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Planning for the impacts of climate change requires accurate projections by Earth system models (ESMs). ESMs, as developed by many research centres, estimate changes to weather and climate as atmospheric greenhouse gases (GHGs) rise, and they inform the influential Intergovernmental Panel on Climate Change (IPCC) reports. ESMs are advancing the understanding of key climate system attributes. However, there remain substantial inter-ESM differences in their estimates of future meteorological change, even for a common GHG trajectory, and such differences make adaptation planning difficult. Until recently, the primary approach to reducing projection uncertainty has been to place an emphasis on simulations that best describe the contemporary climate. Yet a model that performs well for present-day atmospheric GHG levels may not necessarily be accurate for higher GHG levels and vice versa. A relatively new approach of emergent constraints (ECs) is gaining much attention as a technique to remove uncertainty between climate models. This method involves searching for an inter-ESM link between a quantity that we can also measure now and a second quantity of major importance for describing future climate. Combining the contemporary measurement with this relationship refines the future projection. Identified ECs exist for thermal, hydrological and geochemical cycles of the climate system. As ECs grow in influence on climate policy, the method is under intense scrutiny, creating a requirement to understand them better. We hypothesise that as many Earth system components vary in both space and time, their behaviours often satisfy large-scale differential equations (DEs). Such DEs are valid at coarser scales than the equations coded in ESMs which capture finer high-resolution grid-box-scale effects. We suggest that many ECs link to such effective hidden DEs implicit in ESMs and that aggregate small-scale features. An EC may exist because its two quantities depend similarly on an ESM-specific internal bulk parameter in such a DE, with measurements constraining and revealing its (implicit) value. Alternatively, well-established process understanding coded at the ESM grid box scale, when aggregated, may generate a bulk parameter with a common “emergent” value across all ESMs. This single emerging parameter may link uncertainties in a contemporary climate driver to those of a climate-related property of interest. In these circumstances, the EC combined with a measurement of the driver that is uncertain constrains the estimate of the climate-related quantity. We offer simple illustrative examples of these concepts with generic DEs but with their solutions placed in a conceptual EC framework.

Current climate change is highly transient, and Earth is at a significant distance from thermal equilibrium. This well-established finding is predominantly due to the large flux of thermal energy currently entering the oceans. If globally the sources and sinks of atmospheric GHGs were to become zero, this is identical to stating that their concentrations are subsequently invariant. Such a policy would correspond to a full offset of the remaining emissions by both land and ocean CO2 drawdown, as well as any implementation of carbon capture and storage (CCS) methods. The 5th Intergovernmental Panel on Climate Change (IPCC) report defines this as a “constant composition commitment”

Earth system models are complex and represent a large number of processes, resulting in a persistent spread across climate projections for a given future scenario. Owing to different model performances against observations and the lack of independence among models, there is now evidence that giving equal weight to each available model projection is suboptimal. This Perspective discusses newly developed tools that facilitate a more rapid and comprehensive evaluation of model simulations with observations, process-based emergent constraints that are a promising way to focus evaluation on the observations most relevant to climate projections, and advanced methods for model weighting. These approaches are needed to distil the most credible information on regional climate changes, impacts, and risks for stakeholders and policy-makers. ISSN:1758-6798

The transient climate response (TCR), transient warming for a doubling of CO2, contributes the biggest uncertainty to estimates of the carbon budgets consistent with the Paris targets. In the IPCC 5th Assessment Report (AR5), the stated ‘likely’ range of TCR was given as 1.0 to 2.5K, with a central estimate which was broadly consistent with the ensemble mean of the CMIP5 Earth System Models (ESMs) available at the time (1.8 +/- 0.4 K) . Many of the latest CMIP6 ESMs have larger climate sensitivities, with 6 of 23 models having TCR values above 2.5 K, and an ensemble mean TCR of 2.1 +/- 0.4 K. On the face of it, these latest ESM results suggest that the IPCC likely range of TCRE may need revising upwards, which would cast further doubt on the feasibility of the Paris targets.We analyse the CMIP6 models through an emergent constraint approach which relates the value of TCR to the global warming from 1970 onwards. We confirm a consistent emergent constraint on TCR when we apply the same method to CMIP5 model. Our emergent constraint on TCR benefits from both the large range of TCR values across the CMIP6 models, and also from the extension of the historical simulations into a period when the uncertain changes in aerosol forcing have had a far less significant impact on the trend in global warming.We show that rather than increasing the uncertainty in climate sensitivity, the CMIP6 models help to further constrain the likely range of TCR to 1.5-2.2 K. In CMIP6, diagnosed emissions at carbon doubling was found to be independent of TCR, so that a constraint on TCR directly leads to a constrained estimate of TCRE, with a likely range of 1.3 – 2.0 K per EgC.

