ROSSBY-WAVE INTERACTION IN A SHEAR-FLOW WITH CRITICAL LEVELS

An approach is presented for studying Rossby wave interaction in a she ar flow with both regular and singular modes (i.e. those possessing a critical level). The approach relies on a truncated normal mode expans ion of the equations of motion. Such an expansion remains valid in the presence of singular modes, provided that these modes are not conside red individually, but that complete packets are taken into account in the truncated system. Mathematically, this means that the interaction equations need to be integrated with respect to the phase velocity (or , equivalently, the critical level position) of the singular modes. Th e action of two regular modes on a packet of singular modes is treated in detail; in particular, asymptotic results are deduced for the long -term behaviour of the packet. The case of a linear shear is considere d as an illustration: analytical expressions are derived for the norma l modes and their pseudomomentum, and they are used to present explici t results for the evolution of the packet of singular modes.