Observations of wave energy fluxes and swash motions on a low-sloping, dissipative beach


  • Rafael M. C. Guedes,

    Corresponding author
    1. Department of Earth and Ocean Sciences, University of Waikato, Hamilton, New Zealand
    2. Now at MetOcean Solutions Ltd., Raglan, New Zealand
    • Corresponding author: R. M. C. Guedes, MetOcean Solutions Ltd, 5 Wainui Road, Raglan 3225, New Zealand. (r.guedes@metocean.co.nz)

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  • Karin R. Bryan,

    1. Department of Earth and Ocean Sciences, University of Waikato, Hamilton, New Zealand
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  • Giovanni Coco

    1. National Institute of Water and Atmospheric Research, Hamilton, New Zealand
    2. Now at Environmental Hydraulics Institute, “IH Cantabria,” Universidad de Cantabria, Santander, Spain
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[1] Field observations of swash and ocean waves show that runup saturation at infragravity frequencies (<0.05 Hz) can occur under mild offshore energy conditions if the beach slope is sufficiently gentle. Infragravity saturation was observed for higher-frequency (>0.025–0.035 Hz) infragravity waves, where typically less than 5% of the (linear) energy flux was reflected from the beach and where, similar to the sea swell band, the swash energy was independent of offshore wave energy. The infragravity frequency range of saturation was determined by the tide, with saturation extending to lower frequencies at low tide when the local beach face slope over the concave-shaped profile was gentler. Runup was strongly dominated by infragravity frequencies, which accounted on average for 96% of the runup variance, and its energy levels were entirely consistent with strong infragravity wave dissipation observed in the surfzone, particularly when including the nonlinear contributions to the wave energy fluxes. The infragravity wave dissipation was strongly associated with breaking of sea swell waves, which abruptly decreased nonlinear transfers to infragravity frequencies and made infragravity dissipation prevail over forcing within the breaking region. Our observations show evidence of nonlinear interactions involving infragravity and high-frequency, harmonic waves and suggest that these harmonics could play a role in the wave energy balance near the shoreline on low-sloping, dissipative beaches.

1. Introduction

[2] Energy spectra of wave-driven shoreline oscillations (runup) can be dominated by low-frequency infragravity motions (0.004–0.05 Hz), below the sea swell frequency range (0.05–0.4 Hz) that normally dominates the offshore wave spectrum [e.g., Guza and Thornton, 1982; Guza et al., 1984; Holman and Bowen, 1984; Holland et al., 1995; Raubenheimer et al., 1995; Ruessink et al., 1998; Ruggiero et al., 2004; Sénéchal et al., 2011]. This frequency downshift implies dissipation of (steeper) waves at sea swell frequencies, which occurs mostly due to breaking in shallow water [e.g., Thornton and Guza, 1983], and energy transfer to gently sloping infragravity waves [e.g., Henderson et al., 2006], for which breaking is less likely to occur. Infragravity swash motions can provide the main mechanism for sediment transport on low-sloping, dissipative beaches [Butt and Russell, 2000], and therefore, predicting their occurrence and magnitude is a critical component of coastal change models. Yet, we do not have a clear understanding of the processes that control the infragravity wave transformations between the very shallow surfzone and swash zone as only few field observations [e.g., Holland et al., 1995; Raubenheimer et al., 1995] have linked cross-shore wave evolution patterns in shallow water to infragravity swash motions.

[3] Low-frequency infragravity waves can be excited by interactions of short waves at sea swell frequencies in the shoaling region. Longuet-Higgins and Stewart [1962] showed that a pair of short waves with closely spaced frequencies f1 and f2 and wave numbers k1 and k2 excite group-bound, out-of-phase infragravity waves with frequency and wave number f1 − f2 and k1 − k2, due to low-frequency modulations of mass and momentum fluxes associated with the wave groups. As the water depth decreases and the short wave pair shoals, the interaction becomes nearly resonant, since the bound wave frequency and wave number approach those satisfying the dispersion relationship, causing energy to be more easily transferred from the short to the infragravity waves [e.g., Battjes et al., 2004]. This transfer has been demonstrated in field observations, where statistically significant phase coupling involving pairs of sea swell and infragravity waves at their difference frequency has been observed using bispectral analysis [e.g., Hasselmann et al., 1963; Elgar and Guza, 1985; Herbers et al., 1994; Ruessink, 1998a; Sheremet et al., 2002].

[4] Within the surfzone, the sea swell wave amplitude becomes depth-limited [e.g., Thornton and Guza, 1983; Howd et al., 1991; Raubenheimer et al., 1996] and the groupiness of short waves drastically decreases [List, 1991] due to wave breaking. The group-bound long waves are believed to be “released” and propagate onshore as free waves [Longuet-Higgins and Stewart, 1962; Janssen et al., 2003], eventually reflecting from the beach. Infragravity forcing may also take place near the edge of the surfzone due to group-induced temporal and spatial variations in the breakpoint position [Symonds et al., 1982; Lippmann et al., 1997]. Nearshore infragravity energy increases with offshore sea swell wave energy [Holman, 1981; Guza and Thornton, 1982, 1985]. However, recent observations have shown that a decrease in infragravity energy in very shallow water might be caused by bottom friction [e.g., Henderson and Bowen, 2002] and nonlinear energy transfers from low-frequency back to higher-frequency motions [e.g., Henderson et al., 2006; Thomson et al., 2006], including infragravity breaking [Battjes et al., 2004; van Dongeren et al., 2007].

[5] Swash motions can be dominated by infragravity frequencies, particularly when dissipation influences the sea swell wave energy range more than the infragravity energy range. Such conditions are usually met on gently sloping, dissipative beaches, characterized by low values of a nondimensional beach steepness parameter, the Iribarren number ξ0 [Iribarren and Nogales, 1949; Battjes, 1974]

display math(1)

where β is the beach slope (for a planar beach), H0 is the deep water wave height, and L0 is the deep water wavelength. However, the roles of offshore conditions (H0 and L0) and surfzone conditions (β) in controlling infragravity swash are still debated. For example, Guza and Thornton [1982] measured swash motions under mildly dissipative conditions (β and H0 roughly between 0.03 and 0.05 and between 0.5 and 1.5 m, respectively, ξ0 ≈ 0.3–1.4) and found that whereas vertical swash excursions (runup) at sea swell wave frequencies were saturated (independent of H0), infragravity runup RIg increased linearly with H0, at a rate of 0.7. Similar linear dependence was found by Ruessink et al. [1998] for a site exposed to highly dissipative conditions (β ≈ 0.017, H0 ≈ 0.5–5 m, ξ0 ≈ 0.05–0.3), although a much smaller coefficient of proportionality, 0.18, was observed. Holman and Sallenger [1985], analyzing a data set obtained under broader (and overall more reflective) environmental conditions (ξ0 ≈ 0.5–4), found RIg (normalized by H0) to be linearly dependent on ξ0, suggesting that beach slope and wavelength may also be important. The beach slope effect was confirmed by Ruggiero et al. [2004] who observed alongshore changes in RIg under highly dissipative conditions (ξ0 ≈ 0.05–0.25) to be linearly dependent on β. Alternatively, Stockdon et al. [2006] examined infragravity runup for a data set composed of measurements from 10 different field experiments that spanned a range of ξ0 and found RIg to be best predicted using a parameter dependent only on H0 and L0 (and no significant linear relationship with β). More recently, Guza and Feddersen [2012] showed that infragravity runup may also depend on incident wave directional and frequency spreading.

