New perspectives on superparameterization for geophysical turbulence

Proc. Natl. Acad. Sci. U.S.A.

2014

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Understanding the complexity of anisotropic turbulent processes in engineering and environmental fluid flows is a formidable challenge with practical significance because energy often flows intermittently from the smaller scales to impact the largest scales in these flows. Conceptual dynamical models for anisotropic turbulence are introduced and developed here which, despite their simplicity, capture key features of vastly more complicated turbulent systems. These conceptual models involve a large-scale mean flow and turbulent fluctuations on a variety of spatial scales with energy-conserving wave-mean-flow interactions as well as stochastic forcing of the fluctuations. Numerical experiments with a six-dimensional conceptual dynamical model confirm that these models capture key statistical features of vastly more complex anisotropic turbulent systems in a qualitative fashion. These features include chaotic statistical behavior of the mean flow with a sub-Gaussian probability distribution function (pdf) for its fluctuations whereas the turbulent fluctuations have decreasing energy and correlation times at smaller scales, with nearly Gaussian pdfs for the large-scale fluctuations and fat-tailed non-Gaussian pdfs for the smaller-scale fluctuations. This last feature is a manifestation of intermittency of the small-scale fluctuations where turbulent modes with small variance have relatively frequent extreme events which directly impact the mean flow. The dynamical models introduced here potentially provide a useful test bed for algorithms for prediction, uncertainty quantification, and data assimilation for anisotropic turbulent systems.wave-mean interaction | stochastic model U nderstanding the complexity of anisotropic turbulence processes over a wide range of spatiotemporal scales in engineering shear turbulence (1-3) as well as climate atmosphere ocean science (4-6) is a grand challenge of contemporary science. This is especially important from a practical viewpoint because energy often flows intermittently from the smaller scales to affect the largest scales in such anisotropic turbulent flows. The typical features of such anisotropic turbulent flows are the following (2-4):(A) The large-scale mean flow is usually chaotic but more predictable than the smaller-scale fluctuations. The overall single point probability distribution function (pdf) of the flow field is nearly Gaussian whereas the mean flow pdf is subGaussian, in other words, with less extreme variability than a Gaussian random variable. The goal here is to develop the simplest conceptual dynamical model for anisotropic turbulence that captures all of the features in (A)-(D) in a transparent qualitative fashion. In contrast with deterministic models of turbulence which are derived by Galerkin truncation of the Navier-Stokes equation (7) and do not display all of the features in (A)-(D), the conceptual models developed here are low-dimensional stochastic dynamical systems; the nonlinear interactions between the large-scale meanflow component and the sm...

Section: Concluding Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

Conceptual dynamical models for turbulence

Majda

Proc. Natl. Acad. Sci. U.S.A.

2014

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“…Superparameterization (SP) is a multiscale computational approach that has been successfully applied to modeling atmospheric dynamics, and that shows promise for more general applications (Tao et al, 2009;Randall et al, 2013;Majda and Grooms, 2014). Grooms et al (2014) have developed an ensemble Kalman filter framework for use with SP.…”

Section: Discussionmentioning

confidence: 99%

“…This framework was developed in the context of stochastic SP, a variant of SP that reduces computational cost by replacing the small scale simulations of SP with quasilinear stochastic models (Grooms and Majda, 2013;Majda and Grooms, 2014). Stochastic SP has only been developed for idealized turbulence models Majda, 2013, 2014a, b;Grooms et al, 2015), and is not yet implemented in global atmosphere, ocean, or climate models.…”

Section: Introductionmentioning

confidence: 99%

A framework for variational data assimilation with superparameterization

Grooms

Nonlin. Processes Geophys.

2015

Abstract. Superparameterization (SP) is a multiscale computational approach wherein a large scale atmosphere or ocean model is coupled to an array of simulations of small scale dynamics on periodic domains embedded into the computational grid of the large scale model. SP has been successfully developed in global atmosphere and climate models, and is a promising approach for new applications, but there is currently no practical data assimilation framework that can be used with these models. The authors develop a 3D-Var variational data assimilation framework for use with SP; the relatively low cost and simplicity of 3D-Var in comparison with ensemble approaches makes it a natural fit for relatively expensive multiscale SP models. To demonstrate the assimilation framework in a simple model, the authors develop a new system of ordinary differential equations similar to the two-scale Lorenz-'96 model. The system has one set of variables denoted {Y i }, with large and small scale parts, and the SP approximation to the system is straightforward. With the new assimilation framework the SP model approximates the large scale dynamics of the true system accurately.

“…As the forecast model for both filtering methods, we use the stochastic superparameterization multiscale method (32,33), which is a seamless, multiscale, coarse-grid method using conditional Gaussian statistics for the unresolved small scales. The forecast model uses only 128 grid points whereas the full resolution uses 8,192 grid points, which yields about 250 times cheaper computational savings (considering the saving in the time step).…”

Section: Application To Multiscale Data Assimilationmentioning

confidence: 99%

State estimation and prediction using clustered particle filters

Proc. Natl. Acad. Sci. U.S.A.

Majda

2016

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Particle filtering is an essential tool to improve uncertain model predictions by incorporating noisy observational data from complex systems including non-Gaussian features. A class of particle filters, clustered particle filters, is introduced for high-dimensional nonlinear systems, which uses relatively few particles compared with the standard particle filter. The clustered particle filter captures non-Gaussian features of the true signal, which are typical in complex nonlinear dynamical systems such as geophysical systems. The method is also robust in the difficult regime of high-quality sparse and infrequent observations. The key features of the clustered particle filtering are coarse-grained localization through the clustering of the state variables and particle adjustment to stabilize the method; each observation affects only neighbor state variables through clustering and particles are adjusted to prevent particle collapse due to high-quality observations. The clustered particle filter is tested for the 40-dimensional Lorenz 96 model with several dynamical regimes including strongly nonGaussian statistics. The clustered particle filter shows robust skill in both achieving accurate filter results and capturing nonGaussian statistics of the true signal. It is further extended to multiscale data assimilation, which provides the large-scale estimation by combining a cheap reduced-order forecast model and mixed observations of the large-and small-scale variables. This approach enables the use of a larger number of particles due to the computational savings in the forecast model. The multiscale clustered particle filter is tested for one-dimensional dispersive wave turbulence using a forecast model with model errors.data assimilation | particle filter | non-Gaussian | uncertainty quantification D ata assimilation or filtering combines numerical forecast models with observational data to provide the best statistical estimation and prediction of complex systems. Due to the high dimensionality of complex nonlinear systems such as geophysical systems, accurate and efficient estimation and prediction of such complex systems are formidable tasks. They require enormously large computational resources to run forecast models and observations can be sparse and infrequent, as in oceanography. As a Monte Carlo approach, ensemble-based methods (1, 2) combined with covariance inflation and localization are indispensable tools because they allow computationally cheap, low-dimensional ensemble state approximations for the systems. They have performed well for operational applications such as numerical weather prediction (3, 4). Nevertheless, because the ensemble-based methods approximate the forecast distribution using Gaussian statistics, when the ensemble is not sufficiently large these methods can lead to inaccurate estimation and prediction when the true signal has non-Gaussian statistics, which are typical for a wide range of systems. Particle filtering captures non-Gaussian features using different weights for different...