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Sökning: L773:1364 503X > (2020-2023)

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1.
  • Abrahao, FS, et al. (författare)
  • Emergence and algorithmic information dynamics of systems and observers
  • 2022
  • Ingår i: Philosophical transactions. Series A, Mathematical, physical, and engineering sciences. - : The Royal Society. - 1471-2962. ; 380:2227, s. 20200429-
  • Tidskriftsartikel (refereegranskat)abstract
    • One of the challenges of defining emergence is that one observer’s prior knowledge may cause a phenomenon to present itself as emergent that to another observer appears reducible. By formalizing the act of observing as mutual perturbations between dynamical systems, we demonstrate that the emergence of algorithmic information does depend on the observer’s formal knowledge, while being robust vis-a-vis other subjective factors, particularly: the choice of programming language and method of measurement; errors or distortions during the observation; and the informational cost of processing. This is called observer-dependent emergence (ODE). In addition, we demonstrate that the unbounded and rapid increase of emergent algorithmic information implies asymptotically observer-independent emergence (AOIE). Unlike ODE, AOIE is a type of emergence for which emergent phenomena will be considered emergent no matter what formal theory an observer might bring to bear. We demonstrate the existence of an evolutionary model that displays the diachronic variant of AOIE and a network model that displays the holistic variant of AOIE. Our results show that, restricted to the context of finite discrete deterministic dynamical systems, computable systems and irreducible information content measures, AOIE is the strongest form of emergence that formal theories can attain.This article is part of the theme issue ‘Emergent phenomena in complex physical and socio-technical systems: from cells to societies’.
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2.
  • Andrighetto, Giulia, et al. (författare)
  • A research agenda for the study of social norm change
  • 2022
  • Ingår i: Philosophical Transactions. Series A. - : ROYAL SOC. - 1364-503X .- 1471-2962. ; 380:2227
  • Forskningsöversikt (refereegranskat)abstract
    • Social norms have been investigated across many disciplines for many years, but until recently, studies mainly provided indirect, implicit and correlational support for the role of social norms in driving behaviour. To understand how social norms, and in particular social norm change, can generate a large-scale behavioural change to deal with some of the most pressing challenges of our current societies, such as climate change and vaccine hesitancy, we discuss and review several recent advances in social norm research that enable a more precise underpinning of the role of social norms: how to identify their existence, how to establish their causal effect on behaviour and when norm change may pass tipping points. We advocate future research on social norms to study norm change through a mechanism-based approach that integrates experimental and computational methods in theory-driven, empirically calibrated agent-based models. As such, social norm research may move beyond unequivocal praising of social norms as the missing link between self-interested behaviour and observed cooperation or as the explanation for (the lack of) social tipping. It provides the toolkit to understand explicitly where, when and how social norms can be a solution to solve large-scale problems, but also to recognize their limits.This article is part of the theme issue 'Emergent phenomena in complex physical and socio-technical systems: from cells to societies'.
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3.
  • Argatov, Ivan (författare)
  • Controlling the adhesive pull-off force via the change of contact geometry
  • 2021
  • Ingår i: Philosophical Transactions. Series A. - : The Royal Society. - 1364-503X .- 1471-2962. ; 379:2203
  • Tidskriftsartikel (refereegranskat)abstract
    • A first-order asymptotic analysis of the Griffith energy balance in the Johnson-Kendall-Roberts model of adhesive contact under non-symmetric perturbation of the contact geometry is presented. The pull-off force is evaluated in explicit form. A particular case of adhesive contact between a relatively stiff sphere and an elastic half-space is considered under the assumption that the sphere geometry is changed by the application of an arbitrary lateral normal surface loading. The effect of the sphere Poisson's ratio on controlling the adhesive pull-off force is considered.This article is part of a discussion meeting issue 'A cracking approach to inventing new tough materials: fracture stranger than friction'.
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4.
