Effects of a submerged bar and trench on weakly nonlinear surface gravity waves
Authors: Li Yan 
Published: March 2022

&lt;p&gt;It is known that the kurtosis of wave elevations can reach a maximum near the top of abrupt depth transitions (Zeng, &lt;em&gt;et al.&lt;/em&gt; 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, &lt;em&gt;et al.&lt;/em&gt;, 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, &lt;em&gt;et al.&lt;/em&gt; 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 &amp;#8216;mean&amp;#8217; 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 &lt;em&gt;et al.&lt;/em&gt; (2020) and numerically by Laurence &lt;em&gt;et al.&lt;/em&gt; (2021) and Zhang &lt;em&gt;et al.&lt;/em&gt; (2021).&lt;/p&gt;&lt;p&gt;&lt;strong&gt;&lt;span&gt;References&lt;/span&gt;&lt;/strong&gt;&lt;/p&gt;&lt;p&gt;&lt;span&gt;Lawrence, C., Trulsen, K. &amp; Gramstad, O. 2021 Statistical properties of wave kinematics in long-crested irregular waves propagating over non-uniform bathymetry. &lt;em&gt;Phys. Fluids &lt;/em&gt;33 (4), 046601.&lt;/span&gt;&lt;/p&gt;&lt;p&gt;&lt;span&gt;Li, Y., Draycott, S., Zheng, Y., Lin, Z., Adcock, T.A.A. &amp; Van Den Bremer, T.S. 2021b Why rogue waves occur atop abrupt depth transitions. &lt;em&gt;J. Fluid Mech.&lt;/em&gt; 919, R5.&lt;/span&gt;&lt;/p&gt;&lt;p&gt;&lt;span&gt;Trulsen, K., Raust&amp;#248;l, A., Jorde, S. &amp; B&amp;#230;verfjord Rye, L. 2020 Extreme wave statistics of long-crested irregular waves over a shoal. &lt;em&gt;J. Fluid Mech.&lt;/em&gt; 882, R2.&lt;/span&gt;&lt;/p&gt;&lt;p&gt;&lt;span&gt;Zeng, H. &amp; Trulsen, K. 2012 Evolution of skewness and kurtosis of weakly nonlinear unidirectional waves over a sloping bottom. &lt;em&gt;Nat. Hazards Earth Syst. Sci. &lt;/em&gt;12 (3), 631&amp;#8211;638.&lt;/span&gt;&lt;/p&gt;&lt;p&gt;&lt;span&gt;Zhang, J. &amp; Benoit, M. 2021 Wave&amp;#8211;bottom interaction and extreme wave statistics due to shoaling and de-shoaling of irregular long-crested wave trains over steep seabed changes. &lt;em&gt;J. Fluid Mech.&lt;/em&gt; 912, A28.&lt;/span&gt;&lt;/p&gt;