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A purely data‐driven and transformer‐based model with a novel self‐attention mechanism (3D‐Geoformer) is used to make predictions by adopting a rolling predictive manner similar to that in dynamical coupled models. The 3D‐Geoformer yields a successful prediction of the 2021 second‐year cooling conditions that followed the 2020 La Niña event, including covarying anomalies of surface wind stress and three‐dimensional (3D) upper‐ocean temperature, the reoccurrence of negative subsurface temperature anomalies in the eastern equatorial Pacific and a corresponding turning point of sea surface temperature (SST) evolution in mid‐2021. The reasons for the successful prediction with interpretability are explored comprehensively by performing sensitivity experiments with modulating effects on SST due to wind and subsurface thermal forcings being separately considered in the input predictors for prediction. A comparison is also conducted with physics‐based modeling, illustrating the suitability and effectiveness of 3D‐Geoformer as a new platform for El Niño and Southern Oscillation studies. Plain Language Summary The tropical Pacific experienced the prolonged cooling conditions during 2020–2022 (often called a triple La Niña), which exerted great impacts on the weather and climate globally. However, physics‐derived coupled models still have difficulty in accurately making long‐lead real‐time predictions for sea surface temperature (SST) evolution in the tropical Pacific. With the rapid development of deep learning‐based modeling, purely data‐driven models provide an innovative way for SST predictions. Here, a transformer‐based deep learning model is used to evaluate its performance in predicting the evolution of SST in the tropical Pacific during 2020–2022 and explore process representations that are important for SST evolution during 2021, including subsurface thermal effect and surface wind forcing on SST, the crucial factors determining the second‐year prolonged La Niña conditions and turning point of SST evolution. A comparison is made between the completely differently constructed physics‐derived dynamical coupled model and the pure‐data driven deep learning model, showing they both can be used for predictions of SST evolution in the 2021 second‐year cooling conditions. This indicates that it is necessary to adequately represent the thermocline feedback in predictive models, either in dynamical coupled models or purely data‐driven models, so that El Niño and Southern Oscillation predictions can be improved. Key Points A transformer‐based deep learning model is used for El Niño‐Southern Oscillation multivariate prediction in a rolling predictive manner The purely data‐driven model successfully predicts the 2021 second‐year La Niña and turning point of temperature evolution in mid‐2021 Applications of purely data‐driven model for process representations and understanding are demonstrated as in dynamical coupled models

The Last Glacial Maximum (LGM; 21 ka BP) was the most recent glacial period when the global ice sheet volume was at a maximum. Therefore, the LGM can be used to investigate atmospheric dynamics under a climate that differed significantly from the present. This study quantitatively compares pollen records of boreal summer (June–July–August) precipitation with the PMIP3 LGM simulations. The data-model comparison shows an overall agreement on a drier than pre-industrial East Asian summer monsoon (EASM) climate. Nevertheless, 17 out of 55 records show a regional precipitation increase that is also simulated over the additional land mass area due to sea level drop. The thermodynamic and dynamic responses are analyzed to explain a drier LGM EASM as a combination of these two antagonistic mechanisms. Relatively low atmospheric moisture content was the main thermodynamic control on the lower LGM (relative to pre-industrial levels) EASM precipitation amounts in both the reconstructions and the models. In contrast, two dynamic processes in relation to stationary eddy activity act to increase EASM precipitation regionally in records and simulations, respectively. Precipitation increase in records is explained by dynamic enhancement of the horizontal moisture transport, while dynamic enhancement of the vertical moisture transport leads to simulated precipitation increase over the specific region where landmass was exposed during LGM along continental coastlines of China due to significant drop in sea level (relative to pre-industrial levels). Overall, the opposing effects of thermodynamic and dynamic processes on precipitation during the LGM provide a means to reconcile the spatial heterogeneity of recorded precipitation changes in sign, although data-model comparison suggests that the simulated dynamic wetting mechanism is too weak relative to the thermodynamic drying mechanism over data-model disagreement regions.

