Abstract

This study explores Eulerian and Lagrangian circulation during weak winds at two inner-shelf locations off the Southern California coast where the shoreline, shelf, wind, and wave characteristics differ from those in previous studies. In agreement with recent observational studies, wave-driven Eulerian offshore flow just outside the surf zone, referred to as undertow, is a substantial component of the net cross-shore circulation during periods of weak winds. Drifter observations show onshore surface flow, likely due to light onshore winds, and a consistent decrease in onshore velocity of roughly 4 cm s−1 within a few hundred meters of the surf zone. Undertow is examined as a possible explanation for the observed Lagrangian decelerations. Model results suggest that, even when waves are small, undertow can decrease the velocity of shoreward-moving drifters by >2 cm s−1, roughly half the observed deceleration. The coastal boundary condition also has the potential to contribute to the observed decelerations. Subtracting predicted Stokes drift velocities from the Lagrangian drifter observations improves the agreement between the drifter observations and coincident Eulerian ADCP observations.

1. Introduction

When surface gravity wave propagation is viewed in a reference frame rotating with the earth, the Coriolis force destroys the exact quadrature between horizontal and vertical wave orbital velocities. This leads, in the wave-averaged momentum equation, to a Reynolds stress with vertical shear (Hasselmann 1970). For a linear, steady-state model with negligible turbulent mixing, the wave-driven stress generates (for an alongshore uniform domain) a mean Eulerian current, referred to as undertow, exactly equal and opposite to the Lagrangian Stokes (Stokes 1847) drift.

Moored velocity profile data collected outside the surf zone off the Massachusetts and North Carolina coasts show wave-driven undertow is the dominant component of the depth-averaged cross-shelf flow during all wind conditions (Lentz et al. 2008). Both the observed velocities and predicted undertow are greatest (1–3 cm s−1) near the surface and decrease with depth. That surface-intensified velocity profile seaward of the surf zone differs from the parabolic undertow profiles found in the surf zone, where turbulent mixing is significantly enhanced (e.g., Reniers et al. 2004). Wave-driven undertow observations off the Oregon coast, characterized by a steep narrow shelf and a relatively rocky coastline, which contrast the East Coast study sites, are similar to the Lentz et al. (2008) measured and modeled values (Kirincich et al. 2009). However, wave-driven cross-shore circulation off the Oregon coast is not substantial compared to the local wind-driven currents.

In this study, coincidently sampled Eulerian and Lagrangian observations of circulation just beyond the surf zone are analyzed. The addition of Lagrangian data is a significant expansion of the Lentz et al. (2008) and Kirincich et al. (2009) studies, which are purely Eulerian. Study sites are at two locations off the Southern California coast where shoreline, shelf, wind, and wave characteristics differ from those in the aforementioned studies (Fig. 1). The first site, Santa Barbara, lies almost directly east of Point Conception along a south-facing coastline where direct effects of the prevailing northwesterly winds, as well as large swells from the North Pacific, are limited (e.g., Brink and Muench 1986; Winant et al. 2003). The northern Channel Islands are located ~40 km offshore and block nearly all swell from the south. The continental shelf is steep and narrow. The most significant difference between Santa Barbara and the previous study sites is the wave regime and shelf slope combination. In Santa Barbara, locally generated short period wind waves (Dorman and Winant 1995) move over a steep shelf.

Fig. 1.

(a) Mooring locations at the Santa Barbara (SB) and Huntington Beach (HB) sites, located along the Southern California coast. Locations of NDBC buoys are shown with triangles. (b) Trajectories of ~300 drifters deployed offshore of the mooring location (gray circle) at SB. (c) Trajectories of all drifters deployed at HB. In (b),(c) black trajectories are those that moved shoreward of the 5-m (10-m) isobath at SB (HB) during low wind conditions and are used in the deceleration analysis. Gray circles indicate mooring locations. Principal axis coordinate systems are indicated.

Fig. 1.

(a) Mooring locations at the Santa Barbara (SB) and Huntington Beach (HB) sites, located along the Southern California coast. Locations of NDBC buoys are shown with triangles. (b) Trajectories of ~300 drifters deployed offshore of the mooring location (gray circle) at SB. (c) Trajectories of all drifters deployed at HB. In (b),(c) black trajectories are those that moved shoreward of the 5-m (10-m) isobath at SB (HB) during low wind conditions and are used in the deceleration analysis. Gray circles indicate mooring locations. Principal axis coordinate systems are indicated.

The second site, Huntington Beach (located between Los Angeles and San Diego), is characterized by a sandy, straight, southwest-facing coastline and a steep narrow shelf. During summer (austral winter), Huntington Beach is impacted by long-period, small-amplitude Southern Hemisphere swell and small-amplitude, short-period wind waves. The most significant difference between Huntington Beach and previous study sites is the mixed-wave regime and steep coastal bathymetry combination.

The observations are presented in section 2. During weak winds, shoreward-moving drifters in shallow water (<10-m total depth) show a consistent decrease in onshore velocity just outside the surf zone. This primary result is detailed in section 3. Possible explanations for the observed Lagrangian decelerations, including cross-shore variations in undertow and Stokes drift profiles, are discussed in section 4. Conclusions are given in section 5.

