On intrinsic uncertainties in earth system modelling

Various types of uncertainty plague climate change simulation, which is, in turn, a crucialelement of Earth System modelling. This fact was recognized for example in the Third Assessment Report (TAR) of the Intergovernmental Panel on Climate Change (IPCC, Houghton et al. (2001)), where the authors indicate that for the period between 1990 and 2100 an increase of the global mean temperature around 1.4-5.8°C is to be expected (Houghton et al. 2001). The width of this span as well as the fact that the authors did not give a number concerning the most probable value or a probability distribution shows clearly the large uncertainty. This uncertainty does not only arise due to the different scenarios of future development concerning greenhouse gas emissions for example, but follows to large degree from the wide range of results from different models as well. The chain of these uncertainties of imponderables in the analysis of the Earth System (Schellnhuber and Wenzel 1998), which includes the climate system as well as the anthroposphere, reaches from uncertainties about the existence of critical thresholds, to ignorance of the exact state of today's climate, and ultimately to a lack of knowledge concerning climate-relevant processes, some of which are visible as uncertainties in climate models. Many attempts have been made to reduce these uncertainties by gaining a conceptual understanding of processes, e.g. of El Ni~no / Southern Oscillation (ENSO) (Jin 1997, e.g.) or of the Atlantic overturning (Stommel 1961; Rahmstorf 1996, e.g.), by developing methods to identify critical thresholds in the climate system (Alley et al. 2003; Rial et al. 2004, e.g.), or by implementing an increasing number of processes in a model, resulting in high resolution general circulation models (GCMs), e.g. ECHAM5/MPI-OM (Jungclaus et al. 2006) or HadCM3 (Gordon et al. 2000) and many more. Nevertheless, the much larger part of uncertainties is inevitable in the process of modelling as well as in our understanding of the Earth System. In this thesis we will structure this conglomeration of uncertainties climate research is confronted with. We will address several types of uncertainty and apply methods of dynamical systems theory on a trendsetting field of climate research, i.e. the Indian monsoon ...thesi

Predicting climate tipping as a noisy bifurcation: A review

Copyright © 2011 World Scientific PublishingElectronic version of an article published in International Journal of Bifurcation and Chaos, Vol. 21 (2), pp. 399 – 423. DOI: 10.1142/S0218127411028519. Copyright © World Scientific Publishing Company. www.worldscientific.com/worldscinet/ijbcThere is currently much interest in examining climatic tipping points, to see if it is feasible to predict them in advance. Using techniques from bifurcation theory, recent work looks for a slowing down of the intrinsic transient responses, which is predicted to occur before an instability is encountered. This is done, for example, by determining the short-term autocorrelation coefficient ARC(1) in a sliding window of the time-series: this stability coefficient should increase to unity at tipping. Such studies have been made both on climatic computer models and on real paleoclimate data preceding ancient tipping events. The latter employ reconstituted time-series provided by ice cores, sediments, etc., and seek to establish whether the actual tipping could have been accurately predicted in advance. One such example is the end of the Younger Dryas event, about 11 500 years ago, when the Arctic warmed by 7°C in 50 yrs. A second gives an excellent prediction for the end of "greenhouse" Earth about 34 million years ago when the climate tipped from a tropical state into an icehouse state, using data from tropical Pacific sediment cores. This prediction science is very young, but some encouraging results are already being obtained. Future analyses will clearly need to embrace both real data from improved monitoring instruments, and simulation data generated from increasingly sophisticated predictive models

