Session Ci - Vortex Dynamics.
ORAL session, Sunday, November 23
309, Moscone Center
Weakly nonlinear, nonaxisymmetric wave propagation on concentrated vortex cores is considered under the assumption of inviscid, incompressible flow. A formulation of the type first used by Leibovich et al.~is used, in which the linear-theory perturbation is modulated by a slowly varying complex amplitude function having finite amplitude. The only place where the full details of the above authors' analysis are presented is an internal memorandum by Ma~(H.-Y. Ma, Rept.~FDA-82-10, Sibley School of Engineering, Cornell U.) which contains several inconsistencies, the major one being the absence of a quadratic amplitude term at second order. This type of term is present in the later papers of Leibovich, Yang, Brown and Patel, but in a different form than used by us. The solvability condition at third order yields a cubically nonlinear Schrödinger equation (NLS) in one (slow) space dimension for the envelope function, which means that solitary waves are possible for certain wavenumber ``windows.'' We examine these in detail for the families of linear disturbances which exist on the well-known Q-vortex model flow. Most interestingly, we find a set of singularities in the coefficient of the nonlinear term in NLS, which occur at discrete values of the axial wavenumber. These may serve physically as a wavenumber selection mechanism in nonlinear phenomena such as those leading to vortex breakdown.