  • Bhatnagar, Akshay, et al. (författare)
  • Rate of formation of caustics in heavy particles advected by turbulence
  • 2022
  • Ingår i: Philosophical Transactions. Series A. - : The Royal Society. - 1364-503X .- 1471-2962. ; 380:2219
  • Tidskriftsartikel (refereegranskat)abstract
    • The rate of collision and the relative velocities of the colliding particles in turbulent flows are a crucial part of several natural phenomena, e.g. rain formation in warm clouds and planetesimal formation in protoplanetary discs. The particles are often modelled as passive, but heavy and inertial. Within this model, large relative velocities emerge due to formation of singularities (caustics) of the gradient matrix of the velocities of the particles. Using extensive direct numerical simulations of heavy particles in both two (direct and inverse cascade) and three-dimensional turbulent flows, we calculate the rate of formation of caustics, J as a function of the Stokes number (St). The best approximation to our data is J similar to exp(-C/St), in the limit St -> 0 where C is a non-universal constant. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.
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5.
  • Bostan, Lars, et al. (författare)
  • Upscaling of lignin precursor melt spinning by bicomponent spinning and its use for carbon fibre production
  • 2021
  • Ingår i: Philosophical Transactions. Series A. - : Royal Society Publishing. - 1364-503X .- 1471-2962. ; 379:2209
  • Tidskriftsartikel (refereegranskat)abstract
    • Upscaling lignin-based precursor fibre production is an essential step in developing bio-based carbon fibre from renewable feedstock. The main challenge in upscaling of lignin fibre production by melt spinning is its melt behaviour and rheological properties, which differ from common synthetic polymers used in melt spinning. Here, a new approach in melt spinning of lignin, using a spin carrier system for producing bicomponent fibres, has been introduced. An ethanol extracted lignin fraction from LignoBoost process of commercial softwood kraft black liquor was used as feedstock. After additional heat treatment, melt spinning was performed in a pilot-scale spinning unit. For the first time, biodegradable polyvinyl alcohol (PVA) was used as a spin carrier to enable the spinning of lignin by improving the required melt strength. PVA-sheath/lignin-core bicomponent fibres were manufactured. Afterwards, PVA was dissolved by washing with water. Pure lignin fibres were stabilized and carbonized, and tensile properties were measured. The measured properties, tensile modulus of 81.1 ± 3.1 GPa and tensile strength of 1039 ± 197 MPa, are higher than the majority of lignin-based carbon fibres reported in the literature. This new approach can significantly improve the melt spinning of lignin and solve problems related to poor spinnability of lignin and results in the production of high-quality lignin-based carbon fibres. This article is part of the theme issue 'Bio-derived and bioinspired sustainable advanced materials for emerging technologies (part 2)'. © 2021 The Author(s).
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6.
  • Chafik, Léon, 1985-, et al. (författare)
  • Observed mechanisms activating the recent subpolar North Atlantic Warming since 2016
  • 2023
  • Ingår i: Philosophical Transactions. Series A. - 1364-503X .- 1471-2962. ; 381:2262
  • Tidskriftsartikel (refereegranskat)abstract
    • The overturning circulation of the subpolar North Atlantic (SPNA) plays a fundamental role in Earth's climate variability and change. Here, we show from observations that the recent warming period since about 2016 in the eastern SPNA involves increased western boundary density at the intergyre boundary, likely due to enhanced buoyancy forcing as a response to the strong increase in the North Atlantic Oscillation since the early 2010s. As these deep positive density anomalies spread southward along the western boundary, they enhance the North Atlantic Current and associated meridional heat transport at the intergyre region, leading to increased influx of subtropical heat into the eastern SPNA. Based on the timing of this chain of events, we conclude that this recent warming phase since about 2016 is primarily associated with this observed mechanism of changes in deep western boundary density, an essential element in these interactions.This article is part of a discussion meeting issue 'Atlantic overturning: new observations and challenges'.
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7.
