2.1. Facilities

The experiments were conducted at the rebuilt (2017) Glass Channel Facility in the Hydraulics Laboratory at the Scripps Institution of Oceanography (SIO). The stainless steel and glass tank (see figure 2a) is 32 m long, 0.5 m wide and 1 m deep. Wave packets were generated using a computer-controlled electro-mechanical wave maker installed at one end, while a beach of $9^{\circ }$ slope, covered with a thick fibrous mat, was installed on the other end, to absorb and dissipate any waves that reached the end of the channel, minimizing wave reflections. Each run corresponded to the generation and measurement of a single wave packet. The tank was filled with fresh water to a working depth of 0.50 m.

Figure 2.

Diagram of the experimental set-up. (a) Conceptual diagram of the wave tank, including location of the breaking region and the seven wave gauges. (b) Close-up of the breaking region. In these experiments, the measured trajectories of 18 nearly neutrally buoyant particles were used to estimate the total surface transport induced by focusing non-breaking and breaking wave packets. The images used to measure particle displacement were taken with the downward-looking camera. The side-looking camera was used to measure wave height at breaking ($h$) and to obtain the breaking locations $x_b$. A hydrophone (not pictured) was used to measure the active breaking duration for the breaking packets and was located at the bottom of the tank, roughly $50\ \text {cm}$ down-channel from the breaking location.

Note, throughout this paper we will make use of the archived data from Lenain et al. (Reference Lenain, Pizzo and Melville2019), which, when necessary to distinguish, we denote as LPM, while the data from this experiment are denoted SGLP. Wave packet spectral parameters and measurements for these experiments are provided in table 1.

Table 1.

Table of wave packet spectral parameters and measurements; $f_c$ is the packet centre frequency, $\varDelta$ is the corresponding bandwidth, $S_*$ is the breaking threshold, $S$ is the slope, $h$ is the wave height at breaking, $c_{gs}$ is the spectrally weighted group velocity, $\delta F_b$ is the energy flux lost due to breaking, $\tau$ is the duration of active breaking, $b$ is the breaking dissipation, $x_{02}-x_{01}$ is the width of the averaging window used when calculating the net surface transport and $\langle \overline {\delta x}\rangle$ is the mean surface transport.

2.2. Focusing packet generation

The wave packets were generated using a dispersive focusing technique. That is, by exploiting the fact that longer waves travel faster than shorter waves in deep water, and tuning the relative phases of the packet components, focusing packets may be created (Longuet-Higgins Reference Longuet-Higgins1974; Rapp & Melville Reference Rapp and Melville1990).

Following Drazen et al. (Reference Drazen, Melville and Lenain2008), in these experiments we generate wave packets of the form

(2.1)\begin{equation} \eta(x,t) = \sum_n^N a_n\cos(k_n(x-x_b)-\omega_n(t-t_b)), \end{equation}

where $a_n$ are the amplitudes of the $N$ Fourier components, and $k_n$ and $\omega _n$ are their wavenumbers and radial frequencies, respectively, related by the dispersion relation $\omega _n^2 = gk_n\tanh (k_nd)$ in water of depth $d$, where g is gravitational acceleration. The variables $x_b$ and $t_b$ are the prescribed focusing location and time, respectively, as predicted by linear theory. Thus, according to (2.1), all of the components constructively interfere at $(x_b,t_b)$. Furthermore, the wave train described by (2.1) is windowed, so that only one wave packet is generated.

Furthermore, the spectrum of the wave packet is localized in frequency space, so that it has a well-defined bandwidth and central frequency $f_c$. The normalized bandwidth $\varDelta$ is defined as

(2.2)\begin{equation} \varDelta \equiv \frac{f_N-f_1}{f_c}, \end{equation}

so that the dimensionful bandwidth $f_c\varDelta$ contains all of the Fourier components which contribute to the wave packet at focusing. Note, the definition presented in (2.2) assumes that the spectral distribution is uniform.

2.3. Surface elevation and hydrophone measurements

The height of the free surface $\eta (x,t)$ was measured with seven resistance wire wave gauges placed along the channel at positions 1.76, 4.76, 5.38–7.76, 13.76, 15.76, 17.76 and 19.76 m from the mean position of the wave paddle. (The third wave gauge was moved upstream in some cases to avoid being near the breaking location of the narrow-banded packets). The wave gauge signals were subsequently amplified using an amplifier from the Danish Hydraulics Institute (Model 80-74G) and digitized. Furthermore, we conducted experiments in which no particle locations were measured, but instead we recorded the duration of active breaking $\tau _{br}$, which was estimated from the acoustic signal of the breaking event, measured by an hydrophone placed at the bottom of the tank.

