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On large-scale oceanic wind-drift currents

Christian Puntini, Luigi Roberti, Eduard Stefanescu

TL;DR

This work develops a geometry-preserving, double-asymptotic model for wind-driven oceanic drift currents on a rotating sphere, deriving a leading-order Ekman-type system and a first-order correction by expanding in the thin-shell parameter $\varepsilon$ and inverse Rossby number $\mathscr{R}$. The authors prove existence and uniqueness of the leading-order Ekman spiral, obtain a closed-form expression for the surface deflection angle, and provide explicit solutions for five depth-dependent eddy-viscosity profiles, enabling direct comparison with observations. The results yield detailed insight into how viscosity profiles shape surface deflection and Ekman transport, reproducing and explaining deviations from the classical $45^{\circ}$ deflection and aligning with mid-latitude measurements. Overall, the framework offers a rigorous, large-scale alternative to $f$-plane models, with practical implications for understanding wind-driven upper-ocean dynamics and transport.

Abstract

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.

On large-scale oceanic wind-drift currents

TL;DR

This work develops a geometry-preserving, double-asymptotic model for wind-driven oceanic drift currents on a rotating sphere, deriving a leading-order Ekman-type system and a first-order correction by expanding in the thin-shell parameter and inverse Rossby number . The authors prove existence and uniqueness of the leading-order Ekman spiral, obtain a closed-form expression for the surface deflection angle, and provide explicit solutions for five depth-dependent eddy-viscosity profiles, enabling direct comparison with observations. The results yield detailed insight into how viscosity profiles shape surface deflection and Ekman transport, reproducing and explaining deviations from the classical deflection and aligning with mid-latitude measurements. Overall, the framework offers a rigorous, large-scale alternative to -plane models, with practical implications for understanding wind-driven upper-ocean dynamics and transport.

Abstract

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 -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.
Paper Structure (20 sections, 4 theorems, 244 equations, 25 figures)

This paper contains 20 sections, 4 theorems, 244 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Governing equations
  3. The equations of motion
  4. Boundary conditions
  5. Scaling and non-dimensionalisation
  6. Asymptotics
  7. The leading-order problem
  8. The problem at first order
  9. Some analytical results
  10. The leading-order problem
  11. Existence and uniqueness of the solution, and the Ekman spiral
  12. The surface deflection angle
  13. The problem at first order
  14. Explicit solutions to the leading-order problem
  15. Constant eddy viscosity
  16. ...and 5 more sections

Key Result

Theorem 4.1

Let $W_{\rm w} \in \mathbb{C}\setminus\{0\}$ be arbitrary, $\theta \in (-\frac{\pi}{2},\frac{\pi}{2})\setminus\{0\}$, and suppose that $m\in C^1((z_0,0))\cap C([z_0,0])$ with $m(z) > 0$ for each ${z\in [z_0,0]}$. Then, there exists a unique function $W\in C^2((z_0,0)) \cap C([z_0,0])$ that solves od where $M(z) = |W(z)|$, we have that, for all $z\in (z_0,0)$, that is, the solution behaves as a cl

Figures (25)

  • Figure 1: The classical spherical coordinate system.
  • Figure 2: Schematic depiction of the flow configuration.
  • Figure 3: 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 4: Depiction of the solution \ref{['W constant visc']} (the blue curve) for $\theta=45^\circ$ and $z_0=-4$. The latitude-dependent turning rate $\sqrt{|\sin\theta|}$ is equal to its decay rate---one the features of the classical Ekman spiral. The surface deflection angle of approximately $45^\circ$ between direction of the wind (in red, not in scale) and the surface current is clearly visible. The spiral traced out by the velocity field is also projected (in black) onto the bottom plane for clarity.
  • Figure 5: The ratio $|W(0)/W_{\rm w}|$ between the intensity of the surface current and that of the wind for the case of constant eddy viscosity.
  • ...and 20 more figures

Theorems & Definitions (9)

  • Definition 3.1
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Definition 4.3
  • Lemma 4.4
  • Theorem 4.5
  • proof