3.3.1. Breaking region development and evolution after jet impact

After jet impact, significant run-to-run variations of the profiles are found in some spatial regions while in other regions the profiles are quite repeatable. The run-to-run variations are to be expected since at jet impact the flow begins transition to a temporally evolving and spatially non-uniform turbulent flow. An illustration of the run-to-run variation of the surface profiles after jet impact is given in figure 8(a) in which 10 profiles, one from each realisation of the strong breaker, all recorded $\tilde t = 171\pm 12.3$ ms after jet impact, are shown along with the ensemble-averaged profile and a measure of SD, $n_{sd}$, see figure 8(b) for definition. Starting from the right and proceeding to the left (downstream in laboratory coordinates), the profiles are at first well aligned; the first local height maximum is the original crest of the breaker and the second maximum is both the current wave crest point (the highest point on the profile) and the top of the first splash zone. The local minimum, referred to below as an indentation, between these two maximums is the point where the upper surface of the plunging jet is now submerging under the splash zone. Farther downstream of the indentation, the run-to-run variation in surface height increases with the maximum variation occurring near the leading edge (called the toe) of the splash zone (also called the turbulent breaking region). In the region downstream of the toe, the profiles are nearly the same in each run. It should be noted that the entire region shown in the plot is downstream of the jet impact location, $\tilde x =0$.

Figure 8.

(a) Breaker profiles in $\tilde y\lambda _0^{-1}$–$\tilde x\lambda _0^{-1}$ coordinates from 10 realisations of the strong breaker (solid black lines) all measured $171 \pm 12.3$ ms after jet impact. The thick coloured line is the ensemble-averaged profile, where the colour contour along this line indicates the SD of the normal distance $n$, $n_{sd}=\sqrt {\langle [n-\langle n\rangle ]^2\rangle }/\lambda _{gc}$, where $n$ is defined in (b) as the distance between the average profile and the individual profiles measured at each point along the average profile in the direction of its local normal and $\lambda _{gc}=1.7$ cm, is the wavelength of the gravity-capillary wave with minimum phase speed. A second measure of profile variability, based on the profile local arc length, is also defined in (b) as the SD of the difference between the local arc length of the ensemble-averaged profile ($s_m$) and the corresponding local arc length ($s$) of the profile of a single realisation of the breaker non-dimensionalised by $s_m$, $s_{sd}=\sqrt {\langle [s/s_m-1]^2\rangle }$.

Details of the evolution of the ensemble-averaged profile and the SD profile histories of the crest region for the weak, moderate and strong breakers are more clearly seen in the sequences of profiles in figure 9(a–c), respectively. The plots are shown in a reference frame moving with the speed of the crest point at the moment of jet impact, $\langle u^i_c\rangle$ (see table 2), and cover a streamwise width of $0.4\lambda _0$ and a time interval of $0.55f_0^{-1}$. Further details of the plots are given in the figure caption. In the moving reference frame of these plots, profile features that move faster (slower) than $\langle u^i_c\rangle$, move to the left (right) as time increases. The profiles in figure 9(a–c) are coloured locally according to the corresponding local value of $n_{sd}$. The horizontal and vertical positions of the indentations, the crest point and the toe of the splash, all extracted from the profiles in figure 9, are plotted vs time ($\tilde t f_0$) in figure 10. Finally, for comparison, a profile history from a single realisation of the strong breaker is given in figure 9(d). Comparison of the ensemble-averaged (figure 9c) and single realisation (figure 9d) profile histories helps to demonstrate the repeatability of the process, particularly the space–time locations of the indentations and splashes.

Figure 9.

