Abstract
The simplest geometry of the domain, for which internal wave attractors were for the first time investigated both experimentally and numerically, has the shape of a trapezium with one vertical wall and one inclined lateral wall, characterized by two parameters. Using the symmetries of such a geometry we give an exact solution for the coordinates of the wave attractors with one reflection from each of the lateral boundaries and an integer amount n of reflections from each of the horizontal boundaries. The area of existence for each (n,1) attractor has the form of a triangle in the (d,τ) parameter plane, and the shape of this triangle is explicitly given with the help of inequalities or vertices. The expression for the Lyapunov exponents and their connection to the focusing parameters is given analytically. The corresponding direct numerical simulations with low viscosity fully support the analytical results and demonstrate that in bounded domains (n,1) wave attractors can be effective transformers of the global forcing into traveling waves. The saturation time from the state of rest to the final wave regime depends almost linearly on the number of cells, n.
Highlights
Internal waves are ubiquitous in the oceans and astrophysical objects
Let us consider the most simple and typical configurations of a domain filled with a stratified fluid, suitable for the description of the basic features of internal or inertial waves that are somehow initiated inside the domain
We derived the exact expressions for the calculation of the coordinates of wave attractors with one reflection from a lateral wall and n reflections from a horizontal wall
Summary
Internal waves are ubiquitous in the oceans and astrophysical objects. The importance of taking internal and inertial waves into account is illustrated by their role in supporting the vertical mixing and energy transport [1]. Such a kind of geometrical billiard can not be found under the traditional rule of specular wave reflection from walls where the angle of reflection is equal to the angle of incidence, measured with respect to the normal to the boundary. This addresses the saturation time-scale under which attractors establish themselves after turning on the forcing
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