Parametric excitation of buoyancy oscillation and formation of internal boundary layers in a stably stratified fluid

Abstract

Resonant excitation of buoyancy oscillation by strained vertical vortices in a stably stratified fluid, which leads to formation of new internal boundary layers, is investigated theoretically. Any two-dimensional steady vorticity distributions, uniform in the vertical direction, are allowable under the Boussinesq approximation, i.e., at the zeroth order of Froude number (Fr) expansions. But, such an inviscid solution shows algebraic divergence at the first order. This anomaly is attributable to the resonance between the vertical oscillation due to a gravitational restoring force and the rotational motion of a fluid particle along a closed streamline. Weak viscous and diffusive effects are incorporated in order to remove the singularity and to obtain stationary solutions. Internal boundary layers of thickness proportional to Re−1/3(1+Pr−1)1/3 are formed, inside of which vertical velocity variations of magnitude N2 Fr Re1/3(1+Pr−1)−1/3 are observed. Here, the parameters Re, Pr, and N2 denote the Reynolds number, Prandtl number, and the normalized Brunt–Väisälä frequency, respectively.