Evolution of the ensemble-averaged crest profiles of the weak, moderate and strong breakers are presented in panels (a–c), respectively. Each line is the average profile at an instant in time, $\tilde t_k$, after jet impact ($k=0$) and the colour indicates the local value of the non-dimensional SD of the local normal distance, $n_{sd}$, see figure 8 its for definition. Each successive profile is plotted $\mathrm {d}\kern 0.05em y = 10$ mm above the previous profile and the temporal separation between profiles is $\tilde t_{k+1}- \tilde t_k = 0.0123$ s. The ninth breaker profile is marked by (i) and occurs at $\tilde{t} = 0.1107$ s; subsequent profiles marked by Roman numerals are 10 profiles apart. The profiles are shown in a reference frame moving with the speed of the crest point at the moment of jet impact, $\langle u^i_c\rangle$. The green squares mark the location of the highest point on each profile and the magenta upside down triangles are the local minimums of height in the indentations. A similar set of surface height profiles, but for only one realisation of the weakest breaker and plotted in the laboratory reference frame, was presented in Erinin et al. (Reference Erinin, Wang, Liu, Towle, Liu and Duncan2019). Panel (d) is a profile evolution plot for one realisation of the strong breaker.

Figure 10.

Plots of the dimensionless coordinates of several features of the breaking wave crest are plotted vs $\tilde t f_0$ in the laboratory reference frame. (a,b) The horizontal coordinates of the leading edge of the turbulent region (toe, $\tilde x_{toe} \lambda _0^{-1}$) and the local minima of the three indentations ($\tilde x_k \lambda _0^{-1}$), respectively. (c,d) The horizontal ($\tilde x_c \lambda ^{-1}$) and vertical ($\tilde y_c \lambda ^{-1}$) coordinates of the crest point, respectively. The time range in each plot is from the moment of jet impact to $\approx$730 ms after jet impact, the same time interval as the profiles shown in figure 9. The vertical red dashed lines labelled (i) to (vi) in (c) are plotted at the times of the profiles in figure 9 with the same labels.

The repeatability of the early phase of the breaking process, including the jet impact and several cycles of splash generation and impact is demonstrated by the structure of each of the ensemble-averaged profile histories as well as the qualitative similarity of the profile histories of the three breaking waves in figure 9. Despite the fact that profiles are taken from the period in which the flow transitions from a laminar to a turbulent flow, many features of the profile histories are repeatable enough in space and time in each realisation to survive the ensemble averaging. The first repeatable feature is, of course, the average crest profile at the moment of jet impact, which is the first (lowest) profile. These are the same profiles of the crest region that are shown in figure 3(d). The $n_{sd}$ magnitude along the first profile is essentially zero for each of the three breakers. Immediately after impact, an indentation forms between the plunging jet's upper surface and the splash generated by the first impact. This indentation quickly slows down and connects to the zone of very high $n_{sd}$ (between profiles (iii) and (iv) and centred near the horizontal position $-$0.2) that is associated with the bursting of bubbles entrapped under the plunging jet at the moment of impact, see Movie 4 included as supplementary material. The $n_{sd}$ level in this region increases monotonically with increasing breaking intensity, see § 3.3.2 for more details. The first indentation is also discussed in a number of studies including the experiments in Bonmarin (Reference Bonmarin1989) and the two-dimensional numerical computations in Iafrati (Reference Iafrati2009). In Bonmarin (Reference Bonmarin1989), the idea of air entrainment via this indention is discussed along with a few surface profiles, while in Iafrati (Reference Iafrati2009), analysis of the surface profile contours show that the indentation is deep and closes near the surface, entrapping a pocket of air. The LIF method in the present study only records the full depth of the indentation if the laser light rays are falling along the centreline of the cross-stream shape of the indentation and if the camera's line of sight is aligned with the indentation across the width of the tank. With a fixed camera, as in the present measurements, this alignment would only occur at one instant in time during a given breaking event. Since the cameras are stationary and not pointed across the tank width, the measured profiles indicate the location of the indentation, but not the deep centre of the cross section. In Reference Erinin, Liu, Wang, Liu and DuncanPart 2, it will be shown that a sheet of droplets is ejected at the instant in time when the indentation disappears in the profile sequence and that this disappearance is the result of the rapid rise of the bottom of the thin nearly vertical air crater at the indentation site.

