## 1. Introduction

The solitary wave has long been the working paradigm for both experimental and numerical simulations of tsunamis (e.g., Ippen and Kulin 1954), persisting as the common performance benchmark for numerical models (e.g., Madsen et al. 2008; Ma et al. 2012; Grilli et al. 2002; and many others) even after the emergence of more realistic models that account, for example, for “N-waves” (Tadepalli and Synolakis 1994) and solitary wave fission (i.e., the disintegration of a leading wave into a train of solitary waves; Madsen and Mei 1969). Because tsunamis reach deep-water phase speeds of the order of 200 m s^{−1} (Geist et al. 2006) and flow velocities near 2–5 m s^{−1} when impacting the coast (Fritz et al. 2012), the term “solitary” was also used in the “soliton” sense—implying a general insensitivity to the oceanographic and sedimentary background. Nevertheless, the need for more realistic forecasting models has eventually encouraged efforts to include the effects of the oceanographic background. Recent studies have considered tsunami interaction with tides (Kowalik et al. 2006) and vorticity (Constantin and Johnson 2008), as well as more accurate and realistic initial conditions needed to better reproduce tsunami fission (Matsuyama et al. 2007).

In accordance with the soliton paradigm, the interaction between tsunamis and wind-wave fields has been approximated as affecting only the wind waves (e.g., Longuet-Higgins 1987; Zhang and Melville 1990). As a slowly varying current, the tsunami can force wave steepening, a description that applies for a scale separation between the current and wave of the order *γ* is the ratio of the characteristic spatial scales and *μ* is the wave steepness. In deep water, the soliton paradigm is justified by the significant scale separation (100 m to 100 km; 10 to 200 m s^{−1}) as well as the intrinsic stability of the soliton as a coherent structure (Osborne 2010).

In the nearshore, however, it is conceivable that the effect of the wind-wave fields on the tsunami could become significant as the characteristic length and speed of the tsunami decrease. From the point of view of soliton dynamics, the balance between dispersion and nonlinearity is broken, with the solitary wave eventually breaking in shallow water. Field observations (e.g., Aida et al. 1964; Madsen et al. 2008) and numerical simulations (e.g., Madsen et al. 2008) also show that the scale gap between the tsunami and wind waves decreases significantly even for relatively short swells. For example, the 2004 Indian Ocean tsunami in the Strait of Malacca had a height of 5 m in 14-m water depth (Madsen et al. 2008); the assumption of a KdV solitary wave shape yields a characteristic length of 240 m (based on Goring 1979). For 10-s, 2-m swell waves, one obtains

The possibility of nontrivial interactions between tsunamis and background swell over a sloping bathymetry in shallow water (at swell scale) was investigated in a series of laboratory experiments by Kaihatu and El Safty (2011) and Kaihatu et al. (2012). Remarkably, images recorded by overhead cameras observing a solitary wave shoaling over a random wave field suggest that background waves can accelerate the solitary wave-breaking process. Using wavelet analysis, Kaihatu and El Safty (2011) showed that the energy of the short-wave band increased when the solitary wave and the random wave field were superposed. While this process may be connected to the acceleration of breaking seen in the overhead imagery, the interaction mechanism forcing the early breaking is not understood. Although the phenomenon was observed in the laboratory (with all the implied scaling limitations when representing tsunami dynamics), understanding the mechanism responsible for the wave–tsunami coupling could be important for improving tsunami forecasting skill of models as well as their interaction with other aspects of the environment (e.g., sediment and sediment transport).

This study focuses on the analysis of the experimental data (described in section 2) in an attempt to identify the location of the initial breaking event and evaluate possible mechanisms for tsunami–swell interaction (section 3). The results are validated numerically in section 4 and summarized in section 5, where the future directions of research are also discussed.

