A class of time-dependent, irrotational flows in which the velocity potential has the form 1/2Az2 (where z is a complex coordinate and A is a complex function of the time) are of special interest in relation to the flow in the jet of a plunging breaker. In this paper we calculate the trajectories of marked particles, and make use of the result to construct a semi-Lagrangian representation of the flow. Special attention is paid to the limit when the free-surface profile tends to an infinitely thin hyperbola, and to another special case (the 4“5°-rotor”) in which the axes rotate with uniform angular velocity, and the angle between them is constant at 45°.