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We introduce a minimization formulation for the determination of afinite-dimensional, time-dependent, orthonormal basis that captures directionsof the phase space associated with transient instabilities. While theseinstabilities have finite lifetime they can play a crucial role by eitheraltering the system dynamics through the activation of other instabilities, orby creating sudden nonlinear energy transfers that lead to extreme responses.However, their essentially transient character makes their description aparticularly challenging task. We develop a minimization framework that focuseson the optimal approximation of the system dynamics in the neighborhood of thesystem state. This minimization formulation results in differential equationsthat evolve a time-dependent basis so that it optimally approximates the mostunstable directions. We demonstrate the capability of the method for twofamilies of problems: i) linear systems including the advection-diffusionoperator in a strongly non-normal regime as well as the Orr-Sommerfeld/Squireoperator, and ii) nonlinear problems including a low-dimensional system withtransient instabilities and the vertical jet in crossflow. We demonstrate thatthe time-dependent subspace captures the strongly transient non-normal energygrowth (in the short time regime), while for longer times the modes capture theexpected asymptotic behavior.

A minimization principle for the description of time-dependent modes associated with transient instabilities