Asymptotical analysis of internal gravity wave dynamics in stratified medium

Abstract

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 horizontallyhomogeneous 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 concervation of the some invariants along the characteristics (rays).

Analytical Approximations of Dispersion Relations for Internal Gravity Waves Equation with Shear Flows

“…Analytical representations of dispersion relations make it possible to study the IGW dynamics in a stratified medium with flows and slowly varying parameters [1,31,32]. Horizontal heterogeneity and non-stationarity have a significant impact on the IGW propagation in the world ocean [4,5,[12][13][14][15].…”

“…The phase functions (model integrals) of asymptotic solutions are expressed in terms of various special functions: Fresnel integrals, Airy functions, Pearcey integrals, and others. The specific choice of the phase function (model integrals) is completely determined by the analytical properties of the dispersion relations [8,9,31,32]. The obtained analytical solutions of dispersion relations allow one to efficiently calculate the main phase characteristics of the excited IGW fields and, in addition, to qualitatively analyze the obtained solutions, which is important for the correct statement of more complicated mathematical models of wave dynamics in real natural stratified media (world ocean, Earth's atmosphere).…”