A superrotating atmosphere, one in which the angular momentum of the atmosphere exceeds the solid body rotation of the planet, occurs on Venus and Titan. However, it may have occurred on the Earth in the hot house climates of the Early Cenozoic and some climate models have transitioned abruptly to a superrotating state under the more extreme global warming scenarios. Applied to the Earth, the transition to superrotation causes the prevailing easterlies at the equator to become westerlies and accompanying large changes in global circulation patterns. Although current thinking is that this scenario is unlikely, it shares features of other global tipping points in that it is a low probability, high risk event.More than anything though, this tipping point serves as an ideal example to test some spatial early warning methods. I’ll show some preliminary results how the critical spatial modes and time scales change through the transition to superrotation using an idealized general circulation model (GCM), Isca.

The transient climate response (TCR) is the metric of temperature sensitivity that is most relevant to warming in the next few decades, and contributes the biggest uncertainty to estimates of the carbon budgets consistent with the Paris targets (Arora et al., 2019). In the IPCC 5th Assessment Report (AR5), the stated likely range of TCR was given as 1.0 to 2.5 K, with a central estimate which was broadly consistent with the ensemble mean of the CMIP5 Earth System Models (ESMs) available at the time (1.8 ± 0.4 K). Many of the latest CMIP6 ESMs have larger climate sensitivities, with 6 of 23 models having TCR values above 2.5 K, and an ensemble mean TCR of 2.1 ± 0.4 K. On the face of it, these latest ESM results suggest that the IPCC likely range of TCR may need revising upwards, which would cast further doubt on the feasibility of the Paris targets. Here we show that rather than increasing the uncertainty in climate sensitivity, the CMIP6 models help to further constrain the likely range of TCR to 1.5–2.2 K, with a central estimate of 1.82 K. We reach this conclusion through an emergent constraint approach which relates the value of TCR to the global warming from 1970 onwards. We confirm a consistent emergent constraint on TCR when we apply the same method to CMIP5 models (Jiménez-de-la Cuesta and Mauritsen, 2019). Our emergent constraint on TCR benefits from both the large range of TCR values across the CMIP6 models, and also from the extension of the historical simulations into a period when the uncertain changes in aerosol forcing have had a far less significant impact on the trend in global warming.

Climate-related risks are dependent not only on the warming trend from GHGs, but also on the variability about the trend. However, assessment of the impacts of climate change tends to focus on the ultimate level of global warming1, only occasionally on the rate of global warming, and rarely on variability about the trend. Here we show that models that are more sensitive to GHGs emissions (that is, higher equilibrium climate sensitivity (ECS)) also have higher temperature variability on timescales of several years to several decades2. Counter-intuitively, high-sensitivity climates, as well as having a higher chance of rapid decadal warming, are also more likely to have had historical ‘hiatus’ periods than lower-sensitivity climates. Cooling or hiatus decades over the historical period, which have been relatively uncommon, are more than twice as likely in a high-ECS world (ECS = 4.5 K) compared with a low-ECS world (ECS = 1.5 K). As ECS also affects the background warming rate under future scenarios with unmitigated anthropogenic forcing, the probability of a hyper-warming decade—over ten times the mean rate of global warming for the twentieth century—is even more sensitive to ECS. Projections of temperature often focus on maximum warming levels, with variability less often considered. Investigating decadal variability in models shows those with higher equilibrium climate sensitivity also have a higher chance of super warming, and hiatus periods.

The prospect of finding generic early warning signals of an approaching tipping point in a complex system has generated much interest recently. Existing methods are predicated on a separation of timescales between the system studied and its forcing. However, many systems, including several candidate tipping elements in the climate system, are forced periodically at a timescale comparable to their internal dynamics. Here we use alternative early warning signals of tipping points due to local bifurcations in systems subjected to periodic forcing whose timescale is similar to the period of the forcing. These systems are not in, or close to, a fixed point. Instead their steady state is described by a periodic attractor. For these systems, phase lag and amplification of the system response can provide early warning signals, based on a linear dynamics approximation. Furthermore, the Fourier spectrum of the system's time series reveals harmonics of the forcing period in the system response whose amplitude is related to how nonlinear the system's response is becoming with nonlinear effects becoming more prominent closer to a bifurcation. We apply these indicators as well as a return map analysis to a simple conceptual system and satellite observations of Arctic sea ice area, the latter conjectured to have a bifurcation type tipping point. We find no detectable signal of the Arctic sea ice approaching a local bifurcation.

