Dynamics of non-harmonic internal gravity wave packets in stratified media

In the paper taking the assumption of the slowness of the change of the parameters of the vertically stratified medium in the horizontal direction and in time, the evolution of the non-harmonic wave packages of the internal gravity waves has been analyzed. The concrete form of the wave packages can be expressed through some model functions and is defined by the local behavior of the dispersive curves of the separate modes near to the corresponding special points. The solution of this problem is possible with the help of the modified variant of the special-time ray method offered by the authors (the method of geometrical optics), the basic difference of which consists that the asymptotic representation of the solution may be found in the form the series of the non-integer degrees of some small parameter. At that the exponent depends on the concrete form of representation of this package. The obvious kind of the representation is determined from the principle of the localness and the asymptotic behavior of the solution in the stationary and the horizontally-homogeneous case. The phases of the wave packages are determined from the corresponding equations of the eikonal, which can be solved numerically on the characteristics (rays). Amplitudes of the wave packages are determined from the laws of conservation of the some invariants along the characteristics (rays).Comment: 6 page

Dynamics of the internal gravity waves in the heterogeneous and nonstationary stratified mediums

In the present paper in the assumption of the slowness of variation of the vertically stratified medium parameters in the horizontal direction within the time we have analyzed the evolution of the non-harmonic wave trains of the internal gravity waves. The particular form of the wave train can be expressed through some special functions, for example, Airy functions, Fresnel integrals, Pearsy integrals, etc., and is determined by the local behavior of the dispersion curves of the separate modes near to the corresponding singular points. The solution of this problem is possible using the modified version of the space-time ray-tracing method offered by the authors (the method of the ray optics), the fundamental difference of which consists, that the asymptotic notation of such a solution should be searched for in the form of the series using the non-integral degrees of some small parameter, the asymptotic forms of the solution at analysis of evolution of the non-harmonic wave trains present in the stratified non-stationary horizontally-non-uniform mediums is searched in the form of the series using the non-integral degrees of some small parameter, at that the exponent depends on the concrete type of the wave train notation. The particular form of the notation is determined from the asymptotical behavior of the solution in the stationary horizontally-homogeneous event. The phase of the wave train will be determined from the corresponding eikonal equation, which can be solved numerically using the characteristics (rays). The amplitude of the wave train is determined from some law of preservation along the characteristics (rays).Comment: 57 pages, 11 figure

Non-local boundary conditions and internal gravity wave generation

This work focuses on the mathematical modeling of wave dynamics in a stratified medium. Non-local absorbing boundary conditions are considered based on the two following assumptions: (i) a linear theory can be applied at large distances from perturbation sources; and (ii) there are no other sources of wave disturbance outside the mixing zone in the stratified medium. The boundary conditions considered in this paper allowed us to describe the diverging internal gravity waves generated by the mixing region in a stratified medium.Comment: 23 pages, 1 figur

Fundamental problems of modeling the dynamics of internal gravity waves with applications to the Arctic Basin

In this paper, we consider fundamental problems of the dynamics of internal gravity waves. We present analytical and numerical algorithms for calculating the wave fields for a set of values of the parameters, as observed in the ocean. We show that our mathematical models can describe the wave dynamics of the Arctic Basin, taking into account the actual physical characteristics of sea water, topography of its floor, etc. The numerical and analytical results show that the internal gravity waves have a significant effect on underwater sea objects in the Arctic Basin.Comment: 26 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:physics/0609236, arXiv:1110.613

Internal gravity waves in a stratified medium of non-uniform depth

The problem of reconstructing non-harmonic internal gravity wave packets generated by a source moving in a stratified ocean is considered. The uniform asymptotic form of the internal gravity waves field generated by a source moving above the smoothly varying floor is constructed. The solution is proposed in terms of wave modes, propagating independently at the adiabatic approximation, and described as a non-integral degree series of a small parameter characterizing the stratified medium. A specific form of the wave packets, which can be parameterized in terms of model functions (Airy functions), depends on a local behavior of the dispersion curves of individual wave mode. A modified space-time ray method was proposed, which belongs to the class of geometrical optics methods. The key point of the proposed technique is the possibility to derive the asymptotic representation of the solution in terms of a non-integral degree series of the some small parameter.Comment: 33 pages, 18 figure

