  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.
