The primary objective of this article is to analyse a set of canonical spatial patterns that approximate the daily rainfall across the Indian region, as identified in the companion article where we developed a discrete representation of the Indian summer monsoon rainfall using state variables with spatio-temporal coherence maintained using a Markov random field prior. In particular, we use these spatio-temporal patterns to study the variation of rainfall during the monsoon season. First, the 10 patterns are divided into three families of patterns distinguished by their total rainfall amount and geographic spread. These families are then used to establish ‘active’ and ‘break’ spells of the Indian monsoon at the all-India level. Subsequently, we characterize the behaviour of these patterns in time by estimating probabilities of transition from one pattern to another across days in a season. Patterns tend to be ‘sticky’: the self-transition is the most common. We also identify most commonly occurring sequences of patterns. This leads to a simple seasonal evolution model for the summer monsoon rainfall. The discrete representation introduced in the companion article also identifies typical temporal rainfall patterns for individual locations. This enables us to determine wet and dry spells at local and regional scales. Last, we specify sets of locations that tend to have such spells simultaneously and thus come up with a new regionalization of the landmass.
It is well known that periodic forcing of a nonlinear system, even of a two-dimensional autonomous system, can produce chaotic responses with sensitive dependence on initial conditions if the forcing induces sufficient stretching and folding of the phase space. Quasiperiodic forcing can similarly produce chaotic responses, where the transition to chaos on changing a parameter can bring the system into regions of strange non-chaotic behaviour. Although it is generally acknowledged that the timings of Pleistocene ice ages are at least partly due to Milankovitch forcing (which may be approximated as quasiperiodic, with energy concentrated near a small number of frequencies), the precise details of what can be inferred about the timings of glaciations and deglaciations from the forcing is still unclear. In this paper, we perform a quantitative comparison of the response of several low-order nonlinear conceptual models for these ice ages to various types of quasiperiodic forcing. By computing largest Lyapunov exponents and mean periods, we demonstrate that many models can have a chaotic response to quasiperiodic forcing for a range of forcing amplitudes, even though some of the simplest conceptual models do not. These results suggest that pacing of ice ages to forcing may have only limited determinism.
The Mid-Pleistocene Transition, the shift from 41 kyr to 100 kyr glacial-interglacial cycles that occurred roughly 1 Myr ago, is often considered as a change in internal climate dynamics. Here we revisit the model of Quaternary climate dynamics that was proposed by Saltzman and Maasch (1988). We show that it is quantitatively similar to a scalar equation for the ice dynamics only when combining the remaining components into a single delayed feedback term. The delay is the sum of the internal times scales of ocean transport and ice sheet dynamics, which is on the order of 10 kyr. We find that, in the absence of astronomical forcing, the delayed feedback leads to bistable behaviour, where stable large-amplitude oscillations of ice volume and an equilibrium coexist over a large range of values for the delay. We then apply astronomical forcing. We perform a systematic study to show how the system response depends on the forcing amplitude. We find that over a wide range of forcing amplitudes the forcing leads to a switch from small-scale oscillations of 41 kyr to large-amplitude oscillations of roughly 100 kyr without any change of other parameters. The transition in the forced model consistently occurs near the time of the Mid-Pleistocene Transition as observed in data records. This provides evidence that the MPT could have been primarily a forcing-induced switch between attractors of the internal dynamics. Small additional random disturbances make the forcing-induced transition near 800 kyr BP even more robust. We also find that the forced system forgets its initial history during the small-scale oscillations, in particular, nearby initial conditions converge prior to transitioning. In contrast to this, in the regime of large-amplitude oscillations, the oscillation phase is very sensitive to random perturbations, which has a strong effect on the timing of the deglaciation events.
We propose a representation of the Indian summer monsoon rainfall in terms of a probabilistic model based on a Markov Random Field, consisting of discrete state variables representing low and high rainfall at grid-scale and daily rainfall patterns across space and in time. These discrete states are conditioned on observed daily gridded rainfall data from the period 2000-2007. The model gives us a set of 10 spatial patterns of daily monsoon rainfall over India, which are robust over a range of user-chosen parameters as well as coherent in space and time. Each day in the monsoon season is assigned precisely one of the spatial patterns, that approximates the spatial distribution of rainfall on that day. Such approximations are quite accurate for nearly 95% of the days. Remarkably, these patterns are representative (with similar accuracy) of the monsoon seasons from 1901 to 2000 as well. Finally, we compare the proposed model with alternative approaches to extract spatial patterns of rainfall, using empirical orthogonal functions as well as clustering algorithms such as K-means and spectral clustering.
