The problem of accurate detection of climate response to slow external forcing in 19-21 centuries is complicated by the presence of internal climate variability, which can also exhibit slow (decadal and multidecadal) large-scale dynamics, and also by the fact that there is only one observed climate realization available. At the same time, state-of-the-art Earth system models (ESMs) exhibit different spatiotemporal content on slow time scales, and their ability to estimate forced and internal climate variability needs further verification, especially given a relatively poor (short) observational statistics with respect to slow time scales.Here we present a method called ensemble linear dynamical mode (E-LDM) decomposition [1] which addresses the problem of forced signal and internal variability detection from small ensembles of ESM simulations. The method is based on the general assumption that the forced response is the same in all ensemble members and the internal variability is uncorrelated, while both of them can be essentially represented by a low-dimensional set of spatial patterns and corresponding forced and internal time series with certain time scales; the patterns, the time series and their time scales are optimized via the Bayesian framework. We compare the E-LDM method with other state-of-the-art methods of forced signal detection on synthetic and ESM-simulated data, and also discuss its applicability to the problem of intercomparison of ESMs and their verification with respect to real data. This research was supported by the state assignment of the Institute of Applied Physics of the Russian Academy of Sciences (Project No. FFUF-2022-0008). 1. Gavrilov, A., Kravtsov, S., Buyanova, M., Mukhin, D., Loskutov, E., & Feigin, A. (2023). Forced response and internal variability in ensembles of climate simulations: identification and analysis using linear dynamical mode decomposition. Climate Dynamics, 1–28. https://doi.org/10.1007/S00382-023-06995-1.

Advanced numerical climate models are known to exhibit biases in simulating some features of El Nino-Southern Oscillation (ENSO), which is a key mode of interannual climate variability. In this study we analyze how two fundamental features of observed ENSO - asymmetry between hot and cold states and phase-locking to the annual cycle - are reflected in two different versions of the INMCM Earth system model (state-of-the-art Earth system model participating in the Coupled Model Intercomparison Project). We identify the above ENSO features using the conventional empirical orthogonal functions (EOF) analysis, which is applied to both observed and simulated upper ocean heat content (OHC) data in the tropical Pacific. We obtain that the observed tropical Pacific OHC variability is described well by two leading EOF-modes, which roughly reflect the fundamental recharge-discharge mechanism of ENSO. These modes exhibit strong seasonal cycles associated with ENSO phase locking while the revealed nonlinear dependencies between amplitudes of these cycles reflect ENSO asymmetry.We also assess and compare the predictability of observed and simulated ENSO based on linear inverse modelling. We find that the improved INMCM6 model has significant benefits in simulating described features of observed ENSO as compared with the previous INMCM5 model. The improvements of the INMCM6 model providing such benefits are discussed. We argue that proper cloud parameterization scheme is crucial for accurate simulation of ENSO dynamics with numerical climate models.

In this article the title was incorrectly given as ‘‘The Method of Periodic Principal Components for the Dynamic Spectrum of Radio Pulsars and Faraday Rotation of Nine Pulse Components of PSR B0329#’’ but it should have been ‘‘The Method of Periodic Principal Components for the Dynamic Spectrum of Radio Pulsars and Faraday Rotation of Nine Pulse Components of PSR B0329+54’’.

In this work, we present a new diagnostic tool for El Nino Southern Oscillation (ENSO) simulations in Earth System Models (ESMs) based on the analysis of upper ocean heat content data. It allows us to identify the seasonally dependent structure of temperature anomalies in the equatorial Pacific Ocean in the form of a dominant spatio-temporal pattern. We demonstrate the results of applying a tool to analysis of real data as well as climate simulations in two versions of the Institute of Numerical Mathematics ESM. We find that the latest version of the model, with improved parameterizations of clouds, large-scale condensation, and aerosols, provides significantly better reproduction of ENSO-related structure of anomalies, as well as the phase locking of ENSO to the annual cycle. We recommend to use the tool for diagnostic analysis of ESMs regarding simulation of climate phenomena with strong seasonality.

The phase variables describing the interannual variability of the El Niño Southern Oscillation (ENSO) are reconstructed from the tropical Pacific ocean heat content data, and an optimal non-linear stochastic model of the evolution of these variables is constructed in the form of a discrete map. It is shown that a significant ENSO atmospheric predictor localized in the Northern Hemisphere correlates with oceanic predictors constructed using this model. In addition, atmospheric anomalies is found that correlate with the residual part of the ENSO, which is not described by the indicated oceanic variables.

