Ocean Wave Prediction Using Large-Scale Phase-Resolved Computations

Guangyu Wu, Yuming Liu, and Dick K.P. Yue Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA

{wgy, yuming, yue}@mit.edu

Abstract

We apply direct, large-scale, phase-resolved computations to study the evolution of nonlinear ocean wavefields. Starting from a nonlinear wavefield that matches a prescribed wave spectrum, a highly-efficient wave simulation tool is applied to simulate the long-time nonlinear evolution of this wavefield using high-performance computers and to obtain its statistical properties such as directional spreading function, spectral slope, and exceeding probability of crest and trough heights as the wavefield reaches a quasi-stationary state. The characteristics of computational results are compared well with field measurements in spite of the different wave conditions in simulations and field observations. The importance of capturing nonlinear wave interactions in ocean wavefield predictions is illustrated.

1. Introduction

The prediction of ocean surface wavefield evolution has so far been primarily achieved using phase-averaged models such as WAM (in deep water) and SWAN (in littoral zones). It is however known that the prediction of wave statistics often deviates significantly from field measurements and laboratory experiments (Komen, et al., 1994). This is associated with the inherent oversimplification in these models that assume ocean wave motion to be Gaussian, homogeneous, and stationary. To obtain a more reliable and accurate prediction of ocean wave evolution, we develop a new-generation tool based on direct large-scale phase-resolved wavefield computations for which the above assumptions are not applied. The new tool provides a complement in the near future and an alternative in the long-term to the existing models. This study is of critical importance to marine exploration and naval operations. The new generation of ocean wave prediction model, which is called SNOW (simulation of nonlinear oceanwavefields), directly solves the primitive field equation

subject to nonlinear free surface and bottom boundary conditions. Unlike the existing phase-averaged models, SNOW preserves the phases of wave components, and thus obtains a phase-resolved description of the free surface profile and wave dynamics of the wavefield. Based on the use of a pseudo-spectral approach for the treatment of nonlinear free-surface boundary conditions, SNOW is computationally highly efficient with exponential convergence and nearly a linear effort with respect to the number of wave modes and nonlinear interaction order. More importantly, SNOW achieves almost linear scalability on state-of-the-art high performance computing (HPC) platforms. To study the nonlinear evolution of an ocean wavefield using SNOW, an iterative algorithm is applied first to generate an initial large-scale nonlinear ocean wavefield that matches a prescribed wave spectrum. Using this wavefield as initial conditions, we perform SNOW computations (with a resolution of O(5)m in space and O(0.1)s in time) to obtain a phase-resolved solution for nonlinear evolution of this wavefield in a period of O(1) hour. Based on the phase-resolved description of the wavefield, the statistics and its evolution of the wavefield can be obtained. By comparing the results with existing field measurements, SNOW predictions are shown to capture reliably the key characteristics of nonlinear ocean wavefield evolution. In particular, the SNOW solution predicts the presence of a distinct bi-modal angular distribution for short waves in the evolution of a directional wavefield. This result agrees well with recent buoy and radar data of realistic ocean wavefields.

2. Methodology

SNOW is a highly-efficient computational tool developed at the Massachusetts Institute of Technology (MIT) over the past two decades for understanding and prediction of nonlinear ocean surface wavefields. It is based on a pseudo-spectral method and incorporates direct/local models to account for the effects of nonlinear wave-wave interactions, wave-current interactions,

