8,978 research outputs found

    The prediction of future from the past: an old problem from a modern perspective

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    The idea of predicting the future from the knowledge of the past is quite natural when dealing with systems whose equations of motion are not known. Such a long-standing issue is revisited in the light of modern ergodic theory of dynamical systems and becomes particularly interesting from a pedagogical perspective due to its close link with Poincar\'e's recurrence. Using such a connection, a very general result of ergodic theory - Kac's lemma - can be used to establish the intrinsic limitations to the possibility of predicting the future from the past. In spite of a naive expectation, predictability results to be hindered rather by the effective number of degrees of freedom of a system than by the presence of chaos. If the effective number of degrees of freedom becomes large enough, regardless the regular or chaotic nature of the system, predictions turn out to be practically impossible. The discussion of these issues is illustrated with the help of the numerical study of simple models.Comment: 9 pages, 4 figure

    Standing Swells Surveyed Showing Surprisingly Stable Solutions for the Lorenz '96 Model

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    The Lorenz '96 model is an adjustable dimension system of ODEs exhibiting chaotic behavior representative of dynamics observed in the Earth's atmosphere. In the present study, we characterize statistical properties of the chaotic dynamics while varying the degrees of freedom and the forcing. Tuning the dimensionality of the system, we find regions of parameter space with surprising stability in the form of standing waves traveling amongst the slow oscillators. The boundaries of these stable regions fluctuate regularly with the number of slow oscillators. These results demonstrate hidden order in the Lorenz '96 system, strengthening the evidence for its role as a hallmark representative of nonlinear dynamical behavior.Comment: 10 pages, 8 figure

    Data assimilation in a multi-scale model

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    Data assimilation for multi-scale models is an important contemporary research topic. Especially the role of unresolved scales and model error in data assimilation needs to be systematically addressed. Here we examine these issues using the Ensemble Kalman filter (EnKF) with the two-level Lorenz-96 model as a conceptual prototype model of the multi-scale climate system. We use stochastic parameterization schemes to mitigate the model errors from the unresolved scales. Our results indicate that a third-order autoregressive process performs better than a first-order autoregressive process in the stochastic parameterization schemes, especially for the system with a large time-scale separation.Model errors can also arise from imprecise model parameters. We find that the accuracy of the analysis (an optimal estimate of a model state) is linearly correlated to the forcing error in the Lorenz-96 model. Furthermore, we propose novel observation strategies to deal with the fact that the dimension of the observations is much smaller than the model states. We also propose a new analog method to increase the size of the ensemble when its size is too small

    Evaluating Data Assimilation Algorithms

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    Data assimilation leads naturally to a Bayesian formulation in which the posterior probability distribution of the system state, given the observations, plays a central conceptual role. The aim of this paper is to use this Bayesian posterior probability distribution as a gold standard against which to evaluate various commonly used data assimilation algorithms. A key aspect of geophysical data assimilation is the high dimensionality and low predictability of the computational model. With this in mind, yet with the goal of allowing an explicit and accurate computation of the posterior distribution, we study the 2D Navier-Stokes equations in a periodic geometry. We compute the posterior probability distribution by state-of-the-art statistical sampling techniques. The commonly used algorithms that we evaluate against this accurate gold standard, as quantified by comparing the relative error in reproducing its moments, are 4DVAR and a variety of sequential filtering approximations based on 3DVAR and on extended and ensemble Kalman filters. The primary conclusions are that: (i) with appropriate parameter choices, approximate filters can perform well in reproducing the mean of the desired probability distribution; (ii) however they typically perform poorly when attempting to reproduce the covariance; (iii) this poor performance is compounded by the need to modify the covariance, in order to induce stability. Thus, whilst filters can be a useful tool in predicting mean behavior, they should be viewed with caution as predictors of uncertainty. These conclusions are intrinsic to the algorithms and will not change if the model complexity is increased, for example by employing a smaller viscosity, or by using a detailed NWP model
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