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

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 Brünta-Väissälä frequency N 2( 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 − 1 2 A(r, z, z 0) exp i(tω(r, z, 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 ω 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 N 2 has a unique maximum, these asymptotics are justified for the case of internal waves propagating in a waveguide layer, when as ¦z¦ → ∞, N 2(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.