NON-LINEAR OPTIMAL PERTURBATIONS IN SUBCRITICAL INSTABILITIES

摘要

Non-linear optimal perturbations are defined here as those of minimum energy leading to subcritical instability. We show that a necessary condition for an ini?tial perturbation to be a non-linear optimal is that the initial perturbation energy growth is zero. The fulfillment of this condition does not depend on the disturb?ance amplitude but only on the linearized operator as long as the non-linearity conserves energy. Saddle point solutions and linear optimal perturbations lead?ing to maximum transient growth both satisfy the non-linear optimality condi?tion. We discuss these issues on low-dimensional models of subcritical transition for which non-linear optimals and the minimum threshold energy are computed.