Fully nonlinear simulations of unidirectional extreme waves provoked by strong depth transitions: The effect of slope

Recent experimental and numerical studies of surface gravity waves propagating over a sloping bottom have shown that an increase in the probability of extreme waves can be triggered by depth variations in sufficiently shallow waters. This phenomenon is studied here by means of a boundary element method with fast multipole acceleration to solve the fully nonlinear water wave equations. We focus on the case of a random, unidirectional wave field with prescribed statistical properties propagating over a submerged slope and consider different depth variations, including a step. Validation is provided by comparing with experiments by Trulsen, Zeng, and Gramstad [Phys. Fluids 24, 097101 (2012)]. Strongly non-Gaussian statistics are observed in a region localized near the depth transition, beyond which the statistics settle rapidly to the steady statistical state of finite-depth random wave fields. Using a harmonic separation technique, we show that the second-order terms are responsible for the change in the statistical properties near the depth transition. We explore in detail the effects of peak frequency, significant wave height, the inclination of the slope, and the depth of the shallower water side on the kurtosis, skewness, and the excess probability of the crest height, including their spatial distributions.

Figure 1

Side view of the two-dimensional numerical wave tank.

Figure 2

Demonstration of the fast multipole method (FMM) and its three constituent processes: moment concentration (process I), moment translation (process II), and moment expansion (process III).

Figure 3

Three spectra corresponding to our cases A1–A3 and cases 1–3 in TZG12's experiments (see Table 2). The markers show the lower-bound and upper-bound cutoffs chosen for the frequency interval ($[0.5{\omega }_p,3{\omega }_p]$).

Figure 4

The variance ${\sigma }^2$, kurtosis ${\lambda }_4$, and skewness ${\lambda }_3$ for case A1 (case 1 in TZG12). The gray-shaded strip is the 95% confidence interval of the statistical moments obtained from our numerical simulation. The red dots and error bars are obtained from TZG12. The blue dotted line shows the linear signal ${\eta }_{\mathrm{odd}}$ obtained from phase inversion. The dashed vertical lines denote the beginning and end of the slope.

Figure 5

The variance ${\sigma }^2$, kurtosis ${\lambda }_4$, and skewness ${\lambda }_3$ for case A2 (case 2 in TZG12). The legend is the same as in Fig. 4.

Figure 6

The variance ${\sigma }^2$, kurtosis ${\lambda }_4$, and skewness ${\lambda }_3$, for case A3 (case 3 in TZG12). The legend is the same as in Fig. 4.

Figure 7

Exceedance probability at the locations of maximum skewness and kurtosis for different peak periods (cases A1–A3).

Figure 8

Energy spectra of ${\eta }_0$, ${\eta }_{\mathrm{odd}}$, and ${\eta }_{\mathrm{even}}$ for cases A1–A3 at the locations where the skewness and kurtosis reach their maxima. The insets examine the spectra near $\omega =2{\omega }_p$.

Figure 9

Probability density functions (PDFs) of the normalized surface elevation for cases A1–A3 at locations just before the slope (top row) and after the slope where the skewness and kurtosis reach maxima (bottom row). The triangles denote the complete nonlinear signal ${\eta }_0$, the circles denote the linear signal ${\eta }_{\mathrm{odd}}$ obtained from phase inversion, and the continuous line is a Gaussian distribution.

Figure 10

The variance ${\sigma }^2$, kurtosis ${\lambda }_4$, and skewness ${\lambda }_3$ for different significant wave heights (cases B2, B4, and B6). The dashed vertical lines denote the beginning and the end of the slope.

Figure 11

Exceedance probability of the amplitude of the Hilbert envelope of the free surface at the locations of maximum skewness and kurtosis for cases B2 ($H_s=0.04\mathrm{m}$), B4 ($H_s=0.06\mathrm{m}$), and B6 ($H_s=0.08\mathrm{m}$).

Figure 12

Scaling analysis of the maximum values of skewness ${\lambda }_3$ and excess kurtosis ${\lambda }_4-3$ with respect to steepness $k_{p1}{a}_{c1}$.

Figure 13

Exceedance probability of the amplitude of the Hilbert envelope of the free surface at the locations of maximum skewness and kurtosis for different slopes (cases C1–C7). The solid line without markers denotes the Rayleigh distribution.

Figure 14

Exceedance probability of the amplitude of the Hilbert envelope of the free surface for four values of $A/\sigma$, where $A$ is the wave envelope. The error bar presents a $95\%$ confidence interval of the exceedance probability obtained from bootstrapping. The horizontal red dashed line is the exceedance probability of a Rayleigh distribution for the given value of $A/\sigma$.

Figure 15

The variance ${\sigma }^2$, kurtosis ${\lambda }_4$, and skewness ${\lambda }_3$ for different shallower side depths (cases D1–D3). The dashed vertical lines denote the beginning and the end of the slope.

Figure 16

Exceedance probability of the amplitude of the Hilbert envelope of the free surface at the locations of maximum skewness and kurtosis for cases D1–D3. The solid lines without markers denote the Rayleigh distribution.

Figure 17

Partial time series of the surface elevation for different mesh densities at probes 4–6, which are located at 143.41 m, 146.43 m, and 149.41 m, respectively. The time window is chosen to be $t=795.0$–811.3 s, when large waves propagate over the probes.

Figure 18

Partial spatial profiles of the surface elevation for different mesh densities at $t=876.1$ s, chosen so that large waves propagate through this zone at this time. The interval of $x$ is chosen to be 140–155 m, covering the sloped bottom and two regions of constant depth it connects. The dashed vertical lines denote the beginning and the end of the slope.

Figure 19

Statistical moments as a function of the length of time over which they are computed normalized by the peak period $T_p$.

Figure 20

The variance ${\sigma }^2$, kurtosis ${\lambda }_4$, and skewness ${\lambda }_3$ for case A2 for different numbers or random wave components $M$. The shaded strips correspond to the 95% confidence intervals obtained from our numerical simulations. The dashed vertical lines denote the beginning and the end of the slope.