Four state-of-the-science numerical weather prediction (NWP) models were used to perform mountain wave (MW)-resolving hindcasts over the Drake Passage of a 10-day period in 2010 with numerous observed MW cases. The Integrated Forecast System (IFS) and the Icosahedral Nonhydrostatic (ICON) model were run at Δ x ≈ 9 and 13 km globally. The Weather Research and Forecasting (WRF) Model and the Met Office Unified Model (UM) were both configured with a Δ x = 3-km regional domain. All domains had tops near 1 Pa ( z ≈ 80 km). These deep domains allowed quantitative validation against Atmospheric Infrared Sounder (AIRS) observations, accounting for observation time, viewing geometry, and radiative transfer. All models reproduced observed middle-atmosphere MWs with remarkable skill. Increased horizontal resolution improved validations. Still, all models underrepresented observed MW amplitudes, even after accounting for model effective resolution and instrument noise, suggesting even at Δ x ≈ 3-km resolution, small-scale MWs are underresolved and/or overdiffused. MW drag parameterizations are still necessary in NWP models at current operational resolutions of Δ x ≈ 10 km. Upper GW sponge layers in the operationally configured models significantly, artificially reduced MW amplitudes in the upper stratosphere and mesosphere. In the IFS, parameterized GW drags partly compensated this deficiency, but still, total drags were ≈6 times smaller than that resolved at Δ x ≈ 3 km. Meridionally propagating MWs significantly enhance zonal drag over the Drake Passage. Interestingly, drag associated with meridional fluxes of zonal momentum (i.e., ) were important; not accounting for these terms results in a drag in the wrong direction at and below the polar night jet.

Proxy‐based reconstructions of long‐term Atlantic tropical cyclone (TC) variability reveal low‐frequency oscillations in regional TC landfalls over the Common Era. However, the limited spatial coverage and increased uncertainty of the proxy records complicates assessments of this feature. Here we present a new multi‐ensemble set of synthetic TCs downscaled from the Last Millennium Reanalysis project, which is based on sea surface temperatures that more accurately reflect past conditions. Throughout ensemble members, there are coherent multi‐centennial shifts in landfalls with persistent intervals of increased (decreased) occurrence along the eastern US concurrent with inverse activity in the southwest Caribbean and Gulf of Mexico, associated with basin‐scale redistributions of storm tracks. The emergent TC‐dipole from modeled climate provides context and support for its presence within proxy‐reconstructions. Furthermore, dipole recurrence across ensembles demonstrates that it arises from sea surface temperature‐informed climate processes. However, timing differences between ensembles indicate that transient atmospheric variability influences dipole position.

Numerical weather, climate, or Earth system models involve the coupling of components. At a broad level, these components can be classified as the resolved fluid dynamics, unresolved fluid dynamical aspects (i.e., those represented by physical parameterizations such as subgrid-scale mixing), and nonfluid dynamical aspects such as radiation and microphysical processes. Typically, each component is developed, at least initially, independently. Once development is mature, the components are coupled to deliver a model of the required complexity. The implementation of the coupling can have a significant impact on the model. As the error associated with each component decreases, the errors introduced by the coupling will eventually dominate. Hence, any improvement in one of the components is unlikely to improve the performance of the overall system. The challenges associated with combining the components to create a coherent model are here termed physics–dynamics coupling. The issue goes beyond the coupling between the parameterizations and the resolved fluid dynamics. This paper highlights recent progress and some of the current challenges. It focuses on three objectives: to illustrate the phenomenology of the coupling problem with references to examples in the literature, to show how the problem can be analyzed, and to create awareness of the issue across the disciplines and specializations. The topics addressed are different ways of advancing full models in time, approaches to understanding the role of the coupling and evaluation of approaches, coupling ocean and atmosphere models, thermodynamic compatibility between model components, and emerging issues such as those that arise as model resolutions increase and/or models use variable resolutions.

We present results from an ensemble of eight climate models, each of which has carried out simulations of the early Eocene climate optimum (EECO, ∼ 50 million years ago). These simulations have been carried out in the framework of the Deep-Time Model Intercomparison Project (DeepMIP; http://www.deepmip.org, last access: 10 January 2021); thus, all models have been configured with the same paleogeographic and vegetation boundary conditions. The results indicate that these non-CO2 boundary conditions contribute between 3 and 5 ∘C to Eocene warmth. Compared with results from previous studies, the DeepMIP simulations generally show a reduced spread of the global mean surface temperature response across the ensemble for a given atmospheric CO2 concentration as well as an increased climate sensitivity on average. An energy balance analysis of the model ensemble indicates that global mean warming in the Eocene compared with the preindustrial period mostly arises from decreases in emissivity due to the elevated CO2 concentration (and associated water vapour and long-wave cloud feedbacks), whereas the reduction in the Eocene in terms of the meridional temperature gradient is primarily due to emissivity and albedo changes owing to the non-CO2 boundary conditions (i.e. the removal of the Antarctic ice sheet and changes in vegetation). Three of the models (the Community Earth System Model, CESM; the Geophysical Fluid Dynamics Laboratory, GFDL, model; and the Norwegian Earth System Model, NorESM) show results that are consistent with the proxies in terms of the global mean temperature, meridional SST gradient, and CO2, without prescribing changes to model parameters. In addition, many of the models agree well with the first-order spatial patterns in the SST proxies. However, at a more regional scale, the models lack skill. In particular, the modelled anomalies are substantially lower than those indicated by the proxies in the southwest Pacific; here, modelled continental surface air temperature anomalies are more consistent with surface air temperature proxies, implying a possible inconsistency between marine and terrestrial temperatures in either the proxies or models in this region. Our aim is that the documentation of the large-scale features and model–data comparison presented herein will pave the way to further studies that explore aspects of the model simulations in more detail, for example the ocean circulation, hydrological cycle, and modes of variability, and encourage sensitivity studies to aspects such as paleogeography, orbital configuration, and aerosols.

