3.1. Description of solitary wave shoaling, breaking and overturning

We now examine solitary wave shoaling, breaking (vertical wave face) and overturning (figure 2) for two contrasting cases: (i) a gentler slope ( $s=1/30$ ) with a larger initial wave height ( $H_0/h_0=0.6$ ), and (ii) a steeper slope ( $s=1/10$ ) with a smaller initial wave height ( $H_0/h_0=0.2$ ). The solitary waves propagate to the right, shoaling and overturning over the bathymetry, with significant qualitative differences in their evolution (figure 2). The first waves of the sequence (leftmost curves in figure 2, $1.6\,\tilde {t}$ prior to the subsequent wave) are shoaling and have amplified wave heights relative to $H_0/h_0$ : for case (i), $\eta _{{pk}}/h_0 = 0.694$ , and for case (ii), $\eta _{{pk}}/h_0 = 0.231$ , where $\eta _{{pk}}/h_0$ is the peak of the free-surface elevation. The shoaling solitary waves change shape asymmetrically (figure 2), characteristic of shoaling solitary waves (e.g. Grilli et al. Reference Grilli, Svendsen and Subramanya1997; Mostert & Deike Reference Mostert and Deike2020; Feddersen et al. Reference Feddersen, Hanson, Mostert and Fincham2024). We define the front and rear faces of the solitary wave relative to the $+x$ -propagation direction. In both cases, the first wave front face has steepened through shoaling, whereas the rear face may steepen or flatten. For case (i), the front-face minimum slope of the first wave is $\mathrm{min}(\partial \eta / \partial x) = -0.712$ , whereas the rear-face maximum slope is $\mathrm{max}(\partial \eta / \partial x) = 0.330$ , both steeper than the initial condition maximum slope $0.283$ . In contrast, for case (ii), the front-face $\mathrm{min}(\partial \eta / \partial x) = -0.285$ is smaller than in case (i), but has steepened five times the initial condition maximum slope $0.056$ . The rear-face $\mathrm{max}(\partial \eta / \partial x) = 0.050$ is smaller than the initial condition, indicating a qualitative difference in wave shoaling for the two example cases.

Figure 2.

Two time sequences of $+x$ -propagating solitary wave shoaling, breaking and overturning as a function of $x/h_0$ and $z/h_0$ . The two cases (see figure 1) (i) and (ii) are presented: (a) $s = 1/30$ and $H_0/h_0 = 0.6$ ; (b) $s = 1/10$ and $H_0/h_0 = 0.2$ . The solid black curve represents the free surface, blue represents water, and the brown region represents the bathymetry. The second wave is the earliest time when the front face is vertical, representing breaking. The leftmost curve (lightest blue) represents shoaling and is at a time $\Delta \tilde {t} = -1.6$ prior to the second wave. The subsequent four waves illustrate wave overturning and are evenly spaced in time for case (i) $\Delta \tilde {t} = 0.329$ , and case (ii) $\Delta \tilde {t} = 0.188$ .

Figure 2. Long description

The figure contains two panels showing time sequences of solitary wave evolution over a planar slope. The horizontal axis is non-dimensional distance, x/h_0, and the vertical axis is non-dimensional elevation, z/h_0. Water is shaded blue, the bathymetry is shaded brown, and black curves denote successive free-surface profiles. In both panels, the leftmost profile shows the shoaling wave before breaking, the second profile corresponds to the onset of breaking with a vertical wave front, and the remaining four profiles show progressive wave overturning. Panel (a) shows a larger solitary wave (H_0/h_0 = 0.6) propagating over a gentle slope (s = 1/30). Panel (b) shows a smaller solitary wave (H_0/h_0 = 0.2) propagating over a steeper slope (s = 1/10). The free-surface profiles are evenly spaced in time following breaking.

The second solitary wave in the sequence (figure 2) is at the earliest moment when the front face has an essentially vertical slope, which is denoted as the break point (e.g. Grilli et al. Reference Grilli, Svendsen and Subramanya1997; Varing et al. Reference Varing, Filipot, Grilli, Duarte, Roeber and Yates2021). At this time, the solitary wave height for case (i) has further increased to $\eta _{{pk}}/h_0 = 0.749$ , whereas for case (ii), it has decreased slightly to $\eta _{{pk}}/h_0 = 0.225$ . Similarly, the case (i) rear-face slope has steepened relative to the shoaling time step to $\mathrm{max}(\partial \eta / \partial x) = 0.352$ , whereas for case (ii), it has further reduced to $\mathrm{max}(\partial \eta / \partial x) = 0.047$ . These shape differences in wave shoaling to breaking between cases (i) and (ii) continue through to the process of overturning.

The third to fifth waves in figure 2 show the wave overturning process as the free surface becomes multi-valued and an overturning jet is ejected forwards, descending towards the free surface. The sixth wave shows the last time step prior to the overturning jet impacting the surface below (figure 2). For both cases, the maximum surface elevation decreases somewhat relative to the break point time to $\eta _{{pk}}/h_0 = 0.707$ and $\eta _{{pk}}/h_0 = 0.212$ for cases (i) and (ii), respectively, consistent with previous solitary wave studies (e.g. Grilli et al. Reference Grilli, Svendsen and Subramanya1997). The rear-face maximum slope is essentially unchanged from the break point time, with $\mathrm{max}(\partial \eta / \partial x) = 0.352$ for case (i), and $\mathrm{max}(\partial \eta / \partial x) = 0.047$ for case (ii). For case (i), the jet impacts on the wave’s front face over the sloping bathymetry, whereas for case (ii), the jet impacts at the still-water level over the shallow flat section. Clear qualitative differences in the shape of the overturn and its jet are evident between the two cases. For example, in case (i), the overturn jet is thin relative to the angled overturn, whereas in case (ii), the overturn jet is far thicker relative to the horizontally oriented overturn.

