  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
Poincar\'e wave packets. This work enlightens the role of wave polarization's
geometry in various geophysical and astrophysical fluid waves, beyond the
shallow water model.