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., Comment: 12 pages, 4 figures, accepted for publication in Dynamics and Statistics of the Climate System. V2 with rewritten abstract to conform to journal style and references no longer corrupted

The notion that small changes can have large consequences in the climate or ecosystems has become popular as the concept of tipping points. Typically, tipping points are thought to arise from a loss of stability of an equilibrium when external conditions are slowly varied. However, this appealingly simple view puts us on the wrong foot for understanding a range of abrupt transitions in the climate or ecosystems because complex environmental systems are never in equilibrium. In particular, they are forced by diurnal variations, the seasons, Milankovitch cycles and internal climate oscillations. Here we show how abrupt and sometimes even irreversible change may be evoked by even small shifts in the amplitude or time scale of such environmental oscillations. By using model simulations and reconciling evidence from previous studies we illustrate how these phenomena can be relevant for ecosystems and elements of the climate system including terrestrial ecosystems, Arctic sea ice and monsoons. Although the systems we address are very different and span a broad range of time scales, the phenomena can be understood in a common framework that can help clarify and unify the interpretation of abrupt shifts in the Earth system.

We introduce a novel bridge between the familiar gauge field theory approaches used in many areas of modern physics such as quantum field theory and the SLOCC protocols familiar in quantum information. Although the mathematical methods are the same the meaning of the gauge group will be different. The measure we introduce, `twist', is constructed as a Wilson loop from a correlation induced holonomy. The measure can be understood as the global asymmetry of the bipartite correlations in a loop of three or more qubits; if the holonomy is trivial (the identity matrix), the bipartite correlations can be globally untwisted using general local qubit operations, the gauge group of our theory, which turns out to be the group of Lorentz transformations familiar from special relativity. If it is not possible to globally untwist the bipartite correlations in a state globally using local operations, the twistedness is given by a non-trivial element of the Lorentz group, the correlation induced holonomy. We provide several analytical examples of twisted and untwisted states for three qubits, the most elementary non-trivial loop one can imagine., Comment: 13 pages, 3 figures, title changed, results and content remain unchanged

It has been widely debated whether Arctic sea-ice loss can reach a tipping point beyond which a large sea-ice area disappears abruptly. The theory of dynamical systems predicts a slowing down when a system destabilises towards a tipping point. In simple stochastic systems this can result in increasing variance and autocorrelation, potentially yielding an early warning of an abrupt change. Here we aim to establish whether the loss of Arctic sea ice would follow these conceptual predictions, and which trends in sea ice variability can be expected from pre-industrial conditions toward an Arctic that is ice-free during the whole year. To this end, we apply a model hierarchy consisting of two box models and one comprehensive Earth system model. We find a consistent and robust decrease of the ice volume's annual relaxation time before summer ice is lost because thinner ice can adjust more quickly to perturbations. Thereafter, the relaxation time increases, mainly because the system becomes dominated by the ocean water's large heat capacity when the ice-free season becomes longer. Both trends carry over to the autocorrelation of sea ice thickness in time series. These changes are robust to the nature and origin of climate variability in the models and hardly depend on the balance of feedbacks. Therefore, characteristic trends can be expected in the future. As these trends are not specific to the existence of abrupt ice loss, the prospects for early warnings seem very limited. This result also has implications for statistical indicators in other systems whose effective mass changes over time, affecting the trend of their relaxation time. However, the robust relation between state and variability would allow an estimate of sea-ice variability from only short observations. This could help one to estimate the likelihood and persistence of extreme events in the future., Comment: Discussion Paper, currently (Jan. 2016) under review in The Cryosphere