3.2. Definition of key analysis times

We now explicitly define three salient time points in the evolution of a solitary wave that will be used in the subsequent analysis. We use a case with $s = 1/15$ and $H_0/h_0 = 0.6$ as a demonstration (figure 3). The first wave in figure 3 shows the time, denoted $\tilde {t}_{{t}}$ , when the wave peak crosses the toe of the slope at $x/h_0=0$ . In discussing figure 2, we have already referenced times of wave breaking and the overturning jet impact. Here, we quantify those definitions. A kinematic parameter $B$ for wave breaking is the ratio of horizontal water particle velocity at the crest ( $u_c$ ) to the local crest celerity $c$ :

(3.1) \begin{align} B = \frac {u_c}{c}. \end{align}

Recent studies have shown that when $B\gt 0.85$ , wave breaking will eventually occur across deep and shallow water, and different breaker types (Barthelemy et al. Reference Barthelemy, Banner, Peirson, Fedele, Allis and Dias2018; Derakhti et al. Reference Derakhti, Kirby, Banner, Grilli and Thomson2020). However, as noted in Varing et al. (Reference Varing, Filipot, Grilli, Duarte, Roeber and Yates2021), this threshold does not define the onset time of wave breaking, but is a necessary precursor condition for it. Varing et al. (Reference Varing, Filipot, Grilli, Duarte, Roeber and Yates2021) defined the time of breaking onset $\tilde {t}_{{b}}$ as the earliest time that the front face of the solitary wave becomes vertical or $|\mathrm{min}(\partial \eta / \partial x)| \to \infty$ . However, the finite temporal output resolution of the model precludes capturing the exact time when this occurs. We define the time of breaking onset $\tilde {t}_{{b}}$ as the time step previous to that when the wave’s free surface becomes multi-valued (figure 3, green diamond). Finally, we define the time of jet impact $\tilde {t}_{{I}}$ , when the overturning jet collides with the free surface ahead of the wave (figure 3, blue diamond). Again, due to finite output time resolution, we do not capture the exact moment when the overturning jet impacts. Because the FNPF model assumes irrotational flow, it remains valid only up to the moment just before the overturning jet contacts the free surface below. Thus the output time just prior to impact defines the time of the overturning jet impact.

Figure 3.

Three time points of the $+x$ -propagating solitary wave as a function of $x/h_0$ and $z/h_0$ for a case with $s=1/15$ and $H_0/h_0=0.6$ . The coloured diamonds indicate the maximum surface elevation, and the legend indicates the times. The first solitary wave (left to right) shows the time $\tilde {t}_{{t}}$ (yellow diamond) when the wave peak crosses the toe of the slope. The second wave shows the time of wave breaking $\tilde {t}_{{b}}$ (green diamond) where the front-face slope is vertical. The third wave shows the time of wave impact $\tilde {t}_{{I}}$ (blue diamond) when the overturning jet is about to impact the surface below. The solid black curve represents the free surface, blue represents water, and the brown region represents the bathymetry.

Figure 3. Long description

The figure shows three snapshots of a solitary wave propagating from left to right over a planar slope. The horizontal axis is non-dimensional distance, x/h_0, and the vertical axis is non-dimensional elevation, z/h_0. Water is shaded blue, the bathymetry is shaded brown, and black curves denote the free surface. The leftmost wave shows the instant when the wave crest crosses the toe of the slope (t_t), marked by a yellow diamond. The middle wave shows the onset of breaking (t_b), identified by a vertical front face and marked by a green diamond. The rightmost wave shows the instant immediately before the overturning jet impacts the water surface (t_I), marked by a blue diamond. The simulation corresponds to a beach slope of s = 1/15 and an initial solitary-wave height of H_0/h_0 = 0.6.

3.3. Statistics of solitary wave shoaling and overturning

We now examine the solitary wave statistics as a function of non-dimensional time $\tilde {t}$ for the same two cases (i) and (ii) as in figures 1 and 2. The solitary wave’s peak elevation is $\eta _{{pk}}/h_0$ (figure 4 b), with horizontal position $x_{{pk}}/h_0$ (figure 4 a). For both cases (i) and (ii), $x_{{pk}}/h_0$ varies largely linearly with $\tilde {t}$ (figure 4 a), with some noticeable downward curvature, indicating slowing, particularly for case (ii). Solitary wave deceleration is more evident at lower slopes and smaller (less nonlinear) $H_0/h_0$ . The weak deceleration of the solitary wave is consistent with previous model simulations over rapidly varying bathymetry (Guyenne & Grilli Reference Guyenne and Grilli2006; Feddersen et al. Reference Feddersen, Hanson, Mostert and Fincham2024). The fit solitary wave speeds $\tilde {C}$ from the beginning of the simulation to $\tilde {t}_0$ are consistent in close agreement with low-order solitary wave theory (2.7). For cases (i) and (ii), the fits $\tilde {C} = 1.250$ and $\tilde {C} = 1.096$ are consistent with the low-order solitary wave speed (2.7).

Figure 4.

