A minimization principle for the description of time-dependent modes associated with transient 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 by either 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 neighborhood 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 crossflow. 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 behavior.