Statistics of near-inertial waves over a background flow via quantum and statistical mechanics

We revisit the interaction of an initially uniform near-inertial wave (NIW) field with a steady background flow, with the goal of predicting the subsequent organisation of the wave field. To wit, we introduce an exact analogy between the Young–Ben Jelloul (YBJ) equation and the quantum dynamics of a charged particle in a steady electromagnetic field, whose potentials are expressed in terms of the background flow. We derive the time-averaged spatial distributions of wave kinetic energy, potential energy and Stokes drift in two asymptotic limits. In the ‘strongly quantum’ limit, where the background flow is weak compared with wave dispersion, we compute the wave statistics by extending a strong-dispersion expansion initially introduced by YBJ. In the ‘quasi-classical’ limit, where the background flow is strong compared with wave dispersion, we compute the wave statistics by leveraging the equilibrium statistical mechanics of classical systems. We compare our predictions with numerical simulations of the YBJ equation, using an instantaneous snapshot from a two-dimensional turbulent flow as the steady background flow. The agreement is very good in both limits. In particular, we quantitatively describe the preferential concentration of NIW energy in anticyclones. We predict weak NIW concentration in both asymptotic limits of weak and strong background flow, and maximal anticyclonic concentration for background flows of intermediate strength, providing theoretical underpinning to observations reported by Danioux, Vanneste and Bühler (2015 J . Fluid Mech ., vol. 773, R1).