On the large-time asymptotics of Green's function for internal gravity waves

✍ Scribed by V.A. Borovikov

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 1022 KB
• Volume: 26
• Category: Article
• ISSN: 0165-2125
• DOI: 10.1016/s0165-2125(97)00012-7

✦ Synopsis

We study Green's function for the internal gravity waves in a horizontally uniform fluid. The exact representation of Green's function is found for the half-space z > 0 under the assumptions that the square of Briinta-V&still frequency N'(z) = cont. z, and the boundary condition at z = 0 is zero. The large-time asymptotics of this function contain two terms of the form t~'/~A(r, z, ZO) exp i(to(r, z, ~0)). This result suggests that in the general case the large-time asymptotics of Green's function contain analogous terms. The analogs of the eikonal equation for o and the transport equation for A are obtained and solved. This enables us to formulate an algorithm for calculating the asymptotics of Green's function in the general case. Under an additional assumption that N2 has a unique maximum, these asymptotics are justified for the case of internal waves propagating in a waveguide layer, when as 1.~1 + 03, N2(z) + 0. In order to prove this we employ the well-known representation of Green's function as a sum of normal modes as obtained by the method of separation of variables. The appropriate eigenvalues and eigenfunctions are replaced by their WKB-asymptotics, and the Poisson summation formula is used. The resulting integrals are evaluated by applying the stationary phase method.