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We show that a known condition for having rough basin boundaries in bistable 2D maps holds for high-dimensional bistable systems that possess a unique nonattracting chaotic set embedded in their basin boundaries. The condition for roughness is that the cross-boundary Lyapunov exponent $\lambda_x$ {\bfac on the nonattracting set} is not the maximal one. Furthermore, we provide a formula for the generally noninteger co-dimension of the rough basin boundary, which can be viewed as a generalization of the Kantz-Grassberger formula. This co-dimension that can be at most unity can be thought of as a partial co-dimension, and, so, it can be matched with a Lyapunov exponent. We show {\bfac in 2D noninvertible- and 3D invertible minimal models,} that, formally, it cannot be matched with $\lambda_x$. Rather, the partial dimension $D_0^{(x)}$ that $\lambda_x$ is associated with in the case of rough boundaries is trivially unity. Further results hint that the latter holds also in higher dimensions. This is a peculiar feature of rough fractals. Yet, $D_0^{(x)}$ cannot be measured via the uncertainty exponent along a line that traverses the boundary. Indeed, one cannot determine whether the boundary is a rough or a filamentary fractal by measuring fractal dimensions. Instead, one needs to measure both the maximal and cross-boundary Lyapunov exponents numerically or experimentally.

Dynamical systems are often subject to forcing or changes in their governing parameters and it is of interest to studyhow this affects their statistical properties. A prominent real-life example of this class of problems is the investigationof climate response to perturbations. In this respect, it is crucial to determine what the linear response of a system isas a quantification of sensitivity. Alongside previous work, here we use the transfer operator formalism to study theresponse and sensitivity of a dynamical system undergoing perturbations. By projecting the transfer operator onto asuitable finite dimensional vector space, one is able to obtain matrix representations which determine finite Markovprocesses. Further, using perturbation theory for Markov matrices, it is possible to determine the linear and nonlinearresponse of the system given a prescribed forcing. Here, we suggest a methodology which puts the scope on theevolution law of densities (the Liouville/Fokker-Planck equation), allowing to effectively calculate the sensitivity andresponse of two representative dynamical systems.

Providing efficient and accurate parameterizations for model reduction is a key goal in many areas of science and technology. Here, we present a strong link between data-driven and theoretical approaches to achieving this goal. Formal perturbation expansions of the Koopman operator allow us to derive general stochastic parameterizations of weakly coupled dynamical systems. Such parameterizations yield a set of stochastic integrodifferential equations with explicit noise and memory kernel formulas to describe the effects of unresolved variables. We show that the perturbation expansions involved need not be truncated when the coupling is additive. The unwieldy integrodifferential equations can be recast as a simpler multilevel Markovian model, and we establish an intuitive connection with a generalized Langevin equation. This connection helps setting up a parallelism between the top-down, equation-based methodology herein and the well-established empirical model reduction (EMR) methodology that has been shown to provide efficient dynamical closures to partially observed systems. Hence, our findings, on the one hand, support the physical basis and robustness of the EMR methodology and, on the other hand, illustrate the practical relevance of the perturbative expansion used for deriving the parameterizations.\ud Parameterizations aim to reduce the complexity of high-dimensional dynamical systems. Here, a theory-based and a data-driven approach for the parameterization of coupled systems are compared, showing that both yield the same stochastic multilevel structure. The results provide very strong support to the use of empirical methods in model reduction and clarify the practical relevance of the proposed theoretical framework.

Geoengineering can control only some climatic variables but not others, resulting in side-effects. We investigate in an intermediate-complexity climate model the applicability of linear response theory (LRT) to the assessment of a geoengineering method. This application of LRT is twofold. First, our objective (O1) is to assess only the best possible geoengineering scenario by looking for a suitable modulation of solar forcing that can cancel out or otherwise modulate a climate change signal resulting from a rise in CO2 alone. Here we consider only the cancellation of the expected global mean surface air temperature. It is a straightforward inverse problem for this solar forcing, and, considering an infinite time period, we use LRT to provide the solution in the frequency domain in closed form. We provide procedures suitable for numerical implementation that apply to finite time periods too. Second, to be able to use LRT to quantify side-effects, the response with respect to uncontrolled observables, such as regional must be approximately linear. Our objective (O2) here is to assess the linearity of the response. We find that under geoengineering in the sense of (O1) the asymptotic response of the globally averaged temperature is actually not zero. This is due to an inaccurate determination of the linear susceptibilities. The error is due to a significant quadratic nonlinearity of the response. This nonlinear contribution can be easily removed, which results in much better estimates of the linear susceptibility, and, in turn, in a fivefold reduction in the global average surface temperature under geoengineering. This correction dramatically improves also the agreement of the spatial patterns of the predicted and of the true response. However, such an agreement is not perfect and is worse in the case of the precipitation patterns, as a result of greater degree of nonlinearity., Comment: 23 pages, 14 figures

