The Necessity of Instantaneous Optimals in Stationary Turbulence

Timothy DelSole George Mason University, Fairfax, Virginia, and Center for Ocean–Land–Atmosphere Studies, Calverton, Maryland

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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 similar conservation principle. It is shown that the growing instantaneous optimals imply the existence of growing finite-time singular vectors. Moreover, the average projection on the growing instantaneous optimals must be sufficient to balance the average projection on all other eddies. In contrast to most other types of optimal perturbations, the phase space spanned by the growing instantaneous optimals is independent of the norm used to measure the initial amplitude. This paper also proves that growing instantaneous optimals must exist and play a significant role in nonlinear vacillation phenomena. The argument put forward here follows essentially from statistical equilibrium and conservation of energy, and is independent of any closure theory of turbulence.

Corresponding author address: Timothy DelSole, Center for Ocean–Land–Atmosphere Studies, 4041 Powder Mill Rd., Suite 302, Calverton, MD 20705-3106. Email: delsole@cola.iges.org

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 similar conservation principle. It is shown that the growing instantaneous optimals imply the existence of growing finite-time singular vectors. Moreover, the average projection on the growing instantaneous optimals must be sufficient to balance the average projection on all other eddies. In contrast to most other types of optimal perturbations, the phase space spanned by the growing instantaneous optimals is independent of the norm used to measure the initial amplitude. This paper also proves that growing instantaneous optimals must exist and play a significant role in nonlinear vacillation phenomena. The argument put forward here follows essentially from statistical equilibrium and conservation of energy, and is independent of any closure theory of turbulence.

Corresponding author address: Timothy DelSole, Center for Ocean–Land–Atmosphere Studies, 4041 Powder Mill Rd., Suite 302, Calverton, MD 20705-3106. Email: delsole@cola.iges.org

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