2.4. Surface transport measurements

To measure the surface transport, the surface of the water was seeded with 18 slightly buoyant coloured particles of $11.8\ \text {mm} \pm 0.1\ \text {mm}$ diameter. The size of the particles was chosen for their visibility before and after breaking, and because they should follow the flow streamlines without excessive slip (Melling Reference Melling1997). The mass densities of the particles were calculated from their buoyant forces. Buoyancy was measured by releasing the particles at depth in a small tank filled with water at rest and recording their rise velocities with a high-speed camera (Model Phantom M320s), giving a mean particle material density of $0.9960\ \text {g}\ \text {cm}^{-3}$ with a maximum density of $0.9974\ \text {g}\ \text {cm}^{-3}$ and a minimum density of $0.9937\ \text {g}\ \text {cm}^{-3}$. Starting at rest and at depth, the particles were observed to rise at speeds $O(10^{-2})\ \text {m}\ \text {s}^{-1}$ (with corresponding Reynolds number $O(10)$). Typical speeds observed during breaking are $O(1)\ \text {m}\ \text {s}^{-1}$ (Rapp & Melville Reference Rapp and Melville1990; Melville et al. Reference Melville, Veron and White2002; Drazen & Melville Reference Drazen and Melville2009). Therefore, the magnitudes of particle rise velocities are small in comparison with the speed of the flow near the surface at breaking, so we expect the effects due to buoyancy to negligibly affect the particle trajectories during breaking.

The particles were painted to distinguish between them, but to accommodate 18 particles, some colours were repeated. Particles of the same colour were placed sufficiently far apart to distinguish them.

Before the wave packet was generated, the particles were released and remained at rest almost completely submerged. The wave maker generated a wave packet and two cameras placed near the breaking region were simultaneously triggered. A downward-looking camera, mounted above the channel, took pictures at a prescribed sampling frequency, and was used to record the particle locations before and after the passing of the wave packet. A side looking camera was used to identify the location of maximum focusing $x_b$, and, in the case of wave breaking, to measure the height of the breaking wave $h$. The cameras sampled for less than one minute, while the wave gauges sampled for three minutes. After the wave gauges stopped recording, the channel was left undisturbed for 12 min to allow time for the water to come to a state of rest, at which point the next run was conducted.

This operation was repeated with the initial positions of the particles being moved upstream and downstream of the breaking location so that the Lagrangian transport was measured across the entire region of significant transport.

2.5. Imagery

Images of the breaking region were taken with two colour cameras (see figure 2b): a downward-looking camera and a side-looking camera. The downward-looking camera was a Nikon D810 SLR ($7360 \times 4912\ \textrm {px}$, $35.9 \times 24 \ \text {mm}$ full frame FX format CMOS sensor with $4.88\ \mathrm {\mu }\text {m}$ pixel size) that was mounted above the channel and equipped with a 14 mm $f/$2.8$D$ Nikon lens. The camera took $104$ images per run, sampling at $3\text { Hz}$ for the first 80 images, followed by an image every $600\ \text {ms}$ for the remaining 24 images. The side-looking camera was a JaiPulnix AB800CL colour camera ($3296 \times 2472\ \textrm {px}$, $18.13 \times 13.6\ \text {mm}$ 4/3 inch format CCD with $5.5\ \mathrm {\mu }\text {m}$ pixel size) with a $14\ \text {mm}$ $f/$2.8$L$ Canon lens. This camera sampled at 10 Hz. The camera was placed adjacent to the channel and pointed at the wave-breaking region.

3.1. Characteristic wave variables

Following Rapp & Melville (Reference Rapp and Melville1990), we characterize the focusing wave packets through the linear prediction of the maximum slope at focusing $S$, the normalized bandwidth $\varDelta$, the central frequency $f_c$ and the focusing location according to linear theory $x_b$. In this study we examine the effects of the two variables $S$ and $\varDelta$ on surface transport.

3.2. The linear prediction of the maximum slope, $S$ and the wave packet bandwidth, $\varDelta$

The strength of focusing or breaking is measured with the quantity $S$, the linear prediction of the maximum slope at focusing, (which we also refer to as the slope) and has been traditionally equated with a sum of the slopes $\sum _n a_nk_n$ of the Fourier components in (2.1). This parameter corresponds qualitatively to the strength of the breaking event, with larger $S$ generally implying a stronger breaking event, which is characterized by a greater wave height at breaking $h$ (the definition of which is illustrated in figure 1(b) of Drazen et al. Reference Drazen, Melville and Lenain2008).

Following Tian et al. (Reference Tian, Perlin and Choi2010), we seek to measure this quantity, instead of using the a priori prescribed value, in order to avoid any assumptions regarding the transfer function of our wave paddle. As was noted by Tian et al. (Reference Tian, Perlin and Choi2010), the sum $\sum _n a_n k_n$ of the Fourier coefficients at the first wave gauge does not converge due to the amplification of the shorter wave components. This is remedied by excluding Fourier components that lie outside of the bandwidth $f_c\varDelta$.

