An algorithm for computing the deflection angle of surface ocean currents relative to the wind direction

11 The angle between the wind stress that overlies the ocean and the resulting current 12 at the ocean surface is calculated for a two-layer ocean with uniform eddy viscosity in 13 the lower layer and for several assumed eddy viscosity profiles in the upper layer. The 14 calculation of the deflection angle is greatly simplified by transforming the linear, second 15 order, vertical structure equation to its associated nonlinear, first order, Riccati equation. 16 Though the transformation to a Riccati equation can be used as an alternate numeri17 cal scheme, its main advantage is that it yields analytic expressions for particular eddy 18 viscosity profiles. 19

1 Introduction 20 For wind-driven surface ocean currents, various ranges of the deflection angle are recorded (see 21 Röhrs and Christensen, 2015). Predictions for the deflection angle are only available for spe- flows. The WKB approximation consists of a rapidly oscillating complex exponential multiplied 27 by a slowly varying amplitude, and requires that the properties of the medium vary more slowly 28 than the solution (see the discussion in Holmes, 2013). In particular, the eddy viscosity should 29 vary gradually with depth, an assumption that limits the applicability of the WKB approach. 30 In this paper we derive a uniformly valid formula for the deflection angle that, rather than 31 relying on solving a second-oder boundary-value problem on an interval of infinite length, only 32 requires the solution of a first-order initial-value problem, with a suitable Riccati equation, on 33 a finite interval.

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Note that the Riccati equation arises in many different fields of physics and engineering, e.g.

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control theory, statistical thermodynamics, quantum mechanics, cosmology (see the survey in 36 Schuh, 2014). In light of this, its relevance to the study of wind-driven currents is perhaps not 37 that surprising. The non-dimensional linear governing equations for steady wind-driven ocean currents in the 40 non-equatorial Northern Hemisphere are (see Dritschel et al., 2020)

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where the complex vector ψ = u + iv represents the horizontal velocity field, z is the upward 46 pointing vertical variable (with the free surface at z = 0) scaled on (2τ /ρ)/f (where τ is the applied wind stress at the ocean's surface, ρ is the water density and f is the constant 48 Coriolis parameter) and K(z) is the vertical (depth-dependent) non-dimensional eddy viscosity 49 (that equals the dimensional eddy viscosity scaled on τ /f ). Since the turbulence is practically 50 confined to a near-surface ocean layer, it is reasonable to assume that below a certain depth h 51 the eddy viscosity is equal to the molecular viscosity of sea water, normalised so that  Let us now justify the algorithm described in Section 2. 78 for which the general solution is a linear combination of the linearly-independent functions 79 e ±(1+i)z . If we denote by ψ ± the solutions of (1) with 84 for some complex constant C determined by the boundary condition (2). Differentiating (10) 85 and evaluating the outcome and equation (10) at z = 0, we find 87 taking (2) into account. It is known (see Constantin, 2020) that ψ(z) = 0 for all z ≤ 0.
119 120 To ensure the regularity of Q(z) the values of a and h have to satisfy ah > −1. The differential 121 equation (17) is separable and can be straightforwardly integrated, yielding 140 equals µ at the surface and decreases/increases with depth, according to whether µ > 1 or µ ∈ (0, 1), respectively. In this case the general solution of (5) is available in terms of the where C 1 and C 2 are chosen such that their ratio satisfies the boundary condition (6), while 145 ζ = 2h(1 − i)/(|µ − 1|) and x = µ + z(µ − 1)/h ranges between 1 and µ. Note that (6) becomes: which determines the ratio C 1 /C 2 purely in terms of ζ; notably the solution q(z) depends 148 only on this ratio.

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No data was used in this theoretical paper