Unsteady Ekman--Stokes dynamics: implications for surface-wave induced drift of floating marine litter

We examine Stokes drift and wave-induced transport of ﬂoating marine litter on the sur- 15 face of a rotating ocean with a turbulent mixed layer. Due to Coriolis–Stokes forcing and 16 surface wave stress, a second-order Eulerian-mean ﬂow forms, which must be added to 17 the Stokes drift to obtain the correct wave-induced Lagrangian velocity. We show that 18 this wave-driven Eulerian-mean ﬂow can be expressed as a convolution between the un- 19 steady Stokes drift and an ‘Ekman–Stokes kernel’. Using this convolution we calculate 20 the unsteady wave-driven contribution to particle transport. We report signiﬁcant dif- 21 ferences in both direction and magnitude of transport when the Eulerian-mean Ekman– 22 Stokes velocity is included. 23 ﬂow We develop a model that enables estimation of this induced Eulerian-mean ﬂow from measurements or predictions the wave ﬁeld and our model to data. Accounting for the wave-induced Eulerian-mean ﬂow sig- 32 niﬁcantly alters predictions of transport of ﬂoating marine litter by waves.

Stokes velocity is included. . This letter focuses on one of these processes: surface waves. 49 As a particle undergoes its periodic motion beneath surface waves, it experiences 50 a Lagrangian-mean velocity in the waves' direction known as Stokes drift (Stokes, 1847).   We consider a homogeneous (constant-density), incompressible ocean of constant depth d, described by horizontal coordinates x and y, and a vertical coordinate z measured upwards from the undisturbed water level. The governing equations, divided through by the (constant) density ρ, are where z = η(x, y, t) denotes the free surface elevation, u is the three-dimensional ve-  We assume the wave steepness is small, α ≡ kA 1, where A is the peak wave 99 amplitude of η and k the peak wavenumber, and solve (1) to O(α 2 ) using a Stokes ex-100 pansion u = u 1 + u 2 + · · · , where the subscript denotes the order in α. We focus on 101 deep-water waves (kd 1). Integrating the O(α 2 ) equations over a wave period, we obtain the wave-averaged mean flow equations (e.g. Huang, 1979; Suzuki & Fox-Kemper, 2016) where the overbar denotes a time average, u L = u+u S is the Lagrangian (or particle- (3) Examining the viscous but non-rotating case, Longuet-Higgins (1953) showed that vor-  In the Stokes layer, vertical gradients dominate over horizontal ones. It follows from (2b) that the vertical velocity component and pressure gradient can be neglected. Introducing the complex notation U = u+iv as in Huang (1979), we obtain the Ekman-Stokes equations We solve (4) by Laplace transform, assuming that the Stokes drift U S has a timeindependent vertical structure exp(2kz) corresponding to a quasi-monochromatic wave field, but an otherwise arbitrary time dependence. Denoting the Laplace transform by a tilde, with where γ is a real number such that the contour path of integration is in the region of convergence ofg(s), we find that when U(t = 0) = 0 This is the sum of a particular solution -the second term -which can be interpreted  (6)).

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A special case of (6) occurs if U S approaches a steady value U S as t → ∞. Theñ U tends to the time-independent solution (cf. Seshasayanan & Gallet (2019)) where D ≡ δ E /δ S is the fixed ratio of Ekman to Stokes depths. In the limit D → 0 + , 151 equation (7) approaches −U S exp(2kz): up to an inertial oscillation this is the so-called 152 'anti-Stokes' Eulerian-mean flow, predicted by Hasselmann (1970) to be induced by pe-153 riodic waves in a rotating, inviscid ocean. Viscosity acts to reduce the shear in the anti-

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Stokes flow, so that a nonzero Lagrangian-mean velocity remains. 156 We now use the Laplace convolution theorem to write the unsteady solution for the Ekman-Stokes mean flow as a function of time for arbitrary Stokes drift as

Ekman-Stokes kernel
where * denotes convolution in time and The convolution kernel K(z, t), which we will term the Ekman-Stokes kernel, can be evaluated by deforming the integration contour involved in the inverse Laplace transform Limit Behaviour Theory to obtain (see supplementary material) where ± denotes the sum of the plus and minus terms and the complementary error function erfc(x) = 1 − erf(x). An equivalent form emphasising dependence on wave parameters uses the scaled error function erfcx(t) = e t 2 erfc(t) and reads The Ekman-Stokes kernel K captures the (Eulerian-mean) flow response to the Stokes 160 Several limits of the kernel are of interest; they are given in dimensional terms in Table 1. The limits ν → 0 + and f → 0 + are best understood by rewriting (11) in terms of the dimensionless parameters D = δ E /δ S , ζ = 2kz and τ = f t to obtain When D 1, e.g. because f → 0 + , the Coriolis-Stokes sum term in (12)  Bottom: Wave roses for US, U, and UL, with radial distance representing the fraction of time during which the velocity has a given direction, and colour indicating magnitude in m/s.

Buoy data 201
We use half-hourly records for the San Nicolas Island buoy (33.22 • N, 119.88 • W) obtained from CDIP (the Coastal Data Information Project) and estimate the Stokes drift using the formula where θ p is the peak wave direction, H s the significant wave height, and ω p the peak fre-202 quency calculated from the peak period T p . By making a quasi-monochromatic approx-203 imation, we assume the wavenumber spectrum is peaked about k = mean(k p ) = mean(ω 2 p /g), 204 to leading order. We integrate (11) using the Stokes drift (13) by a trapezoidal rule with   Third, while we have used a quasi-monochromatic assumption in our model, the 265 Ekman-Stokes kernel can in principle be applied to broad-banded spectra using an ad-266 ditional integration over frequency. For typical broad-banded spectra, the near-surface 267 Stokes drift is more strongly sheared than for a monochromatic wave corresponding to 268 the forcing of the Eulerian-mean flow whose magnitude is strengthened and whose di-270 rection becomes more aligned with that of the Stokes drift, as we have confirmed in pre-271 liminary computations. We emphasise that the wave stress is proportional to the fifth 272 moment of the frequency spectrum and hence ill-defined for most empirical spectra, whose  The authors thank two referees whose comments have improved this paper and en-