a. Field deployment and data acquisition

Current meters, sonar altimeters, and a pressure gauge were mounted on two frames deployed in the surf zone near the Scripps Institution of Oceanography pier, on the southern California coast. One frame (Fig. 1, top right) contained a Marsh–McBirney biaxial electromagnetic current meter (EMC1) with a 4-cm-diameter spherical probe (Aubrey and Trowbridge 1985; Guza et al. 1988), 3 SonTek acoustic Doppler OCEAN probes (5-MHz transmitter; Cabrera et al. 1987; Lohrmann et al. 1994, 1995; Voulgaris and Trowbridge 1998), and a sonar altimeter (ALT1; Gallagher et al. 1996). Two of the acoustic Doppler sensors (AD2D and AD4D) were pointed down, and one was rotated from upward (AD3U) to downward (AD3D) looking during the deployment. A Setra pressure gauge (PRES) was deployed adjacent to a frame leg. A second frame (Fig. 1, top left) displaced about 5 m alongshore from the first frame contained a downward-looking SonTek acoustic Doppler OCEAN probe (AD5D), a sonar altimeter (ALT2), and a MAVS acoustic travel time current meter (ATT1; Williams et al. 1987). Cables connected the sensors to shore-based data acquisition computers and power supplies.

The sensing volumes of EMC1, AD4D, AD3U, ATT1, and AD5D were approximately 75 cm above the seafloor, and the sensing volumes of AD3D and AD2D were approximately 40 and 25 cm above the seafloor, respectively (Fig. 1). The sensors were aligned (±2°) to the frames before deployment, and the frames were aligned (±5°) with the shoreline by sighting along the cross bars with a handheld compass.

Data were acquired for 3072-s (51.2 min) periods every hour for 2 weeks during November 1999. All samples from both frames were controlled by a common shore-based clock. The instruments were deployed in water depths that ranged from 75 to 275 cm owing primarily to tidal fluctuations. Smaller depth changes caused by erosion and accretion occurred over several days (±10 cm) and over tidal periods (±1 cm). Significant wave heights (4 times the standard deviation of sea surface elevation fluctuations) ranged from 37 to 132 cm. The deployment location was in the surf zone most of the time (Fig. 1), and wave heights often were limited by breaking. The ratio γ of significant wave height (H_sig) to water depth (h) ranged from 0.21 to 0.64. The frequency f_p of the power spectral primary peak ranged from 0.055 to 0.160 Hz. Mean wave directions ranged from 0° to 15° relative to shore normal. Maximum 51.2-min mean cross-shore (U), alongshore (V), and vertical (W) currents were 20, 40, and 5 cm s⁻¹, respectively. Instantaneous horizontal velocities greater than 300 cm s⁻¹ were observed.

b. Current meters and data reduction

The electromagnetic current meter measures the cross-shore and alongshore velocity in a volume within approximately 1 diameter (4 cm) of the spherical probe. Laboratory studies suggest the spherical electromagnetic sensors may be sensitive to free stream turbulence and wakes behind the probes (Aubrey and Trowbridge 1985). However, field studies have shown no evidence of large distortions owing to the complex flow field, although substantial errors in velocity measurements may occur when the sensor is near the free surface or seabed (Guza et al. 1988). Antialiasing filters in the electromagnetic current meter attenuated EMC1 signal levels above about 1.5 Hz.

The acoustic travel time current meter measures the average cross-shore, alongshore, and vertical velocity along the 10-cm-long acoustic path between two 12-cm diameter rings separated 7 cm in the vertical. Previous studies have shown these current meters to be accurate, even at low flow speeds (Williams et al. 1987). The acoustic travel time sensor (ATT1) had a maximum sample rate of 4 Hz.

Acoustic Doppler current meters transmit short acoustic pulses that are scattered back by reflectors in the water within the sample volume. For the parameters used here, the acoustic Doppler current meters measure the velocity within a cylindrical sample volume approximately 1.8-cm long and 1.2-cm diameter centered about 18 cm from the transducer. Using information about the instrument orientation and measurements along three beams, the average phase differences between several successive returns are converted into cross-shore, alongshore, and vertical velocities (Lhermitte and Serafin 1984; Cabrera et al. 1987; Brumley et al. 1991; Lhermitte and Lemmin 1994; Zedel et al. 1996; Voulgaris and Trowbridge 1998; and references therein). Bubbles and suspended sediment in the surf zone are strong reflectors, and the signal-to-noise ratio of the backscattered acoustic pulses usually is high. In contrast, the electromagnetic and acoustic travel time current meters do not require scatterers, and therefore would work equally well in clear water. The acoustic Doppler current meters, the altimeters, and the pressure gauge were sampled at rates ranging from 2 to 16 Hz.

