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How strong are the Rossby vortices

The Rossby wave instability, associated with density bumps in differentially rotating discs, may arise in several different astrophysical contexts, such as galactic or protoplanetary discs. While the linear phase of the instability has been well studied, the nonlinear evolution and especially the saturation phase remain poorly understood. In this paper, we test the non-linear saturation mechanism analogous to that derived for wave-particle interaction in plasma physics. To this end we perform global numerical simulations of the evolution of the instability in a two-dimensional disc. We confirm the physical mechanism for the instability saturation and show that the maximum amplitude of vorticity can be estimated as twice the linear growth rate of the instability. We provide an empirical fitting formula for this growth rate for various parameters of the density bump. We also investigate the effects of the azimuthal mode number of the instability and the energy leakage in the spiral density waves. Finally, we show that our results can be extrapolated to 3D discs.