2. Observations

Eulerian and Lagrangian ocean current data were collected within 3 km of the offshore edge of the surf zone in water depths mostly ranging from 2 to 20 m off the coasts of Santa Barbara (SB) and Huntington Beach (HB), California (Fig. 1). Eulerian velocity profiles and surface wave information are from bottom-mounted upward-looking acoustic Doppler current profilers (ADCPs) manufactured by RD Instruments (San Diego, California). SB data are from a 600-kHz ADCP located ~0.5 km from the shoreline in a water depth of 10 m (Fig. 1b). Data were collected from 18 November 2007 through 17 November 2008, with quarterly data gaps of a few days owing to instrument recovery and redeployment. Current vectors from the SB ADCP were recorded as 4-min averages in 0.35-m vertical bins (the instrument sampled 90 pings per ensemble at a rate of 0.95 Hz). Significant wave height Hsig, dominant wave period, and direction of the dominant waves were collected every 2 h through bursts of 2400 samples at a rate of 2 Hz (so surface wave frequencies are not aliased) and analyzed using RD Instruments WavesMon v3.05 software. HB data were collected from July through October 2006 as part of the HB06 experiment (e.g., Spydell et al. 2009). HB velocity profiles are from a 1200-kHz ADCP located ~1 km from the shoreline, also in a water depth of 10 m (Fig. 1c). Velocity was recorded as 3-min averages in 0.5-m vertical bins (the instrument sampled 60 pings per ensemble at a rate of 1.0 Hz). An ADCP on a nearby mooring (directly seaward on the 15-m isobath) burst sampled to record dominant wave information.

All ADCP velocity data were processed with bin locations defined relative to the time-varying ocean surface to maximize accuracy in near-surface values. Data within ~2 m of the water surface were discarded because of contamination from side-lobe reflections. The 3- and 4-min-average velocity data were linearly interpolated onto a 20-min time base. To produce “subtidal” frequency velocities, tidal components were removed using T_TIDE (Pawlowicz et al. 2002), and the detided data were low-pass filtered with a half-amplitude cutoff of ⅓ h−1. Along- and cross-shore directions were defined that align with the major and minor principal axis components of the subtidal (low-pass filtered) velocities, respectively (u is positive alongshore equatorward and υ positive onshore, Figs. 1b,c).

The Lagrangian observations were collected with Microstar drifters manufactured by Pacific Gyre (Oceanside, California) that record their position with GPS every 10 min. The drifters have drogues that extend from 0.5 to 1.5 m beneath the surface and follow the horizontal motion of water in this depth range to within 0.1% of the wind speed (1 cm s−1 for 10 m s−1 winds). Details of drifter design, operation, and water-following characteristics are given by Ohlmann et al. (2005). At SB, 12 drifters were deployed once each week for 50 weeks during the ADCP sampling time. Drifters were released from three stations aligned in the cross-shore direction within 2 km of the shoreline (Fig. 1b). At HB, 15 drifters were deployed on 15 days from 14 September through 11 October 2006. HB deployments occurred within 3 km of the shoreline in either cross-shore or grid configurations (Fig. 1c). At both locations drifters were deployed in the morning and retrieved before sunset.

As the focus of this study is on the deceleration of shoreward moving drifters just beyond the surf zone and the possible role of surface gravity waves in the decelerations, the drifter data are conditionally sampled. Only tracks of drifters that begin seaward of the 10-m isobath (gray tracks in Figs. 1b,c) and sample shoreward of the 5- and 10-m isobaths in SB and HB, respectively, are considered. These are the regions where the decelerations occur (section 3a). In addition, only drifter tracks that sample these regions during weak winds are considered, so the influence of waves can be best isolated. Drifter tracks that meet both criteria are the black trajectories in Figs. 1b,c. The conditional sampling scheme filters about 50% of the SB drifter tracks: those deployed inshore of the 10-m isobath (not shown) and those deployed beyond the 10-m isobath that do not move shoreward of the 5-m isobath. All available drifter trajectories are considered in the Eulerian versus Lagrangian velocity comparisons (section 3c). Drifter velocities are computed as a first difference in position.

Spectral wave data from National Data Buoy Center (NDBC) buoys 46216 and 46230 (located 18 and 4 km from the SB and HB ADCP moorings, respectively; Fig. 1) are the basis of the Stokes drift calculations. The NDBC buoys give half-hourly measurements of wave energy and direction in nine frequency bands. These spectral data are then placed within four frequency bands (<0.083; 0.83–0.125; 0.125–0.182; >0.182 s−1) by combining the low frequency observations. Reducing the number of frequency bands reduces the number of required computations in the Lentz et al. (2008) model runs. It also increases degrees of freedom in some of the spectral wave angle regressions (described below) by partitioning the available observations among fewer bins. Reducing the number of frequency bands has negligible influence on the computed Stokes drift quantities because the wave bands that are combined are low frequencies that make relatively small contributions to Stokes drift. Summing over frequency bands, the onshore component of the Lagrangian Stokes drift velocity is (e.g., LeBlond and Mysak 1978)

 
formula

where is significant wave height of the waves in frequency band n; ωn is angular wave frequency (rad s−1); kn is wavenumber (rad m−1); is the direction the waves are traveling toward, measured counterclockwise from the positive y direction; h is water depth; and z is the vertical coordinate, with z = 0 at the water surface.

The procedure for determining at the ADCP mooring locations differs between study sites. At SB, is computed from Hsig measured by the ADCP, colored by normalized spectral energy coincidently observed at NDBC buoy 46216. Here at the ADCP is determined by binning the ADCP wave data by dominant wave frequency (four bins) and computing the mean for each of the wave bands. This scheme is implemented because buoy 46216 is located well offshore and westward of the SB mooring site (Fig. 1a) and does not measure an that is characteristic of the nearshore wave regime. Propagating wave data from buoy 46216 to the SB mooring using conservation of total wave energy results in considerably larger Hsig values than measured by the ADCP.

At HB, NDBC buoy 46230 is located a few kilometers directly offshore of the ADCP mooring (Fig. 1a). Given the close proximity, from buoy 46230 are propagated to the HB ADCP site using conservation of wave energy flux. Refraction in the shoaling transformation is accounted for through relationships between coincidently observed at NDBC buoy 46230 and at an ADCP mooring located ~1 km seaward of the HB study site. The wave angle relationships are developed as follows: Dominant wave information measured by the ADCP is binned by wave frequency (four bins). Within each spectral bin a linear least squares regression between coincidently measured by the ADCP and buoy 46230 is determined. The fits (r2 values range from 0.21 for the 13-s band to 0.56 for the 4-s band) are then used to determine at the ADCP from at buoy 46230, and the change in wave angle is used to account for refraction in the shoaling transformation. This scheme for determining wave angle was not used at SB because observed wave angles at buoy 46216 and the SB ADCP were not correlated (r2 < 0.1) in any of the frequency bands.

