The path taken by dense turbulent outflows usually requires the numerical solution of along-flow equations for mass, tracers, and momentum and cannot easily be predicted. Instead, we consider the consequences of two simple assumptions. First, there is a quadratic turbulent bottom drag. Second, the outflow is assumed to be approximately in local equilibrium so that a best fit formula from atmospheric and ocean surface layer observations plus large-eddy simulations, used by Zilitinkevich and Mironov <1996>, can be used to predict the local thickness. (No energy budget for turbulent bottom layers is known, which is a constant difficulty for numerical models of such layers.) The equilibrium solution is approximately equivalent, for most oceanic conditions, to a constant bulk Richardson or Froude number. It is shown that dense turbulent overflows follow a simple trajectory, in which the rate of depth increase is a constant, until the level of turbulence drops sufficiently that the equilibrium solution becomes invalid. This result is independent of the detailed thermodynamics, entrainment or detrainment, and the quadratic drag coefficient (but does depend on the assumption of quadratic drag). Trajectories for the major overflow regions give reasonable results when compared with the limited available data. An argument is given as to why entrainment should only occur over limited regions, with detrainment elsewhere. ¿ 2001 American Geophysical Union