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 the Berry curvature, which describes the local geometry of wave polarization relations and is 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 atom physics. Here, we use a variational method with a vectorial Wentzel–Kramers–Brillouin ansatz to derive ray- tracing equations for geophysical waves and to reveal the contribution of the Berry curvature. We detail the case of shallow-water wave packets and propose a new interpretation of their oscillating motion around the equator. Our result shows a mismatch with the textbook scalar approach for ray tracing, by predicting a larger eastward velocity for Poincaré wave packets. This work enlightens the role of the geometry of wave polarization in various geophysical and astrophysical fluid waves, beyond the shallow-water model.

从福柯摆的进动到量子霍耳效应以及拓扑绝缘体的存在，几何相（例如Berry相）已被证明是理解众多物理现象的有力概念。贝里相位由称为贝里曲率的量生成，贝里曲率描述了波偏振关系的局部几何形状，并且已知会出现在多分量波包的运动方程中。在凝聚态，光学和冷原子物理学中已经观察到矢量场的射线传播中的这种几何贡献。在这里，我们使用矢量Wentzel–Kramers–Brillouin ansatz的变分方法来推导地球物理波的射线追踪方程式，并揭示Berry曲率的贡献。我们详细介绍了浅水波包的情况，并提出了关于其在赤道附近的振荡运动的新解释。我们的结果表明，通过预测庞加莱波包的更大的东向速度，与教科书中用于射线追踪的标量方法不匹配。这项工作启发了波极化几何学在浅水模型之外的各种地球物理和天体流体波中的作用。