4.2.1 Summary of preliminary results

The evolution of wave packets, as described, often results in a breakdown of the numerical solver. We establish that this is a result of the physical phenomenon of wave breaking. As this is not captured in our model, the solver cannot resolve the wave form.

We simulate wave packets with initial conditions from table 1, especially parameter values about where the waves appear to transition from non-breaking to breaking. The waves with greatest steepness $S$  (2.8), wave height-to-depth ratio $H$  (2.9), kinematic $K$  (2.10), and, by inference, energetic $B$  (2.15) measures are determined for each simulation. Comparisons of the different measures are shown in figure 3, with more simulations performed than reported in Herterich & Dias (Reference Herterich and Dias2017a ,Reference Herterich and Dias b ).

Figure 3.

Comparison of steepness $S$  (2.8), wave height-to-depth ratio $H$  (2.9), kinematic $K$  (2.10), and energetic $B$  (2.15) wave-breaking measures. We distinguish between non-breaking (○) and breaking ( $\times$ ) waves. Each marker represents a simulation with initial conditions from table 1. (a) Height-to-depth ratio and kinematic. (b) Energetic and steepness. The dashed lines are accepted critical values.

As discussed, there is often inconsistency in both geometric measures; non-breaking waves which exceed typical criteria values, and vice versa, as often reported. The critical ranges are summarized in table 2. There are many non-breaking waves that significantly exceed the critical values. However, in the kinematic case, the maximum (0.951) is still below the theoretical critical value of 1. We will analyse these waves in more detail in the subsequent sections of the paper. However, the lower bounds for breaking by the kinematic and energetic measures are consistent with the literature (Wu & Nepf Reference Wu and Nepf2002; Barthelemy et al. Reference Barthelemy, Banner, Peirson, Fedele, Allis and Dias2018).

There is a clear linear trend in the relationships among geometric ( $S$ and $H$ ), kinematic ( $K$ ) and energetic ( $B$ ) measures for non-breaking waves. The kinematic and energetic measures best adhere to the accepted criteria. The dimensionless amplitude  $a_{0}$ and number of waves  $N_{w}$ are the biggest factors related to wave breaking, as discussed in Herterich & Dias (Reference Herterich and Dias2017a ). These influence the wave–wave interactions that amplify and steepen waves, similar to the flat case (Carbone et al. Reference Carbone, Dutykh, Dudley and Dias2013). For example, the amplitude above which breaking usually occurs can be up to 50 % higher for $N_{w}=1$ (dimensionless amplitude $a_{0}>0.063$ ) than $N_{w}>1$ (dimensionless amplitude $a_{0}>0.043$ ).

Table 2.

Geometric criteria $S$ (2.8) and $H$ (2.9), kinematic criteria $K$ (2.10) and energetic criteria $B$ (2.15), for wave breaking of simulated wave packets approaching a wall (figures 1 and 2). The initial parameter values are from the ranges given in table 1.

4.2.2 Breaking strength

The breaking strength is related to the onset of breaking via the energetic measure $B$  (2.15), as discussed in § 2.2.4. The breaking strength parameter also takes the form $b=b(\text{d}B/\text{d}t)$  (2.17). We observe the quantity $\text{d}B/\text{d}t$ to increase with $B$  (figure 4). As $B$ is a measure of energy flux, we expect more energetic fluxes to have larger breaking strengths. The values of $\text{d}B/\text{d}t$ obtained are of similar order of magnitude to the $\unicode[STIX]{x1D6E4}\propto \,\text{d}B/\text{d}t$ in figure 5 of Derakhti et al. (Reference Derakhti, Banner and Kirby2018). Note that we neglect the scaling in the proportionality for $\unicode[STIX]{x1D6E4}$ as our simulations are already dimensionless.

Figure 4.

