The effect of a depth-dependent bubble distribution on the normal modes of internal waves: quasistatic approximation

Abstract

We consider internal and surface waves propagating horizontally in the fluid waveguide, formed by an underlying density stratification. As well as the usual basic density stratification, we are concerned with the effect of a nearly stationary depth-dependent distribution of bubbles. While our previous work was restricted to the case of a locally monodisperse mixture, in this paper we show that by using a quasistatic approximation (where attention is confined to those modes whose typical frequencies are much less than the natural frequency for bubble oscillations), we can extend that work to the case of more general discrete and continuous bubble distributions. The equations of motion are formulated in terms of the usual fluid variables and the void fraction of bubbles. Then, to leading order in the Boussinesq approximation, we obtain the usual equation for internal wave modes, but the value of the buoyancy frequency in the fluid is replaced by an effective buoyancy frequency which takes account of the bubble distribution. Indeed, in some circumstances the bubble distribution could be entirely responsible for the density stratification, and hence can give rise to its own internal wave field. Two physical factors are shown to affect the buoyancy frequency in the mixture: the effective stratification due to the bubbles adds to the effect of stratification in the liquid, while the compressibility of the mixture due to the bubbles reduces the buoyancy frequency. Having in mind possible applications of our model to oceanic situations and in accordance with existing observational evidence that the void fraction profile in the ocean decays exponentially with depth, we obtain an explicit description of the normal modes for several simple models of the basic density profile and show that bubble distributions, when present, may considerably change the properties of the internal waveguide.