Non-linear dynamics induced by linear wave interactions in multi-layered flows

Using simple kinematics, we propose a general theory of linear wave interactions between the interfacial waves of a two dimensional (2D), inviscid, multi-layered fluid system. The strength of our formalism is that one does not have to specify the physics of the waves in advance. Wave interactions may lead to instabilities, which may or may not be of the familiar "normal-mode" type. Contrary to intuition, the underlying dynamical system describing linear wave interactions is found to be non-linear. Specifically, a saw-tooth jet profile with three interfaces possessing kinematic and geometric symmetry is explored. Fixed points of the system for different ranges of a Froude number like control parameter $\gamma$ are derived, and their stability evaluated. Depending upon the initial condition and $\gamma$, the dynamical system may reveal transient growth, weakly positive Lyapunov exponents, as well as different non-linear phenomena such as formation of periodic and pseudo-periodic orbits. All these occur for those ranges of $\gamma$ where normal-mode theory predicts neutral stability. Such rich non-linear phenomena is not observed in 2D dynamical system resulting from the $2$-wave problem, which only reveals stable and unstable nodes.