The nonlinearity ofsurface gravity waves refers to their deviations from asinusoidal shape. In the fields ofphysical oceanography andcoastal engineering, the two categories of nonlinearity are skewness and asymmetry. Wave skewness and asymmetry occur when waves encounter an opposingcurrent or a shallow area.^[1][2] As waves shoal in the nearshore zone, in addition to their wavelength and height changing, their asymmetry and skewness also change.^[3] Wave skewness and asymmetry are often implicated inocean engineering andcoastal engineering for the modelling ofrandomsea states, in particular regarding the distribution ofwave height,wavelength and crest length. For practical engineering purposes, it is important to know the probability of these wave characteristics in seas and oceans at a given place and time. This knowledge is crucial for the prediction ofextreme waves, which are a danger for ships andoffshore structures. SatellitealtimeterEnvisat RA-2 data shows geographically coherent skewness fields in the ocean and from the data has been concluded that large values of skewness occur primarily in regions of largesignificant wave height.^[4]

At the nearshore zone, skewness and asymmetry ofsurface gravity waves are the main drivers forsediment transport.^[5]

Sinusoidal waves (orlinear waves) are waves having equal height and duration during the crest and the trough, and they can be mirrored in both the crest and the trough. Due to Non-linear effects, waves can transform from sinusoidal to a skewed and asymmetric shape.

Inprobability theory andstatistics, skewness refers to a distortion or asymmetry that deviates from anormal distribution. Waves that are asymmetric along the horizontal axis are called skewed waves. Asymmetry along the horizontal axis indicates that the wave crest deviates from the wave trough in terms of duration and height. Generally, skewed waves have a short and high wave crest and a long and flat wave trough.^[6] A skewed wave shape results in larger orbital velocities under the wave crest compared to smaller orbital velocities under the wave trough. For waves having the same velocity variance, the ones with higher skewness result in a larger netsediment transport.^[7][8]

Waves that are asymmetric along the vertical axis are referred to as asymmetric waves. Wave asymmetry indicates the leaning forward or backward of the wave, with a steep front face and a gentle rear face. A steep front correlates with an upward tilt, a steep back is correlated with a downward tilt. The duration and height of the wave-crest equal the duration and height of the wave-trough. An asymmetric wave shape results in a larger acceleration between trough and crest and a smaller acceleration between crest and trough.

Skewness (Sk) and asymmetry (As) are measures of the wave nonlinearity and can be described in terms of the following parameters:^[9]

${\displaystyle Sk=\frac{⟨{\eta }^3⟩}{⟨{\eta }^2{⟩}^{\frac{3}{2}}}}$

${\displaystyle As=\frac{⟨\mathcal{H}(\eta {)}^3⟩}{⟨{\eta }^2{⟩}^{\frac{3}{2}}}}$

In which:

• ${\displaystyle \eta }$ is the zero-mean wave surface elevation
• ${\displaystyle \mathcal{H}}$ is theHilbert transform
• ${\displaystyle ⟨\cdot ⟩}$ the angle brackets indicate averaging over many waves

Values for the skewness are positive with typical values between 0 and 1, where values of 1 indicate high skewness. Values for asymmetry are negative with typical values between -1.5 and 0, where values of -1.5 indicate high asymmetry.

The Ursell number, named afterFritz Ursell,^[10] relates the skewness and asymmetry and quantifies the degree of sea surface elevation nonlinearity. Ruessink et al.^[11] defined the Ursell number as:

${\displaystyle Ur=\frac{3}{8}\frac{H_{m0}k}{(kh{)}^3}}$,

where${\displaystyle H_{m0}}$ is the localsignificant wave height,${\displaystyle k}$ is the localwavenumber and${\displaystyle h}$ is the mean water depth.

The skewness and asymmetry at a certain location nearshore can be predicted^[12] from the Ursell number by:

${\displaystyle Sk=B\ast \cos (\psi )}$

${\displaystyle As=B\ast \sin (\psi )}$

${\displaystyle \text{where}B=\frac{0.857}{1+\exp (\frac{-0.471-\log (Ur)}{0.297})},\text{and}\psi =-{90}^{\circ }+{90}^{\circ }\tanh (\frac{0.8150}{U{r}^{0.672}})}$

For small Ursell numbers, the skewness and asymmetry both approach zero and the waves have a sinusoidal shape, and thus waves having small Ursell numbers do not result in net sediment transport. For${\displaystyle Ur\sim 1}$, the skewness is maximum and the asymmetry is small and the waves have a skewed shape. For large Ursell numbers, the skewness approaches 0 and the asymmetry is maximum, resulting in an asymmetric wave shape. In this way, if the wave shape is known, the Ursell number can be predicted and consequently the size and direction of sediment transport at a certain location can be predicted.^[13]

The nearshore zone is divided into the shoaling zone,surf zone and swash zone. In the shoaling zone, the wave nonlinearity increases due to the decreasing depth and the sinusoidal waves approaching the coast will transform into skewed waves. As waves propagate further towards the coast, the wave shape becomes more asymmetric due towave breaking in the surf zone until the waves run up on the beach in the swash zone.

Skewness and asymmetry are not only observed in the shape of the wave, but also in the orbital velocity profiles beneath the waves. The skewed and asymmetric velocity profiles have important implications forsediment transport in shallow conditions, where it both affects the bedload transport as the suspended load transport. Skewed waves have higher flow velocities under the crest of the waves than under the trough, resulting in a net onshoresediment transport as the high velocities under the crest are much more capable of moving large sediments.^[14] Beneath waves with high asymmetry, the change from onshore to offshore flow is more gradual than from offshore to onshore, where sediments are stirred up during peaks in offshore velocity and are transported onshore because of the sudden change in flow direction.^[15] The local sediment transport generatesnearshore bar formation and provides a mechanism for the generation of three-dimensional features such asrip currents and rhythmic bars.

Two different approaches exist to include wave shape in models: thephase-averaged approach and thephase-resolving approach. With the phase-averaged approach, wave skewness and asymmetry are included based on parameterizations.^[16] Phase-averaged models incorporate the evolution of wave frequency and direction in space and time of the wave spectrum. Examples of these kinds of models areWAVEWATCH3 (NOAA) and SWAN (TU Delft). WAVEWATCH3 is a globalwave forecasting model with a focus on the deep ocean. SWAN is a nearshore model and mainly has coastal applications. Advantages of phase-averaged models are that they compute wave characteristics over a large domain, they are fast and they can be coupled to sediment transport models, which is an efficient tool to studymorphodynamics.

• Infragravity waves
• Wind waves
• Wind stress
• Sediment transport

• Gravity waves