Shallow waves in a two-layer vortex fluid under a lid

A mathematical model of the vortex motion of an ideal two-layer fluid in a narrow straight channel is considered. The fluid motion in the Eulerian–Lagrangian coordinate system is described by quasilinear integrodifferential equations. Transformations of a set of the equations of motion which make it possible to apply the general method of studying integrodifferential equations of shallow-water theory, which is based on the generalization of the concepts of characteristics and the hyperbolicity for systems with operator functionals, are found. A characteristic equation is derived and analyzed. The necessary hyperbolicity conditions for a set of equations of motion of flows with a monotone-in-depth velocity profile are formulated. It is shown that the problem of sufficient hyperbolicity conditions is equivalent to the solution of a certain singular integral equation. In addition, the case of a strong jump in density (a heavy fluid in the lower layer and a quite lightweight fluid in the upper layer) is considered. A modeling that results in simplification of the system of equations of motion with its physical meaning preserved is carried out. For this system, the necessary and sufficient hyperbolicity conditions are given.