A statistical perspective on transient growth

The theory of transient growth investigates how linear mechanisms can cause temporary amplification of disturbances even when the linearized system is asymptotically stable as defined by its eigenvalues. This growth is traditionally quantified by finding the initial disturbance that generates the maximum response, in terms of energy gain, at the peak time of its evolution. In this presentation, we introduce a novel statistical perspective on transient growth which seeks statistics of the energy gain in terms of those of the initial disturbance. We derive a formula for the expected energy and two-point spatial correlation of the growing disturbance as a function of the two-point spatial correlation of the initial disturbance. We apply our analysis to Poisseuille and Couette flow and show that the characteristic length scale of the initial disturbances, encapsulated by the spatial decay within the correlation function, has a significant impact on the expected growth. Specifically, large-scale initial disturbances produce orders-of-magnitude-larger expected growth than smaller scales, indicating that the length scale of incoming disturbances may be key in determining whether transient growth leads to transition for a particular flow. Additionally, we consider the probability of observing various levels of growth given a distribution of initial conditions.