In this paper, we study periodic solutions in distribution of mean-field stochastic differential equations. We introduce the notion of upper and lower solutions of mean-field stochastic differential equations. With the help of the comparison principle, we prove the existence of periodic solutions in distribution.

The mid-Holocene (MH; 6 ka) is one of the benchmark periods for the Paleoclimate Modeling Intercomparison Project (PMIP) and provides a unique opportunity to study monsoon dynamics and orbital forcing (i.e., mostly precession) that differ significantly from the present day. We conducted a data–model comparison along with a dynamic analysis to investigate monsoonal (i.e., East Asian summer monsoon; EASM) precipitation changes over East Asia during the MH. We used the three phases of the PMIP simulations for the MH, and quantitatively compare model results with pollen-based climate records. The data–model comparison shows an overall increase in the summer monsoon precipitation, except a local decrease during the MH. Decomposition of the moisture budget into thermodynamic and dynamic components allows us to assess their relative role in controlling EASM precipitation during the MH, and to investigate the precipitation changes obtained from pollen records in terms of physical processes. We show that the dynamic effect, rather than the thermodynamic effect, is the dominant control in increased EASM precipitation during the MH in both the proxy records and models. The dynamic increase in precipitation results mainly from the enhancement of horizontal monsoonal moisture transport that is caused by intensified stationary eddy horizontal circulation over East Asia. In addition, a cloud-related cooling effect reduced the thermodynamic contribution to the increase in EASM precipitation during the MH.

Many applications dealing with the mathematical formulation associated with real-order systems bring long memory complexity, and the subsequent system analysis seems very puzzling and challenging. The settling of random initial-time often limits simple mathematical analysis of many real-order nonlinear dynamic and control systems. This paper formulates some new theories of nonlinear non-autonomous real-order systems attached with random initial-time in the sense of Caputo derivative. A fundamental stability question often arises: can we find reasonable conditions that provide limiting behavior of non-trivial trajectories of such systems to some constant equilibrium vector? We introduce some new comparison results that deal with the construction of potential Metzler matrix, non-negative matrix and non-negative system that enables new order-dependent criteria for concluding non-trivial trajectory responses to constant zero equilibrium vector embedded within such systems. A novel general theorem is put forward which, in short, states that if a relative asymptotic stable system is constructed, it is possible to conclude the limiting behavior of the original system. In the end, some new emerging applications dealing with the problems of equilibrium analysis of dynamic models are discussed by using the proposed results. An engineering control problem demonstrating observer design to a new class of systems is proposed including the use of time-varying linear state feedback control law to illustrate the importance of the theoretical results.

Bayesian model selection (BMS) is a powerful method for determining the most likely among a set of competing hypotheses about the mechanisms that generated observed data. BMS has recently found widespread application in neuroimaging, particularly in the context of dynamic causal modelling (DCM). However, so far, combining BMS results from several subjects has relied on simple (fixed effects) metrics, e.g. the group Bayes factor (GBF), that do not account for group heterogeneity or outliers. In this paper, we compare the GBF with two random effects methods for BMS at the between-subject or group level. These methods provide inference on model-space using a classical and Bayesian perspective respectively. First, a classical (frequentist) approach uses the log model evidence as a subject-specific summary statistic. This enables one to use analysis of variance to test for differences in log-evidences over models, relative to inter-subject differences. We then consider the same problem in Bayesian terms and describe a novel hierarchical model, which is optimised to furnish a probability density on the models themselves. This new variational Bayes method rests on treating the model as a random variable and estimating the parameters of a Dirichlet distribution which describes the probabilities for all models considered. These probabilities then define a multinomial distribution over model space, allowing one to compute how likely it is that a specific model generated the data of a randomly chosen subject as well as the exceedance probability of one model being more likely than any other model. Using empirical and synthetic data, we show that optimising a conditional density of the model probabilities, given the log-evidences for each model over subjects, is more informative and appropriate than both the GBF and frequentist tests of the log-evidences. In particular, we found that the hierarchical Bayesian approach is considerably more robust than either of the other approaches in the presence of outliers. We expect that this new random effects method will prove useful for a wide range of group studies, not only in the context of DCM, but also for other modelling endeavours, e.g. comparing different source reconstruction methods for EEG/MEG or selecting among competing computational models of learning and decision-making.