## 2. Observations and data analysis

### a. Laboratory experiment

The laboratory experiment was conducted during March 2010 at the Tsunami Wave Basin (48.8 m long, 26.5 m wide, and 2.1 m deep) at Oregon State University. Details of the experiment are given in Kaihatu and El Safty (2011) and Kaihatu et al. (2012). The bathymetry profile (Fig. 1) was piecewise linear, with a 0.75-m depth flat section for 0 m ≤ *x* ≤ 10 m, a slope of *x* ≤ 17.5 m, and a slope of *x* ≤ 25 m (*x* is the cross-shore coordinate, with the origin at the location of the wave maker). Free-surface elevation data were collected at a sampling rate of 50 Hz at 22 locations from *x* = 7.35 m to *x* = 23.18 m using wire resistance sensors. Overhead video imagery was also recorded from two web cameras.

The experiment performed four tests (runs W1 to W4 in Table 1) with the same solitary wave shoaling alternatively over undisturbed water (run S) and random wave fields (runs SW1 to SW4). The random waves were generated based on a Texel–Marsden–Arsloe (TMA) spectrum (Bouws et al. 1985) using default values for the free parameters for spectral shape. Intrinsic constraints in the mechanics of generating solitary waves in the laboratory, as well as strong seiching resulting from the solitary wave runup, limited the duration of runs that included the solitary waves to 4 min. However, 6- and 12-min runs of each of the four random wave conditions with and without a solitary wave were also recorded separately (runs SW1 to SW4 and W1 to W4) to allow for statistical analysis.

Wave parameters for short random waves, where

The characteristics of the random wave fields were chosen to preserve the values of nondimensional parameters important for reproducing prototype processes (Table 1). The nonlinearity of the solitary wave (*h* is the characteristic depth) corresponds to a tsunami of 5-m amplitude in 14-m water depth (e.g., the 2004 Indian Ocean tsunami in the Strait of Malacca; Madsen et al. 2008). The random wave tests include two runs (W1 and W3) with dispersive waves characterized by *k* is the characteristic wavenumber, and two weakly dispersive runs (W2 and W4) with

The analysis presented here is motivated by Fig. 2. A careful comparison of the overhead video of the solitary wave alone and in the presence of random waves seems to indicate that random waves accelerate the solitary wave-breaking process (Kaihatu and El Safty 2011). However, the interpretation of the images in Fig. 2 is subjective, and the exact moment of breaking depends on the type of breaking process and the definition of the instantaneous breaking event.

Time series from the experiment (e.g., Fig. 3) suggest that the transformation of the solitary wave in all runs is characterized by the peaking and steepening of the wave front, similar to plunging breakers in random waves (e.g., Whitham 1974; Peregrine 1983; confirmed by visual inspection at the site). However, because the instruments cannot detect a vertical surface corresponding to a vertical wave front, a weaker breaking criterion is needed based on observing the overall evolution of the frontal steepness.

Therefore, the breaking point is defined here as the position of the maximum frontal slope; this is also used by Kaihatu and El Safty (2011) and Kaihatu et al. (2012) in their analysis of the wave evolution characteristics of these experiments. This definition is subject to the ambiguity of defining the wave slope itself; therefore, it seems prudent to use several slope definitions and derive conclusions based on the consistency of the results.

### b. Analysis methods

#### 1) Energy flux estimate

*T*is a characteristic time,

*ϕ*is the velocity potential,

*x*is the wave propagation direction, and

*ρ*is the density. Equation (1) can be approximated based on the information about the free-surface elevation

*η*. For random waves, the energy flux of short random waves was estimated using the linear approximation in discrete form:where

*j*;

*j*in the Fourier decomposition of the free-surface

*η*; and

*n*is the total mode number and

*h*is the local depth.

*a*and

*c*are the amplitude and phase velocity of the solitary wave.

