Self-similarity and the evaluation of long-time nonhydrostatic effects in compositionally driven gravity flows in deep surroundings

In the study of compositionally driven gravity currents involving one or more homogeneous fluid layers, it has been customary to adopt the hydrostatic assumption for the pressure field in each layer which, in turn, leads to a depth-independent horizontal velocity field in each of these layers and significant simplifications to the governing equations. Under this hydrostatic paradigm, each layer will then have its motion governed by the well-known reduced dimension shallow-water equations. For the so-called 11/2-layer or reduced gravity shallow-water equations, similarity solutions for fixed volume gravity currents released in rectangular geometry have been found. Very few attempts have been made to evaluate contributions arising from the possible loss of hydrostatic balance in the context of the problems treated using the classic shallow-water approach. Where such attempts have been pursued, they have usually been carried out in a time-independent context or using layer-averaged equations and very small amplitude disturbances. The vast majority of these studies into nonhydrostatic effects do not include any relevant numerical work to assess these effects. In this paper, we develop an approach for evaluating nonhydrostatic contributions to the flow field for bottom gravity currents in deep surroundings and rectangular geometry. Our approach makes no assumptions on the amplitudes of the disturbances and does not depend on layer-averaging in the governing equations. We seek asymptotic expansions of the solutions to the Euler equations for a shallow fluid by using the small parameter δ², where δ is the aspect ratio of the flow regime. At leading order the equations enforce hydrostatic balance while those obtained at first order retain certain nonhydrostatic effects which we evaluate. Our method for evaluation of these first-order contributions employs the self-similar nature of the solution to the leading-order equations in the new first-order equations without any vertical averaging procedures being employed.