[6] There is also mounting evidence that infragravity swash can be saturated, or independent of offshore wave conditions [Ruessink et al., 1998; Ruggiero et al., 2004; Sénéchal et al., 2011] in a similar way to swash at incident frequencies [e.g., Miche, 1951; Huntley et al., 1977; Guza and Thornton, 1982; Holman and Sallenger, 1985; Raubenheimer and Guza, 1996]. Infragravity swash saturation is consistent with breaking of infragravity waves and has typically been observed under highly energetic offshore wave conditions (when the long waves are steeper). Ruessink et al. [1998] suggested the lowest frequency for which saturation occurs (fs) to be related to ξ0, with fs within the infragravity range for ξ0 roughly less than 0.3. The authors observed different patterns for RIg when ξ0 < 0.3 and linked these differences to infragravity saturation. Ruggiero et al. [2004] found infragravity saturation to extend into somewhat lower frequencies for alongshore regions where the beach face slope was gentler (for equal offshore wave conditions). Infragravity swash saturation has been typically inferred from the characteristics of the swash spectra and their relationship with offshore wave conditions.

[7] This work explores the exchanges of energy between the inner surfzone and the swash using new in situ observations of cross-shore wave evolution in very shallow water coupled with video observations of wave breaking patterns and swash motions, obtained on a low-sloping dissipative beach. Although swash observations are common, and infragravity saturation has been documented in previous studies, the swash infragravity levels have not been linked to the flux of energy from the surfzone, and how and where the energy transfer between incident and infragravity frequencies occurs has not been fully explored. We use simultaneous measurements of water pressure and velocity obtained at different cross-shore distances from the shoreline (controlled by changes in tide) to estimate cross-shore linear and nonlinear wave energy fluxes and nonlinear energy transfers among frequencies, based on equations described by Henderson et al. [2006] and Sheremet et al. [2002]. Our findings indicate that infragravity dissipation is strong within the surfzone, increases with infragravity wave frequency, and plays a pivotal role in controlling the runup spectrum. The equations used to evaluate the wave energy balance are described in section 2. Section 3 details the field experiment and the analysis performed, while results are shown in section 4. Discussion and conclusions are presented in sections 5 and 6, respectively.

2. Theory

[8] In order to investigate the possible surfzone control on infragravity swash observations, an energy balance between infragravity growth and dissipation was evaluated from simultaneous observations of pressure and velocity in the surfzone. In addition, the cross-shore locations and frequency components involved with such growth and dissipation were determined using bispectrum analysis.

2.1. Energy Balance

[9] Henderson et al. [2006] proposed a conservative, depth-integrated, alongshore-uniform energy balance for infragravity waves in shallow water

display math(2)

to describe cross-shore changes in infragravity net energy flux F(f), based primarily on the energy transfer W(f) to infragravity waves from waves at other frequencies, where f is the cyclic frequency. The cross-shore coordinate x is defined as positive seaward.

[10] The net energy flux F(f) at infragravity frequency f is calculated following Henderson et al. [2006] as

display math(3)

where Cf (a, b) denotes the cospectrum of real-valued variables a and b, h is the mean water depth, η is the sea surface elevation around the mean water level, and u is the cross-shore velocity (positive seaward). M and Sxx represent the slowly varying part of the sea swell mass flux and cross-shore component of the sea swell radiation stress, respectively, and are given by

display math(4)


display math(5)

where g is the acceleration due to gravity and primes (′) denote band-passed filtering between 0.05 and 0.4 Hz. Note that equations (3) and (5) have been scaled to be dimensionally consistent with equation (2) of Sheremet et al. [2002]. The first term on the right hand side of (3) corresponds to the linear component FL(f) of the net energy flux and assuming cross-shore propagation, can be decomposed into shoreward propagating math formula and seaward propagating math formula components [Sheremet et al., 2002]:

display math(6)

with math formula (the sign convention adopted here for the two components is the opposite of that in Sheremet et al. [2002], where u is positive shoreward). The second and third terms on the right hand side of (3) are nonlinear corrections which have similar magnitudes when integrated over infragravity frequencies [Henderson et al., 2006]. The nonlinear component of the net energy flux FNL(f) is defined here as the sum of these two terms.

[11] The nonlinear energy transfer term in (2), assuming nearly shore-normal shallow water waves, is defined as [Henderson et al., 2006]

display math(7)

with positive and negative values indicating energy transfer to and from waves with frequency f, respectively. The cross-shore gradient of the cross-shore velocity math formula is calculated from the shallow water mass conservation equation with the wave mass flux term neglected:

display math(8)

where t is time and all other symbols have been previously defined.

2.2. Forced Waves

[12] Following previous studies [e.g., Elgar and Guza, 1985; Herbers et al., 1994; Ruessink, 1998a; Sheremet et al., 2002] the source of wave energy associated with forced infragravity waves is investigated in terms of the digital bispectrum B(f1, f2) [Elgar and Guza, 1985]

display math(9)

where E[ ] represents the expected value, math formula are complex Fourier coefficients at the nth frequency component, and the asterisk indicates complex conjugate. The bispectrum measures the statistical dependence among three waves with frequencies (f1, f2, f1 + f2) [Hasselmann et al., 1963]. B(f1, f2) vanishes if the wave triads are independent and have random phases, such as in a linear wave field. On the other hand, nonlinear coupling between two primary waves and a forced, secondary wave yields a B(f1, f2) which is statistically different than zero. A relative measure of the phase coupling between the wave triads can be obtained by the normalized magnitude and phase of the bispectrum, defined as the bicoherence b(f1, f2) and biphase θ(f1, f2) and calculated respectively as [Kim and Powers, 1979]

display math(10)


display math(11)

where Im{} and Re{} represent the imaginary and real parts, respectively. Different combinations of the math formula coefficients have been used to normalize the bispectrum in (10) and calculate b(f1, f2) [e.g., Herbers et al., 1994; Ruessink, 1998a]. The normalization factor adopted here follows Kim and Powers [1979] and Elgar and Guza [1985] and ensures 0 ≤ b ≤ 1. The 95% significance level on zero bicoherence b95% is calculated as [Haubrich, 1965]

display math(12)

where DoF is the number of degrees of freedom.

3. Methods

3.1. Description of Field Experiment and Data Reduction

[13] The field data were collected between 8 and 9 November 2010 at Ngarunui Beach, Raglan, an exposed, dissipative beach located on the west coast of New Zealand's North Island (Figure 1). Ngarunui is about 2 km in length, constrained by an inlet to the north and a headland to the south and frequently characterized by the presence of groundwater seepage above the swash zone [Huisman et al., 2011]. The beach is composed by fine-medium, black sand [Sherwood and Nelson, 1979] (the median grain size at the measurement positions was about 400 µm) and a gentle slope (the average slope βmean over the intertidal region was about 0.014, see Figure 2). The field site commonly experiences energetic offshore wave conditions with average offshore significant wave height H0 and mean spectral period Tm of 2.0 m and 7.0 s, respectively [Gorman et al., 2003]. The tides are semidiurnal and typically range between 1.8 (neap) and 2.8 m (spring tides) within the adjacent estuary [Heath, 1976] (the tidal range on the open coast was as high as 3.1 m during the period of the experiment).

Figure 1.

(a) Field site location with locations where cameras were mounted and ADCP was deployed indicated by the black circle and the triangle, respectively. White circles with black crosses show locations of bench marks (also shown in Figure 1b)). Bathymetry was digitized and interpolated from the New Zealand Nautical Chart NZ4421. (b) Mosaic composed by 20 min time-exposure (averaged) images obtained using two cameras on 9 November at 15:40 (DST), rectified to a plan view using known geometric transformations. Bright areas indicate regions of preferred wave breaking. White squares and dashed lines show locations where ADVs were deployed and time stacks were defined, respectively. Contour lines show intertidal bathymetry measured with the RTK with white line highlighting the beach contour corresponding to the highest mean water level (measured at location of UoW ADV) under which waves were measured. X and Y are the cross-shore and alongshore coordinates of the local grid defined for this study, respectively.

