Lie group analysis of gravity currents

We consider shallow water theory to study the self-similar gravity currents that describe the motion of a heavy fluid flowing into another lighter ambient fluid. Gratton and Vigo investigated the shallow water theory representing the self-similar gravity currents by using dimensional analysis [J. Gratton, C. Vigo, Self-similarity gravity currents with variable inflow revisited: Plane currents, J. Fluid. Mech. 258 (1994) 77–104]. But in this study, the self-similarity solutions of the one-layer shallow-water equations representing gravity currents are investigated by using Lie group analysis and it is shown that Lie group analysis is the generalization of the dimensional analysis for investigating the self-similarity solutions of the one-layer shallow-water equations. Applying Lie group theory, reduced equations of the shallow water equations are found. Therefore, it becomes possible to obtain the similarity forms depending on the Lie group parameters and also the self-similarity solutions for the special values of these group parameters.