Rainfall in South America follows a dipole-like pattern that is driven by atmospheric waves associated with the jet stream.

Tipping elements in the climate system are large-scale subregions of the Earth that might possess threshold behavior under global warming with large potential impacts on human societies. Here, we study a subset of five tipping elements and their interactions in a conceptual and easily extendable framework: the Greenland Ice Sheets (GIS) and West Antarctic Ice Sheets, the Atlantic meridional overturning circulation (AMOC), the El–Niño Southern Oscillation and the Amazon rainforest. In this nonlinear and multistable system, we perform a basin stability analysis to detect its stable states and their associated Earth system resilience. By combining these two methodologies with a large-scale Monte Carlo approach, we are able to propagate the many uncertainties associated with the critical temperature thresholds and the interaction strengths of the tipping elements. Using this approach, we perform a system-wide and comprehensive robustness analysis with more than 3.5 billion ensemble members. Further, we investigate dynamic regimes where some of the states lose stability and oscillations appear using a newly developed basin bifurcation analysis methodology. Our results reveal that the state of four or five tipped elements has the largest basin volume for large levels of global warming beyond 4 °C above pre-industrial climate conditions, representing a highly undesired state where a majority of the tipping elements reside in the transitioned regime. For lower levels of warming, states including disintegrated ice sheets on west Antarctica and Greenland have higher basin volume than other state configurations. Therefore in our model, we find that the large ice sheets are of particular importance for Earth system resilience. We also detect the emergence of limit cycles for 0.6% of all ensemble members at rare parameter combinations. Such limit cycle oscillations mainly occur between the GIS and AMOC (86%), due to their negative feedback coupling. These limit cycles point to possibly dangerous internal modes of variability in the climate system that could have played a role in paleoclimatic dynamics such as those unfolding during the Pleistocene ice age cycles.

A novel idea for an optimal time delay state space reconstruction from uni- and multivariate time series is presented. The entire embedding process is considered as a game, in which each move corresponds to an embedding cycle and is subject to an evaluation through an objective function. This way the embedding procedure can be modeled as a tree, in which each leaf holds a specific value of the objective function. By using a Monte Carlo ansatz, the proposed algorithm populates the tree with many leafs by computing different possible embedding paths and the final embedding is chosen as that particular path, which ends at the leaf with the lowest achieved value of the objective function. The method aims to prevent getting stuck in a local minimum of the objective function and can be used in a modular way, enabling practitioners to choose a statistic for possible delays in each embedding cycle as well as a suitable objective function themselves. The proposed method guarantees the optimization of the chosen objective function over the parameter space of the delay embedding as long as the tree is sampled sufficiently. As a proof of concept, we demonstrate the superiority of the proposed method over the classical time delay embedding methods using a variety of application examples. We compare recurrence plot-based statistics inferred from reconstructions of a Lorenz-96 system and highlight an improved forecast accuracy for map-like model data as well as for palaeoclimate isotope time series. Finally, we utilize state space reconstruction for the detection of causality and its strength between observables of a gas turbine type thermoacoustic combustor.

Understanding the variability of low-level atmospheric circulation regimes is key for understanding the dynamics of monsoon systems. The South American Monsoon is characterized by strong year-long trade winds that are channeled southward into the South American Low-Level Jet after crossing the Amazon basin, which in turn is elementary for the moisture transport to Southern South America. In this study, we utilize streamflow wind networks, a type of climate networks that tracks the local flow of the wind field, together with the analysis of composites of wind, precipitation, and geopotential height fields, to investigate the variability of the South American low-level circulation. The streamflow wind networks are used here as they are able to directly track the wind flow and encode its spatiotemporal characteristics in their topology. We focus on intraseasonal variations in terms of active and break monsoon phases on the one hand, and on the interannual variability in terms of the impacts of the El Niño-Southern Oscillation on the other hand. Our findings highlight the importance of the South American Low-Level Jet, its spatial position and variability. Our study reveals the relation of the active and break regimes to anomalous high- and low-pressure systems over the southern Atlantic that are connected to Rossby wave trains from the southern Pacific, as well as the impact of these regimes on the cross-equatorial low-level flow. In addition, the streamflow networks that we use demonstrate significant shifts of the dominant wind flow pattern during El Niño and La Niña episodes.