We examine the relationship between the mean and the variability of Arctic sea-ice coverage and volume in a large range of climates from globally ice-covered to globally ice-free conditions. Using a hierarchy of two column models and several comprehensive Earth system models, we consolidate the results of earlier studies and show that mechanisms found in simple models also dominate the interannual variability of Arctic sea ice in complex models. In contrast to predictions based on very idealised dynamical systems, we find a consistent and robust decrease of variance and autocorrelation of sea-ice volume before summer sea ice is lost. We attribute this to the fact that thinner ice can adjust more quickly to perturbations. Thereafter, the autocorrelation increases, mainly because it becomes dominated by the ocean water's large heat capacity when the ice-free season becomes longer. We show that these changes are robust to the nature and origin of climate variability in the models and do not depend on whether Arctic sea-ice loss occurs abruptly or irreversibly. We also show that our climate is changing too rapidly to detect reliable changes in autocorrelation of annual time series. Based on these results, the prospects of detecting statistical early warning signals before an abrupt sea-ice loss at a "tipping point" seem very limited. However, the robust relation between state and variability can be useful to build simple stochastic climate models and to make inferences about past and future sea-ice variability from only short observations or reconstructions.

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.

We generalize a method of detecting an approaching bifurcation in a time series of a noisy system from the special case of one dynamical variable to multiple dynamical variables. For a system described by a stochastic differential equation consisting of an autonomous deterministic part with one dynamical variable and an additive white noise term, small perturbations away from the system's fixed point will decay slower the closer the system is to a bifurcation. This phenomenon is known as critical slowing down and all such systems exhibit this decay-type behaviour. However, when the deterministic part has multiple coupled dynamical variables, the possible dynamics can be much richer, exhibiting oscillatory and chaotic behaviour. In our generalization to the multi-variable case, we find additional indicators to decay rate, such as frequency of oscillation. In the case of approaching a homoclinic bifurcation, there is no change in decay rate but there is a decrease in frequency of oscillations. The expanded method therefore adds extra tools to help detect and classify approaching bifurcations given multiple time series, where the underlying dynamics are not fully known. Our generalisation also allows bifurcation detection to be applied spatially if one treats each spatial location as a new dynamical variable. One may then determine the unstable spatial mode(s). This is also something that has not been possible with the single variable method. The method is applicable to any set of time series regardless of its origin, but may be particularly useful when anticipating abrupt changes in the multi-dimensional climate system.

We examine several aspects of the ocean-atmosphere system over the last 30 000 years, by carrying out simulations with prescribed ice sheets, atmospheric CO2 concentration, and orbital parameters. We use the GENIE-1 model with a frictional geostrophic ocean, dynamic sea ice, an energy balance atmosphere, and a land-surface scheme with fixed vegetation. A transient simulation, with boundary conditions derived from ice-core records and ice sheet reconstructions, is compared with equilibrium snapshot simulations, including the Last Glacial Maximum (21 000 years before present; 21 kyrBP), mid-Holocene (6 kyrBP) and pre-industrial. The equilibrium snapshot simulations are all very similar to their corresponding time period in the transient simulation, indicating that over the last 30 000 years, the model's ocean-atmosphere system is close to equilibrium with its boundary conditions. However, our simulations neglect the transfer of fresh water from and to the ocean, resulting from the growth and decay of ice sheets, which would, in reality, lead to greater disequilibrium. Additionally, the GENIE-1 model exhibits a rather limited response in terms of its Atlantic Meridional Overturning Circulation (AMOC) over the 30 000 years; a more sensitive AMOC would also be likely to lead to greater disequilibrium. We investigate the method of accelerating the boundary conditions of a transient simulation and find that the Southern Ocean is the region most affected by the acceleration. The Northern Hemisphere, even with a factor of 10 acceleration, is relatively unaffected. The results are robust to changes to several tunable parameters in the model. They also hold when a higher vertical resolution is used in the ocean.

We present a minimal spatial model of vegetation carbon, soil carbon and soil water storage and the exchange of energy, water and carbon with the atmosphere. The efficient numerical terrestrial scheme (ENTS) is designed for long time period simulations and large ensemble studies in Earth system models of intermediate complexity (EMICs). ENTS includes new parameterisations of vegetation fractional cover and roughness length as functions of vegetation carbon, and a relationship between soil carbon storage and soil water holding capacity. We make and justify the approximation that when the solar forcing is a diurnal average, as in our EMIC, the land radiation balance equilibrates with the atmosphere within a few days. This allows us to solve directly for equilibrium land temperature, making ENTS very computationally efficient and avoiding problems of numerical instability that beset many land surface schemes. We tune the carbon cycle parameters towards observed values of global carbon storage in vegetation and soil and estimated global fluxes of net photosynthesis, vegetation respiration, leaf litter and soil respiration. When the model is forced with long term monthly mean fields of NCEP reanalysis climate data, we find ENTS yields broadly accurate patterns of vegetation and soil carbon storage, vegetation fraction, surface albedo, land temperature and evaporation.

