A particle representation of tropical shallow water waves

Pao K Wang 

The linearized shallow water equations on a general β-plane reduce to a single meridional structure equation for the meridional velocity. We show that this equation is exactly the Schrödinger equation for a “charged” particle minimally coupled to a four-vector potential whose analogue magnetic field, B_W=-(β/c_g ) x ̂, is uniform and westward. No ad hoc term is required: every term is generated by the minimal coupling. The analogue magnetic length is the equatorial Rossby radius of deformation, the Hermite index of the Matsuno solution is the Landau level index, and the Matsuno dispersion relation is recovered exactly as the Landau spectrum of the constructed Hamiltonian rather than merely resembling it. The zonal wavenumber is the canonical momentum, the gauge-invariant kinematic momentum being k+β/2ω. Because the vector potential depends on frequency, the eigenvalue problem is implicit, and this is the origin of the Rossby branch.