We study the response to perturbations in the thermodynamic limit of a network of coupled identical agents undergoing a stochastic evolution which, in general, describes non-equilibrium conditions. All systems are nudged towards the common centre of mass. We derive Kramers-Kronig relations and sum rules for the linear susceptibilities obtained through mean field Fokker-Planck equations and then propose corrections relevant for the macroscopic case, which incorporates in a self-consistent way the effect of the mutual interaction between the systems. Such an interaction creates a memory effect. We are able to derive conditions determining the occurrence of phase transitions specifically due to system-to-system interactions. Such phase transitions exist in the thermodynamic limit and are associated with the divergence of the linear response but are not accompanied by the divergence in the integrated autocorrelation time for a suitably defined observable. We clarify that such endogenous phase transitions are fundamentally different from other pathologies in the linear response that can be framed in the context of critical transitions. Finally, we show how our results can elucidate the properties of the Desai-Zwanzig model and of the Bonilla-Casado-Morillo model, which feature paradigmatic equilibrium and non-equilibrium phase transitions, respectively., 27 pages, 4 figures, final accepted version

The Earth System Model Evaluation Tool (ESMValTool) is a community diagnostics and performance metrics tool designed to improve comprehensive and routine evaluation of Earth System Models (ESMs) participating in the Coupled Model Intercomparison Project (CMIP). It has undergone rapid development since the first release in 2016 and is now a well-tested tool that provides end-to-end provenance tracking to ensure reproducibility. It consists of an easy-to-install, well documented Python package providing the core functionalities (ESMValCore) that performs common pre-processing operations and a diagnostic part that includes tailored diagnostics and performance metrics for specific scientific applications. Here we describe large-scale diagnostics of the second major release of the tool that supports the evaluation of ESMs participating in CMIP Phase 6 (CMIP6). ESMValTool v2.0 includes a large collection of diagnostics and performance metrics for atmospheric, oceanic, and terrestrial variables for the mean state, trends, and variability. ESMValTool v2.0 also successfully reproduces figures from the evaluation and projections chapters of the Intergovernmental Panel on Climate Change (IPCC) Fifth Assessment Report (AR5) and incorporates updates from targeted analysis packages, such as the NCAR Climate Variability Diagnostics Package for the evaluation of modes of variability the Thermodynamic Diagnostic Tool (TheDiaTo) to evaluate the energetics of the climate system, as well as parts of AutoAssess that contains a mix of top-down performance metrics. The tool has been fully integrated into the Earth System Grid Federation (ESGF) infrastructure at the Deutsches Klima Rechenzentrum (DKRZ) to provide evaluation results from CMIP6 model simulations shortly after the output is published to the CMIP archive. A result browser has been implemented that enables advanced monitoring of the evaluation results by a broad user community at much faster timescales than what was possible in CMIP5.

Constructing efficient and accurate parameterizations of sub-grid scale processes is a central area of interest in the numerical modelling of geophysical fluids. Using a modified version of the two-level Lorenz '96 model, we present here a proof of concept of a scale-adaptive parameterization constructed using statistical mechanical arguments. By a suitable use of the Ruelle response theory and of the Mori-Zwanzig projection method, it is possible to derive explicitly a parameterization for the fast variables that translates into deterministic, stochastic and non-markovian contributions to the equations of motion of the variables of interest. We show that our approach is computationally parsimonious, has great flexibility, as it is explicitly scale-adaptive, and we prove that it is competitive compared to empirical ad-hoc approaches. While the parameterization proposed here is universal and can be easily analytically adapted to changes in the parameters' values by a simple rescaling procedure, the parameterization constructed with the ad-hoc approach needs to be recomputed each time the parameters of the systems are changed. The price of the higher flexibility of the method proposed here is having a lower accuracy in each individual case.

