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.

Abstract. Climate sensitivity to CO2 remains the key uncertainty in projections of future climate change. 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. Equilibrium climate sensitivity (ECS) is vital for understanding longer-term climate change and stabilisation targets. In the IPCC 5th Assessment Report (AR5), the stated “likely” ranges (16 %–84 % confidence) of TCR (1.0–2.5 K) and ECS (1.5–4.5 K) were broadly consistent with the ensemble of CMIP5 Earth system models (ESMs) available at the time. However, many of the latest CMIP6 ESMs have larger climate sensitivities, with 5 of 34 models having TCR values above 2.5 K and an ensemble mean TCR of 2.0±0.4 K. Even starker, 12 of 34 models have an ECS value above 4.5 K. On the face of it, these latest ESM results suggest that the IPCC likely ranges 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 constrain the likely range of TCR to 1.3–2.1 K, with a central estimate of 1.68 K. We reach this conclusion through an emergent constraint approach which relates the value of TCR linearly to the global warming from 1975 onwards. This is a period when the signal-to-noise ratio of the net radiative forcing increases strongly, so that uncertainties in aerosol forcing become progressively less problematic. We find a consistent emergent constraint on TCR when we apply the same method to CMIP5 models. Our constraints on TCR are in good agreement with other recent studies which analysed CMIP ensembles. The relationship between ECS and the post-1975 warming trend is less direct and also non-linear. However, we are able to derive a likely range of ECS of 1.9–3.4 K from the CMIP6 models by assuming an underlying emergent relationship based on a two-box energy balance model. Despite some methodological differences; this is consistent with a previously published ECS constraint derived from warming trends in CMIP5 models to 2005. Our results seem to be part of a growing consensus amongst studies that have applied the emergent constraint approach to different model ensembles and to different aspects of the record of global warming.
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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 years. 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 relationship between ECS and a theoretically informed metric of global temperature variability. This metric of variability can also be calculated from observational records of global warming, 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.

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 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 analyse a large number of Earth System Models (ESMs) and find that there is some evidence that the temperatures of extreme events are rising faster than local background rises in mean summer temperatures. We find this to be true for almost all land regions when analysing the SSP585 scenario and for the decades from now until the end of the 21st Century. We find strong correlations between the level of acceleration and upward trends in sensible heat fluxes. In the few tropical regions where there is less correlation, we instead find a link to background latent plus sensible heat, which acts as a proxy for overall available energy. We then study extreme acceleration in the contemporary period, in both ESMs and ERA5 data. We find in these circumstances particularly strong evidence of faster warming of extreme events, but only for selected regions. We suggest this is a consequence of highly regional effects such as aerosols. Our analysis hints that as atmospheric composition changes move towards alteration by greenhouse gases only, there will be a more general globally-applicable occurrence of high-temperature extremes, with their mean increase more than the more general background warming levels.  

Abstract. Climate change is predicted to lead to major changes in terrestrial ecosystems. However, substantial differences in climate model projections for given scenarios of greenhouse gas emissions continue to limit detailed assessment. Here we show, using a traditional Köppen–Geiger bioclimate classification system, that the latest CMIP6 Earth system models actually agree well on the fraction of the global land surface that would undergo a major change per degree of global warming. Data from “historical” and “SSP585” model runs are used to create bioclimate maps at various degrees of global warming and to investigate the performance of the multi-model ensemble mean when classifying climate data into discrete categories. Using a streamlined Köppen–Geiger scheme with 13 classifications, global bioclimate classification maps at 2 and 4 K of global warming above a 1901–1931 reference period are presented. These projections show large shifts in bioclimate distribution, with an almost exclusive change from colder, wetter bioclimates to hotter, drier ones. Historical model run performance is assessed and examined by comparison with the bioclimatic classifications derived from the observed climate over the same time period. The fraction (f) of the land experiencing a change in its bioclimatic class as a function of global warming (ΔT) is estimated by combining the results from the individual models. Despite the discrete nature of the bioclimatic classification scheme, we find only a weakly saturating dependence of this fraction on global warming f =1-e-0.14ΔT, which implies about 13 % of land experiencing a major change in climate per 1 K increase in global mean temperature between the global warming levels of 1 and 3 K. Therefore, we estimate that stabilizing the climate at 1.5 K rather than 2 K of global warming would save over 7.5 million square kilometres of land from a major bioclimatic change.
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AbstractMany scenarios for limiting global warming to 1.5°C assume planetary‐scale carbon dioxide removal sufficient to exceed anthropogenic emissions, resulting in radiative forcing falling and temperatures stabilizing. However, such removal technology may prove unfeasible for technical, environmental, political, or economic reasons, resulting in continuing greenhouse gas emissions from hard‐to‐mitigate sectors. This may lead to constant concentration scenarios, where net anthropogenic emissions remain non‐zero but small, and are roughly balanced by natural carbon sinks. Such a situation would keep atmospheric radiative forcing roughly constant. Fixed radiative forcing creates an equilibrium “committed” warming, captured in the concept of “equilibrium climate sensitivity.” This scenario is rarely analyzed as a potential extension to transient climate scenarios. Here, we aim to understand the planetary response to such fixed concentration commitments, with an emphasis on assessing the resulting likelihood of exceeding temperature thresholds that trigger climate tipping points. We explore transients followed by respective equilibrium committed warming initiated under low to high emission scenarios. We find that the likelihood of crossing the 1.5°C threshold and the 2.0°C threshold is 83% and 55%, respectively, if today's radiative forcing is maintained until achieving equilibrium global warming. Under the scenario that best matches current national commitments (RCP4.5), we estimate that in the transient stage, two tipping points will be crossed. If radiative forcing is then held fixed after the year 2100, a further six tipping point thresholds are crossed. Achieving a trajectory similar to RCP2.6 requires reaching net‐zero emissions rapidly, which would greatly reduce the likelihood of tipping events.