Figure 2.

Beach profile averaged over the region where the three ADVs were deployed (see Figure 1b). Cross-shore positions of NiwaInn ADV (X = 220 m) and NiwaOut and UoW ADVs (X = 270 m) are shown by the white squares. Horizontal lines show the shallowest and deepest mean water levels (measured at location of UoW ADV) at which waves were measured in the surfzone. Vertical arrows point to the mean position of each swash run with their length corresponding to the respective value of (alongshore-averaged) Rs. Black and gray arrows are associated with data obtained on days 8 and 9, respectively. Horizontal bars at the top of lowermost and uppermost arrows highlight the cross-shore extension of the swash (Rmean ± 2σR) for these two time series.

[14] Simultaneous time series of pressure and velocity were recorded using three acoustic Doppler velocimeters (ADVs) in the intertidal region and an acoustic Doppler current profiler (ADCP) offshore in about 17 m water depth (Figure 1 shows the location of the instruments). Two of the ADVs (Nortek Vector) were deployed along a cross-shore transect extending over 50 m (NiwaInn ADV, X = 220 m and NiwaOut ADV, X = 270 m, where X is the cross-shore coordinate of the grid defined for this study) and collected data at 8 Hz. The acoustic sensors were pointed downward and measured velocity at ∼0.05 m above the bed. The third ADV (UoW ADV; Sontek Triton) was mounted at the same cross-shore location of NiwaOut but 50 m farther in the alongshore (northward) direction Y and measured at a sample rate of 4 Hz. However, the probe was upward looking and measured orbital velocity at ∼0.95 m above the bed. All three instruments collected 20 min long time series every half-an-hour, over three partial tidal cycles (when the sensors were submerged). After a quality control (following Elgar et al. [2005]) to remove bad time series or individual data points (i.e., those recorded when any of the sensors was in too shallow or out of the water), a total of 38, 47, and 36 simultaneous time series of pressure and velocity remained for NiwaInn, NiwaOut, and UoW ADVs, respectively. The measurements spanned cross-shore distances ΔXS from the shoreline (defined here as the mean swash position for each data run) from 50 m (h = 0.55 m) to 184 m (h = 2.60 m). The ADCP (Teledyne Workhorse Sentinel 600 kHz) collected hourly, 20 min long, simultaneous time series of near-bottom pressure and near-surface velocity. The measurements were taken at 2 Hz and spanned the entire period when the ADVs were deployed. In addition, time series of (nearshore) mean water level h were collected every 5 min by averaging 2 min long pressure records, obtained at 4 Hz using a data logger collocated on the frame used to deploy UoW ADV.

[15] The morphology of the intertidal region of the beach was surveyed using a real-time kinematic (RTK) GPS. The GPS receiver was installed on a Quad bike that traveled over the beach around low tide, yielding dense coverage of the intertidal morphology. Analysis of an overlapping area surveyed on the two different days that included the position of the ADVs suggests that intertidal morphology changes were minimal, with a root-mean-square difference in elevation of 0.02 m. A local grid was defined by translating and rotating the coordinate system of the survey, so as to have the origin at the location of a bench mark on the beach, and cross-shore coordinates X parallel to the line formed by NiwaInn and NiwaOut ADVs and increasing offshore (see Figure 1b).

[16] High-resolution images of the beach (1528 × 2016 pixels) were collected continuously at 2 Hz during daylight hours. The images were acquired using two digital cameras, mounted at the southern end of the beach (Figure 1a) at about 95 m above mean sea level. The combined field of view of the cameras spanned most of the subaerial beach and the surfzone. Figure 1b shows a mosaic created using time-averaged images defined from both cameras, rectified to a planview using known geometric transformations.

[17] Swash oscillations were measured at two alongshore locations on the beach using the video images. Two cross-shore lines (Niwa and UoW lines) were defined on the images at the alongshore locations where the respective ADVs were deployed, as shown in Figure 1b (conversions between pixel and ground coordinates were made using the colinearity equations described by Holland et al. [1997] with corrections for lens distortions). Time stacks with cross-shore pixel resolution of ∼0.2 m were defined over these lines by interpolating the intensities on each selected image frame at the locations of these two lines (see Aagaard and Holm [1989] and Guedes et al. [2011] for more detailed description of the technique). The period selected for creating the time stacks included the daylight hours when the ADVs were collecting data and resulted in 32, 30 min long, 2 Hz time stacks defined over each line (e.g., Figure 3b). The swash was defined as the most shoreward moving-edge of water identifiable on the time stacks (see Figure 3b) and was manually digitized. This definition has been shown by Holman and Guza [1984] and Holland et al. [1995] to be consistent with swash measurements obtained using resistance wires deployed near the seabed. Finally, the digitized swash positions from the time stacks were converted into time series of vertical runup elevation by mapping each cross-shore swash position to an elevation Z, which was accurately known from the (interpolated) RTK survey. The gentle slope of the beach resulted in high vertical pixel resolution over the intertidal region (∼0.003 m).

Figure 3.

(a) Mosaics composed by 20 min variance images obtained using the two cameras deployed at Ngarunui Beach, rectified to plan views using known geometric transformations. Contour lines show intertidal bathymetry measured with the RTK DGPS survey system. Horizontal dashed lines show one of the two locations where time stacks were defined (Niwa line). Inner and outer squares show positions of NiwaInn and NiwaOut ADVs, respectively. White circles with crosses show two of the bench marks on the beach. (b) Time stacks created over the alongshore location highlighted by the dashed line in Figure 3a. Black solid lines show digitized swash locations, when available. (c) Probability of breaking Pbreak estimated from the time stacks in Figure 3b, as a function of cross-shore position. Vertical dashed lines in Figures 3b and 3c highlight the cross-shore positions of the two ADVs shown in Figure 3a. Left, middle, and right figures are associated with data obtained near low tide (8 November, 17:10, η = −1.54 m), middle tide (8 November, 14:40, η = −0.08 m), and high tide (8 November, 11:40, η = 1.51 m) (times are in DST).

[18] The cross-shore wave breaking structure was defined by using the probability of wave breaking Pbr(x) which can be extracted from video images. The method was modified from Guedes et al. [2011] who used time stacks to approximate Pbr(x) as the probability of pixel intensity being greater than a threshold at any cross-shore location. Here individual breaking waves were identified by taking the gradient of the pixel intensities in the time stacks over time, math formula, and locating sequences of positive followed by negative intensity gradients that typically characterized a broken wave. The result was an estimate of the number of breaking waves Nbr(x). The probability of breaking Pbr(x) was calculated by dividing Nbr(x) by the total number of individual wave crests observed during the same periods in 17 m water depth, which was defined from the ADCP pressure series as the length of the time series divided by the mean spectral period. Figure 3c shows examples of Pbr(x) obtained from the Niwa time stack near low, middle, and high tide. Though this technique does not account for nonlinear interactions such as harmonic decomposition [e.g., Elgar et al., 1997] which potentially increased the number of individual waves in the shoreward direction (see section 5) and thus the values of Pbr (since they were calculated using the number of offshore waves), Pbr(x) was consistent with the patterns visually observed from the time stacks (see Figure 3).

3.2. Analysis

[19] The hydrodynamic data were analyzed in the frequency domain using spectral analysis. Runup and pressure surfzone spectra with frequency resolution df of 0.0039 Hz were estimated from Fourier transforms of the time series, segmented into 256 s, 75% overlapping sections, that were linearly detrended and tapered with a Hanning window. Offshore spectra with the same frequency resolution were estimated from the velocity time series collected by the ADCP at three different depth cells located just below the sea surface (spectra from the three cells were averaged). Linear wave theory was used to convert the pressure (ADVs) and velocity data (ADCP) into sea surface elevation.

