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The theory of transient growth describes how linear mechanisms can causetemporary amplification of disturbances even when the linearized system isasymptotically stable as defined by its eigenvalues. This growth istraditionally quantified by finding the initial disturbance that generates themaximum response, in terms of energy gain, at the peak time of its evolution.While this bounds the growth, it can vastly overstate the growth of a realdisturbance. In this paper, we introduce a statistical perspective on transientgrowth that models statistics of the energy amplification of the disturbances.We derive a formula for the mean energy amplification in terms of the two-pointspatial correlation of the initial disturbance. We also derive an accurateapproximation of the probability density function of the energy of the growingdisturbance, from which confidence bounds on the growth can be obtained.Applying our analysis to Poisseuille flow yields a number of observations.First, the mean gain can be drastically smaller than the maximum, especiallywhen the disturbances are broadband in wavenumber content. In these cases, itis exceedingly unlikely to achieve near-optimal growth due to the exponentialbehavior which we observe in the probability density function. Second, thecharacteristic length scale of the initial disturbances has a significantimpact on the expected growth; specifically, large-scale initial disturbancesproduce orders-of-magnitude-larger expected growth than smaller scales,indicating that the length scale of incoming disturbances may be key indetermining whether transient growth leads to transition for a particular flow.Finally, while the optimal growth scales quadratically with Reynolds number, weobserve that the mean energy amplification scales only linearly for certainreasonable choices of the initial correlations.

Beyond optimal disturbances: a statistical framework for transient growth