Long Internal Waves of Large Amplitude in a Two-Layer Fluid

Abstract

Long internal waves of large amplitude are studied for a two-layer fluid with a rigid upper boundary. Steady periodic waves are calculated numerically assuming that mean velocities for each layer are equal. In the limit of long wavelength, the waves asymptote to either of a non-dissipative internal bore or a solitary wave of elevation or depression. The conditions for the appearance of each solution are obtained numerically, and one of the conditions is complemented by an analysis based on the conservations of mass, momentum and wave energy. Then it is shown that the interface for the solitary wave of large amplitude is not allowed to cross a critical level similarly to the case of small amplitude. Here the critical level is defined by the distance \(\sqrt{\Delta}h/(1+\sqrt{\Delta})\) from the bottom, and h is the interval between two boundaries, Δ=ρ 2 /ρ 1 and ρ 1 and ρ 2 are densities of upper and lower layers. This result gives a criterion for the largest amplitude of a solitary wave.