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wave-breaking and bottom dissipations, and wave refraction and diffraction by rapidly-varying bottom topography, and wind input (e.g., Mei, Stiassnie & Yue, 2005). SNOW follows the evolution of a large number of wave modes and accounts for their nonlinear interactions up to an arbitrary order in the wave steepness (e.g., Dommermuth & Yue, 1987). The validity and efficacy of SNOW has been established in the study of basic mechanisms of nonlinear wave-wave interactions in the presence of atmospheric forcing (Dommermuth & Yue 1988), long-short waves (Zhang, Hong & Yue, 1993), finite depth and depth variations (Liu & Yue, 1998), submerged/floating bodies (Liu, Dommermuth & Yue, 1992), and viscous dissipation (Wu, Liu & Yue, 2006). Recently, large-scale SNOW simulations have been applied to study the features and occurrence statistics of extreme waves in open seas (Wu, Liu & Yue, 2005). To efficiently simulate the nonlinear evolution of large-scale three-dimensional directional wavefields, SNOW is parallelized using a Message Passing Interface (MPI) library on current HPC platforms to maximize its performance. By effectively decomposing the computational domain and taking advantage of fast transforms between physical and spectral domains, SNOW achieves almost linear scalability when a large number of, O(2,000), processors are employed. Figure 1 shows the performance of SNOW computations on two typical HPC platforms: IBM P5+ and Cray XT3. Plotted are the number of time steps of SNOW computations finished in 5 minutes of wall clock time versus the number of processors used for three different problem sizes. It shows that, in general, SNOW has good scalability on both systems, while for the same problem using the same number of processors, the SNOW simulation is much faster on the Cray XT3 than on the IBM P5+. In particular for the problem with 4,096×4,096 wave components using 2,048 processors, the simulation on the Cray XT3 is about 5 times faster than that on the IBM P5+. This suggests that further improvement on parallelization of SNOW code is needed when executed on the IBM P5+. In this study, we apply SNOW simulations to investigate the nonlinear evolution of wave spectrum and statistics for various ocean wavefields. For each simulation, we specify the initial conditions by matching a nonlinear wavefield to a prescribed wave spectrum using an iterative algorithm. Briefly, this iterative algorithm starts from a linear wavefield generated by discretizing the prescribed spectrum into a large number of wave components and applying random phase to each wave component. Due to nonlinear wave-wave interactions, the spectrum of such a linear wavefield changes apparently after even a short period of nonlinear evolution. Therefore, this algorithm iteratively modifies

the initial linear wavefield until the spectrum of the nonlinear wavefield developed after a certain period (e.g., 10 peak periods) of evolution matches the prescribed spectrum. The nonlinear evolution of spectrum and statistics can then be obtained by simulating this nonlinear wavefield using SNOW for a long time.

Figure 1. Scalability of SNOW simulations on IBM P5+ and Cray XT3. Plotted is the number of time steps of SNOW

computations completed in 5 minutes wall clock time versus the number of computer processors for nonlinear wavefield

evolution with Nx×Ny wave components.

Typically, Joint North Sea Wave Project (JONSWAP) spectrum and cos2-type directional spreading function (for directional wavefield) are used to generate the initial nonlinear wavefield. The non-dimensionalized JONSWAP spectrum in wavenumber form can be written as:

2

212 exp

24

5exp42

PP

k kk

P

kS k

kk (1)

And the cos2-type directional spreading function is generally in a form:

22 cos , / 2

0 / 2

forD

for

(2)

where k is the wavenumber magnitude, the wave component propagation direction, parameter =0.07 for k kp and 0.09 for k>kp. The following parameters can usually be varied to represent different wave conditions: peak wavenumber kp, Phillips parameter , peak enhancement coefficient , and angular spreading width

. To demonstrate the effectiveness of the iterative algorithm in generating nonlinear initial wavefield conditions, we consider a directional wave spectrum in

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the form given by Eqs. (1 and 2) with parameters kp=8,=0.0081, =3.3, and = . Figure 2 compares the

desired wave spectrum with the one computed from the generated nonlinear ocean wavefield. It shows that the algorithm does generate a nonlinear ocean wavefield with wave spectrum matching the prescribed one very well. This result is obtained using 16 processors on the Cray XT3. Twenty iterations are performed and the wall clock time for the computation is about 10 hours.

Figure 2.Comparison of the wavenumber spectrum contours of (a) the prescribed wave spectrum; and (b) the spectrum of

a nonlinear ocean wavefield generated by the iterative algorithm using SNOW simulations.