Because the study domain is seaward of the surf zone, dissipation due to wave breaking is neglected as in prior studies (e.g., Lentz et al. 2008). To correspond with drifter observations, υst(z) is averaged from 0.5- to 1.5-m depth, the depth range of drifter drogues. This depth-averaged value is subsequently indicated as υst(1 m). Wave quantities and υst(1 m) values are determined hourly.

Wind data were recorded at the Santa Barbara airport, located ~18 km west of SB, and at a mooring on the 20-m isobath ~2 km west of the HB ADCP (not shown). Wind velocity was recorded every 20 min at SB and every 1 h at HB. Gaps in the wind data less than 4 h in duration were filled by linear interpolation, and the hourly wind data were linearly interpolated to the times of the ADCP and drifter velocity observations. Wind stress was computed from the interpolated wind velocity observations following Smith (1988). To isolate wave-driven flow, only ocean velocity data recorded during periods of low winds (wind stress magnitude |τ| < 0.05 N m−2) are considered. This low wind threshold gives similar results to the cutoff |τ| < 0.03 N m−2 used by Lentz et al. (2008) and Kirincich et al. (2009) but allows more data to be retained.

To further isolate the effects of surface wave forcing, Eulerian velocity profiles during low wind and large wave forcing are examined relative to the mean velocity profiles during low wind and small wave forcing, following Lentz et al. (2008) and Kirincich et al. (2009). The observed flow during low wind and small wave conditions is assumed to represent the effects of other processes (e.g., pressure gradient forcing) not correlated with wave and wind forcing, and not directly measured as part of this study. Small-wave and large-wave forcing regimes are based on υst(1 m) as follows. First, mean values of υst(1 m) for the low wind period, defined as , and their associated standard deviations, defined as συst, are calculated from hourly values at each ADCP mooring site (Table 1). Mean quantities are then computed for the case of small (large) waves using times when vst(1 m) is more than 0.5 συst below (above) (Table 1).

Table 1.

Significant wave height and Stokes drift statistics by site (rows) during periods of low winds (|τ| < 0.05 N m−2; column 1), low winds and small waves (column 2), and low winds and large waves (column 3). The mean ±1 standard deviation and sample size n (computed as the number of samples multiplied by the 1-h sampling frequency and converted to days) are indicated. Quantitative definitions for the three regimes are given at the end of section 2. Stokes drift values at an indicated depth of 1 m are depth averaged between 0.5 and 1.5 m beneath the sea surface.

Significant wave height and Stokes drift statistics by site (rows) during periods of low winds (|τ| < 0.05 N m−2; column 1), low winds and small waves (column 2), and low winds and large waves (column 3). The mean ±1 standard deviation and sample size n (computed as the number of samples multiplied by the 1-h sampling frequency and converted to days) are indicated. Quantitative definitions for the three regimes are given at the end of section 2. Stokes drift values at an indicated depth of 1 m are depth averaged between 0.5 and 1.5 m beneath the sea surface.
Significant wave height and Stokes drift statistics by site (rows) during periods of low winds (|τ| < 0.05 N m−2; column 1), low winds and small waves (column 2), and low winds and large waves (column 3). The mean ±1 standard deviation and sample size n (computed as the number of samples multiplied by the 1-h sampling frequency and converted to days) are indicated. Quantitative definitions for the three regimes are given at the end of section 2. Stokes drift values at an indicated depth of 1 m are depth averaged between 0.5 and 1.5 m beneath the sea surface.

3. Results

a. Drifter decelerations

Drifters typically sample the portion of the diurnal cycle characterized by weak onshore winds (Fig. 2) and mostly move shoreward. A consistent decrease in the onshore velocity of shoreward moving drifters approaching the surf zone during low wind conditions is observed at both locations. For drifters deployed at SB beyond the 10-m isobath that move into waters shallower than 5 m, onshore velocity is mostly constant (3–11 cm s−1) between the 12- and 5-m isobaths (Fig. 3a). Once drifters reach water shallower than 4 or 5 m, their onshore velocity consistently decreases. Onshore velocity inshore of the 3-m isobath is always <6 cm s−1. Average onshore velocity of all shoreward moving drifters that meet the sampling criteria decreases from 7 cm s−1 offshore of the 5-m isobath to 3 cm s−1 inshore of the 3-m isobath, a velocity decrease of 4 cm s−1 over a cross-shore distance of roughly 100 m.

Fig. 2.

Wind stress values during times of drifter observations and low wind conditions (black) and before and after drifter sampling times (gray lines): (top) the alongshore component, positive eastward, and (bottom) the across-shore component, positive shoreward, for the (left) SB and (right) HB sites.

Fig. 2.

Wind stress values during times of drifter observations and low wind conditions (black) and before and after drifter sampling times (gray lines): (top) the alongshore component, positive eastward, and (bottom) the across-shore component, positive shoreward, for the (left) SB and (right) HB sites.

Fig. 3.

Onshore component of drifter velocity vs water depth during low wind conditions, for all drifters that move shoreward of the (a) 5-m isobath at SB and (b) 10-m isobath at HB. The last position recorded by a drifter prior to recovery (gray dots) is generally within 50 and 20 m of the surf zone at SB and HB, respectively. To avoid Lagrangian sampling biases, means and 95% confidence intervals (connected black dots and vertical error bars) of onshore velocities are computed at the center of 1-m depth bins from drifter velocities spatially interpolated to those locations. The number of independent observations used in error bar calculations is equal to the number of sampling days to account for correlated drifter velocities on a given calendar day.

Fig. 3.

