Wave equilibration process of a non-equilibrium sea-state in shallow water after strong depth variation
Authors: Zhang Jie , Benoit Michel , Ma Yuxiang 
Published: March 2022

&lt;p&gt;Freak waves are extreme events in the ocean, usually defined as waves with crest-to-trough excursion higher than twice the significant wave height of the ambient sea-state. In recent decades, freak waves have attracted considerable attention of oceanographers, as they seem to manifest with unexpected high frequencies and resulted in numerous tragedies. &amp;#160;&lt;/p&gt;&lt;p&gt;The dominant formation mechanisms of freak waves are still an open question, and various hypotheses have been put forward. The modulation instability (MI) is the most well-known prototype of freak wave in deep water. However, some researchers argue that the role played by MI is insignificant in real ocean, where the sea-state could be stabilized by the wave directionality, finite spectral width and shallow water depth. The non-equilibrium dynamics (NED) induced by significant depth variations could explain freak waves occurring in coastal areas where the MI is absent.&lt;/p&gt;&lt;p&gt;The NED manifests when an incident quasi-equilibrium sea-state undergoes a rapid depth decrease, and propagates in the new shallower water depth. Before reaching the new equilibrium state after depth transition, the wave evolution is characterized by strong non-Gaussian behavior. Previous studies mainly focused on the NED effects over an extent of a couple of wavelengths after the depth variation, showing the local enhancement of skewness and kurtosis (third- and fourth-order moments of the free surface elevation), the excitation of bound super-harmonics, and the intensified occurrence probability of freak waves. However, very few studies discuss how NED fades away at larger scale, and what is the equilibrium state established in the shallower region.&lt;/p&gt;&lt;p&gt;The present work numerically investigates the experiments reported by Trulsen et al. (J. Fluid Mech., vol. 882, R2, 2020) using a fully nonlinear potential wave model. We extend the analysis of NED to a longer spatial extent in the shallow water area, from O(&lt;em&gt;L&lt;sub&gt;p&lt;/sub&gt;&lt;/em&gt;) to O(10&lt;sup&gt;2 &lt;/sup&gt;&lt;em&gt;L&lt;sub&gt;p&lt;/sub&gt;&lt;/em&gt;), with &lt;em&gt;L&lt;sub&gt;p&lt;/sub&gt;&lt;/em&gt; being the spectral peak wavelength. It is found that the NED affects the shallow water wave evolution in two spatial scales: (i) in the shorter scale O(&lt;em&gt;L&lt;sub&gt;p&lt;/sub&gt;&lt;/em&gt;), as reported in the literature, the sea-state undergoes fast changes of wave statistics (skewness, kurtosis and probability of freak waves); (ii) in the longer scale O(10&lt;sup&gt;2 &lt;/sup&gt;&lt;em&gt;L&lt;sub&gt;p&lt;/sub&gt;&lt;/em&gt;), the NED results in strong modulation of the spectral shape. The analysis of the wave height distribution shows that, in the long scale, the sea-state is still non-Gaussian, and the freak wave occurrence probability is lower than the linear expectation. So the NED effects could &amp;#8220;protect&amp;#8221; the structures and ships from freak waves, at long distances from abrupt depth transitions. The output of the current work allows for a better assessment of the risk of freak waves in coastal areas, with practical benefits for coastal engineers and oceanographers.&lt;/p&gt;