The splash generated by the plunging jet impact forms a region of high $n_{sd}$ but the motion of its leading edge, the toe, is highly repeatable, see the region of the splash zone between profiles (i) and (ii) and to the left of the first indentation in figure 9(a–c). Another two cycles of splash impact and generation are also visible with the second splash impact occurring between profiles (ii) and (iii) and the third between profiles (iii) and (iv). The impact–splash cycles are particularly prominent in the profiles from the two stronger breakers. After the initial jet impact, the leading edge of the turbulent breaking region accelerates downstream in all three profile histories and goes through cycles of accelerating and decelerating between profiles (ii) and (iv) as the splashes form and impact the wave face. The speed of the front splash region is close to the initial phase speed of the wave at several times, between surface profile (ii) and (v), but then decelerates by the time it reaches surface profile (vi). The horizontal position of the leading edge (toe) of the breaking region, $\langle \tilde x_\mathrm {toe}\rangle \lambda _0^{-1}$, is plotted vs $\tilde t f_0$ in figure 10(a). The toe position increases monotonically in time and, at any time, increases with breaker strength. There is evidence of some surging forward and backward in some of the curves, but the amplitude is small.

In a manner similar to the formation of the first indentation, with each splash impact an indentation forms between the plunging splash and the secondary splash that forms ahead of it. Each of these indentations at first moves at or near the speed $\langle u_c^i\rangle$ and quickly slows down and falls behind the crest. The streamwise positions of these indentations, $\langle \tilde x_k\rangle$, were extracted from the ensemble-averaged profiles and are plotted in figure 10(b). The streamwise trajectory of the first indentation is nearly the same for the three breakers. The three trajectories of the second and third indentations are spread out in time and streamwise position, with the trajectories starting later in time and farther from the initial jet impact point as the breaker strength is increased.

The horizontal and vertical positions of the crest point (the highest point on each profile), $\langle \tilde x_c\rangle \lambda _0^{-1}$ and $\langle \tilde y_c\rangle \lambda _0^{-1}$, respectively, are plotted vs $\tilde t f_0$ in figure 10(c,d), respectively. The crest point is also depicted as the green squares in the profile history plots in figure 9. The three curves of horizontal position in subplot (c) follow a similar trend with regions of nearly linear increase connected by jumps downstream at 0.201$f_0^{-1}$, 0.437$f_0^{-1}$, and 0.575$f_0^{-1}$. Comparison with the profiles in figure 9(a–c), indicates that the jumps in $\langle \tilde x_c\rangle \lambda _0^{-1}$ occur as the crest point jumps across an indentation from the main crest to the first splash, from the first to the second splash, etc. The jumps occur at increasingly later times as the breaker intensity is increased. If one imagines a smooth curve going through the dataset for each wave, one can see that the curves for the three waves would be nearly the same, as are the raw data curves between each of the jumps. In addition, the local curve slope is higher for the overall curve than for the regions between jumps. The speed of the crest point in each of the linear segments, which are labelled by uppercase Roman numerals in figure 10(c), are listed in table 3. These speeds are substantially less than $\langle u^i_c\rangle$ and $c_0 =1.358$ m s$^{-1}$. The variable $\langle u^p/c\rangle$ in line one is the speed from a least-squares fit of a straight line to the data for each breaker over the entire range of the plot. These speeds are only a little less than $c_0$. As seen in figure 10(d), the height of the crest point, $\langle \tilde y_c\rangle \lambda _0^{-1}$, decreases dramatically after jet impact, approximately 0.05$\lambda _0$ (a little more than 50 % of the maximum crest height, see last line of table 2) in a time interval $\Delta \tilde tf_0^{-1}=0.8$. It is thought that this decay is due partially to the conversion of wave energy to energy in the turbulence and due to the wave crest moving past the peak of the wave packet envelope. On top of this decaying curve is an oscillation with decaying amplitude; each local minimum occurs at the jump points in figure 10(c), where the crest point moves from one ripple to the next. The overall slopes and shapes of the curves are similar in the three breakers, but the decay occurs later in time as the breaker intensity is increased. This general behaviour of $x_c$ and $y_c$ is also seen in Bonmarin (Reference Bonmarin1989) (figures 23, 24, 26 and 28) where the positions of the breaking crest and the first and second splash-ups are plotted vs time for a single plunging breaking event.

Table 3.