Practical implementations of quantum cryptography use attenuated laser pulses as the signal source rather than single photons. The channels used to transmit are also lossy. Here we give a simple derivation of two beam-splitting attacks on quantum cryptographic systems using laser pulses, either coherent or mixed states with any mean photon number. We also give a simple derivation of a photon-number splitting attack, the most advanced, both in terms of performance and technology required. We find bounds on the maximum disturbance for a given mean photon number and observed channel transmission efficiency for which a secret key can be distilled. We start by reviewing two incoherent attacks that can be used on single photon quantum cryptographic systems. These results are then adapted to systems that use laser pulses and lossy channels., Comment: to appear in J. Mod. Opt

Global phase factors of topological origin, resulting from cyclic local $\rm{SU}$ evolution, called topological phases, were first described in [Phys. Rev. Lett. {\bf 90}, 230403 (2003)], in the case of entangled qubit pairs. In this paper we investigate topological phases in multi-qubit systems as the result of cyclic local $\rm{SU(2)}$ evolution. These phases originate from the topological structure of the local $\rm{SU(2)}$-orbits and are an attribute of most entangled multi-qubit systems. We discuss the relation between topological phases and SLOCC-invariant polynomials and give examples where topological phases appear. A general method to find the values of the topological phases in an $n$-qubit system is described and a complete list of these phases for up to seven qubits is given., Comment: Minor changes; journal reference added

We analyze the effects of quantum correlations, such as entanglement and discord, on the efficiency of phase estimation by studying four quantum circuits that can be readily implemented using NMR techniques. These circuits define a standard strategy of repeated single-qubit measurements, a classical strategy where only classical correlations are allowed, and two quantum strategies where nonclassical correlations are allowed. In addition to counting space (number of qubits) and time (number of gates) requirements, we introduce mixedness as a key constraint of the experiment. We compare the efficiency of the four strategies as a function of the mixedness parameter. We find that the quantum strategy gives square-root of N enhancement over the standard strategy for the same amount of mixedness. This result applies even for highly mixed states that have nonclassical correlations but no entanglement., 10 pages, 6 figures

We explore a geometric approach to generating local SU(2) and $SL(2,\mathbb{C})$ invariants for a collection of qubits inspired by lattice gauge theory. Each local invariant or 'gauge' invariant is associated to a distinct closed path (or plaquette) joining some or all of the qubits. In lattice gauge theory, the lattice points are the discrete space-time points, the transformations between the points of the lattice are defined by parallel transporters and the gauge invariant observable associated to a particular closed path is given by the Wilson loop. In our approach the points of the lattice are qubits, the link-transformations between the qubits are defined by the correlations between them and the gauge invariant observable, the local invariants associated to a particular closed path are also given by a Wilson loop-like construction. The link transformations share many of the properties of parallel transporters although they are not undone when one retraces one's steps through the lattice. This feature is used to generate many of the invariants. We consider a pure three qubit state as a test case and find we can generate a complete set of algebraically independent local invariants in this way, however the framework given here is applicable to mixed states composed of any number of $d$ level quantum systems. We give an operational interpretation of these invariants in terms of observables., Comment: 9 pages, 3 figures

Recently, it was demonstrated by Son et al., Phys. Rev. Lett. \textbf{102}, 110404 (2009), that a separable bipartite continuous variable quantum system can violate the Clauser-Horne-Shimony-Holt (CHSH) inequality via operationally local transformations. Operationally local transformations are parameterized only by local variables, however in order to allow violation of the CHSH inequality a maximally entangled ancilla was necessary. The use of the entangled ancilla in this scheme caused the state under test to become dependent on the measurement choice one uses to calculate the CHSH inequality thus violating one of the assumptions used in deriving a Bell inequality, namely the `free will' or `statistical independence' assumption. The novelty in this scheme however is that the measurement settings can be external free parameters. In this paper we generalize these operationally local transformations for multipartite Bell inequalities (with dichotomic observables) and provide necessary and sufficient conditions for violation within this scheme. Namely, a violation of a multipartite Bell inequality in this setting is contingent on whether an ancillary system admits any realistic, local hidden variable model i.e. whether the ancilla violates the given Bell inequality. These results indicate that violation of a Bell inequality performed on a system does not necessarily imply that the system is nonlocal. In fact the system under test may be completely classical. However, nonlocality must have resided somewhere, this may have been in the environment, the physical variables used to manipulate the system or the detectors themselves provided the measurement settings are external free variables., 6 pages