Estimating climate response to observed and projected increases in atmospheric greenhouse gases usually requires averaging among multiple independent simulations of computationally expensive global climate models to filter out the internal climate variability. Studies have shown that advanced pattern recognition methods allow one to obtain accurate estimates of the forced climate signal from just a handful of such climate realizations. The accuracy of these methods for a fixed ensemble size, however, decreases with an increasing magnitude of the low-frequency, decadal and longer internal climate variability. Here we generalize a previously developed Bayesian methodology of Linear Dynamical Mode (LDM) decomposition for spatially extended time series to enable joint identification and analysis of forced signal and internal variability in ensembles of climate simulations, a methodology dubbed here an ensemble LDM, or ELDM. The new ELDM method is shown to outperform its pattern-recognition competitors by more accurately isolating the forced signal in small ensembles of both toy- and state-of-the-art climate-model simulations. It is able to do so by explicitly recognizing a non-random structure of the internal variability, identified by the ELDM algorithm alongside the optimal forced-signal estimate, which allows one to study possible dynamical connections between the two types of variability. The optimal ELDM filtering provides a unique opportunity for objective intercomparison of decadal and longer climate variability across different global climate models-a task that proved difficult due to uncertainties associated with the noisy character and limited length of historical climate simulations combined with parameter uncertainties of alternative signal-detection methods.

In this work, we demonstrate the performance of a new data-driven method [1] for modeling cross-interacting processes with different time scales represented by time series with different sampling steps. The model is a generalization of a nonlinear stochastic model of evolution operator based on neural networks, designed for the case of time series with constant sampling step, and has complex structure. First, it describes each process by its own stochastic evolution operator with its own time step. Second, it considers possible nonlinear connections within each pair of processes in both directions. These connections are parameterized asymmetrically, depending on which process is faster and which process is slower. They make this model essentially different from the set of independent stochastic models constructed individually for each time scale. All evolution operators and connections are trained and optimized using Bayesian framework, forming a multi-scale stochastic model.We demonstrate the capabilities of the model using two climate data examples. The first example is a climate spatio-temporal data generated by a global climate model running in the middle 19th century external conditions. A second example is spatially distributed reanalysis data describing current climate evolution. In both examples we investigate first how correctly the multiscale model reconstructs the individual evolution of both subsystems, and second whether it reveals the connection between processes that is not captured by models built individually for each process.The work was supported by the Russian Science Foundation (Grant No. 20-62-46056).[1] Gavrilov et al. (2021). Data-driven stochastic model for cross-interacting processes with different time scales, EGU General Assembly 2021, online, 19–30 Apr 2021, EGU21-4199.

In this work the interval stability concept is used for study the stability of the Earth’s climate system during last 2.6 million years. For the study we used an empirical model of the Earth’s climate in the Pleistocene [1] which was constructed on the basis of LR04 stack data. By means of this model, we analysed dependence of the Pleistocene climate interval stability on time, and compare the results with the behavior of local linear stability. We find that the global climate resilience to any perturbations decreases throughout the Pleistocene and the nonlinearity plays significant role in this phenomenon. The work was carried out within the framework of the Program for the Development of the Regional Scientifc and Educational Mathematical Center "Mathematics of Future Technologies", project #075-02-2020-1483/1.1. Mukhin, D., Gavrilov, A., Loskutov, E. et al. Bayesian Data Analysis for Revealing Causes of the Middle Pleistocene Transition. Sci Rep 9, 7328 (2019). https://doi.org/10.1038/s41598-019-43867-3

Abstract The loss of autocorrelations of tropical sea surface temperatures (SST) during late spring, also called the spring predictability barrier (SPB), is a factor that strongly limits the predictability of El Nino Southern Oscillation (ENSO), and especially the statistical SST‐based ENSO forecasts starting from the winter‐spring season. Recent studies show that Pacific atmospheric circulation anomalies in winter‐spring may have a long‐term impact on the summer tropical climate via the SST footprint. Here, we infer an index based on sea level pressure (SLP) data from February to March in a single area surrounding Hawaii, and show that this area is the most informative part of the large SLP pattern initiating the SST footprinting mechanism. We then construct a statistically optimal linear model of the Nino 3.4 index taking this atmospheric index as a forcing. We find that this forcing efficiently lowers the SPB and provides significant improvements of interseasonal Niño 3.4 forecasts.

