The fully nonlinear barotropic vorticity equation is integrated in time to study the development of a Rossby wave critical level. The initial conditions consist of a hyperbolic tangent shear flow and a steady forced wave at the northern boundary; a radiation condition is used at the southern boundary. Linear, quasi-linear and nonlinear integrations are made, and the results are compared with previous studies. For small values of the perturbation amplitude, such that nonlinear interactions are important only in the critical layer, a steady state is obtained in the outer domain, in which the Reynolds stress vanishes both above and below the critical level, and the forced wave is totally reflected as suggested by previous analytical nonlinear steady-state solutions; the approach to that steady state and the structure of the critical layer, however, are quite different from the quasi-linear integrations performed by other authors. Some conclusions are drawn with respect to the forcing of the equatori... Abstract The fully nonlinear barotropic vorticity equation is integrated in time to study the development of a Rossby wave critical level. The initial conditions consist of a hyperbolic tangent shear flow and a steady forced wave at the northern boundary; a radiation condition is used at the southern boundary. Linear, quasi-linear and nonlinear integrations are made, and the results are compared with previous studies. For small values of the perturbation amplitude, such that nonlinear interactions are important only in the critical layer, a steady state is obtained in the outer domain, in which the Reynolds stress vanishes both above and below the critical level, and the forced wave is totally reflected as suggested by previous analytical nonlinear steady-state solutions; the approach to that steady state and the structure of the critical layer, however, are quite different from the quasi-linear integrations performed by other authors. Some conclusions are drawn with respect to the forcing of the equatori...