The four-vector potential of equatorial shallow water waves from Hamilton’s principle

Pao K Wang 

Linear shallow water waves on an equatorial β-plane can be represented as the states of a charged particle minimally coupled to a four-vector potential whose analogue magnetic field is uniform and proportional to β/c_g. There the potential is written down and verified rather than derived. We show that it need not be assumed. In the Eulerian variational principle for rotating flow the Coriolis coupling is already a minimal coupling, the frame velocity playing the part of a vector potential, and the quadratic Lagrangian for linear shallow water motion is that of a two-dimensional charged particle in a uniform field acting in the plane of parcel displacement. Eliminating the zonal displacement amplitude, which enters algebraically, returns the four-vector potential identically. The βk/ω term then originates in a single integration by parts, the vertical wavenumber is forced to vanish, and the potential’s frequency dependence is an artifact of the reduction.