3. Results

The understanding of characteristics of the nonlinear ocean wavefield is of great interest to Navy ship design and naval operation. Due to the randomness of the ocean wavefield, it is usually described from a statistical point of view, using quantities such as spectrum, directional spreading, and wave height probability. However, limited by available measurement techniques, the true feature of the nonlinear ocean wavefield is still unclear. On the other hand, theoretical work is only available for more simplified problems, such as linear or second-order and narrow-banded wavefields. In this study, we apply SNOW simulations to obtain the evolution of large-scale nonlinear ocean wavefields and study the spectral and statistical characteristics of such wavefields. In the following, we show the simulation results of a wavefield generated from a spectrum with kp=128, =0.0081, =3.3, and =4 /9. The results are obtained after this nonlinear wavefield is evolved for 150Tp (Tp is the peak wave period). Each simulation of this typical nonlinear ocean wavefield evolution takes about 20 hours of wall clock time when using 256 processors on the Cray XT3.

3.1. Directional Spreading Function

It has been believed for a long time that the directional spreading of an ocean wavefield is unimodal. That is, for any given frequency or wavenumber, the wave energy is focused in the main propagating direction while decreasing monotonically when deviating from this direction. Therefore, unimodal directional spreading functions such as the one in Eq. (2) are commonly assumed in existing studies. However, as measurement techniques and resolution are improved, recent field measurements indicate that the wave energy for short waves (large wavenumber/frequency) demonstrates a bi-modal feature, i.e., most of the wave energy is in two directions symmetric about the main propagating direction of the whole wavefield (Ewans 1998, Hwang & Wang, 2001). In order to verify the existence and investigate the generation mechanism of such a bi-modal directional spreading feature, we compute the normalized direction spreading function for different wavenumber from the nonlinear ocean wavefield at t=150 Tp obtained from our SNOW simulation. The results are shown in Figure 3, compared with two field measurements. It shows that the directional spreading changes from a unimodal function for wavenumber near the peak wavenumber to a bi-modal function for wavenumber much larger than the peak wavenumber. Note that these features agree with field measurements very well although the wavefield in our SNOW simulation has totally different wave conditions from those of field measurements. More importantly, since wind forcing is not included in our simulation and the SNOW simulation is initialized with a uniform unimodal directional spreading function for all wavenumber, we can conclude that the bi-modal directional spreading feature is a consequence of long-time nonlinear wave-wave interactions. In order to verify the existence and investigate the generation mechanism of such a bi-modal directional spreading feature, we compute the normalized direction spreading function for different wavenumber from the nonlinear ocean wavefield at t=150 Tp obtained from our SNOW simulation. The results are shown in Figure 3, compared with two field measurements. It shows that the directional spreading changes from a unimodal function for wavenumbers near the peak wavenumber to a bi-modal function for wavenumbers much larger than the peak wavenumber. Note that these features agree with field measurements very well although the wavefield in our SNOW simulation has totally different wave conditions from those of field measurements. More importantly, since wind forcing is not included in our simulation and the SNOW simulation is initialized with a uniform unimodal directional spreading function for all wavenumbers, we can conclude that the bi-modal

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directional spreading feature is a consequence of long-time nonlinear wave-wave interactions.

Figure 3. Dependence of directional spreading function on wavenumber: ———: k/kp=1; – – –: k/k =3; – · – · –: k/kp=5; · · · · ·: k/kp=7. Plotted are the results from (a) SNOW simulation (Hs=10m, Tp=12sec); (b) field observation using buoy (Ewans 1998, Hs=0.54~4.2m, Tp=3.3~6.9sec); and (c) field observation

using radar (Hwang & Wang, 2001, Hs=1.5m, Tp=6.3sec).