(a) The rate of change of $B$  (2.15) for breaking waves, an indicator of the breaking strength for all waves. In (b,c) we group the results in (a) with respect to initial wave amplitude and period: (b) $0.04<a_{0}<0.06$ ( $\times$ ) and $0.06<a_{0}<0.1$ (▫); (c) $P_{0}<210$ ( $\times$ ) and $210<P_{0}<315$ (▫). The ranges for the types of breaking (spilling and plunging) are from Wu & Nepf (Reference Wu and Nepf2002), converted from a condition on $K$ to $B$ , via (2.15).

Classifying the initial waves that correspond to each simulation in figures 3 and 4 is relatively simple. In general, all measures $S$ , $H$ , $K$ , $B$ and $\text{d}B/\text{d}t$ increase with initial amplitude $a_{0}$ and decrease with period $P_{0}$ , as expected. This is in agreement with experiments (Goda Reference Goda1975, Reference Goda2010). The initial wave profile has an effect on the final breaking strength. Waves with greater initial amplitudes are more likely to result in plunging (figure 4 b). However, for wave periods it is less clear (figure 4 c). There are similar numbers of waves that exhibit plunging as well as spilling for all initial wave periods. Note that many long waves break up into wave trains with smaller periods.

4.3.1 Summary of preliminary results

Each wave in the wave packet is amplified at the wall. However, we consider a single parameter to represent the runup in each simulation (from table 1), the maximum runup amplification,

(4.1) $$\begin{eqnarray}R_{max}=\max (R(t)).\end{eqnarray}$$

In general, $R_{max}$ increases with the amplitude of the initial wave packet. As we do not know the subsequent wave evolution for breaking waves, after the onset of breaking, we do not consider these waves here. Typically, waves break for $a_{0}>0.04$ (Herterich & Dias Reference Herterich and Dias2017a ). The runup amplification steadies out with respect to  $N_{w}$ for $N_{w}\geqslant 3$ , as for a flat bathymetry (Viotti et al. Reference Viotti, Carbone and Dias2014a ).

Increasing the bathymetry slope (figure 1) from flat to 1 % almost doubles the runup amplification, with $R_{max}>10$ in all cases. Dimensionally, this means that a wave packet of initial amplitude 3 m can overtop a 30 m cliff in ocean-type conditions. In the laboratory, that would correspond to a 3 cm wave overtopping a 30 cm cliff.

4.3.2 Tidal effects

The tide affects the absolute depth over a bathymetry profile. While the bathymetry clearly is important in runup amplification, changes in the tide can increase or decrease its influence. Decreasing the free-surface height by a constant of 5 m increases $R_{max}$ by approximately 1 (figure 5). Furthermore, the runup amplification increases with increasing period, from 4 to $9+$ , until $P_{0}=140$ , when increases are at a slower rate.

Figure 5.

Maximum runup amplification $R_{max}$  (4.1) against initial wave period $P_{0}$ for different tidal levels with the same underlying Aran Islands bathymetry (figure 2). The ocean depth is taken to be $h=75~\text{m}$ (▫), 80 m (○) and 85 m ( $\times$ ). All wave packets are initialized dimensionally with respect to the new depths (numbers in table 1 are with respect to $h=80~\text{m}$ ), and with $N_{w}=3$ and $a_{0}=0.04$ .

Figure 6.

Runup amplification, $R(t)$ , as a function of the dimensionless time $t$ for three different bathymetries: flat, flat followed by a 1 : 200 slope, flat followed by a 1 : 100 slope. The 1 % slope represents the Aran Islands bathymetry (figure 2). In all simulations $P_{0}=140,N_{w}=3,a_{0}=0.04$ .