We investigate the decadal-to-centennial ENSO variability based on nonlinear data-driven stochastic modeling. We construct data-driven model of yearly Niño-3.4 indices reconstructed from paleoclimate proxies based on three different sea-surface temperature (SST) databases at the time interval from 1150 to 1995 [1]. The data-driven model is forced by the solar activity and CO2 concentration signals. We find the persistent antiphasing relationship between the solar forcing and Niño-3.4 SST on the bicentennial time scale. The dynamical mechanism of such a response is discussed.The work was supported by the Russian Science Foundation (Grant No. 20-62-46056)1. Emile-Geay, J., Cobb, K. M., Mann, M. E., & Wittenberg, A. T. (2013). Estimating Central Equatorial Pacific SST Variability over the Past Millennium. Part II: Reconstructions and Implications, Journal of Climate, 26(7), 2329-2352.

The problem of modeling interaction between processes with different time scales is very important in geoscience. In this report, we propose a new form of empirical evolution operator model based on the analysis of multiple time series representing processes with different time scales. We assume that the time series are given on the same time interval.To construct the model, we extend the previously developed general form of nonlinear stochastic model based on artificial neural networks and designed for the case of time series with constant sampling interval [1]. This sampling interval is related to the main time scale of the process under consideration, which is described by the deterministic component of the model, while the faster time scales are modeled by its stochastic component, possibly depending on the system’s state. This model also includes slower processes in the form of weak time-dependence, as well as external forcing. The structure of the model is optimized using Bayesian approach [1]. The model has proven its efficiency in a number of applications [2-4].The idea of modeling time series with different time scales is to formulate the above-described model individually for each time scale, and then to include the parameterized influence of the other time scales in it. Particularly, the influence of “slower” time series is included in the form of parameter trends, and the influence of “faster” time series is included by time-averaging their statistics. The algorithm and first results of comparison between the new model and the model without cross-interactions will be discussed.The work was supported by the Russian Science Foundation (Grant No. 20-62-46056).1. Gavrilov, A., Loskutov, E., & Mukhin, D. (2017). Bayesian optimization of empirical model with state-dependent stochastic forcing. Chaos, Solitons & Fractals, 104, 327–337. http://doi.org/10.1016/j.chaos.2017.08.0322. Mukhin, D., Kondrashov, D., Loskutov, E., Gavrilov, A., Feigin, A., & Ghil, M. (2015). Predicting Critical Transitions in ENSO models. Part II: Spatially Dependent Models. Journal of Climate, 28(5), 1962–1976. http://doi.org/10.1175/JCLI-D-14-00240.13. Gavrilov, A., Seleznev, A., Mukhin, D., Loskutov, E., Feigin, A., & Kurths, J. (2019). Linear dynamical modes as new variables for data-driven ENSO forecast. Climate Dynamics, 52(3–4), 2199–2216. http://doi.org/10.1007/s00382-018-4255-74. Mukhin, D., Gavrilov, A., Loskutov, E., Kurths, J., & Feigin, A. (2019). Bayesian Data Analysis for Revealing Causes of the Middle Pleistocene Transition. Scientific Reports, 9(1), 7328. http://doi.org/10.1038/s41598-019-43867-3

We use the data-driven stochastic model for reconstruction of variability of the North Atlantic Oscillation (NAO) and its response to anthropogenic forcing. We apply the data-driven model to both the data produced by INM RAS Climate Model and NCEP / NCAR reanalysis data. The data-driven model reproduces well the characteristic statistical properties of the NAO index, such as skewness. We predict the NAO variability in the 21st century under various scenarios of anthropogenic CO2 emissions using data-driven model. The study was supported by the Russian Science Foundation (grant No. 9-42-04121).

