Statistically accurate low-order models for uncertainty quantification in turbulent dynamical systems

Abstract

A framework for low-order predictive statistical modeling and uncertainty quantification in turbulent dynamical systems is developed here. These reduced-order, modified quasilinear Gaussian (ROMQG) algorithms apply to turbulent dynamical systems in which there is significant linear instability or linear nonnormal dynamics in the unperturbed system and energy-conserving nonlinear interactions that transfer energy from the unstable modes to the stable modes where dissipation occurs, resulting in a statistical steady state; such turbulent dynamical systems are ubiquitous in geophysical and engineering turbulence. The ROMQG method involves constructing a low-order, nonlinear, dynamical system for the mean and covariance statistics in the reduced subspace that has the unperturbed statistics as a stable fixed point and optimally incorporates the indirect effect of non-Gaussian third-order statistics for the unperturbed system in a systematic calibration stage. This calibration procedure is achieved through information involving only the mean and covariance statistics for the unperturbed equilibrium. The performance of the ROMQG algorithm is assessed on two stringent test cases: the 40-mode Lorenz 96 model mimicking midlatitude atmospheric turbulence and two-layer baroclinic models for high-latitude ocean turbulence with over 125,000 degrees of freedom. In the Lorenz 96 model, the ROMQG algorithm with just a single mode captures the transient response to random or deterministic forcing. For the baroclinic ocean turbulence models, the inexpensive ROMQG algorithm with 252 modes, less than 0.2% of the total, captures the nonlinear response of the energy, the heat flux, and even the one-dimensional energy and heat flux spectra.dynamical systems with many instabilities | nonlinear response and sensitivity | reduced-order modified quasilinear Gaussian closure T urbulent dynamical systems are characterized by both a large dimensional phase space and a large dimension of instabilities (i.e., a large number of positive Lyapunov exponents on the attractor). They are ubiquitous in many complex systems with fluid flow such as, for example, the atmosphere, ocean, and coupled climate system, confined plasmas, and engineering turbulence at high Reynolds numbers. In these, linear instabilities are mitigated by energy-conserving nonlinear interactions that transfer energy to the linearly stable modes where it is dissipated, resulting in a statistical steady state. Uncertainty quantification (UQ) in turbulent dynamical systems is a grand challenge where the goal is to obtain statistical estimates such as the change in mean and variance for key physical quantities in the nonlinear response to changes in external forcing parameters or uncertain initial data. These key physical quantities are often characterized by the degrees of freedom that carry the largest energy or variance, and an even more ambitious grand challenge is to develop truncated low-order models for UQ for a reduced set of important variables with the largest variance. Thi...

Statistical energy conservation principle for inhomogeneous turbulent dynamical systems

“…This strategy for imperfect models using such an energy principle has recently been tested on the Lorenz 96 (L-96) system (19) for a family of low-order closure models (5) where Corollary 1 above is derived in a different explicit fashion for these models for homogeneous statistics. Other examples of using low-order closure for UQ include the modified quasilinear Gaussian closure (6,7) and applications include the L-96 model and linearly unstable two-layer baroclinic turbulence, which is another complex turbulent dynamical system satisfying all of the assumptions of the Theorem. Other complex turbulent dynamical systems where the Theorem should prove useful for UQ include the following turbulent dynamical systems, which satisfy the assumptions of the Theorem:…”

“…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 (5)(6)(7).…”

Effective control of complex turbulent dynamical systems through statistical functionals

“…The simplest prototype example of a turbulent dynamical system to illustrate and verify the statistical control strategy is because of Lorenz and called the L-96 model (25). It is widely used as a test model for algorithms for prediction, filtering, and low-frequency climate APPLIED MATHEMATICS response (13) as well as algorithms for uncertainty quantification (19,26). The L-96 model is a discrete periodic model given by the following system:…”

“…As illustrated, adding even small random forcing in the system can greatly increase the variability in the zero mode and thus, vastly change the entire energy spectrum to a more active state. The dynamical equation for the statistical energy in [26] in this homogeneous case can be derived as…”

Blended particle filters for large-dimensional chaotic dynamical systems

“…For the blended particle filters developed below for a state vector u ∈ R N , there are two subspaces that typically evolve adaptively in time, where u = ðu 1 ; u 2 Þ, u j ∈ R Nj , and N 1 + N 2 = N, with the property that N 1 is low dimensional enough so that the non-Gaussian statistics of u 1 can be calculated from a particle filter whereas the evolving statistics of u 2 are conditionally Gaussian given u 1 . Statistically nonlinear forecast models with this structure with high skill for uncertainty quantification have been developed recently by Sapsis and Majda (16)(17)(18)(19) and are used below in the blended filters.…”