Is Weather Chaotic? Coexisting Attractors, Multistability, and Predictability

Since Lorenz’s 1963 study and 1972 presentation, the statement “weather is chaotic’’ has been well accepted. Such a view turns our attention from regularity associated with Laplace’s view of determinism to irregularity associated with chaos. In contrast to single type chaotic solutions, recent studies using a generalized Lorenz model (Shen 2019a, b; Shen et al. 2019) have focused on the coexistence of chaotic and regular solutions that appear within the same model, using the same modeling configurations but different initial conditions. The results suggest that the entirety of weather possesses a dual nature of chaos and order with distinct predictability. Furthermore, Shen et al. (2021a, b) illustrated the following two mechanisms that may enable or modulate attractor coexistence: (1) the aggregated negative feedback of small-scale convective processes that enable the appearance of stable, steady-state solutions and their coexistence with chaotic or nonlinear limit cycle solutions; and (2) the modulation of large-scale time varying forcing (heating). Recently, the physical relevance of findings within Lorenz models for real world problems has been reiterated by providing mathematical universality between the Lorenz simple weather and Pedlosky simple ocean models, as well as amongst the non-dissipative Lorenz model, and the Duffing, the Nonlinear Schrodinger, and the Korteweg–de Vries equations (Shen 2020, 2021). We additionally compared the Lorenz 1963 and 1969 models. The former is a limited-scale, nonlinear, chaotic model; while the latter is a closure-based, physically multiscale, mathematically linear model with ill-conditioning. To support and illustrate the revised view, this short article elaborates on additional details of monostability and multistability by applying skiing and kayaking as an analogy, and provides a list of non-chaotic weather systems. We additionally address the influence of the revised view on real-world model predictions and analyses using hurricane track predictions as an illustration, and provide a brief summary on the recent deployment of methods for multiscale analyses and classifications of chaotic and non-chaotic solutions.

The Maximum Likelihood Ensemble Filter with State Space Localization

New method for ensemble data assimilation that incorporates state space covariance localization, global numerical optimization, and implied Bayesian inference, is presented. The method is referred to as the MLEF with State Space Localization (MLEF-SSL) due to its similarity with the Maximum Likelihood Ensemble Filter (MLEF). One of the novelties introduced in MLEF-SSL is the calculation of a reduced-rank localized forecast error covariance using random projection. The Hessian preconditioning is accomplished via Cholesky decomposition of the Hessian matrix, accompanied with solving triangular system of equations instead of directly inverting matrices. For ensemble update the MLEF-SSL system employs resampling of posterior perturbations. The MLEF-SSL was applied to Lorenz model II and compared to Ensemble Kalman Filter with state space localization and to MLEF with observation space localization. The observations include linear and nonlinear observation operators, each applied to integrated and point observations. Results indicate improved performance of MLEF-SSL, particularly in assimilation of integrated nonlinear observations. Resampling of posterior perturbations for ensemble update also indicates a satisfactory performance. Additional experiments were conducted to examine the sensitivity of the method to the rank of random matrix and to compare it to truncated eigenvectors of the localization matrix. The two methods are comparable in application to low-dimensional Lorenz model, except that the new method outperforms the truncated eigenvector method in case of severe rank reduction. The random basis method is simple to implement and may be more promising for realistic high-dimensional applications.

STUDYING THE INTERCONNECTION OF FOOD, ENERGY AND WATER RESOURCES USING THE THREE-SECTORAL LORENTZ MODEL

The work is devoted to the problem of creating new methods for complex modeling and risk management, which will allow to study synergistic interactions between sources of risks of various origins under conditions of uncertainty. The paper proposes an approach to the study of the relationship between food, water and energy resources using the three-sectoral Lorenz model, combining in a single structure similarly described sectors of the economy, each of which is considered in terms of the productivity level, the workplaces number and the structural disturbances level. As a mathematical modeling result, the conditions of the deterministic chaos origin in the minimum economic development model were determined and possible reasons of the global economy growing vulnerability to small changes in management parameters were identified. The problem of determining effective controls for minimizing the total structural violations on selected time interval is considered. As a result of model experiments, the trajectories of control parameters changes were determined, which make it possible to reduce the structural violations number. This is achieved through changes in the ratio of supply and demand levels for products, demand and supply for workplaces creation. The influence of random perturbations on the deterministic attractors stochastic deformation of the Lorentz model is considered. It is shown that, under random perturbations, the trajectories of the stochastic system leave a deterministic attractor and form around it a certain bundle with the corresponding probabilistic distribution. The further model complicating possibility by taking into account other sectors of the economy using the Lorenz model in a complex form is considered. In this case the task of studying n sectors of economy is reduced to considering the behavior of an ensemble of n coupled oscillators that generate oscillations with frequencies ωn, respectively. Collective synchronization of oscillator data can be investigated using Kuramoto’s model. The problem of managing socio-economic development under the chaotic modes origin conditions is reduced for a complex model to controlling the frequency of a nonzero mean field generated by coupled oscillators.

State, global and local parameter estimation using local ensemble Kalman filters: applications to online machine learning of chaotic dynamics

<div>In a recent methodological paper, we have shown how a (local) ensemble Kalman filter can be used to learn both the state and the dynamics of a system in an online framework. The surrogate model is fully parametrised (for example, this could be a neural network) and the update is a two-step process: (i) a state update, possibly localised, and (ii) a parameter update consistent with the state update. In this framework, the parameters of the surrogate model are assumed to be global. <br><br>In this presentation, we show how to extend the method to the case where the surrogate model, still fully parametrised, admits both global and local parameters (typically forcing parameters). In this case, localisation can be applied not only to the state update, but also to the local parameters update. This results in a collection of new algorithms, depending on the localisation method (covariance localisation or domain localisation) and on whether localisation is applied to the state update, or to both the state and local parameter update. The algorithms are implemented and tested with success on the 40-variable Lorenz model. Finally, we show a two-dimensional illustration of the method using a multi-layer Lorenz model with radiance-like non-local observations.</div>