The Deflection Angle of Surface Ocean Currents From the Wind Direction

Wind stress drives the upper ocean circulation in nonequatorial regions by means of an interplay with the vertical turbulent friction and the Coriolis force, generating horizontal wind drift currents which spiral and decay with depth. Classical Ekman theory—applied almost universally in oceanography—predicts that the angle between the vectors of the surface current and surface wind is 45◦, if the coefficient of vertical turbulent mixing is constant. However, observations show that the deflection angle is usually close to 30◦ in arctic regions and larger than 45◦ in some low-latitude areas, because the vertical turbulent mixing varies with depth. In contrast to Ekman's classical solution, the solutions that are available for depth-dependent eddy viscosity are quite involved and, as in data-driven studies, it is difficult to sort out spurious correlations that obscure the underlying structure. We propose a perturbative approach, providing a formula for the deviation of the deflection angle from the 45◦ reference value and discussing its implications.


Introduction
As the wind blows across the ocean, it moves its near-surface waters as a result of its frictional drag on the surface. In equatorial regions, where the Coriolis effect due to the Earth's rotation vanishes, the resulting wind drift current moves in the same direction as the wind (see Boyd, 2018). However, in nonequatorial regions the Coriolis effect, arising because of the Earth's spin around its polar axis, is of overriding importance: The wind-driven stress (acting in the direction of the wind) is balanced not only by frictional forces opposing it but also by the Coriolis force. The outcome is a deflection of the surface ocean current from the direction of the wind, to the right in the Northern Hemisphere, and to the left in the Southern Hemisphere (see Marshall & Plumb, 2016).
The deflection of surface currents was first noticed by the Norwegian explorer F. Nansen during the arctic Fram expedition (1893)(1894)(1895)(1896). Nansen observed that the ice, rather than moving in the same direction as the wind, was moving consistently to the right of the wind direction. He discerned that this was due to the effect of the Earth's rotation and reasoned that, as the depth increased, each successive layer of water, moving over the one below it like a wind, would produce an increasing deviation until, at a certain depth, the flow direction would be opposite to that at the surface (Huntford, 2002). Nansen turned to the Swedish physicist V. W. Ekman to give theoretical support for his observations and reasoning. Ekman's subsequent theory of wind-driven currents stands more than a century after its introduction (Ekman, 1905) as the basis for our understanding of wind-driven ocean circulation. While Ekman theory is also relevant for the air flow in the atmospheric planetary boundary layer (see Constantin & Johnson, 2019a), our concern will be wind drift ocean currents. These surface currents regulate the global climate by transporting the heat stored in the upper ocean, thus moderating the extremes of temperature on our planet. The oceans cover over 70% of the Earth's surface and water can absorb and release large amounts of heat without a large increase in temperature. IPCC 5th Assessment Report from 2014 estimates that most of the excess heat trapped by greenhouse gases has been stored in upper 75 m of the oceans, while the ocean heat content in 2018 was larger than any other year since systematic observations were made (see Cheng et al., 2019).

Journal of Geophysical Research: Oceans
10.1029/2019JC015454 current from the wind is typically larger during the day than during the night (see Krauss, 1993). These discrepancies are ascribed to Ekman's oversimplified hypotheses, in particular, the assumption of constant eddy viscosity. While other factors may also play a role (e.g., nonuniform densities, as pointed out by Cronin & Kessler, 2009), a depth-dependent eddy viscosity is typically essential for realistic predictions. Field data show that the class of relevant eddy viscosities feature quite different behaviors (the intensity could increase or decrease with depth, but specific regions present nonmonotonic types with multiple local extrema). Since the available explicit solutions for nonconstant eddy viscosity are very scarce (see the discussion in Constantin & Johnson, 2019a), one has to rely on case-by-case approximations. This type of approach makes it very difficult to identify the important factors that control the size of the deflection angle. We propose an alternative method: By developing a perturbative approach, we provide a generally valid formula that predicts the deviation of the deflection angle from the classical 45 • reference value. In particular, our approach invalidates the premise that an eddy viscosity that increases with depth produces deflection angles less than 45 • , while an eddy viscosity that decreases with depth yields deflection angles larger than 45 • . The influence of the depth variations of the eddy viscosity on the deflection angle is subtler, in the form of a weighted average quantified by the formula (22).

