In a previous paper it was shown that in steep standing waves on water the collapse of a rounded cavity in the wave trough can produce quite high local vertical accelerations, initiating the growth of a strong vertical jet. In one example given here, the acceleration exceeds 100g. The main purpose of the present paper is to follow the subsequent development of the jet by means of a boundary-integral time-stepping technique. It is found that the jets tend to bifurcate into two halves, each forming a plunging or spilling breaker. The transition of the rising wave trough into a thin jet is here compared with an asymptotic flow in which the free surface is given by a quartic equation.