Statistical energy conservation principle for inhomogeneous turbulent dynamical systems

Abstract

Understanding the complexity of anisotropic turbulent processes over a wide range of spatiotemporal scales in engineering shear turbulence as well as climate atmosphere ocean science is a grand challenge of contemporary science with important societal impact. In such inhomogeneous turbulent dynamical systems there is a large dimensional phase space with a large dimension of unstable directions where a large-scale ensemble mean and the turbulent fluctuations exchange energy and strongly influence each other. These complex features strongly impact practical prediction and uncertainty quantification. A systematic energy conservation principle is developed here in a Theorem that precisely accounts for the statistical energy exchange between the mean flow and the related turbulent fluctuations. This statistical energy is a sum of the energy in the mean and the trace of the covariance of the fluctuating turbulence. This result applies to general inhomogeneous turbulent dynamical systems including the above applications. The Theorem involves an assessment of statistical symmetries for the nonlinear interactions and a self-contained treatment is presented below. Corollary 1 and Corollary 2 illustrate the power of the method with general closed differential equalities for the statistical energy in time either exactly or with upper and lower bounds, provided that the negative symmetric dissipation matrix is diagonal in a suitable basis. Implications of the energy principle for low-order closure modeling and automatic estimates for the single point variance are discussed below.mean | fluctuations | interaction statistical symmetries U nderstanding the complexity of anisotropic turbulent processes over a wide range of spatiotemporal scales in engineering shear turbulence as well as climate atmosphere ocean science is a grand challenge of contemporary science (1-4) with important societal impact. In such inhomogeneous turbulent dynamical systems, there is a large dimensional phase space with a large dimension of unstable directions where a large-scale ensemble mean and the turbulent fluctuations exchange energy and strongly influence each other. These complex features strongly impact practical prediction and uncertainty quantification. A systematic energy conservation principle is developed here in a Theorem that precisely accounts for the statistical energy exchange between the mean flow and the related turbulent fluctuations. This statistical energy is a sum of the energy in the mean and the trace of the covariance of the fluctuating turbulence. This result applies to general inhomogeneous turbulent dynamical systems including the above applications. The Theorem involves an assessment of statistical symmetries for the nonlinear interactions and a selfcontained treatment is presented below. Corollary 1 and Corollary 2 illustrate the power of the method with general closed differential equalities for the statistical energy in time either exactly or with upper and lower bounds provided that the negative symmetric dissipation ...

Effective control of complex turbulent dynamical systems through statistical functionals

“…According to general recent theory (10,11), the time rate of change of the statistical energy has a tendency to decay subject to forcing by the product of the current statistical mean, u (t), and the forcing control, F (t). C) Statistical linear response theory to define the control.…”

“…B) Only an estimate of the statistical energy at the initial time of control and not any details of the forcing history are needed to set up the effective statistical control in i. C) Control of statistical energy by i automatically gives control bounds on the mean and variance of the random state at spatial locations (10). For a climate mitigation scenario, the random state could be the mean and variance of the temperature at spatial locations; in general, this control bound is key information and provides important bounds for uncertainty quantification (19)(20)(21).…”

“…The statistical energy principle can be stated as follows. Under suitable general assumptions (10,11), assume that D = −dI , with d > 0; then, the turbulent dynamical system [1] satisfies the closed statistical energy equation for E = 1/2ū ·ū + 1/2trR,…”

“…The goal here is to propose an effective statistical control strategy for complex turbulent dynamical systems based on a recent statistical energy principle (10,11) and statistical linear response theory (12)(13)(14). We illustrate the potential practical efficiency and verify this effective statistical control strategy on the 40D Lorenz 1996 (L-96) model in forcing regimes with various types of fully turbulent dynamics with nearly one-half of the phase space unstable.…”

“…A) Goal and statistical energy. The statistical energy, E , is the sum of the energy of the statistical mean and the trace of the statistical covariance (10,11). A turbulent dynamical system is subjected to poorly known external forcing, and the goal of the statistical control strategy is to find an effective deterministic feedback control to drive the statistical energy measured at some time back to a small neighborhood of a prescribed statistical steady state with energy, E∞, in a finite time with a given cost.…”