Background
For our purposes it is convenient to use a locally valid Cartesian coordinate system with x, y, and z aligned eastward, northward, and upward, respectively, and with the corresponding fluid velocity components denoted by u, v, and w, respectively. In nonequatorial regions, away from surface, bottom, and coastal boundary layers the ocean flow is geostrophic (arising from a balance between the pressure gradient force and the Coriolis force), whereas in the mixed layer near the surface (the upper 200-300 m) one has to account for the wind drift (see Marshall & Plumb, 2016). We consider nonequatorial open ocean regions since equatorial flows present peculiar features that are not encountered elsewhere (the vanishing of the meridional component of the Coriolis force at the equator leads to a breakdown of the geostrophic balance, while its change of sign across the equator produces an effective waveguide, with the equator acting as a fictituous natural boundary that facilitates azimuthal flow propagation-see the discussions in Basu, 2019;Boyd, 2018;Constantin & Johnson, 2019b, 2019cHenry, 2018;Johnson et al., 2001). Since the ratio of vertical speed to horizontal speed is about 10 −4 (see Marshall & Plumb, 2016), we neglect the vertical motion and consider an open ocean region with a horizontal geostrophic flow field (ū,v), subjected to a wind stress ( 1 , 2 ) along its surface. For a homogeneous fluid of density 0 the vertical component of the geostrophic flow is zero and the horizontal geostrophic flow depends on the magnitude of the pressure gradient (see Marshall & Plumb, 2016): where p is the pressure and = 2 sin is the Coriolis parameter, equal to twice the vertical component of the Earth's rotation vector ( ≈ 7.272 × 10 −5 rad s −1 and being the radian frequency of Earth's rotation and the latitude, respectively). Neglecting vertical motion means that the pressure distribution is hydrostatic, so that (ū,v) is depth independent. Within this setting, assuming steady wind conditions and a turbulent mixing parametrized by a constant vertical eddy viscosity coefficient K, the balance between frictional and Coriolis forces can be expressed in the form of the following differential equations for the latitudinal and longitudinal velocity components of the wind drift, as a function of depth: (2) The wind stress at the surface z = 0 is given by and the boundary condition expresses the fact that the wind drift current is insignificant at great depths. Ekman's solution to this problem in the Northern Hemisphere is, in complex variables notation, where d = √ 2K∕ is the depth of the Ekman-layer (the near-surface region of the ocean affected by the movement of wind and frictional influence, typically about 30-200 m deep). The wind-driven horizontal flow component (5) is independent of the geostrophic flow field (ū,v), with the surface current directed 45 • to the right/left of the surface wind stress in the Northern/Southern Hemisphere, with velocities decaying and rotating with depth to the right/left to form a spiral (see Figure 1).
Observational evidence for Ekman's solution (5) is provided by the wind-driven currents in the Southern Ocean and in some coastal straits (see Polton et al., 2013;Roach et al., 2015;Stacey et al., 1986). However, most field studies of wind drift ocean currents report systematic differences (see, e.g., Cronin & Tozuka, 2016). Ekman's basic predictions, that the wind drift current veers at the surface to the right/left of the wind (in the Northern/Southern Hemisphere) and that with increasing depth the current speed is reduced, while the direction rotates farther away from the wind direction following a spiral, are generally accepted but the details are not. It is by now recognized that the assumption of constant eddy viscosity is an extreme simplification and a depth-dependent eddy viscosity K(z) is necessary, so that instead of (2) we have the system: with the boundary conditions (3) and (4); see Cronin and Kessler (2009). We regard the physically relevant eddy viscosities K(z) as perturbations of the asymptotic reference value K 0 = lim where ≪ 1 and the asymptotic rate of convergence is faster than quadratic; that is, there exist constants a, b, c > 0 such that Field data show that quite often the perturbation K 1 actually converges exponentially fast to zero at great depths (see the discussion in Constantin & Johnson, 2019a).