#### 2) Frontal steepness

*h*as the local depth and

*g*as the gravitational acceleration). Because nonlinear behavior is expected to dominate near the breaking point, a KdV approximation (e.g., Whitham 1974),may be more appropriate (e.g.,

#### 3) Wavelet filtering

One of the basic difficulties in comparing observations of the solitary wave alone and in the presence of random wave fields is separating the two wave structures. This is especially true for estimating the steepness of the solitary wave as the superposed waves distort the solitary wave surface (Fig. 3b). Filtering out the random wave signal becomes necessary, but simple frequency filters (e.g., a Fourier filter) are not usable because they do not differentiate between the random wave signal and the bound high-frequency components associated with the steep frontal slope of the solitary wave. The approach used here takes advantage of the intrinsic temporal localization of the solitary wave and uses time–frequency analysis (e.g., wavelet transforms; see Chui 1992; Torrence and Compo 1998; and many others). Time localization allows for separating at least the nonsynchronous, random wave, high-frequency Fourier components from the bound components associated with the solitary wave-breaking process.

*g*and

*G*are the transform pair of functions, and

*τ*and

*s*represent the translation and scaling groups of transformations. The quantity

*χ*is the characteristic function of the pyramid. This procedure does not eliminate the random wave variance inside the pyramid. The exact determination of the contour

*N*= 1024 points. The wavelet transform was performed by using the wavelet script for MATLAB developed by Torrence and Compo (1998). The procedure reconstructs well the original solitary wave for run S (Figs. 5a,b). For the SW1 case, the wavelet filter captures the sharp peak of the solitary wave and preserves the slope of the wave front (Figs. 5c,d). To conclude this discussion, the validity of the method used here hinges on a significant frequency separation between solitary wave and the random wave field, which appears to be satisfied in this case.

## 3. Results

### a. Solitary wave shoaling and breaking

Figure 6 shows the evolution of the solitary wave frontal steepness. For all runs, and regardless of the steepness estimator used (linear or nonlinear, maximum or mean), the evolution of the solitary wave frontal steepness (Fig. 6) shows two maxima, suggesting two individual breaking events. In run S (solitary wave alone), breaking events are sharp and occur in close succession at sensors 18 and 20. This is in marked contrast with the evolution in the presence of random waves, as seen in run SW4, illustrated in Fig. 6. All SW runs behaved similar to SW4 (with the exception of SW3, in which the solitary wave breaks at sensor 8). In the presence of waves (Fig. 6), the first breaking event is “smoother,” with a milder slope (sensor 16), while the second breaking is much weaker and occurs farther onshore (sensor 21). Steepness values grow faster for SW4 than for S before the maximum but stay much lower after that. Overall, the trends of the steepness estimators seem to agree with assertion derived from visual observation (Fig. 2) that the solitary wave breaks earlier in the presence of random waves.

Wave–amplitude evolution (Fig. 7) is not exhibited as a clear indication of the early breaking of the solitary wave in the SW runs. In both S and SW runs, the amplitude peaks at sensor 20, with the exception, again, of the SW3 run. However, there is a subtle difference: for evolution in the presence of random waves, the growth rate of the solitary wave amplitude is noticeably weaker, especially close to the breaking point (SW1 and SW3 show almost no growth; Fig. 7c). This behavior suggests a difference in the mechanisms leading to the solitary wave breaking in the runs S and SW. Alone (run S), the solitary wave appears to break by growing and peaking, much like a regular shoaling wave. In the presence of random waves, the frontal slope grows faster, but the amplitude growth is suppressed.

Random waves clearly have an effect on the solitary wave, but the mechanism for interaction is not clear. Possible nonlinear interactions should have an expression in the evolution of the energy flux associated with the two wave fields. However, the evolution of net fluxes integrated over typical time–frequency bands (Fig. 8) does not show any significant energy exchange. Both waves are subject to breaking dissipation and the solitary wave flux decays faster in the presence of waves (Fig. 8a), but the evolution of the energy flux of the random waves shows no detectable change in the presence of the solitary wave (Fig. 8b). Note that in Fig. 8 the energy fluxes for S and SW4 represent one realization (30-s time series), while those for the random waves alone (run W4) are averaged for 47 realizations (12-min time series divided into 47 segments with 50% overlap). Therefore, discrepancies in the behavior of the two energy fluxes of the random waves are expected.

