Abstract
The angle between the wind stress that overlies the ocean and the resulting current at the ocean surface is calculated for a two-layer ocean with uniform eddy viscosity in the lower layer and for several assumed eddy viscosity profiles in the upper layer. The calculation of the deflection angle is greatly simplified by transforming the linear, second-order, vertical structure equation to its associated nonlinear, first-order, Riccati equation. The transformation to a Riccati equation can be used as an alternate numerical scheme, but its main advantage is that it yields analytic expressions for several eddy viscosity profiles.
Highlights
For wind-driven surface ocean currents, various ranges of the deflection angle are recorded.1 Analytic expressions of the deflection angle are only available for special profiles of vertical eddy viscosities.2–6 Direct numerical calculations of the transport in the Ekman layer were carried out on a sphere,7 while the numerical calculations of the deflection angle for depthdependent eddy viscosity coefficients8 rely on the WKB approach to find accurate approximations for the solution of the secondorder boundary-value problem that governs Ekman flows
We examine how the surface deflection angle θ0 varies with the value of the surface eddy viscosity K(0) and the depth h of the upper layer of variable eddy viscosity, for the three examples provided in Sec
We compare the deflection angle obtained from the associated Riccati equation with the case of constant eddy viscosity in the upper layer that was examined previously in Ref. 6
Summary
For wind-driven surface ocean currents, various ranges of the deflection angle (i.e., the angle between the current at the ocean surface and the overlying wind that forces it) are recorded. Analytic expressions of the deflection angle are only available for special profiles of vertical eddy viscosities. Direct numerical calculations of the transport in the Ekman layer were carried out on a sphere, while the numerical calculations of the deflection angle for depthdependent eddy viscosity coefficients rely on the WKB approach to find accurate approximations for the solution of the secondorder boundary-value problem that governs Ekman flows. Analytic expressions of the deflection angle are only available for special profiles of vertical eddy viscosities.. Direct numerical calculations of the transport in the Ekman layer were carried out on a sphere, while the numerical calculations of the deflection angle for depthdependent eddy viscosity coefficients rely on the WKB approach to find accurate approximations for the solution of the secondorder boundary-value problem that governs Ekman flows. The eddy viscosity coefficient should vary gradually with depth, an assumption that limits the applicability of the WKB approach. We derive a uniformly valid formula for the deflection angle that, rather than relying on solving a second-order boundary-value problem on an infinite interval, only requires the solution of a first-order initial-value (non-linear) Riccati equation, on a finite interval. Its relevance to the problem of wind-driven ocean currents is, perhaps, not surprising
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