Post-impact horizontal speeds of the crest point and the toe of the three breakers. The speeds $\langle u^P_c \rangle$ are from the slopes of straight lines fitted to the entire range of data in figure 10(c). The speeds $\langle u^I_c\rangle$ to $\langle u^{IV}_c\rangle$ are from the slopes of straight lines fitted to the nearly linear segments of the $\langle \tilde x_c\rangle$–$\tilde t f_0$ curves in figure 10(c). The horizontal speed of the splash front, $\langle u_{toe} \rangle$, is measured immediately following jet impact from $\tilde t = [0\ \mathrm {to}\ 0.14]$ using the data shown in figure 10(e).

3.3.2. Wave field evolution after jet impact

In this subsection, the surface profile measurements over the entire measurement region in time ($-500\ {\rm ms} \leq \tilde t \leq 2500\ {\rm ms}$) and space ($-0.3\ {\rm m} \leq \tilde x \le 1.0\ {\rm m}$) are presented and discussed. Data for the ensemble averaged surface height, the local surface normal distance SD ($n_{sd}$), and the local arc length SD ($s_{sd}$) on the $\tilde x \lambda _0^{-1}-\tilde t f_0$ plane are shown in figures 11–13, respectively. Each figure contains subplots for the weak, moderate, and strong breakers in (a–c), respectively. The profile and $n_{sd}$ data in figure 9 are from a diagonal band (tilted toward increasing $\tilde t f_0$ and $\tilde x \lambda _0^{-1}$) starting at the jet impact location in the contour plots in figures 11 and 12. The definitions of the local surface normal distance ($n$) and the local arc length of the mean and instantaneous surface profiles ($s_m$ and $s$), respectively, are given in figure 8(b). These two measures of SD were chosen to emphasise different features of the run-to-run variations of the profile shape, as is discussed in the following.

Figure 11.

Contour maps of the non-dimensional mean surface height $\langle \tilde y\rangle \lambda _{gc}^{-1}$ from 10 breaker realisations on the $\tilde x \lambda _0^{-1} \mathit {-} \, \tilde t f_0$ plane for the weak, moderate and strong breakers are given in panels (a), (b) and (c), respectively. The wavelength $\lambda _{cg} = 1.71$ cm is the wavelength of the gravity-capillary wave with minimum phase speed. The contour maps are shown in the laboratory reference frame for the full measurement region, $\approx$1300 mm in streamwise distance and $\approx$3000 ms in time and have a spatial and temporal resolution of $\Delta x = 0.165$ mm and $\Delta t = 12.3$ ms, respectively. The position and time of the maximum height of the wave crest point and the jet tip impact and are indicated by the green square and the red triangle, respectively. The call-outs A–E and the magenta downward triangles are discussed in the text. During the time approaching the maximum crest elevation, the back face of the breaking wave crest becomes partially obscured from the camera view and is not measured, as indicated by the white region just to the right of the green square in panels (a) and (b).

Figure 12.

Contour maps of the non-dimensional SD of the local surface normal distance $n_{sd}$ generated from 10 realisations on the $\tilde x \lambda _0^{-1} - \tilde t f_0$ plane for the (a) weak, (b) moderate and (c) strong plunging breakers. See figure 8(b) for definitions of $n_{sd}$. Plot details are given in the caption to figure 11.

Figure 13.

Contour maps of the non-dimensional SD of the local arc length $s_{sd}$ generated from 10 realisations on the $\tilde x \lambda _0^{-1} - \tilde t f_0$ plane for the (a) weak, (b) moderate and (c) strong plunging breakers. See figure 8(b) for definitions of $s_{sd}$. The dashed orange line in all three plots is the horizontal trajectory of a surface particle according to linear wave theory modified to include a large drift current, see the text for a discussion. The spatial resolution is $\Delta x = 2.740$ mm, other plot details are given in the caption to figure 11.