Methods
To simplify notation, we use the complex function and perform the change of variable to write the system (6) more compactly as the complex-valued second-order differential equation Journal of Geophysical Research: Oceans The boundary conditions (3) and (4) are transformed to respectively. Writing we set where Ψ 0 is the explicit classical Ekman solution Inserting (13) in (9), one obtains Dividing by and letting → 0, this yields a linear, nonhomogeneous second-order differential equation for the perturbation , namely, to be solved with homogeneous boundary and asymptotic conditions: The solution to (15) and (16) can be expressed as the convolution where for t < 0 the Green's function w(s) = G(s, t) provides a solution to The solution to (18) is of the form Using the continuity of w and the jump condition in the derivative at s = t, we find so that the solution to (15) and (16) is given by At s = 0 one finds so that (14) yields The complex-variables formula d d arg(1 + )| =0 = Im( ) enables us to express the change of the deflection angle due to the perturbation as A positive/negative value of the integral ) dt corresponds to an increase/decrease of the deflection angle from the reference value ∕4 (see Figure 2).

Discussion
The formula (22) for the change of the deflection angle due to the perturbation provides insight into the way the depth variation of the eddy viscosity influences the deflection angle.

A Slowly Varying Eddy Viscosity Profile
Let k(s) = a + b e μs with a > 0, > 0, and b > −a. Due to (10) and (12), we have since (8)  . In this case we can compute the integral on the right side of (22) as 0 ∫ −∞ e 2 t (a + be μt ) sin

Journal of Geophysical Research: Oceans
10.1029/2019JC015454 using twice integration by parts in the last step and relying on the relation that emerges because the antiderivative of the integrand is 1 2 √ 2 e 2 t sin(2 t). Consequently, for b > 0, corresponding to an eddy viscosity (23) that decays slowly with depth, the formula (22) predicts a deflection angle in excess of 45 • . On the other hand, for b < 0, corresponding to an eddy viscosity (23) that increases slowly with depth, the formula (22) predicts a deflection angle less than 45 • . Actually, consulting tables of special functions, one can estimate with accuracy the values of the deflection angle for eddy viscosities of type (23). Indeed, note that if we set then we transform (9) into the general Bessel equation . The general solution of (25) can expressed in terms of the (complex valued) Bessel functions of order , namely, J and Y (see Abramowitz & Stegun, 1964). Since the boundary condition Ψ → 0 as s → −∞ is transformed to Φ → 0 as x → 0, we discard the solution Y (which diverges as x → 0). Varying the parameters a, b, and , by using the Maple mathematical package one can produce deflection angles within the range from 30 • to 50 • .

General Considerations
To investigate the formula (22) in general, note that the change of variables (8) preserves the monotonicity properties, and, as s decreases from 0 toward −∞, the function e 2 t k(t) sin changes sign alternately, being positive for s ∈ I 0 ∶= (− 8 , 0), negative on the intervals I 2 +1 ∶= (− 8 − 2 (2 + 1), 3 8 − 2 (2 + 1)) and positive on I 2 +2 ∶= (− 8 − 2 (2 + 2), 3 8 − 2 (2 + 2)) for integers j ≥ 0. Moreover, we have since the antiderivative of the integrand is 1 2 √ 2 e 2 t sin(2 t). The sequence {E m } m ≥ 0 of the absolute values of these integral expressions converges exponentially fast to 0 and is strictly decreasing starting with m ≥ 1 but The test case when the perturbation k is constant on each of the intervals I m shows that a decreasing eddy viscosity does not necessarily yield the key factor being the relative size of the values assigned to the intervals I 0 and I 1 . Note that I 0 corresponds to the upper eighth part of the Ekman layer of depth d = , while I 1 extends within the lower half of the Ekman layer. This analysis shows that an eddy viscosity that increases/decreases with depth in the Ekman layer does not always induce a deflection angle that lags/exceeds ∕4. In this context, we point out that this implication holds for a linear increase with depth (see Madsen, 1977)-a case study that led to the expectation that this is a universally valid feature (see the discussion in Krauss, 1993). Also, the generally 10.1029/2019JC015454 invalid implication holds for the case discussed in section 4.1 because of the slow and gradual variation with depth. On the other hand, if the eddy viscosity at the top of the Ekman layer (within I 0 ) is lower than toward the middle of the layer, then the negative contribution from I 1 dominates the positive one from I 0 , with a negative integral as the typical outcome. This explains why in arctic regions the deflection angle is below ∕4, since the ice cover tends to quell the turbulence near the ocean surface.
With regard to the observed day-night changes in the deflection angle, note that solar heating quenches turbulence throughout the Ekman layer during the day, while turbulence becomes stronger at night due to nocturnal convection, as the vertical extent of the Ekman layer (typically less than 100 m) coincides with the depth above which the downward flux of solar energy exceeds 1% of that entering at the sea surface (see Woods, 2002). The nocturnal increase of the eddy viscosity occurs across the upper half of the Ekman layer (the change in the lower half being rather insignificant). This amplifies the contribution from the region I 1 , leading thus typically to an overall negative integral in (22) with the result that the deflection angle is smaller during the night. The same argument applies to the large seasonal variations of the deflection angle observed at some locations. For example, records of surface velocities measured with high-frequency radars reveal that in the Tsushima Strait the deflection angle is 17.7-27.3 • in winter and increases to 48.5-67.3 • in summer (see Yoshikawa & Masuda, 2009).

