Turbulent Breakup Without Turbulence

Plunging breaking waves entrain air pockets below the ocean surface, which fragment in the turbulent agitation of the water. The size distribution of the resulting bubbles has been interpreted along different lines, notably one initiated by Garrett & al. (2000), invoking a Kolmogorov type of turbulence below the wave. That interpretation predicts, for bubbles larger than a millimeter (the so-called Hinze length-scale), a power law distribution of the sizes d as n(d)~d^-β, with β=10/3 (there is another exponent for smaller bubbles). The turbulent cascade picture, and notably its stationary character is however highly questionable in wave breaking, admittedly the paradigm of a transient phenomenon, as well as the relevance of a reasoning relying on the unicity of its stirring intensity, known to be broadly distributed over orders of magnitude.

We invoke here a `maximal randomness' principle (the analogue of Boltzmann's Stosszahlansatz in the kinetic theory of gases) to describe the distribution of the fragments sizes which, when applied to the appropriate function and coupled with an original conservation law provides new, unifying results, including for ocean bubbles.

This principle, consisting in maximizing an entropy has an origin essentially different from the cascade scenario, assumes randomness, but not necessarily turbulence and predicts that β=3.5, a value not incompatible with recent precise data.