Internal travelling waves in the limit of a discontinuously stratified fluid

We consider internal travelling waves in a perfect stratified fluid, in the singular limit case when smooth stratifications approach a discontinuous t wo-layer profile. Our analysis concerns two-dimensional waves of small ampl itude, propagating in an infinite horizontal strip of finite depth. The pro blems with smooth or discontinuous stratification are formulated as a unify ing spatial evolution problem, where the stratification rho plays the role of a functional parameter. The vector field is not smooth with respect to r ho, but has some weak continuity. When the Froude number is close to a crit ical value, we reduce the problem to one on a center manifold in a neighbor hood of the trivial state independent of rho (for the usual topology). Cons idering a weaker topology, we prove the continuity in rho of the center man ifold. Then the small solutions are described by an ordinary differential e quation in R-2, which depends continuously on rho in the C-k norm.