We study the dynamics of turbulent jets of bubbles created by the injection of a slug flow into a quiescent cavity, both theoretically and experimentally using data from drop tower experiments. The generated bubble jets exhibit a remarkably low degree of coalescence leading to virtually monodisperse suspensions of spherical bubbles. The turbulence of the jet is responsible for the spatial spreading of the bubble distribution in roughly conical jets. We propose a stochastic model for the dispersion of bubbles in the jet, in which the trajectories of individual bubbles can be computed from the stationary solution of a k-epsilon model. Bubble concentration behaves to a good approximation as a passive scalar with a non-homogeneous diffusion. Numerical integration of the model compares well with the experimental data.