4.3.3 Extreme waves

As discussed, the bathymetry plays a crucial role in runup amplification of waves. In figure 6, we show the runup $R(t)$ for a wave packet with dimensionless parameters $(P_{0},N_{w},a_{0})=(140,3,0.04)$ over three different bathymetries: flat, flat followed by a 1 : 200 (0.5 %) slope, flat followed by a 1 : 100 (1 %) slope. The 1 : 100 slope represents the Aran Islands bathymetry as approximated in figure 2, with full wave simulation in Herterich & Dias (Reference Herterich and Dias2017a ). During each simulation, there are three significant sets of peaks (for the three crests in the initial wave packet). In each set, there is an increasing amount of wave disintegration and amplification, similar to the wave evolution in Akrish et al. (Reference Akrish, Rabinovitch and Agnon2016a ,Reference Akrish, Schwartz, Rabinovitch and Agnon b ). This is due to nonlinear wave–wave interactions between incoming and reflecting waves. However, allowing for the bathymetry to vary, the runup is significantly amplified from a maximum $R\approx 4.7$ in the flat case to $R\approx 10.76$ in the Aran Islands bathymetry case. Hence we clearly see an amplification increase from the wave–wave interactions and from the bathymetry slope.

We examine more closely the free surface during wave runup of one of the most amplified waves, with $P_{0}=175$ , $a_{0}=0.043$ and $N_{w}=3$ , over the Aran Islands bathymetry. The third wave (with greatest amplification) in the wave packet smoothly runs up and down the cliff face with no sign of a vertical jet (figure 7). The crest, when it is on the cliff face, has a half-wavelength of over 1.25 (dimensionally 100 m). Here, $R_{max}\approx 11.8$ . We highlight that a cliff of 25 m (marked by a solid vertical line at wall and a dashed horizontal line) would be overtopped by this wave.

Figure 7.

Runup of a wave with initial parameters $P_{0}=175$ , $N_{w}=3$ , $a_{0}=0.043$ . (a) The wave runs up the cliff. (b) The wave is reflected from the cliff. The amplification is $R_{max}\approx 11.8$ . The wave smoothly runs up the face of the cliff, to the right. A cliff of (dimensionless height) 0.3125 (marked by a solid vertical line at the wall and a dashed horizontal line, dimensionally 25 m) would be overtopped by the wave.

We consider now the Aran Islands bathymetry (figure 2) to determine extreme waves that may occur in the near shore. We focus on the wave amplification and location when $H$ is large, as this is directly related to large wave heights.

In all simulations, the majority of non-breaking waves during evolution respect the generally accepted condition $H<0.55$ . However, many exceed this value for small time intervals (figure 3). The maximum value is $H\approx 0.951$ . Although not shown, there is a continuous  $H(t)$ relationship that ascends to and descends from these (non-breaking) maximal values.

Such wave events occur briefly during wave–wall and wave–wave interactions. These waves have the largest non-breaking kinematic ( $K>0.487$ ) and steepness ( $S>0.323$ ) measures. All other non-breaking waves with $H<0.55$ have lower kinematic and steepness measures. Hence, these waves are amplified, steep and near breaking.

The scaled free-surface amplification $R_{a}=\unicode[STIX]{x1D702}/a_{0}$ at the crests tips of maximum  $H$ show that all waves are significantly amplified (figure 8 a); $R_{a}>10$ in one case and $R_{a}>6$ for all. In general, $R_{a}$ increases with  $H$ . The location $x_{a}$ of these waves is evaluated as within  $0.001\unicode[STIX]{x1D706}_{0}$ of the wall for most of the waves (figure 8 b). However, it is actually occurring on the wall face as ${\approx}0.001\unicode[STIX]{x1D706}_{0}$ is a small numerical error when mapping between conformal and physical space with computational domain length $2L=100+2(2+N_{w})\unicode[STIX]{x1D706}_{0}$ . The instances further from the wall ( ${>}0.015\unicode[STIX]{x1D706}_{0}$ ) are wave–wave interactions between an incoming and reflected waves.

We call these amplified, steep, near-breaking waves at the wall extreme waves.

Figure 8.

Amplification and location of non-breaking crests with maximum $H_{w}>0.55$ . (a) Free-surface crest amplification $R_{a}=\unicode[STIX]{x1D702}/a_{0}$ . (b) Distance from the wall ( $L-x_{a}$ ), scaled with the initial wavelength  $\unicode[STIX]{x1D706}_{0}$ .