Southern Pakistan (Sindh) is one of the hottest regions in the world and is highly vulnerable to temperature extremes. In order to improve rural and urban planning, it is useful to gather information about the recurrence of temperature extremes. In this work, return levels of the daily maximum temperature Tmax are estimated, as well as the daily maximum wet-bulb temperature TWmax extremes. We adopt the peaks over threshold (POT) method, which has not yet been used for similar studies in this region. Two main datasets are analyzed: temperatures observed at nine meteorological stations in southern Pakistan from 1980 to 2013, and the ERA-Interim (ECMWF reanalysis) data for the nearest corresponding locations. The analysis provides the 2-, 5-, 10-, 25-, 50-, and 100-year return levels (RLs) of temperature extremes. The 90 % quantile is found to be a suitable threshold for all stations. We find that the RLs of the observed Tmax are above 50 °C at northern stations and above 45 °C at the southern stations. The RLs of the observed TWmax exceed 35 °C in the region, which is considered as a limit of survivability. The RLs estimated from the ERA-Interim data are lower by 3 to 5 °C than the RLs assessed for the nine meteorological stations. A simple bias correction applied to ERA-Interim data improves the RLs remarkably, yet discrepancies are still present. The results have potential implications for the risk assessment of extreme temperatures in Sindh.

We study the response of a simple quasi-geostrophic barotropic model of the atmosphere to various classes of perturbations affecting its forcing and its dissipation using the formalism of the Ruelle response theory. We investigate the geometry of such perturbations by constructing the covariant Lyapunov vectors of the unperturbed system and discover in one specific case - orographic forcing - a substantial projection of the forcing onto the stable directions of the flow. This results into a resonant response shaped as a Rossby-like wave that has no resemblance to the unforced variability in the same range of spatial and temporal scales. Such a climatic surprise corresponds to a violation of the fluctuation-dissipation theorem, in agreement with the basic tenets of nonequilibrium statistical mechanics. The resonance can be attributed to a specific group of rarely visited unstable periodic orbits of the unperturbed system. Our results reinforce the idea of using basic methods of nonequilibrium statistical mechanics and high-dimensional chaotic dynamical systems to approach the problem of understanding climate dynamics., Comment: 20 pages, 12 figures; final revision of the paper

The last decade has seen the success of stochastic parameterizations in short-term, medium-range and seasonal forecasts: operational weather centers now routinely use stochastic parameterization schemes to better represent model inadequacy and improve the quantification of forecast uncertainty. Developed initially for numerical weather prediction, the inclusion of stochastic parameterizations not only provides better estimates of uncertainty, but it is also extremely promising for reducing longstanding climate biases and relevant for determining the climate response to external forcing. This article highlights recent developments from different research groups which show that the stochastic representation of unresolved processes in the atmosphere, oceans, land surface and cryosphere of comprehensive weather and climate models (a) gives rise to more reliable probabilistic forecasts of weather and climate and (b) reduces systematic model bias. We make a case that the use of mathematically stringent methods for the derivation of stochastic dynamic equations will lead to substantial improvements in our ability to accurately simulate weather and climate at all scales. Recent work in mathematics, statistical mechanics and turbulence is reviewed, its relevance for the climate problem demonstrated, and future research directions outlined., Comment: 26 pages, 15 figures. Final published version

Extreme geophysical events are of crucial relevance to our daily life: they threaten human lives and cause property damage. To assess the risk and reduce losses, we need to model and probabilistically predict these events. Parametrizations are computational tools used in Earth system models, which are aimed at reproducing the impact of unresolved scales on resolved scales. The performance of parametrizations has usually been examined on typical events rather than on extreme events. In this paper we consider a modified version of the two-level Lorenz'96 model and investigate how two parametrizations of the fast degrees of freedom perform in terms of the representation of extreme events. One parametrization is constructed following Wilks (2005) and is constructed through an empirical fitting procedure; the other parametrization is constructed through the statistical mechanical approach proposed by Wouters and Lucarini (2012, 2013). The two strategies show different advantages and disadvantages. We discover that the agreement between parametrized models and true model is in general worse when looking at extremes rather than at the bulk of the statistics. The results suggest that stochastic parametrizations should be accurately and specifically tested against their performance on extreme events, as usual optimization procedures might neglect them., 21 pages and 17 figures

The Earth is well-known to be, in the current astronomical configuration, in a regime where two asymptotic states can be realised. The warm state we live in is in competition with the ice-covered snowball state. The bistability exists as a result of the positive ice-albedo feedback. In a previous investigation performed on a intermediate complexity climate model we have identified the unstable climate states (Melancholia states) separating the co-existing climates, and studied their dynamical and geometrical properties. The Melancholia states are ice-covered up to the mid-latitudes and attract trajectories initialised on the basins boundary. In this paper, we study how stochastically perturbing the parameter controlling the intensity of the incoming solar radiation impacts the stability of the climate. We detect transitions between the warm and the snowball state and analyse in detail the properties of the noise-induced escapes from the corresponding basins of attraction. We determine the most probable paths for the transitions and find evidence that the Melancholia states act as gateways, similarly to saddle points in an energy landscape., 5 pages, 4 figures; final PRL accepted version

