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Title: Vorticity Generation by Short-crested Wave Breaking
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Authors: David B. Clark+*, Steve Elgar+, Britt Raubenheimer+
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Affiliation:
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Woods Hole Oceanographic Institution
*Corresponding Author: [email protected]
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Abstract:
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Eddies and vortices associated with breaking waves rapidly disperse pollution, nutrients,
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and terrestrial material along the coast. Although theory and numerical models suggest
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that vorticity is generated near the ends of a breaking wave crest, this hypothesis has not
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been tested in the field. Here we report the first observations of wave-generated vertical
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vorticity (e.g., horizontal eddies), and find that individual short-crested breaking waves
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generate significant vorticity [O(0.01 s-1)] in the surfzone. Left- and right-handed wave
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ends generate vorticity of opposite sign, consistent with theory. In contrast to theory, the
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observed vorticity also increases inside the breaking crest, possibly owing to onshore
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advection of vorticity generated at previous stages of breaking or from the shape of the
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breaking region. Short-crested breaking transferred energy from incident waves to lower
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frequency rotational motions that are a primary mechanism for dispersion near the
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shoreline.
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Introduction
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The tremendous dissipation of wave energy in the surfzone (the region of depth-limited
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breaking near the shoreline) drives vigorous dispersion of bacteria [Bohem et al., 2002],
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larvae [Rilov et al., 2008], sediments, and other suspended material, with surfzone eddy
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diffusivities roughly 10 times larger than those measured seaward of the breaking region
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[Fong and Stacey, 2003; Spydell et al., 2007; Jones et al., 2008; Spydell and Feddersen,
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2009; Spydell et al., 2009; Brown et al., 2009; Clark et al., 2010; Clark et al., 2011].
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Recent studies suggest these high diffusivities are generated primarily by low-frequency
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(0.030-0.001 Hz) quasi-2D horizontal eddies [Spydell and Feddersen, 2009, Clark et al.,
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2010; Clark et al., 2011]. This region of high diffusivity causes rapid dispersion within
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the surfzone over long O(103-104 m) alongshore distances and relatively short O(100 m)
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cross-shore distances, with much slower dispersion seaward of the surfzone, and results
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in material appearing to be “trapped” near the coast [Grant et al., 2005, Clark et al.,
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2010].
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Despite the importance of eddies and dispersion, there have been few, if any direct
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observations of surfzone vorticity, and there are no observations of vertical vorticity (e.g.,
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horizontal eddies) generated by breaking waves (including whitecaps). Moreover, the
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mechanisms that transfer energy from incident swell and sea waves to lower frequency
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eddies are not understood. These mechanisms must be quantified to model the dispersion
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and transport of pollutants, sediments, nutrients, and other tracers near the shoreline.
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Breaking waves (Fig. 1) dissipate energy while transferring momentum that forces water
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in the direction of wave propagation. In the surfzone the time-averaged wave-driven
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forcing raises water levels near the shoreline [Longuet-Higgins and Stewart, 1964] and,
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in the case of obliquely incident waves, drives alongshore currents [Longuet-Higgins,
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1970]. However, the instantaneous forcing is composed of individual breaking waves
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that often are short-crested owing to the directional spread of the wave field [Longuet-
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Higgins, 1957], with crests from different directions interacting to create alongshore
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varying wave amplitudes. Thus, the wave field is spatially inhomogeneous over wave
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time and spatial scales, with the largest amplitude section of the wave breaking first,
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resulting in a breaking region with a finite alongshore extent (Fig. 1).
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Here, the non-uniform forcing from individual short-crested waves that generates the
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rotational motions associated with dispersion is explored. It has been hypothesized
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[Peregrine, 1998; 1999] that vorticity about a vertical axis (termed "vorticity" here) is
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generated at the ends of short-crested breaking waves. Specifically, it is theorized that
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vorticity is generated by along-crest variations in dissipation dEd / dyc where Ed is the
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breaking dissipation and yc is the along-crest direction. Thus, the maximum vorticity
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generation is predicted to occur at the crest end yc = 0, where adjacent regions of breaking
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and non-breaking wave crest form a large differential in forcing (Fig. 2). The direction of
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the crest end is hypothesized to determine the sign of the generated vorticity, with left-
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(Fig. 2A) and right-handed (Fig. 2B) ends generating positive and negative vorticity,
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respectively [Peregrine, 1998; 1999]. In agreement with this theory, numerical model
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simulations suggest that short-crested breaking waves can generate vorticity [Buhler,
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2000; Johnson and Pattiaratchi, 2006, Sullivan et al., 2007; Bruneau et al., 2011], and
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that vorticity variance and dispersion increase with the number of crest ends [Spydell and
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Feddersen, 2009] (via directional spread). However, the magnitude and structure of this
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vorticity have not been quantified, and the theories have not been tested with ocean
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observations.