  • Field, A. R., et al. (författare)
  • Comparing pedestal structure in JET-ILW H-mode plasmas with a model for stiff ETG turbulent heat transport
  • 2023
  • Ingår i: Philosophical Transactions. Series A. - : The Royal Society. - 1364-503X .- 1471-2962. ; 381:2242
  • Tidskriftsartikel (refereegranskat)abstract
    • A predictive model for the electron temperature profile of the H-mode pedestal is described, and its results are compared with the pedestal structure of JET-ILW plasmas. The model is based on a scaling for the gyro-Bohm normalized, turbulent electron heat flux qe/qe,gB resulting from electron temperature gradient (ETG) turbulence, derived from results of nonlinear gyrokinetic (GK) calculations for the steep gradient region. By using the local temperature gradient scale length L-Te in the normalization, the dependence of q(e)/q(e,g)B on the normalized gradients R/L-Te and R/(Lne) can be represented by a unified scaling with the parameter eta(e) = L-ne/L-Te, to which the linear stability of ETG turbulence is sensitive when the density gradient is sufficiently steep. For a prescribed density profile, the value of R/L-Te determined from this scaling, required to maintain a constant electron heat flux qe across the pedestal, is used to calculate the temperature profile. Reasonable agreement with measurements is found for different cases, the model providing an explanation of the relative widths and shifts of the T-e and n(e) profiles, as well as highlighting the importance of the separatrix boundary conditions. Other cases showing disagreement indicate conditions where other branches of turbulence might dominate.This article is part of a discussion meeting issue "H-mode transition and pedestal studies in fusion plasmas'.
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8.
  • Flammini, Francesco, Senior Lecturer, 1978- (författare)
  • Digital twins as run-time predictive models for the resilience of cyber-physical systems : a conceptual framework
  • 2021
  • Ingår i: Philosophical Transactions. Series A. - : ROYAL SOC. - 1364-503X .- 1471-2962. ; 379:2207
  • Tidskriftsartikel (refereegranskat)abstract
    • Digital twins (DT) are emerging as an extremely promising paradigm for run-time modelling and performability prediction of cyber-physical systems (CPS) in various domains. Although several different definitions and industrial applications of DT exist, ranging from purely visual three-dimensional models to predictive maintenance tools, in this paper, we focus on data-driven evaluation and prediction of critical dependability attributes such as safety. To that end, we introduce a conceptual framework based on autonomic systems to host DT run-time models based on a structured and systematic approach. We argue that the convergence between DT and self-adaptation is the key to building smarter, resilient and trustworthy CPS that can self-monitor, self-diagnose and-ultimately-self-heal. The conceptual framework eases dependability assessment, which is essential for the certification of autonomous CPS operating with artificial intelligence and machine learning in critical applications. This article is part of the theme issue 'Towards symbiotic autonomous systems'.
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9.
  • Grennfelt, Peringe, et al. (författare)
  • A chronology of global air quality
  • 2020
  • Ingår i: Philosophical Transactions. Series A. - : The Royal Society. - 1364-503X .- 1471-2962.
  • Tidskriftsartikel (refereegranskat)abstract
    • Air pollution has been recognized as a threat to human health since the time of Hippocrates, ca 400 BC. Successive written accounts of air pollution occur in different countries through the following two millennia until measurements, from the eighteenth century onwards, show the growing scale of poor air quality in urban centres and close to industry, and the chemical characteristics of the gases and particulate matter. The industrial revolution accelerated both the magnitude of emissions of the primary pollutants and the geographical spread of contributing countries as highly polluted cities became the defining issue, culminating with the great smog of London in 1952. Europe and North America dominated emissions and suffered the majority of adverse effects until the latter decades of the twentieth century, by which time the transboundary issues of acid rain, forest decline and ground-level ozone became the main environmental and political air quality issues. As controls on emissions of sulfur and nitrogen oxides (SO2 and NOx) began to take effect in Europe and North America, emissions in East and South Asia grew strongly and dominated global emissions by the early years of the twenty-first century. The effects of air quality on human health had also returned to the top of the priorities by 2000 as new epidemiological evidence emerged. By this time, extensive networks of surface measurements and satellite remote sensing provided global measurements of both primary and secondary pollutants. Global emissions of SO2 and NOx peaked, respectively, in ca 1990 and 2018 and have since declined to 2020 as a result of widespread emission controls. By contrast, with a lack of actions to abate ammonia, global emissions have continued to grow. This article is part of a discussion meeting issue ‘Air quality, past present and future’
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10.