Previous estimates of the power spectrum and of the scaling exponent of the detrended fluctuation analysis of palaeoclimate time series yielded the suggestion that climate fluctuations are scale invariant over a wide range of time scales. Specifically, the last glacial period is characterised by Dansgaard-Oeschger events, with rapid and frequent transitions between stadial and interstadial regimes, and it was suggested that climate changes were then exhibiting multi-fractal dynamics. The present contribution clarifies the interpretation of detrended fluctuation analysis and of power spectra during the last glacial period. We use two simple models exhibiting regime behaviour reminiscent of Dansgaard-Oeshger dynamics: the random telegraph process with additive white Gaussian noise, and Stommel-Cessi’s box model of thermohaline circulation (a study of the Lorenz model is also proposed in the appendix). This analysis then provides a support for interpreting the generalized Hurst exponent h(q) and power-spectrum estimates of two NGRIP (Greenland) ice core records: the oxygen isotope ratio and the calcium ion concentration. We also analyse simulations with a stochastic dynamical system model with ice and astronmical forcings, which we recently proposed to simulate Dansgaard-Oeschger events. Multifractal detrended fluctuation analyses of time series generated by the toy models reveal that their generalized fluctuation functions have a local scaling regime. The generalized Hurst exponent h(q) is lower for q<<0 than for q>0. This dependency of h(q) is named here “apparent multifractality”. It occurs because the behaviour of the autocorrelation function of small fluctuations (within a regime) differs from that of large fluctuations (regime shifts). The generalized Hurst exponents of NGRIP records exhibit a form of apparent multifractality similar to that described in the toy models. The stochastic dynamical system model also captures both the power spectrum of the observations, and the behaviour of h(q). The apparent multifractality of the Dansgaard-Oeschger events records is a consequence of regime switching between stadial and interstadial climates. Neither the local scaling in the power spectrum, nor the output of the multifractal detrended fluctuation analysis implies that the underlying process is scale invariant.
Background. Extreme value theory for chaotic deterministic dynamical systems is a rapidly expanding area of research. Given a system and a scalar observable defined on its state space, extreme value theory studies the asymptotic probability distributions of large values attained by the observable along evolutions of the system. Objective. The aim of this paper is to study the relation between the statistics and predictability of extremes. Methods. Predictability is measured by the mean squared error (MSE), which is estimated from the difference of pairs of forecasts conditional on one of the forecasts exceeding a threshold. Results. Under the assumption that pairs of forecast variables satisfy a linear regression model, we show that the MSE can be decomposed into the sum of three terms: a threshold-independent constant, a mean term that always increases with threshold, and a variance term that can either increase, decrease, or stay constant with threshold. Using the generalised Pareto distribution to model excesses over a threshold, we show that the MSE always increases with threshold at sufficiently high threshold. However, when the forecasts have a negative tail index the MSE can be a decreasing function of threshold at lower thresholds. Conclusions. Our method is illustrated by means of four examples: the tent map, the cusp map, and two low-order models for atmospheric regime transitions and the El Ni\~{n}o-Southern Oscillation phenomenon. These examples clearly show that predictability depends on the observable and the invariant measure of the system.
BackgroundThe removal of non-climatic artefacts, inhomogeneities, from observed station data for the purpose of climate research is an ongoing task. Progress on homogenization algorithms is limited by a lack of suitable test data that are sufficiently realistic and where the 'truth' is known a priori.