Statistics of solitary wave simulations versus non-dimensional time $\tilde {t}$ for case (i) $s = 1/30$ , $H_0/h_0 = 0.6$ , and case (ii) $s = 1/10$ , $H_0/h_0 = 0.2$ : (a) horizontal location of peak water elevation $x_{{pk}}/h_0$ ; (b) peak water elevation $\eta _{{pk}}/h_0$ ; (c) normalised change in energy $\Delta E/E_0$ , and (d) normalised change in volume $\Delta V/V_0$ , where $E_0$ is the initial energy, and $V_0$ is the initial volume of the solitary wave. The coloured diamonds represent three distinct times defined in figure 3: crossing the toe of the slope $\tilde {t}_t$ , wave breaking $\tilde {t}_b$ , and jet impact $\tilde {t}_I$ , as indicated in the legend.

Figure 4. Long description

The figure contains four panels showing wave statistics as functions of non-dimensional time, t, for two solitary-wave cases: a gentle slope (s = 1/30, black curves) and a steeper slope (s = 1/10, gray curves). Panel (a) shows the horizontal position of the wave peak, x_pk/h_0, increasing nearly linearly with time. Panel (b) shows the crest elevation, η_pk/h_0, increasing prior to breaking and decaying thereafter up to the jet impact time. Panel (c) shows the normalized energy change, Δ E/E_0, which remains close to zero throughout both simulations. Panel (d) shows the normalized volume change, Δ V/V_0, which is approximately conserved for both cases. Yellow, green, and blue diamonds identify the three characteristic times defined in Figure 3: the wave crest crossing the toe of the slope (t_t), the onset of breaking (t_b), and jet impact (t_I).

As the solitary wave shoals over the slope, $\eta _{{pk}}/h_0$ gradually increases in both cases, reaching 0.749 at $\tilde {t}_{{b}}$ for case (i), and 0.225 for case (ii) (figure 4 b). Overall, the solitary wave height shoaling ( $\eta _{{pk}}/h_0$ ) develops slightly more slowly than predicted by Green’s law $(h/h_0)^{-1/4}$ , consistent with previous modelling studies (e.g. Grilli et al. Reference Grilli, Subramanya, Svendsen and Veeramony1994; Knowles & Yeh Reference Knowles and Yeh2018; Feddersen et al. Reference Feddersen, Hanson, Mostert and Fincham2024). Following the onset of wave breaking at $\tilde {t}_{{b}}$ , the wave height begins to decrease as the wave overturns, reaching $\eta _{{pk}}/h_0 = 0.707$ for case (i), and $\eta _{{pk}}/h_0 = 0.212$ for case (ii), at $\tilde {t}_{{I}}$ . The decrease in $\eta _{{pk}}/h_0$ throughout wave overturning is consistent with previous solitary wave modelling (e.g. Grilli et al. Reference Grilli, Svendsen and Subramanya1997).

The normalised change in wave energy $\Delta E/E_0$ , where $E_0$ is energy at $\tilde {t}=0$ , remains nearly constant throughout the case (i) and case (ii) simulations (figure 4 c). For case (i) with larger $H_0/h_0$ , $\Delta E/E_0$ increases throughout shoaling to a maximum $2 \times 10^{-3}$ just prior to jet impact at $\tilde {t}_{{I}}$ , indicating that effectively, energy is conserved. For case (ii) with smaller $H_0/h_0$ , $\Delta E/E_0$ is essentially constant during shoaling, with a small increase to $2\times 10^{-4}$ at $\tilde {t}_{{I}}$ . The normalised change in wave volume $\Delta V/V_0$ , where $V_0$ is volume of the solitary wave (not the model domain) at $\tilde {t}=0$ , also varies slightly ( $\lt 2\times 10^{-3}$ for both cases; figure 4 d). This small volume change confirms that the model conserves volume with high fidelity. The occasional scallop-like features in $\Delta E/E_0$ and $\Delta V/V_0$ are due to model regrids. Across all simulations, the $\Delta E/E_0$ and $\Delta V/V_0$ variations are similarly small, and are consistent with prior shoaling and overturning solitary wave simulations of similar $H_0/h_0$ (e.g. Grilli & Subramanya Reference Grilli and Subramanya1996). The near-constant energy and volume demonstrate that the model accurately captures the physics of overturning solitary waves.

Figure 5.

The moment of jet impact for four solitary waves as a function of $x/h_0$ and $z/h_0$ for varying slope $s$ and initial wave height $H_0/h_0$ : (a) $s = 1/15$ , $H_0/h_0 = 0.2$ ; (b) $s = 1/30$ , $H_0/h_0 = 0.2$ ; (c) $s = 1/15$ , $H_0/h_0 = 0.6$ ; and (d) $s = 1/30$ , $H_0/h_0 = 0.6$ . Dark blue represents water, the thin black line represents the water surface, brown represents the bathymetry, and the dashed black line marks the still-water level at $z/h_0 = 0$ .

Figure 5. Long description

The figure contains four panels showing solitary waves at the instant immediately before the overturning jet impacts the water surface. The horizontal axis is non-dimensional distance, x/h_0, and the vertical axis is non-dimensional elevation, z/h_0. Water is shaded blue, the bathymetry is shaded brown, the free surface is outlined in black, and a dashed horizontal line marks the still water level. Panels (a) and (b) show smaller initial waves (H_0/h_0 = 0.2) for slopes s = 1/15 and s = 1/30, respectively. Panels (c) and (d) show larger initial waves (H_0/h_0 = 0.6) for the same slopes. In all cases the wave forms an overturning jet enclosing a cavity. Larger initial wave heights and steeper slopes produce larger overturning jets and air cavities.