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.Using an idealized general circulation model developed for exoplanet research here at Exeter, we simulate the transition from a normal to a superrotating atmospheric state. We look at the changes in typical early warning indicators of tipping which show critical slowing down as well as oscillatory behaviour close to the transition. Inspired by the studies of phase transitions we also look at the critical spatial modes and correlation lengths close to the transition.

Abstract. Several general circulation models (GCMs) have showed bifurcations of their atmospheric state under a broad range of warm climates. These include some of the more extreme global warming scenarios. This bifurcation can cause the transition to a superrotating state, a state where its angular momentum exceeds the solid body rotation of the planet. Here we use an idealized GCM to simulate this transition by altering a single non-dimensional control parameter, the thermal Rossby number. For a bifurcation induced transition there is potential for early warnings and we look for these here. Typically used early warning indicators, variance and lag 1 autocorrelation, calculated for the mean zonal equatorial wind speed, increase and peak just before the transition. The full autocorrelation function taken at multiple lags is also oscillatory, with a period of 25 days preceding the transition. This oscillatory behaviour is reminiscent of a Hopf bifurcation. Motivated by this extra structure, we use a generalised early warning vector technique to diagnose the dominant spatial modes of the horizontal windfield fluctuations. We find a zonal wavenumber zero pattern we call the `precursor' mode, that appears shortly before and disappears soon after the transition. We attribute the increase in the early warning indicators to this spatial precursor mode.
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Abstract. 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.
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Abstract
. The discovery of a near-proportionality between cumulative anthropogenic carbon dioxide emissions and global warming since pre-industrial times is arguably the most important policy-relevant simplification of climate change science in the last 25 years. Unfortunately, the latest CMIP6 Earth System Models continue to diagnose a wide range of carbon budgets for each level of global warming. In this paper we present emergent constraints on the carbon budget as a function of global warming, which combine the available CMIP6 Earth System Model historical simulations and future projections with observational estimates of global warming and anthropogenic CO2 emissions to the present day. We derive estimates for cumulative carbon budgets for the Paris targets of 1.5oC and 2oC of global warming of 812 +/- 121 PgC and 1048 +/- 167 PgC, which are significantly larger than the ensemble mean values from the CMIP6 models of 727 +/- 181 PgC and 911 +/- 247 PgC. Our approach produces estimates of the remaining carbon budgets for 1.5oC and 2.0oC within the uncertainty bounds of values given in the IPCC WG1 6th Assessment report, without requiring individual assumptions about the likely range of a number of climate and carbon cycle sensitivity factors, or implicit assumptions about changes in non-CO2 forcing factors. We infer a reduced mean specific carbon budget in the future (473+/-80 PgC/oC. beyond 1.2oC of global warming) compared to the past (553+/82 PgC/oC ), primarily due to the decline in cooling atmospheric aerosols. However, the linearity between future cumulative emissions and future global warming is found to be maintained at least until 4oC of global warming, and is consistent with an effective Transient Climate Response to Emissions (TCRE) of 2.1 oC per trillion tonnes of carbon (likely range of 1.8 to 2.6 oC/1000PgC), from now onwards.

