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Large Scale Phase-resolved Simulations of Ocean Surface Waves Yuming Liu* and Dick K.P. Yue* Abstract We envision a new-generation wave prediction tool based on direct large-scale nonlinear phase-resolved wavefield simulations that will aug- ment existing phase-averaged approaches to provide heretofore unavail- able modeling and prediction of realistic ocean wavefield evolutions. Upon integration with advanced in situ and/or remote wave sensing technology, the new tool is capable of incorporating such sensed data in phase-resolved reconstruction and forecasting of nonlinear ocean surface waves, providing information that could significantly enhance marine operations and safety. Such a capability also provides a useful framework for assisting in the optimal deployment and utilization of ocean surface sensing systems. Background The accurate prediction of ocean surface wavefield evolutions is a challenging task due to nonlinearities in the wave interactions, the difficulties in modeling wave-breaking dissipation and wind forcing, and, in the context of coastal environment, effects of currents, bottom bathymetry and properties, and the presence of coastlines. Until recently, phase-averaged models such as WAve Prediction Models (WAM; for deep ocean) and Simulating WAves Nearshore models (SWAN; for nearshore * Massachusetts Institute of Technology 171

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172 OCEANOGRAPHY IN 2025 regions) are the mainstay of practical predictions. These models are devel- oped based on the phase-averaged energy-balance equation with physi- cal effects associated with nonlinear wave interactions, wind input, and wave-breaking/bottom dissipations modeled as “source” terms. While much progress has been made over the past decades in the basic approach and in the parameterizations of the model terms, the success has not been uniform, with predictions often falling outside the error band of the observations or in some cases outright failing. Given the basic phase- averaged assumption and the necessary simplifications in the models, further major advances could prove difficult in the present framework. Equally important, these phase-averaged models provide predictions only of the spectral characteristics of the waves, and are not as useful when detailed space-time phased-resolved information is of importance such as in the understanding of extreme wave dynamics. New approach Over the past ten years, we have worked on the development of a new powerful capability, which we call SNOW (simulations of nonlinear ocean wave-field), for predicting the evolution of large-scale nonlinear ocean wavefields using direct physics-based phase-resolved simulations. With rapid development of computational capabilities and, more sig- nificantly, fast algorithms for nonlinear phase-resolved wave simulations, we believe that SNOW could be useful for wave predictions in spatial- temporal scales that would complement and possibly replace phase- averaged models for many practical applications. SNOW is fundamentally different from existing phase-averaged models. It predicts the nonlinear wavefield evolution by direct simula- tion of the wave dynamics including nonlinear wave-wave, wave-current, and wave-bottom interactions, wind input, and wave-breaking dissipa- tion. Where modeling is required, say in the wind forcing or capturing wave-bottom interactions, it can be directly physics-based and generally applied as boundary conditions on the field equation. Since phase infor- mation and wave profile are inherent in the simulation, this provides for opportunities for model calibrations, advances and refinements not possible in the phase-averaged context. In addition, spectral and statisti- cal wave information from SNOW could provide valuable guidance to developing new models and parameterizations in existing approaches such as WAM and SWAN. In terms of the intended spatial-temporal scales, the computational efficiency of an approach like SNOW is paramount. SNOW is based on a highly efficient pseudo-spectral approach, which solves the primitive Euler equations, follows the evolution of a large number (N) of wave

