The viscous Green’s function for internal wave generation and scattering*

Motivated by the goal of understanding the physics of scattering and generation of internal waves (IWs) in a stratified fluid, we look at Boundary Integral Equation approaches that can be used with arbitrary geometries. These require an appropriate Green's function. The inviscid Green's function is well known, for example using the Hurley analytical continuation procedure. but the viscous case is less well understood. We examine its properties asymptotically and numerically, focussing on the case of small viscosity (appropriately nondimensionalized).

We employ Fourier transforms to find the monochromatic Green’s function. The difficulty lies in the inverse Fourier transform. As the viscous dispersion relation for internal waves is biquadratic in wavenumber,  close attention to the appropriate branch in the complex plane is necessary.  For small viscosity, the resulting integrals become highly oscillatory, and we investigate the use of stationary phase integrals and appropriate quadrature methods. A previous method developed to understand anisotropic Brinkman flow is also examined, as these two problems are closely related.

Finally, we use these results to examine IW generation from bodies with zero thickness.