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

TL;DR

The paper uncovers an exact quantum analogy to the YBJ equation for near-inertial waves in a steady background flow, enabling two complementary analyses: a strong-dispersion (quantum) regime with explicit predictions for kinetic energy, potential energy, and Stokes drift, and a strong-advection (classical) regime analyzed via ergodic theory and microcanonical ensembles. In the quantum limit, the NIW distributions are perturbatively expanded around a uniform base state, revealing how energy concentrates in flow features and how Stokes drift opposes the background flow. In the classical limit, NIW statistics are shown to be predominantly ergodic and uniform in space for kinetic energy, while potential energy traces the background-flow kinetic energy and Stokes drift scales with the background velocity; nonergodic trapping near anticyclones provides a refined, small but finite enhancement of energy there. Overall, the results—validated against DNS using a frozen 2D Navier–Stokes background—quantify the transient and long-time organization of NIWs, explain the observed maximal anticyclonic concentration at intermediate flow strengths, and provide a rigorous framework for applying equilibrium statistical mechanics to wave–flow interactions.

Abstract

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 organization 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 to 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 to wave dispersion, we compute the wave statistics by leveraging the equilibrium statistical mechanics of classical systems. We compare our predictions to 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 (Journal of Fluid Mechanics, 773, 2015).

Figures (7)

• Figure 1: A two-layer model with an infinitely deep lower layer. The base state consists of a vertically invariant steady horizontal flow ${\bf U}_g(x,y)$ spanning both layers, together with a flat interface between the two layers. We consider perturbations ${\bf u}(x,y,t)$ to the horizontal velocity in the upper layer only, whose depth is then denoted as $h(x,y,t)$. In line with the rigid-lid approximation, we neglect the fluctuations of the free surface as compared to $h$.
• Figure 2: Steady background flow used in the numerical simulations of the YBJ equation: background streamfunction $\chi(x,y)$ (left), kinetic energy $|\bnabla \chi|^2$ (center) and vorticity field $\Delta\chi$ (right). The normalization is such that $\left< \chi^2 \right>=1$ (see text). In all panels, the black contours correspond to streamlines of the background flow.
• Figure 3: Time-averaged spatial distributions of NIW kinetic energy (left), potential energy (center) and Stokes drift (right). The top row corresponds to the predictions (\ref{['eq:KESDR']}-\ref{['eq:usSDR']}) from the low-$\gamma$ asymptotic expansion. The bottom row corresponds to a numerical simulation in the strong-dispersion regime ($\gamma=0.05$). Isovalues are indicated with black contours with identical levels and colorbars for predictions and observations.
• Figure 4: A narrow wave packet with mean position $\boldsymbol{x}(t)$ and wavevector ${\bf p}(t)$ behaves like a charged classical particle in a steady 2D electromagnetic field.
• Figure 5: Top row: ergodic predictions for the time-averaged NIW potential energy $\overline{|\bnabla M|^2}(\boldsymbol{x})$ (left) and Stokes' drift $\overline{\boldsymbol{u}_s}(\boldsymbol{x})$ (right). Bottom row: same fields extracted from a numerical run with $\gamma=30$. In the left-hand column, black contours indicate isovalues 1, 4, 9 and 16.