Yoonsang Lee

Conceptual dynamical models for turbulence.

Significance 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. These complex features strongly impact practical prediction, uncertainty quantification, and data assimilation strategies in such anisotropic turbulent systems. The large scales in turbulence are chaotic whereas the small scales with low variance have relatively frequent extreme events––intermittency––which can impact the large scales. Here conceptual dynamical models of turbulence are developed which, despite their simplicity, capture many of these key features of anisotropic turbulent systems in a qualitative fashion. The paper is a self-contained treatment of these conceptual models and their properties. 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.

Numerical Schemes for Stochastic Backscatter in the Inverse Cascade of Quasigeostrophic Turbulence

Backscatter is the process of energy transfer from small to large scales in turbulence; it is crucially important in the inverse energy cascades of geophysical turbulence, where the net transfer of energy is from small to large scales. One approach to modeling backscatter in underresolved simulations is to add a stochastic forcing term. This study, set in the idealized context of the inverse cascade of two-layer quasigeostrophic turbulence, focuses on the importance of spatial and temporal correlation in numerical stochastic backscatter schemes when used with low-order finite-difference spatial discretizations. A minimal stochastic backscatter scheme is developed as a stripped-down version of stochastic superparameterization [Grooms and Majda, J. Comput. Phys., 271, (2014), pp. 78--98]. This simplified scheme allows detailed numerical analysis of the spatial and temporal correlation structure of the modeled backscatter. Its essential properties include a local formulation amenable to implementation in finite difference codes and nonperiodic domains, and tunable spatial and temporal correlations. Experiments with this scheme in the idealized context of homogeneous two-layer quasigeostrophic turbulence demonstrate the need for stochastic backscatter to be smooth at the coarse grid scale when used with low-order finite-difference schemes in an inverse-cascade regime. In contrast, temporal correlation of the backscatter is much less important for achieving realistic energy spectra. It is expected that the spatial and temporal correlation properties of the simplified backscatter schemes examined here will inform the development of stochastic backscatter schemes in more realistic models.

State estimation and prediction using clustered particle filters

Significance Particle filtering is an essential tool for the estimation and prediction of complex systems including non-Gaussian features. A class of particle filters, clustered particle filters, is introduced for high-dimensional dynamical systems such as geophysical systems. The proposed method uses relatively few particles compared with the standard particle filter and captures the non-Gaussian features of the true signal, which are typical in complex nonlinear systems. The method is also robust for the difficult regime of high-quality sparse and infrequent observations and does not show any filter divergence in our tests. In the clustered particle filter, coarse-grained localization is implemented through the clustering of state variables and particles are adjusted to stabilize the filter. 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 non-Gaussian statistics. The clustered particle filter shows robust skill in both achieving accurate filter results and capturing non-Gaussian 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.

Multiscale Methods for Data Assimilation in Turbulent Systems

Data assimilation of turbulent signals is an important challenging problem because of the extremely complicated large dimension of the signals and incomplete partial noisy observations which usually mix the large scale mean flow and small scale fluctuations. Due to the limited computing power in the foreseeable future, it is desirable to use multiscale forecast models which are cheap and fast to mitigate the curse of dimensionality in turbulent systems; thus model errors from imperfect forecast models are unavoidable in the development of a data assimilation method in turbulence. Here we propose a suite of multiscale data assimilation methods which use stochastic Superparameterization as the forecast model. Superparameterization is a seamless multiscale method for parameterizing the effect of small scales by cheap local problems embedded in a coarse grid. The key ingredient of the multiscale data assimilation methods is the systematic use of conditional Gaussian mixtures which make the methods efficient by filtering a subspace whose dimension is smaller than the full state. The multiscale data assimilation methods proposed here are tested on a six dimensional conceptual dynamical model for turbulence which mimics interesting features of anisotropic turbulence including two way coupling between the large and small scale parts, intermittencies, and extreme events in the smaller scale fluctuations. Numerical results show that suitable multiscale data assimilation methods have high skill in estimating the most energetic modes of turbulent signals even with infrequent observation times.

Ensemble Filtering and Low-Resolution Model Error: Covariance Inflation, Stochastic Parameterization, and Model Numerics

AbstractThe use of under-resolved models in ensemble data assimilation schemes leads to two kinds of model errors: truncation errors associated with discretization of the large-scale dynamics and errors associated with interactions with subgrid scales. Multiplicative and additive covariance inflation can be used to account for model errors in ensemble Kalman filters, but they do not reduce the model error. Truncation errors can be reduced by increasing the accuracy of the numerical discretization of the large-scale dynamics, and subgrid-scale parameterizations can reduce errors associated with subgrid-scale interactions. Stochastic subgrid-scale parameterizations both reduce the model error and inflate the ensemble spread, so their effectiveness in ensemble assimilation schemes can be gauged by comparing with covariance inflation techniques. The effects of covariance inflation, stochastic parameterizations, and model numerics in two-layer periodic quasigeostrophic turbulence are compared on an f plane and...

Multiscale numerical methods for passive advection-diffusion in incompressible turbulent flow fields

We propose a seamless multiscale method which approximates the macroscopic behavior of the passive advection-diffusion equations with steady incompressible velocity fields with multi-spatial scales. The method uses decompositions of the velocity fields in the Fourier space, which are similar to the decomposition in large eddy simulations. It also uses a hierarchy of local domains with different resolutions as in multigrid methods. The effective diffusivity from finer scale is used for the next coarser level computation and this process is repeated up to the coarsest scale of interest. The grids are only in local domains whose sizes decrease depending on the resolution level so that the overall computational complexity increases linearly as the number of different resolution grids increases. The method captures interactions between finer and coarser scales but has to sacrifice some of interactions between different scales. The proposed method is numerically tested with 2D examples including a successful approximation to a continuous spectrum flow.

Preventing Catastrophic Filter Divergence Using Adaptive Additive Inflation for Baroclinic Turbulence

AbstractEnsemble-based filtering or data assimilation methods have proved to be indispensable tools in atmosphere and ocean science as they allow computationally cheap, low-dimensional ensemble state approximation for extremely high-dimensional turbulent dynamical systems. For sparse, accurate, and infrequent observations, which are typical in data assimilation of geophysical systems, ensemble filtering methods can suffer from catastrophic filter divergence, which frequently drives the filter predictions to machine infinity. A two-layer quasigeostrophic equation, which is a classical idealized model for geophysical turbulence, is used to demonstrate catastrophic filter divergence. The mathematical theory of adaptive covariance inflation by Tong et al. and covariance localization are investigated to stabilize the ensemble methods and prevent catastrophic filter divergence. Two forecast models—a coarse-grained ocean code, which ignores the small-scale parameterization, and stochastic superparameterization (...

Stochastic superparameterization and multiscale filtering of turbulent tracers

Data assimilation or filtering combines a numerical forecast model and observations to provide accurate statistical estimation of the state of interest. In this paper we are concerned with accurate data assimilation of a sparsely observed passive tracer advected in turbulent flows using a reduced-order forecast model. The turbulent flows which contain anisotropic and inhomogeneous structures such as jets are typical in geophysical turbulent flows in atmosphere and ocean science, and passive tracers with a mean gradient can exhibit anisotropic transport with intermittent extreme events, as shown below. Stochastic superparameterization, which is a seamless multiscale method developed for large-scale models of atmosphere and ocean models without scale-gap between the resolved and unresolved scales, generates large-scale turbulent velocity fields using a significantly smaller degree of freedom compared to a direct fine resolution numerical simulation. In a large-scale model of the tracer transport, the tracer...