The bandwidth was determined by minimizing the $L_2$ norm of the difference between the measured and the linear prediction of free surface displacement at wave gauges 2 and 3 (which were chosen because they are upstream of the breaking location). The linear prediction was computed by propagating the free surface measured at the first wave gauge and band-pass filtered using a window of width $f_c \varDelta$ centred on $f_c$. The bandwidth was increased until this metric reached a constant value. Note that the bandwidths described using this method were such that, on average, they contained $95.9\pm 1.8\,\%$ of the total wave packet energy measured at wave gauge 1. Furthermore, the computed bandwidths are linearly proportional to the prescribed bandwidth values.

Because excluding the Fourier components that lie outside of our computed bandwidths improves the accuracy of the linear prediction of the free surface upstream from the breaking location, it will also improve our estimates of the slope at focusing $S$. Therefore, the slope is defined as

(3.1)\begin{equation} S \equiv \max(|\eta_{x}(x,t)|), \end{equation}

where $\eta (x,t)$ is the linear prediction of the free surface displacement and $\eta _{x}(x,t)$ is its spatial derivative. The linear prediction of the free surface displacement was computed from the spectral and phase information of the band-passed wave packet measured at the first wave gauge. The surface displacement and its spatial derivative were computed with a spatial resolution of $1\ \text {cm}$, which is small enough to accurately estimate $S$.

The most important test of our estimates of the slope is its relationship with the wave height at breaking $h$, which is shown in figure 3. The slope is a measure of the strength of breaking, and its correlation with the wave height at breaking is the motivation for its use in these experiments. In that figure, it is shown that the linear prediction of the maximum slope scales approximately linearly with $hk_c$, so that this parameter $S$ characterizes the breaking strength. For comparison, figure 15 of Drazen et al. (Reference Drazen, Melville and Lenain2008) also compares prescribed $S$ values with direct video measurements of $hk_c$. They also observed a linear relationship, but with significantly more scatter. This implies that our method of estimating $S$ provides a more accurate description of the breaking region than the prescribed values. We also note that, while each data point on that figure in Drazen et al. (Reference Drazen, Melville and Lenain2008) represents a single direct video measurement, each data point in figure 3 of this paper represents the mean of 50 to approximately 150 direct video measurements.

Figure 3.

The dimensionless wave height $hk_c$ plotted against the linear slope at focusing $S$ minus the breaking threshold value $S_*$. The black line is a best fit of the form $hk_c = \chi _h(S-S_*)$, where $\chi _h$ is a fitting constant equal to 4.15. There is no significant effect of the bandwidth on the relationship between $hk_c$ and the slope $S$, save for through the weak dependence of $S_*$ on $\varDelta$, which is accounted for by our choice of the independent variable as $S-S_*$. This figure includes data from LPM (triangles) and SGLP (circles).

Wave packets with $S$ values slightly above a breaking threshold $S_*$ exhibit spilling breaking, and as $S$ increases, the packets transition to plunging breakers or multiple breakers. In this study, only single breaking events are considered. Note, we define the breaking threshold as the value $S_*=S$ when the free surface first entrains air, as detected by the side-looking camera. The breaking threshold values are given in table 1 and shown in figure 9.

Note, in § 6 we consider archived data from Drazen et al. (Reference Drazen, Melville and Lenain2008), and recompute their values of $S$ and $\varDelta$ using the definitions given in this section.

3.3. Surface transport measurements

Figure 4 shows measurements of the surface Lagrangian transport $\delta x$ computed from particle displacements as a function of the distance from the breaking location $(x_0-x_b)$ where $x_0$ is the initial position of the particle and $x_b$ is the measured breaking location (this is the location where the free surface first becomes multi-valued) of the wave packet. The figure is organized into panels by the bandwidth and the data points are colour coded by the slope $S$. Each panel in figure 4 contains all the Lagrangian transport measurements of each run for a given bandwidth and central frequency. Every data point in these figures corresponds to a single particle displacement.

Figure 4.

Lagrangian transport $\delta x$ as a function of the initial distance from the breaking location, $(x_0-x_b)$. Each point corresponds to a single particle displacement. Each panel presents data from packets with the same bandwidth $\varDelta$ and central frequency $f_c$; data points are colour coded by slope $S$; $x_b$ is defined as the location where the free surface becomes vertical. For non-breaking packets, the breaking location $x_b$ of the lowest $S$ single breaking packet is used.