Tank seiching could also cause early breaking of the solitary wave and, if prominent, can affect the ability to translate these results to possible predictive applications. Approximating the seiche as a slowly varying current, we would anticipate that *U* and *c* are the characteristic velocities associated with the seiche and solitary wave, if the seiche were significant. If so, the modulation induced by the seiching should result in an increase in the frontal steepness for upstream propagation (

### b. The effect of random waves

*a*(see the appendix)where

*A*,

*C*, and

*a*is the solitary wave amplitude;

*D*is the dissipation rate; and the zero subscript denotes the value of the parameter at

*D*.

Without random waves

In the presence of random waves, the variable radiation stress gradient modulates the behavior of the solitary wave. In the random wave shoaling zone,

One can estimate the dissipation rate induced by the random waves using the 12-min runs of random waves only (run W4). The results based on Eq. (12), shown in Fig. 10b, appear to capture the trend of the observations despite the crudeness of the formulation (e.g., the solitary wave is assumed to remain symmetric in the process) and the different statistics represented by the different curves. An alternative model not accounting for the dissipation induced by random waves (e.g., Synolakis and Skjelbreia 1993) significantly overestimates the shoaling growth rate.

The effect of the radiation stresses also explains the difference between the evolution of the solitary wave height in four SW runs (Fig. 7). Indeed, the height growth is weaker and breaking occurs earlier for runs SW1 and SW3*,* which exhibit a stronger radiation stresses gradient (Fig. 11b). Conversely, for a weak radiation stresses gradient, the effect of the random wave field on the solitary wave is also weak (Fig. 11a).

## 4. Numerical simulations

The early breaking phenomenon exhibited by all SW runs has so far been implicitly treated as statistically significant behavior, despite having only a single realization for each of the four runs. To overcome the scarcity of laboratory observations, we turn to a numerical model to simulate a statistical ensemble of runs. The numerical simulations were conducted using the Non-Hydrostatic WAVE model (NHWAVE) (Ma et al. 2012), a time-domain model capable of accurately describing fully dispersive, nonlinear surface waves in 3D coastal environments, as well as the breaking solitary wave runup and rundown on sloping beaches. The model solves the incompressible Navier–Stokes equations in well-balanced conservative form, with the governing equations discretized by a combined finite volume/finite difference approach with a Godunov-type shock-capturing scheme.

Numerical experiments were conducted using a 10-layer,

For each of the four SW runs, an ensemble of 60 realizations was simulated by superposing the time series recorded at sensor 1 in run S with a random wave field constructed based on the random-phase approximation (random, uniformly distributed initial phases; e.g., Nazarenko 2011) to match the properties in Table 1. The statistical distribution of the solitary wave-breaking point (Fig. 15), obtained by applying the procedure detailed in sections 2–3, clearly shows the early breaking effect induced by the presence of the random wave field, with

## 5. Discussion and conclusions

Overhead video from a small number of laboratory experiments conducted by Kaihatu and El Safty (2011) and Kaihatu et al. (2012) at the Tsunami Wave Basin at Oregon State University suggests that the breaking point of the solitary wave shifts to deeper water if random wave fields are present. In general, this points to the possibility of a measurable interaction between shoaling solitary waves and the background short-wave fields. The mechanism for this interaction has not been studied. By extension, in as much as the solitary wave can be used as a paradigm for tsunami propagation, one would hypothesize that a similar effect should be detectable in the case of shoaling tsunamis.

Understanding the evidence provided by the laboratory experiments posed a number of challenges. Surface elevation data were collected for only a small number of tests, and the early breaking of the solitary wave was established through visual inspection. The goal of this study was to quantify the perception of “early breaking”; to verify the plausibility of this process; to develop a theoretical background for understanding the process; and finally to reconstruct the missing statistics to test the significance of the process.

