The Necessity of Instantaneous Optimals in Stationary Turbulence

Abstract

Abstract An optimal perturbation is an initial condition that optimizes some measure of 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. More precisely, it is shown that eddies in statistically stationary turbulence must project onto a class of amplifying perturbations called instantaneous optimals, which are defined as initial conditions that optimize the rate of change of energy associated with the dynamical system linearized about the time-mean flow. An analogous conclusion holds for potential enstrophy when the latter satisfies a s...