Atmospheric ozone plays an important role on the temperature structure of the atmosphere. However, it has not been included in previous studies on the effect of an increasing solar radiation on the Earth's climate. Here we study the climate sensitivity to the presence/absence of ozone with an increasing solar forcing for the first time with a global climate model. We show that the warming effect of ozone increases both the humidity of the lower atmosphere and the surface temperature. Under the same solar irradiance, the mean surface temperature is 7 K higher than in an analog planet without ozone. Therefore, the moist greenhouse threshold, the state at which water vapor becomes abundant in the stratosphere, is reached at a lower solar irradiance (1572 Wm^-2 with respect to 1647 Wm^-2 in the case without ozone). Our results imply that ozone reduces the maximum solar irradiance at which Earth-like planets would remain habitable., Comment: 14 pages, 7 figures, accepted for publication in Icarus

Comment: 39 pages, 14 figures; improved discussion and referencing; several new/updated figures, We use a simple yet Earth-like atmospheric model to propose a new framework for understanding the mathematics of blocking events. Analysing error growth rates along a very long model trajectory, we show that blockings are associated with conditions of anomalously high instability of the atmosphere. Additionally, the lifetime of a blocking is positively correlated with the intensity of such an anomaly, against intuition. In the case of Atlantic blockings, predictability is especially reduced at the onset and decay of the blocking, while a relative increase of predictability is found in the mature phase, while the opposite holds for Pacific blockings, for which predictability is lowest in the mature phase. We associate blockings to a specific class of unstable periodic orbits (UPOs), natural modes of variability that cover the attractor of the system. The UPOs differ substantially in terms of instability, which explains the diversity of the atmosphere in terms predictability. The UPOs associated to blockings are indeed anomalously unstable, which leads to them being rarely visited. The onset of a blocking takes place when the trajectory of the system hops into the neighbourhood of one of these special UPOs. The decay takes place when the trajectory hops back to the neighbourhood of usual, less unstable UPOs associated with zonal flow. This justifies the classical Markov chains-based analysis of transitions between weather regimes. The existence of UPOs differing in the dimensionality of their unstable manifold indicates a very strong violation of hyperbolicity in the model, which leads to a lack of structural stability. We propose that this is could be a generic feature of atmospheric models and might be a fundamental cause behind difficulties in representing blockings for the current climate and uncertainties in predicting how their statistics will change as a result of climate change.

Carbon dioxide is one of the major contributors to the radiative forcing, increasing both the temperature and the humidity of Earth's atmosphere. If the stellar irradiance increases and water becomes abundant in the stratosphere of an Earth-like planet, it will be dissociated and the resultant hydrogen will escape from the atmosphere. This state is called the moist greenhouse threshold (MGT). Using a global climate model (GCM) of intermediate complexity, we explore how to identify this state for different CO2 concentrations and including the radiative effect of atmospheric ozone for the first time. We show that the moist greenhouse threshold correlates with the inflection point in the water vapor mixing ratio in the stratosphere and a peak in the climate sensitivity. For CO2 concentrations between 560 ppm and 200 ppm, the moist greenhouse threshold is reached at a surface temperature of 320 K. Despite the higher simplicity of our model, our results are consistent with similar simulations without ozone by complex GCMs, suggesting that they are robust indicators of the MGT. We discuss the implications for inner edge of the habitable zone as well as the water loss timescales for Earth analog planets., Comment: 17 pages, 10 figures, accepted for publication in ApJ

The first main result of Baladi et al (2017 Nonlinearity 30 1204–20) is modified as follows: For any θ in the Sobolev space , with and , the map is α-Holder continuous for all where is the strongest volume contraction and is the weakest contraction. This applies to (for all ) for h and g smooth and the Heaviside function, if a is not a critical value of g'.

We investigate the geometrical structure of instabilities in the two-scales Lorenz '96 model through the prism of Lyapunov analysis. Our detailed study of the full spectrum of covariant Lyapunov vectors reveals the presence of a slow manifold in tangent space, composed by a set of vectors with a significant projection on the slow degrees of freedom; they correspond to the smallest (in absolute sense) Lyapunov exponents and thereby to the longer time scales. We show that the dimension of this manifold is extensive in the number of both slow and fast degrees of freedom, and discuss its relationship with the results of a finite-size analysis of instabilities, supporting the conjecture that the slow-variable behavior is effectively determined by a non-trivial subset of degrees of freedom. More precisely, we show that the slow manifold corresponds to the Lyapunov spectrum region where fast and slow instability rates overlap, mixing their evolution into a set of vectors which simultaneously carry information on both scales. We suggest these results may pave the way for future applications to ensemble forecasting and data assimilation in weather and climate models.

