The approach to self-similarity of the solutions of the shallow-water equations representing gravity-current releases

Abstract

Known similarity solutions of the shallow-water equations representing the motion of constant-volume gravity currents are studied in both plane and axisymmetric geometries. It is found that these solutions are linearly stable to small correspondingly symmetric perturbations and that they constitute the large-time limits of the solutions of the initial-value problem. Furthermore, the analysis reveals that the similarity solution is approached in an oscillatory manner. Two initial-value problems are solved numerically using finite differences and in each case the approach to the similarity solution is compared with the analytic predictions.