Time scale interaction in low-order climate models

Over the last decades, the study of climate variability has attracted ample attention. The observation of structural climatic change has led to questions about the causes and the mechanisms involved. The task to understand interactions in the complex climate system is particularly di±cult because of the lack of observational data, spanning a period of time typical for natural climate variability. One way around this problem is to represent the earth s climate in a computer model, as a set of prognostic equations. A disadvantage of this approach is that, if the model under consideration is to faithfully represent the climate system, it has to be large in terms of the number of degrees of freedom. This puts it out of reach of the ordinary analysis of dynamical systems theory. Alternatively, we can impose symmetries, consider limits of physical parameters, exploit perturbation theory and use Galerkin approximation to obtain simplified models of the earth s climate. Such models should highlight some isolated aspects of climate dynamics. A feature these simplified models have in common is the presence of widely different time scales. Throughout this thesis the emphasis is on the question to what extent the slow time scales play a role in the model s dynamics. The slow time scales are related to ocean dynamics and the fast time scales to atmospheric dynamics. The atmosphere model, studied here, was introduced by Edward Lorenz (1984). In chapter 2 a derivation of this model is given and it is shown that the Lorenz-84 model describes the jet stream in the mid-latitude atmosphere, and planetary waves, which can grow if the jet stream becomes dynamically unstable. The Lorenz-84 model is coupled to two different low-order ocean models. In chapter 3, it is coupled to Stommel s two box model. Stommel s model mimics the thermohaline circulation in the North Atlantic ocean. The typical time scale of variability of this circulation is of the order of centuries. This will be the longest time scale in the coupled models. In chapter 4, the Lorenz-84 model is coupled to an ocean model formulated by Leo Maas (1994). A physical description of the coupling is given. Apart from the overturning circulation, Maas model represents a wind driven gyre. There is coupling through exchange of heat at the surface and through wind shear forcing. The latter acts on a time scale of about one year, in between the fast atmospheric time scale and the slow overturning time scale. The simplified models are sets of coupled, nonlinear, ordinary differential equations. These can be analysed with the aid of dynamical systems theory. The emphasis will be on bifurcation analysis. Also, the time scale separation leads to the presence of small parameters in the equations. The consequences for the behaviour of the coupled models are explored by means of singular perturbation theory. In both coupled models, intermittent behaviour is observed. The slow subsystem, i.e. the ocean model, repeatedly pushes the fast subsystem, i.e. the atmosphere model, through a sequence of bifurcations. Thus, the ocean model plays an active role in the coupled system. Secondly, in the Lorenz-Maas model a periodic solution is shown to exist, with a period on the slow, overturning time scale. Along this solution the behaviour of the coupled model is dictated by internal ocean dynamics. Both these phenomena occur near a critical point of the coupled system, in agreement with the general idea that in climate models the slow components can play an active role near such critical points and are passive otherwise

Effects of inflation expectations on macroeconomic dynamics: Extrapolative versus regressive expectations

In this paper we integrate heterogeneous inflation expectations into a simple monetary model. Guided by empirical evidence we assume that boundedly rational agents, selecting between extrapolative and regressive forecasting rules to predict the future inflation rate, prefer rules that have produced low prediction errors in the past. We show that integrating this behavioral expectation formation process into the monetary model leads to the possibility of endogenous macroeconomic dynamics. For instance, our model replicates certain empirical regularities such as irregular growth cycles or inflation persistence. Moreover, we observe multi-stability via a Chenciner bifurcation. --Extrapolative and regressive expectations,dynamic predictor selection,macroeconomic dynamics,nonlinearities and chaos,bifurcation analysis

Nonlinear forecasting of the generalised Kuramoto-Sivashinsky equation

We study the emergence of pattern formation and chaotic dynamics in the one-dimensional (1D) generalized Kuramoto-Sivashinsky (gKS) equation by means of a time-series analysis, in particular a nonlinear forecasting method which is based on concepts from chaos theory and appropriate statistical methods. We analyze two types of temporal signals, a local one and a global one, finding in both cases that the dynamical state of the gKS solution undergoes a transition from high dimensional chaos to periodic pulsed oscillations through low dimensional deterministic chaos with increasing the control parameter of the system. Our results demonstrate that the proposed nonlinear forecasting methodology allows to elucidate the dynamics of the system in terms of its predictability properties

Modelling dryland vegetation patterns : nonlocal dispersal, temporal variability in precipitation and species coexistence

