^{1}, Marc Perlin

^{2}and Wooyoung Choi

^{1,3}

### Abstract

An eddyviscositymodel to describe energy dissipation in two-dimensional breaking waves in deep water is implemented in a numerical model for the evolution of nonlinear surface waves and evaluated with experimental results. In the experiments, to develop a reliable eddyviscositymodel, breaking waves are generated by both energy focusing and modulated wave groups. Local wave parameters prior to and following breaking are defined and then determined. Significant correlations between the pre-breaking and post-breaking parameters are identified and adopted in the eddyviscositymodel. The numerical model detects automatically wave breaking onset based on local surface slope, determines pre-breaking local wave parameters, predicts post-breaking time and length scales, and estimates eddyviscosity to dissipate energy in wave breaking events. Numerical simulations with the model are performed and compared to the experiments. It is found that the model predicts well the total energy dissipation due to breaking waves. In addition, the computed surface elevations after wave breaking agree reasonably well with the measurements for the energy focusing (plunging) wave groups. However, for breaking wave groups due to modulational instability (plunging and spilling), a relatively large discrepancy between the surface elevation predictions and the experimental measurements is observed, in particular, at the downstream wave probe locations. This is possibly due to wave reflection and three-dimensionality in the experiments. To further validate the eddyviscositymodel, the evolution of highly nonlinear irregular waves is studied numerically and the numerical solutions are compared with additional independent laboratory experiments for long-crested irregular waves. It is shown that the numerical model is capable of predicting the wave evolution subsequent to wave breaking.

W.C. and Z.T. gratefully acknowledge the support from the National Research Foundation of Korea funded by the ministry of Education, Science, and Technology through the WCU program (Grant No. R31-2008-000-10045-0). The laboratory experiments at the Institute for Ocean Technology in Newfoundland, Canada were supported by the US Office of Naval Research (ONR) through Grant No. N00014-05-1-0537 and W.C. thanks Okey Nwogu and Lawrence Mak for their help in setting up the experiments.

I. INTRODUCTION

II. EXPERIMENTS

A. Facility

B. Breaking wave generation

C. Surface elevation and profile measurements

III. LOCAL WAVE GEOMETRIES OF BREAKING WAVES

A. Definitions

B. Wave crests approaching breaking

C. Breaking time and horizontal breaking length

D. Falling crest height

IV. NUMERICAL IMPLEMENTATION OF THE EDDYVISCOSITYMODEL

A. Wave breaking criterion

B. Eddyviscosity estimation

C. Numerical simulation results

1. Energy focusing wave groups

2. Wave groups subject to modulational instability

D. Discussion

V. APPLICATION OF THE EDDYVISCOSITYMODEL TO IRREGULAR WAVES

VI. CONCLUSIONS

### Key Topics

- Viscosity
- 60.0
- Eddies
- 55.0
- Eddy viscosity closure
- 54.0
- Numerical modeling
- 30.0
- Ocean waves
- 28.0

## Figures

Illustration of the two-dimensional wave tank (not to scale) and measurement devices.

Illustration of the two-dimensional wave tank (not to scale) and measurement devices.

Example time series of measured surface elevation of (a) a focusing group and (b) a group subject to modulational instability. Both groups lead to wave breaking. An offset of 5 cm is applied in the ordinates to separate measurements from different stations. Locations of the wave station relative to the wavemaker are shown in (a), e.g., 1.83 and 2.53 (m), and correspond to those in (b) also.

Example time series of measured surface elevation of (a) a focusing group and (b) a group subject to modulational instability. Both groups lead to wave breaking. An offset of 5 cm is applied in the ordinates to separate measurements from different stations. Locations of the wave station relative to the wavemaker are shown in (a), e.g., 1.83 and 2.53 (m), and correspond to those in (b) also.

Example images of the surface profiles of steep wave crests due to (a) energy focusing and (b) modulational instability. Both steep crests propagate from left to right and develop into plunging breakers subsequently.

Example images of the surface profiles of steep wave crests due to (a) energy focusing and (b) modulational instability. Both steep crests propagate from left to right and develop into plunging breakers subsequently.

Sketch of a steep wave crest and definitions of local wave parameters. The wave propagation direction is from left to right.

Sketch of a steep wave crest and definitions of local wave parameters. The wave propagation direction is from left to right.

