The Ekman spiral for piecewise-uniform diffusivity

10 We re-visit Ekman’s (1905) classic problem of wind-stress induced ocean currents to 11 help interpret observed deviations from Ekman’s theory, in particular from the predicted 12 surface current deflection of 45◦. While previous studies have shown that such deviations 13 can be explained by a vertical eddy viscosity varying with depth, as opposed to the 14 constant profile taken by Ekman, analytical progress has been impeded by the difficulty 15 in solving Ekman’s equation. Herein, we present a solution for piecewise-constant eddy 16 viscosity which enables a comprehensive understanding of how the surface deflection angle 17 depends on the vertical profile of eddy viscosity. For two layers, the dimensionless problem 18 depends only on the depth of the upper layer and the ratio of layer viscosities. A single 19 diagram then allows one to understand the dependence of the deflection angle on these 20 two parameters. 21


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The motion of the near-surface ocean layer is a superposition of waves, wind-driven currents 23 and geostrophic flows. The basic theory of wind-driven surface currents in the ocean, away 24 from the Equator, is due to Ekman (1905) and constitutes a cornerstone of oceanography 25 (see Vallis, 2017). Ekman dynamics is due to the balance between Coriolis and the frictional 26 forces generated by the wind stress. Its main features, consistent with observations of steady 27 wind-driven ocean currents, are: 28 (i) the surface current is deflected to the right/left of the prevailing wind direction (in the 29 Northern/Southern Hemisphere); 30 (ii) with increasing depth in the boundary layer, the current speed is reduced, and the direc-31 tion rotates farther away from the wind direction following a spiral; 32 (iii) the net transport is at right angles to the wind direction, to the right/left of the wind 33 direction in the Northern/Southern Hemisphere. 34 While near the Equator wind-drift currents move in the same direction as the wind (see the 35 discussion in Boyd, 2018), away from the Equator a deflection of steady wind-driven currents 36 with respect to the prevailing wind direction occurs in a surface boundary layer, whose typ-37 ical depth is tens of metres. Ekman's pioneering solution (see Ekman, 1905), derived for a 38 constant vertical eddy viscosity, captures the general qualitative behaviour, but differences of 39 detail between observations and Ekman theory were recorded in the last decades. While the 40 characteristics (ii)-(iii) hold for any depth-dependent vertical eddy viscosity (see Constantin,41 2020), there is a need to explain the occurrence of surface currents at an angle in the range  This discrepancy is typically ascribed to the effect of a vertical eddy viscosity that varies with 46 depth. The explicit solution found by Madsen (1977), for a vertical eddy viscosity that varies where U (Z) = U + iV is the complex horizontal velocity in the (X, Y )-plane, Z is the depth 75 below the mean surface Z = 0, f is the Coriolis parameter, ρ is the (constant) density, ∇P = 76 ∂P/∂X + i ∂P/∂Y is the horizontal pressure gradient, τ (Z) = τ x + iτ y is the shear stress due 77 to molecular and turbulent processes, and the higher-order terms, representing interactions 78 between the variables, are presumed to be small. Decomposing the horizontal velocity into 79 pressure-driven (geostrophic) and wind-driven (Ekman) components U = U g + U e , we see from 80 (1) that the leading-order geostrophic and wind-driven flows separate, with the linear equation governing the dynamics of the wind-driven flow. By relating the stress vector within the fluid, 82 τ , to the shear profile through a turbulent eddy viscosity coefficient ν(Z), from (2) we obtain Ekman's equations for wind-driven ocean currents Let us now discuss the appropriate boundary conditions. At the surface, the shear stress 85 balances the wind stress, τ 0 : The "bottom" boundary condition expresses the vanishing of the wind-driven current with 87 depth (necessary to keep the total kinetic energy finite), where the flow is essentially geostrophic: Letting τ 0 denote the magnitude of the surface wind stress, we non-dimensionalise the 89 problem by scaling U e on 2τ 0 /ρ and Z on 2τ 0 /ρ/f , since τ 0 /ρ has units of L 2 /T 2 . The 90 factor of 2 is introduced for convenience below. Upon defining a dimensionless eddy viscosity where ψ = uK(0) (cf. equations (14)-(16) in Gill, 1982). The scaling performed does not  In each region, the complex velocity ψ satisfies a simple constant-coefficient equation having exponential solutions where A, B and C are (generally complex) constants. The boundary condition ψ → 0 as 108 z → −∞ has been used to eliminate the growing solution in (13).
Applying the surface boundary condition (15) determines C as The surface current deflection angle, θ 0 , measured clockwise, is determined from But given C above in (19), we have Hence, taking the (negative of the) ratio of the imaginary to real parts of this, we obtain

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First, we examine certain special cases.

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As → 0, the eddy viscosity vanishes in the lower layer, and the flow field ψ must also 128 vanish. In this case, tan θ 0 reduces to which has a non-trivial dependence on h. The maximum value is attained as h → 0; then 130 tan θ 0 → ∞ or θ 0 → 90 • .

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As → ∞, corresponding to an extremely viscous lower layer, tan θ 0 reduces to the inverse  shows that this is consistent with our conclusions.

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The results obtained may help better formulate appropriate parametrisations of eddy vis-