Spatiotemporal patterns of vegetation are a characteristic feature of dryland ecosystems occurring on all continents except Antarctica. The development of an understanding of their ecosystem dynamics is an issue of considerable socio-economic importance as both the livestock and agricultural sectors in dryland economies heavily depend on ecosystem functioning. Mathematical modelling is a powerful tool to disentangle the complex ecosystem dynamics. In this thesis, I present theoretical models to explore the impact of nonlocal seed dispersal and temporal precipitation variability on dryland vegetation patterns and propose several mechanisms that enable species coexistence within vegetation patterns. To do so, I present extensions of the Klausmeier reaction-advection-diffusion model, a well-established model describing the ecohydrological dynamics of vegetation patterns. Model analyses focus on pattern onset at high precipitation values (i.e. on the transition from uniformly vegetated to spatially patterned states) to assess the impact of nonlocal seed dispersal and precipitation seasonality and intermittency, and on comprehensive bifurcation analyses, including results on pattern existence and stability to investigate coexistence of species in the mathematical framework. Results include the inhibition of pattern onset due to long-range seed dispersal and put emphasis on the functional response of plants to low soil moisture levels to understand effects of rainfall intermittency. Moreover, results suggest that coexistence is facilitated by resource heterogeneities induced by the plant’s spatial self-organisation and highlight the importance of considering out-of-equilibrium solutions.UK Engineering and Physical Sciences Research Council (grant EP/L016508/01) Scottish Funding Council Heriot-Watt Universit

Extreme value statistics for dynamical systems with noise

We study the distribution of maxima ( extreme value statistics ) for sequences of observables computed along orbits generated by random transformations. The underlying, deterministic, dynamical system can be regular or chaotic. In the former case, we show that, by perturbing rational or irrational rotations with additive noise, an extreme value law appears, regardless of the intensity of the noise, while unperturbed rotations do not admit such limiting distributions. In the case of deterministic chaotic dynamics, we will consider observables specially designed to study the recurrence properties in the neighbourhood of periodic points. Hence, the exponential limiting law for the distribution of maxima is modified by the presence of the extremal index , a positive parameter not larger than one, whose inverse gives the average size of the clusters of extreme events. The theory predicts that such a parameter is unitary when the system is perturbed randomly. We perform sophisticated numerical tests to assess how strong the impact of noise level is when finite time series are considered. We find agreement with the asymptotic theoretical results but also non-trivial behaviour in the finite range. In particular, our results suggest that, in many applications where finite datasets can be produced or analysed, one must be careful in assuming that the smoothing nature of noise prevails over the underlying deterministic dynamics

Linear and nonlinear dynamics in stratified shear flows

Stably stratified shear flows, in which a less dense layer of fluid lies above and moves counter to a more dense layer below, are ubiquitous in geophysical fluid dynamics. These are often found to be unstable if the non-dimensional Richardson number Ri, quantifying the strength of stratification to shear, is sufficiently low. This is of particular importance in oceanography, where shear instabilities are conjectured to be important in the generation of turbulence in the deep ocean, an area of huge uncertainty in contemporary climate models. The Miles-Howard theorem tells us that for a steady, inviscid, parallel shear flow, if the local Richardson number is everywhere greater than one quarter, the flow is stable to infinitesimal perturbations. Though an important result, the strong restrictions in the applicability of this theorem mean care must be used when applying the criterion of Ri > 1/4 for stability. This thesis explores some of these limitations, beginning with an overview in chapter 1. Chapter 2 explores the infinitesimal restriction of the Miles-Howard theorem, by asking whether finite-amplitude perturbations could lead to significant nonlinear behaviour, in a so-called subcritical instability. It is found that while the classical Kelvin-Helmholtz instability does indeed exhibit subcriticality, nonlinear steady states are found only just above Ri = 1/4. Chapter 3 investigates in detail a hitherto unknown linear instability, which was discovered in chapter 2. Behaving similarly to the classic Holmboe instability, it exists for Ri > 1/4 when viscosity is introduced, and reveals new insights into the possible physical interpretations of stratified shear instability. Chapter 4 revisits the results of chapter 2 but considers two cases of the Prandtl number Pr, the ratio of diffusivity of the momentum to density. When Pr = 0.7, as is approximately the case for air, a simple supercritical instability is found. However, for Pr = 7, corresponding approximately to water, strong subcritical behaviour is observed, and it is demonstrated that finite-amplitude perturbations can trigger Kelvin-Helmholtz-like behaviour well above Ri = 1/4. Chapter 5 considers the time-varying, non-parallel flow of an oblique internal gravity wave incident on a shear layer. Using direct-adjoint looping, it is shown that the disturbances which maximise energy after a certain time, so-called linear optimal perturbations, can be convective-like rolls in the spanwise direction, rather than a shear instability, calling into question the relevance of the classical shear instabilities in oceanography. Chapter 6 concludes the thesis with a discussion of the implications of the results