From the surface height contour maps shown in figure 11, the mean spatiotemporal evolution of the large-scale surface motion of the focusing wave packet can be analysed. The height contour maps for the three breakers are quite similar. Perhaps the most important wave feature in these plots is the trajectory of the main breaking wave crest, the yellow band indicated by A and travelling from right to left as time increases (vertically up in the subplots). The sharp yellow–green transition on the lower side of the breaker crest is near the toe of the turbulent breaking region. In the moments leading up to jet impact, the mean wave crest height reaches a maximum, indicated by the green square. The jet tip impact point is denoted by the red triangle at $(\tilde x \lambda _0^{-1} , \tilde t f_0) = (0,0)$, after which the surface becomes irregular as the wave breaks, see discussion in § 3.3.1. In laboratory coordinates, the jet tip impact point for the weak breaker is located at $5.93$ m downstream from the back face of the wave maker, while the focusing distance parameter for the wave maker motion equation is $x_b = 6.17$ m, see (2.1). After the jet impact point, the breaking crest continues to decrease in amplitude while the amplitude of the crest following the breaker, indicated by D in the subplots, increases. This behaviour is consistent with the expected evolution of a deepwater wavetrain moving at its phase velocity through an amplitude envelope that moves in the same direction at the group velocity. A third crest of smaller height is barely visible in the upper right corner of the subplots. Shortly after jet impact, three indentations form sequentially on the main wave crest. The indentations are seen as faint dark orange lines in the otherwise yellow region at the breaker crest and are identified by the three sets of three magenta downward triangles drawn beginning at the heads of the three arrows at B in the three contour plots. These features were discussed in reference to the crest profile plots in figure 9 where the indentations can be seen more clearly in the sequences of profiles shown in crest-fixed coordinates. The downward magenta triangles for each indentation are the first, midrange and last downward magenta triangles of the corresponding indentation in figure 9(a–c).

Figure 14.

Normalised arc length parameter $\Delta s /\Delta x= (s-s_m)/\Delta x$ and normalised average normal displacement $\bar n/\Delta x$ both over a measurement interval $\Delta x$, see figure 8 for definitions, with the mean profile taken as a straight horizontal line ( $y=0$, length $s_m=\Delta x$) and an individual profile realisation taken as a sine wave, $y = a\sin kx$, where $k=2{\rm \pi} /\lambda$ and $\lambda$ is the wavelength. The measurement interval $\Delta x$ is from $x=0$ to $x=\lambda /2$. The slope of the $\bar n/\Delta x$ curve is $1/{\rm \pi} ^2=0.101$, while the slope of the $\Delta s/\Delta x$ curve is zero at $ak=0$ and asymptotes to a straight line with a slope of approximately 0.52 at $ak = 2.0$.

As mentioned previously, two measures of the local SD of the breaker profiles are presented herein: the normal distance SD, $n_{sd}=\sqrt {\langle [n-\langle n\rangle ]^2\rangle }/\lambda _{gc}$, and the arc length SD, $s_{sd}= \sqrt {\langle [s/s_m -1]^2\rangle }$. Both quantities are dimensionless and are defined in figure 8(b). The normal distance SD was used in the discussion of the crest profile evolution in § 3.3.1. The normal distance was chosen rather than the vertical height as a measure of the difference between two water surface profiles because the height change exaggerates the miss match of the two curves when a small phase difference occurs between them at sharp profile features such as the bottom of the indentations discussed in § 3.3.1. The arc length SD is a measure of the relative shape of two profiles. If the individual and average profiles have the same shape and orientation over $\Delta x$, the difference in $s_{sd}$ is zero no matter the displacement ($\bar {n}$) between the two curves. For the same curves, $n_{sd}$ would equal the displacement. If the shapes of the two curves are different over $\Delta x$ but the curves are on top of one another, the $n_{sd}$ may be small, while $s_{sd}$ can be large. In order to explore this difference numerically, a plot of these two quantities for an interval with an average profile consisting of a straight line (say $y=0$) and an individual profile consisting of a sine wave, $y = a\sin {kx}$, where $k = 2{\rm \pi} /\lambda$ and $\lambda$ is the wavelength, is given in figure 14. The arc length and $\bar n$ are computed over the interval $0\leq x \leq \lambda /2$. In this case, $\bar n$ is due to the shape of the profile within the interval $\Delta x$ and $\bar n = 2a/{\rm \pi}$. As can be seen from the figure, $\bar n$ varies linearly with $ak$ whereas the arc length at first increases more slowly than $\bar n$ but soon increases more rapidly. Thus, compared with $n_{sd}$, $s_{sd}$ is less sensitive to small slope disturbances and more sensitive to large slope disturbances. These differences in the behaviour of the two measures of SD create contour plots that emphasise different features of the surface roughness.

