On the fundamental solution for viscous internal waves and Brinkman flows. Part 1. Two dimensions

TL;DR

This paper derives explicit fundamental solutions for viscous internal waves and Brinkman flows, analyzing their asymptotic behavior, attenuation, and flow structure, with implications for numerical methods and flow confinement.

Contribution

It provides the first rigorous uniform asymptotic expansion for wave-fields at Prandtl numbers greater than or around one, including efficient computation methods.

Findings

Density diffusion attenuates wave amplitude as (1+Pr^{-1})^{-2/3}.

Wave beam width broadens according to (1+Pr^{-1})^{1/3}.

Flow becomes confined to the direction of least resistance with increasing anisotropy.

Abstract

We obtain the viscous and diffusive fundamental solution for monochromatic internal waves in a uniformly stratified medium and for anisotropic Brinkman flow. These solutions take the form of single integrals with logarithmic singularities, and can be computed numerically in an efficient manner for possible use in boundary integral methods. Far-field asymptotic results are obtained, giving solutions valid far from and inside a ``beam'' corresponding to the internal wave angle in the internal wave case, consistent with Thomas & Stevenson (1972). For Prandtl numbers $\text{Pr} \gtrsim O(1)$, the wave field is given by a superposition of wave- and Stokeslet-like terms. Unlike previous studies, a uniform asymptotic expansion of the wave-field for $\text{Pr} \gtrsim O(1)$ can be computed rigorously. Density diffusion attenuates the wave amplitude as to $(1+\text{Pr}^{-1})^{-2/3}$ and broadens the beam width according to $(1+\text{Pr}^{-1})^{1/3}$. Evanescent waves in a stratified medium and anisotropic Brinkman flows have similar behaviour. Anisotropic Brinkman flow is purely real, dominated by a single circulation cell. As anisotropy increases, the flow becomes increasingly confined to the direction with least resistance. The stratified evanescent wave field has near-vertical cells in its real part, and a dominant single circulation cell in its imaginary part.