Global mean surface temperature is widely used in the climate literature as a measure of the impact of human activity on the climate system. While the concept of a spatial average is simple, the estimation of that average from spatially incomplete data is not. Correlation between nearby map grid cells means that missing data cannot simply be ignored. Estimators that (often implicitly) assume uncorrelated observations can be biased when naively applied to the observed data, and in particular, the commonly used area weighted average is a biased estimator under these circumstances. Some surface temperature products use different forms of infilling or imputation to estimate temperatures for regions distant from the nearest observation, however the impacts of such methods on estimation of the global mean are not necessarily obvious or themselves unbiased. This issue was addressed in the 1970s by Ruvim Kagan, however his work has not been widely adopted, possibly due to its complexity and dependence on subjective choices in estimating the dependence between geographically proximate observations. The aim of this work is to present a simple estimator for global mean surface temperature from spatially incomplete data which retains many of the benefits of the work of Kagan, while being fully specified by two equations and a single parameter. The main purpose of the simplified estimator is to better explain to users of temperature data the problems associated with estimating an unbiased global mean from spatially incomplete data, however the estimator may also be useful for problems with specific requirements for reproducibility and performance. The new estimator is based on generalized least squares, and uses the correlation matrix of the observations to weight each observation in accordance with the independent information it contributes. It can be implemented in fewer than 20 lines of computer code. The performance of the estimator is evaluated for different levels of observational coverage using reanalysis data with artificial noise. For recent decades the generalized least squares estimator mitigates most of the error associated with the use of a naive area weighted average. The improvement arises from the fact that coverage bias in the historical temperature record does not arise from an absolute shortage of observations (at least for recent decades), but rather from spatial heterogeneity in the distribution of observations, with some regions being relatively undersampled and others oversampled. The estimator addresses this problem by reducing the weight of the oversampled regions, in contrast to some existing global temperature datasets which extrapolate temperatures into the unobserved regions. The results are almost identical to the use of kriging (Gaussian process interpolation) to impute missing data to global coverage, followed by an area weighted average of the resulting field. However, the new formulation allows direct diagnosis of the contribution of individual observations and sources of error. More sophisticated solutions to the problem of missing data in global temperature estimation already exist. However the simple estimator presented here, and the error analysis that it enables, demonstrate why such solutions are necessary. The 2013 Fifth Assessment Report of the Intergovernmental Panel on Climate Change discussed a slowdown in warming for the period 1998-2012, quoting the trend in the HadCRUT4 historical temperature dataset from the United Kingdom Meteorological Office in collaboration with the Climatic Research Unit of the University of East Anglia, along with other records. Use of the new estimator for global mean surface temperature would have reduced the apparent slowdown in warming of the early 21st century by one third in the spatially incomplete HadCRUT4 product.
A statistical test is presented to address the null hypothesis that sea-level fluctuations in the open ocean are solely due to additive noise in the wind stress. The effect of high-frequency wind-stress variations can be represented as a correlated additive and multiplicative noise (CAM) stochastic model of sea-level variations. The main novel aspect of this article is the estimation of parameters in the discrete CAM model from time series of sea surface height observations. This leads to a statistical test, similar to the red noise [or AR(1)] test for sea surface temperature, which can be used to attribute specific sea-level variability to other effects than wind-stress noise. We demonstrate the performance of this test using altimeter data at several locations in the open ocean.
Englacial radar reflectors in the central West Antarctic Ice Sheet contain information about past dynamics and ice properties. Due to significant data coverage in this area, these isochronous reflectors can be traced over large portions of the ice sheet, but assigning ages to the reflectors for the purpose of studying dynamics requires incorporation of chronologic data from ice cores. To date reflectors in the Marie Byrd Land region, we consider the Byrd ice core, strategically located between the catchments of Thwaites Glacier and the Siple Coast ice streams. We determine ages with uncertainty for four englacial radar reflectors spanning the ice thickness using Bayesian approaches to combine radar observations, an existing Byrd ice core chronology, and a simplified ice flow model. This method returns the marginal probability distribution of depth and age for each of the observed radar reflectors. The results also include inferences of accumulation rate at the Byrd ice core site during the last 30 ka that show a minimum accumulation rate during the Last Glacial Maximum at half the modern rate. The deepest continuous radar reflector is 25.67 ± 1.45 ka, <30% of the estimated age of the oldest ice at the Byrd ice core site despite being located at 70% of the ice depth, limiting the age of radar-interpretable ice in this region. The inferred reflector age profiles at the Byrd ice core site derived here compare favourably with the more recent WAIS Divide ice core record. However, uncertainty in reflector depth due to radar range precision contributes considerably to uncertainty in reflector age in a way that is not readily reducible using currently available ice-penetrating radar systems.
Decline in the Arctic sea ice extent (SIE) is an area of active scientific research with profound socio-economic implications. Of particular interest are reliable methods for SIE forecasting on subseasonal time scales, in particular from early summer into fall, when sea ice coverage in the Arctic reaches its minimum. Here, we apply the recent data-adaptive harmonic (DAH) technique of Chekroun and Kondrashov, (2017), Chaos, 27 for the description, modeling and prediction of the Multisensor Analyzed Sea Ice Extent (MASIE, 2006–2016) data set. The DAH decomposition of MASIE identifies narrowband, spatio-temporal data-adaptive modes over four key Arctic regions. The time evolution of the DAH coefficients of these modes can be modelled and predicted by using a set of coupled Stuart–Landau stochastic differential equations that capture the modes’ frequencies and amplitude modulation in time. Retrospective forecasts show that our resulting multilayer Stuart–Landau model (MSLM) is quite skilful in predicting September SIE compared to year-to-year persistence; moreover, the DAH–MSLM approach provided accurate real-time prediction that was highly competitive for the 2016–2017 Sea Ice Outlook.
