The effect of a depth-dependent bubble distribution on normal modes in the oceanic waveguide: quasistatic approximation.

We consider the effect of a depth-dependent distribution of bubbles on internal and surface waves propagating horizontally in the oceanic waveguide. 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. 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. Then, for a typical oceanic situation, the correction due to the first effect is shown to be more significant than the second correction. 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, and show that bubble distributions, when present, may considerably change the properties of the oceanic waveguide.