[20] Energy fluxes and nonlinear energy transfers were calculated by evaluating equations (3)-(8) with the simultaneous time series of u and p collected by each ADV. A Lanczos filter was used to band-pass u and p to calculate time series of M and Sxx in (4) and (5). Time series of math formula were estimated in (8) by (central) finite-differencing p to evaluate math formula, and approximating math formula by the local bed slope at the location of each ADV. The cospectrum Cf(a, b) in (3), (6), and (7) was estimated using the same parameter settings as for the spectrum.

[21] Linear alongshore energy fluxes (calculated following Henderson et al. [2006] from the cospectrum between p and alongshore velocity v) integrated over infragravity frequencies were always smaller than (at most 46% of) the cross-shore component, supporting the alongshore-uniform assumption in equation (2). Moreover, shear wave contributions to the infragravity velocity variance (calculated using the method of Lippmann et al. [1999]) were at most 63%, with average of 31% for the whole data set. These values are below the threshold defined by Henderson et al. [2006] to exclude cases with energetic shear wave energy (which is omitted in equation (2)).

[22] Bicoherences and biphases were calculated from the ADVs as (10) and (11), after removing seaward propagating infragravity waves from the data, following Elgar and Guza [1985] and Sheremet et al. [2002]. Seaward propagating infragravity waves are free and mix the phase structure between shoreward propagating infragravity waves at equivalent frequencies and sea swell waves. The separation was performed by decomposing the data into surface elevation time series of shoreward propagating η and seaward propagating η+ waves as [Guza et al., 1984; Sheremet et al., 2002]

display math(13)

high-passing η+ to remove outgoing infragravity waves and adding back the two components. The bispectra estimated from these data had the same frequency resolution and DoF as the auto- and cross-spectra calculated using the ADV data. The relatively short time length of the time series was compensated by averaging bicoherence and biphase over several frequency pairs (see section 4.3).

4. Results

4.1. Observations

[23] Offshore sea swell waves in 17 m water depth were mild during the field experiment and decreased in energy from the first to the second day. Offshore significant wave height H0 decreased from about 1.3 to 1.0 m from the first to the third tidal cycle, where H0 was defined using the ADCP spectra as

display math(14)

with Ed(f) denoting the energy density and the term inside the parentheses representing the variance associated with the defined frequency band. Peak period Tp, calculated as the inverse of the peak frequency in Ed(f), decreased from about 12 to 10 s during the same period (see Figure 4a). Offshore peak wave direction Dp, determined from the ADCP, was roughly constant near 270° (W).

Figure 4.

Logarithm of energy density Ed as a function of frequency f and time, with white circles showing the peak frequency for each data run. (a) Offshore Ed obtained from the ADCP, with solid line (right axis) showing the mean local water depth hADCP. Surfzone Ed obtained from the (b) NiwaOut and (c) NiwaInn ADVs, with solid and dashed (black) lines (right axes) showing the mean local water depth hNiwaOut and hNiwaInn, respectively. (d) Swash Ed calculated from the (video) runup time series, obtained over the alongshore location of NiwaOut and NiwaInn ADVs, with solid and short dashed lines (right axis) showing the distance ΔXS from these two ADVs to the shoreline, respectively. Horizontal white dashed line in Figures 4a–4d highlight the frequency used to separate incident and infragravity bands. (e) Time series of wave variance integrated over incident (solid markers) and infragravity (open markers) bands, obtained from: ADCP (circles), NiwaOut ADV (triangles), and video runup data (squares) (incident runup variances ranged between ∼10−4 and 10−3 m2 and are not shown here). Colors represent the mean water depth at the location of NiwaOut ADV.

[24] The beach morphology was relatively alongshore-homogeneous over the intertidal region where the ADVs were deployed. The three instruments were placed near the center of the embayment of a mega-cusp with wavelength about 800 m (Figure 1b), where the intertidal cross-shore profile was concave upward (Figure 2). The local slope tan β increased by a factor of 4 from 0.008 at the lowermost to 0.032 at the uppermost location where swash was measured (tan β was ∼0.01 at the location of the ADVs) and no sandbar was observed over the intertidal region. The Iribarren number ξ0 (1), calculated using the local slope at the mean swash location (tan β) and linear wave theory to estimate L0 from the peak offshore wave period Tp, was characteristic of highly dissipative conditions, with small values changing by a factor of 4 (following tan β) from low tide (ξ0 ≈ 0.1) to high tide (ξ0 ≈ 0.4). A NE-directed (+Y → −Y) alongshore current, calculated by averaging the ADVs' velocity records V, was present, and increased in magnitude from negligible values at high tide (when the instruments were outside the surfzone) to about 0.4–0.5 m/s near the shallowest locations. A seaward component in V was observed, especially for measurements from NiwaInn and NiwaOut ADVs, and was likely caused by the presence of a rip channel that extended diagonally through the beach, from [X, Y] ≈ [150, 500] m passing seaward of the outer ADVs (Figures 1b and 3a). The channel acted as a trough around middle tide, separating an outer breaking region seaward of X ≈ 350 m from an inner breaking region, where the waves broke continuously until they reached the swash zone (Figure 3; see also Figure 5c). In contrast, at high tide the outer breaking region was too deep and breaking occurred mostly over the inner surfzone, shoreward of the channel, and near low tide the channel was shallow enough to be within the inner breaking region.

Figure 5.

Logarithm of (a) shoreward propagating (F) and (b) seaward propagating (F+) components of the linear energy flux as a function of frequency f (left axis) and time. Horizontal dashed lines highlight the frequency used to separate incident and infragravity bands. (c) Probability of breaking Pbreak as a function of cross-shore position X (left axis) and time. White dashed lines indicate cross-shore positions of NiwaOut (270 m) and NiwaInn (220 m) ADVs. Cross-shore positions shoreward of the mean swash locations have been blanked. In Figures 5a–5c, gray solid lines (right axes) represent tide level η.

[25] Figure 4 shows an overview of the wave conditions offshore, in the surf and swash zones during the experiment. Most of the energy in 17 m water depth was within sea swell frequencies (Figure 4a), with little energy (typically corresponding to less than 5% of the total wave variance math formula) observed at infragravity frequencies (see Figure 4e). Consistent with previous observations [e.g., Okihiro and Guza, 1995; Thomson et al., 2006], offshore infragravity energy was somewhat tidally modulated (Figure 4e, open circles), with the variance integrated over infragravity frequencies math formula showing some linear correlation with the tide η, with r2 = 0.23. In contrast to the waves in 17 m water depth, most of the swash energy was constrained within infragravity frequencies, with peaks between 0.01 and 0.02 Hz (Figure 4d), near the infragravity wave peaks observed offshore. The infragravity runup variance math formula was nearly two orders of magnitude larger than the sea swell runup variance math formula, accounting on average to 96% of the total swash variance math formula (similar to observations under highly energetic offshore wave conditions). Moreover, the infragravity swash variance decreased from the first to the second day and was also modulated by the tide, with math formula increasing by an order of magnitude from low to high tide within each tidal cycle (Figure 4e).