Rapidly moving particles within the sample volume can result in successive returns from different scatterers, leading to inaccurate velocity estimates (Cabrera et al. 1987; Voulgaris and Trowbridge 1998). Furthermore, excessive scatterers (e.g., bubbles) near the sample volume can reflect sidelobe energy resulting in noisy velocity estimates. For example, 16-Hz velocity samples in the surf zone (H_sig = 80 cm, h = 160 cm) are noisy (Fig. 2a, dotted curve), probably owing to bubbles from breaking waves, and the correlation (not shown) between successive returns is low.

The acoustic Doppler current meters report the correlation along the three beams, and thus postprocessing can identify potentially inaccurate measurements. The size of statistical fluctuations is approximately inversely proportional to the square root of the number of pulses per sample (i.e., proportional to the square root of the sample frequency s_f; Jenkins and Watts 1968). The correlation threshold used here was 0.3 + 0.4 (s_f/25)^1/2, which decreases as (s_f)^1/2 from the recommended (SonTek 1995) values of 0.7 for s_f = 25 Hz to 0.3 for mean currents (i.e., s_f = 0 Hz). Sequences of samples less than 1-s duration that fell below the threshold were replaced with values linearly interpolated between velocities before and after the incoherent sequence. Sequences of incoherent values longer than 1-s duration were replaced with a 1-s running mean of the values (Fig. 2a, solid curve). The amount of data with low correlations was not dependent on sample rates (not shown) nor on velocity values (e.g., for 0 < time < 20 s in Figs. 2a and 2b, the absolute value of the velocity varied from 0 to 300 cm s⁻¹, but there were no low correlation values, and AD4D is similar to EMC1). The percent of values with low correlations ranged from 0% for waves outside the surf zone (no bubbles) to about 2% for the downward-looking acoustic Doppler sensors for the largest breaking waves (γ > 0.45), which probably injected the most bubbles into the water column. The upward-looking acoustic Doppler (AD3U) could remain operational relatively closer to the surface than the downward-looking sensors because the transducer is submerged whenever the sensing volume (18 cm above the upward-looking transducer) is submerged, and as many as 8% of the values from AD3U were below the correlation threshold for the largest breaking waves.

Sensors sometimes were not submerged at low tide or in the wave troughs. The strength of the backscattered acoustic signal along each beam reported by the acoustic Doppler current meters was used to determine when sensors were out of the water. To avoid near-surface observations where the performance of all the current meters used here may be degraded, if more than 10 samples in a 51.2-min acoustic Doppler record had low signal-to-noise ratios, the time series from the acoustic Doppler, and from acoustic travel time and electromagnetic sensors at the same elevation, were discarded for that 51.2-min period.

The acoustic pulses emitted by the Doppler current meters also can be used to detect the presence of a boundary. After each 51.2-min data collection, the downward-looking acoustic Doppler current meters were used to estimate the distance to the seafloor every 3 s for 6.4 min.

a. Current meter comparisons

Horizontal velocities (corrected for low correlations as described above) observed in the surf zone at 16 Hz with the acoustic Doppler current meters agree with nearby electromagnetic current meter measurements, although some spikes remain in the acoustic Doppler time series (Fig. 2b, solid curve). When the sensors were seaward of the surf zone, differences between velocity time series obtained with acoustic Doppler and electromagnetic current meters are small (Fig. 2c, H_sig = 50 cm, h = 215 cm).

Energy density spectra of cross-shore velocity measured 75 cm above the seafloor with acoustic Doppler and electromagnetic current meters are similar both within the surf zone (H_sig = 80 cm, h = 160 cm, Fig. 3a) and seaward of the surf zone (H_sig = 50 cm, h = 215 cm, Fig. 3b). The horizontal velocity noise floor of the acoustic Doppler sensors within the surf zone was about 10 (cm s⁻¹)²/Hz, and was reached for frequencies above about 4 Hz, similar to the noise floor for the electromagnetic current meter (which had a 2-Hz antialiasing filter) at 1.5 Hz. The horizontal velocity noise floor seaward of the surf zone was lower, approximately 3 (EMC1) to 7 (AD3U, AD4D, AD5D) (cm s⁻¹)²/Hz, and was reached for frequencies above 1 Hz (not shown because the Nyquist frequency was 1 Hz for data shown in Fig. 3b). In both cases, noise levels for vertical velocities measured by the acoustic sensors were about one order of magnitude lower. The noise floors for the surf zone observations are consistent with theory (Cabrera et al. 1987; Brumley et al. 1991) and with laboratory results (Voulgaris and Trowbridge 1998). Occasional spikes in the ATT1 time series caused by a malfunctioning circuit resulted in high noise levels above about 0.3 Hz. For all conditions encountered the coherence between cross-shore velocity time series measured with neighboring sensors was close to 1.0 for frequencies below those affected by the noise floor (the spikes in ATT1 resulted in reduced coherences), and phase differences were less than a few degrees.

