47 research outputs found

    Stochastic and Statistical Methods in Climate, Atmosphere, and Ocean Science

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    Introduction The behavior of the atmosphere, oceans, and climate is intrinsically uncertain. The basic physical principles that govern atmospheric and oceanic flows are well known, for example, the Navier-Stokes equations for fluid flow, thermodynamic properties of moist air, and the effects of density stratification and Coriolis force. Notwithstanding, there are major sources of randomness and uncertainty that prevent perfect prediction and complete understanding of these flows. The climate system involves a wide spectrum of space and time scales due to processes occurring on the order of microns and milliseconds such as the formation of cloud and rain droplets to global phenomena involving annual and decadal oscillations such as the EL Nio-Southern Oscillation (ENSO) and the Pacific Decadal Oscillation (PDO) [5]. Moreover, climate records display a spectral variability ranging from 1 cycle per month to 1 cycle per 100, 000 years [23]. The complexity of the climate system stems in large part from the inherent nonlinearities of fluid mechanics and the phase changes of water substances. The atmosphere and oceans are turbulent, nonlinear systems that display chaotic behavior (e.g., [39]). The time evolutions of the same chaotic system starting from two slightly different initial states diverge exponentially fast, so that chaotic systems are marked by limited predictability. Beyond the so-called predictability horizon (on the order of 10 days for the atmosphere), initial state uncertainties (e.g., due to imperfect observations) have grown to the point that straightforward forecasts are no longer useful. Another major source of uncertainty stems from the fact that numerical models for atmospheric and oceanic flows cannot describe all relevant physical processes at once. These models are in essence discretized partial differential equations (PDEs), and the derivation of suitable PDEs (e.g., the so-called primitive equations) from more general ones that are less convenient for computation (e.g., the full Navier-Stokes equations) involves approximations and simplifications that introduce errors in the equations. Furthermore, as a result of spatial discretization of the PDEs, numerical models have finite resolution so that small-scale processes with length scales below the model grid scale are not resolved. These limitations are unavoidable, leading to model error and uncertainty. The uncertainties due to chaotic behavior and unresolved processes motivate the use of stochastic and statistical methods for modeling and understanding climate, atmosphere, and oceans. Models can be augmented with random elements in order to represent time-evolving uncertainties, leading to stochastic models. Weather forecasts and climate predictions are increasingly expressed in probabilistic terms, making explicit the margins of uncertainty inherent to any prediction

    Overstromingen en Monte Carlo

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    Reduced model-error source terms for fluid flow

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    It is well known that the wide range of spatial and temporal scales present in geophysical flow problems represents a (currently) insurmountable computational bottleneck, which must be circumvented by a coarse-graining procedure. The effect of the unresolved fluid motions enters the coarse-grained equations as an unclosed forcing term, denoted as the ’eddy forcing’. Traditionally, the system is closed by approximate deterministic closure models, i.e. so-called parameterizations. Instead of creating a deterministic parameterization, some recent efforts have focused on creating a stochastic, data-driven surrogate model for the eddy forcing from a (limited) set of reference data, with the goal of accurately capturing the long-term flow statistics. Since the eddy forcing is a dynamically evolving field, a surrogate should be able to mimic the complex spatial patterns displayed by the eddy forcing. Rather than creating such a (fully data-driven) surrogate, we propose to precede the surrogate construction step by a proce- dure that replaces the eddy forcing with a new model-error source term which: i) is tailor-made to capture spatially-integrated statistics of interest, ii) strikes a balance between physical in- sight and data-driven modelling , and iii) significantly reduces the amount of training data that is needed. Instead of creating a surrogate for an evolving field, we now only require a surrogate model for one scalar time series per statistical quantity-of-interest. Our current surrogate mod- elling approach builds on a resampling strategy, where we create a probability density function of the reduced training data that is conditional on (time-lagged) resolved-scale variables. We derive the model-error source terms, and construct the reduced surrogate using an ocean model of two-dimensional turbulence in a doubly periodic square domain

    Reducing data-driven dynamical subgrid scale models by physical constraints

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    Recent years have seen a growing interest in using data-driven (machine-learning) techniques for the construction of cheap surrogate models of turbulent subgrid scale stresses. These stresses display complex spatio-temporal structures, and constitute a difficult surrogate target. In this paper we propose a data-preprocessing step, in which we derive alternative subgrid scale models which are virtually exact for a user-specified set of spatially integrated quantities of interest. The unclosed component of these new subgrid scale models is of the same size as this set of integrated quantities of interest. As a result, the corresponding training data is massively reduced in size, decreasing the complexity of the subsequent surrogate construction

    Quantifying dependencies for sensitivity analysis with multivariate input sample data