Definition of the falling crest height, *H* _{ br } for (a) a spiller and (b) a plunger according to Drazen *et al.* ^{7}

Definition of the falling crest height, *H* _{ br } for (a) a spiller and (b) a plunger according to Drazen *et al.* ^{7}

Temporal evolution of the breaking wave crest height, (negative) height of the wave troughs of (a) a spiller and (b) a plunger due to modulational instability and (c) a plunger due to energy focusing. Triangles represent *H* _{ C }(*t*)/*H* _{ C0 }; circles represent *H* _{ t1 }(*t*)/*H* _{ C0 }, and asterisks depict *H* _{ t2 }(*t*)/*H* _{ C0 }. Here *H* _{ C0 } is the wave crest height at wave breaking, i.e., at *t*−*t* _{ b } = 0. The relative time, *t*−*t* _{ b }, is non-dimensionalized with the local angular wave frequency, *ω* _{ b }, at wave breaking.

Temporal evolution of the breaking wave crest height, (negative) height of the wave troughs of (a) a spiller and (b) a plunger due to modulational instability and (c) a plunger due to energy focusing. Triangles represent *H* _{ C }(*t*)/*H* _{ C0 }; circles represent *H* _{ t1 }(*t*)/*H* _{ C0 }, and asterisks depict *H* _{ t2 }(*t*)/*H* _{ C0 }. Here *H* _{ C0 } is the wave crest height at wave breaking, i.e., at *t*−*t* _{ b } = 0. The relative time, *t*−*t* _{ b }, is non-dimensionalized with the local angular wave frequency, *ω* _{ b }, at wave breaking.

Evolution of normalized *L* _{ C } as wave crests approach breaking. *L* _{ C0 } is *L* _{ C } at wave breaking, where *t*−*t* _{ b } = 0. Note that the wavelength of the plunger due to energy focusing (solid triangles) increased abruptly after breaking. This sudden increase is corresponding to the disappearance of one zero-crossing point due to the trough in front rising above the mean water level, as shown in (c) of Figure 6. This also causes the horizontal crest asymmetry parameter, *R* _{ b }, to decrease to a very small value, as shown later in Figure 8. In this scenario, the wavelength should be redefined with the horizontal location of the wave trough in front rather than the next available zero-crossing point. The relative time, *t*−*t* _{ b }, is non-dimensionalized with the local angular wave frequency, *ω* _{ b }, at wave breaking.

Evolution of normalized *L* _{ C } as wave crests approach breaking. *L* _{ C0 } is *L* _{ C } at wave breaking, where *t*−*t* _{ b } = 0. Note that the wavelength of the plunger due to energy focusing (solid triangles) increased abruptly after breaking. This sudden increase is corresponding to the disappearance of one zero-crossing point due to the trough in front rising above the mean water level, as shown in (c) of Figure 6. This also causes the horizontal crest asymmetry parameter, *R* _{ b }, to decrease to a very small value, as shown later in Figure 8. In this scenario, the wavelength should be redefined with the horizontal location of the wave trough in front rather than the next available zero-crossing point. The relative time, *t*−*t* _{ b }, is non-dimensionalized with the local angular wave frequency, *ω* _{ b }, at wave breaking.

Evolution of the wave asymmetry parameter, *R* _{ b }, as wave crests approach breaking. *t* _{ b } represents the time associated with wave breaking. The relative time, *t*−*t* _{ b }, is non-dimensionalized with the local angular wave frequency, *ω* _{ b }, at wave breaking.

Evolution of the wave asymmetry parameter, *R* _{ b }, as wave crests approach breaking. *t* _{ b } represents the time associated with wave breaking. The relative time, *t*−*t* _{ b }, is non-dimensionalized with the local angular wave frequency, *ω* _{ b }, at wave breaking.

Connection between pre- and post-breaking scales. (a) and (b) show *k* _{ b } *L* _{ br } and *ω* _{ b } *T* _{ br } as a function of *S* _{ b }, respectively; (c) can be used to estimate the breaking crest speed, *C* _{ br } = *L* _{ br }/*T* _{ br }, relative to the local phase speed at breaking onset, *C* _{ b } = *ω* _{ b }/*k* _{ b }. Open triangles indicate results of breaking waves due to energy focusing (essentially plunging breakers), including results of the experiments in Tian *et al.*;^{6} solid circles represent breaking waves due to modulational instability (most are spilling breakers). Solid lines in (a) and (b) represent linear least-squares fits and their equations are provided in Sec. IV B. The slope of the dash line in (c) is one, indicating *C* _{ br } = *C* _{ b }.