3.4. Moment of jet impact

We now examine the overturning jet at the time of impact $\tilde {t}_I$ over four cases spanning two bathymetric slopes $s$ and two initial wave heights $H_0/h_0$ (figure 5). These four cases demonstrate the variety in wave overturning. For a given $H_0/h_0$ , the wave height at $\tilde {t}_I$ is smaller for the steeper-sloped case $s = 1/15$ than for the lower-sloped case $s=1/30$ , consistent with slope- and wave-height-dependent shoaling scaling laws (Knowles & Yeh Reference Knowles and Yeh2018). For steeper-slope cases ( $s=1/15$ ), the jet impacts the free surface above the shallow flat section for $H_0/h_0=0.2$ (figure 5 a). In contrast, for larger $H_0/h_0$ and gentler slopes ( $s=1/30$ ), the jet impacts above the slope at a deeper location for the larger $H_0/h_0$ (figures 5 b–d). The four overturns also have qualitative shape differences. The $s = 1/30$ overturns (figures 5 b,d) exhibit the classic symmetric teardrop shape associated with the parametric overturn form of Longuet-Higgins (Reference Longuet-Higgins1982) (LH82) seen in both observations (e.g. Blenkinsopp & Chaplin Reference Blenkinsopp and Chaplin2008) and models (e.g. Grilli et al. Reference Grilli, Svendsen and Subramanya1997) of wave overturning. In addition, the overturning jet is thinner and inclined at a steeper angle as it impacts upon the wave face, i.e. at $z/h_0\gt 0$ . The $s=1/15$ and $H_0/h_0=0.6$ case has a larger overturn than the $s=1/30$ and $H_0/h_0=0.6$ case (figures 5 c,d). The $s=1/15$ overturning jet is thicker and the jet impacts lower down than for $s=1/30$ , near $z/h_0=0$ . In contrast, the $s = 1/15$ and $H_0/h_0=0.2$ case (figure 5 a) has an asymmetric shape, akin to the cross-section of a foil, deviating from the LH82 form. This results from the fact that the bottom of the overturn is bounded by still, flat water as the jet impacts at $z/h_0=0$ (dashed line in figure 5 a). It also has a thicker overturning jet and is more horizontally oriented. These solitary wave overturning shapes are consistent with those from previous FNPF models (e.g. Grilli et al. Reference Grilli, Svendsen and Subramanya1997), direct numerical simulations (DNS) results (e.g. Mostert & Deike Reference Mostert and Deike2020; Feddersen et al. Reference Feddersen, Hanson, Mostert and Fincham2024), and experiments (e.g. Li & Raichlen Reference Li and Raichlen2003; Feddersen et al. Reference Feddersen, Fincham, Brodie, Young, Spydell, Grimes, Pieszka and Hanson2023).

Figure 6.

(a) The solitary wave at the breaking onset time $\tilde {t}_{{b}}$ (when the wave face first goes vertical) as a function of $x/h_0$ and $z/h_0$ for $s = 1/30$ and $H_0/h_0 = 0.6$ . The non-dimensional wave height at the break point $H_b/h_0$ is designated by the green diamond. (b) The solitary wave overturn at the time of jet impact $\tilde {t}_{{I}}$ as a function of $x/h_0$ and $z/h_0$ . The non-dimensional wave height at jet impact $H_I/h_0$ is indicated by the blue diamond. The red curve denotes the enclosed overturn region. The light blue and grey shaded areas represent the non-dimensional overturn area $A_O/h_0^2$ and jet area $A_J/h_0^2$ , respectively. The dashed red lines indicate the length $L_O$ and width $W_O$ of the overturn. The angle of the overturn relative to the horizontal $\theta _O$ is also displayed. The horizontal dashed line in (a) and (b) represents the still-water level at $z/h_0 = 0$ .

3.5. Definition of wave breaking and overturning parameters

We now quantify the wave geometry at the breaking onset time $\tilde {t}_{{b}}$ and the overturning parameters at the jet impact time $\tilde {t}_{{I}}$ for the $s = 1/30$ and $H_0/h_0 = 0.6$ case (figure 6). At the break point time $\tilde {t}_{{b}}$ , the wave face has become vertical, and the breaking wave height $H_b/h_0$ (green diamond in figure 6 a) corresponds to the maximum surface elevation, as no trough forms in front of the solitary wave. We also define the water depth at the $x$ -location of $H_b/h_0$ as $h_b/h_0$ . At the time of jet impact $\tilde {t}_{{I}}$ , the wave height $H_I/h_0$ is defined as the maximum surface elevation (blue diamond in figure 6 b). Across all cases, $H_I/h_0$ and $H_b/h_0$ are highly correlated, with $H_I/H_b$ generally slightly less than 1, consistent with Grilli et al. (Reference Grilli, Svendsen and Subramanya1997) and Yasuda et al. (Reference Yasuda, Mutsuda and Mizutani1997). The overturn boundary (red curve in figure 6 b) encloses the void and defines the overturn area $A_O/h_0^2$ . The overturning jet is defined as the upper region of water where the free surface is multi-valued in $x/h_0$ with jet area $A_J/h_0^2$ (grey shading in figure 6 b). Following previous studies of overturn area (O’Dea et al. Reference O’Dea, Brodie and Elgar2021; Feddersen et al. Reference Feddersen, Fincham, Brodie, Young, Spydell, Grimes, Pieszka and Hanson2023, Reference Feddersen, Hanson, Mostert and Fincham2024), both $A_O$ and $A_J$ are normalised with $H_I/h_0$ , yielding $A_O/H_I^2$ and $A_J/H_I^2$ . Overturn length $L_O$ , width $W_O$ , and overturn angle $\theta _O$ (figure 6 b) are estimated by fitting the rotated overturn boundary to the Longuet-Higgins (Reference Longuet-Higgins1982) parametric curve:

(3.2) \begin{align} \frac {z'}{W_O} = \pm \frac {3\sqrt {3}}{4} \, \sqrt {\frac {x'}{L_O}} \left ( \frac {x'}{L_O} - 1 \right ), \end{align}

where $x'$ and $z'$ are the longitudinal and transverse coordinates of the overturn, respectively, and $L_O$ and $W_O$ denote its characteristic length and width (figure 6 b). For many $s$ and $H_0/h_0$ cases, the simulated overturn boundary fits this form well, consistent with previous results (e.g. Blenkinsopp & Chaplin Reference Blenkinsopp and Chaplin2008; O’Dea et al. Reference O’Dea, Brodie and Elgar2021; Feddersen et al. Reference Feddersen, Fincham, Brodie, Young, Spydell, Grimes, Pieszka and Hanson2023, Reference Feddersen, Hanson, Mostert and Fincham2024). Some cases with the steepest slope (e.g. $s = 1/10$ ) have asymmetrical overturns resembling a foil (figures 5 a,c). This is a result of the flat still water bounding the bottom of the overturn. For cases where less than 75 % of the asymmetric overturn curve is above the still-water line, we do not fit to the LH82 form, and $W_O/L_O$ and $\theta _O$ are not estimated.

For most cases, at time $\tilde {t}_{{b}}$ , the solitary wave has a vertical front-wave face over the sloping bathymetry (e.g. figure 2 a). However, at time $\tilde {t}_{{b}}$ , some solitary waves have a vertical front-wave face over the flat section (e.g. figure 2 b), which occurs with the steepest slope ( $s=1/10$ ) and smallest wave heights ( $H_0/h_0 = 0.2, 0.3$ ). These waves will also reach the moment of jet impact on top of the shallow flat section, thereby influencing both breaking and overturning parameters. We remove these cases from the subsequent analysis, to eliminate any bias of the shallow flat section. In addition, some cases ( $s = 1/10$ , $H_0/h_0 = 0.4, 0.5, 0.6$ and $s = 1/15$ , $H_0/h_0 = 0.2$ ) reach the break point on the slope, but then overturn on the shallow flat section at $\tilde {t}_{{I}}$ . We remove these cases from the analysis of overturning parameters only, to eliminate any bias induced by the shallow flat section.

3.6. Scaling the breaker depth index

In order to establish the validity of our model solutions and to motivate the upcoming results on wave overturning parameters, we examine the breaker depth index $\gamma _b = H_b/h_b$ in (1.1), where $h_b$ is the still-water depth at the location of the breaker height $H_b$ (figure 6 a). To eliminate bias of the shallow flat section, we estimate $\gamma _b$ from our model simulations, considering only simulations where $h_b$ is on the slope and not on the shallow flat section. We also incorporate data from both laboratory experiments and numerical modelling of Grilli et al. (Reference Grilli, Svendsen and Subramanya1997), specifically the spilling and plunging cases identified as $S_0 \lt 0.3$ , where

(3.3) \begin{align} S_0 = \frac {1.521s}{(H_0/h_0)^{1/2}}. \end{align}

This excludes the surging wave-breaking cases associated with the steep surface-piercing bathymetry, which we do not simulate. Thus any bias induced by the different bathymetries (surface-piercing slope or shallow flat section) is likely removed.

Figure 7.

Breaker depth index $\gamma _b$ from (1.1) versus (a) the solitary wave surf similarity parameter $S_0 = 1.521s/(H_0/h_0)^{1/2}$ in (3.3), and (b) an alternate scaling parameter $\psi _0 = s/(H_0/h_0)^{1/4}$ in (3.5). Experimental and numerical modelling data points for spilling and plunging waves from tables 1 and 2 of Grilli et al. (Reference Grilli, Svendsen and Subramanya1997) are denoted by magenta squares and diamonds, respectively. Model results from this study are circles coloured by slope $s$ . Only cases with the break point located on the slope are included. The black curves in (a) and (b) represent the empirical fits (3.4) and (3.6), respectively.

Figure 7. Long description

The figure contains two scatter plots of breaker depth index, γ_b, for solitary waves. Panel (a) plots γ_b versus the surf similarity parameter, S_0 = 1.521s/(H_0/h_0)^{1/2}, while panel (b) plots γ_b versus the alternate scaling parameter, ψ_0 = s/(H_0/h_0)^{1/4}. Circles show results from the present simulations and are colored by slope, ranging from s = 1/10 to 1/100. Magenta squares and diamonds denote the experimental and numerical results of Grilli et al. (1997), respectively. Black curves show empirical fits. In both panels, the breaker depth index increases monotonically with the scaling parameter. Compared with panel (a), the alternate parameter in panel (b) produces a tighter collapse of the combined simulation and published datasets about the empirical fit.

