Nonlinear interactions between deep-water waves and currents

Abstract

The effects of nonlinearity on a train of linear water waves in deep water interacting with underlying currents are investigated numerically via a boundary-integral method. The current is assumed to be two-dimensional and stationary, being induced by a distribution of singularities located beneath the free surface, which impose sharp and gentle surface velocity gradients. For ‘slowly’ varying currents, the fully nonlinear results confirm that opposing currents induce wave steepening and breaking within the region where a high convergence of rays occurs. For ‘rapidly’ varying currents, wave blocking and breaking are more prominent. In this case reflection was observed when sufficiently strong adverse currents are imposed, confirming that at least part of the wave energy that builds up within the caustic can be released in the form of partial reflection and wave breaking. For bichromatic waves, the fully nonlinear results show that partial wave blocking occurs at the individual wave components in the wave groups and that waves become almost monochromatic upstream of the blocking region.

An analytical model of the evolution of a Stokes wave and its two Benjamin–Feir sidebands on nonuniform unidirectional current

“…Amplitude dispersion effects can considerably alter the location of wave blocking predicted by linear theory, and nonlinear processes can adversely affect the dynamics of the wave field beyond the blocking point. Donato et al (1999), Stocker and Peregrine (1999), and Moreira and Peregrine (2012) conducted fully nonlinear computations to analyze the behavior of a train of water waves in deep water when meeting nonuniform currents, especially in the region where linear solutions become singular. The authors employed spatially periodic domains in numerical study and showed that adverse currents induce wave steepening and breaking.…”

“…Toffoli et al (2013) showed experimentally that an initially stable surface wave can become modulationally unstable and even produce freak or giant waves when meeting negative horizontal current. Onorato et al (2011) suggested an equation for predicting the maximum amplitude A max during the wave evolution of currents in deep water. Their numerical results revealed that the maximum amplitude of the freak wave depends on U/C g , where U is the velocity of the current and C g is the group velocity of the wave packet.…”

“…They mentioned that theoretical values of amplification (Onorato et al, 2011;Toffoli et al, 2013) are essentially overestimated, probably owing to the effects of finite depth and wave breaking.…”

Vorticity effects on nonlinear wave–current interactions in deep water

“…An increase in wave steepness prior to blocking leads to 'rough' water surfaces, which cause significant hazards to boats and ships navigating under these circumstances. For rapidly varying currents, Moreira & Peregrine (2012) showed that part of this wave energy can be released in the form of partial reflection before wave breaking occurs. Depending on its direction, Kharif et al (2008) observed that wind may sustain these steep waves which then evolve into breaking waves.…”

Trapping and instability of directional gravity waves in localized water currents

“…Water waves encountering a counterpropagating current will slow down and their wave energy pile up, enhancing their amplitudes and increasing the statistical risk of extreme events. Opposing currents can lead to the steepening and breaking of waves [3], as well as to instabilities and localization of waves [4]. Variable currents can also focus waves into caustic regions [5,6] and capture waves into localized, opposing currents [7][8][9][10], where large amplitude waves are produced.…”