We discuss the problem of the separation of total correlations in a given quantum state into entanglement, dissonance, and classical correlations using the concept of relative entropy as a distance measure of correlations. This allows us to put all correlations on an equal footing. Entanglement and dissonance, whose definition is introduced here, jointly belong to what is known as quantum discord. Our methods are completely applicable for multipartite systems of arbitrary dimensions. We investigate additivity relations between different correlations and show that dissonance may be present in pure multipartite states., 5 pages, two figures

When a multi-qubit state evolves under local unitaries it may obtain a geometric phase, a feature dependent on the geometry of the state's projective Hilbert space. A correction term to this geometric phase in addition to the local subsystem phases may appear from correlations between the subsystems. We find this correction term can be characterized completely either by the entanglement or completely by the classical correlations for several classes of entangled state. States belonging to the former set are W states and their mixtures, while members of the latter set are cluster states, GHZ states and two classes of bound entangled state. We probe the structures of these states more finely using local invariants and suggest the cause of the entanglement correction is a gauge field like $SL(2,\mathbb{C})$ invariant recently introduced named twist., Comment: 21 pages, 2 figures. Contribution to the Proceedings of the 40th Symposium on Mathematical Physics, Torun, Poland, 25-28 June 2008

We have used the Grid ENabled Integrated Earth system modelling (GENIE) framework to undertake a systematic search for bi-stability of the ocean thermohaline circulation (THC) for different surface grids and resolutions of 3-D ocean (GOLDSTEIN) under a 3-D dynamical atmosphere model (IGCM). A total of 407,000 years were simulated over a three month period using Grid computing. We find bi-stability of the THC despite significant, quasi-periodic variability in its strength driven by variability in the dynamical atmosphere. The position and width of the hysteresis loop depends on the choice of surface grid (longitude-latitude or equal area), but is less sensitive to changes in ocean resolution. For the same ocean resolution, the region of bi-stability is broader with the IGCM than with a simple energy-moisture balance atmosphere model (EMBM). Feedbacks involving both ocean and atmospheric dynamics are found to promote THC bi-stability. THC switch-off leads to increased import of freshwater at the southern boundary of the Atlantic associated with meridional overturning circulation. This is counteracted by decreased freshwater import associated with gyre and diffusive transports. However, these are localised such that the density gradient between North and South is reduced tending to maintain the THC off state. THC switch-off can also generate net atmospheric freshwater input to the Atlantic that tends to maintain the off state. The ocean feedbacks are present in all resolutions, across most of the bi-stable region, whereas the atmosphere feedback is strongest in the longitude–latitude grid and around the transition where the THC off state is disappearing. Here the net oceanic freshwater import due to the overturning mode weakens, promoting THC switch-on, but the atmosphere counteracts this by increasing net freshwater input. This increases the extent of THC bi-stability in this version of the model.

When an entangled state evolves under local unitaries, the entanglement in the state remains fixed. Here we show the dynamical phase acquired by an entangled state in such a scenario can always be understood as the sum of the dynamical phases of its subsystems. In contrast, the equivalent statement for the geometric phase is not generally true unless the state is separable. For an entangled state an additional term is present, the mutual geometric phase, that measures the change the additional correlations present in the entangled state make to the geometry of the state space. For $N$ qubit states we find this change can be explained solely by classical correlations for states with a Schmidt decomposition and solely by quantum correlations for W states., Comment: 4 pages, 1 figure, improved presentation, results and conclusions unchanged from v1. Accepted for publication in PRA

A new Earth System Model of Intermediate Complexity, GENIE-1, is used to simulate the most recent glacial-interglacial cycle by prescribing orbital forcing, atmospheric CO2 concentration, and the time evolution of ice sheet extent and orography. A series of experiments investigates uncertainty in the amplitude, frequency and location of prescribed meltwater pulses (MWPs) associated with Heinrich events in the North Atlantic and layers enriched in ice rafted debris around Antarctica. Associated with each MWP is a flux into the ocean of very light glacial oxygen isotope ratios, which serve as a tracer of the melt water. Additionally accounted for are temperature-related changes in the fractionation of stable oxygen isotopes between water and calcite. Modelled forwards from 120 000 years ago, simulated oxygen isotope records can thus be directly compared with measurements in calcite taken from International Marine Global Change Study (IMAGES) and Ocean Drilling Program (ODP) sediment cores at three locations representative of the North and South Atlantic, and the South Pacific. During the period of simulation corresponding to Marine Isotope Stage 3, the best agreement between the simulated oxygen isotope record in the North Atlantic and core measurements is found in the experiment that includes MWPs around Antarctica as well as into the North Atlantic. This challenges previous assumptions about the dominant role of northern ice sheets in glacial sea-level variability.