An analysis of the climate system is usually complicated by its very high dimensionality and its nonlinearity which impedes spatial and time scale separation. An even more difficult problem is to obtain separate estimates of the climate system’s response to external forcing (e.g. anthropogenic emissions of greenhouse gases and aerosols) and the contribution of the climate system’s internal variability into recent climate trends. Identification of spatiotemporal climatic patterns representing forced signals and internal variability in global climate models (GCMs) would make it possible to characterize these patterns in the observed data and to analyze dynamical relationships between these two types of climate variability. In contrast with real climate observations, many GCMs are able to provide ensembles of many climate realizations under the same external forcing, with relatively independent initial conditions (e.g. LENS [1], MPI-GE [2], CMIP ensembles of 20th century climate). In this report, a recently developed method of empirical spatio-temporal data decomposition into linear dynamical modes (LDMs) [3] based on Bayesian approach, is modified to address the problem of self-consistent separation of the climate system internal variability modes and the forced response signals in such ensembles. The LDM method provides the time series of principal components and corresponding spatial patterns; in application to an ensemble of realizations, it determines both time series of the internal variability modes of current realization and the time series of forced response (defined as signal shared by all realizations). The advantage of LDMs is the ability to take into account the time scales of the system evolution better than some other linear techniques, e.g. traditional empirical orthogonal function decomposition. Furthermore, the modified ensemble LDM (E-LDM) method is designed to determine the optimal number of principal components and to distinguish their time scales for both internal variability modes and forced response signals. The technique and results of applying LDM method to different GCM ensemble realizations will be presented and discussed. This research was supported by the Russian Science Foundation (Grant No. 18-12-00231). [1] Kay, J. E., Deser, C., Phillips, A., Mai, A., Hannay, C., Strand, G., Arblaster, J., Bates, S., Danabasoglu, G., Edwards, J., Holland, M. Kushner, P., Lamarque, J.-F., Lawrence, D., Lindsay, K., Middleton, A., Munoz, E., Neale, R., Oleson, K., Polvani, L., and M. Vertenstein (2015), The Community Earth System Model (CESM) Large Ensemble Project: A Community Resource for Studying Climate Change in the Presence of Internal Climate Variability, Bulletin of the American Meteorological Society, doi: 10.1175/BAMS-D-13-00255.1, 96, 1333-1349 [2] Maher, N., Milinski, S., Suarez-Gutierrez, L., Botzet, M., Dobrynin, M., Kornblueh, L., Kröger, J., Takano, Y., Ghosh, R., Hedemann, C., Li, C., Li, H., Manzini, E., Notz, N., Putrasahan, D., Boysen, L., Claussen, M., Ilyina, T., Olonscheck, D., Raddatz, T., Stevens, B. and Marotzke, J. (2019). The Max Planck Institute Grand Ensemble: Enabling the Exploration of Climate System Variability. Journal of Advances in Modeling Earth Systems, 11, 1-21. https://doi.org/10.1029/2019MS001639 [3] Gavrilov, A., Kravtsov, S., Mukhin, D. (2020). Analysis of 20th century surface air temperature using linear dynamical modes. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(12), 123110. https://doi.org/10.1063/5.0028246

The global climate system is an aggregate of a huge number of interacting components, each having an intrinsic time scale. Such a complex dynamical system demonstrates nontrivial behavior and can exhibit a variety of possible modes of evolution. Gradual change of the parameters of the global climate system can lead to transitions (e.g., the Mid-Pleistocene Transition or to abrupt climate changes) from the observed to a new mode. In this work, we investigate the stability of the global climate system against strong sudden perturbations in the last 2.5 million years. This case is fundamentally different from the small perturbations case: in particularly, the system response cannot be described by a linearized evolution operator. To estimate the climate system’s nonlinear stability during the last 2.5 million years, we use a nonlinear data-driven model of climate dynamics in Pleistocene [1] and basin stability criterion [2]. Our results indicate that the stabilityof the Pleistocene climate to large perturbations decreases with time: past climates being much more stable compared to the present one. This work was supported by RFBR grant 19-02-00502. 1. D. Mukhin, A. Gavrilov, E. Loskutov, J. Kurths, A. Feigin. “Bayesian Data Analysis for Revealing Causes of the Middle Pleistocene Transition”. ScientificReports, 9 7328 (2019). 2. V. Klinshov, S. Kirillov, J. Kurths, V. Nekorkin. “Interval stability for complex systems”. New Journal of Physics, v. 20, p. 043040.

Purpose. Purpose of this article is to review recent results (over the past three years) obtained at the Institute of Applied Physics (IAP RAS) relating of applications of the method for constructing optimal empirical models to climatic systems. Methods. This method, developed by the authors of the article, includes the construction of reduced models of the system under study in the form of random dynamical systems. In combination with Bayesian optimization of the model structure, this method allows us to reconstruct statistically justified laws underlying the observed dynamics. Results. The article describes results of applying this method to modeling three climatic subsystems corresponding to different time scales: the Pleistocene climate characterized by glacial cycles, El Niño – Southern Oscillation in the modern climate – a phenomenon with a scale of the order of a year, and the climate of the tropical Pacific Ocean on a centennial scale. Conclusions. Based on the presented results, it can be concluded that the method used for constructing optimal models is a useful tool for verifying the mechanisms underlying the observed climatic variability, e.g., analyzing the response of the system to external signals.

In this work we investigate connection between mid-latitude atmosphere and principal modes of global sea surface temperature (SST). For this purpose, we apply the linear dynamical mode decomposition to the monthly SST anomalies: the obtained modes describe the Pacific Decadal Oscillation and the El-Niño Southern Oscillation (ENSO). We analyze the regression of atmospheric characteristics on the time-shifted modes and detect bidirectional connections between ENSO and the mid-latitude atmosphere up to one-year delay. The work was carried out within the framework of the Program for the Development of the Regional Scientifc and Educational Mathematical Center "Mathematics of Future Technologies", project #075-02-2020-1483/1.