The total horizontal (U² + V²) velocity variances in the wind–wave frequency band (0.05 ≤ f ≤ 0.30 Hz) determined from time series obtained with the different current meters usually differ by less than 10% (Fig. 4a). The relationship between horizontal velocity variances U²_z1 and U²_z2 for linear waves with radian frequency ω = 2πf at elevations z₁ and z₂ above the seafloor is (e.g., Mei 1983)

View Expanded

where the wavenumber k is given by the dispersion relationship

ω²gkkh,(2)

and g is gravitational acceleration. Thus, linear theory predicts that wave-orbital horizontal velocities decrease only slightly over the vertical in these shallow depths, with a larger attenuation for high frequencies. The roughly 5% decrease in horizontal velocity variance between z₂ = 75 and z₁ = 25 cm above the seafloor is accounted for in Fig. 4 by using (1) and (2) to increase spectral levels to those 75 cm above the seafloor before integrating over the wind–wave frequency band. There is some scatter and a bias toward overestimates in the variances from the electromagnetic current meter for the strongest flows (Fig. 4a, cf. diamonds with the diagonal line), consistent with previous field studies (Guza et al. 1988). Variances estimated with upward- (AD3U) and downward- (AD4D) looking acoustic Doppler current meters are nearly equal at low flows (Fig. 4a, cf. filled squares with diagonal line), with slightly higher variances estimated with the upward-looking sensor for the most energetic flows. Relative to the downward-looking acoustic Doppler (AD4D) measurements 75 cm above the seafloor, the acoustic travel time current meter (ATT1) has a slight (less than 10%) bias toward underestimation of horizontal variance, as do the acoustic Doppler sensors (AD2D, AD3D) located closer to the seafloor. Horizontal velocity variance estimated with the two acoustic Doppler sensors (AD5D, AD4D) located 75 cm above the seafloor, but separated approximately 5 m alongshore are within a few percent for all conditions (Fig. 4a, cf. triangles with the diagonal line).

According to linear theory, at 75-cm elevation above the seafloor in 1.25-m water depth the horizontal velocity variance at f = 0.10 Hz is about 45 times the vertical velocity variance, and thus the measured vertical velocities may have small errors owing to sensor tilts of a few degrees. Although there is some scatter, vertical velocity variances measured 75 cm above the seafloor by upward- (AD3U) and downward- (AD4D, AD5D) looking acoustic Doppler current meters agree within about 20% (Fig. 4b, cf. filled squares and triangles with diagonal lines). The acoustic travel time current meter (ATT1) measured somewhat larger vertical velocity variance, perhaps owing to spikes in the time series. The relationship between vertical velocity variances W²_z1 and W²_z2 at elevations z₁ and z₂ above the seafloor is given by

View Expanded

Thus, vertical velocities (3) are attenuated more strongly over the water column than horizontal velocities (1), and 25 cm above the seafloor in 1.25-m depth the horizontal velocity variance at f = 0.01 Hz is almost 400 times the vertical velocity variance. Consequently, the measured vertical velocities at 25-cm elevation likely are corrupted by tilts of a few degrees in the vertical alignment of the sensors.

Mean horizontal currents measured with the different sensors are similar (Figs. 5a and 5b). Rip currents, which were visible from the neighboring pier, were observed occasionally to meander near the instrument frames, resulting in alongshore inhomogeneities in the mean flow field, and thus differences in mean horizontal currents observed at the two sensor frames, separated about 5 m alongshore (Figs. 5a and 5b, cf. triangles with diagonal lines). In contrast, mean cross- and alongshore currents measured with ATT1 and AD5D (located on the same instrument frame) are similar (Figs. 5a and 5b, cf. asterisks with diagonal lines). Mean vertical currents measured with the different sensors are less than about 4 cm s⁻¹. However, only the acoustic travel time current meter (ATT1), known to be accurate in low steady flows (Williams et al. 1987), measured approximately zero vertical flow (less than about 1 cm s⁻¹, which is not distinguishable from 0 for the instrument calibrations used here) for all conditions. Deviations from zero mean flows may be the result of flow blockage. The mean vertical flows measured by the downward-looking acoustic Doppler sensors (AD2D, AD3D, AD4D, and AD5D) differ from each other by less than 1 cm s⁻¹ (cf. filled circles, open circles, and triangles with solid lines in Fig. 5c), whereas mean vertical flows measured with the upward-looking AD3U are scattered relative to the downward-looking AD4D (cf. squares with solid line in Fig. 5c).