A new Earth system model, GENIE-1, is presented which comprises a 3-D frictional geostrophic ocean, phosphate-restoring marine biogeochemistry, dynamic and thermodynamic sea-ice, land surface physics and carbon cycling, and a seasonal 2-D energy-moisture balance atmosphere. Three sets of model climate parameters are used to explore the robustness of the results and for traceability to earlier work. The model versions have climate sensitivity of 2.8–3.3°C and predict atmospheric CO2 close to present observations. Six idealized total fossil fuel CO2 emissions scenarios are used to explore a range of 1,100–15,000 GtC total emissions and the effect of rate of emissions. Atmospheric CO2 approaches equilibrium in year 3000 at 420–5,660 ppmv, giving 1.5–12.5°C global warming. The ocean is a robust carbon sink of up to 6.5 GtC year−1. Under ‘business as usual’, the land becomes a carbon source around year 2100 which peaks at up to 2.5 GtC year−1. Soil carbon is lost globally, boreal vegetation generally increases, whilst under extreme forcing, dieback of some tropical and sub-tropical vegetation occurs. Average ocean surface pH drops by up to 1.15 units. A Greenland ice sheet melt threshold of 2.6°C local warming is only briefly exceeded if total emissions are limited to 1,100 GtC, whilst 15,000 GtC emissions cause complete Greenland melt by year 3000, contributing 7 m to sea level rise. Total sea-level rise, including thermal expansion, is 0.4–10 m in year 3000 and ongoing. The Atlantic meridional overturning circulation shuts down in two out of three model versions, but only under extreme emissions including exotic fossil fuel resources.

In the context of two-particle interferometry, we construct a parallel transport condition that is based on the maximization of coincidence intensity with respect to local unitary operations on one of the subsystems. The dependence on correlation is investigated and it is found that the holonomy group is generally non-Abelian, but Abelian for uncorrelated systems. It is found that our framework contains the L\'{e}vay geometric phase [2004 {\it J. Phys. A: Math. Gen.} {\bf 37} 1821] in the case of two-qubit systems undergoing local SU(2) evolutions., Comment: Minor corrections; journal reference added

Equilibrium climate sensitivity—which remains the largest uncertainty in climate projections—is constrained to a ‘likely’ range of 2.2–3.4 K by taking into account the variability of global temperature about long-term historical warming. Equilibrium climate sensitivity (ECS) is the long-term change in global mean surface temperature predicted to occur in response to an instantaneous doubling of atmospheric carbon dioxide concentrations. It is an inherently artificial metric, but is nonetheless an important tool when comparing climate models, and a key point of policy discussion. The seemingly intractable range of ECS estimates complicates policy making because the response of the real climate system to the lowest and highest predicted temperature change would translate into radically different policy options. Peter Cox and colleagues now constrain climate models by their ability to simulate observed variations in climate, and conclude that ECS has a central estimate of 2.8 degrees Celsius (°C), which sits towards the middle to lower end of current estimates, and a range of 2.2–3.4 °C. Importantly, their approach allows them to almost exclude ECS estimates above 4.5 °C or below 1.5 °C. Equilibrium climate sensitivity (ECS) remains one of the most important unknowns in climate change science. ECS is defined as the global mean warming that would occur if the atmospheric carbon dioxide (CO2) concentration were instantly doubled and the climate were then brought to equilibrium with that new level of CO2. Despite its rather idealized definition, ECS has continuing relevance for international climate change agreements, which are often framed in terms of stabilization of global warming relative to the pre-industrial climate. However, the ‘likely’ range of ECS as stated by the Intergovernmental Panel on Climate Change (IPCC) has remained at 1.5–4.5 degrees Celsius for more than 25 years1. The possibility of a value of ECS towards the upper end of this range reduces the feasibility of avoiding 2 degrees Celsius of global warming, as required by the Paris Agreement. Here we present a new emergent constraint on ECS that yields a central estimate of 2.8 degrees Celsius with 66 per cent confidence limits (equivalent to the IPCC ‘likely’ range) of 2.2–3.4 degrees Celsius. Our approach is to focus on the variability of temperature about long-term historical warming, rather than on the warming trend itself. We use an ensemble of climate models to define an emergent relationship2 between ECS and a theoretically informed metric of global temperature variability. This metric of variability can also be calculated from observational records of global warming3, which enables tighter constraints to be placed on ECS, reducing the probability of ECS being less than 1.5 degrees Celsius to less than 3 per cent, and the probability of ECS exceeding 4.5 degrees Celsius to less than 1 per cent.