  • Kaynak, O., et al. (författare)
  • Towards symbiotic autonomous systems
  • 2021
  • Ingår i: Philosophical Transactions. Series A. - : The Royal Society. - 1364-503X .- 1471-2962. ; 379:2720
  • Tidskriftsartikel (refereegranskat)abstract
    • Starting in the second half of the last century, the widespread use of computers has significantly changed the lifestyles of humankind. Since then, with the explosive and perhaps disruptive developments in digital technology, the internet and Internet of Things and artificial intelligence have promoted a growing interaction among humans and machines, with technology capable of extending human capabilities (augmentation) and making devices seamlessly interact with humans. In the longer term, there is a growing consensus that this will create symbiotic autonomous systems (SASs). Although this is a trend that will see maturity possibly in the next decade, research and industry interest are already strong, resulting in several concrete results ready to hit the market. What matters most in the context of SASs is the varying degrees of autonomy they have, their capability to evolve (e.g. to learn and adapt), and their ability to interact with their environment, between themselves, and with humans. Hence, SAS is a class of advanced information and control technology that has gradually evolved, based on cybernetics, information theory, artificial intelligence, bionics, neurophysiology and computer science. At the same time, it has also promoted the joint development and mutual penetration of multiple fields such as microelectronics, life sciences and automation technology. In all future areas, SASs will become a seamless extension of our body and mind and an integral part of our life.
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11.
  • Khrennikov, Andrei, 1958- (författare)
  • Coherent decision making stimulated within the social laser : open quantum systems framework
  • 2023
  • Ingår i: Philosophical Transactions. Series A. - : Royal Society. - 1364-503X .- 1471-2962. ; 381:2252
  • Forskningsöversikt (refereegranskat)abstract
    • We start with a brief review on social laser theory which is newly framed with the notion of an infon-social energy quantum carrying coarse-grained information content. Infons are the excitations of the quantum social-information field. Humans are analogues of atoms-social atoms absorbing and emitting infons. Another new development is coupling of the social laser with the decision making model based on open quantum systems. The role of the environment for social atoms is played by the strong coherent social-information field-the output of social lasing. We analyse a simple quantum master equation generating decision jumps towards the coherent decision directed by the social laser beam. As an illustrative example, we analyse the possibility to create a societal benefit laser, i.e. directed for societal benefit.This article is part of the theme issue 'Thermodynamics 2.0: Bridging the natural and social sciences (Part 1)'.
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13.
  • Liu, Bing, et al. (författare)
  • Passenger flow anomaly detection in urban rail transit networks with graph convolution network-informer and Gaussian Bayes models
  • 2023
  • Ingår i: Philosophical Transactions. Series A. - : The Royal Society. - 1364-503X .- 1471-2962. ; 381:2254
  • Tidskriftsartikel (refereegranskat)abstract
    • Passenger flow anomaly detection in urban rail transit networks (URTNs) is critical in managing surging demand and informing effective operations planning and controls in the network. Existing studies have primarily focused on identifying the source of anomalies at a single station by analysing the time-series characteristics of passenger flow. However, they ignored the high-dimensional and complex spatial features of passenger flow and the dynamic behaviours of passengers in URTNs during anomaly detection. This article proposes a novel anomaly detection methodology based on a deep learning framework consisting of a graph convolution network (GCN)-informer model and a Gaussian naive Bayes model. The GCN-informer model is used to capture the spatial and temporal features of inbound and outbound passenger flows, and it is trained on normal datasets. The Gaussian naive Bayes model is used to construct a binary classifier for anomaly detection, and its parameters are estimated by feeding the normal and abnormal test data into the trained GCN-informer model. Experiments are conducted on a real-world URTN passenger flow dataset from Beijing. The results show that the proposed framework has superior performance compared to existing anomaly detection algorithms in detecting network-level passenger flow anomalies. This article is part of the theme issue 'Artificial intelligence in failure analysis of transportation infrastructure and materials'.
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14.