The contour plots of $n_{sd}$ and $s_{sd}$ are given in figures 12 and 13, respectively. In these plots, the call-outs $A$ through $D$, the points of maximum crest height and jet impact, and the downward magenta triangles marking the locations of the indentations are identical to those in the corresponding subplots in figure 11. The most striking feature of all of the $n_{sd}$ and $s_{sd}$ contour plots is the sharp boundary between the low-SD-level region in the lower left (downstream, i.e. in the direction of wave travel) and the high-SD-level region of the front face of the wave crest. This boundary is slightly ahead, down and to the left, of the sharp blue–yellow transition in the panels in figure 11 and corresponds to the leading edge (toe) of the breaking region. The same boundary is visible in the $n_{sd}$ colour contours on each of the ensemble-averaged profiles shown in crest-fixed coordinates in figure 9. The lower left regions of the $n_{sd}$ and $s_{sd}$ contour plots correspond to the water surface upstream of the breaker and the low values of $n_{sd}$ and $s_{sd}$ in these regions are a testament to the repeatability of the ten realisations of the non-breaking sections of the wave profile in $\tilde x$–$\tilde t$ coordinates.

The $n_{sd}$ contour plots in figure 12 show a number of interesting features. In the case of the weak breaker, figure 12(a), one can see three diffuse bright elongated regions along the locations of the three indentations in the height contour plots at positions $B$. The bright regions along the indentation locations become less distinct as the breaker intensity increases, see figure 12(b,c). A small region of high $n_{sd}$ contour levels is found at location $E$, in all three subplots. This location is in line with but well past the end of the first indentation. The $n_{sd}$ levels in these small regions, approximately 2.07, 3.20 and 3.43 mm in the weak, moderate and strong breakers, respectively, are the highest in each contour map. These dimensional values correspond to $0.025 H^i$, $0.036 H^i$ and $0.038 H^i$ and $0.039\langle r^i_y\rangle$, $0.049\langle r^i_y\rangle$ and $0.049\langle r^i_y\rangle$, for the weak, moderate and strong breakers, respectively. See figure 4 for the definitions and table 2 for values of $H^i$ and $\langle r^i_y\rangle$ for each breaker. Observations of the diffuse-light movies indicates that the patch of high-SD level corresponds to the location where the large air bubbles entrapped under the jet at impact escape from the back face of the wave well past the end of the trajectory of the first indentation, see Movie 4 given as supplementary material. The closing of the indentation crater, as discussed above, occurs well before the emergence of the bubbles entrapped under the plunging jet. This large bubble bursting region was also noted by other investigators in experiments, including Bonmarin (Reference Bonmarin1989) and Blenkinsopp & Chaplin (Reference Blenkinsopp and Chaplin2007), and in numerical calculations, including Wang et al. (Reference Wang, Yang and Stern2016) and Mostert et al. (Reference Mostert, Popinet and Deike2022).