Because of experimental constraints, the breaking criterion had to be formulated in terms of surface elevation evolution. The instantaneous breaking point was defined as the position corresponding to the solitary-wave slope reaching a maximum value (defined both as an average and a local value). For combined solitary wave/random wave runs, an additional difficulty was posed by the need to separate the solitary wave from the random wave signal. This difficulty was overcome by using a filter based on the time–frequency analysis (wavelet transform). The solitary wave signal was reconstructed by identifying its signature in the time–frequency domain and then reconstructing the time-domain signal using the inverse wavelet transform. The filtered data preserved the slope and peak of the solitary wave well enough to allow for estimating the frontal steepness even in the presence of the random waves.

The analysis based on the evolution of the maximum and mean steepness estimates confirms the visual observations (Kaihatu and El Safty 2011; Kaihatu et al. 2012). Moreover, it suggests that early breaking is accompanied by a suppressed amplitude growth. While breaking is clearly identifiable in the evolution of the energy fluxes associated with the solitary wave and the random waves, there is little evidence of a transfer of energy between them. The breaking process appears to have more in common with the process of wave propagation through a random flow perturbation than with wave–wave interaction processes. Indeed, a simple modification of the KdV model to include the radiation stresses forcing due to the random wave field compares well with the observed behavior of the solitary wave and explains differences between the four runs based on the characteristics of the random wave fields alone. The tank seiching was shown to be negligible for the solitary wave. The statistical ensemble, reconstructed using the NHWAVE model, confirms the significance of the random wave effect on the solitary wave shoaling.

We believe that the results of this study point to a potentially significant oceanographic process that has so far been ignored. They suggest that systematic research into the interaction between tsunami waves in their various realizations [N-waves in Tadepalli and Synolakis (1994); soliton fission in Madsen and Mei (1969); undular bores in Grue et al. (2008); etc.] is necessary for increasing the accuracy of tsunami forecasting. Laboratory experiments that further investigate this interaction, at a larger scale, are presently underway.

## Acknowledgments

The experimental work was funded by NSF CMMI Grant 0936579 entitled “NEESR Payload: Determining the Added Hazard Potential of Tsunamis by Interaction with Ocean Swell and Wind Waves.” Thanks go to the staff of the NEES Tsunami Wave Basin at Oregon State University; to Dr. Sungwon Shin, Tim Maddux, and Dan Cox; and to Ms. Melora Park, Ms. Linda Frayler, Mr. Jason Killian, and Mr. Adam Ryan. Thanks to Hoda El Safty, Brianna Schilling, Kyle Outten, and Belynda Alonzo (all at Texas A&M University at the time of the experiment) for their assistance in designing the experiment and in processing the data in a manner that helped facilitate their use for this study. We thank Dongyu Feng particularly for his generous help.

The data analysis effort (Drs. Tian and Sheremet) was supported by NSF Grant CMMI-120814, “Interaction of Tsunamis with Short Waves and Bottom Sediment—Numerical and Physical Modeling.”

We are grateful to the anonymous reviewers for their advice in correcting errors in the initial KdV formulation of the solitary wave dissipation induced by the random waves.

## APPENDIX

### The KdV Equation with Radiation Stresses Forcing

Here, we present a formulation for the effect of random waves on solitary wave propagation based on the conceptual model of a wave propagating through a random flow, turbulent background. Because this is a fundamentally statistic model, the derivation presented below is not rigorous, the model will be applied eventually to a handful of realizations, and laboratory scaling may or may not be meaningful for field applications, we regard this model as a first, crude step toward understanding this process. Obviously, further work is required to establish a consistent theoretical model.

*x*is the position,

*t*is the time,

*g*is the gravitational acceleration, and

*x*and

*t*denote partial derivatives; for example,

*A*,

*C*, and

*L*and

*a*characteristic spatial scale and height, and transform to a new reference frame moving with the velocity

*τ*,

*ξ*), Eq. (A1) becomesNeglecting the terms of order

*τ*, and reverting to the original coordinates finally yields the equation

*η*at the leading order (e.g., Grimshaw 1971, 1979; Grimshaw et al. 2010, 2014):Substituting Eq. (A10) into Eq. (A9) yields for Eq. (A9) the form

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