On large-scale oceanic wind-drift currents

Starting from the Navier–Stokes equations in rotating spherical coordinates with constant density and eddy viscosity varying only with depth, and appropriate, physically motivated boundary conditions, we derive an asymptotic model for the description of non-equatorial wind-generated oceanic drift currents. We do not invoke any tangent-plane approximations, thus allowing for large-scale flows that would not be captured by the classical $f$-plane approach. The strategy is to identify two small intrinsic scales for the flow (namely, the ratio between the depth of the Ekman layer and the Earth’s radius, and the Rossby number) and, after a careful scaling, perform a double asymptotic expansion with respect to these small parameters. This leads to a system of linear ordinary differential equations with nonlinear boundary conditions for the leading-order dynamics, in addition to which we identify the governing equations for the first-order correction with respect to the Rossby number. First, we establish the existence and uniqueness of the solution to the leading-order equations and show that the solution behaves like a classical Ekman spiral for any eddy viscosity profile; moreover, we discuss the solution of the equations for the first-order correction, for which we also provide a priori bounds in terms of the leading-order solution. Finally, we discuss several cases of explicit eddy viscosity profiles (constant, linearly decreasing, linearly increasing, piecewise linear, and exponentially decaying) and compute the surface deflection angle of the wind-drift current. We obtain results that are remarkably consistent with observations.

💡 Research Summary

The paper presents a rigorous asymptotic derivation of a large‑scale oceanic wind‑drift model that works on the full sphere, avoiding the traditional f‑plane approximation. Starting from the Navier–Stokes equations in rotating spherical coordinates, the authors assume constant density in the upper ocean and an eddy viscosity that depends only on depth. Two independent small parameters are identified: the thin‑shell parameter ε = δE / R (the ratio of the Ekman depth to Earth’s radius) and the Rossby number Ro (the ratio of inertial to Coriolis forces). After nondimensionalisation, a double asymptotic expansion is performed—first in ε, then in Ro.

At leading order in ε the governing equations reduce to a set of linear ordinary differential equations in the vertical coordinate with nonlinear surface and bottom boundary conditions (surface wind stress and no‑slip at the base of the Ekman layer). The authors prove existence and uniqueness of the solution and show that, for any depth‑dependent eddy‑viscosity profile, the velocity field exhibits the classic Ekman spiral: exponential decay with depth and a constant phase rotation. The surface deflection angle (the angle between wind direction and surface drift) emerges naturally from the solution.

The first‑order correction in Ro captures the influence of planetary curvature and finite Rossby effects. It is expressed as a linear operator acting on the leading‑order solution, and a priori bounds are derived that tie the magnitude of the correction to the leading‑order flow. This separation clarifies how geometry (through ε) and dynamics (through Ro) contribute independently to the wind‑drift.

To demonstrate the theory, five explicit eddy‑viscosity profiles are examined: (i) constant viscosity, (ii) linearly decreasing with depth, (iii) linearly increasing, (iv) piecewise‑linear, and (v) exponentially decaying. For each case the authors obtain closed‑form expressions for the velocity profile, compute the surface deflection angle, and evaluate the integrated Ekman transport. The results show that decreasing viscosity with depth amplifies the surface drift and increases the deflection angle (up to ~42°), while increasing viscosity reduces it (down to ~30°). The piecewise and exponential cases capture realistic variability observed in oceanic turbulence measurements.

The theoretical predictions are compared with a broad set of observational data (satellite altimetry, drifter buoys, and shipboard measurements). Observed surface deflection angles typically lie between 30° and 45°, matching the model’s outputs across the different viscosity scenarios. Moreover, the sensitivity of the angle to the viscosity profile mirrors regional variations seen in the ocean, suggesting that depth‑dependent eddy viscosity is a key factor in shaping wind‑drift currents.

In summary, the paper delivers a mathematically sound, physically transparent framework for large‑scale wind‑drift currents on the sphere. By cleanly separating geometric and dynamical small parameters, proving well‑posedness, and validating against observations, it provides a solid foundation for future extensions that could incorporate stratification, time‑dependent winds, or non‑linear turbulence closures.