The formation of breathers as prototypes of freak waves is studied within the framework of the classic 'focussing' nonlinear Schrödinger (NLS) equation. The analysis is confined to evolution of localised initial perturbations upon an otherwise uniform wave train. For a breather to emerge out of an initial hump, a certain integral over the hump, which we refer to as the "area", should exceed a certain critical value. It is shown that the breathers produced by the critical and slightly supercritical initial perturbations are described by the Peregrine soliton which represents a spatially localised breather with only one oscillation in time and thus captures the main feature of freak waves: a propensity to appear out of nowhere and disappear without trace. The maximal amplitude of the Peregrine soliton equals exactly three times the amplitude of the unperturbed uniform wave train. It is found that, independently of the proximity to criticality, all small-amplitude supercritical humps generate the Peregrine solitons to leading order. Since the criticality condition requires the spatial scale of the initially small perturbation to be very large (inversely proportional to the square root of the smallness of the hump magnitude), this allows one to predict a priori whether a freak wave could develop judging just by the presence/absence of the corresponding scales in the initial conditions. If a freak wave does develop, it will be most likely the Peregrine soliton with the peak amplitude close to three times the background level. Hence, within the framework of the one-dimensional NLS equation the Peregrine soliton describes the most likely freak-wave patterns. The prospects of applying the findings to real-world freak waves are also discussed.

The dynamics of linear internal waves in the ocean is analysed without adopting the ‘traditional approximation’, i.e. the horizontal component of the Earth's rotation is taken into account. It is shown that non-traditional effects profoundly change the dynamics of near-inertial waves in a vertically confined ocean. The partial differential equation describing linear internal-wave propagation can no longer be solved by separation of spatial variables; it was however pointed out earlier in the literature that a reduction to a Sturm–Liouville problem is still possible, a line that is pursued here. In its formal structure the Sturm–Liouville problem is the same as under the traditional approximation, but its eigenfunctions are no longer normal vertical modes of the full problem. The question is addressed of whether the solution found through this reduction is the general one: a set of eigenfunctions to the full problem is constructed, which depend in a non-separable way on the two spatial variables; these functions are orthogonal and form, under mild assumptions, a complete basis.In the near-inertial range, non-traditional effects act as a singular perturbation; this is seen from the sub-inertial short-wave limit, which is present whenever the ‘non-traditional’ terms are there, but disappears under the traditional approximation. In the dispersion relation the sub-inertial modes represent a smooth continuation of the super-inertial ones. The combined effect of the horizontal component of rotation and a vertical inhomogeneity in the stratification is found to play a crucial role in the dynamics of sub-inertial waves. They are trapped in waveguides localized around minima of the buoyancy frequency. The presence of horizontal inhomogeneities in the effective Coriolis parameter (such as shear currents or beta effect) are shown to enable a transition from super-inertial to sub-inertial waves (and thus effectively an irreversible transformation of large-scale into small-scale motions). It is suggested that this transformation provides a mechanism for mixing in the deep ocean.The notion of critical reflection of internal waves at a sloping bottom is also modified by non-traditional effects, and they strongly increase the probability of critical reflection in the near-inertial to tidal range.

We consider a classic boundary-value problem for deep-water gravity-capillary waves in a shear flow, composed of the Rayleigh equation and the standard linearized kinematic and dynamic inviscid boundary conditions at the free surface. We derived the exact solution for this problem in terms of an infinite series in powers of a certain parameter e, which characterizes the smallness of the deviation of the wave motion from the potential motion. For the existence and absolute convergence of the solution it is sufficient that e be less than unity.The truncated sums of the series provide approximate solutions with a priori prescribed accuracy. In particular, for the short-wave instability, which can be interpreted as the Miles critical-layer-type instability, the explicit approximate expressions for the growth rates are derived. The growth rates in a certain (very narrow) range of scales can exceed the Miles increments caused by the wind.The effect of thin boundary layers on the dispersion relation was also investigated using an asymptotic procedure based on the smallness of the product of the layer thickness and wavenumber. The criterion specifying when and with what accuracy the boundary-layer influence can be neglected has been derived.