Understanding under what conditions it is possible to construct equivalent ensembles is key to advancing our ability to connect microscopic and macroscopic properties of non-equilibrium statistical mechanics. In the case of fluid dynamical systems, one issue is to test whether different models for viscosity lead to the same macroscopic properties of the fluid systems in different regimes. Such models include, besides the standard choice of constant viscosity, cases where the time symmetry of the evolution equations is exactly preserved, as it must be in the corresponding microscopic systems, when available. Here a time-reversible dynamics is obtained by imposing the conservation of global observables. We test the equivalence of reversible and irreversible ensembles for the case of a multiscale shell model of turbulence. We verify that the equivalence is obeyed for the mean values of macroscopic observables, up to an error that vanishes as the system becomes more and more chaotic., 11 pages, 9 figures

Constructing accurate, flexible, and efficient parametrizations is one of the great challenges in the numerical modelling of geophysical fluids. We consider here the simple yet paradigmatic case of a Lorenz 84 model forced by a Lorenz 63 model and derive a parametrization using a recently developed statistical mechanical methodology based on the Ruelle response theory. We derive an expression for the deterministic and the stochastic component of the parametrization and we show that the approach allows for dealing seamlessly with the case of the Lorenz 63 being a fast as well as a slow forcing compared to the characteristic time scales of the Lorenz 84 model. We test our results using both standard metrics based on the moments of the variables of interest as well as Wasserstein distance between the projected measure of the original system on the Lorenz 84 model variables and the measure of the parametrized one. By testing our methods on reduced phase spaces obtained by projection, we find support to the idea that comparisons based on the Wasserstein distance might be of relevance in many applications despite the curse of dimensionality., 22 pages, 9 figures

The understanding of the statistical properties and of the dynamics of multistable systems is gaining more and more importance in a vast variety of scientific fields. This is especially relevant for the investigation of the tipping points of complex systems. Sometimes, in order to understand the time series of given observables exhibiting bimodal distributions, simple one-dimensional Langevin models are fitted to reproduce the observed statistical properties, and used to investing-ate the projected dynamics of the observable. This is of great relevance for studying potential catastrophic changes in the properties of the underlying system or resonant behaviours like those related to stochastic resonance-like mechanisms. In this paper, we propose a framework for encasing this kind of studies, using simple box models of the oceanic circulation and choosing as observable the strength of the thermohaline circulation. We study the statistical properties of the transitions between the two modes of operation of the thermohaline circulation under symmetric boundary forcings and test their agreement with simplified one-dimensional phenomenological theories. We extend our analysis to include stochastic resonance-like amplification processes. We conclude that fitted one-dimensional Langevin models, when closely scrutinised, may result to be more ad-hoc than they seem, lacking robustness and/or well-posedness. They should be treated with care, more as an empiric descriptive tool than as methodology with predictive power.

We study temporally persistent and spatially extended extreme events of temperature anomalies, i.e. heat waves and cold spells, using large deviation theory. To this end, we consider a simplified yet Earth-like general circulation model of the atmosphere and numerically estimate large deviation rate functions of near-surface temperature in the mid-latitudes. We find that, after a re-normalisation based on the integrated auto-correlation, the rate function one obtains at a given latitude by looking, locally in space, at long time averages agrees with what is obtained, instead, by looking, locally in time, at large spatial averages along the latitude. This is a result of scale symmetry in the spatial-temporal turbulence and of the fact that advection is primarily zonal. This agreement hints at the universality of large deviations of the temperature field. Furthermore, we discover that the obtained rate function is able to describe spatially extended and temporally persistent heat waves or cold spells, if we consider temporal averages of spatial averages over intermediate spatial scales. Finally, we find out that large deviations are relatively more likely to occur when looking at these spatial averages performed over intermediate scales, thus pointing to the existence of weather patterns associated to the low-frequency variability of the atmosphere. Extreme value theory is used to benchmark our results., Comment: 33 pages, 9 figures