A Piecewise Linear Eddy Viscosity Profile
The considerations made in section 4.2 invalidate the premise that deflection angles less/larger than 45 • occur for eddy viscosities that increase/decrease with depth. We illustrate the situation by considering for K 0 > 0 > z 0 and a > 0, the eddy viscosity profile since dz ds = K(z) K 0 by (8). In this case the right side of formula (22) can be computed as since e bs 0 ∕K 0 = K 0 a and the antiderivative of e (2 +b∕K 0 )t sin ) is 1 4 2 + (2 + b∕K 0 ) 2 e (2 +b∕K 0 )t

[
(2 + b∕K 0 ) sin The expression (28) in quite intricate. Nevertheless, let us we fix a > K 0 > 0, so that the eddy viscosity profile decreases with depth (b > 0). In the limit z 0 → −∞, the leading order term in (28) is the first one, which is positive. On the other hand, the limit z 0 → 0 yields b → ∞ and s 0 → 0, so that the leading order term in (28) is the last one, which is negative. In light of the formula (22), this shows that an eddy viscosity of type (26), decreasing with depth, does not ensure a deflection angle in excess of 45 • . Regarding the range of the deflection angles corresponding to the eddy viscosity profile (26), note that the change of variables transforms (9) into the Bessel equation The general solution of the above Bessel equation can be expressed as a linear combination of the (complex valued) Bessel functions of the first and second kind, of order 0 (see Abramowitz & Stegun, 1964). Using the Maple mathematical package one can obtain deflection angles as large as 55 • , as well as angles below 30 • , for suitable choices of the parameters a and z 0 .

Comparison With the WKB Approach and Future Developments
The WKB approach proved very successful for gaining insight into the dynamics of currents for eddy viscosities that vary with depth (see Wenegrat & McPhaden, 2016, for ocean flows and Grisogno, 1995, for flows in the atmosphere). This multiple-scales approach relies on the assumption that the dependence on the fast scale is exponential (see the discussion in Holmes, 2013;Johnson, 2005), yielding an approximation consisting of a rapidly oscillating complex exponential multiplied by a slowly varying amplitude. The WKB requires that the properties of the medium vary more slowly than the solution (see Wenegrat & McPhaden, 2016); in particular, the eddy viscosity should vary gradually with depth (see Grisogno, 1995). These constraints on the validity of the WKB expansion make it difficult to identify within this framework the equivalent of the weighted average (22) that predicts the deviation of the deflection angle from the constant eddy viscosity reference case 45 • . For the specific question of whether the deflection angle is lags/exceeds the reference value 45 • , the perturbative method presented in this paper is more general and also provides the surprisingly simple and effective predictive formula (22). However, for adequate settings, a WKB approach offers a more comprehensive insight into the underlying dynamical mechanisms. On the other hand, it is of interest to enhance the physical realism by studying the effects of stratification and of time dependence (see the discussions in Lewis & Belcher, 2004;Price & Sundermeyer, 1999;Rio & Hernandez, 2003). The method that we presented is amenable to address this added complexity (this is work in progress), but the development of a time-varying WKB approach appears to be very challenging, especially regarding the accuracy of the approximation.