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    We present a novel method for quantifying dependencies in multivariate datasets, based on estimating the RĂ©nyi entropy by minimum spanning trees (MSTs). The length of the MSTs can be used to order pairs of variables from strongly to weakly dependent, making it a useful tool for sensitivity analysis with dependent input variables. It is well-suited for cases where the input distribution is unknown and only a sample of the inputs is available. We introduce an estimator to quantify dependency based on the MST length, and investigate its properties with several numerical examples. To reduce the computational cost of constructing the exact MST for large datasets, we explore methods to compute approximations to the exact MST, and find the multilevel approach introduced recently by Zhong et al. (2015) to be the most accurate. We apply our proposed method to an artificial testcase based on the Ishigami function, as well as to a real-world testcase involving sediment transport in the North Sea. The results are consistent with prior knowledge and heuristic understanding, as well as with variance-based analysis using Sobol indices in the case where these indices can be computed

    Towards data-driven dynamic surrogate models for ocean flow

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    Coarse graining of (geophysical) flow problems is a necessity brought upon us by the wide range of spatial and temporal scales present in these problems, which cannot be all represented on a numerical grid without an inordinate amount of computational resources. Traditionally, the effect of the unresolved eddies is approximated by deterministic closure models, i.e. so-called parameterizations. The effect of the unresolved eddy field enters the resolved-scale equations as a forcing term, denoted as the’eddy forcing’. Instead of creating a deterministic parameterization, our goal is to infer a stochastic, data-driven surrogate model for the eddy forcing from a (limited) set of reference data, with the goal of accurately capturing the long-term flow statistics. Our surrogate modelling approach essentially builds on a resampling strategy, where we create a probability density function of the reference data that is conditional on (time-lagged) resolved-scale variables. The choice of resolved-scale variables, as well as the employed time lag, is essential to the performance of the surrogate. We will demonstrate the effect of different modelling choices on a simplified ocean model of two-dimensional turbulence in a doubly periodic square domain

    Towards a turbulence closure based on energy modes

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    A new approach to parameterizing subgrid-scale processes is proposed: The impact of the unresolved dynamics on the resolved dynamics (i.e., the eddy forcing) is represented by a series expansion in dynamical spatial modes that stem from the energy budget of the resolved dynamics. It is demonstrated that the convergence in these so-called energy modes is faster by orders of magnitude than the convergence in Fourier-type modes. Moreover, a novel way to test parameterizations in models is explored. The resolved dynamics and the corresponding instantaneous eddy forcing are defined via spatial filtering that accounts for the representation error of the equations of motion on the low-resolution model grid. In this way, closures can be tested within the high-resolution model, and the effects of different parameterizations related to different energy pathways can be isolated. In this study, the focus is on parameterizations of the baroclinic energy pathway. The corresponding standard closure in ocean models, the Gent-McWilliams (GM) parameterization, is also tested, and it is found that the GM field acts like a stabilizing direction in phase space. The GM field does not project well on the eddy forcing and hence fails to excite the model's intrinsic low-frequency variability, but it is able to stabilize the model

    Mitigation of Large Power Spills by a Storage Device

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    The unpredictable nature of wind energy makes its integration to the electric grid highly challenging. However, these challenges can be addressed by incorporating storage devices (batteries) in the system. We perform an overall assessment of a single domestic power system with a wind turbine supported by an energy storage device. The aim is to investigate the best operation mode of the storage device such that the occurrence of large power spills can be minimized. For estimating the small probability of large power spills, we use the splitting technique for rare-event simulations. An appropriate Importance Function for splitting is formulated such that it reduces the work-load of the probability estimator as compared to the conventional Crude Monte Carlo probability estimator. Simulation results show that the ramp constraints imposed on the charging/discharging rate of the storage device plays a pivotal role in mitigating large power spills. It is observed that by employing a new charging strategy for the storage device large power spills can be minimized further. There exists a trade-off between reducing the large power spills versus reducing the average power spills

    Simulation-based assessment of the stationary tail distribution of a stochastic differential equation

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    A commonly used approach to analyzing stochastic differential equations (SDEs) relies on performing Monte Carlo simulation with a discrete-time counterpart. In this paper we study the impact of such a time-discretization when assessing the stationary tail distribution. For a family of semi-implicit Euler discretization schemes with time-step h > 0, we quantify the relative error due to the discretization, as a function of h and the exceedance level x. By studying the existence of certain (polynomial and exponential) moments, using a sequence of prototypical examples, we demonstrate that this error may tend to 0 or ÂĄ. The results show that the original shape of the tail can be heavily affected by the discretization. The cases studied indicate that one has to be very careful when estimating the stationary tail distribution using Euler discretization schemes
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