Linear response theory has developed into a formidable set of tools for studying the forced behaviour of a large variety of systems - including out of equilibrium ones. In this paper we provide a new angle on the problem, by studying under which conditions it is possible to perform predictions on the response of a given observable of a system to perturbations, using one or more other observables of the same system as predictors, and thus bypassing the need to know all the details of the acting forcing. Thus, we break the rigid separation between forcing and response, which is key in linear response theory, and revisit the concept of causality. As a result, the surrogate Green functions one constructs for predicting the response of the observable of interest may have support that is not necessarily limited to the nonnegative time axis. This implies that not all observables are equally good as predictands when a given forcing is applied, as result of the properties of their corresponding susceptibility. In particular, problems emerge from the presence of complex zeros. We derive general explicit formulas that, in absence of such pathologies, allow one to reconstruct the response of an observable of interest to N independent forcings by using as predictors N other observables. We provide a thorough test of the theory and of the possible pathologies by using numerical simulations of the paradigmatic Lorenz '96 model. Our results are potentially relevant for problems like the reconstruction of data from proxy signals, like in the case of paleoclimate, and, in general, the analysis of signals and feedbacks in complex systems where our knowledge on the system is limited, as in neurosciences. Our technique might also be useful for reconstructing the response to forcings of a spatially extended system in a given location by looking at the response in a separate location., 26 pages, 7 Figures; Revised Version

The classical approach for studying atmospheric variability is based on defining a background state and studying the linear stability of the small fluctuations around such a state. Weakly non-linear theories can be constructed using higher order expansions terms. While these methods have undoubtedly great value for elucidating the relevant physical processes, they are unable to follow the dynamics of a turbulent atmosphere. We provide a first example of extension of the classical stability analysis to a non-linearly evolving quasi-geostrophic flow. The so-called covariant Lyapunov vectors (CLVs) provide a covariant basis describing the directions of exponential expansion and decay of perturbations to the non-linear trajectory of the flow. We use such a formalism to re-examine the basic barotropic and baroclinic processes of the atmosphere with a quasi-geostrophic beta-plane two-layer model in a periodic channel driven by a forced meridional temperature gradient $\Delta T$. We explore three settings of $\Delta T$, representative of relatively weak turbulence, well-developed turbulence, and intermediate conditions. We construct the Lorenz energy cycle for each CLV describing the energy exchanges with the background state. A positive baroclinic conversion rate is a necessary but not sufficient condition of instability. Barotropic instability is present only for few very unstable CLVs for large values of $\Delta T$. Slowly growing and decaying hydrodynamic Lyapunov modes closely mirror the properties of the background flow. Following classical necessary conditions for barotropic/baroclinic instability, we find a clear relationship between the properties of the eddy fluxes of a CLV and its instability. CLVs with positive baroclinic conversion seem to form a set of modes for constructing a reduced model of the atmosphere dynamics., Comment: 50 pages, 10 figures

The climate is a forced and dissipative nonlinear system featuring non-trivial dynamics of a vast range of spatial and temporal scales. The understanding of the climate's structural and multiscale properties is crucial for the provision of a unifying picture of its dynamics and for the implementation of accurate and efficient numerical models. We present some recent developments at the intersection between climate science, mathematics, and physics, which may prove fruitful in the direction of constructing a more comprehensive account of climate dynamics. We describe the Nambu formulation of fluid dynamics, and the potential of such a theory for constructing sophisticated numerical models of geophysical fluids. Then, we focus on the statistical mechanics of quasi-equilibrium flows in a rotating environment, which seems crucial for constructing a robust theory of geophysical turbulence. We then discuss ideas and methods suited for approaching directly the non-equilibrium nature of the climate system. First, we describe some recent findings on the thermodynamics of climate and characterize its energy and entropy budgets, and discuss related methods for intercomparing climate models and for studying tipping points. These ideas can also create a common ground between geophysics and astrophysics by suggesting general tools for studying exoplanetary atmospheres. We conclude by focusing on non-equilibrium statistical mechanics, which allows for a unified framing of problems as different as the climate response to forcings, the effect of altering the boundary conditions or the coupling between geophysical flows, and the derivation of parametrizations for numerical models., Comment: 44 pages, 22 figures. Major revision with respect to previous version. A glossary of terms is included at the end

We consider a smooth one-parameter family $t \to f_t$ of diffeomorphisms with compact transitive Axiom A attractors. Our first result (corrected) is that for any function $G$ in the Sobolev space $H^r_p$, with $p>1$ and $0