Despite major advances in climate science over the last 30 years, persistent uncertainties in projections of future climate change remain. Climate projections are produced with increasingly complex models which attempt to represent key processes in the Earth system, including atmospheric and oceanic circulations, convection, clouds, snow, sea-ice, vegetation and interactions with the carbon cycle. Uncertainties in the representation of these processes feed through into a range of projections from the many state-of-the-art climate models now being developed and used worldwide. For example, despite major improvements in climate models, the range of equilibrium global warming due to doubling carbon dioxide still spans a range of more than three. Here we review a promising way to make use of the ensemble of climate models to reduce the uncertainties in the sensitivities of the real climate system. The emergent constraint approach uses the model ensemble to identify a relationship between an uncertain aspect of the future climate and an observable variation or trend in the contemporary climate. This review summarises previous published work on emergent constraints, and discusses the huge promise and potential dangers of the approach. Most importantly, it argues that emergent constraints should be based on well-founded physical principles such as the fluctuation-dissipation theorem. It is hoped that this review will stimulate physicists to contribute to the rapidly developing field of emergent constraints on climate projections, bringing to it much needed rigour and physical insights., 39 pages, 7 figures, submitted to Reviews of Modern Physics

Ecology and Biogeography in Sri Lanka. Edited by C.H. Fernando. Dr W. Junk: 1984. Pp.505. Dfl. 265, 110, 67.50. Biogeography and Ecology of the Seychelles Islands. Edited by D.R. Stoddart. Dr W. Junk: 1984. Pp.691. Dfl. 300, 115, 76.50.

In a very interesting paper Wolda (1969) describes a remarkable cline in gene frequency in the snail Cepaea nemoralis (L.), which has remained fairly constant over a period of 12 years. He asks the question, if the environment is stationary, how can one distinguish between those cases in which the gene frequency stays constant from a regulating form of selection, and those cases in which it stays constant because of a balance resulting from changes in the direction of selection from time to time. On p. 631 he raises various difficulties, and he says 'I hope some statistician will accept the challenge'. This problem has a fairly simple solution in the terms stated. If the selective forces are not tending to produce an equilibrium, either by heterozygote advantage or by any of the other methods known (Williamson 1958) then there will be some selective forces whose action will tend to move the gene frequency continuously in one direction, which we may call 'up' and others which will tend to move it in what we may call 'down'. In the stationary environment postulated by Wolda, the net expected change of gene frequency will be the resultant of these two, and will, in all real cases, be either up or down. That is, despite fluctuations that will take place from time to time, there will be a net movement of the gene frequency in one way or the other. This change will not usually be linear, and in any case the change at any one time will be Markovian, depending on a previous point reached, so it is not valid to test this by a regression of gene frequency against time. However, there is a simple way to test the hypothesis. Let the number of observations be k + 1, taken at time to to tk, and the frequency observed by qo to qk. Calculate successive, nested, variances vi based on the values from qo to qi inclusive, with i from 1 to k. Then, by the Central Limit theorem, if the polymorphism is a balanced one the vi will show no tendency to increase and will approach a fixed value asymptotically. On the other hand, if the polymorphism is not governed, then the series vi will tend to increase. The vi are, of course, not independent, but a persistent increase in their values is nevertheless valid evidence against the hypothesis that one is observing a balanced polymorphism in a stationary environment. This method will not work unless the environment is stationary. When it is not, the problems are much more acute, and can be solved only with a knowledge either of the nature of the selective forces, or of the values of the relevant environmental variables, or both. These considerations can be ignored in examining Wolda's data, because here the variance is remarkably stationary, and we can say there is every indication that he is working with a balanced polymorphism. Without knowing how this system is brought about though, we cannot start to estimate the size of the selective forces from the observed