Connection between pre- and post-breaking scales. (a) and (b) show *k* _{ b } *L* _{ br } and *ω* _{ b } *T* _{ br } as a function of *S* _{ b }, respectively; (c) can be used to estimate the breaking crest speed, *C* _{ br } = *L* _{ br }/*T* _{ br }, relative to the local phase speed at breaking onset, *C* _{ b } = *ω* _{ b }/*k* _{ b }. Open triangles indicate results of breaking waves due to energy focusing (essentially plunging breakers), including results of the experiments in Tian *et al.*;^{6} solid circles represent breaking waves due to modulational instability (most are spilling breakers). Solid lines in (a) and (b) represent linear least-squares fits and their equations are provided in Sec. IV B. The slope of the dash line in (c) is one, indicating *C* _{ br } = *C* _{ b }.

Falling crest height, *k* _{ b } *H* _{ br }, as a function of (a) local steepness, *S* _{ b }, and (b) the crest asymmetry parameter, *R* _{ b }. Open symbols indicate results of breaking waves due to energy focusing (essentially plunging breakers), including results of the experiments in Tian *et al.*;^{6} circles represent breaking waves due to modulational instability (open circles for plungers and solid ones for spillers).The solid line in (b) represents a linear least-squares fit and its equation is provided in Sec. IV B.

Falling crest height, *k* _{ b } *H* _{ br }, as a function of (a) local steepness, *S* _{ b }, and (b) the crest asymmetry parameter, *R* _{ b }. Open symbols indicate results of breaking waves due to energy focusing (essentially plunging breakers), including results of the experiments in Tian *et al.*;^{6} circles represent breaking waves due to modulational instability (open circles for plungers and solid ones for spillers).The solid line in (b) represents a linear least-squares fit and its equation is provided in Sec. IV B.

A comparison of the long time integration of the surface variance, , for the four energy focusing wave groups (Group EF 1 is a non-breaking one, the rest are breaking ones). Here, *T* = 40.95 s. According to linear wave theory, the integration is proportional to the total energy passing a wave station. Symbols are experimental measurements and solid lines are numerical results. Dash lines indicate the breaking regions of the major breaking events in experiments; Dash-dot lines are for those predicted in the simulations. Note that the shown breaking regions are for the major breaking events only; locations of secondary breaking prior and/or subsequent to the major ones are not indicated.

A comparison of the long time integration of the surface variance, , for the four energy focusing wave groups (Group EF 1 is a non-breaking one, the rest are breaking ones). Here, *T* = 40.95 s. According to linear wave theory, the integration is proportional to the total energy passing a wave station. Symbols are experimental measurements and solid lines are numerical results. Dash lines indicate the breaking regions of the major breaking events in experiments; Dash-dot lines are for those predicted in the simulations. Note that the shown breaking regions are for the major breaking events only; locations of secondary breaking prior and/or subsequent to the major ones are not indicated.

(a)–(d) present the surface elevation at wave stations along the tank. Wave group EF 1 is non-breaking, the rest are breaking cases. Solid lines are measured and dash lines are predicted. Locations of the wave stations are provided in the figure, e.g., *x* = 1.83 m. For clarity, an offset of 7.5 cm is applied to the ordinate to separate the surface elevations at different stations.

(a)–(d) present the surface elevation at wave stations along the tank. Wave group EF 1 is non-breaking, the rest are breaking cases. Solid lines are measured and dash lines are predicted. Locations of the wave stations are provided in the figure, e.g., *x* = 1.83 m. For clarity, an offset of 7.5 cm is applied to the ordinate to separate the surface elevations at different stations.

Comparison of the long time integration of the surface variance, , for the wave groups subject to modulational instability. Here, *T* = 50 s. According to linear wave theory, the integration is proportional to the total energy passing a wave station. Symbols are experimental measurements and solid lines are numerical results. Note that for the BFI 1712 comparison, once breaking ensued, it continued to the downstream extent of the tank. Regions in which wave breaking was observed in the experiments are highlighted with the dash lines. Note that the decrease of the integration (total energy) is mainly due to two factors. One is energy dissipation in breaking events; the other is that, since the integration is limited from 0 to 50 s, not all generated waves upstream have arrived at the downstream wave station yet (see Figure 14).

Comparison of the long time integration of the surface variance, , for the wave groups subject to modulational instability. Here, *T* = 50 s. According to linear wave theory, the integration is proportional to the total energy passing a wave station. Symbols are experimental measurements and solid lines are numerical results. Note that for the BFI 1712 comparison, once breaking ensued, it continued to the downstream extent of the tank. Regions in which wave breaking was observed in the experiments are highlighted with the dash lines. Note that the decrease of the integration (total energy) is mainly due to two factors. One is energy dissipation in breaking events; the other is that, since the integration is limited from 0 to 50 s, not all generated waves upstream have arrived at the downstream wave station yet (see Figure 14).