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Methods
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The vorticity generated by short-crested breaking waves was measured in an ocean
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surfzone using a novel circular array of current meters. Time-dependent Boussinesq
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simulations of surfzone waves and currents [Feddersen et al., 2011; Clark et al., 2011]
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indicate that a circular-shaped array of current meters reduces wave noise relative to that
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from a rectangular array. An array with 10 acoustic Doppler current meters arranged in a
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10-m diameter circle was deployed for 12 hours on a long, straight ocean beach in Duck,
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NC. The 10-m diameter was large enough to capture many individual crest ends, but not
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so large as to encompass multiple ends at one time. The array was deployed in 1.6-m
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mean water depth on the crest of an alongshore sandbar, and was always within the
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surfzone. The current meter sample volumes were 0.8 m above the sandy seafloor. Waves
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were normally incident and directionally spread, and the wave conditions seaward of the
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surfzone were nearly constant. At the array the significant wave height varied with the
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tidally fluctuating water depth (Fig. 3B), the directional spread varied inversely with
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depth from 16° to 22° (possibly by wave current interaction [Henderson et al., 2006]), the
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mean period was 8 s, and the hourly mean alongshore currents were weak (< 0.08 m/s).
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Video observations were used to identify times when short-crested breaking occurred
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within or near the array. The crest end is defined at the location where the forward
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projecting edge of the wave first intersects the wave trough (plunging wave, where
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dissipation begins), or where the turbulent water on the front face of the wave first
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reaches its maximum extent in front of the wave (spilling wave, where dissipation
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appears to develop fully). A short-crested breaking event occurs at time t0 when the crest
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bisects the array, and the only crests considered are those with a single well-developed
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breaking region within 20 m alongshore of the array.
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The mean (in space) vertical vorticity 𝜔 within the array is estimated using Kelvin’s
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circulation theorem ω = A −1 
∫ u • dl , where A is the area inside the array, u is the
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horizontal velocity vector estimated from the current meters, and l is the closed path
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around the perimeter of the array. The vorticity generated by a short-crested wave
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Δω = ω (t0 + Δt) − ω (t0 ) is the change in vorticity between the observation at time t0
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(when the crest bisects the array) and the observation at Δt later. The observations are
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grouped by the along-crest position yc of the array center relative to the crest end (Fig.
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2C), with yc > 0 in the breaking region and yc < 0 in the non-breaking region of the crest.
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Results and Discussion
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The variation in short-crested vorticity over a tidal cycle (Fig. 3) is estimated from the
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average of waves with 5 ≤ yc ≤ 10 m and 3 ≤ Δt ≤ 18 s. These averaging parameters
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span the maximum Δω (discussed below), and the range of values is used to increase the
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number of waves averaged. The observed Δω is positive for left-handed and negative for
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right-handed wave ends, in agreement with theory [Peregrine, 1998; 1999], and the
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temporal variations in left- and right-handed Δω are similar (Fig. 3A). The Δω
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magnitudes are maximum near low tide and minimum near high tide. At low tide the
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array is farther inside the surfzone, with a larger region of breaking offshore, and at high
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tide the array is near the outer edge of the surfzone where waves begin to break. The
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variation in vorticity may be caused by greater total wave energy dissipated on the
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sandbar at low tide, or Δω may be cross-shore variable with greater Δω in the inner
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surfzone, possibly owing to a longer time history of breaking (see below).
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There were similar numbers of left- and right-handed crests, and the mean wave direction
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was near zero relative to shore normal. Under these conditions short-crested breaking is
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expected to produce vorticity variation, but little or no mean vorticity. Consistent with
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expectations, hourly mean vorticity (not shown) is on average 20% of the vorticity
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standard deviation (Fig. 3A). During low tide individual waves produce Δω with
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magnitude equal to the vorticity standard deviation (Fig. 3A, t = 16 hours), suggesting
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that short-crested breaking is a significant source of vorticity variation in the surfzone.
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The O(0.01 s-1) Δω observed here (Fig. 3A) is similar to the modeled near-surface
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vorticity immediately after a single breaking event in the open ocean [Sullivan et al.,
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2007], and to the modeled vorticity generated near a surfzone crest end [Johnson and
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Pattiaratchi, 2006].
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The structure of short-crested vorticity is explored by examining Δω over a range of Δt
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and yc (Fig. 4). Waves are averaged over low tides (Fig. 3, 6 < t < 8 and 13 < t < 18
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hours) when Δω is greatest, and Δω from left-handed waves is combined with - Δω
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from right-handed waves. Before the short-crested breaking event ( Δt < 0) the Δω are
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noisy with no significant change, whereas after the event ( Δt > 0) there is a rapid increase
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in Δω at all yc ≥ 0 (Fig. 4) with a maximum Δω = 0.018 s-1 at yc = 5 m and Δt = 5 s. For
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yc < 0 (not shown) there is no significant change in Δω owing to the breaking event at
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Δt = 0.