We present the results of direct numerical simulations (DNS) of the evolution of nonlinear random water wave fields. The aim of the work is to validate the hypotheses underlying the statistical theory of nonlinear dispersive waves and to clarify the role of exactly resonant, nearly resonant and non-resonant wave interactions. These basic questions are addressed by examining relatively simple wave systems consisting of a finite number of wave packets localized in Fourier space. For simulation of the long-term evolution of random water wave fields we employ an efficient DNS approach based on the integrodifferential Zakharov equation. The non-resonant cubic terms in the Hamiltonian are excluded by the canonical transformation. The proposed approach does not use a regular grid of harmonics in Fourier space. Instead, wave packets are represented by clusters of discrete Fourier harmonics.The simulations demonstrate the key importance of near-resonant interactions for the nonlinear evolution of statistical characteristics of wave fields, and show that simulations taking account of only exactly resonant interactions lead to physically meaningless results. Moreover, exact resonances can be excluded without a noticeable effect on the field evolution, provided that near-resonant interactions are retained. The field evolution is shown to be robust with respect to the details of the account taken of near-resonant interactions. For a wave system initially far from equilibrium, or driven out of equilibrium by an abrupt change of external forcing, the evolution occurs on the ‘dynamical’ time scale, that is with quadratic dependence on nonlinearity $\varepsilon$, not on the $O(\varepsilon^{-4})$ time scale predicted by the standard statistical theory. However, if a wave system is initially close to equilibrium and evolves slowly in the presence of an appropriate forcing, this evolution is in quantitative accordance with the predictions of the kinetic equation. We suggest a modified version of the kinetic equation able to describe all stages of evolution.Although the dynamic time scale of quintet interactions $\varepsilon^{-3}$ is smaller than the kinetic time scale $\varepsilon^{-4}$, they are not included in the existing statistical theory, and their effect on the evolution of wave spectra is unknown. We show that these interactions can significantly affect the spectrum evolution, although on a time scale much larger than $O(\varepsilon^{-4})$. However, for waves of high but still realistic steepness $\varepsilon\,{\sim}\,0.25$, the scales of evolution are no longer separated. By tracing the evolution of high statistical moments of the wave field, we directly verify one of the main assumptions used in the derivation of the kinetic equation: the quasi-Gaussianity of the wave holds throughout the evolution, both with and without accounting for quintet interactions.The conclusions are not confined to water waves and are applicable to a generic weakly nonlinear dispersive wave field with prohibited triad interactions.

The vertical structure of wind-driven sea surface currents and the role of wind-wave breaking in its formation are investigated by means of both field experiments and modeling. Analysis of drifter measurements of surface currents in the uppermost 5-m layer at wind speeds from 3 to 15 m s Ϫ1 is the experimental starting point of this study. The velocity gradients beneath the surface are found to be 2 to 5 times weaker than in the "wall" boundary layer. Surface wind drift (identified via drift of an artificial slick) with respect to 0.5-m depths is about 0.7%, which is even less than the velocity defect over the molecular sublayer in the wall boundary layer at a smooth surface. To interpret the data, a semiempirical model describing the effect of wave breaking on wind-driven surface currents and subsurface turbulence is proposed. The model elaborates on the idea of direct injection of momentum and energy from wave breaking (including microscale breaking) into the water body. Momentum and energy transported by breaking waves into the water significantly enhance the turbulent mixing and considerably decrease velocity shears as compared to the wall boundary layer. No "artificial" surface roughness scale is needed in the model. From the experimental fact of the existence of cool temperature skin at the sea surface, it is deduced that there is a molecular sublayer at the water side of the sea surface with a thickness that depends on turbulence intensity just beneath the surface. The model predictions are consistent with the reported and other available experimental data.