Nonlinear Dynamics of Wind-Drift Currents at Mid-Latitudes

TL;DR

The paper addresses nonlinear wind-driven upper-ocean currents at mid-latitudes by deriving the leading-order dynamics on the $f$-plane using a thin-shell asymptotic expansion. The authors obtain an explicit Lagrangian solution in which the horizontal flow comprises a mean Ekman drift, near-inertial trochoidal oscillations, and a background geostrophic current, with the surface wind-drift current uniquely fixed by the surface-stress boundary condition; the trochoidal components are governed by a dispersion relation $c=\frac{f}{k}+2k$ and the depth-averaged Ekman transport is $\mathcal{I}_{Ek}=\frac{d(0)}{\lambda \sqrt{2}}e^{-i\frac{\pi}{4}}$. They show the deflection angle between the drift and wind relative to the geostrophic flow remains below $45^\circ$, aligning with observations, and discuss limitations of the flat-surface assumption and possible extensions to variable eddy viscosity and surface-wave effects. This work provides a rigorous analytical framework for wind-driven upper-ocean nonlinear dynamics at mid-latitudes, clarifying boundary-condition roles and highlighting the significance of near-inertial motions beyond classical Ekman theory.

Abstract

Starting from the Navier-Stokes equation in the $f$-plane approximation, we provide an exact and explicit solution of the governing equations at leading order for fluid flows in the upper layer of the ocean at mid-latitudes, driven by a wind stress. Such a solution highlights the presence of a mean Ekman current superimposed to trochoidal oscillations and a background geostrophic current.

Figures (4)

• Figure 1: The Cartesian coordinate system on the rotating Earth. Here, $\bf{r}$ represents the position vector of a point located at latitude $\theta$ and longitude $\varphi$. Z represents the null island.
• Figure 2: Monthly-averaged ocean wind speed and direction vectors, with vector lengths proportional to the reference scale (in $\mathrm{m\, s^{-1}}$), based on observations from NASA's QuikSCAT satellite. Image credit: NOAA.
• Figure 3: Schematic depiction of the surface currents: in red the wind velocity; in blue the geostrophic current, in black the surface current with the angle $\Theta-\Psi$ to the right of the vector of the wind velocity relative to the geostrophic current (olive green)
• Figure 4: The wind-drift current. In blue the geostrophic current, in black the wind-driven surface current, and in red the Ekman spiral. The $u, v$-axes, and the $z$-axis are not to scale.