  • Lv, Zhihan, Dr. 1984-, et al. (författare)
  • Automatic identification of pavement cracks in public roads using an optimized deep convolutional neural network model
  • 2023
  • Ingår i: Philosophical Transactions. Series A. - : The Royal Society. - 1364-503X .- 1471-2962. ; 381:2254
  • Tidskriftsartikel (refereegranskat)abstract
    • The current study aims to improve the efficiency of automatic identification of pavement distress and improve the status quo of difficult identification and detection of pavement distress. First, the identification method of pavement distress and the types of pavement distress are analysed. Then, the design concept of deep learning in pavement distress recognition is described. Finally, the mask region-based convolutional neural network (Mask R-CNN) model is designed and applied in the recognition of road crack distress. The results show that in the evaluation of the model's comprehensive recognition performance, the highest accuracy is 99%, and the lowest accuracy is 95% after the test and evaluation of the designed model in different datasets. In the evaluation of different crack identification and detection methods, the highest accuracy of transverse crack detection is 98% and the lowest accuracy is 95%. In longitudinal crack detection, the highest accuracy is 98% and the lowest accuracy is 92%. In mesh crack detection, the highest accuracy is 98% and the lowest accuracy is 92%. This work not only provides an in-depth reference for the application of deep CNNs in pavement distress recognition but also promotes the improvement of road traffic conditions, thus contributing to the progression of smart cities in the future.This article is part of the theme issue 'Artificial intelligence in failure analysis of transportation infrastructure and materials'.
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15.
  • Menzel, A., et al. (författare)
  • Extremal states and coupling properties in electroelasticity
  • 2022
  • Ingår i: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. - : The Royal Society. - 1364-503X .- 1471-2962. ; 380:2234
  • Tidskriftsartikel (refereegranskat)abstract
    • Electroelastic materials possess properties most attractive for the design of smart devices and systems such as actuators and sensors. Typical polymers show changes in shape under the action of an electric field, and vice versa, together with fast actuation times, high strain levels and low elastic moduli. This paper deals with an Ogden model inspired framework for large deformation electroelasticity which, as a special case, can also be reduced to the modelling of transversely isotropic elasticity. Extremal (local) states are elaborated based on a coaxiality analysis, i.e. extremal states of energy are considered at fixed deformation and changing direction of electric field, respectively, fixed electric field and changing principal directions of deformation. This analysis results in extremal states when stresses and strain commutate, respectively, dielectric displacements and electric field are aligned. In order to further elaborate electromechanical coupling properties, the sensitivity of stresses with respect to electric field is analysed. This sensitivity is represented by a third-order tensor which, in general, depends on deformation and electric field. To illustrate this third-order tensor, a decomposition into deviators is adopted. Related norms of these deviators, together with the electromechanical coupling contribution to the augmented energy, are investigated for different states under homogeneous deformation and changing electric field direction. The analysis is considered to contribute to a better understanding of electromechanical coupling properties and extremal states in large deformation electroelasticity and by that, as a long-term goal, may contribute to the improved design of related smart devices and systems. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.
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16.
  • Metzler, Holger (författare)
  • A decrease in the age of respired carbon from the terrestrial biosphere and increase in the asymmetry of its distribution
  • 2023
  • Ingår i: Philosophical Transactions A: Mathematical, Physical and Engineering Sciences. - 1364-503X .- 1471-2962. ; 381
  • Tidskriftsartikel (refereegranskat)abstract
    • We provide here a model-based estimate of the transit time of carbon through the terrestrial biosphere, since the time of carbon uptake through photosynthesis until its release through respiration. We explored the consequences of increasing productivity versus increasing respiration rates on the transit time distribution and found that while higher respiration rates induced by higher temperature increase the transit time because older carbon is respired, increases in productivity cause a decline in transit times because more young carbon is available to supply increased metabolism. The combined effect of increases in temperature and productivity results in a decrease in transit times, with the productivity effect dominating over the respiration effect. By using an ensemble of simulation trajectories from the Carbon Data Model Framework (CARDAMOM), we obtained time-dependent transit time distributions incorporating the twentieth century global change. In these simulations, transit time declined over the twentieth century, suggesting an increased productivity effect that augmented the amount of respired young carbon, but also increasing the release of old carbon from high latitudes. The transit time distribution of carbon becomes more asymmetric over time, with more carbon transiting faster through tropical and temperate regions, and older carbon being respired from high latitude regions.
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17.