Onshore component of drifter velocity vs water depth during low wind conditions, for all drifters that move shoreward of the (a) 5-m isobath at SB and (b) 10-m isobath at HB. The last position recorded by a drifter prior to recovery (gray dots) is generally within 50 and 20 m of the surf zone at SB and HB, respectively. To avoid Lagrangian sampling biases, means and 95% confidence intervals (connected black dots and vertical error bars) of onshore velocities are computed at the center of 1-m depth bins from drifter velocities spatially interpolated to those locations. The number of independent observations used in error bar calculations is equal to the number of sampling days to account for correlated drifter velocities on a given calendar day.

For drifters deployed at HB that move onshore of the 10-m isobath during low winds, onshore velocity varies between 15 and −5 cm s−1 (Fig. 3b). The average onshore velocity of those drifters decreases from 7 cm s−1 just beyond the 12-m isobath to ≤4 cm s−1 inshore of the 10-m isobath and remains low to inshore of the 8-m isobath where the drifters were mostly recovered. The observations in the shallowest waters are from late September and mid-October during times of very small waves and thus a narrow surf zone (Fig. 4b).

Fig. 4.

Time series of swell (gray, wave periods > 8 s) and wind wave (black, wave periods from 4–8 s) height at (a) SB and (b) HB. Vertical markers along the time axes identify days of drifter deployments.

Fig. 4.

Time series of swell (gray, wave periods > 8 s) and wind wave (black, wave periods from 4–8 s) height at (a) SB and (b) HB. Vertical markers along the time axes identify days of drifter deployments.

The deceleration of shoreward moving drifters at HB occurs in deeper water, and the depth at which deceleration begins is more variable than at SB. This likely reflects the more energetic and variable wave regime at HB (Fig. 4), where long-period swell produces a surf zone with an outer edge that is deeper and more variable in location than at SB. Some drifters were observed moving offshore within rip currents during times of large breakers. Rip currents may explain the offshore (seaward) drifter velocities just beyond the surf zone at HB (i.e., negative values shown in Fig. 3b). Overall, during low wind conditions, shoreward-moving drifters at HB and SB consistently slow down as they approach the outer edge of the surf zone. In section 4, undertow and Stokes drift are investigated as possible contributors to the observed drifter decelerations.

b. Eulerian flow associated with waves

Observed Eulerian velocity profiles associated with wave forcing at the SB and HB moorings are isolated by subtracting the mean low wind, small wave ADCP velocity profiles (representing any background flow not due to wind or waves) from the mean low wind, large wave profiles, following Lentz et al. (2008). The mean low wind, small wave velocity profile at SB extends from −1.0 (at 2-m depth) to 0.1 cm s−1 (at 7-m depth) and is nearly linear. At HB, the profile extends from −0.4 (2-m depth) to 0.7 cm s−1 (6-m depth) and is slightly concave downward. Both profiles tend toward zero at the bottom. The mean Eulerian flow associated with wave forcing at the SB mooring location is then −0.5 ± 0.5 cm s−1 in the topmost ADCP bin considered (~2-m depth) and decreases approximately linearly with depth to 0 ± 0.5 cm s−1 just above the bottom (Fig. 5a). At the HB mooring, the mean flow associated with wave forcing also has a velocity near −0.5 cm s−1 in the topmost ADCP bin considered (2-m depth) and decreases with depth to 0 cm s−1 below 5 m (Fig. 5b).

Fig. 5.

Large black dots show mean velocity profiles (positive toward shore) computed from ADCP data during low wind conditions, after subtracting the mean low wind, small wave profiles, for (a) SB and (b) HB. Small black dots show the 95% confidence intervals (calculated from standard error of the mean). Thin black lines show corresponding mean profiles obtained from the Lentz et al. (2008) model, forced with observed wave characteristics during times of observations.

Fig. 5.

Large black dots show mean velocity profiles (positive toward shore) computed from ADCP data during low wind conditions, after subtracting the mean low wind, small wave profiles, for (a) SB and (b) HB. Small black dots show the 95% confidence intervals (calculated from standard error of the mean). Thin black lines show corresponding mean profiles obtained from the Lentz et al. (2008) model, forced with observed wave characteristics during times of observations.

Mean theoretical undertow profiles computed from results of the Lentz et al. (2008) undertow model, forced with hourly spectral wave data during the low-wind and large-wave regime, are in qualitative agreement with the ADCP observations (Fig. 5). At SB, mean modeled undertow at 2-m depth is −0.5 cm s−1, the same as the observed velocity associated with wave forcing. At HB, the modeled value (−0.8 cm s−1) is slightly larger than observed. For the time-varying depth-averaged cross-shore velocity associated with waves, the coefficient of determination (r2) between the observations and the Lentz et al. (2008) model results is 0.67 at SB, significant at the 95% confidence level. A statistically significant relationship does not exist at HB.

c. Eulerian and Lagrangian velocity comparisons

The Lagrangian drifter velocity observations include Stokes drift, but the Eulerian ADCP velocities do not. To determine the extent to which the difference between coincidently sampled Lagrangian and Eulerian velocities can be explained by Stokes drift, the velocities are compared with and without subtraction of predicted υst(1 m) from the Lagrangian data. Comparisons are performed with ADCP velocity averaged over the top three depth bins with reliable data during times when drifters sample within 200 and 400 m of the SB and HB ADCPs, respectively. Beyond these distances, correlation coefficients between the drifter and ADCP observations decrease dramatically (not shown).

To avoid Lagrangian sampling biases associated with velocity—because slow moving drifters spend more time “colocated” with the ADCPs—each drifter track is treated as one independent observation in the comparisons. The time during which a drifter is within the required distance of an ADCP mooring is first determined. Next, the average drifter velocity for that time is computed and compared with the corresponding time average velocity computed with the 4-min (3-min), unfiltered, not detided ADCP velocity record at SB (HB). Comparisons are made with 250 drifter tracks sampled on 42 days at SB and with 71 drifter tracks sampled on 12 days at HB.

