The evolution of large ocean waves: the role of local and rapid spectral changes

Abstract

This paper concerns the formation of large-focused or near-focused waves in both unidirectional and directional sea-states. When the crests of wave components of varying frequency superimpose at one point in space and time, a large, transient, focused wave can occur. These events are believed to be representative of the largest waves arising in a random sea and, as such, are of importance to the design of marine structures. The details of how such waves form also offer an explanation for the formation of the so-called freak or rogue waves in deep water. The physical mechanisms that govern the evolution of focused waves have been investigated by applying both the fully nonlinear wave model of Bateman et al . (Bateman et al . 2001 J. Comput. Phys . 174 , 277–305) and the Zakharov's evolution equation (Zakharov 1968 J. Appl. Mech. Tech. Phys . 9 , 190–194). Aspects of these two wave models are complementary, and their combined use allows the full nonlinearity to be considered and, at the same time, provides insights into the dominant physical processes. In unidirectional seas, it has been shown that the local evolution of the wave spectrum leads to larger maximum crest elevations. In contrast, in directional seas, the maximum crest elevation is well predicted by a second-order theory based on the underlying spectrum, but the shape of the largest wave is not. The differences between the evolution of large waves in unidirectional and directional sea-states have been investigated by analysing the results of Bateman et al . (2001) using a number of spectral analysis techniques. It has been shown that during the formation of a focused wave event, there are significant and rapid changes to the underlying wave spectrum. These changes alter both the amplitude of the wave components and their dispersive properties. Importantly, in unidirectional sea-states, the bandwidth of the spectrum typically increases; whereas, in directional sea-states it decreases. The changes to the wave spectra have been investigated using Zakharov's equation (1968). This has shown that the third-order resonant effects dominate changes to both the amplitude of the wave components and the dispersive properties of the wave group. While this is the case in both unidirectional and directional sea-states, the consequences are very different. By examining these consequences, directional sea-states in which large wave events that are higher and steeper than second-order theory would predict have been identified. This has implications for the types of sea-states in which rogue waves are most likely to occur.

The evolution of large non-breaking waves in intermediate and shallow water. I. Numerical calculations of uni-directional seas

“…In this latter case the energy transfers occur in a random sea and are characterized as being extremely slow, a small percentage of the total energy being transferred over hundreds of wave cycles. In contrast, Johannessen & Swan (2003) and Gibson & Swan (2007) showed that, in a deep-water-focused wave group, with the phasing far from random but the surface profile believed to be representative of an extreme wave event, the local nonlinearity is such that the third-order resonant interactions can be responsible for highly localized and very rapid energy transfers. In this case, significant energy transfers can occur over 5-10 wave cycles immediately preceding the extreme wave event.…”

“…In an earlier study, Gibson & Swan (2007) applied the ZE model to describe the evolution of a number of deep-water wave cases. In outlining the theory they also demonstrated the validity of comparisons between the BST and ZE models.…”

“…The essence of the paper concerns the importance of the reduced water depth in unidirectional seas; it seeks to determine whether the consequent weakening of the dispersive properties of a wave field leads to a fundamental change in the nature of large wave events. In particular, it will establish whether recent *Author for correspondence (c.swan@imperial.ac.uk). advances in the description of large deep-water waves, not least the local and rapid energy transfers leading to large departures from established wave solutions (Gibson & Swan 2007), are equally appropriate to intermediate and shallow-water depths, and whether they lead to different and unexpected wave events.…”

What kind of a wave is Hokusai's Great wave off Kanagawa ?

“…Linear waves when superposed add together linearly, so that their combined amplitude is the sum of the amplitudes of the components, but for nonlinear waves this is no longer true, and when two solitons coincide the combined height may be far greater than the sum; this has been postulated as a possible mechanism for the occurrence of freak waves. [44][45][46] In Hokusai's image we see the trailing slope of another large wave at the right of the picture, as if perhaps there is a group of freak waves, and at the left behind the great wave we glimpse a smaller wave that might be one of those colliding to produce the freak wave. Although he produced the image long before there existed any scientific understanding of freak waves, and Hokusai may well have exaggerated the wave height for artistic effect, it is possible that he could have painted The great wave off Kanagawa after hearing sailors tell of such a monster.…”

Systematic study of rogue wave probability distributions in a fourth‐order nonlinear Schrödinger equation

“…Realistic fully nonlinear computations wave by wave over large areas are very challenging, but initial attempts have been made to simulate the ocean surface using the full Euler equation both on large scales [Tanaka, 2001] and over smaller areas [Gibbs and Taylor, 2005;Gibson and Swan, 2007]. However, in this paper, we rely on series expansions, resulting in a sequence of approximations.…”