A hydrodynamical theory of conservative bounded density currents

The Benjamin (1968) analysis of a two-fluid density current is extended to include the effect of vorticity within the current. In the case of constant vorticity, the density-current depth is shown to lie between the limits of half and two-thirds of the channel depth. More general vorticity distributions are also considered, namely those that have: (i) a maximum in the upper and lower regions of the density current; and (ii) a maximum in the middle of the density current. In the former, as in the case of constant vorticity, density-current structures exist, whereas in the latter, deep overturning circulations predominate which can cause a ‘blocking’ of the upstream inflow. A generalized propagation formula which includes the effects of finite depth and rear inflow into the density current is established and the uniqueness issue is considered.

The analysis is further extended to a three-fluid system, composed of physically distinct component flows, namely, a density current, an overturning updraught region in upper levels ahead of the density current and an updraught in which the fluid ascends without overturning to its outflow level. Two types of behaviour are identified. First, a symmetric mode in which the density current and the overturning updraught have the same depth and, second, an asymmetric mode with solutions restricted to a certain parameter range. A special case in which the fluids have the same density illustrates the basic dynamics of the problem and also the nature of the vertical transport of momentum.