Wave nonlinearity

The nonlinearity of surface gravity waves refers to their deviations from a sinusoidal shape. In the fields of physical oceanography and coastal engineering, the two categories of nonlinearity are skewness and asymmetry. Wave skewness and asymmetry occur when waves encounter an opposing current 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 in ocean engineering and coastal engineering for the modelling of random sea states, in particular regarding the distribution of wave 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 of extreme waves, which are a danger for ships and offshore structures. Satellite altimeter Envisat 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 large significant wave height. ^[4]

At the nearshore zone, skewness and asymmetry of surface gravity waves are the main drivers for sediment transport. ^[5]

Skewness and asymmetry

a) sinusoidal, b) skewed and c) asymmetric wave shape

Sinusoidal waves (or linear 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.

Skewed waves

In probability theory and statistics, skewness refers to a distortion or asymmetry that deviates from a normal 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 net sediment transport. ^{[7] [8]}

Asymmetric waves

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.

Mathematical description

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

${\displaystyle Sk={\frac {\langle \eta ^{3}\rangle }{\langle \eta ^{2}\rangle ^{\frac {3}{2}}}}}$

Skewness (top) and Asymmetry (bottom) plotted against the Ursell number on a logarithmic scale.

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

In which:

• ${\displaystyle \eta }$ is the zero-mean wave surface elevation
• ${\displaystyle {\mathcal {H}}}$ is the Hilbert transform
• ${\displaystyle \langle \cdot \rangle }$ 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.

Ursell number

The Ursell number, named after Fritz 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 local significant wave height, ${\displaystyle k}$ is the local wavenumber 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:

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

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

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

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]

Impact on sediment transport

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 to wave 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 for sediment 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 onshore sediment 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 generates nearshore bar formation and provides a mechanism for the generation of three-dimensional features such as rip currents and rhythmic bars.

Models including wave skewness and asymmetry

Two different approaches exist to include wave shape in models: the phase-averaged approach and the phase-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 are WAVEWATCH3 (NOAA) and SWAN (TU Delft). WAVEWATCH3 is a global wave 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 study morphodynamics.

See also

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

References

1. Elgar, Steve, and R. T. Guza. "Observations of bispectra of shoaling surface gravity waves." Journal of Fluid Mechanics 161.1 (1985): 425-448.
2. van de Ven, Maartje (2018). "The effects of currents on wave nonlinearities".{{cite journal}}: Cite journal requires |journal= (help)
3. Doering, J.C.; Bowen, A.J. (1995-09-01). "Parametrization of orbital velocity asymmetries of shoaling and breaking waves using bispectral analysis". Coastal Engineering. 26 (1–2): 15–33. doi:10.1016/0378-3839(95)00007-X. ISSN   0378-3839.
4. Gómez-Enri, J.; Gommenginger, C. P.; Srokosz, M. A.; Challenor, P. G.; Benveniste, J. (2007-06-01). "Measuring Global Ocean Wave Skewness by Retracking RA-2 Envisat Waveforms". Journal of Atmospheric and Oceanic Technology. 24 (6): 1102–1116. Bibcode:2007JAtOT..24.1102G. doi: 10.1175/JTECH2014.1 . ISSN   0739-0572.
5. Ruessink, B. G.; Berg, T. J. J. van den; Rijn, L. C. van (2009). "Modeling sediment transport beneath skewed asymmetric waves above a plane bed". Journal of Geophysical Research: Oceans. 114 (C11). Bibcode:2009JGRC..11411021R. doi:10.1029/2009JC005416. ISSN   2156-2202. S2CID   129854001.
6. Elgar, Steve, and R. T. Guza. "Shoaling gravity waves: Comparisons between field observations, linear theory, and a nonlinear model." Journal of fluid mechanics 158 (1985): 47-70.
7. Dibajnia, Mohammad; Watanabe, Akira (1998-11-01). "Transport rate under irregular sheet flow conditions". Coastal Engineering. 35 (3): 167–183. doi:10.1016/S0378-3839(98)00034-9. ISSN   0378-3839.
8. Dugdale, Hannah L.; Macdonald, David W.; Pope, Lisa C.; Johnson, Paul J.; Burke, Terry (2008). "Reproductive skew and relatedness in social groups of European badgers, Meles meles". Molecular Ecology. 17 (7): 1815–1827. doi:10.1111/j.1365-294X.2008.03708.x. ISSN   1365-294X. PMID   18371017. S2CID   17970642.
9. Elgar, S. (December 1987). "Relationships involving third moments and bispectra of a harmonic process". IEEE Transactions on Acoustics, Speech, and Signal Processing. 35 (12): 1725–1726. doi:10.1109/TASSP.1987.1165090. ISSN   0096-3518.
10. Ursell, F. (October 1953). "The long-wave paradox in the theory of gravity waves". Mathematical Proceedings of the Cambridge Philosophical Society. 49 (4): 685–694. Bibcode:1953PCPS...49..685U. doi:10.1017/S0305004100028887. ISSN   1469-8064. S2CID   121889662.
11. Ruessink, B.G.; Ramaekers, G.; Van Rijn, L.C. (2012-07-01). "On the parameterization of the free-stream non-linear wave orbital motion in nearshore morphodynamic models". Coastal Engineering. 65: 56–63. doi:10.1016/j.coastaleng.2012.03.006. ISSN   0378-3839.
12. Ruessink, B. G.; Michallet, H.; Abreu, T.; Sancho, F.; A, D. A. Van der; Werf, J. J. Van der; Silva, P. A. (2011). "Observations of velocities, sand concentrations, and fluxes under velocity-asymmetric oscillatory flows". Journal of Geophysical Research: Oceans. 116 (C3). Bibcode:2011JGRC..116.3004R. doi:10.1029/2010JC006443. hdl: 2164/2592 . ISSN   2156-2202. S2CID   67785214.
13. Doering, J.C.; Bowen, A.J. (1995-09-01). "Parametrization of orbital velocity asymmetries of shoaling and breaking waves using bispectral analysis". Coastal Engineering. 26 (1–2): 15–33. doi:10.1016/0378-3839(95)00007-X. ISSN   0378-3839.
14. Ribberink, J. S.; Werf, J. J. van der; O'Donoghue, T.; Hassan, W. N. M. (2008-01-01). "Sand motion induced by oscillatory flows: Sheet flow and vortex ripples". Journal of Turbulence. 9: N20. Bibcode:2008JTurb...9...20R. doi:10.1080/14685240802220009. S2CID   122442269.
15. Ruessink, B. G.; Michallet, H.; Abreu, T.; Sancho, F.; A, D. A. Van der; Werf, J. J. Van der; Silva, P. A. (2011). "Observations of velocities, sand concentrations, and fluxes under velocity-asymmetric oscillatory flows". Journal of Geophysical Research: Oceans. 116 (C3). Bibcode:2011JGRC..116.3004R. doi:10.1029/2010JC006443. hdl: 2164/2592 . ISSN   2156-2202. S2CID   67785214.
16. Ruessink, B.G.; Ramaekers, G.; Van Rijn, L.C. (2012-07-01). "On the parameterization of the free-stream non-linear wave orbital motion in nearshore morphodynamic models". Coastal Engineering. 65: 56–63. doi:10.1016/j.coastaleng.2012.03.006. ISSN   0378-3839.