Climate networks have proven to be a valuable method to investigate spatial connectivity patterns of the climate system. However, so far such networks have mostly been applied to scalar observables. In this study, we propose a new method for constructing networks from atmospheric wind fields on two-dimensional isobaric surfaces. By connecting nodes along a spatial environment based on the local wind flow, we derive a network representation of the low-level circulation that captures its most important characteristics. In our approach, network links are placed according to a suitable statistical null model that takes into account the direction and magnitude of the flow. We compare a simulation-based (numerically costly) and a semi-analytical (numerically cheaper) approach to determine the statistical significance of possible connections, and find that both methods yield qualitatively similar results. As an application, we choose the regional climate system of South America and focus on the monsoon season in austral summer. Monsoon systems are generally characterized by substantial changes in the large-scale wind directions, and therefore provide ideal applications for the proposed wind networks. Based on these networks, we are able to reveal the key features of the low-level circulation of the South American Monsoon System, including the South American Low-Level Jet. Networks of the dry and the wet season are compared with each other and their differences are consistent with the literature on South American climate.

Many high-dimensional complex systems exhibit an enormously complex landscape of possible asymptotic states. Here, we present a numerical approach geared towards analyzing such systems. It is situated between the classical analysis with macroscopic order parameters and a more thorough, detailed bifurcation analysis. With our machine learning method, based on random sampling and clustering methods, we are able to characterize the different asymptotic states or classes thereof and even their basins of attraction. In order to do this, suitable, easy to compute, statistics of trajectories with randomly generated initial conditions and parameters are clustered by an algorithm such as DBSCAN. Due to its modular and flexible nature, our method has a wide range of possible applications in many disciplines. While typical applications are oscillator networks, it is not limited only to ordinary differential equation systems, every complex system yielding trajectories, such as maps or agent-based models, can be analyzed, as we show by applying it the Dodds-Watts model, a generalized SIRS-model, modeling social and biological contagion. A second order Kuramoto model, used, e.g. to investigate power grid dynamics, and a Stuart-Landau oscillator network, each exhibiting a complex multistable regime, are shown as well. The method is available to use as a package for the Julia language.world applications: the Dodds-Watts model of social and biological contagion, a network of second order Kuramoto oscillators, used e.g. to model power grids and a network of Stuart-Landau oscillators, of importance for many chemical and biological systems, will follow in section 3. Lastly, these results and the performance and applicability of the presented method will be discussed in section 4.

When predicting complex systems one typically relies on differential equation which can often be incomplete, missing unknown influences or higher order effects. By augmenting the equations with artificial neural networks we can compensate these deficiencies. We show that this can be used to predict paradigmatic, high-dimensional chaotic partial differential equations even when only short and incomplete datasets are available. The forecast horizon for these high dimensional systems is about an order of magnitude larger than the length of the training data.

Abstract. Earth system models (ESMs) are the primary tools for investigating future Earth system states at timescales from decades to centuries, especially in response to anthropogenic greenhouse gas release. State-of-the-art ESMs can reproduce the observational global mean temperature anomalies of the last 150 years. Nevertheless, ESMs need further improvements, most importantly regarding (i) the large spread in their estimates of climate sensitivity, i.e., the temperature response to increases in atmospheric greenhouse gases; (ii) the modeled spatial patterns of key variables such as temperature and precipitation; (iii) their representation of extreme weather events; and (iv) their representation of multistable Earth system components and the ability to predict associated abrupt transitions. Here, we argue that making ESMs automatically differentiable has a huge potential to advance ESMs, especially with respect to these key shortcomings. First, automatic differentiability would allow objective calibration of ESMs, i.e., the selection of optimal values with respect to a cost function for a large number of free parameters, which are currently tuned mostly manually. Second, recent advances in machine learning (ML) and in the number, accuracy, and resolution of observational data promise to be helpful with at least some of the above aspects because ML may be used to incorporate additional information from observations into ESMs. Automatic differentiability is an essential ingredient in the construction of such hybrid models, combining process-based ESMs with ML components. We document recent work showcasing the potential of automatic differentiation for a new generation of substantially improved, data-informed ESMs.