The Connection between Bubble Size Spectra and Energy Dissipation Rates in the Upper Ocean

Abstract

A formula for the maximum size of a bubble for which surface tension forces can prevent bubble breakup by inertial forces, combined with the observed sizes of air bubbles in breaking waves, implies an energy dissipation rate. One dataset from the surf zone gives a dissipation rate of the order of 0.1 W kg−1, but the large number of small bubbles, and the bubble size spectrum generally, are puzzling. A simple dimensional cascade argument suggests that injected air beneath a breaking wave is rapidly broken up by turbulence, producing an initial size spectrum proportional to (radius)−10/3 before modification by dissolution and rising under buoyancy. This spectral slope is comparable with data from the surf zone. The cascade argument does, however, predict that for a constant dissipation rate there is a rapid accumulation of a large number of bubbles at the scale at which surface tension prevents further breakup; it is possible that the observed size spectrum reflects the range of turbulent energy dissipation rates rather than the result of a cascade. If so, an estimate of about 40 W kg−1 is obtained for the dissipation rate implied by the surf zone dataset. Once an initial size spectrum is formed by the rapid action of differential pressure forces, it will evolve subject to dissolution and buoyancy. It is shown that the former will tend to flatten the size spectrum at small scales, whereas the latter will tend to steepen the time-averaged spectrum observed at large scales. The slope change and transition radius predicted by a very simple model are in reasonable agreement with observations.