Two free waves propagating in a parallel shear flow generate a critica
l layer when their nonlinear interaction induces a perturbation whose
phase velocity matches the basic-state velocity somewhere in the flow
domain. The condition necessary for this to occur may be interpreted a
s a resonance condition for a triad formed by the two waves and a (sin
gular) mode of the continuous spectrum associated with the shear. The
formation of the critical layer is investigated in the case of freely
propagating Rossby waves in a two-dimensional inviscid flow in a beta-
channel. A weakly nonlinear analysis based on a normal-mode expansion
in terms of Rossby waves and modes of the continuous spectrum is devel
oped; it leads to a system of amplitude equations describing the evolu
tion of the two Rossby waves and of the modes of the continuous spectr
um excited during the interaction. The assumption of weak nonlinearity
is not however self-consistent: it breaks down because nonlinearity a
lways becomes strong within the critical layer, however small the init
ial amplitudes of the Rossby waves. This demonstrates the relevance of
nonlinear critical layers to monotonic, stable, unforced shear flows
which sustain wave propagation. A nonlinear critical-layer theory is d
eveloped that is analogous to the well-known theory for forced critica
l layers. Differences arise because of the presence of the Rossby wave
s: the vorticity in the critical layer is advected in the cross-stream
direction by the oscillatory velocity field due to the Rossby waves.
An equation is derived which governs the modification of the Rossby wa
ves that results from their interaction; it indicates that the two Ros
sby waves are undisturbed at leading order. An analogue of the Stewart
son-Warn-Warn analytical solution is also considered.