[26] Similarly to the waves in 17 m water depth, most of the wave energy in the surfzone was within sea swell frequencies (Figures 4b, 4c, and 4e). The variances integrated over both sea swell math formula and infragravity frequencies math formula decreased from the deepest to the shallowest locations (and from the first to the third tidal cycle). However, the sea swell waves experienced greater dissipation, which resulted in math formula accounting for an increasing proportion of the total wave variance math formula from the deepest (h ≈ 2.60 m, math formula) to the shallowest locations (h ≈ 0.55 m, math formula). Over the inner surfzone region between NiwaInn ADV and the upper boundary of the swash zone, the sea swell waves were nearly completely dissipated, as evidenced by the strong infragravity dominance observed in the runup spectra. Coherences and phases estimated between the swash and ADVs time series (not shown) show phases consistent with standing wave patterns in the infragravity band, with strongest coherence levels observed for very low frequencies (0.01–0.02 Hz), where the swash peak was observed.

4.2. Linear Energy Fluxes

[27] The linear component of the wave energy flux was predominantly shoreward throughout the surfzone (compare Figures 5a and 5b). The shoreward propagating linear energy flux math formula integrated over sea swell frequencies, math formula, accounted for roughly 98–99% of the (total) sea swell energy flux math formula, consistent with the nearly total sea swell dissipation observed in the runup spectra. At infragravity frequencies, the contribution of the shoreward component math formula ranged from 90% when the ADVs were farthest from the shoreline to 70% when they were closest, since math formula decreased more than math formula in the shoreward direction. The dissipation of math formula was clearly accelerated within the breaking region for both sea swell and infragravity frequencies (Figure 5c).

[28] In contrast to math formula, which mimicked the surfzone spectrum (compare Figures 4b and 5a), the seaward propagating linear energy flux math formula was dominated by infragravity frequencies (Figure 5b) and strongly resembled the patterns observed in the runup spectrum (see Figure 4d). The infragravity component of the seaward propagating flux math formula accounted on average for 74% of the outgoing linear energy flux and increased by more than a factor of 4 from low to high tide within each tidal cycle (not shown), consistent with the infragravity modulation observed offshore (math formula was correlated with the tidally modulated beach slope at the mean swash location tan β, r2 = 0.44, whereas its correlation with the tide itself was lower, with r2 = 0.32).

[29] Most of the infragravity energy observed in both the runup and outgoing energy flux spectra were within very low infragravity frequencies (<0.02 Hz). In order to investigate the infragravity behavior in detail, the infragravity band was further divided into three frequency bands Ig1 (0.004 < f < 0.025 Hz), Ig2 (0.025 < f < 0.035 Hz), and Ig3 (0.035 < f < 0.05 Hz), and bulk reflection coefficients R2 were calculated as [Elgar et al., 1994; Sheremet et al., 2002]

display math(15)

where lf and hf represent the low- and high-frequency cutoff for each of these bands, respectively. Figure 6a shows R2 calculated for the three bands plotted as a function of the Iribarren number ξ0. For the three infragravity bands, R2 was smaller than one, consistent with shoreward dissipation of the linear infragravity energy flux, and increased with ξ0 (which was calculated using Tp and changed mostly due to tidal-induced changes in tan β). However, the dissipation was clearly greater for Ig2 and Ig3, with the bulk reflection coefficient R2 typically a factor of 3 to 4 larger for the lowest frequency band Ig1. R2 was around 0.05 for both Ig2 and Ig3 for ξ0 < ∼0.3, suggesting almost complete dissipation for these infragravity bands, although an increase in R2 was observed for higher ξ0. The increase in R2 with decreasing frequency and increasing tan β is consistent with the evidence of standing wave patterns observed for these conditions (from coherences and phases between the swash the ADVs).

Figure 6.

Reflection coefficient R2 of partitioned infragravity bands, shown by the colors, as a function of (a) Iribarren number ξ0 (equation (1)) and (b) normalized bed slope parameter βH (equation (16)). Data for each frequency band have been grouped in Figure 6a by 0.02 ξ0 intervals, with the circles and bars representing the average and standard deviation associated with each group for the three infragravity bands.

[30] A good parameterization for R2 was obtained using a normalized bed slope parameter βH proposed by Battjes et al. [2004], which accounts for changes in water depth within a wavelength (for individual frequencies) as

display math(16)

where hx is a “characteristic” value of bed slope, ω is the radian frequency 2πf, and H is the wave height, taken as the local significant wave height Hs. Here hx has been approximated by the slope at the mean swash location tan β, ω was taken as the central radian frequency of each infragravity band, and Hs was calculated as (14) using the pressure data from the ADVs. The bulk reflection coefficient R2 was strongly linearly correlated with βH (Figure 6b) with r2 = 0.82. It should be noticed that the normalized bed slope βH is directly related to the Iribarren number as ξ0 = βH (2π)1/2 when the deep water wave length L0 in (1) is estimated for each infragravity frequency band (rather than using the peak offshore wave period).

4.3. Nonlinear Energy Fluxes and Energy Transfers

[31] The net energy flux F(f) at infragravity frequencies was dominated by its linear component FL(f) (Figures 7a–7d). However, the nonlinear terms FNL(f) increasingly contributed to F(f) from the breaking to the shoaling regions and from the first to the third tidal cycle (when the offshore wave energy was decreasing), especially for the lowest frequency band Ig1 (Figure 7d). In contrast to FL(f), FNL(f) was mostly seaward-directed (positive values), except for the shallowest locations (Figure 7b), and its increasing contribution near middle-high tide resulted in seaward net fluxes for some frequencies in F(f) that were not observed in FL(f) (compare Figures 7a and 7c). Similar to FNL(f), the wave energy transfer W(f) was positive during middle-high tide at infragravity frequencies (Figure 7e), indicating energy transfer to infragravity waves from waves at other frequencies. At the shallowest locations within each tidal cycle, when the ADVs were well within the surfzone (see Figure 5c), W(f) was typically negative at infragravity frequencies, consistent with nonlinear energy transfer from infragravity waves.

Figure 7.

Net wave energy flux components and wave energy transfer as a function of frequency f and time. (a) Linear component of the net wave energy flux FL. (b) Nonlinear component of the net wave energy flux FNL at infragravity frequencies. (c) Total net wave energy flux F at infragravity frequencies. Positive and negative values in Figures 7a–7c indicate seaward and shoreward net fluxes, respectively. (d) Contribution of the nonlinear component to net wave energy flux at infragravity frequencies. White dashed lines delimit shoaling and breaking regions within each tidal cycle (defined here as those where Pbreak < 0.05 and Pbreak > 0.05, respectively). (e) Energy transfer W to (positive values) and from (negative values) waves with frequency f. Contours highlight transitions between positive and negative values. In Figures 7a–7e, data from NiwaOut ADV are shown, and gray solid lines (right axes) represent mean local water depth hNiwaOut.

[32] The cross-shore structure of F(f) and W(f) was investigated by integrating over the infragravity band (following Henderson et al. [2006]):

display math(17)

and plotting against the tidally modulated, mean distance from the shoreline ΔXS (Figure 8). The observed net energy flux integrated over infragravity frequencies FIg typically diverged (increased seaward) when the ADVs were offshore of the inner surfzone, indicating a net energy gain (blue region), and converged when the ADVs were inside the surfzone, indicating a net energy loss (red region). The cross-shore region of energy gain extended closer and closer to the shoreline from the first (Figure 8a) to the third tidal cycle (Figure 8c) as the offshore wave energy decreased and the surfzone became narrower. In addition, FIgXS) was normally higher during ebb tide (triangles) compared to the data obtained during flood tide (circles) within a given tidal cycle. The total nonlinear energy transfer to infragravity waves WIg was always positive over the cross-shore regions where infragravity gain was observed, consistent with observations of Henderson et al. [2006]. On the other hand, positive values of WIg were also observed over regions of infragravity dissipation (e.g., Figure 8a, 130 m < ΔXS < 165 m), suggesting a disagreement with the theoretical wave energy balance in (2).

Figure 8.

