Energy balances and front speed conditions of two-layer models for gravity currents produced by lock release

The exchanges of energy for an inviscid gravity current, which is released from a lock and then propagates over a horizontal boundary, are considered. The investigation uses the two-layer shallow-water formulation and tests four different closure Fr (Froude number) conditions for the speed at the front. The energies in the current and in the ambient are calculated from accurate numerical solutions of the initial-value problem. Fundamental corresponding analytical control-volume energy balances are presented and the consequences are discussed. Both the numerical and analytical results indicate that, in general, the increase of kinetic energy of the shallow-water system cannot fully recover the decay of potential energy during the propagation. It is shown that this imbalance of energy in the present lock-release time-dependent problem is fully equivalent with the "dissipation" predicted by Benjamin's (J Fluid Mech 31: 209-248, 1968) classical analysis of the steady-state current. The connection between the nose Fr conditions and this dissipation are elucidated. In particular, it is shown that the use of "energy-conserving" (i.e. zero dissipation) boundary conditions in the shallow-water predictions would produce, in general, unrealistic currents. The implications on reliable modeling and the physical interpretations of the results are discussed.