For these combined spilling and plunging model and experimental results, $\gamma _b$ has a consistent exponential relationship with $S_0$ , albeit with some scatter (figure 7 a), consistent with Grilli et al. (Reference Grilli, Svendsen and Subramanya1997). The FNPF model’s ability to reproduce the results of Grilli et al. (Reference Grilli, Svendsen and Subramanya1997) further validates its accuracy. The Grilli et al. (Reference Grilli, Svendsen and Subramanya1997) $\gamma _b$ parametrisation

(3.4) \begin{align} \gamma _b = 0.841\,{\rm e}^{6.421S_0} \end{align}

fits the combined data set well (solid line in figure 7 a), with high skill $r^2 = 0.937$ and root mean square error (RMSE) $0.193$ . Note that this parametrisation (3.4) also represents well the surging wave breaking simulations with $S_0\gt 0.3$ (Grilli et al. Reference Grilli, Svendsen and Subramanya1997). We also scale $\gamma _b$ with an alternative parameter

(3.5) \begin{align} \psi _0 = \frac {s}{(H_0/h_0)^{1/4}}, \end{align}

with exponent $1/4$ in the denominator. The $\psi _0$ parameter in (3.5) collapses the $\gamma _b$ data onto an exponential curve (figure 7 b) with less scatter than does $S_0$ . The exponential fit of $\psi _0$ to $\gamma _b$ is

(3.6) \begin{align} \gamma _b = 0.871\,{\rm e}^{11.874\psi _0}, \end{align}

which has higher skill $r^2 = 0.990$ and lower (by more than a half) RMSE at $0.078$ (figure 7 b) than (3.4). The $1/4$ exponent in (3.5) minimised the $\mathrm{RMSE}$ over a range of candidate exponents. Note that the fit (3.6) applies only to the spilling and plunging cases and not to the surging cases ( $S_0\gt 0.3$ ) across which the $S_0$ -based scaling (3.4) does apply. In practice, both $S_0$ and $\psi _0$ scalings are highly effective with high skill in estimating $\gamma _b$ across spilling and plunging cases. Here, we primarily show the results with $\psi _0$ to motivate our upcoming use of it to scale wave overturning parameters.

3.7. Scaling wave overturning parameters

The dependence of wave overturning parameters upon $s$ and $H_0/h_0$ has not previously been quantified. Here, we only consider cases where the jet impacts on the sloping bathymetry (e.g. figures 5 b,c,d), removing any bias of the shallow flat section. This results in cases with $S_0\lt 0.2$ and $\psi _0\lt 0.1$ . We examine first the dependence of $A_O/H_I^2$ on both $S_0$ and $\psi _0$ (figure 8). The non-dimensional overturn area $A_O/H_I^2$ spans an order of magnitude (from 0.04 to 0.43) and increases approximately linearly with $S_0$ and $\psi _0$ (figure 8 a). The area $A_O/H_I^2$ does not collapse as well using $S_0$ as with $\psi _0$ , and with $S_0$ , each $s$ -grouping (colour in figure 8 a) has its own $S_0$ scaling. This indicates that the scatter is not random. In contrast, $A_O/H_I^2$ is effectively scaled by $\psi _0$ , and each $s$ -grouping collapses similarly (figure 8 b), motivating our usage of $\psi _0$ . In the subsequent analysis, the overturning parameters are scaled more effectively with $\psi _0$ than with $S_0$ . Thus we focus on $\psi _0$ in the analysis of overturning parameters. A linear least squares fit between $A_O/H_I^2$ and $\psi _0$ yields the relation

(3.7) \begin{align} A_O/H_I^2 = 5.319\psi _0 - 0.043, \end{align}

which has high-skill $r^2 = 0.990$ and small $\mathrm{RMSE} = 0.013$ . This quantifies how $s$ and $H_0/h_0$ affect overturn area and allow its parametrisation.

Figure 8.

Non-dimensional overturn area $A_O/H_I^2$ versus (a) the solitary wave surf similarity parameter $S_0 = 1.521s/(H_0/h_0)^{1/2}$ in (3.3), and (b) the scaling parameter $\psi _0 = s/(H_0/h_0)^{1/4}$ in (3.5). Data points are coloured by slope $s$ . The black line in (b) represents the empirical fit (3.7).

Figure 8. Long description

The figure contains two scatter plots of non-dimensional overturn area, A_O/H_I^2, for solitary waves. Panel (a) plots A_O/H_I^2 versus the surf similarity parameter, S_0 = 1.521s/(H_0/h_0)^{1/2}, while panel (b) plots A_O/H_I^2 versus the alternate scaling parameter, ψ_0 = s/(H_0/h_0)^{1/4}. Circles show results from the present simulations and are colored by slope, ranging from s = 1/15 to 1/100. Black curves show empirical fits. In both panels, the nondimensional overturn area increases approximately linearly with the scaling parameter. Compared with panel (a), the alternate parameter in panel (b) produces a tighter collapse of the combined simulation and published datasets about the empirical fit.

Figure 9.

Overturning parameters at the time of jet impact ( $\tilde {t}_{{I}}$ ) as a function of the scaling parameter $\psi _0 = s/(H_0/h_0)^{1/4}$ in (3.5): (a) non-dimensional jet area $A_J/H_I^2$ ; (b) wave overturn aspect ratio $W_O/L_O$ ; and (c) wave overturn angle $\theta _O$ . Data points are coloured by slope $s$ . Only cases with symmetric overturns are fit to the LH82 parametric form and shown in (b) and (c).

Figure 9. Long description

The figure contains three scatter plots of various overturning parameters for solitary waves. Panel (a) shows the non-dimensional jet area, A_J/H_I^2, panel (b) shows the overturn aspect ratio, W_O/L_O, and panel (c) shows the overturn angle, θ_O, each plotted as a function of the alternate scaling parameter, ψ_0 = s/(H_0/h_0)^{1/4}. Circles show results from the present simulations and are colored by slope, ranging from s = 1/15 to 1/100. Black curves show empirical fits. In all panels, the points collapse tightly along the empirical fits.