(top) Net wave energy flux (left axes) and (bottom) wave energy transfer (right axes) integrated over infragravity frequencies (FIg and WIg, respectively), as a function of distance from the shoreline ΔXS. (a) First, (b) second, and (c) third tidal cycles. Notice the different scale for FIg in Figure 8a. Circles and triangles show data obtained during flood and ebb tide. Black and white markers in Figures 8a and 8c indicate NiwaOut ADV shoreward and seaward of the breakpoint position, respectively (breakpoint positions are unknown during the second tidal cycle shown in Figure 8b, which occurred during night time). Gray markers in Figure 8a indicate NiwaOut ADV seaward of inner breaking region but with a few set waves breaking offshore at outer breaking region (e.g., Figure 3b, right). Red and blue patches highlight cross-shore region where infragravity loss (FIg increasing shoreward) and infragravity gain (FIg increasing seaward) were observed. Data from NiwaOut ADV are shown.

[33] The energy balance in equation (2) was evaluated by calculating cross-shore changes in the (infragravity) net energy flux between simultaneous time series from NiwaOut and NiwaInn ADVs, ΔFIg = FIg,NiwaOut − FIg,NiwaInn, and plotting against the nonlinear energy transfer to infragravity frequencies, integrated over the two ADVs

display math(18)

with dx representing the cross-shore distance between the two instruments and math formula evaluated using the trapezoidal rule. The observed ΔFIg and math formula were correlated (see Figure 9) with r2 = 0.65. Positive and negative values for both energy transfer and changes in energy flux typically occurred when the inner ADV was outside (open circles and triangles) and inside the breaking region (solid circles and triangles), respectively. However, ΔFIg and math formula did not balance (the slope of their linear regression line was 3.4), in disagreement with observations of Henderson et al. [2006] on a steeper barred beach under predominantly low energy conditions. In order to allow measurements in very shallow water, the velocities measured by the two Niwa ADVs were close (∼0.05 m) to the bed and possibly in the boundary layer. However, the velocity measurements from the higher sensor (UoW ADV) which is at the same cross-shore location, but 50 m alongshore, suggest that the potential boundary layer influence is small.

Figure 9.

Change in total infragravity energy flux ΔFIg between NiwaOut and NiwaInn ADVs as a function of nonlinear transfer to (positive values) and from (negative values) infragravity motions WIgINT. Circles, asterisks, and triangles are associated with data obtained during first, second, and third tidal cycles, respectively. Colors indicate the distance from the shoreline ΔXS. Solid and open markers (circles and triangles) indicate NiwaInn ADV shoreward and seaward of the breakpoint position, respectively (breakpoint positions are unknown during the second tidal cycle (asterisks), which occurred during night time). Dashed line indicates agreement with the conservative energy balance.

[34] The bispectra observations showed consistent evidence of nonlinear energy transfers across the shoaling and breaking regions involving swell and harmonic frequencies. Figures 10a and 10d show example bicoherences b(f1, f2) and biphases θ(f1, f2) calculated from NiwaOut ADV on 8 November at 22:40 DST, near high tide. Within the sea swell band, strong bicoherence (b > 0.7) can be observed involving the incident wave peak fp (f2 ≈ 0.085 Hz) with itself and its higher harmonics (f1 = nf2, where n is a positive integer), indicating nonlinear coupling between wave triads with frequencies [nf2, f2, nf2 + f2] (Figure 10a). The pattern was repeated for the two higher harmonics as well (note the bicoherence peaks extending over f2 ≈ 0.17 Hz and f2 ≈ 0.26 Hz) and were similar to previous bispectra observations in shallow water [e.g., Elgar and Guza, 1985; Sénéchal et al., 2002]. The biphases associated with these nonlinear interactions typically ranged between 0° and 70° (e.g., Figure 10b), suggesting waves with shapes changing from skewed (sharp peak and wide trough) to asymmetric (steep front face and gentle back face) forms, respectively [see Elgar and Guza, 1985]. The cross-shore evolution of bicoherences and biphases involving the incident peak fp and its harmonics was examined by averaging, for each bispectrum, b(f1, f2) and (the modulus of) θ(f1, f2) over the area highlighted by the dark gray trapezium in Figures 10a (math formula) and 10b (math formula), which extended from [f1, f2] = [0.05–0.4, 0.07–0.09] Hz (the biphases θ(f1, f2) were averaged only over pairs for which b(f1, f2) > b95%, in order to avoid the random values normally associated with wave triads with nonsignificant bicoherences). Figures 10c and 10d show math formula and math formula calculated from NiwaOut ADV, as a function of the distance from the mean shoreline location ΔXS. The nonlinear coupling of sea swell wave triads clearly decreased from the deepest locations outside of the surfzone (math formula) toward the shallowest locations within the inner surfzone region (math formula) and was followed by a systematic evolution from skewed to highly asymmetric wave shapes.

Figure 10.

Example (a) bicoherence b(f1, f2), and (b) biphase θ(f1, f2), calculated from a time series obtained by NiwaOut ADV at 22:40 (DST), during the second tidal cycle near high tide. Values associated with frequency pairs where no significant bicoherence (at the 95% confidence level) was observed have been blanked. (c) Bicoherences and (d) the modulus of biphases, averaged over the dark gray area on Figures 10a and 10b to include frequency pairs involving coupling between sea swell (and higher-frequency) modes, as a function of distance from the shoreline ΔXS. (e) Bicoherences and (f) the modulus of biphases, averaged over the light gray area on Figures 10a and 10b to include frequency pairs involving coupling between infragravity (and higher-frequency) modes, as a function of distance from the shoreline ΔXS. Circles, asterisks, and triangles are associated with data obtained during first, second, and third tidal cycles, respectively. Black and white markers in Figures 10c–10f indicate NiwaOut ADV shoreward and seaward of the breakpoint position, respectively (breakpoint positions are unknown during the second tidal cycle which occurred during night time). Gray markers in Figures 10c–10f indicate NiwaOut ADV seaward of inner breaking region but with a few set waves breaking offshore at outer breaking region (e.g., Figure 3b, right). Biphases were averaged only over triplets with bicoherence significantly different than zero (at the 95% confidence level). Data from NiwaOut ADV are shown.

[35] Significant bicoherence peaks involving infragravity modes were also observed in the bispectra (Figure 10a). The nonlinear coupling took place not only between infragravity waves with frequency f2 and modes close to the incident peak f1fp (and their sum), but also for modes f1 at higher frequencies, within the range of higher harmonics. Averaged bicoherences and biphases involving (at least) one infragravity mode (math formula and math formula, respectively) were also calculated for each bispectrum over the region defined by [f1, f2] = [0.004–0.5, 0.004–0.05] Hz (light gray area in Figures 10a and 10b). The infragravity bicoherences math formula showed a consistent cross-shore behavior over the three tidal cycles with values sharply decreasing shoreward as the waves started to break, and slightly increasing again in very shallow water (Figure 10e). On the other hand, the infragravity biphases math formula steadily decreased from ∼150° at h = 2.6 m to near 50° at the shallowest locations around h = 0.6 m (Figure 10f). The (deep water) biphase of wave triads associated with pairs of higher-frequency incident waves f1, f1 + f2 and a forced infragravity wave f2 is θ(f1, f2) = 180° [Elgar and Guza, 1985]. The decreasing shoreward pattern for math formula is consistent with the increasing phase shift from π between the wave groups and the forced waves with decreasing depth, caused by the trough of the bound wave increasingly lagging behind the group envelope. Similar shoreward phase decrease has been observed in the field [Elgar and Guza, 1985] and in the laboratory [Battjes et al., 2004; van Dongeren et al., 2007], and according to Janssen et al. [2003] it allows energy transfer between the short-wave group and the forced bound wave.

