A mathematical framework for the analysis of particle-driven gravity currents

Recent studies have modelled the flow of particle-driven gravity currents o ver horizontal boundaries using either shallow-water equations or simple 'b ox' (integral) models. The shallow-water equations are typically integrated numerically, whereas box models admit analytical solutions. However, the t heoretical validity of the latter models has not been fully established. In this paper a novel mathematical technique is developed which permits the d erivation of analytical solutions to the shallow-water model for gravity cu rrent motion. These solutions, confirmed by comparison with the results of numerical integration, are in good agreement with experimental observations . They also indicate why the simplified box models have been so successful. Moreover, they reveal how the internal dynamics of particle-driven flows a re different from gravity currents arising solely due to compositional dens ity differences. While compositionally driven gravity currents, which have a fixed density difference between the intruding and ambient fluids, may be modelled using similarity solutions to the governing equations, particle-d riven gravity currents do not possess such solutions because their density is progressively reduced by particle sedimentation. Instead the new analysi s determines ho ct their behaviour progressively diverges from the similari ty solution. By a change of independent variables, it is possible to develo p convergent series expansions for each of the dependent variables which ch aracterize the motion. It is suggested that this approach may find applicat ion to a number of other problems in which the dynamics are initially gover ned by a simple dynamical balance which is progressively lost as extra phys ical effects begin to influence the motion.