Next, we examine the non-dimensional jet area $A_J/H_I^2$ (figure 9 a), which spans an order of magnitude from $0.012$ to $0.266$ , and increases with $\psi _0$ , following a consistent scaling. A second-degree polynomial least squares fit provides the functional relationship

(3.8) \begin{align} A_J/H_I^2 = 37.072\psi _0^2 - 0.587\psi _0 + 0.020, \end{align}

with high-skill $r^2 = 0.997$ and low $\mathrm{RMSE} = 0.005$ . The ratio $A_J/A_O$ is always smaller than 1, but as $A_J$ is quadratic whereas $A_O$ is linear with $\psi _0$ , the ratio $A_J/A_O$ increases for larger $\psi _0$ . We next examine the aspect ratio $W_O/L_O$ (figure 9 b), which varies weakly, increasing from $0.30$ to $0.44$ with increasing $\psi _0$ . This trend indicates the overturn shape transitioning from an elongated ellipse towards a more circular form as $\psi _0$ increases. The parameter $W_O/L_O$ exhibits more scatter than $A_O/H_I^2$ or $A_J/H_I^2$ , particularly for the smallest $\psi _0$ and thus smallest overturns, which may result from noise in the parametric fit to the LH82 form. A linear fit between $W_O/L_O$ and $\psi _0$ yields

(3.9) \begin{align} W_O/L_O = 1.661\psi _0 + 0.298, \end{align}

with skill $r^2 = 0.943$ and $\mathrm{RMSE} = 0.010$ . Finally, we examine the overturn angle $\theta _O$ (figure 9 c), which decreases from $52.7^\circ$ to $22.4^\circ$ with increasing $\psi _0$ . This quantifies how the overturn orientation rotates towards the horizontal as $\psi _0$ increases, as seen in the example overturns (figure 5). The more horizontal overturns are geometrically linked to the larger $A_J/H_I^2$ and the cross-shore extent of the overturning jet. The relationship between $\theta _O$ and $\psi _0$ is fit well by a quadratic

(3.10) \begin{align} \theta _O = -5746.4\psi _0^2 + 225.2\psi _0 + 48.4, \end{align}

with $r^2 = 0.982$ and $\mathrm{RMSE} = 1.200^\circ$ . The high skill of the fits for $A_O/H_I^2$ , $A_J/H_I^2$ , $W_O/L_O$ and $\theta _O$ quantifies the dependence of the overturn parameters on $s$ and $H_0/h_0$ , and allows their parametrisation.

3.8. Rate of wave steepening

Wave overturning is the integrated result of the entirety of wave shoaling, thus the wave overturning parameters could be expected to be related to wave shoaling parameters. Here, we examine a metric of wave shoaling: the time rate of wave steepening. In figure 10, solitary wave shoaling to breaking is schematised. Previously, we defined the non-dimensional break point time $\tilde {t}_{{b}}$ as the time step previous to when the wave profile becomes multi-valued. We also define the non-dimensional time $\tilde {t}_{{s}}$ as the first time step where $|\mathrm{min}(\partial \eta / \partial x)| \geqslant 0.5$ ( $26.6^\circ$ ), where the wave face is sufficiently steep such that it cannot be described as weakly nonlinear. We examine the time dependence of $|\mathrm{min}(\partial \eta / \partial x)|$ until breaking for four example cases, highlighting the effect of $s$ and $H_0/h_0$ (figure 11).

Figure 10.

Two snapshots of a $+x$ -propagating and shoaling solitary wave as a function of $x/h_0$ and $z/h_0$ , showing the evolution of the steepening wave front for the $s = 1/15,\ H_0/h_0 = 0.6$ case. The first profile is at the non-dimensional time $\tilde {t}_{{s}}$ when $|\mathrm{min}(\partial \eta / \partial x)|=0.5$ , and the second profile is at the non-dimensional break point time $\tilde {t}_{{b}}$ . The dashed red lines highlight the local slope of the front face at $\tilde {t}_{{s}}$ and $\tilde {t}_{{b}}$ .

For the $s = 1/15,\ H_0/h_0 = 0.2$ case (red curve in figure 11 a), the front-face slope is largely constant near $|\mathrm{min}(\partial \eta / \partial x)| = 0.056$ up to crossing the toe of the slope ( $\tilde {t}_{\mathrm{t}}=14.4$ ), consistent with conserved solitary wave shape over flat bathymetry. As the solitary wave propagates up the slope, the rate of steepening increases rapidly, becoming super-exponential. The solitary wave reaches $|\mathrm{min}(\partial \eta / \partial x)| = 0.5$ at $\tilde {t}_{{s}} = 23.0$ . Thereafter, solitary wave steepness continues to grow even more rapidly, reaching $|\mathrm{min}(\partial \eta / \partial x)|=55.3$ at $\tilde {t}_{{b}}=23.9$ .

Figure 11.

Magnitude of the front-face minimum wave slope $|\mathrm{min}(\partial \eta / \partial x)|$ versus non-dimensional time $\tilde {t}$ for (a) $s = 1/15$ and (b) $s = 1/50$ . The red and blue curves represent $H_0/h_0 = 0.2$ and $H_0/h_0 = 0.6$ , respectively. Coloured diamonds mark key times in the wave evolution: crossing the toe of the slope ( $\tilde {t}_{{t}}$ ), front-face wave slope $|\mathrm{min}(\partial \eta / \partial x)| = 0.5$ ( $\tilde {t}_{{s}}$ ), and wave breaking ( $\tilde {t}_{{b}}$ ), as indicated in the legend.

