Wave self-focusing into an incompressible inviscid rotating fluid for the cylinder geometry

Inertial waves are driven by the Coriolis force in rotating fluids, they are valuable to study since rotating fluids occur very often in geophysics and astrophysics. The linear behavior of these waves has been extensively studied in various situations (in rotating and stratified media etc. . . ) but these waves can exhibit also nonlinear properties like electromagnetic waves in traditional non-linear optics.We are interested, here, by inertial wave-wave interaction into an incompressible inviscid rotating fluid for the cylinder geometry. Some previous studies have been devoted to possible focusing of inertial waves but mainly in the spirit of focusing toward "attractors"for those waves but not for their focusing driven by non linear processes. We analyze couplings of inertial waves and in particular their self interaction that can induce self-focusing in suitable conditions.To examine the possibility of wave focusing we shall mainly deal with a Ginzburg-Landau equation which is obtained by reduction of the Navier–Stokes equation using an asymptotic analysis.

We compute the wave coupling terms at different orders in wave amplitude, relying on an approximated associated Beltrami property. At third order it is shown that wave self-coupling can lead to wave self-focusing.