallow water wave packets and propose a new interpretation to the equatorial oscillation and the bending of rays in mid-latitude area. Our result shows a mismatch with the textbook scalar approach for ray tracing, by predicting a larger eastward velocity for Poincare wave packets. This work enlightens the role of wave polarizations geometry in various geophysical and astrophysical fluid waves, beyond the shallow water model.
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