A nonlinear Schr\"odinger equation for water waves on finite depth with constant vorticity

A nonlinear Schr\"odinger equation for the envelope of two dimensional surface water waves on finite depth with non zero constant vorticity is derived, and the influence of this constant vorticity on the well known stability properties of weakly nonlinear wave packets is studied. It is demonstrated that vorticity modifies significantly the modulational instability properties of weakly nonlinear plane waves, namely the growth rate and bandwidth. At third order we have shown the importance of the coupling between the mean flow induced by the modulation and the vorticity. Furthermore, it is shown that these plane wave solutions may be linearly stable to modulational instability for an opposite shear current independently of the dimensionless parameter kh, where k and h are the carrier wavenumber and depth respectively

Wave modelling - the state of the art

This paper is the product of the wave modelling community and it tries to make a picture of the present situation in this branch of science, exploring the previous and the most recent results and looking ahead towards the solution of the problems we presently face. Both theory and applications are considered. The many faces of the subject imply separate discussions. This is reflected into the single sections, seven of them, each dealing with a specific topic, the whole providing a broad and solid overview of the present state of the art. After an introduction framing the problem and the approach we followed, we deal in sequence with the following subjects: (Section) 2, generation by wind; 3, nonlinear interactions in deep water; 4, white-capping dissipation; 5, nonlinear interactions in shallow water; 6, dissipation at the sea bottom; 7, wave propagation; 8, numerics. The two final sections, 9 and 10, summarize the present situation from a general point of view and try to look at the future developments

Hamiltonian model for coupled surface and internal waves in the presence of currents

We examine a two dimensional fluid system consisting of a lower medium bounded underneath by a flatbed and an upper medium with a free surface. The two media are separated by a free common interface. The gravity driven surface and internal water waves (at the common interface between the media) in the presence of a depth-dependent current are studied under certain physical assumptions. Both media are considered incompressible and with prescribed vorticities. Using the Hamiltonian approach the Hamiltonian of the system is constructed in terms of 'wave' variables and the equations of motion are calculated. The resultant equations of motion are then analysed to show that wave-current interaction is influenced only by the current profile in the 'strips' adjacent to the surface and the interface. Small amplitude and long-wave approximations are also presented.Comment: 33 pages, 1 figur

Nonlinear effects in resonant layers in solar and space plasmas

The present paper reviews recent advances in the theory of nonlinear driven magnetohydrodynamic (MHD) waves in slow and Alfven resonant layers. Simple estimations show that in the vicinity of resonant positions the amplitude of variables can grow over the threshold where linear descriptions are valid. Using the method of matched asymptotic expansions, governing equations of dynamics inside the dissipative layer and jump conditions across the dissipative layers are derived. These relations are essential when studying the efficiency of resonant absorption. Nonlinearity in dissipative layers can generate new effects, such as mean flows, which can have serious implications on the stability and efficiency of the resonance

Conservation Laws and Bounds on the Efficiency of Wind-Wave Growth

We examine two means by which wind can impart energy to waves: sheltering and deposition of material upwards from windward surface shear. The shear driven deposition is shown to be the more efficient process. Lengthening of waves to match the wind speed is shown to be very inefficient and consume a large fraction of the energy imparted by the wind. The surface shear provides a low energy sink that absorbs most of the momentum from the wind. These produce bounds on the efficiency of wave growth. The results here are computed in a model independent and perturbation free fashion by a careful consideration of conservation laws. By combining these effects we can place bounds on the rates waves can grow in a given fetch and the relative amount of shear flow versus the, relatively small, Stokes drift that must arise

Toward Regional Characterizations of the Oceanic Internal Wavefield

Many major oceanographic internal wave observational programs of the last 4 decades are reanalyzed in order to characterize variability of the deep ocean internal wavefield. The observations are discussed in the context of the universal spectral model proposed by Garrett and Munk. The Garrett and Munk model is a good description of wintertime conditions at Site-D on the continental rise north of the Gulf Stream. Elsewhere and at other times, significant deviations in terms of amplitude, separability of the 2-D vertical wavenumber - frequency spectrum, and departure from the model's functional form are noted. Subtle geographic patterns are apparent in deviations from the high frequency and high vertical wavenumber power laws of the Garrett and Munk spectrum. Moreover, such deviations tend to co-vary: whiter frequency spectra are partnered with redder vertical wavenumber spectra. Attempts are made to interpret the variability in terms of the interplay between generation, propagation and nonlinearity using a statistical radiative balance equation. This process frames major questions for future research with the insight that such integrative studies could constrain both observationally and theoretically based interpretations