Studies of the observed record of global warming suggest that the Earth’s climate sensitivity is at the lower end of the range produced by the CMIP6 Earth System Models (Jimenez and Mauritsen, 2019; Nijsse et al. 2020; Tokarska et al. 2020). However, studies based on top-of-the-atmosphere fluxes often suggest the opposite (Brown et al. 2017; Sherwood et al. 2020).Earthshine estimates (Goode et al. 2021) and satellite measurements of the planetary albedo from CERES (Loeb et al. 2018) both indicate that the Earth has darkened significantly over the past two decades. Planetary darkening is also simulated in CMIP6 historical simulations , but the models with the highest climate sensitivities tend to fit the observed decline in planetary albedo much better. Observed planetary darkening therefore favours higher climate sensitivities, but constraints based on ground-based global warming records favour lower climate senstivities.We explore this apparent paradox by calculating the contributions to changes in global warming that arise from diagnosable changes in planetary albedo and effective global emissivity, in both models and observational records. Differences between low and high sensitivity models are found to be predominantly due to the rate at which the modelled planetary albedo declines, which can in principle be due to a combination of forcing and feedbacks. However, planetary darkening in higher sensitivity models is primarily due to reductions in cloud cover, which results in a positive SW cloud feedback.By contrast, the planetary darkening seen in the CERES satellite record is driven not by reductions in cloud cover, but instead by the darkening of clouds, and to a lesser extent by the darkening of clear skies. This suggest that darkening in CERES is driven by reductions in aerosols, which leads to reductions in negative aerosol forcing.  Planetary darkening in CERES therefore seems to be due primarily to changes in aerosol forcing.Our proposed resolution to ‘The paradox of the darkening planet and the Earth’s climate sensitivity’ is therefore that climate sensitivity is indeed towards the lower end of the CMIP6 model range (as suggested by observed records of global warming), and that higher sensitivity models get the rate of planetary darkening ‘right’ but by the wrong mechanism (i.e. as a cloud forcing rather than as an aerosol feedback).  We will back this up by comparing the spatial patterns of planetary albedo change from models and the CERES satellite data, and finish by discussing possible implications for the time-varying aerosol precursor fields that are used to drive the CMIP6 simulations.

Climate change is predicted to lead to major changes in terrestrial
ecosystems. However, significant differences in climate model projections for
given scenarios of greenhouse gas emissions, continue to hinder detailed
assessment. Here we show, using a traditional Koppen-Geiger bioclimate
classification system, that the latest CMIP6 Earth System Models actually agree
very well on the fraction of the global land-surface that will undergo a
significant change per degree of global warming. Data from historical and
ssp585 model runs are used to create bioclimate maps at various degrees of
global warming, and to investigate the performance of the ensemble mean when
classifying climate data into discrete categories. Using a streamlined scheme
with 13 classifications, global bioclimate classification maps at 2K and 4K of
global warming above a 1901-1931 reference period are presented. These
projections show large shifts in bioclimate distribution, with an almost
exclusive change from colder, wetter bioclimates to hotter, dryer ones.
Historical model run performance is assessed and examined by comparison with
the bioclimatic classifications derived from the observed climate over the same
time period. The fraction of the land experiencing a change in its bioclimatic
class as a function of global warming is estimated by combining the results
from the individual models. Despite the discrete nature of the bioclimatic
classification scheme, we find only a weakly-saturating dependence of this
fraction on global warming which implies about 12 pct of land experiencing a
significant change in climate, per 1K increase in global mean temperature
between the global warming levels of 1 and 3K. Therefore, we estimate that
stabilising the climate at 1.5K rather than 2K of global warming, would save
over 7 million square kilometres of land from a major bioclimatic change.

Abstract. 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 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 measure now and another of major importance for in 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 Partial Differential Equations (PDEs). Such PDEs are valid at coarser scales than the equations coded in ESMs which capture finer high resolution gridbox--scale effects. We suggest that many ECs link to such an effective hidden PDE that is implicit in most or all ESMs. An EC may exist because its two quantities depend similarly on an ESM--specific internal bulk parameter in such a PDE, and with measurements constraining and revealing its (implicit) value. Alternatively, well--established process understanding coded at the ESM gridbox--scale, when aggregated, may generate a bulk parameter with a common ``emergent'' value across all ESMs. This single parameter may link uncertainties in a contemporary climate driver to those of a climate--related property of interest, the EC constraining the latter by measurements of the former. We offer illustrative examples of these concepts with generic differential equations and their solutions, placed in a conceptual EC framework.
.

Abstract. 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.
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. <p>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.</p><p>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.</p><p>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. </p>
.

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.

