Supervisor : Frédéric Dias (Professor, Centre Borelli)
Guest Lab : Centre Borelli, ENS Paris-Saclay
Sponsor: ERC Advanced Grant HIGHWAVE
Duration: 5-6 months – PhD level salary
Possibility of extension to a PhD: Yes

HIGHWAVE is a cutting-edge mathematical project that uses real-time raw data harvested in situ on the west coast of Ireland by the project team to develop new models and new algorithms for breaking waves on the ocean surface. These new models will provide information about air and water exchange in oceanic environments, boulder deposits, erosion, and structural damage.

As discussed in a recent book [1], two questions remain partially unresolved: what causes a wave to break (breaking onset), and how much wave energy is lost during breaking (breaking dissipation). The field measurement of breaking waves is one of the greatest challenges in fluid mechanics. As stated in Ref [2], “advances have been made in various areas [of wave breaking field measurement], but consensus among researchers is limited. The slow rate of progress is a result of the difficult nature of measurements in the field, as well as the costs associated with conducting these experiments.”

What are the challenges? Determining the actual onset of breaking and breaking severity of real-world waves is a challenge. The ability of numerical codes to predict the short-scale surface evolution and the energy dissipation involved in breaking regions, spray sheets, and turbulence has not yet been validated and remains a challenge. Wave breaking field measurement is an issue, with a preference for instruments that are far from the breaking zone. Stereo video is a possibility, but it depends highly on weather conditions and brightness. HIGHWAVE attacks all these challenges, both theoretically and experimentally.

The work packages (WPs) of the project are summarized below:

The present internship deals with WP1.  A new breaking threshold of 0.85 was found for the parameter B computed as the ratio of the wave crest speed to the fluid speed [3]. This threshold has since been independently confirmed [4]. However, whilst the breakthrough is considerable, the threshold is no longer valid when waves start to strongly feel the bottom [5]. A new generic breaking onset threshold will be looked for, using machine learning (ML) algorithms. Since the breaking strength parameter b appears to be directly related to the rate of change of B [6], we will also check the link between b and the newly found threshold parameter valid for a wider range of wave breaking mechanisms.

The key goals, together with the proposed methodology, include:

  • Universal threshold for wave breaking onset that is robust to the range of mechanisms that contribute to wave breaking (the approach here is to combine Machine Learning classification algorithms with numerical simulations based on the full water wave equations to find the breaking onset threshold)
  • Prediction of breaking strength across the whole range of wave breaking mechanisms (the approach here is to combine Machine Learning tools with field measurements to find the link between breaking strength and breaking threshold parameters).

State-of-the-art ML algorithms will be used for the first time in wave breaking to determine crucial parameters such as the breaking threshold parameter and the breaking strength parameter. The methodology will be designed to be generic so that it can be applied to other areas of wave dynamics.

The research group of Professor Dias has a dozen of members (mostly PhD students and postdocs) spread over ENS Paris-Saclay and University College Dublin. A field trip to the west coast of Ireland will be part of the internship if the sanitary situation allows it.

Contact details: Professor Frederic Dias (


[1] A. Babanin, Breaking and dissipation of ocean surface waves, Cambridge University Press (2011)

[2] M. Perlin, W. Choi, and Z. Tian, Breaking waves in deep and intermediate waters, Annu. Rev. Fluid Mech. 45, 115–145 (2013)

[3] X. Barthelemy, M.L. Banner, W.L. Peirson, F. Fedele, M. Allis, and F. Dias, On a unified breaking onset threshold for gravity waves in deep and intermediate depth water, J. Fluid Mech. 841, 463–488 (2018)

[4] B.R. Seiffert, G. Ducrozet, and F. Bonnefoy, Simulation of breaking waves using the high-order spectral method with laboratory experiments: Wave-breaking onset, Ocean Modelling 119, 94–104 (2017); B.R. Seiffert, and G. Ducrozet, Simulation of breaking waves using the high-order spectral method with laboratory experiments: wave-breaking energy dissipation, Ocean Dynamics 68, 65–89 (2018)

[5] J. Herterich and F. Dias, Extreme long waves over a varying bathymetry. J. Fluid Mech.

878, 481–501 (2019); doi : 10.1017/jfm.2019.618

[6] M. Derakhti, M.L. Banner, and J.T. Kirby, Predicting the breaking strength of gravity water waves in deep and intermediate depth, J. Fluid Mech. 848, R2 (2018)