Quantum geometry of the rotating shallow water model

TL;DR

This work develops a quantum-geometric framework for the rotating shallow water equations (RSWE) on the $f$-plane by computing the full quantum geometric tensor (QGT) for the three linear wave bands. Using a pseudospin-1 Hamiltonian ${\cal H}_{\chi}=-\mathbf{d}\cdot\mathbf{S}$ with $\mathbf{d}=(k_x,k_y,f)$, it derives the real part (Fubini–Study metric) and the imaginary part (Berry curvature) of each band, showing a transverse metric on a direction sphere and a monopole-like Berry field with Chern numbers ${\cal C}_m=2m$ for the Poincaré bands. The metric components are $g^{(m)}_{ij}=\frac{C_m}{d^2}(\delta_{ij}-\hat{d}_i\hat{d}_j)$ and the Berry curvatures are $F^{(m)}_{ij}=m\epsilon_{ijk}d_k/d^3$, which combine into $Q^{(m)}_{ij}=C_m\partial_i\hat{\mathbf{d}}\cdot\partial_j\hat{\mathbf{d}}+i\frac{m}{2}\hat{\mathbf{d}}\cdot(\partial_i\hat{\mathbf{d}}\times\partial_j\hat{\mathbf{d}})$. A practical route to measure these quantities is geometric spectroscopy, where parametric driving induces interband transitions whose integrated rates isolate FS metric and Berry curvature components. The results illuminate how geometry and topology jointly govern RSWE wave dynamics and suggest extensions to inhomogeneous backgrounds and experimental realizations 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 Poincaré bands. Finally, we describe a feasible route to probing this geometry in rotating-tank experiments via weak, time-periodic parametric driving.

Figures (1)

• Figure 1: The $\beta$-plane on a rotating planet. The angular velocity $\boldsymbol{\Omega}$ is aligned with the rotation axis; the tangent plane is at a latitude $\vartheta$ with respect to the equatorial plane; the Coriolis parameter is $f=2\Omega \sin\vartheta$, which becomes $f=f_0+\beta y$ in the $\beta$-plane approximation; and $(x,y)$ are the local Cartesian coordinates on the tangent plane.