The Lagrangian transport is noticeably asymmetric, with the maximum of the transport located close to the breaking location $x_b$. It is sharply peaked for packets with higher $S$ and $\varDelta$ values. The low transport values slightly downstream of $x_b$ in figures 4(b) and 4(d) correspond to particles that have been pushed downward into the water column by energetic plunging breakers. The slopes of the breaking wave packets with bandwidths $\varDelta = 0.77$ and $\varDelta = 0.91$ were in the range $0.32 - 0.33$ and $0.38 - 0.41$, respectively, while the slopes of breaking wave packets with bandwidths $\varDelta = 1.05$ and $\varDelta = 1.15$ were in the range $0.30 - 0.39$, and $0.38 - 0.46$, respectively. The narrowness of the range of breaking slopes measured in the two smaller bandwidth wave packets is due to the technical limitations of generating steep wave packets with narrow bandwidths in a wave tank of finite length. Wave packets with small bandwidth need more distance to focus and de-focus than packets with larger bandwidth. To counteract this effect, for small bandwidths, packets with large slope values must be generated with an initial large steepness which can violate our assumption that the wave field is linear at the first wave gauge. Therefore, the initial steepness of the small bandwidth wave packets has been limited, reducing the range of S values for these breaking packets.

In figure 5, the averaged surface Lagrangian transport $\overline {\delta x}$ for each slope and bandwidth combination is shown. The averaged Lagrangian transport $\overline {\delta x}$ is computed by averaging the Lagrangian transport $\delta x$ over a running window as follows:

(3.2)\begin{equation} \overline{\delta x}(x_0) = \frac{4}{\lambda_c}\int_{x_0-\lambda_c/8}^{x_0+\lambda_c/8} \delta x(\tilde{x}) \,\text{d}\tilde{x}, \end{equation}

where $\tilde {x}$ is a dummy variable. This is the same method as employed in Lenain et al. (Reference Lenain, Pizzo and Melville2019).

Figure 5.

Averaged Lagrangian transport $\overline {\delta x}$ as a function of the distance from the breaking location, both variables being normalized by the central wavenumber $k_c$. The averaged Lagrangian transport $\overline {\delta x}$ is defined in (3.2). Each panel presents data from packets with the same $\varDelta$ and $f_c$. The breaking location $x_b$ is the same as in figure 4. The effect of $S$ on both the amplitude of the transport and the width of the breaking region is clearly illustrated here, particularly in (c,d), as increasing $S$ noticeably increases both of these values. Qualitative differences in the shape of the average transport curves is noticeable between wave packets with different bandwidths but similar $S$ values, implying that the Lagrangian transport depends both on $\varDelta$ and $S$. (Note, the sudden increase in average transport far downstream from the breaking region in panel (b) is caused by four data points (visible in figure 4b), so we do not consider it statistically significant.)

In figure 5 the dependence of the mass transport on the slope is apparent, and some effects of the bandwidth can be seen. Within figures 5(c) and 5(d), greater average transport and wider regions of significant breaking-induced transport are observed in steeper wave packets. There are also qualitative differences observed between breaking wave packets with different bandwidths but similar slopes. For instance, in figure 5(b), the averaged surface transport $\overline {\delta x}(x_0)$ of the breaking packets generally decrease monotonically from their maxima. The slope values are ranging from $S=0.38$ to $S=0.41$. In figure 5(c), the curve showing the averaged transport profile of the wave packet of the highest slope $S=0.39$ has two maxima separated by a distance of roughly 7 radians (i.e. approximately 2 m).

To illustrate the effects of bandwidth on the average transport, the average transport curves of two wave packets of similar slope and differing bandwidths are plotted in figure 6. From this figure, it can be seen that an increase in the bandwidth $\varDelta$ corresponds to a decrease in the width of the region of significant breaking-induced transport, as well as a decrease in the average transport.

Figure 6.

Averaged Lagrangian transport $\overline {\delta x}$ of two wave packets with similar slopes but different bandwidths. The average transport and the width of the region of significant breaking-induced transport are observed to decrease as the bandwidth is increased. Both axes are normalized by the central wavenumber $k_c$, which is the same for both of the wave packets in this figure.

To estimate the overall surface transport induced by the passing of a wave packet, we define the mean surface transport $\langle \overline {\delta x}\rangle$ as the integral of the averaged surface transport $\overline {\delta x}$ over the region of significant transport, the limits of which are denoted $x_{01}$ and $x_{02}$, such that

(3.3)\begin{equation} \langle\overline{\delta x}\rangle \equiv \frac{1}{x_{02}-x_{01}} \int_{x_{01}}^{x_{02}}\overline{\delta x}(x_0)\,\text{d}\kern0.7pt x_0. \end{equation}

The limits of the region of significant transport are defined as the first points upstream and downstream of the breaking location where the average transport curve becomes flat.