There is as yet no theoretical framework to guide the search for emergent constraints. As a result, there are significant risks that indiscriminate data-mining of the multidimensional outputs from GCMs could lead to spurious correlations and less than robust constraints on future changes. To mitigate against this risk, Cox et al (hereafter CHW18) proposed a theory-motivated emergent constraint, using the one-box Hasselmann model to identify a linear relationship between ECS and a metric of global temperature variability involving both temperature standard deviation and autocorrelation ($\Psi$). A number of doubts have been raised about this approach, some concerning the theory and the application of the one-box model to understand relationships in complex GCMs which are known to have more than the single characteristic timescale. We illustrate theory driven testing of emergent constraints using this as an example, namely we demonstrate that the linear $\Psi$-ECS proportionality is not an artifact of the one-box model and rigorously features to a good approximation in more realistic, yet still analytically soluble conceptual models, namely the two-box and diffusion models. Each of the conceptual models predict different power spectra with only the diffusion model's pink spectrum being compatible with observations and the complex CMIP5 GCMs. We also show that the theoretically predicted $\Psi$-ECS relationship exists in the \texttt{piControl} as well as \texttt{historical} CMIP5 experiments and that the differing gradients of the proportionality are inversely related to the effective forcing in that experiment.
We present a 5-Myr probabilistic stack, which we name the Prob-stack. It is constructed from 180 globally distributed benthic δ18O records using a profile hidden Markov model (HMM). Benthic stacks have been extensively employed to estimate ages in marine sediment cores and lead–lag relationships between paleoclimate proxies. Because this stack is probabilistic, it incorporates the variability among multiple records used in its construction and the uncertainty in alignments of records to the stack that have not been included in previous studies. It represents the uncertainty stemming from these two sources using a normal distribution with mean and variance in benthic δ18O at each point in the stack. In doing so, Prob-stack captures the time-varying mean and variance of benthic δ18O in the global ocean, which allows for better assessment of the uncertainty in records' relative ages, such as lead–lag relationships. Algorithms are included.
It is a common practice to use a simple model to explain the mechanisms or processes that occur in a much more complex, complete and computationally expensive model. Many such examples can be found in climate change research. This paper uses two illustrative examples to show how we can quantitatively relate the mechanisms or processes observed in a simple climate model to similar mechanisms in a more complex one. A simple model can only explain a more complex solution’s mechanisms if outcomes are tested over a broad range of inputs. By carefully sampling the full set of inputs for both the simple and complex models, we can robustly compare the process or mechanistic outcomes, statistically, between them. Thus, by examining the similarity or differences in the relationship between the inputs and outputs. The method can reject an incorrect simple model. The examples are, first, analytic and numerical solutions to the heat equation and, second, the 1948 Stommel model of horizontal ocean circulation and a more complex quasi-geostrophic ocean model. We quantitatively state how similar the simple model’s mechanisms are to the mechanisms in the more complex representation. In addition, when a simple solution may be correct, we give the percentage of the variance of the complex model’s outcomes that is explained by the simple response along with an uncertainty estimate. We successfully tested a methodology for robustly quantifying how the physics encapsulated by a simple model of a process may exhibit itself in another, more complex formulation. Suggestions are given as a guide for use of the methodology with more complex and realistic models.
Understanding the impact of future heatwaves and the development of effective adaptation strategies requires knowledge of both the changes in heatwave temperatures and their durations. We develop a framework, utilising extreme value theory, which allows for the effect of a covariate on both the marginal quantiles and the temporal dependence structure of daily maximum temperatures enabling the changes in heatwave temperatures (marginal effects) to be identified separately from duration changes (dependence effects). To characterize future heatwave changes we use global mean temperature anomalies as a covariate to provide the metric for climate change. Future daily maximum temperatures and global mean temperature changes are provided by 13 general circulation models (GCMs) from the CMIP5 archive forced with predicted future emissions of radiative forcing agents from the RCP8.5 scenario. For Orleans, central France, we find that for all GCMs temporal dependence is unaffected by greenhouse gas induced climate change indicating that durations of heatwaves that exceed time varying high thresholds (i.e. the 1-year level) will not change in the future. However, all GCMs project significant changes in the temperature margins with events similar to the 2003 European heatwave increasing by 1.3°C to 2.7°C and (8.0°C to 18.7°C) for a 1°C (5°C) increase in global temperature. Collectively our results indicate there could be a significant increase in heatwave risk as the world warms with heatwaves increasing in temperature significantly faster than the global mean and local average temperatures.
