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In the context of linear stability analysis, considering unsteady base flowsis notoriously difficult. A generalisation of modal linear stability analysis,allowing for arbitrarily unsteady base flows over a finite time, is thereforerequired. The recently developed optimally time-dependent (OTD) modes form aprojection basis for the tangent space. They capture the leading amplificationdirections in state space under the constraint that they form an orthonormalbasis at all times. The present numerical study illustrates the possibility todescribe a complex flow case using the leading OTD modes. The flow underinvestigation is an unsteady case of the Blasius boundary layer, featuringstreamwise streaks of finite length and relevant to bypass transition. Itcorresponds to the state space trajectory initiated by the minimal seed; such atrajectory is unsteady, free from any spatial symmetry, and shadows thelaminar-turbulent separatrix for a finite time only. The finite-timeinstability of this unsteady base flow is investigated using the 8 leading OTDmodes. The analysis includes the computation of finite-time Lyapunov exponentsas well as instantaneous eigenvalues, and of the associated flow structures.The reconstructed instantaneous eigenmodes are all of outer type. They mapunambiguously the spatial regions of largest instantaneous growth. Other flowstructures, previously reported as secondary, are identified with this methodas relevant to streak switching and to streamwise vortical ejections. Thedynamics inside the tangent space features both modal and non-modalamplification. Non-normality within the reduced tangent subspace, quantified bythe instantaneous numerical abscissa, emerges only as the unsteadiness of thebase flow is reduced.

Instability of the optimal edge trajectory in the Blasius boundary layer