Root-mean-square differences between the Lagrangian drifter and Eulerian ADCP measurements for along- and cross-shore velocity components at both SB and HB are between 3.0 and 3.3 cm s−1 (Fig. 6). Mean differences are near 1 cm s−1 in the alongshore direction but >2 cm s−1 in the cross-shore direction. The drifters show greater onshore and equatorward (alongshore) flow than the ADCPs. This result is consistent with the typical direction of wave propagation and suggests Stokes drift does contribute to the observed difference between Eulerian and Lagrangian velocities.

Fig. 6.

Coincidently sampled Eulerian and Lagrangian velocities for each time a drifter was within (a),(b) 200 m of the SB ADCP or (c),(d) 400 m of the HB ADCP. (a),(c) Alongshore velocity, positive equatorward. (b),(d) Cross-shore velocity, positive onshore. Solid and dotted lines are best (least squares) fit and 1:1 lines, respectively. Rms and mean difference values have units of cm s−1.

Fig. 6.

Coincidently sampled Eulerian and Lagrangian velocities for each time a drifter was within (a),(b) 200 m of the SB ADCP or (c),(d) 400 m of the HB ADCP. (a),(c) Alongshore velocity, positive equatorward. (b),(d) Cross-shore velocity, positive onshore. Solid and dotted lines are best (least squares) fit and 1:1 lines, respectively. Rms and mean difference values have units of cm s−1.

Mean velocity differences decrease by 0.4 and 0.5 cm s−1 (to values of 1.8 and 1.6 cm s−1) in the cross-shore direction at SB and HB, respectively, when predicted υst(1 m) values are removed from the Lagrangian velocities. Subtracting υst(1 m) changes the alongshore differences by 0.3 and 0.4 cm s−1 at SB and HB, respectively. Differences in sampling location between instruments, both horizontally and vertically, likely explain the remaining mean difference in velocity components. When the ADCP velocities are linearly extrapolated upward from the top three ADCP bins to the drifter drogue depth, mean differences at HB further decrease by 0.9 and 1.3 cm s−1 (to values of 0.3 and −0.1 cm s−1) in the along- and cross-shore directions, respectively. Upward extrapolation further decreases the SB differences only slightly (0.1–0.2 cm s−1). The fact that the ADCP and Lagrangian velocity measurements are in close agreement after accounting for υst(1 m) suggests 1) the drifters follow water parcels well and 2) the wave forcing estimates that are subsequently used to drive the Lentz et al. (2008) model are accurate.

4. Discussion

a. Comparison with previous studies of wave-forced flow

The observed circulation just outside the surf zone at the SB and HB mooring locations agrees with the model of wave-driven flow used in earlier studies even though the wave forcing is substantially weaker than observed by Lentz et al. (2008) and Kirincich et al. (2009) off the Oregon, Massachusetts, and North Carolina coasts. Background (low wind and small wave) velocity profiles removed to isolate wave-driven flow are computed by Lentz et al. (2008) and Kirincich et al. (2009) as averages during times when Hsig < 0.75 and 0.70 m, respectively. The background flow profile removed from SB observations corresponds to times of Hsig < 0.38 m, and SB observations are characterized by = 0.8 m in the large-wave regime (Table 1). The smaller water velocities associated with waves at SB are consistent with the much weaker wave forcing. Undertow predicted by the Lentz et al. (2008) model agrees well with observed velocities associated with the weak wave forcing environment at SB (Fig. 5a).

At sites with mixed short and long period waves, such as HB, it is better to define the strength of wave forcing based on Stokes drift than on significant wave height. In the Lentz et al. study, short period wind waves dominate and wave forcing is defined based on Hsig. In our study, wave forcing is defined based on υst(z) because of the mixed wave regime at HB, where short period wind waves and long period swell both contribute to Hsig. If the wave regime were defined based on Hsig, then υst(z) and the observed mean velocity profile during the small-wave regime at HB would be greater than during the large-wave regime. This is due to the long period southerly swell of varying amplitude that contributes to Hsig at HB (Fig. 4). Such waves are characterized by large phase speed c. The depth-averaged Stokes drift and modeled undertow velocity offshore of the surf zone are both proportional to c−1 (for constant Hsig/h; e.g., Lentz et al. 2008). Long-period swell with a given Hsig contributes little to Stokes drift or undertow when compared to a short-period wind wave with the same Hsig. At the 10-m isobath where the ADCP is located, the phase speed of a 20-s wave is twice the phase speed of a 3-s wave and the 20-s wave contributes only half as much to the depth-averaged Stokes drift. Therefore, for sites with mixed wave regimes, wave forcing is better characterized by vst(z), which takes into account both significant wave height and wave period, than by Hsig. Accurate prediction of Stokes drift and undertow requires consideration of amplitude for the steep, short period (≤8 s) waves that have the greatest contribution to υst(z) and the undertow, not just for the dominant waves.

b. Could surface waves contribute to the drifter deceleration?

Because the undertow profiles are surface intensified and directed offshore (e.g., Fig. 5), undertow outside the surf zone could contribute to drifter deceleration. At first glance, the Eulerian velocities associated with wave forcing at the SB and HB ADCP locations (mean < 1 cm s−1) seem too small to explain the observed drifter decelerations of several centimeters per second (section 3b). However, the ADCP observations are at the 10-m isobath, and the observed drifter decelerations mainly occur in shallower water. To understand the influence of wave forcing on the observed drifter decelerations, the Lentz et al. (2008) model is used to predict υst(1 m) and the undertow (also averaged between 0.5 and 1.5 m beneath the surface) for the water depths where the observed decelerations occur (Fig. 3).

