Victor I. Shrira

Long Nonlinear Surface and Internal Gravity Waves in a Rotating Ocean

Nonlinear dynamics of surface and internal waves in a stratified ocean under the influence of the Earths rotation is discussed. Attention is focussed upon guided waves long compared to the ocean depth. The effect of rotation on linear processes is reviewed in detail as well as the existing nonlinear models describing weakly and strongly nonlinear dynamics of long waves. The influence of rotation on small-scale waves and two-dimensional effects are also briefly considered. Some estimates of the influence of the Earths rotation on the parameters of real oceanic waves are presented and related to observational and numerical data.

Near-inertial waves in the ocean : beyond the 'traditional approximation'

The dynamics of linear internal waves in the ocean is analysed without adopting the ‘traditional approximation’, i.e. the horizontal component of the Earths 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.

Role of non-resonant interactions in the evolution of nonlinear random water wave fields

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

On the irreversibility of internal-wave dynamics due to wave trapping by mean flow inhomogeneities. Part 1. Local analysis

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Surface waves on shear currents: solution of the boundary-value problem

, not on the

A model of water wave 'horse-shoe' patterns

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Numerical modelling of water-wave evolution based on the Zakharov equation

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

On two approaches to the problem of instability of short-crested water waves

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Evolution of kurtosis for wind waves

is smaller than the kinetic time scale

On the Vertical Structure of Wind-Driven Sea Currents

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