3.2. Spectral Slope

Another controversial point in the understanding of wave spectrum properties is the decay slope of the spectrum for short waves with wavenumbers much larger than the peak wavenumber. It is in general believed that there is an equilibrium range in the wave spectrum within which the wave spectral shape follows a power law. The value of the power is inconclusive (Komen, et al., 1994), although recent field measurements indicate that the

spectral shape is proportional to k 2.5 for wave components with a wavenumber between 2kp and 5kp(Hwang, et al., 2000). The questions is (i) whether such equilibrium range exists or not; and (ii) if yes, what is the power of the spectral shape with respect to the wavenumber remaining unsolved. To study these problems, we compute the omni-directional wavenumber spectrum from the nonlinear ocean wavefield at t=150Tp obtained from our SNOW simulation. The result compared with field measurement is shown in Figure 4s. It shows that the omni-directional wave spectrum obtained from the SNOW simulation does follow a k 2.5 shape for wavenumber between 2kp and 5kp, which agrees very well with the field measurements. Again, since the SNOW simulation does not include wind forcing and is initialized with the JONSWAP spectrum as shown in Eq. (1), which does not have a k 2.5 shape in the above wavenumber range, we can conclude that the nonlinear wave-wave interaction is the key mechanism in forming such a spectral feature.

Figure 4. Comparison of the omni-directional wavenumber spectra obtained from (a) SNOW simulation (Hs=10m,

Tp=12sec); and (b) field measurement (Hwang et al., 2000, Hs=1.5m, Tp=6.3sec). Plotted curves are: – · – · –: SNOW

simulation results; –x–x–: radar measurements; –o–o–: buoy measurements; ———: line of k 2.5.

3.3. Exceeding Probability of Crest and Trough Heights

Wave statistics of the ocean wavefield are often an important factor considered in ship design. Based on the assumptions of linear Gaussian process and narrow-banded ocean wavefield, it is known that crest and trough heights have the same distribution given by the well-known Rayleigh distribution. However, recent field measurements indicate that not only the Gaussian

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assumption is questionable, but also the wave statistics deviate from Rayleigh distribution significantly for large crest and trough heights (Mori & Yasuda, 2002). We compute the exceeding probability of crest and trough heights from the nonlinear ocean wavefield at t=150Tp obtained from our SNOW simulation. The results are shown in Figure 5, compared with field measurements. It shows that both distributions of crest and trough heights start to deviate from the Rayleigh distribution at about R/ rms=2. Here R is the height of wave crest or trough and rms is the standard deviation of the wave elevation. Specifically, the Rayleigh distribution tends to under-predict the occurrence of large crest and to over-predict the occurrence of large trough. In particular, significant difference is observed between the computational results and the Rayleigh distribution for R/ rms>4, which is generally considered a criteria for extreme waves. All of these features of the direct SNOW computed wavefield agree well with the field measurements although the wavefield conditions are not exactly the same in the SNOW simulation and the measured wavefield.

Figure 5. Comparison of exceeding probability of crest and trough heights obtained from (a) SNOW simulation (Hs=10m,

Tp=12sec); and (b) field measurement (Mori et al., 2002,Hs=4.73m, Tp =11sec). Plotted are: : trough probability; :

crest probability; – – –: Rayleigh distribution.

4. Conclusion

We applied direct large-scale, phase-resolved computations of nonlinear ocean wave dynamics to study the spectral and statistical features in the evolution of nonlinear ocean wavefields. A highly-efficient wave simulation tool, SNOW, was applied to simulate the evolution of a nonlinear wavefield that initially matches a prescribed wave spectrum. The directional spreading function, spectral slope, and the exceeding probability of crest and trough heights are computed from the simulated nonlinear wavefield after a long-time evolution. The computational results are compared with field measurements, showing reasonably good agreement in spite of the different wave conditions in our simulations and those in field observations. It is illustrated that it is critical to capture nonlinear wave-wave interactions in understanding and modeling the ocean surface wavefield. The validity and efficacy of SNOW simulations in reliably predicting nonlinear ocean wavefield evolution was demonstrated and verified.

Acknowledgements

We gratefully acknowledge the supports of the DoD High Performance Computing Modernization Program and the Office of Naval Research for this study.

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