5. Discussion

5.1. Swash Motions

[36] The energy density in the swash was consistent with changes to infragravity energy fluxes between NiwaOut and NiwaInn ADVs. We predicted infragravity energy fluxes within the swash zone Fp(f) by linear-fitting the net energy fluxes simultaneously observed from the two ADVs (for each infragravity frequency component), and extrapolating the lines to a cross-shore position where the water depth hswash was representative of that in the swash zone (defined here as 0.05 m). Predicted runup energy density Ed,p(f) was then defined as

display math(19)

and compared with the energy density Ed(f) directly estimated from the runup time series. Figure 11 shows time series of infragravity variances integrated from Ed(f), as well as from Ed,p(f) estimated using the linear FL(f) (Figure 11a) and the total F(f) net energy fluxes (Figure 11b). Measured (black lines) and predicted (gray lines) runup variances were similar, with the increasing trend from low (↓tan β) to high tide (↑tan β) generally well described. However, best results were obtained using F(f), which included the nonlinear energy flux terms (Figure 11b). Though one may argue that Ed,p(f) depends on the somewhat arbitrary parameter hSwash, and the shoaling approximation is inaccurate in the swash zone, the variance derived from the linear net flux FL(f) differed from the measured runup variance between the 2 days (Figure 11a), whereas F(f) yielded more similar temporal changes for predicted and measured runup variances (which were correlated, r2 = 0.79), for any hSwash chosen. This is consistent with the observed changes in the relative contribution of the nonlinear energy flux FNL(f) from the first to the second day (Figure 7d).

Figure 11.

Time series of swash variance at infragravity frequencies (black lines with circles) and wave variance at infragravity frequencies extrapolated at the shoreline (gray lines with circles) using (a) the linear component of the net wave energy flux FL, and (b) the total net wave energy flux F between NiwaOut and NiwaInn ADVs. Extrapolated energy fluxes were converted to energy density using linear wave theory and assuming water depth of 0.05 m (the same depth at which the FL and F were extrapolated at).

[37] Our observations suggest that sea swell and most notably infragravity saturation may also occur under relatively mild offshore wave conditions if the beach slope near the shoreline is sufficiently gentle. The swash energy density Ed(f) (on a log-log scale) decayed with frequency at the (negligible) sea swell band at a linear rate of about f−4 (Figure 12a), and its temporal variability was very small compared to that observed at infragravity frequencies (Figure 12b). The f−4 energy decay rate for incident swash frequencies is very similar to that observed by Huntley et al. [1977] and several subsequent studies, which suggest energy levels for these frequencies that are independent of offshore wave conditions or “saturated.” However, the f−4 “roll-off” clearly extended into the highest infragravity band Ig3 as well (Figure 12a), suggesting saturation also for this band. The saturation is consistent with the very low reflection coefficients R2 (typically <0.05) found for Ig3, and also for Ig2 when ξ0 < ∼0.3 (Figure 6a) and agrees with previous observation under highly energetic offshore wave conditions [e.g., Ruessink et al., 1998; Ruggiero et al., 2004; Sénéchal et al., 2011]. However, such conditions did not occur during our observations, suggesting that the saturation within the infragravity band was controlled by the very gentle slope near the shoreline. Note that R2 did tend to increase for the three infragravity frequency bands with the Iribarren number ξ0 (Figure 6a), which changed mostly due to changes in the beach slope at the mean swash location tan β. Our findings support the hypothesis of Ruessink et al. [1998] in which infragravity saturation might be associated with ξ0. The increase in R2 for ξ0 > ∼0.3 observed for the two highest infragravity bands (especially Ig2) is remarkably similar to the value of ξ0 = 0.27 suggested by the authors as the threshold below which infragravity saturation occurs. However, the Iribarren number (calculated using the peak offshore wave period) did not collapse the wave reflection coefficient R2 across frequencies, whereas the normalized bed slope parameter βH did.

Figure 12.

(a) Average of swash energy density obtained on the first day (black thin solid line), second day (gray thin solid line), and over the entire period (black thick solid line) plotted on a log-log scale as a function of frequency f. Black dashed line shows the best fit of the saturated (linear on the log-log scale) band of the swash spectra (calculated between 0.03 Hz < f < 0.3 Hz). (b) Variance σ2 of swash energy density as a function of f on a log scale (line styles are associated with the same periods defined for Figure 12a). Vertical gray dashed lines in Figures 12a and 12b delimit the frequency regions corresponding to the sea swell and three infragravity bands adopted in this study. Data have been averaged alongshore.

5.2. Nonlinear Infragravity Energy Flux Contributions

[38] The nonlinear component of the net energy flux FNL(f) was observed to have a greater contribution to the net energy flux F(f) outside of the breaking region (see Figure 7). Within the surfzone, the bulk infragravity net energy flux FIg (linear + nonlinear components) was very similar to its linear component FL,Ig, suggesting that the nonlinear component could be neglected in very shallow water on low-sloping dissipative beaches. This is in contrast to observations of Henderson et al. [2006] in very shallow water on an intermediate beach (Duck), where FNL,Ig was found to be important. The contrasting patterns might be a result of contrasting morphologic and hydrodynamic conditions, since their observations took place on a steeper barred beach with a nearshore slope ∼0.07, with predominantly smaller offshore waves (significant wave heights in 8 m depth between 0.3 and 1.2 m).

5.3. Forced Waves

[39] The bispectral observations suggest that significant nonlinear energy transfer took place between incident swell waves and their harmonics between 2.6 and 0.6 m water depth (Figures 10a–10d). The patterns are somewhat similar to field observations described by Elgar and Guza [1985] between h = 9 and 1 m, in which shoaling of sea swell waves caused the wave shape to change from skewed to asymmetric forms, as a result of continuous energy transfer to phase-coupled waves at harmonic frequencies, and changes in the phase structure among the nonlinearly coupled waves. Field observations have shown that the harmonics can eventually be released as they propagate over a sandbar, resulting in an increasing number of individual wave crests shoreward of the sandbar crest [e.g., Elgar et al., 1997; Masselink, 1998; Sénéchal et al., 2002; Guedes et al., 2011]. We observed a similar shoreward increase in the number of individual wave crests, even for a more complex bathymetry lacking a well-defined sandbar. Figure 13a shows the number of individual wave crests per minute fCrest, estimated from NiwaOut ADV using a modified version of the zero-crossing method [see Guedes et al., 2011], as a function of the cross-shore distance from the shoreline ΔXS. The frequency of observed individual wave crests fCrest increased by a factor of 2 from the farthest to the shallowest locations, with the rate of change in the shoreward direction getting higher at about ΔXS ≈ 110 m (h ≈ 1.2 m). The harmonic release can also be observed in the time stacks as the divergence of an individual wave into two wave crests, which propagate toward the shoreline with different wave speeds (Figure 14), and might be associated with the shoreward decrease in the nonlinear coupling involving sea swell and harmonic waves observed in Figure 10c.

Figure 13.

(a) Frequency of wave crests estimated as the number of wave maxima per minute, (b) Infragravity bicoherence (see Figure 10), and (c) groupiness factor GF, as a function of the cross-shore distance from the mean shoreline location ΔXS. Solid lines in Figure 13a are best fit lines calculated from data shoreward and seaward of ΔXS = 110 m. Solid curves in Figures 13b and 13c are second-degree polynomials fitted to the data. Data from NiwaOut ADV are shown.

Figure 14.

Time stack created over the Niwa line on 9 November around 08:05 DST. The harmonic release can be observed as the divergence of an incident wave into two wave crests which propagate toward the shoreline with different wave speeds (notice the different slope of the two waves highlighted in the figure).