The El Ni\~no-Southern Oscillation (ENSO) is a mode of interannual variability in the coupled equatorial Pacific coupled atmosphere/ocean system. El Ni\~no describes a state in which sea surface temperatures in the eastern Pacific increase and upwelling of colder, deep waters diminishes. El Ni\~no events typically peak in boreal winter, but their strength varies irregularly on decadal time scales. There were exceptionally strong El Ni\~no events in 1982-83, 1997-98 and 2015-16 that affected weather on a global scale. Widely publicized forecasts in 2014 predicted that the 2015-16 event would occur a year earlier. Predicting the strength of El Ni\~no is a matter of practical concern due to its effects on hydroclimate and agriculture around the world. This paper discusses the frequency and regularity of strong El Ni\~no events in the context of chaotic dynamical systems. We discover a mechanism that limits their predictability in a conceptual "recharge oscillator" model of ENSO. Weak seasonal forcing or noise in this model can induce irregular switching between an oscillatory state that has strong El Ni\~no events and a chaotic state that lacks strong events, In this regime, the timing of strong El Ni\~no events on decadal time scales is unpredictable.
Equilibrium climate sensitivity (ECS) is a key predictor of climate change. However, it is not very well constrained, either by climate models or by observational data. The reasons for this include strong internal variability and forcing on many time scales. In practise this means that the 'equilibrium' will only be relative to fixing the slow feedback processes before comparing palaeoclimate sensitivity estimates with estimates from model simulations. In addition, information from the late Pleistocene ice age cycles indicates that the climate cycles between cold and warm regimes, and the climate sensitivity varies considerably between regime because of fast feedback processes changing relative strength and time scales over one cycle. In this paper we consider climate sensitivity for quite general climate dynamics. Using a conceptual Earth system model of Gildor and Tziperman (2001) (with Milankovich forcing and dynamical ocean biogeochemistry) we explore various ways of quantifying the state-dependence of climate sensitivity from unperturbed and perturbed model time series. Even without considering any perturbations, we suggest that climate sensitivity can be usefully thought of as a distribution that quantifies variability within the 'climate attractor' and where there is a strong dependence on climate state and more specificially on the 'climate regime' where fast processes are approximately in equilibrium. We also consider perturbations by instantaneous doubling of CO_2 and similarly find a strong dependence on the climate state using our approach.
Dynamical systems methodology is a mature complementary approach to forward simulation which can be used to investigate many aspects of climate dynamics. With this paper, a review is given on the methods to analyse deterministic and stochastic climate models and show that these are not restricted to low-dimensional toy models, but that they can be applied to models formulated by stochastic partial differential equations. We sketch the numerical implementation of these methods and illustrate these by showing results for two canonical problems in climate dynamics.
Compatible finite elements provide a framework for preserving important structures in equations of geophysical fluid dynamics, and are becoming important in their use for building atmosphere and ocean models. We survey the application of compatible finite element spaces to geophysical fluid dynamics, including the application to the nonlinear rotating shallow water equations, and the three-dimensional compressible Euler equations. We summarise analytic results about dispersion relations and conservation properties, and present new results on approximation properties in three dimensions on the sphere, and on hydrostatic balance properties.
Research on the possibility of future abrupt climate change has been popularized under the term 'tipping points' and has often been motivated by using simple, low-dimensional concepts. These include the iconic fold bifurcation, where abrupt change occurs when a stable equilibrium is lost, and early warning signals of such a destabilization that can be derived based on a simple stochastic model approach. In this paper, we review the challenges and limitations that are associated with this view, and we discuss promising research paths to explore the causes and the likelihood of abrupt changes in future climate. We focus on several climate system components and ecosystems that have been proposed as candidates for tipping points, with an emphasis on ice sheets, the Atlantic Ocean circulation, vegetation in North Africa and Arctic sea ice. In most example cases, multiple equilibria found in simple models do not appear in complex models or become more difficult to find, while the potential for abrupt change still remains. We also discuss how the low-dimensional logic of current methods to detect and interpret the existence of multiple equilibria can fail in complex models. Moreover, we highlight promising methods to detect abrupt shifts and to obtain information about the mechanisms behind them. These methods include linear approaches such as statistical stability indicators and radiative feedback analysis as well as non-linear approaches to detect dynamical transitions and infer the causality behind events. Given the huge complexity of comprehensive process-based climate models and the non-linearity and regional peculiarities of the processes involved, the uncertainties associated with the possible future occurrence of abrupt shifts are large and not well quantified. We highlight the potential of data mining approaches to tackle this problem and finally discuss how the scientific community can collaborate to make efficient progress in understanding abrupt climate shifts.
