Wave attractors in a smooth convex enclosed geometry

The equations governing monochromatic internal or inertial waves in an enclosed two-dimensional basin lead in two dimensions to a hyperbolic equation in spatial dimensions. The wave rays have a fixed slope with respect to gravity or the rotation axis, depending on the frequency of the wave as compared to the strength of the stratification and the rotation rate of the fluid. This angle is conserved on the wave’s reflection at the boundary. The slope of the rays is denoted by κ. Depending on κ and the geometry, either all wave rays are periodic (standing wave), no characteristic is periodic, or there is a limited number of periodic orbits to which wave rays are ‘attracted’. In this study, the boundary is formed by the convex part of a third degree curve. Depending on a parameter ε, this curve varies between a circle (ε=0) and a triangle (ε=2), for ε<2 it is completely smooth. The axis of symmetry (z-axis) is parallel to the direction of gravity/rotation axis, which are taken antiparallel. For the triangle and the circle the behaviour is well known: the corners of the triangle can attract wave rays, for the circle either all wave rays are periodic, or no wave ray is periodic, so that attractors do not exist. In the (κ,ε)-parameter space, investigation of the strength of convergence of characteristics yields Arnol’d tongues, stemming from ε=0, broadening for increasing ε and finally all converging to κ=3 for ε=2. Tongues with attractors are bounded by values of κ for which wave rays either connect the top and bottom of the boundary or connect its critical points, where the wave ray is directly reflected back onto itself. As compared to nonsmooth geometries, corners are a degenerate form of critical points. Only for κ=3 all wave rays return back onto themselves for all values of ε due to an additional symmetry, resulting in standing wave behaviour. If the symmetry of the curve with respect to the z-axis is removed by rotating it, the ordering of the periods of successive attractors changes and there is no standing wave mode. A general criterion, based on first order perturbation theory, is derived that states whether attractors exist for geometries that are small perturbations of the circle. For the geometry under consideration, first order perturbation theory is inconclusive and second order perturbation theory is used to verify the existence of the strongest attractor and to describe the Arnol’d tongue for small values of ε.