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We revisit the classical but as yet unresolved problem of predicting the strength of breaking 2-D and 3-D gravity water waves, as quantified by the amount of wave energy dissipated per breaking event. Following Duncan ( J. Fluid Mech. , vol. 126, 1983, pp. 507–520), the wave energy dissipation rate per unit length of breaking crest may be related to the fifth moment of the wave speed and the non-dimensional breaking strength parameter  $b$ . We use a finite-volume Navier–Stokes solver with large-eddy simulation resolution and volume-of-fluid surface reconstruction (Derakhti & Kirby, J. Fluid Mech. , vol. 761, 2014 a , pp. 464–506; J. Fluid Mech. , vol. 790, 2016, pp. 553–581) to simulate nonlinear wave evolution, with a strong focus on breaking onset and postbreaking behaviour for representative cases of wave packets with breaking due to dispersive focusing and modulational instability. The present study uses these results to investigate the relationship between the breaking strength parameter $b$ and the breaking onset parameter $B$ proposed recently by Barthelemy et al. ( J. Fluid Mech. , vol. 841, 2018, pp. 463–488). The latter, formed from the local energy flux normalized by the local energy density and the local crest speed, simplifies, on the wave surface, to the ratio of fluid speed to crest speed. Following a wave crest, when $B$ exceeds a generic threshold value at the wave crest (Barthelemy et al. 2018), breaking is imminent. We find a robust relationship between the breaking strength parameter $b$ and the rate of change of breaking onset parameter $\text{d}B/\text{d}t$ at the wave crest, as it transitions through the generic breaking onset threshold ( $B\sim 0.85$ ), scaled by the local period of the breaking wave. This result significantly refines previous efforts to express $b$ in terms of a wave packet steepness parameter, which is difficult to define robustly and which does not provide a generically accurate forecast of the energy dissipated by breaking. 

We examine how Eulerian statistics of wave breaking and associated turbulence dissipation rates in a field of intermittent events compare with those obtained from sparse Lagrangian sampling by surface following drifters. We use a polydisperse two-fluid model with large-eddy simulation (LES) resolution and volume-of-fluid surface reconstruction (VOF) to simulate the generation and evolution of turbulence and bubbles beneath short-crested wave breaking events in deep water. Bubble contributions to dissipation and momentum transfer between the water and air phases are considered. Eulerian statistics are obtained from the numerical results, which are available on a fixed grid. Next, we sample the LES/VOF model results with a large number of virtual surface-following drifters that are initially distributed in the numerical domain, regularly or irregularly, before each breaking event. Time-averaged Lagrangian statistics are obtained using the time series sampled by the virtual drifters. We show that convergence of statistics occurs for signals that have minimum length of approximately 1000–3000 wave periods with randomly spaced observations in time and space relative to three-dimensional breaking events. We further show important effects of (i) extent of measurements over depth and (ii) obscuration of velocity measurements due to entrained bubbles, which are the two typical challenges in most of the available in situ observations of upper ocean wave breaking turbulence. An empirical correction factor is developed and applied to the previous observations of Thomson et al. Applying the new correction factor to the observations noticeably improves the inferred energy balance of wind input rates and turbulence dissipation rates. Finally, both our simulation results and the corrected observations suggested that the total wave breaking dissipation rates have a nearly linear relation with active whitecap coverage.

 

On Dec. 22, 2018, at approximately 20:55–57 local time, Anak Krakatau volcano, located in the Sunda Straits of Indonesia, experienced a major lateral collapse during a period of eruptive activity that began in June. The collapse discharged volcaniclastic material into the 250 m deep caldera southwest of the volcano, which generated a tsunami with runups of up to 13 m on the adjacent coasts of Sumatra and Java. The tsunami caused at least 437 fatalities, the greatest number from a volcanically-induced tsunami since the catastrophic explosive eruption of Krakatau in 1883 and the sector collapse of Ritter Island in 1888. For the first time in over 100 years, the 2018 Anak Krakatau event provides an opportunity to study a major volcanically-generated tsunami that caused widespread loss of life and significant damage. Here, we present numerical simulations of the tsunami, with state-of the-art numerical models, based on a combined landslide-source and bathymetric dataset. We constrain the geometry and magnitude of the landslide source through analyses of pre- and post-event satellite images and aerial photography, which demonstrate that the primary landslide scar bisected the Anak Krakatau volcano, cutting behind the central vent and removing 50% of its subaerial extent. Estimated submarine collapse geometries result in a primary landslide volume range of 0.22–0.30 km³, which is used to initialize a tsunami generation and propagation model with two different landslide rheologies (granular and fluid). Observations of a single tsunami, with no subsequent waves, are consistent with our interpretation of landslide failure in a rapid, single phase of movement rather than a more piecemeal process, generating a tsunami which reached nearby coastlines within ~30 minutes. Both modelled rheologies successfully reproduce observed tsunami characteristics from post-event field survey results, tide gauge records, and eyewitness reports, suggesting our estimated landslide volume range is appropriate. This event highlights the significant hazard posed by relatively small-scale lateral volcanic collapses, which can occuren-masse, without any precursory signals, and are an efficient and unpredictable tsunami source. Our successful simulations demonstrate that current numerical models can accurately forecast tsunami hazards from these events. In cases such as Anak Krakatau’s, the absence of precursory warning signals together with the short travel time following tsunami initiation present a major challenge for mitigating tsunami coastal impact.

 

We investigate the validity and robustness of the Barthelemy et al. (2018,https://doi.org/10.1017/jfm.2018.93) wave‐breaking onset prediction framework for surface gravity water waves in arbitrary water depth, including shallow water breaking over varying bathymetry. We show that the Barthelemy et al. (2018) breaking onset criterion, which they validated for deep and intermediate water depths, also segregates breaking crests from nonbreaking crests in shallow water, with subsequent breaking always following the exceedance of their proposed generic breaking threshold. We consider a number of representative wave types, including regular, irregular, solitary, and focused waves, shoaling over idealized bed topographies including an idealized bar geometry and a mildly to steeply sloping planar beach. Our results show that the new breaking onset criterion is capable of detecting single and multiple breaking events in time and space in arbitrary water depth. Further, we show that the new generic criterion provides improved skill for signaling imminent breaking onset, relative to the available kinematic or geometric breaking onset criteria in the literature. In particular, the new criterion is suitable for use in wave‐resolving models that cannot intrinsically detect the onset of wave breaking.

 

This paper documents development of a multiple‐Graphics Processing Unit (GPU) version of FUNWAVE‐Total Variation Diminishing (TVD), an open‐source model for solving the fully nonlinear Boussinesq wave equations using a high‐order TVD solver. The numerical schemes of FUNWAVE‐TVD, including Cartesian and spherical coordinates, are rewritten using CUDA Fortran, with inter‐GPU communication facilitated by the Message Passing Interface. Since FUNWAVE‐TVD involves the discretization of high‐order dispersive derivatives, the on‐chip shared memory is utilized to reduce global memory access. To further optimize performance, the batched tridiagonal solver is scheduled simultaneously in multiple‐GPU streams, which can reduce the GPU execution time by 20–30%. The GPU version is validated through a benchmark test for wave runup on a complex shoreline geometry, as well as a basin‐scale tsunami simulation of the 2011 Tohoku‐oki event. Efficiency evaluation shows that, in comparison with the CPU version running at a 36‐core HPC node, speedup ratios of 4–7 and above 10 can be observed for single‐ and double‐GPU runs, respectively. The performance metrics of multiple‐GPU implementation needs to be further evaluated when appropriate.