Spectral simulation is used to test the perturbation expansion of Davey [J. Fluid Mech.14, 336 (1962)] applied to two‐dimensional, streamwise‐oriented vortices in curved channel flow and rotating channel flow over a wide range of parameters. The power series dependence on Reynolds number implied by the perturbation expansion provides an excellent description of the amplitudes of all ten independent Fourier modes resolved in the numerical simulations. Weakly nonlinear analysis, associated with truncating the perturbation expansion, is developed for rotating channel vortices and compared to the full simulations. For both channel geometries, significant increases in the accuracy of weakly nonlinear theory occurs with increasing vortex wavenumber; this is explained by examining spatial energy spectra.