In this presentation, we will attempt to bring together topics which are intrinsically connected, but usually belong to different communities and therefore are discussed separately. These are modulational instability and wave-height probability, respectively.
The breaking of deep-water waves, as well as rogue waves, are anomalous waves in a sense of their height with respect to the neighbouring waves in a wave train/field. There are essential indications that both processes have a similar genesis and result from the modulational instability of nonlinear wave trains. If so, distinction between the two phenomena has to be made and, since the wave breaking is much more frequent, it can be further employed for investigating dynamics of the modulational instability in oceanic wave fields. Furthermore, the modulational instability is usually associated with two-dimensional wave trains. There exists argument, both analytical and experimental, that this kind of instability is impaired or even suppressed in three-dimensional (directional) wave systems, and this argument needs to be answered quantitatively: i.e whether the instability is active or not in the typical directional wave fields.
This discussion leads us to the short-term wave statistics, i.e. probability of crests/heights of individual waves. The typical approach is to treat all possible waves in the ocean or at a particular location as a single ensemble for which some comprehensive solution can be found. We will argue, however, that the probability distributions in different physical circumstances should be different, and by combining them together the inevitable scatter is introduced. The scatter and the accuracy will not improve by increasing the bulk data quality and quantity, and it hides the actual distribution of extreme events. The stable and unstable wave trains/fields have to be separated and their probability distributions treated differently. If the wave trains/fields in the wave records are stable, distributions for the second-order waves should serve well. If modulational instability is active, the rare extreme events not predicted by the second-order theory should become possible. This depends on wave steepness, bandwidth and directionality. Mean steepness also defines the wave breaking and therefore the upper limit for wave heights in this group of conditions. Under hurricane-like circumstances, the instability gives way to direct wind forcing, and yet another statistics is to be expected.
In the presentation, results of two-dimensional numerical and laboratory experiments featuring wave breaking due to modulational instability will be described. Then, experimental evidences which relates the wave breaking in oceanic conditions to such features of two-dimensional breaking waves due to modulational instability will be demonstrated. These will be followed by discussion of direct measurements of such instability-caused breaking in a directional wave tank with directional spread and mean steepness typical of those in the field. The modulational instability appears to be active in such conditions, and therefore conclusions on what is a possible maximal height of an individual wave due to such evolution of nonlinear wave trains can be proposed. Role of the wind forcing on the modulational instability of water waves will also be outlined.
Alexander V. Babanin is Professor in Ocean Engineering and Director of the Centre for Disaster Management and Public Safety at the University of Melbourne, Australia. Areas of expertise, research and teaching, are wind-generated waves, maritime, coastal and Arctic engineering, air-sea interactions, ocean turbulence and ocean dynamics, climate, environmental instrumentation and remote sensing of the ocean. These include extreme Metocean conditions, from tropical cyclones to Arctic and Antarctic environments. 360+ career total publications.