Mean observed wave parameters at SB and HB during the large wave forcing regimes are used to force the Lentz et al. (2008) undertow model outside the surf zone for h < 15 m. The spectral wave parameters from the ADCP locations are propagated into shallower water by assuming conservation of wave energy flux. Dissipation due to breaking waves is ignored, consistent with previous studies (e.g., Lentz et al. 2008) as the study domain is outside the surf zone and only times of weak winds are considered. Both a constant sloping bottom and realistic bathymetry are considered (Figs. 7b,e). Eddy viscosity in the model is vertically constant at 10−4 m2 s−1, a “small” value within the range of eddy viscosities considered by Lentz et al. (2008). However, the modeled drifter decelerations are not qualitatively sensitive to the choice of eddy viscosity magnitude over a wide range (10−5–10−1 m2 s−1) or to eddy viscosity profile shape.

Fig. 7.

Predicted Lagrangian cross-shore drifter velocity (positive onshore) as a function of cross-shore distance for (a) SB and (d) HB. The predicted Lagrangian velocity is the sum of the predicted Eulerian undertow velocity and the estimated Stokes drift velocity, averaged from 0.5- to 1.5-m depth. Eulerian undertow velocity is calculated from the Lentz et al. (2008) model using realistic bathymetry (black curves) and a constant sloping bottom (gray lines) for (b) SB and (e) HB. Model forcing is with observed mean large wave conditions for the different wave bands (labeled by period) during light winds for (c) SB and (f) HB. Model eddy viscosity is 10−4 m2 s−1. The vertical dashed line represents the outer edge of the surf zone.

Fig. 7.

Predicted Lagrangian cross-shore drifter velocity (positive onshore) as a function of cross-shore distance for (a) SB and (d) HB. The predicted Lagrangian velocity is the sum of the predicted Eulerian undertow velocity and the estimated Stokes drift velocity, averaged from 0.5- to 1.5-m depth. Eulerian undertow velocity is calculated from the Lentz et al. (2008) model using realistic bathymetry (black curves) and a constant sloping bottom (gray lines) for (b) SB and (e) HB. Model forcing is with observed mean large wave conditions for the different wave bands (labeled by period) during light winds for (c) SB and (f) HB. Model eddy viscosity is 10−4 m2 s−1. The vertical dashed line represents the outer edge of the surf zone.

In the deeper part of the model domain (h ≥ 5 m), the modeled Stokes drift and undertow cancel to produce no net cross-shore wave-driven Lagrangian flow within the depth range from 0.5 to 1.5 m covered by drifter drogues (Figs. 7a,d). Onshore of the 5-m isobath at SB, the undertow overcomes the Stokes drift, resulting in a net offshore Lagrangian velocity that increases to 2 cm s−1 as h decreases to 3 m (Figs. 7a,b). The mean observed onshore drifter velocity at SB decreases by a comparable value (4 cm s−1; Fig. 3).

The modeled Lagrangian velocity change at SB is forced by cross-shore variations in bathymetry that alter the shape of both the undertow profile and the Stokes drift profile within the 0.5–1.5-m depth range. The modeled undertow profile transitions from a Hasselmann (1970) profile offshore (Fig. 8a), where the undertow is equal and opposite to the Stokes drift except in the surface and bottom wave-driven boundary layers, to a more parabolic profile near the surf zone (Fig. 8b), as in Figs. 1 and 3a of Lentz et al. (2008). The Stokes drift profile is more vertically uniform in shallower water (Figs. 8a,b). This simple process model suggests the combined effect of these transitions in the wave-driven flow plays a primary role in the observed drifter decelerations at SB.

Fig. 8.

Velocity profiles from the model runs in Fig. 7 with realistic bathymetry. Vertical profiles of cross-shore Stokes drift υst(z) (dotted–dashed), predicted undertow (dashed), and their sum (solid; the estimated Lagrangian velocity) for locations (a) 0.8 and (b) 0.23 km from the Santa Barbara shore and (c) 0.8 and (d) 0.27 km from the Huntington Beach shore. The shallow profile at each site is located ~70 m outside the surf zone.

Fig. 8.

Velocity profiles from the model runs in Fig. 7 with realistic bathymetry. Vertical profiles of cross-shore Stokes drift υst(z) (dotted–dashed), predicted undertow (dashed), and their sum (solid; the estimated Lagrangian velocity) for locations (a) 0.8 and (b) 0.23 km from the Santa Barbara shore and (c) 0.8 and (d) 0.27 km from the Huntington Beach shore. The shallow profile at each site is located ~70 m outside the surf zone.

The modeled Lagrangian velocity change at HB is slightly larger and more abrupt than at SB, though again not as large as the observed change (Figs. 7a,d). The sum of υst(1 m) and undertow (i.e., the predicted Lagrangian velocity) at HB changes by nearly 3 cm s−1 as h decreases from 5 to 3 m (Figs. 7d,e). Mean observed velocities decrease from 6 to 2 cm s−1, a slightly larger decrease than the model shows. Transient rip currents may contribute to the larger drifter decelerations at HB; rip currents explain occasional offshore drifter velocities observed at HB just beyond the surf zone (section 3a). Rip currents are not included in the Lentz et al. model. The change in modeled velocity profiles with water depth at HB is qualitatively similar to the change at SB, in spite of different bathymetry and wave forcing. The insensitivity of the modeled decelerations to the exact bathymetry and wave forcing suggests undertow and Stokes drift do contribute to drifter decelerations at HB, though other factors such as rip currents enhance the deceleration.