  • Sallee, J. B., et al. (författare)
  • Southern ocean carbon and heat impact on climate
  • 2023
  • Ingår i: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. - 1364-503X .- 1471-2962. ; 381:2249
  • Tidskriftsartikel (refereegranskat)abstract
    • The Southern Ocean greatly contributes to the regulation of the global climate by controlling important heat and carbon exchanges between the atmosphere and the ocean. Rates of climate change on decadal timescales are therefore impacted by oceanic processes taking place in the Southern Ocean, yet too little is known about these processes. Limitations come both from the lack of observations in this extreme environment and its inherent sensitivity to intermittent processes at scales that are not well captured in current Earth system models. The Southern Ocean Carbon and Heat Impact on Climate programme was launched to address this knowledge gap, with the overall objective to understand and quantify variability of heat and carbon budgets in the Southern Ocean through an investigation of the key physical processes controlling exchanges between the atmosphere, ocean and sea ice using a combination of observational and modelling approaches. Here, we provide a brief overview of the programme, as well as a summary of some of the scientific progress achieved during its first half. Advances range from new evidence of the importance of specific processes in Southern Ocean ventilation rate (e.g. storm-induced turbulence, sea-ice meltwater fronts, wind-induced gyre circulation, dense shelf water formation and abyssal mixing) to refined descriptions of the physical changes currently ongoing in the Southern Ocean and of their link with global climate.This article is part of a discussion meeting issue 'Heat and carbon uptake in the Southern Ocean: the state of the art and future priorities'.
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20.
  • Tessin, Allyson, et al. (författare)
  • Benthic phosphorus cycling within the Eurasian marginal sea ice zone
  • 2020
  • Ingår i: Philosophical Transactions. Series A. - : The Royal Society. - 1364-503X .- 1471-2962. ; 378:2181
  • Tidskriftsartikel (refereegranskat)abstract
    • The Arctic Ocean region is currently undergoing dramatic changes, which will likely alter the nutrient cycles that underpin Arctic marine ecosystems. Phosphate is a key limiting nutrient for marine life but gaps in our understanding of the Arctic phosphorus (P) cycle persist. In this study, we investigate the benthic burial and recycling of phosphorus using sediments and pore waters from the Eurasian Arctic margin, including the Barents Sea slope and the Yermak Plateau. Our results highlight that P is generally lost from sediments with depth during organic matter respiration. On the Yermak Plateau, remobilization of P results in a diffusive flux of P to the seafloor of between 96 and 261 mu molm(-2)yr(-1). On the Barents Sea slope, diffusive fluxes of P are much larger (1736-2449 mu molm(-2)yr(-1)), but these fluxes are into near-surface sediments rather than to the bottom waters. The difference in cycling on the Barents Sea slope is controlled by higher fluxes of fresh organic matter and active iron cycling. As changes in primary productivity, ocean circulation and glacial melt continue, benthic P cycling is likely to be altered with implications for P imported into the Arctic Ocean Basin. This article is part of the theme issue 'The changing Arctic Ocean: consequences for biological communities, biogeochemical processes and ecosystem functioning'.
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22.
  • Varner, Ruth K., et al. (författare)
  • Permafrost thaw driven changes in hydrology and vegetation cover increase trace gas emissions and climate forcing in Stordalen Mire from 1970 to 2014
  • 2022
  • Ingår i: Philosophical Transactions. Series A. - : The Royal Society. - 1364-503X .- 1471-2962. ; 380:2215
  • Tidskriftsartikel (refereegranskat)abstract
    • Permafrost thaw increases active layer thickness, changes landscape hydrology and influences vegetation species composition. These changes alter belowground microbial and geochemical processes, affecting production, consumption and net emission rates of climate forcing trace gases. Net carbon dioxide (CO2) and methane (CH4) fluxes determine the radiative forcing contribution from these climate-sensitive ecosystems. Permafrost peatlands may be a mosaic of dry frozen hummocks, semi-thawed or perched sphagnum dominated areas, wet permafrost-free sedge dominated sites and open water ponds. We revisited estimates of climate forcing made for 1970 and 2000 for Stordalen Mire in northern Sweden and found the trend of increasing forcing continued into 2014. The Mire continued to transition from dry permafrost to sedge and open water areas, increasing by 100% and 35%, respectively, over the 45-year period, causing the net radiative forcing of Stordalen Mire to shift from negative to positive. This trend is driven by transitioning vegetation community composition, improved estimates of annual CO2 and CH4 exchange and a 22% increase in the IPCC's 100-year global warming potential (GWP_100) value for CH4. These results indicate that discontinuous permafrost ecosystems, while still remaining a net overall sink of C, can become a positive feedback to climate change on decadal timescales.This article is part of a discussion meeting issue ‘Rising methane: is warming feeding warming? (part 2)’.
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