The provision of accurate methods for predicting the climate response to anthropogenic and natural forcings is a key contemporary scientific challenge. Using a simplified and efficient open-source general circulation model of the atmosphere featuring O($10^5$) degrees of freedom, we show how it is possible to approach such a problem using nonequilibrium statistical mechanics. Response theory allows one to practically compute the time-dependent measure supported on the pullback attractor of the climate system, whose dynamics is non-autonomous as a result of time-dependent forcings. We propose a simple yet efficient method for predicting - at any lead time and in an ensemble sense - the change in climate properties resulting from increase in the concentration of CO$_2$ using test perturbation model runs. We assess strengths and limitations of the response theory in predicting the changes in the globally averaged values of surface temperature and of the yearly total precipitation, as well as in their spatial patterns. The quality of the predictions obtained for the surface temperature fields is rather good, while in the case of precipitation a good skill is observed only for the global average. We also show how it is possible to define accurately concepts like the the inertia of the climate system or to predict when climate change is detectable given a scenario of forcing. Our analysis can be extended for dealing with more complex portfolios of forcings and can be adapted to treat, in principle, any climate observable. Our conclusion is that climate change is indeed a problem that can be effectively seen through a statistical mechanical lens, and that there is great potential for optimizing the current coordinated modelling exercises run for the preparation of the subsequent reports of the Intergovernmental Panel for Climate Change., Comment: 45 pages, 8 figures

We discuss applications of a recently developed method for model reduction based on linear response theory of weakly coupled dynamical systems. We apply the weak coupling method to simple stochastic differential equations with slow and fast degrees of freedom. The weak coupling model reduction method results in general in a non-Markovian system, we therefore discuss the Markovianization of the system to allow for straightforward numerical integration. We compare the applied method to the equations obtained through homogenization in the limit of large time scale separation between slow and fast degrees of freedom. We numerically compare the ensemble spread from a fixed initial condition, correlation functions and exit times from a domain. The weak coupling method gives more accurate results in all test cases, albeit with a higher numerical cost.

The climatology of major sources and pathways of moisture for three locales along the Hindu-Kush-Himalayan region are examined, by use of Lagrangian methods applied to the ERA-Interim dataset, over the period from 1980 to 2016 for both summer (JJA) and winter (NDJ) periods. We also investigate the major flooding events of 2010, 2013, and 2017 in Pakistan, Uttarakhand, and Kathmandu, respectively, and analyse a subset of the climatology associated with the 20 most significant rainfall events over each region of interest. A comparison is made between the climatology and extreme events, in the three regions of interest, during the summer monsoon period. For Northern Pakistan and Uttarakhand, the Indus basin plays the largest role in moisture uptake. Moisture is also gathered from Eastern Europe and Russia. Extreme events display an increased influence of sub-tropical weather systems, which manifest themselves through low-level moisture transport, predominantly from the Arabian sea and along the Gangetic plain. In the Kathmandu region, it is found that the major moisture sources come from the Gangetic plain, Arabian Sea, Red Sea, Bay of Bengal, and the Indus basin. In this case, extreme event pathways largely match those of the climatology, although an increased number of parcels originate from the western end of the Gangetic plain. These results provide insights into the rather significant influence of mid-latitudinal weather systems, even during the monsoon season, in defining the climatology of the Hindu-Kush-Himalaya region, as well as how extreme precipitation events in this region represent atypical moisture pathways. We propose a detailed investigation of how such water pathways are represented in climate models for the present climate conditions and in future climate scenarios, as this may be extremely relevant for understanding the impacts of climate change on the cryosphere and hydrosphere of the region.

An extensive analysis of an optically thin, dry atmosphere at different values of the thermal Rossby number R o and of the Taylor number F f is performed with a general circulation model by varying the rotation rate Ω and the surface drag τ in a wide parametric range. By using nonequilibrium thermodynamics diagnostics such as material entropy production, efficiency, meridional heat transport and kinetic energy dissipation we characterize in a new way the different circulation regimes. Baroclinic circulations feature high mechanical dissipation, meridional heat transport, material entropy production and are fairly efficient in converting heat into mechanical work. The thermal dissipation associated with the sensible heat flux is found to depend mainly on the surface properties, almost independent from the rotation rate and very low for quasi-barotropic circulations and regimes approaching equatorial super-rotation. Slowly rotating, axisymmetric circulations have the highest meridional heat transport. At high rotation rates and intermediate-high drag, atmospheric circulations are zonostrophic with very low mechanical dissipation, meridional heat transport and efficiency. When τ is interpreted as a tunable parameter associated with the turbulent boundary layer transfer of momentum and sensible heat, our results confirm the possibility of using the Maximum Entropy Production Principle as a tuning guideline in the range of values of Ω . This study suggests the effectiveness of using fundamental nonequilibrium thermodynamics for investigating the properties of planetary atmospheres and extends our knowledge of the thermodynamics of the atmospheric circulation regimes.

