Mode Structure of Internal Gravity Waves Generated by Localized Sources in a Stratified Ocean With Shears Flows

The aim of the work is to study the mode structure of solutions describing the generation of internal gravity waves in stratified media with model distributions of the buoyancy frequency and background shear currents, which allows us to determine the main qualitative characteristics of the behavior of dispersion relations at small wave numbers depending on the mode number. The problem of constructing solutions describing the generation of linear internal gravity waves in a layer of a stratified medium of finite depth with model distributions of the buoyancy frequency and background shear current is considered. Under the assumption of the Miles–Howard stability for the Richardson number, the corresponding dispersion dependences are studied. It is shown that, depending on the parameters of the linear shear current, the dispersion curves of the wave modes can have qualitatively different asymptotic representations at small wave numbers. The dispersion curves of a finite number of modes describing waves with a limited length, at small values of the wave number, admit expansions in even powers of a small parameter. The dispersion curves of the remaining modes, corresponding to waves with an arbitrarily large length, are expanded in a series in odd powers of small wave numbers. The phase structure of the wave fields is studied depending on the mode number and the main characteristics of the shear currents. Analytical estimates are obtained that make it possible, depending on the parameters of the model flow, to find the number of the wave mode that divides the entire existing set of wave modes into limited and long-wave ones.