Wave-Current Interaction as a Spatial Dynamical System: Analogies with Rainbow and Black Hole Physics

We study the hydrodynamic phenomenon of waves blocking by a countercurrent with the tools of dynamical systems theory. We show that, for a uniform background velocity and within the small wavelength approximation, the stopping of gravity waves is described by a stationary saddle-node bifurcation due to the spatial resonance of an incident wave with the converted blueshifted wave. We explain why the classical regularization effect of interferences avoids the height singularity in complete analogy with the intensity of light close to the principal arc of a rainbow. The application to the behavior of light near a gravitational horizon is discussed.