Interseasonal forecasting of El Niño Southern Oscillation (ENSO), which is traditionally based on data of tropical sea surface temperatures (SST), is in high demand due to the impacts of ENSO on regional climatic conditions around the world as well as the global climate. Improvements in the quality of data in recent decades have led to the active use of statistical ENSO models, which compete with physical models in predictive power. The main disadvantage of statistical forecasts is the pronounced seasonal growth of uncertainty when predicting the upcoming summer-fall ENSO conditions from winter-spring months (so called the spring predictability barrier (SPB)). Recent studies show that Pacific atmospheric circulation anomalies in winter-spring may have a long-term impact on the summer tropical climate via the SST footprint. Here, we infer an index based on sea level pressure (SLP) data from February-March in a single area surrounding Hawaii, and show that this area is the most informative part of the large SLP pattern initiating the SST footprinting mechanism. We define the Hawaiian index (HI) as the mean SLP anomalies in the region (130N-190N, 1500W-1600W) averaged over February-March and demonstrate that the statistical AR model of the Niño 3.4 index taking the HI as a forcing is better in the Bayesian sense and delivers significantly better multimonth predictions. In fact, the HI forcing in the model substantially lowers the SPB and hence increases the predictability of the whole June-May ENSO cycle for forecasts starting in spring. Thus, we can recommend that modelers test the HI as an additional predictor in statistical ENSO models.This research was supported by the Russian Science Foundation (Contract 19-42-04121)

According to recent study [1], the current state-of-the-art climate models lack the substantial part of internal multidecadal climate signal which is observed in the 20th century surface air temperature reanalysis data as a global stadium wave (GSW). In the presented work we further investigate this phenomenon using the recently developed method [2] of empirical spatio-temporal data decomposition into linear dynamical modes (LDMs). The important property of LDMs is their ability to take into account the time scales of the system evolution (they are extracted from observed dataset by the Bayesian optimization technique) better than some other linear techniques, e.g. traditional empirical orthogonal function decomposition. Like any linear decomposition, it provides the time series of principal components and corresponding spatial patterns.We modify the initially developed LDM decomposition to make it possible to take into account a prescribed external forcing (like CO2 emissions, sun activity etc.) and then find part of variability which may be considered as an internal climate dynamics decomposed into set of modes with different time scales, and hence may be helpful in GSW interpretation. The results of applying the method to the 20th century surface air temperature with different ways of forcing inclusion will be presented and discussed.1. Kravtsov, S., Grimm, C., & Gu, S. (2018). Global-scale multidecadal variability missing in state-of-the-art climate models. Npj Climate and Atmospheric Science, 1(1), 34. https://doi.org/10.1038/s41612-018-0044-62. Gavrilov, A., Seleznev, A., Mukhin, D., Loskutov, E., Feigin, A., & Kurths, J. (2018). Linear dynamical modes as new variables for data-driven ENSO forecast. Climate Dynamics. https://doi.org/10.1007/s00382-018-4255-7

We investigate the Middle Pleistocene Transition (MPT) - a rapid change in the periodicity of the Pleistocene glacial cycles from 41 kyr to about 100 kyr, which occurred about a million years ago - using the data-driven model [1]. Here we estimate stability of the model using a novel concept of interval stability [2-4], referring to the behavior of the perturbed model during a finite time interval. In a few words we define the class of 'safe' perturbations after which the system (our data-driven model) returns back to the initial dynamical regime and 'unsafe' perturbation of minimal amplitude needed to disrupt the system. We demonstrate that the MPT is likely associated with decreasing of the climate system's interval stability to rapid disturbances (millennial and shorter). This confirms the statement made in the paper [1] that the main factor in the onset of the long-period glacial cycles is strongly nonlinear oscillations induced by the short-scale variability. 1. D. Mukhin, A. Gavrilov, E. Loskutov, J. Kurths, A. Feigin. Bayesian Data Analysis for Revealing Causes of the Middle Pleistocene Transition. Scientific Reports, 9 7328 (2019). 2. P. Menck, J. Heitzig, N. Marwan, J. Kurths. How basin stability complements the linear-stability paradigm. Nature Phys, 9 89–92 (2013). 3. V. Klinshov, V. Nekorkin, J. Kurths. Stability threshold approach for complex dynamical systems. New Journal Physics, 18 013004 (2016). 4. V. Klinshov, S. Kirillov, J. Kurths, V. Nekorkin. Interval stability for complex systems. New Journal Physics, 18 013004 (2018).