In the remaining areas of the $n_{sd}$ contour maps, the $n_{sd}$ level is lower than in the breaking crest region but exhibits some interesting well-organised features. The magnitude of $n_{sd}$ decreases in the trough ($C$) between the breaking crest and following wave crest, increases again at the following wave crest ($D$) to a lower level than on the breaker crest and falls off again after the second crest. The pattern of light blue small-scale features in the $n_{sd}$ contour plots in the area bounded by the breaking crest on the lower left edge and approximately by the location corresponding to that of the red dashed wavy line between $E$ and $D$ in the $s_{sd}$ contour plots in figure 13 is likely the manifestation of the breaker-generated turbulent flow. The slow downstream mean motion of the wavy boundary is consistent with the idea of turbulent vortical flow features moving with the fluid rather than the wave crest. The relatively high level of $n_{sd}$ at the following wave crest may be associated with an interaction between the flow field of the long wavelength carrier wave of the packet and the short waves generated by the turbulence (Phillips Reference Phillips1981; Longuet-Higgins Reference Longuet-Higgins1987; Miller, Shemdin & Longuet-Higgins Reference Miller, Shemdin and Longuet-Higgins1991), or the modification of the turbulence by the carrier wave flow field (Tucker & Reynolds Reference Tucker and Reynolds1968; Reynolds & Tucker Reference Reynolds and Tucker1975; Kevlahan & Hunt Reference Kevlahan and Hunt1996, Reference Kevlahan and Hunt1997; Hadzic, Hanjalic & Laurence Reference Hadzic, Hanjalic and Laurence2001). A larger-scale faint wavelike structure is visible in the upper right portion of the plot ($F$) where as many as five waves measuring approximately 120 mm in wavelength can be seen. This wavy feature is clearly visible in the plot for the weak breaker. The source of these waves is unknown, but may be associated with the interaction of the waves with the tank walls.

Contour plots of $s_{sd}$, are shown in figure 13. The bright regions of the $s_{sd}$ contour plots are located in the same general areas as those in the $n_{sd}$ contour plots and consist of many small bright spots and short streaks. Some of the highest average contour levels are at the leading edge of the breaking region, just to the left of the each of the three indentations and at the location of large bubble bursting that was also prominent in the $n_{sd}$ contour maps. The faint vertical lines at $\tilde x \lambda _0^{-1} \approx 0.18$ and 0.55 are due to the slight mismatching of the profiles at the boundaries of the three images from which each profile is extracted. The local arc length seems to be particularly sensitive to errors in profile continuity at these boundaries. The collections of bright spots and streaks form striations with swaying motion. The three brightest of these striations issue from the three indentations (the first (right most) is labelled $S1$), and striations with similar shape and lower intensity are constantly produced all along the breaking crest region. The striations persist for the entire time span of the plot, about 2.7 wave periods, and sway to the left on the back face of the two wave crests and to the right on the front face of the following wave crest. With the exception of the striation marked by the call-out $S_0$ and issuing from approximately the end of the rightmost arrow at call-out $B$, there is, on top of the swaying motions, a general motion downstream (to the left). A visual inspection of some of the LIF images indicates that many of the bright spots and streaks that make up these striations are signatures of short wavelength ripples and individual bubbles that are in the light sheet during one or more frames of one or more of the LIF movies. These signatures seem to be strong enough to survive the ensemble averaging. This supports the hypothesis that the striations are collections of signatures of surface features generated at the breaking crest and moving downstream with the water surface particle velocity. The faint rightmost striation ($S_0$), which has only a small downstream drift, is thought to be in the region upstream of the breaking zone where the surface drift is caused primarily by nonlinear (non-breaking) wave motion. On the other hand, the surface drift affecting the surface features making up the other striations is thought to include a strong component due to the intense surface current created by the breaking process. The surface features downstream (toward the lower left in the plot) of the breaking zone may be generated by the impacts of droplets ejected with downstream trajectories from the breaking zone, as seen, for example, in Movie 4 given in the supplementary material.

Figure 15.

Plots of $(n_{sd})_{max}$ and $(s_{sd})_{ave}$ vs $Q^i$ are given in (a) and (b), respectively. The maximum value $(n_{sd})_{max}$ is the average value $n_{sd}$ in a region that is $11\Delta t=135.3$ ms by $301\delta x= 49.7$ mm centred at the peak value of $n_{sd}$ in the large bubble popping region on the back face of the breaker. The average value $(s_{sd})_{ave}$ is the average of $s_{sd}$ over a rectangular region defined by the corners $(\tilde x\lambda ^{-1}$, $\tilde t f_0) =(0, 0)$ and $(0.85,1.0)$. The solid line in each plot is a least-squares fit of a straight line to the three data points: $(n_{sd})_{max} = 4.33\times 10^{-5}Q^i + 0.0289$ in (a) and $(s_{sd})_{ave}= 5.79 \times 10^{-6}Q^i + 0.0411$ in (b).