Internal gravity waves from a non-local perturbation source

The internal gravity waves far field exited by a non-local perturbation sources was considered. A separate wave mode asymptomatic presentation was constructed, describing the wave field key features depending on the source geometry.Comment: 9 pages, 3 figure

Green modified function of the equation of the internal gravity waves in the stratum of the stratified medium with constant average flow

In the present paper construction of the modified function of Green equation for internal gravity waves in the stratum of the stratified medium at presence of constant average flows is considered, properties of the corresponding spectral problems, the modified eigenfunctions and eigenvalues are investigated. Usage of the modified function of Green equation can give in some physically interesting events more friendly representations of the solutions for the fields of the internal gravity waves, including the wave fields disturbed by the non-local disturbing bodies.Comment: 46 pages, 4 figure

Evaluation of the linear theory satisfiability limits in propagation of the internal gravity waves

A problem of finding the linear theory satisfaction limits in propagation of the internal gravity waves is considered. It is evident that internal gravity waves excitation, propagation in actual practice is highly nonlinear phenomenon. However with some reasonable assumptions it is possible to linearize equations of internal waves generation and propagation. It is shown that in interesting for us wavelength range we can use linear approximation during study of internal gravity waves dynamics. Similarly it is easy to evaluate also influence of other corrections to the linear theory of internal gravity waves generation and propagation, and the obtained results indicate adequacy and supportability of linear model wave dynamics.Comment: 15 page

The dynamics of internal gravity waves in the ocean: theory and applications

In this paper we consider fundamental processes of the disturbance and propagation of internal gravity waves in the ocean modeled as a vertically stratified, horizontally non-uniform, and non-stationary medium. We develop asymptotic methods for describing the wave dynamics by generalizing the spatiotemporal ray-tracing method (a geometrical optics method). We present analytical and numerical algorithms for calculating the internal gravity wave fields using actual ocean parameters such as physical characteristics of the sea water, topography of its floor, etc. We demonstrate that our mathematical models can realistically describe the internal gravity wave dynamics in the ocean. Our numerical and analytical results show that the internal gravity waves have a significant impact on underwater objects in the ocean.Comment: 51 pages, 8 figure

General problems of the internal gravity waves linear theory

The internal gravity waves are the oscillations present in the gravitational field of the stratified medium, that is the mediums which density raises with the depth change. If the equilibrium state of the component volume of this medium is disturbed, for example, upward, then it will become more heavy, than the medium surrounding it, and Archimedian forces will cause its motion back to its equilibrium position . The main parameter of any oscillation system is the oscillation frequency, and it is determined by the ratio of two factors - the restoring forces seeking to return the disturbed system to its equilibrium position and the inertial forces. For the internal waves the restoring forces are proportional to the vertical gradient of density of the liquid, and the inertial forces are proportional to the density itself. The oscillating frequency typical for the internal gravitational waves is the value N(z)=({-g/ro(z)}* {d{ro(z)}/d{z}})^{-1/2} called the buoyancy frequency or Brunt-Vaisala frequency. Here ro(z) - the density as the function of the depth z, g - the acceleration of the gravity, the sign "-" originates due to the fact, that the density raises with the increasing depth and consequently {d{ro}/dz}<0. The paper has considered the planar internal gravitational waves in the exponentially stratified medium, that is in the medium with the constant distribution of Brunt-Vaisala frequency in depth, analyzed the problem of the waves reflection from the planar boundaries, defined the Green's function for the equation of the internal gravity waves in the exponentially stratified medium of the of the endless depth, and also outlined the information on the main properties of the internal gravity waves in the stratified layer of the terminated depth and in the stratified mediums with average shift streams.Comment: 52 page