Figure 11. Long description

The figure displays the magnitude of the front-face minimum wave slope |min(∂ η / ∂ x)| as a function of non-dimensional time t for (a) s = 1/15 and (b) s = 1/50. In each panel, the red curve corresponds to H_0 / h_0 = 0.2, while the blue curve corresponds to H_0 / h_0 = 0.6. The horizontal axis t ranges from approximately 0 to 40 in both panels, and the vertical axis ranges on a log-scale, from 0 to 10^2.

For the other three example cases, the evolution of $|\mathrm{min}(\partial \eta / \partial x)|$ is qualitatively similar, just with different initial values and rates of steepening (figure 11). For the $s = 1/15$ and $H_0/h_0 = 0.6$ case (blue curve in figure 11 a), initial $|\mathrm{min}(\partial \eta / \partial x)| = 0.28$ is larger than for the $H_0/h_0=0.2$ case. Due to being larger, solitary wave shoaling commences earlier, reaching $|\mathrm{min}(\partial \eta / \partial x)| = 0.5$ at $\tilde {t}_{{s}} = 13.5$ . The rate of steepening also increases, leading to breaking at $\tilde {t}_{{b}}=15.8$ . A distinction between the $H_0/h_0=0.2$ and $H_0/h_0=0.6$ cases is worth noting. At a fixed value of $|\mathrm{min}(\partial \eta / \partial x)|$ , the $H_0/h_0=0.2$ case has a larger rate of wave steepening than the $H_0/h_0=0.6$ case. Furthermore, for $H_0/h_0=0.2$ , the time to go from $\tilde {t}_{{s}}$ to $\tilde {t}_{{b}}$ is 0.9, whereas for $H_0/h_0=0.6$ it is 2.3. Thus a smaller $H_0/h_0$ results in more rapid wave steepening just prior to breaking. We next examine the two $H_0/h_0$ cases for a shallower slope $s=1/50$ (figure 11 b). For $s=1/50$ , the shoaling and breaking times are later, with overall lower rates of steepening relative to $s=1/15$ . For $H_0/h_0=0.2$ , the solitary wave reaches $|\mathrm{min}(\partial \eta / \partial x)| = 0.5$ at $\tilde {t}_{{s}} = 36.3$ . Wave steepening continues to grow, and $|\mathrm{min}(\partial \eta / \partial x)| = 23.1$ is reached at $\tilde {t}_{{b}}=38.3$ . Thus for $H_0/h_0=0.2$ , the rate of wave steepening is overall slower for $s=1/50$ than for $s=1/15$ . Analogous features are seen in the $s=1/50$ and $H_0/h_0=0.6$ case.

Figure 12.

Non-dimensional wave steepening rate $\partial _{\tilde {t}}|{\min(\partial \eta / \partial x)}|$ versus the shoaling scaling parameter $\psi _1 = s/(H_0/h_0)^{3/2}$ in (3.11). Data points are coloured by bathymetric slope $s$ . The solid line is the fit (3.12).

Examining the time series of $|\mathrm{min}(\partial \eta / \partial x)|$ across the four cases, it seems likely that the super-exponential curve could be parametrised. However, in order to connect the rate of wave steepening to overturn geometry, we opt for a simpler parameter, defining the rate of wave steepening $\partial _{\tilde {t}}|\mathrm{min}(\partial \eta / \partial x)|$ as the non-dimensional time derivative of $|\mathrm{min}(\partial \eta / \partial x)|$ , evaluated at $\tilde {t}_{{s}}$ (where $|\mathrm{min}(\partial \eta / \partial x)|=0.5$ ). We estimate $\partial _{\tilde {t}}|\mathrm{min}(\partial \eta / \partial x)|$ for all cases with $s \leqslant 1/10$ using a centred difference. We scale the rate of wave steepening $\partial _{\tilde {t}}|\mathrm{min}(\partial \eta / \partial x)|$ using $s$ and $H_0/h_0$ with another scaling parameter,

(3.11) \begin{align} \psi _1 = \frac {s}{(H_0/h_0)^{3/2}}, \end{align}

with form similar to $\psi _0$ in (3.5), only with denominator exponent $3/2$ , that minimised the $\mathrm{RMSE}$ . The $\partial _{\tilde {t}}|\mathrm{min}(\partial \eta / \partial x)|$ value varies from 0.05 to 0.56, and collapses onto a consistent increasing quadratic function of $\psi _1$ (figure 12). Thus for larger $\psi _1$ , the solitary waves steepen more rapidly for a fixed $|\mathrm{min}(\partial \eta / \partial x)|$ . A least squares fit yields the relation

(3.12) \begin{align} \partial _{\tilde {t}}|\mathrm{min}(\partial \eta / \partial x)| = 2.476\psi _1^2 + 0.684\psi _1 + 0.047, \end{align}

with high-skill $r^2 = 0.986$ and low $\mathrm{RMSE} = 0.013$ , showing that the rate of wave steepening can be well predicted given $s$ and $H_0/h_0$ . The $\psi _1$ denominator exponent $3/2$ is larger than the $1/4$ of $\psi _0$ , indicating that $\partial _{\tilde {t}}|\mathrm{min}(\partial \eta / \partial x)|$ has a stronger dependence upon initial wave height $H_0/h_0$ than does e.g. the overturning area $A_O/H_I^2$ . That both the rate of wave steepening $\partial _{\tilde {t}}|\mathrm{min}(\partial \eta / \partial x)|$ and wave overturning parameters ( $A_O/H_I^2$ , $A_J/H_I^2$ , $W_O/L_O$ , $\theta _O$ ) have similar dependence upon $s$ and $H_0/h_0$ demonstrates that the geometry of wave overturning is linked to the process of wave shoaling.