Internal gravity waves excited by a body moving in a stratified fluid

Abstract

A problem of steady linear internal gravity wave generated by a thin rapidly moving solid body is considered. A linearized boundary no-flow condition and Boussinesq approximation are used. The solution of this problem is expressed in terms of the Green's function of the internal waves equation. The asymptotic representations of the solution in the far zone and the near zone are considered and its range of applicability is estimated. Numerical results are presented.

A modified Green's function for the internal gravitational wave equation in a layer of a stratified medium with a constant shear flow

“…[1][2][3][4][5][6] As a rule, the problem of constructing Green's function for the IGW equation is considered in this formulation, which makes it possible to describe the wave fields excited in the case of the motion of a point source of perturbations in a stratified medium with an arbitrary density distribution throughout its depth. Even within the limits of linear models, the resulting equations in the form of multiple quadratures are quite unusual.…”

“…In the general case of wave generation by arbitrary non-local sources of perturbations, the solution for all the components of the wave fields is expressed in terms of Green's function for the IGW equation and its asymptotic representations. [1][2][3][4][5][6] Green's function for the IGW equation, when there are mean shear flows in a layer −H < z < 0 of a stratified medium, 1,2 is considered next, namely, (1.1) where N(z) is the Väisälä-Brunt frequency, V 1 and V 2 are the components of the flow velocity V = {V 1 , V 2 , 0} for a certain level z and z is the immersion depth of the point source.…”

Uniform far field asymptotics of internal gravity waves from a source moving in a stratified fluid layer with smoothly varying bottom

“…The difficulty of this problem consist in the fact that when H=H(x,y) (H-function of bottom shape) the partial differential equations describing the internal waves do not admit the separations of variables. An exact analytic solution of this problem can be derived, for example by separation of variables, only in the case when the density distribution and the shape of the bottom are described by quite simple model functions [2]. For an arbitrary bottom shape and stratification, it is possible to construct only asymptotic representations of the solution in the near and far regions; however, for describing the internal wave field between these regions an exact numerical solution of the problem is required [2].…”

“…If the ocean depth varies slowly as compared with the characteristic length of the internal waves, which is reasonably true for a real ocean, then the method of geometric optics (WKB method) can be used for solving the problem of internal wave propagation above a varying bottom [1,6]. Using the asymptotic representation of the wave field at large distances, from the source [2], we can solve the problem of constructing the uniform asymptotic of the internal waves by a modification of the geometric optics method, namely by the "vertical modes -horizontal rays" method which does not assume that the medium parameters vary slowly with the vertical coordinate [1,3]. In [2] the uniform asymptotic of the far field of the internal waves were constructed for the constant depth case, and it was shown that the far internal waves field is a sum of individual modes each of which is enclosed within its own Mach cone, the asymptotic form of each mode near the corresponding wave front being expressed in terms of certain special function -the Airy function and its derivative, Fresnel integrals [2,5].…”

“…Using the asymptotic representation of the wave field at large distances, from the source [2], we can solve the problem of constructing the uniform asymptotic of the internal waves by a modification of the geometric optics method, namely by the "vertical modes -horizontal rays" method which does not assume that the medium parameters vary slowly with the vertical coordinate [1,3]. In [2] the uniform asymptotic of the far field of the internal waves were constructed for the constant depth case, and it was shown that the far internal waves field is a sum of individual modes each of which is enclosed within its own Mach cone, the asymptotic form of each mode near the corresponding wave front being expressed in terms of certain special function -the Airy function and its derivative, Fresnel integrals [2,5]. Therefore in investigating the problem of internal gravity waves propagation in a stratified fluid with slowly varying bottom must be sought not in the form of usual WKB expansion in harmonic waves but in the form of waves of a special types -Airy and Fresnel waves.…”