Mean wave direction (Fig. 6a) and directional spread (Fig. 6b) estimated from the covariance of U with V (Kuik et al. 1988) are similar for the time series acquired with the different current meters. The increased directional spread estimated by ATT1 is an artifact of the spikes caused by the malfunctioning circuit, leading to reduced covariance.

Third moments of wave-orbital velocities are important to sediment transport (Bowen 1980; Bailard 1981; and many others). Cross-shore velocity skewness (the mean of the cube of the demeaned cross-shore velocity time series normalized by the cross-shore velocity variance raised to the 3/2 power) and asymmetry [the mean of the cube of the Hilbert transform of the demeaned cross-shore velocity time series normalized by the cross-shore velocity variance raised to the 3/2 power (Elgar and Guza 1985)] estimated from time series acquired with the different current meters agree well [Fig. 7, average root-mean-square differences relative to AD4D are 0.03 (skewness) and 0.04 (asymmetry)]. The spikes in time series acquired with ATT1 occur in pairs, one positive and one negative, and thus cancel in odd moments (e.g., mean, skewness, and asymmetry).

b. Nonlinearities

Linear wave theory often is used to convert between bottom pressure, sea surface elevation, and wave-orbital velocity. For example, the ratios of the variance of pressure (P²_zp) at a location z_p above the bottom to the variance of horizontal (U²_zu + V²_zu) and vertical (W²_zu) velocities at a location z_u above the bottom are, respectively,

View Expanded

Similarly, the ratio of sea surface elevation variance (η²) to pressure variance is

View Expanded

Nearshore and surf zone significant wave heights, estimated by applying linear theory transfer functions to bottom pressure or near-bottom wave-orbital velocity spectra and integrating the resulting sea surface elevation spectra over the wind–wave frequency band, differ by less than 10% from the wave heights obtained with surface-piercing wave staffs (Guza and Thornton 1980; and references therein). However, the effects of nearly resonant triad nonlinear interactions can become strong in shallow water, especially at frequencies corresponding to harmonics of the power spectral primary peak (f_p). For the conditions here, the relationship between the spectral density of wave-orbital horizontal velocities and bottom pressure fluctuations at f_p is within 10% of linear theory for the full range of observed wave height to water depth ratios γ (Fig. 8a). However, when γ is larger than about 0.35, linear theory tends to underpredict the observed transfer function between horizontal velocity and pressure fluctuations at harmonic frequencies (2f_p, 3f_p), with the deviations increasing as γ increases. Similarly, the observed relationship between vertical velocity and pressure fluctuations at the spectral primary peak frequency is consistent with linear theory, but motions at harmonic frequencies increasingly deviate from linear theory with increasing γ (Fig. 8b). The deviations from linear theory observed here may be associated with 20%–30% deviations from linear theory of wavenumbers at harmonic frequencies, similar to those observed with arrays of pressure gauges in the surf zone (HESG).

Phases between pressure, cross-shore velocity, and vertical velocity fluctuations are consistent (within about 4°) with linear theory for γ less than about 0.3, but deviate increasingly with increasing γ (Fig. 9). Deviations from 90° of the phase between U and W correspond to nonzero covariance of U with W (〈UW〉). Above the bottom boundary layer nonzero 〈UW〉 can occur if the seafloor slopes (Chu and Mei 1970), if there is a cross-shore gradient in wave amplitude owing to breaking-induced dissipation or bottom friction (Mei 1983; Deigaard and Fredsoe 1989), or if there are depth-varying mean currents (Peregrine 1976). The vertical variation of 〈UW〉 in these cases is discussed by Rivero and Arcilla (1995), but has not been measured in the surf zone. Consistent with nonzero 〈UW〉 owing to dissipation-induced cross-shore gradients in wave amplitude, the deviations of the observed U–W phase (asterisks in Fig. 9) from the quadrature predicted by linear theory for a flat bottom, as well as values of 〈UW〉 (not shown), increase with increasing γ, with 〈UW〉 about 3% of 〈UU〉 for the highest values of γ. The value of 〈UW〉 is close to zero when wave breaking is minimal (γ ≈ 0.2, not shown), implying bottom slope effects are negligible 75 cm above the seafloor, consistent with Chu and Mei (1970). The 〈UW〉 are positive in the coordinate system used here (e.g., positive U is onshore directed, and positive W is upward flow). Further study is needed to determine if flow blockage distorted the measured vertical flows.

c. Seafloor location

The acoustic Doppler current meters can operate in a mode where the distance to the nearest strongly reflecting boundary, in this case the seafloor, is measured. The distance to the seafloor estimated with acoustic Doppler current meters located 25–100 cm above the bed deviated by less than 5 cm from estimates made with a sonar altimeter designed to operate in the surf zone (Gallagher et al. 1996; Fig. 10).