The model results suggest the observed drifter decelerations arise partly from combined changes in υ st(z) and undertow profile shapes as the water becomes shallow. As discussed in detail by Lentz et al. (2008), cross-shore variations in both water depth and eddy viscosity can contribute to changes in undertow profile shape through the nondimensional ratio δE/h, where δE is an Ekman boundary layer depth scale related to eddy viscosity. For h > 5 m at SB and HB, modeled vertical profiles of υst(z) (dashed–dotted curves in Figs. 8a,c) and undertow (dashed curves) are nearly equal and opposite at each depth in the water column. This results in a negligible net wave-driven flow in a Lagrangian frame (solid curves in Figs. 8a,c). The balance breaks down in the thin top and bottom wave boundary layers due to wave-induced stresses. The relatively large shear at the bottom boundary is probably due to wave streaming in the model (e.g., Longuet-Higgins 1953; Xu and Bowen 1994; Zou et al. 2006; Lentz et al. 2008). As the ratio δE/h increases, the curvature of the undertow profile changes sign, and the profile becomes parabolic above the bottom boundary layer (dashed curves in Figs. 8b,d; see also Lentz et al. 2008, their Fig. 3a). In very shallow water (a few meters deep), the middepth undertow maximum influences the 0.5 to 1.5 m depth range. Because the Stokes drift profile becomes more vertically uniform in shallow water (dotted–dash curves in Figs. 8b,d), the sum of the undertow and Stokes drift profiles is offshore in the upper water column, yielding net offshore wave-driven Lagrangian velocities in the 0.5–1.5-m depth range, consistent with the observed drifter decelerations (solid lines in Figs. 8b,d).

Although the correct formulation for the surface boundary layer condition in the model is uncertain and the boundary condition sets the slope of the undertow profile at the surface (Lentz et al. 2008), that uncertainty does not substantially affect the predicted drifter deceleration near the surf zone. The surface boundary condition depends on the choice of eddy viscosity, and the maximum offshore Lagrangian velocities just outside the surf zone within the 0.5–1.5-m depth range (near 3 cm s−1; Figs. 7a,c) are not sensitive to the choice of eddy viscosity or its cross-shore structure. This is because the shallow water depth near the surf zone leads to a parabolic undertow profile for an extensive range of eddy viscosities (10−5–10−1 m2 s−1, not shown). In deeper water the predicted undertow is sensitive to the eddy viscosity, in agreement with Lentz et al. (2008). As the eddy viscosity increases to 10−3 m2 s−1, the undertow profile in deep water becomes parabolic and resembles the shallow water undertow profile, leading to a net offshore Lagrangian velocity near the surface (not shown). Depth-dependent eddy viscosity profiles with bottom roughness heights between 1 and 10 cm (e.g., Lentz 1995) give qualitatively similar results with slightly weaker undertow velocities in the 0.5–1.5-m depth range.

c. Initial onshore motion of drifters

Ideally, both the initial onshore motion of the drifters and their consistent deceleration just outside the surf zone would be explained, perhaps by considering momentum balances to identify dominant forcings. The drifter observations were collected as a part of studies focused on dispersion and the fate and transport of pollutants, not on dynamical balances. Measurements of horizontal pressure, density gradients, or gradients in wind stress on the spatial scales necessary for determining the momentum balance during each drifter deployment are not available. Plausible reasons for the initial onshore motion of the drifters include flows driven by along-shelf pressure gradients, winds, or wave–current interactions. The most likely reason is cross-shore wind forcing (e.g., Tilburg 2003; Fewings et al. 2008). Although only times of weak winds are considered, the mean onshore wind velocity during times of drifter observations is not zero but 2 ± 2 m s−1 for both SB and HB (Fig. 2). The mean alongshore components are equatorward (2 ± 3 m s−1 and 2 ± 2 m s−1, respectively), the wrong direction to cause onshore surface Ekman transport. Estimating the onshore surface current velocity as 2%–3% of the onshore wind speed (e.g., Ralph and Niiler 1999) gives predicted wind-driven surface velocities of 5–7 cm s−1 and 4–6 cm s−1, respectively. The mean observed onshore drifter velocities are indeed near 6 cm s−1 at both SB and HB. Though all drifters did not always move shoreward during weak wind periods, light onshore winds likely explain the initial onshore drifter motion.

d. Other possible contributors to drifter deceleration

The coastal boundary condition—no cross-shore flow at the coast—may play a role in the decelerations (e.g., Austin and Lentz 2002; Tilburg 2003). However, the way in which the coastal boundary influences circulation on the 100-m cross-shelf scale of the observed decelerations is unclear. Cross-shore pressure gradients (not resolved by these observations) may contribute to changing the cross-shore flow structure. When onshore winds force flows over the inner shelf, the result can be a surface layer moving onshore, an overturning circulation, and a lower layer moving offshore (Tilburg 2003; Fewings et al. 2008). As the drifters move into very shallow water, it is possible that the bottoms of their drogues extend into an offshore-moving bottom layer. In this way, the coastal boundary condition could act to decelerate onshore drifter motion, possibly explaining the component of deceleration not attributable to waves.

If cross-shore wind causes an onshore surface flow of ~6 cm s−1 in the deeper part of the study area (section 4c), that onshore flow has to decrease to zero approaching the coast. The coastal boundary condition must lead to a decrease in drifter velocity even if waves are not present. The horizontal scale over which that deceleration occurs will depend on the (unknown) eddy viscosity (Lentz 1995). Nevertheless, the Lentz et al. (2008) process model suggests wave-driven undertow and Stokes drift can also contribute to the observed deceleration of drifters as they approach the surf zone.

The observed drifter decelerations just beyond the surf zone are likely not directly driven by wind. The wind forcing likely does not change substantially over the 100-m spatial scale of the decelerations. Furthermore, the observed decelerations mostly occur during midday or early afternoon, times of constant or increasing onshore wind velocity associated with the diurnal sea breeze (Fig. 2).

Drifters may experience a small (1–2 cm s−1 or less) slip through waves (e.g., Ohlmann et al. 2005; Poulain et al. 2009). However, slip should not change substantially over the 100-m cross-shore length scale considered here and thus should have minimal influence on the observed decelerations. Drifter slip is a possible explanation for the −0.1 to 1.6 cm s−1 discrepancy between the ADCP and drifter velocities that remains after accounting for υst(1 m) and the difference in depth between the measurements.

