  We present experiments on large air cavities spanning a wide range of sizes
relative to the Hinze scale $d_\mathrm{H}$, the scale at which turbulent
stresses are balanced by surface tension, disintegrating in turbulence. For
cavities with initial sizes $d_0$ much larger than $d_\mathrm{H}$ (probing up
to $d_0 / d_\mathrm{H} = 8.3$), the size distribution of bubbles smaller than
$d_\mathrm{H}$ follows $N(d) \propto d^{-3/2}$, with $d$ the bubble diameter.
The capillary instability of ligaments involved in the deformation of the large
bubbles is shown visually to be responsible for the creation of the small ones.
Turning to dynamical, three-dimensional measurements of individual break-up
events, we describe the break-up child size distribution and the number of
child bubbles formed as a function of $d_0 / d_\mathrm{H}$. Then, to model the
evolution of a population of bubbles produced by turbulent bubble break-up, we
propose a population balance framework in which break-up involves two physical
processes: an inertial deformation to the parent bubble that sets the size of
large child bubbles, and a capillary instability that sets the size of small
child bubbles. A Monte Carlo approach is used to construct the child size
distribution, with simulated stochastic break-ups constrained by our
experimental measurements and the understanding of the role of capillarity in
small bubble production. This approach reproduces the experimental time
evolution of the bubble size distribution during the disintegration of large
air cavities in turbulence.
