Long Internal Waves in a Two-Layer Fluid

Abstract

Steady long periodic internal waves of small amplitude in a two-layer fluid with a rigid upper boundary are examined under the assumption of no difference between mean horizontal velocities in each layer. When the depth ratio in quiescent state is close to a certain value σ_c determined by density ratio, the Korteweg-de Vries equation does not describe well such waves. For this case an equation including both quadratic and cubic nonlinear terms as well as dispersive term is derived. By taking the long-wave limit of periodic solutions to this equation, a shock like solution as well as a solitary wave solution of elevation or depression is obtained. One of the depth ratios at two flat regions for the shock like solution is larger than σ_c, and the other is smaller than σ_c. The velocity of this solution is independent of its amplitude. The amplitude of the solitary wave solution has a certain upper limit.