We devise a stochastic model for the effects of breaking waves and fit its distribution functions to laboratory and field data. This is used to represent the space–time structure of momentum and energy forcing of the oceanic boundary layer in turbulence-resolving simulations. The aptness of this breaker model is evaluated in a direct numerical simulation (DNS) of an otherwise quiescent fluid driven by an isolated breaking wave, and the results are in good agreement with laboratory measurements. The breaker model faithfully reproduces the bulk features of a breaking event: the mean kinetic energy decays at a rate approaching $t^{-1}$, and a long-lived vortex (eddy) is generated close to the water surface. The long lifetime of this vortex (more than 50 wave periods) makes it effective in energizing the surface region of oceanic boundary layers. Next, a comparison of several different DNS of idealized oceanic boundary layers driven by different surface forcing (i.e. constant current (as in Couette flow), constant stress, or a mixture of constant stress plus stochastic breakers) elucidates the importance of intermittent stress transmission to the underlying currents. A small amount of active breaking, about 1.6% of the total water surface area at any instant in time, significantly alters the instantaneous flow patterns as well as the ensemble statistics. Near the water surface a vigorous downwelling–upwelling pattern develops at the head and tail of each three-dimensional breaker. This enhances the vertical velocity variance and generates both negative- and positive-signed vertical momentum flux. Analysis of the mean velocity and scalar profiles shows that breaking effectively increases the surface roughness $z_o$ by more than a factor of 30; for our simulations $z_o/\lambda \,{\approx}\, 0.04$ to 0.06, where $\lambda$ is the wavelength of the breaking wave. Compared to a flow driven by a constant current, the extra mixing from breakers increases the mean eddy viscosity by more than a factor of 10 near the water surface. Breaking waves alter the usual balance of production and dissipation in the turbulent kinetic energy (TKE) budget; turbulent and pressure transports and breaker work are important sources and sinks in the budget. We also show that turbulent boundary layers driven by constant current and constant stress (i.e. with no breaking) differ in fundamental ways. The additional freedom provided by a constant-stress boundary condition permits finite velocity variances at the water surface, so that flows driven by constant stress mimic flows with weakly and statistically homogeneous breaking waves.