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Yuming Liu and Dick K.P. Yue 173 modes, and accounts for their nonlinear interactions up to an arbitrary high order (M). Significantly, SNOW obtains exponential convergence and near linear computational effort with respected to N and M. Thus the scalar computation time is linearly proportional to the domain size and evolution time. Of equal importance, SNOW is highly paralleliz- able on modern high-performance computing (HPC) platforms, achieving almost linear scalability with the number of processors (utilizing O(103) processors to date). At present, we are capable of direct simulations of an ocean wavefield of O(103) km2 propagating over a distance of O(101~2) km (utilizing N~ O(103~4) per dimension, and M = 3~4). With further algo- rithm development and speedup, in conjunction with increases in HPC capabilities, in the foreseeable future (likely by 2025), SNOW is expected to be able to provide routine simulations of wavefields of O(104~5) km2 propagating over distances of O(102~3 km). The main technical challenge here is computational, associated with algorithmic speedup and refine- ments in the context of massively parallel SNOW calculations. The real challenge however is likely not technical/computational, but scientific, in the modeling and capturing the myriad physics associated with the evolution of the wavefield, and in the availability of concurrent high- resolution measurements for calibration and validation, all in the phase- resolved context. Sample results To date, we have used SNOW to obtain nonlinear wave-wave inter- actions in deep water and finite depth including current and complex bathymetry and bottom properties, with relatively simple phenomeno- logical models for wind forcing and wave breaking dissipation. In a particular project to provide realistic/representative wavefields for ship motion analyses, we have computed an ensemble (the MITWAVE dataset) of 3D wavefields (of typical domain size of 30 km × 30 km) based on initial JONSWAP spectra. Figure 1 shows the distribution of exceeding prob- ability of crest heights from MITWAVE wavefields with various spectrum parameters (spreading angle Θ and peak enhancement factor γ) compared with linear and second-order phase-averaged theoretical predictions. SNOW simulations have been used to identify and characterize the occur- rence statistics and dynamical properties of extreme (rogue) wave events. Among other findings, we confirm that linear (Rayleigh) theory signifi- cantly under predicts the probability of large rogue wave events. Finally, we show an application where SNOW uses WAMOS II radar data to first reconstruct and then provide a forecast of the wavefield. Figure 2 shows the comparisons between SNOW and WAMOS radar data at the initial and a subsequent time corresponding to t~Tp.

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174 OCEANOGRAPHY IN 2025 10 0 10 0 γ=1, Θ=80° γ=1, Θ=18° 10 –1 10 –1 Exceeding Probability P(R) Exceeding Probability P(R) 10 –2 10 –2 10 –3 10 –3 10 –4 10 –4 10 –5 10 –5 0 1 2 3 4 0 1 2 3 4 R/ηrms R/ηrms 10 0 10 0 γ=5, Θ=80° γ=5, Θ=18° 10 –1 10 –1 Exceeding Probability P(R) Exceeding Probability P(R) 10 –2 10 –2 10 –3 10 –3 10 –4 10 –4 10 –5 10 –5 0 1 2 3 4 0 1 2 3 4 R/ηrms R/ηrms FIGURE 1  Comparison of exceeding probability of crest heights for various Jon- swap wave spectrum parameters. The results are obtained from phase-resolved SNOW simulations in a domain of 30 km × 30 km after an evolution time of t/Tp = 100 for wavefields with significant wave height Hs = 10 m, peak period Tp = 12 Yue_Fig1.eps s and four combinations of enhancement parameter γ and spreading angle Θ: γ = 1.0 and Θ = 80° (top left), γ = 5.0 and Θ = 80° (bottom left), γ = 1.0 and Θ = 18° (top type converted to shapes, all replaced with vector right), and γ = 5.0 and Θ = 18° (bottom right). Plotted are the results by SNOW simulation (bullets); Rayleigh linear distribution (solid line); and Tayfun second- order distribution (dashed line).

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Yuming Liu and Dick K.P. Yue 175 FIGURE 2  Comparisons of SNOW reconstructed ocean wavefield (top right) and WAMOS II radar sensed wavefield (top left) at time t = 0 as well as SNOW forecasted wavefield (bottom right) and radar sensed wavefield (bottom left) at t = 10 s. The domain of SNOW reconstructed and forecasted wavefield is 1 km × 1 Yue_Fig2.eps km. The wavefield has a significant wave height Hs = 7.0 m and peak period Tp bitmap images North Sea. = 10 s. The radar is fixed on an offshore platform in

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