Quantum geometry of the rotating shallow water model

TL;DR

This paper develops a quantum-geometric framework for the rotating shallow water equations, revealing their topological and geometric properties, and proposes experimental methods to observe these features.

Contribution

It introduces a quantum geometric tensor analysis of RSWE, linking wave polarization, Berry curvature, and topological invariants, with explicit symmetry-guided formulas.

Findings

Derived compact expressions for the quantum geometric tensor of RSWE.

Identified Berry curvature and Chern numbers in the wave bands.

Proposed experimental approach to measure quantum geometry in rotating tanks.

Abstract

The rotating shallow water equations (RSWE) are a mainstay of atmospheric and oceanic modeling, and their wave dynamics has close analogues in settings ranging from two-dimensional electron gases to active-matter fluids. While recent work has emphasized the topological character of RSWE wave bands, here we develop a complementary quantum-geometric description by computing the full quantum geometric tensor (QGT) for the linearized RSWE on an $f$-plane. The QGT unifies two pieces of band geometry: its real part defines a metric that quantifies how rapidly wave polarization changes with parameters, while its imaginary part is the Berry curvature that controls geometric phases and topological invariants. We obtain compact, symmetry-guided expressions for all three bands, highlighting the transverse structure of the metric and the monopole-like Berry curvature that yields Chern numbers for the Poincar\'e bands. Finally, we describe a feasible route to probing this geometry in rotating-tank experiments via weak, time-periodic parametric driving.