(a) and (b) present the measured (solid lines) and the predicted (dash lines) surface elevation at different wave stations along the tank. Locations of wave stations are the same as those shown in Figure 12. For clarity, an offset of 5 cm is applied to the ordinate to separate the surface elevations at different stations.

(a) and (b) present the measured (solid lines) and the predicted (dash lines) surface elevation at different wave stations along the tank. Locations of wave stations are the same as those shown in Figure 12. For clarity, an offset of 5 cm is applied to the ordinate to separate the surface elevations at different stations.

Evolution of long-crested irregular waves characterized by the JONSWAP spectrum with *H* _{ s } = 0.12 m, *f* _{ p } = 1 Hz, and *γ* = 3.3: (a) Comparison between the measured (solid lines) and the predicted (dashed lines) surface elevations at 20 different wave stations along the tank. For clarity, an offset of 10 cm is applied to the ordinate to separate the surface elevations at different stations; (b) Detailed comparison at the 1st, 10th, and 20th wave stations located at *x*/*λ* _{ p } = 0, 6.92, and 14.62 (from bottom to top).

Evolution of long-crested irregular waves characterized by the JONSWAP spectrum with *H* _{ s } = 0.12 m, *f* _{ p } = 1 Hz, and *γ* = 3.3: (a) Comparison between the measured (solid lines) and the predicted (dashed lines) surface elevations at 20 different wave stations along the tank. For clarity, an offset of 10 cm is applied to the ordinate to separate the surface elevations at different stations; (b) Detailed comparison at the 1st, 10th, and 20th wave stations located at *x*/*λ* _{ p } = 0, 6.92, and 14.62 (from bottom to top).

Comparison between the measured (solid lines) and the predicted (dashed lines) surface elevations at the 1st, 10th, and 20th wave stations wave stations (from bottom to top) along the tank for the evolution of long-crested irregular waves characterized by the JONSWAP spectrum with *f* _{ p } = 1 Hz. Other physical parameters are (a) *H* _{ s } = 0.14 m and *γ* = 3.3; (b) *H* _{ s } = 0.12 m and *γ* = 20; and (c) *H* _{ s } = 0.14 m and *γ* = 20.

Comparison between the measured (solid lines) and the predicted (dashed lines) surface elevations at the 1st, 10th, and 20th wave stations wave stations (from bottom to top) along the tank for the evolution of long-crested irregular waves characterized by the JONSWAP spectrum with *f* _{ p } = 1 Hz. Other physical parameters are (a) *H* _{ s } = 0.14 m and *γ* = 3.3; (b) *H* _{ s } = 0.12 m and *γ* = 20; and (c) *H* _{ s } = 0.14 m and *γ* = 20.

Determination of a critical local surface slope for wave breaking prediction in the numerical simulations. In the numerical simulations, initial conditions are generated with linear wave theory and surface elevations of five non-breaking, focusing wave groups (i.e., W1G1, W2G1, W3G1, W4G1, and W5G1). Characteristics of these five non-breaking wave groups and surface elevation measurements are discussed in detail in Tian *et al.* ^{6,22} Solid symbols: wave breaking cases (numerical failure); open symbols: non-breaking groups. Circles for W1G1; diamonds for W2G1; squares for W3G1; triangles for W4G1; and down-pointing triangles for W5G1. The abscissa, Gain, indicates the multiplier of the magnitude of the non-breaking wave groups.

Determination of a critical local surface slope for wave breaking prediction in the numerical simulations. In the numerical simulations, initial conditions are generated with linear wave theory and surface elevations of five non-breaking, focusing wave groups (i.e., W1G1, W2G1, W3G1, W4G1, and W5G1). Characteristics of these five non-breaking wave groups and surface elevation measurements are discussed in detail in Tian *et al.* ^{6,22} Solid symbols: wave breaking cases (numerical failure); open symbols: non-breaking groups. Circles for W1G1; diamonds for W2G1; squares for W3G1; triangles for W4G1; and down-pointing triangles for W5G1. The abscissa, Gain, indicates the multiplier of the magnitude of the non-breaking wave groups.

## Tables

Specified parameters for the energy focusing wave groups. Note that EF 1 is a non-breaking wave group while the remainder are breaking groups.

Specified parameters for the energy focusing wave groups. Note that EF 1 is a non-breaking wave group while the remainder are breaking groups.

Specified parameters for the wave groups subject to modulational instability. Note that BFI 1710 is a non-breaking group while the rest are breaking groups.

Specified parameters for the wave groups subject to modulational instability. Note that BFI 1710 is a non-breaking group while the rest are breaking groups.

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