One of the most relevant weather regimes in the mid-latitudes atmosphere is the persistent deviation from the approximately zonally symmetric jet to the emergence of blocking patterns. Such configurations are usually connected to exceptional local stability properties of the flow which come along with an improved local forecast skills during the phenomenon. It is instead extremely hard to predict onset and decay of blockings. Covariant Lyapunov Vectors (CLVs) offer a suitable characterization of the linear stability of a chaotic flow, since they represent the full tangent linear dynamics by a covariant basis which explores linear perturbations at all time scales. Therefore, we assess whether CLVs feature a signature of the blockings. As a first step, we examine the CLVs for a quasi-geostrophic beta-plane 2-layer model in a periodic channel baroclinically driven by a meridional temperature gradient $\Delta T$. An orographic forcing enhances the emergence of localized blocked regimes. We detect the blocking events with a Tibaldi-Molteni scheme adapted to the periodic channel. When blocking occurs, the global growth rates of the fastest growing CLVs are significantly higher. Hence, against intuition, the circulation is globally more unstable in blocked phases. Such an increase in the finite time Lyapunov exponents with respect to the long term average is attributed to stronger barotropic and baroclinic conversion in the case of high temperature gradients, while for low values of \Delta T, the effect is only due to stronger barotropic instability. In order to determine the localization of the CLVs we compare the meridionally averaged variance of the CLVs during blocked and unblocked phases. We find that on average the variance of the CLVs is clustered around the center of blocking. These results show that the blocked flow affects all time scales and processes described by the CLVs., Comment: 23 pages, 10 figures

Multistability is a ubiquitous feature in systems of geophysical relevance and provides key challenges for our ability to predict a system's response to perturbations. Near critical transitions small causes can lead to large effects and - for all practical purposes - irreversible changes in the properties of the system. The Earth climate is multistable: present astronomical and astrophysical conditions support two stable regimes, the warm climate we live in, and a snowball climate, characterized by global glaciation. We first provide an overview of methods and ideas relevant for studying the climate response to forcings and focus on the properties of critical transitions. Following an idea developed by Eckhardt and co. for the investigation of multistable turbulent flows, we study the global instability giving rise to the snowball/warm multistability in the climate system by identifying the climatic edge state, a saddle embedded in the boundary between the two basins of attraction of the stable climates. The edge state attracts initial conditions belonging to such a boundary and is the gate facilitating noise-induced transitions between competing attractors. We use a simplified yet Earth-like climate model constructed by coupling a primitive equations model of the atmosphere with a simple diffusive ocean. We refer to the climatic edge states as Melancholia states. We study their dynamics, their symmetry properties, and we follow a complex set of bifurcations. We find situations where the Melancholia state has chaotic dynamics. In these cases, the basin boundary between the two basins of attraction is a strange geometric set with a nearly zero codimension, and relate this feature to the time scale separation between instabilities occurring on weather and climatic time scales. We also discover a new stable climatic state characterized by non-trivial symmetry properties., 43 pages; 15 figures. Revised version with additional figures, extended discussion, and new results on the boundary between the basins of attraction

In this diagnostic study we analyze changes of rainfall seasonality and dry spells by the end of the twenty-first century under the most extreme IPCC5 emission scenario (RCP8.5) as projected by twenty-four coupled climate models contributing to Coupled Model Intercomparison Project 5 (CMIP5). We use estimates of the centroid of the monthly rainfall distribution as an index of the rainfall timing and a threshold-independent, information theory-based quantity such as relative entropy (RE) to quantify the concentration of annual rainfall and the number of dry months and to build a monsoon dimensionless seasonality index (DSI). The RE is projected to increase, with high inter-model agreement over Mediterranean-type regions—southern Europe, northern Africa and southern Australia—and areas of South and Central America, implying an increase in the number of dry days up to 1month by the end of the twenty-first century. Positive RE changes are also projected over the monsoon regions of southern Africa and North America, South America. These trends are consistent with a shortening of the wet season associated with a more prolonged pre-monsoonal dry period. The extent of the global monsoon region, characterized by large DSI, is projected to remain substantially unaltered. Centroid analysis shows that most of CMIP5 projections suggest that the monsoonal annual rainfall distribution is expected to change from early to late in the course of the hydrological year by the end of the twenty-first century and particularly after year 2050. This trend is particularly evident over northern Africa, southern Africa and western Mexico, where more than 90% of the models project a delay of the rainfall centroid from a few days up to 2weeks. Over the remaining monsoonal regions, there is little inter-model agreement in terms of centroid changes.