To explore the wavy striations in the $s_{sd}$ contour plots further, the trajectory given by

(3.1)\begin{equation} \tilde x = \tilde x_0 + a \cos(\omega (\tilde t-\tilde t_0))+U_d (\tilde t-\tilde t_0) \end{equation}

was fitted by eye to the first striation ($S_1$) affected by a strong drift current in the three $s_{sd}$ contour plots. In this equation, the first term is a spatial phase shift, the second term mimics the orbital motion of the waves (with the carrier wave frequency $\omega$, amplitude $a$ and time shift $t_0$), and the third term mimics the motion due to a surface drift of constant speed $U_d$. The values of $x_0$, $t_0$, $a$, $T = 2{\rm \pi} /\omega$ and $U_d$ from the qualitative fitting are given in table 4 and its caption and the resulting trajectories are plotted as the dashed wavy red curves in the three panels of figure 13. The trajectories are qualitatively a good fit to the striation labelled $S1$ in each panel, and the striations further downstream (fitted curves not shown), for at least the first wave period after breaking. In later time, the striations seem to drift downstream at a rate slower than $U_d$. The computed values of $U_d$, 0.13, 0.18 and 0.21 m s$^{-1}$ for the weak, moderate and strong breakers, respectively, are substantial and approximately consistent with the fluid speeds in the wakes of breaking waves as reported in Duncan (Reference Duncan1981, Reference Duncan1983), Rapp & Melville (Reference Rapp and Melville1990) and Melville, Veron & White (Reference Melville, Veron and White2002). The coincidence of these trajectories and the patterns in the contour plots lend further support to the idea that the arc length SD is responding to small surface features, such as bubbles, that move along with the surface drift current. According to the fitting, this drift current increases with breaker strength.

Table 4.

Parameters obtained by qualitative (visual) fitting of the drift velocity equation (3.1) to the arc length SD contour plots in figure 13. The wave period, $T= 1.35$ s, and wave amplitude, $a=0.07$ m, used in the fitting function were the same for all three waves. The results are plotted as the wavy dashed red line in each figure.

Table 5.

The SD in millimetres of the position of the wave crest point at the time of jet formation, $x^{\, f}_{c}$ and $y^{\, f}_{c}$, and jet impact, $x^{i}_{c}$ and $y^{\, i}_{c}$, before and after breaker profile alignment.

Comparisons of the magnitudes of $n_{sd}$ and $s_{sd}$ from one wave to another are also of interest. In the case of $n_{sd}$, the comparison is difficult because the contour maps consists of large fields of low-level signal with a few isolated areas of high-level signal. Average values over the entire measurement field or even over only the breaking wave crest have yielded very low levels that are not consistent with the qualitative appearance of the contour maps. Here, only a simple quantitative measure of $n_{sd}$ is chosen, a stable peak value obtained by averaging over a small region centred at the location with the maximum value, marked by $E$ in each panel. This area, measuring $0.156f_0^{-1}$ (135.3 ms) by $0.042\lambda _0$ (49.7 mm), essentially covers the yellow contour level region in the plot for the strong breaker, figure 13(c). The local maximum values obtained in this way are called $(n_{sd})_{max}$. Since the high $s_{sd}$ regions in the panels in figure 13 cover a large area, an average value is calculated over the large rectangle defined by the corners $(\tilde x\lambda ^{-1}$, $\tilde t f_0$) =(0, 0) and (0.85,1.0). This average value is given the symbol $(s_{sd})_{ave}$. Plots of $(n_{sd})_{max}$ and $(s_{sd})_{ave}$ vs $Q^i$ are given in figure 15(a,b), respectively, where the three data points in each plot appear to be form straight lines with positive slope, see figure caption for details. It should be noted that $(n_{sd})_{max}/\overline {\langle r_y^i\rangle }$ and $(s_{sd})_{ave}$ vary from 0.034 to 0.056 and from 0.0538 to 0.064, respectively, from the weak to the strong breaker.