Shoreward-moving drifters off the Oregon coast also show deceleration to near-zero onshore velocity, though on longer time scales (days to weeks) in much deeper water (hundreds of meters, Austin and Barth 2002). The Oregon drifters are drogued much deeper (10–20 m beneath the surface) than the drifters in this study and generally terminate their onshore motion near the 200-m isobath, located tens of kilometers from the surf zone. The Oregon observations focus on large-scale drifter displacements (tens to hundreds of kilometers) during pronounced downwelling circulation and are likely influenced by very different physics than the drifter observations presented here. The deceleration of the Oregon drifters is attributed to turbulence-induced shutdown of the cross-shelf Ekman transport onshore of the downwelling front (Austin and Barth 2002). The turbulence is due to wind-driven mixing and low density stratification, not processes involving surface gravity waves. Wind-driven turbulent shutdown of the cross-shelf circulation cannot explain the drifter decelerations in our study, where strong winds are absent. A change in cross-shelf circulation caused by increased turbulent mixing driven by processes other than wind (e.g., waves) could contribute to the drifter decelerations. However, as stated above, the modeled drifter decelerations in Figs. 7 and 8 are insensitive to the cross-shelf structure of the turbulent mixing.

Recent work suggests a strong cross-shelf density gradient can also prevent transport between the inner shelf and surf zone. An elevated band of near-surface chlorophyll α (Chl), sampled on 12 October 2006 at the HB study site, moves in the onshore direction but does not enter the surf zone (Omand et al. 2011). Near-surface surf zone water warmer than near-surface water just beyond the surf zone sets up a density barrier to onshore transport into the surf zone. Warm water from a nearby marsh that drains into the surf zone during ebb tide plays a key role in setting up the temperature gradient and is given as a possible reason that the intense surface Chl patch did not enter the surf zone (Omand et al. 2011). Such a density gradient could also contribute to the observed drifter decelerations (there are no HB drifter observations on 12 October). However, it is also possible that the wave-driven undertow plays a role in preventing the Chl patch from moving into the surf zone. The Chl study shows a single instance of deceleration in near-surface onshore motion just beyond the surf zone during an ebbing tide. The drifter observations show similar decelerations during tidal phases other than ebb, and at a location other than HB. Perhaps the wave-forced undertow, coastal boundary condition, and cross-shore temperature gradient all play a role in the barrier to surf zone entry observed by Omand et al. (2011) with Chl as a tracer. Existence of the cross-shore temperature gradient at HB on 12 October could be the reason why the onshore motion of the Chl patch actually stops, whereas the drifters typically slow but do not stop moving entirely.

5. Conclusions and summary

This study explores the cross-shelf circulation outside the surf zone in Southern California with both Eulerian and Lagrangian observations. During weak wind forcing, mean observed Eulerian velocities associated with wave forcing at the 10-m isobath are <1 cm s−1. The velocities reach ~2 cm s−1 near the surface during the largest wave event observed (Hsig = 1.2 m). Recent studies suggest that wave-driven undertow can be a significant part of the cross-shore Eulerian circulation just beyond the surf zone (Lentz et al. 2008; Kirincich et al. 2009). The observed velocities, though small, agree with predicted undertow magnitudes and reflect weak wave forcing (both smaller wave amplitudes and longer wave periods) compared with past studies.

Due to the dependence of the Stokes drift velocity profile υst(z) on wave period, short-period wind waves with amplitude < Hsig can have a greater contribution to Stokes drift than long-period swell with amplitude Hsig. Such is the case at HB, where υst(z) and predicted undertow during the small-wave regime are greater than during the large-wave regime, when wave regime is defined solely on the basis of Hsig. At study sites with mixed long and shortperiod waves, such as the HB site, the strength of wave forcing should be characterized by Stokes drift rather than significant wave height.

The Lagrangian velocity of shoreward moving drifters during times of weak wind decreases by roughly 4 cm s−1 just beyond the outer edge of the surf zone at two southern California locations. Predicted Lagrangian velocities based on the Lentz et al. (2008) wave-forced model suggest the depth-dependent wave-driven undertow and Stokes drift explain roughly half of the observed deceleration. The deceleration in the model comes from changes in the relative shapes of the Stokes drift and the wave-driven Eulerian return flow (undertow) profiles. The changes arise from the decrease in water depth relative to turbulent boundary layer depth and from the Stokes drift profile being more vertically uniform in shallower water. The boundary condition of zero cross-shore flow at the coast likely contributes to the remainder of the drifter deceleration. The deceleration of near-surface drifters approaching the surf zone suggests that surface waves can prevent material—larvae, harmful algae, or pollutants—in the upper water column on the inner shelf from being transported into the surf zone.

These new Lagrangian observations of deceleration outside the surf zone emphasize that cross-shore transport to and from the surf zone, important for a variety of applied problems, is not well understood. An observational experiment focused on cross-shore transport just beyond the surf zone at a site with substantially stronger wave forcing would be a logical next step toward understanding how wave-driven undertow influences cross-shore transport. Such an experiment should include Eulerian current profile and wave observations in water depths where the observed drifter decelerations occur; coincident measurements of wind, pressure and density gradients, and turbulent mixing profiles; use of drifters drogued at various depths; and perhaps dye releases. Complete dynamical balances that include the role of waves could then be assessed as a function of cross-shore distance in the region of observed drifter decelerations.

Acknowledgments

Steve Lentz provided his undertow model and associated Matlab codes and very helpful comments on the manuscript. Marlene Noble provided the HB ADCP and wind data, and Libe Washburn provided the SB ADCP data. David Salazar and Kirk Ireson are acknowledged for their tireless efforts in the field. Funding for this work is from the U.S. Department of Interior (M05AC12301), NOAA through the University of California (04-078.02SB), the state of California and Johnson Ohana Charitable Foundation through Heal the Ocean (SB080078), and the National Science Foundation (OCE-0957948).

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