Effects of a submerged bar and trench on weakly nonlinear surface gravity waves

It is known that the kurtosis of wave elevations can reach a maximum near the top of abrupt depth transitions (Zeng, et al. 2012), which can be explained due to a mechanism of the interaction between free and second-order bound waves due to a depth transition (Li, et al., 2021). The horizontal velocity at a still surface can show significantly different statistics from that of surface displacement for an irregular train of random long-crested waves atop a submerged bar (Trulsen, et al. 2020). Motivated by the latter, this work focuses on effects of a submerged bar and trench on the main properties of weakly nonlinear surface gravity waves in two dimensions. The analysis is based on a novel theoretical framework that allows for narrow-banded surface waves experiencing a step-type seabed with two sudden depth transitions, correct to the second order in wave steepness. Such a seabed is modeled both as a submerged trench and bar. To reveal the fundamental physics, the evolution of a wavepacket that experiences abrupt depth transitions are examined in detail; (a) we show the differences of the release of free waves at second order in wave steepness both for the super-harmonic and sub-harmonic or ‘mean’ contents between a submerged bar and trench; (b) we also show the differences between the spatial distributions of horizontal velocity field induced by a narrowband wavepacket over a bar and a trench; (c) furthermore, we examine which parameters affect the release of free waves and the distributions of the horizontal velocity. The novel physics has implications for wave statistics for long-crested irregular waves experiencing a submerged bar as investigated experimentally by Trulsen et al. (2020) and numerically by Laurence et al. (2021) and Zhang et al. (2021).

References

Lawrence, C., Trulsen, K. & Gramstad, O. 2021 Statistical properties of wave kinematics in long-crested irregular waves propagating over non-uniform bathymetry. Phys. Fluids 33 (4), 046601.

Li, Y., Draycott, S., Zheng, Y., Lin, Z., Adcock, T.A.A. & Van Den Bremer, T.S. 2021b Why rogue waves occur atop abrupt depth transitions. J. Fluid Mech. 919, R5.

Trulsen, K., Raustøl, A., Jorde, S. & Bæverfjord Rye, L. 2020 Extreme wave statistics of long-crested irregular waves over a shoal. J. Fluid Mech. 882, R2.

Zeng, H. & Trulsen, K. 2012 Evolution of skewness and kurtosis of weakly nonlinear unidirectional waves over a sloping bottom. Nat. Hazards Earth Syst. Sci. 12 (3), 631–638.

Zhang, J. & Benoit, M. 2021 Wave–bottom interaction and extreme wave statistics due to shoaling and de-shoaling of irregular long-crested wave trains over steep seabed changes. J. Fluid Mech. 912, A28.