A minimization principle for the description of modes associated with finite time instabilities
We introduce a minimization formulation for the determination of a finite dimensional, time dependent, orthonormal basis that captures directions of the phase space associated with transient instabilities. While these instabilities have finite lifetime, they can play a crucial role either by altering the system dynamics through the activation of other instabilities or by creating sudden nonlinear energy transfers that lead to extreme responses. However, their essentially transient character makes their description a particularly challenging task. We develop a minimization framework that focuses on the optimal approximation of the system dynamics in the neighbourhood of the system state. This minimization formulation results in differential equations that evolve a time dependent basis so that it optimally approximates the most unstable directions. We demonstrate the capability of the method for two families of problems: (i) linear systems, including the advection diffusion operator in a strongly non normal regime as well as the Orr Sommerfeld/Squire operator, and (ii) nonlinear problems, including a low dimensional system with transient instabilities and the vertical jet in cross flow. We demonstrate that the time dependent subspace captures the strongly transient non normal energy growth (in the short time regime) while for longer times the modes capture the expected asymptotic behaviour.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences