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# Wave breaking of unstable sinusoidal wave

## Initial configuration

The initial conditions used by many authors correspond to a sine wave of large amplitude in a periodic domain (Cokelet, 1979; Vinje and Brevig, 1981; Abadie et ah, 1998; Chen et ah, 1991; Abadie, 2001; Lubin et ah, 2006; Iafrati, 2009; Hu et ah, 2012). This is an artificial condition, since breaking is not due to bathymetric variations but caused by the instability of the initial wave, which has already demonstrated its effectiveness in describing all kinds of wave breaking types. Initial quantities (free surface, velocity and pressure fields) are calculated from the linear theory (Airy wave) (Craik, 2004). The initial wave being of large amplitude, is outside the range of validity of the analytical solution, which causes the velocity field and the free surface to be unstable. Therefore, this wave is initially steep, so it can not propagate in the periodic domain and will break more or less rapidly.

The overturning motion is controlled by only two initial parameters, namely the steepness and dispersion parameters, which makes this approach very interesting to study all types of breaking.

The numerical domain is periodic in the direction of the flow and is one wavelength long. The initial wave will propagate from left to right, from where it will leave the domain to re-enter the left limit. The initial quantities are calculated from the linear solution of Airy. A free slip boundary condition is imposed at the lower and lateral limits, and an open boundary condition at the upper limit.

This initial condition allows the use of a numerical domain of reasonable size, since it is limited to one wavelength, which requires less discretization points. However, this approach has some limitations. Indeed, the wave breaks in a medium of constant depth, on a flat bottom, and without pre-existing turbulence or currents since no prior breaking waves exist. The use of a periodic domain simulates an infinite number of waves breaking at the same time, which does not happen in reality.

These early works were original because many authors treated the problem under conditions of a non-viscous fluid (Longuet-Higgins and Cokelet, 1976; Cokelet, 1979; Peregrine et al., 1980; Vinje and Brevig, 1981; Longuet-Higgins, 1982; New et al., 1985; Skyner, 1996). In addition, many works using Navier-Stokes equations considered ratios of densities and viscosities not taking into account the characteristics of air and water (Abadie, 1998; Abadie, 1998; Chen et al., 1999; Yasuda et ah, 1999; Abadie, 2001; Guignard, 2001; Guignard et ah, 2001; Biausser and Suivi, 2003). For example, (Abadie, 1998) showed that it was necessary to add an artificial viscosity to eliminate parasitic disturbances observed near the free surface. In other works (Sakai et ah, 1986; Takikawa et ah, 1997; Lin and Liu, 1998; Lin and Liu, 1998; Bradford, 2000; Iafrati and Mascio, 2001; Watanabe and Saeki, 2002; Iafrati and Campana, 2003), the real air and water physical properties are used.

In addition, very few studies have considered three-dimensional numerical domains (Christensen, 1996; Watanabe and Saeki, 1999; Christensen and Deigaard, 2001; Lubin et ah, 2002; Lubin et ah, 2003; Watanabe et ah, 2005; Christensen,2006).

The first step of the study was to vary the two initial parameters to check the validity ranges of the approach versus different types of breaking. For a set of steepness values, the whole spectrum of breaker types has been simulated (Lubin et ah, 2006). What has been called a “weak plunging breaker”, or weakly breaking, is actually the limit between plunging waves and spilling waves. A small jet was observed to be ejected from the crest of the wave and impacted at the top of the wave, very close to the crest. The spilling breaking wave usually involves a much more complex process of vorticity generation, due to disturbances appearing on the front side of the wave that steepens (Duncan et ah, 1999). However, the whole process of plunging breaking has been precisely simulated, with a particular focus on the generation of splash-up and air entrainment (Lubin et ah, 2006; Lubin and Glockner, 2015; Lubin et ah, 2019).

## Splash-up and large vortical structures

Many authors (Kiger and Duncan, 2012) have analyzed in detail the generation of the splash- up and the entry of the impacting free-falling jet in the front face of the wave. (Peregrine, 1981) discussed “splashes” in waterfalls and breaking waves and (Peregrine, 1983) then presented three possible cases of splash-up generation.

To discuss this point, (Lubin, 2004; Lubin et ah, 2006) have investigated the splash-up generation process. Splash-ups contain water from both the impacting jet and the front face of the wave. In all the simulations performed for this study, it has been found that in the very first moments of the impact, the jet bounces, regardless of the position of the plunge point or the angle between the jet and the front face of the wave. Then a surprising fact has been noticed: the jet does not enter the wave, regardless of the breaking force. The water of the impacting jet separates into two parts, one feeding the upper part of the splash- up, the other wrapping the tube create by the overturning motion of the breaking wave (Figure 1). It has been observed that the jet transmits its momentum by percussion, while the air cavity enveloped by the free-falling jet is entrained and pushed into the water by the weight of the crest of the falling wave. This result was confirmed by experimental Bonmarin (1989) and numerical studies (Abadie et ah, 1998; Yasuda et ah, 1999; Lubin et ah, 2006; Narayanaswamy and Dalrymple, 2002; Dalrymple and Rogers, 2006) which also highlighted this phenomenon of quasi-total reflection of water jet impacting the front face of the wave (Kiger and Duncan, 2012).

The impacting jet pushes the water from the front face of the wave, making the splash up grow in size and rise above the crest of the incident wave. A large amount of vorticity is then generated by rotating the pockets entrained during the process of successive splash-ups generation, as observed during some experiments (Miller, 1976; Bonmarin, 1989). Depending on the configuration of the splash-up, several vortical structures will then be created: the со- and counter-rotating vortices. Their number and their size are important parameters, as the dissipation of energy is related to their behavior during wave breaking.

As detailed previously (Section 2.2) (Zhang and Sunamura, 1990) classified the generation of oblique and horizontal structures. Oblique structures were not observed in the numerical simulations, this being probably due to the use of a periodic domain, which did not allow enough time for this phenomenon to appear. Indeed, as shown by (Nadaoka et al., 1989), horizontal structures are mainly present in the breaking bore, whereas oblique structures are observed only well after the crest of the breaking wave.

Once the splash-up reaches its highest point, the splash-up falls back partly onto the impacting jet, and partly towards the front, generating some other successive splash-ups. The falling back part will be responsible for generating a dipole composed of two structures counter-rotating. This structure carries a large amount of momentum.

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