Abstract. Climate sensitivity to CO2 remains the key uncertainty in projections of future climate change. 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. Equilibrium climate sensitivity (ECS) is vital for understanding longer-term climate change and stabilisation targets. In the IPCC 5th Assessment Report (AR5), the stated “likely” ranges (16 %–84 % confidence) of TCR (1.0–2.5 K) and ECS (1.5–4.5 K) were broadly consistent with the ensemble of CMIP5 Earth system models (ESMs) available at the time. However, many of the latest CMIP6 ESMs have larger climate sensitivities, with 5 of 34 models having TCR values above 2.5 K and an ensemble mean TCR of 2.0±0.4 K. Even starker, 12 of 34 models have an ECS value above 4.5 K. On the face of it, these latest ESM results suggest that the IPCC likely ranges 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 constrain the likely range of TCR to 1.3–2.1 K, with a central estimate of 1.68 K. We reach this conclusion through an emergent constraint approach which relates the value of TCR linearly to the global warming from 1975 onwards. This is a period when the signal-to-noise ratio of the net radiative forcing increases strongly, so that uncertainties in aerosol forcing become progressively less problematic. We find a consistent emergent constraint on TCR when we apply the same method to CMIP5 models. Our constraints on TCR are in good agreement with other recent studies which analysed CMIP ensembles. The relationship between ECS and the post-1975 warming trend is less direct and also non-linear. However, we are able to derive a likely range of ECS of 1.9–3.4 K from the CMIP6 models by assuming an underlying emergent relationship based on a two-box energy balance model. Despite some methodological differences; this is consistent with a previously published ECS constraint derived from warming trends in CMIP5 models to 2005. Our results seem to be part of a growing consensus amongst studies that have applied the emergent constraint approach to different model ensembles and to different aspects of the record of global warming.
.

. <p>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.</p><p>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.</p>
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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.

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.

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 years. 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 relationship between ECS and a theoretically informed metric of global temperature variability. This metric of variability can also be calculated from observational records of global warming, 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.

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.

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.

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.

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.

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.

Abstract. The prospect of finding generic early warning signals of an approaching tipping point in a complex system has generated much recent interest. 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 find alternative early warning signals of tipping points due to local bifurcations in systems subjected to periodic forcing whose time scale 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. We show that the phase lag and amplification of the system response provide early warning signals, based on a linear dynamics approximation. Furthermore, the power spectrum of the system's time series reveals the generation of harmonics of the forcing period, the size of which are proportional to how nonlinear the system's response is becoming with nonlinear effects becoming more prominent closer to a bifurcation. We apply these indicators 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.
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Global phase factors of topological origin, resulting from cyclic local SU evolution, called topological phases, were first described in, in the case of entangled qubit pairs. In this paper we investigate topological phases in multiqubit systems as the result of cyclic local SU(2) evolution. These phases originate from the topological structure of the local SU(2) orbits and are an attribute of most entangled multiqubit systems. We discuss the relation between topological phases and stochastic local operations and classical communication (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. © 2012 American Physical Society.

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 Lévay geometric phase (2004 J. Phys. A: Math. Gen. 37 1821) in the case of two-qubit systems undergoing local SU(2) evolutions. © 2011 IOP Publishing Ltd.

We explore a geometric approach to generating local SU(2) and SL(2,C) invariants for a collection of qubits inspired by lattice gauge theory. Each local invariant or "gauge" invariant is associated with 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 with 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 with a particular closed path, are also given by a Wilson looplike 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 generating local unitary invariants for mixed states composed of any number of d-level quantum systems. We give an operational interpretation of these invariants in terms of observables. © 2011 American Physical Society.

We introduce a bridge between the familiar gauge field theory approaches used in many areas of modern physics such as quantum field theory and the stochastic local operations and classical communication protocols familiar in quantum information. Although the mathematical methods are the same, the meaning of the gauge group is 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 using local operations, the twistedness is given by a nontrivial 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 nontrivial loop one can imagine. © 2011 American Physical Society.

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 ffiffiffiffi N p 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.

Recently, it was demonstrated by Son, Phys. Rev. Lett.PRLTAO0031-900710. 1103/PhysRevLett.102.110404 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 parametrized 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. © 2010 the American Physical Society.

When a multi-qubit state evolves under local unitaries it may obtain a geometric phase, a feature dependent on the geometry of the state 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 that this correction term can be characterized completely either by the entanglement or 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 that the cause of the entanglement correction is a recently introduced gauge field-like SL(2,)-invariant called twist. © 2009 World Scientific Publishing Company.

When an entangled state evolves under local unitaries, the entanglement in the state remains fixed. Here we show that 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 that this change can be explained solely by classical correlations for states with a Schmidt decomposition and solely by quantum correlations for W states. © 2007 the American Physical Society.

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 beamsplitting 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. © 2003 Taylor & Francis Group, LLC.