[40] The cross-shore evolution of infragravity waves observed within the surfzone in very shallow water is a shoreward continuation of the progressive evolution that has been observed in deeper water by Elgar and Guza [1985]. The infragravity biphases observed around 2.5 m water depth are close to the biphase of π for a bound wave produced by difference interactions, and their cross-shore evolution agrees with field [e.g., Elgar and Guza, 1985; Masselink, 1995] and laboratory observations [e.g., Janssen et al., 2003; Battjes et al., 2004] of increasing phase lag among interacting frequencies and forced bound waves in the shoreward direction. In fact, Elgar and Guza [1985] observed biphases for selected triads involving infragravity modes to decrease from about 170° to 140° for h between 6.0 and 2.0 m (see their Figure 12). Their shoreward-most observations were at similar depths and had similar biphases to our seaward-most observations (Figure 10f), suggesting that the phase shift progressively evolves toward very shallow water. This evolution is consistent with laboratory observations of van Dongeren et al. [2007], in which short waves propagating around the long wave crest in very shallow water converged on the crest as a result of long wave-induced changes in local depth and currents.

[41] The breaking of waves at sea swell frequencies affected the evolution of waves at infragravity frequencies. Nonlinear wave coupling involving infragravity waves was clearly reduced as the short waves started to break (Figure 10e) and followed by a shoreward decrease in both the infragravity net energy flux and nonlinear wave energy transfer to infragravity waves (Figures 7-9). These observations are consistent with the abrupt decrease of nonlinear forcing within the surfzone and agree with observations reported in previous studies [e.g., Ruessink, 1998a, 1998b; Henderson and Bowen, 2002; Sheremet et al., 2002]. Figures 13b and 13c show the averaged infragravity bicoherences math formula and groupiness factor GFXS), respectively, a relative measure of wave groupiness proposed by List [1991] and calculated from the ADV pressure time series. GFXS) decreased in the shoreward direction for ΔXS > 110 m, as the waves started to break, which is possibly associated with the reduction in the infragravity coupling math formula and nonlinear energy transfer WIg observed over the same region (see Figures 8 and 10). On the other hand, the (surfzone) region shoreward from ΔXS ≈ 110 m, where the rate of harmonic release increased (see the fCrestXS) slope change in Figure 13a), experienced a shoreward increasing trend for both GFXS) and math formula. The surfzone wave groupiness observed here is also consistent with List [1991] who found significant wave height variability to persist within the saturated surfzone. Although the change in the slope of fCrestXS) at h ≈ 1.2 m is not understood, we hypothesize that difference interactions among free waves at higher harmonic frequencies might have occurred within the surfzone. Figure 15a shows the nonlinear energy transfer W(f1 + f2, f1) to (positive) and from (negative values) infragravity waves with frequency f2 by triplets with frequencies f1, f2, and f1 + f2, calculated (following Henderson et al. [2006]) as

display math(20)

where math formula denotes the co-bispectrum. Significant nonlinear energy transfers can be observed for triads involving an infragravity and at least one harmonic wave (region between dashed lines), in contrast to observations of Henderson et al. [2006] on a steeper barred beach, where most transfers occurred through triad interactions with swell frequencies (defined for 0.05 Hz < [f1 + f2, f1] < 0.15 Hz). For the deeper cross-shore locations, positive and negative infragravity transfers involving harmonics largely cancelled each other, possibly in part due to the small number of degrees of freedom in the analysis, and the predominantly positive mean energy transfer math formula was small when compared to the mean energy transferred from swell waves (Figure 15b). On the other hand, average nonlinear energy transfers from infragravity to swell frequencies in shallow water (ΔXS < 110 m) were of similar magnitude to average transfers to harmonic frequencies (Figure 15b). These observations suggest that harmonic waves released in very shallow water could play a key role in the infragravity energy balance under dissipative conditions.

Figure 15.

(a) Example nonlinear energy transfers W(f1 + f2, f1) (equation (20)) to motions at infragravity frequency f2 by triads with frequencies (f1, f2, f1 + f2), calculated from a time series obtained by NiwaOut ADV at 10:10 (DST), during the first tidal cycle near high tide. Diagonal dashed lines mark region below which f1 + f2 < 0.5 Hz (top line) and 0.15 Hz (bottom line). Horizontal solid line along f1 = 0.05 Hz delimits region above which triads consist of only one infragravity wave. (b) Nonlinear energy transfers math formula to/from infragravity waves averaged over the regions in W(f1 + f2, f1) where (gray circles) both f1 + f2 and f1 are “swell” frequencies (0.05–0.15 Hz); and (black asterisks) at least one frequency (f1 + f2) in the triads is within the “harmonic” range (0.15–0.50 Hz), as a function of the cross-shore distance from the mean shoreline location ΔXS. Positive and negative math formula indicate net energy transfer to and from infragravity waves, respectively, from all triads within each defined region.

6. Conclusions

[42] Swash oscillations on a low-sloping, dissipative beach were largely dominated by low-frequency (f < 0.025 Hz) infragravity waves. Incident waves at sea swell frequencies were nearly fully dissipated through the surfzone by breaking, which yielded runup saturated at the incident band. In addition, swash energy at a higher-frequency infragravity band (0.035 Hz < f < 0.05 Hz) also appeared to be saturated, similar to observations under high-energy offshore wave conditions, albeit offshore wave condition during our observations were mild. Our observations show that the frequency distribution (and total energy) of infragravity waves in infragravity-dominated swash motions was entirely consistent with the gradual dissipation of infragravity energy observed in the shallow surfzone.

[43] Infragravity energy dissipation was strongly associated with breaking of sea swell waves within the surfzone. Sea swell wave breaking abruptly decreased nonlinear transfers to infragravity motions, making infragravity losses (e.g., energy transfers from infragravity waves) prevail over forcing and infragravity fluxes to decrease in the shoreward direction. Infragravity wave reflection increased with decreasing infragravity frequency but was less than unity for all frequency bands. Tidal modulations in the dissipation pattern associated with changes to the beach face slope over the concave profile were largely responsible for shifting the zone of infragravity runup saturation to higher and lower frequencies. The strong control of beach slope on infragravity-dominated runup is consistent with field observations of Ruggiero et al. [2004].

[44] Our observations suggest that the pathway of energy transfer between incident and infragravity frequencies as the waves progressed across the surfzone and into the swash zone involved higher-order harmonics. We showed evidence of nonlinear difference interactions involving infragravity and high-frequency, secondary harmonic waves within the inner surfzone. Our findings suggest that these nonlinear harmonic waves could play a role in the wave energy balance near the shoreline on low-sloping, dissipative beaches. This may be in contrast to steeper barred beaches, where transfers of infragravity energy to and from swell waves have been shown to be more important.

[45] Most infragravity forcing clearly occurred outside the surf zone, and energy levels at sea swell frequencies in the swash zone were very small, suggesting that bore-bore capture of sea swell waves [e.g., Mase, 1995] may not be an important mechanism of infragravity generation on very gentle dissipative beaches.


[46] R.G. and G.C. were funded by the (New Zealand) Foundation for Research, Science and Technology. R.G. also acknowledges support from MetOcean Solutions Ltd (New Zealand), and G.C. acknowledges funding from the “Cantabria Campus Internacional Augusto Gonzalez Linares Program.” The field experiment would not have been possible without the support of Cliff Hart, Iain MacDonald, Rod Budd, George Payne, and Dirk Immenga. Thanks are due to Wing Yan Man for extensively helping with the runup digitization. Thanks are also due to Alex Sheremet for commenting on an earlier draft of this manuscript. This work was greatly improved by suggestions from two anonymous reviewers.