The Deflection Angle of Surface Ocean Currents From the Wind Direction

Abstract

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.

Frictional effects in wind-driven ocean currents

“…In section 4, we have derived the formula (25) for the deviation of the deflection angle from the 45 • reference value, for small perturbations of a constant leading-order eddy viscosity. This formula improves the recent result in Bressan and Constantin (2019) and, applied to the case of a piecewise linear eddy viscosity, invalidates the speculation that an eddy viscosity that increases/decreases with depth produces deflection angles less/larger than 45 • . One can actually pursue a general investigation of the formula (25).…”

“…The formula (25) is analogous to a formula derived recently in Bressan and Constantin (2019) in terms of the implicit variable t(z) = K 0 z 0 [K(s)] −1 ds. Despite their similarity -due to a typographical error, a multiplicative minus factor is actually missing on both sides of formula (22) in Bressan and Constantin (2019) -the formula (25) is advantageous since it is expressed in terms of the physical depth variable z, rather than implicitly (as in Bressan and Constantin 2019).…”

“…This invalidates the expectation (arising from the case study Madsen 1977) of a linear decrease with depth (see the discussion in Krauss 1993) that an eddy viscosity that increases/decreases with depth produces a deflection angle that lags/exceeds 45 • . Note that the approach in Bressan and Constantin (2019) also applies to (26), but the conclusion is less precise due to the intricate nature of the implicit alternative to the formula (25).…”

“…In this paper, we will establish the general validity of the characteristics (ii) and (iii) of Ekman flows by adapting and simplifiying the approach used in Constantin and Johnson (2019a) for flows in the atmospheric boundary layer. With regard to the deflection angle, for eddy viscosities that are small perturbations of a constant, we simplify the perturbative approach proposed in Bressan and Constantin (2019) to derive a formula that predicts the deviation of the deflection angle from the classical 45 • reference value. In particular, this formula invalidates the speculation (see the discussion in Krauss 1993) that eddy viscosities that increase/decrease with depth produce deflection angles less/larger than 45 • .…”

A Sturm–Liouville Problem Arising in the Atmospheric Boundary-Layer Dynamics

“…Note that while recently it was established that in general the Ekman flow pattern is spiralling and the horizontal speed is monotone with height (see [7]), there remains the important issue of the deflection angle. Apart from some special, explicit solutions, there is little to go by, whether in the context of atmospheric flows or regarding wind-generated ocean currents (even though for the latter, a perturbative approach, developed in [2,4] and applicable to small perturbations of a constant eddy viscosity, can be used to predict whether the deflection angle is smaller or bigger than the 45 • of the classical solution, valid for constant eddy viscosity). For this reason, it is important to identify new classes of height-dependent eddy viscosities that permit explicit calculations.…”