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The generation of vorticity at yc = 10 and 15 m (Fig. 4, blue and black curves) is not
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predicted by theory, and may result from onshore advection of vorticity generated at
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previous stages of breaking or from the shape of the breaking region. For a directionally
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spread wave field on an ocean beach the breaking region of a wave rarely maintains its
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alongshore length, but starts as an initially narrow region of breaking in the outer
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surfzone and spreads alongshore as it propagates shoreward. This spreading can be seen
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in the triangular region of foam left in the wake of a breaking wave (Fig. 1), where the
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foam indicates the region of forcing over the breaking history of the wave. The cross-
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shore integrated forcing is greatest near the alongshore center of the triangle, and
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decreases towards the alongshore ends. It is possible that this alongshore spreading of the
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breaking generates vorticity along the entire breaking crest, except at the center of the
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triangle where the alongshore gradient in cross-shore integrated forcing is zero. Vorticity
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generated by triangular breaking regions is consistent with the increased Δω observed in
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the inner surfzone (low tide, Fig. 3A), where waves have a longer time history of
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breaking.
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The time delay of the initial increase in Δω (Fig. 4), which occurs over 0 < Δt < 5 s at all
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yc > 0, may be caused by near-surface forcing propagating downwards through the water
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column and the roughly 2.5 s for a wave to cross the array. The subsequent slow rise to
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maximum Δω at Δt = 20 s for yc = 10 and 15 m is consistent with onshore advection of
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vorticity generated at previous stages of breaking, or the slow spin-up of larger scale
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vorticity owing to the triangular shape of the breaking region (Fig. 1). The Δω generated
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by a short-crested wave decays over 20 to 60 s, with the most rapid decay at yc = 0, and
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some evidence of persistent Δω over longer times at yc = 5 m. The Δω decay may be
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caused by the advection of the generated vorticity out of the sample region, the reduction
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of Δω through bottom friction, or the reversal of local vorticity by subsequent breaking
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events of opposite sign.
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Short-crested waves that crossed the array in roughly 2.5 s produced vorticity that
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persisted for 40 to 60 s (Fig. 4), much longer than the 8 s mean period of the waves.
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Thus, momentum from waves with relatively high frequencies (0.12 Hz mean frequency)
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is being transferred into rotational motions with much lower frequencies. These lower-
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frequency rotational motions have been indicated as a primary cross-shore dispersion
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mechanism [Spydell and Feddersen, 2009; Clark et al., 2010; Clark et al., 2011] and a
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primary alongshore dispersion mechanism when alongshore currents are weak [Spydell
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and Feddersen, 2009].
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Acknowledgements: We thank Levi Gorrell, Danik Forsman, Christen Rivera-Erick,
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Regina Yopak, and Seth Zippel for their hard work in the field and the staff of the US
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Army Corps of Engineers Field Research Facility, Duck, NC for excellent logistical
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support. Funding was provided by a National Security Science and Engineering Faculty
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Fellowship, the Office of Naval Research, and a Woods Hole Oceanographic Institution
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Postdoctoral Fellowship.
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Fig. 1. Photograph of breaking waves (propagating toward the shore from lower-right to
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upper-left) showing the triangular patches of residual white foam marking the location
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where breaking occurred. As the waves break, they transfer momentum to the water
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column and generate vorticity. The initially small breaking region on the lower right
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expands as the wave moves toward shore on the upper left. This pattern is typical in the
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surfzone, with the shape of the triangle varying with wave conditions.
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breaking force
breaking crest
A left-handed
positve
vorticity
B right-handed
negative
vorticity
residual
foam
crest
end
C array position relative to the crest end
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-15 m
-10 m
-5 m
0m
5m
10 m
15 m
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Fig. 2. Schematic (looking down from above) of negative and positive vorticity
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generated by (A) left- and (B) right-handed ends of breaking waves. Solid black arrows
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indicate the instantaneous forcing (owing to breaking) on the water column in the
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direction of wave propagation, and the curved arrows indicate the direction of fluid
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rotation for the resulting positive (red) and negative (blue) vorticity. (C) Schematic of the
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vorticity array (black circle) position yc relative to the crest-end with yellow arrows
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indicating the direction of wave propagation (left-handed example).
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A
0.02
Δ ω (1/s)
0.01
0
−0.01
left−handed
right−handed
all vort std.
−0.02
B
0.7
1.8
0.6
1.6
0.5
1.4
1.2
6
8
10
12 14
t (hours)
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Hsig (m)
depth (m)
2
0.4
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Fig. 3. (A) Vorticity generated by (red curve) left- and (blue curve) right-handed short-
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crested waves versus time, with vertical bars indicating the error in the mean for each bin,
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and total vorticity standard deviation (green curve) versus time. (B) Water depth (black
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curve) and significant wave height Hsig (magenta curve, 4 times the standard deviation of
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the sea-surface-elevation fluctuations) versus time.
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15
−3
x 10
0m
5m
10 m
15 m
15
Δ ω (1/s)
10
5
0
−5
−60
−40
−20
0
Δt (s)
20
40
60
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Fig. 4. Change in vorticity associated with short-crested breaking versus time at four
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along-crest positions (see legend). Time Δ𝑡 = 0 is when the crest bisects the array.
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Locations -15 to -5 m (outside the breaking crest, Fig. 2C) are not shown because there is
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no significant change in vorticity.
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