  In the context of linear stability analysis, considering unsteady base flows
is notoriously difficult. A generalisation of modal linear stability analysis,
allowing for arbitrarily unsteady base flows over a finite time, is therefore
required. The recently developed optimally time-dependent (OTD) modes form a
projection basis for the tangent space. They capture the leading amplification
directions in state space under the constraint that they form an orthonormal
basis at all times. The present numerical study illustrates the possibility to
describe a complex flow case using the leading OTD modes. The flow under
investigation is an unsteady case of the Blasius boundary layer, featuring
streamwise streaks of finite length and relevant to bypass transition. It
corresponds to the state space trajectory initiated by the minimal seed; such a
trajectory is unsteady, free from any spatial symmetry, and shadows the
laminar-turbulent separatrix for a finite time only. The finite-time
instability of this unsteady base flow is investigated using the 8 leading OTD
modes. The analysis includes the computation of finite-time Lyapunov exponents
as well as instantaneous eigenvalues, and of the associated flow structures.
The reconstructed instantaneous eigenmodes are all of outer type. They map
unambiguously the spatial regions of largest instantaneous growth. Other flow
structures, previously reported as secondary, are identified with this method
as relevant to streak switching and to streamwise vortical ejections. The
dynamics inside the tangent space features both modal and non-modal
amplification. Non-normality within the reduced tangent subspace, quantified by
the instantaneous numerical abscissa, emerges only as the unsteadiness of the
base flow is reduced.
