The Necessity of Optimal Perturbations

Abstract

An optimal perturbation is an initial condition that optimizes amplitude growth over a prescribed time in a linear system. Previous studies have argued that optimal perturbations play an important role in turbulence. Two basic questions related to this theory are whether optimal perturbations necessarily grow in all turbulent background flows, and whether the turbulent flow necessarily excites optimal perturbations at the rate required to account for the observed eddy variance. This paper shows that both questions can be answered in the affirmative for statistically steady turbulence. The argument put forward here is independent of any closure theory of turbulence. In essence, the result follows from the fact that statistical equilibrium and conservation of energy require the presence of optimal perturbations (with respect to the energy norm).