Spanning Trees with Many Leaves in Graphs without Diamonds and Blossoms

It is known that graphs on n vertices with minimum degree at least 3 have spanning trees with at least n/4+2 leaves and that this can be improved to (n+4)/3 for cubic graphs without the diamond K_4-e as a subgraph. We generalize the second result by proving that every graph with minimum degree at least 3, without diamonds and certain subgraphs called blossoms, has a spanning tree with at least (n+4)/3 leaves, and generalize this further by allowing vertices of lower degree. We show that it is necessary to exclude blossoms in order to obtain a bound of the form n/3+c. We use the new bound to obtain a simple FPT algorithm, which decides in O(m)+O^*(6.75^k) time whether a graph of size m has a spanning tree with at least k leaves. This improves the best known time complexity for MAX LEAF SPANNING TREE.Comment: 25 pages, 27 Figure

Irreversible ocean thermal expansion under carbon dioxide removal

n the Paris Agreement in 2015 countries agreed on holding global mean surface air warming to "well below 2 degrees C above pre-industrial" levels, but the emission reduction pledges under that agreement are not ambitious enough to meet this target. Therefore, the question arises of whether restoring global warming to this target after exceeding it by artificially removing CO2 from the atmosphere is possible. One important aspect is the reversibility of ocean heat uptake and associated sea level rise, which have very long (centennial to millennial) response timescales. In this study the response of sea level rise due to thermal expansion to a 1% yearly increase of atmospheric CO2 up to a quadrupling of the pre-industrial concentration followed by a 1% yearly decline back to the pre-industrial CO2 concentration is examined using the University of Victoria Earth System Climate Model (UVic ESCM). We find that global mean thermosteric sea level (GMTSL) continues to rise for several decades after atmospheric CO2 starts to decline and does not return to pre-industrial levels for over 1000 years after atmospheric CO2 is restored to the pre-industrial concentration. This finding is independent of the strength of vertical sub-grid-scale ocean mixing implemented in the model. Furthermore, GMTSL rises faster than it declines in response to a symmetric rise and decline in atmospheric CO2 concentration partly because the deep ocean continues to warm for centuries after atmospheric CO2 returns to the pre-industrial concentration. Both GMTSL rise and decline rates increase with increasing vertical ocean mixing. Exceptions from this behaviour arise if the overturning circulations in the North Atlantic and Southern Ocean intensify beyond pre-industrial levels in model versions with lower vertical mixing, which leads to rapid cooling of the deep ocean

The Sensitivity of the Proportionality between Temperature Change and Cumulative CO2 Emissions to Ocean Mixing

The ratio of global mean surface air temperature change to cumulative CO2 emissions, referred to as transient climate response to cumulative CO2 emissions (TCRE), has been shown to be approximately constant on centennial time scales. The mechanisms behind this constancy are not well understood, but previous studies suggest that compensating effects of ocean heat and carbon fluxes, which are governed by the same ocean mixing processes, could be one cause for this approximate constancy. This hypothesis is investigated by forcing different versions of the University of Victoria Earth System Climate Model, which differ in the ocean mixing parameterization, with an idealized scenario of 1% annually increasing atmospheric CO2 until quadrupling of the preindustrial CO2 concentration and constant concentration thereafter. The relationship between surface air warming and cumulative emissions remains close to linear, but the TCRE varies between model versions, spanning the range of 1.2°–2.1°C EgC−1 at the time of CO2 doubling. For all model versions, the TCRE is not constant over time while atmospheric CO2 concentrations increase. It is constant after atmospheric CO2 stabilizes at 1120 ppm, because of compensating changes in temperature sensitivity (temperature change per unit radiative forcing) and cumulative airborne fraction. The TCRE remains approximately constant over time even if temperature sensitivity, determined by ocean heat flux, and cumulative airborne fraction, determined by ocean carbon flux, are taken from different model versions with different ocean mixing settings. This can partially be explained with temperature sensitivity and cumulative airborne fraction following similar trajectories, which suggests ocean heat and carbon fluxes scale approximately linearly with changes in vertical mixing

The influence of non-CO2 forcings on cumulative carbon emissions budgets

Carbon budgets provide a useful tool for policymakers to help meet the global climate targets, as they specify total allowable carbon emissions consistent with limiting warming to a given temperature threshold. Non-CO _2 forcings have a net warming effect in the Representative Concentration Pathways (RCP) scenarios, leading to reductions in remaining carbon budgets based on CO _2 forcing alone. Carbon budgets consistent with limiting warming to below 2.0 °C, with and without accounting for the effects of non-CO _2 forcings, were assessed in inconsistent ways by the Intergovernmental Panel on Climate Change (IPCC), making the effects of non-CO _2 forcings hard to identify. Here we use a consistent approach to compare 1.5 °C and 2.0 °C carbon budgets with and without accounting for the effects of non-CO _2 forcings, using CO _2 -only and RCP8.5 simulations. The median allowable carbon budgets for 1.5 °C and 2.0 °C warming are reduced by 257 PgC and 418 PgC, respectively, and the uncertainty ranges on the budgets are reduced by more than a factor of two when accounting for the net warming effects of non-CO _2 forcings. While our overall results are consistent with IPCC, we use a more robust methodology, and explain the narrower uncertainty ranges of carbon budgets when non-CO _2 forcings are included. We demonstrate that most of the reduction in carbon budgets is a result of the direct warming effect of the non-CO _2 forcings, with a secondary contribution from the influence of the non-CO _2 forcings on the carbon cycle. Such carbon budgets are expected to play an increasingly important role in climate change mitigation, thus understanding the influence of non-CO _2 forcings on these budgets and their uncertainties is critical

A Human Development Framework for CO2 Reductions

Although developing countries are called to participate in CO2 emission reduction efforts to avoid dangerous climate change, the implications of proposed reduction schemes in human development standards of developing countries remain a matter of debate. We show the existence of a positive and time-dependent correlation between the Human Development Index (HDI) and per capita CO2 emissions from fossil fuel combustion. Employing this empirical relation, extrapolating the HDI, and using three population scenarios, the cumulative CO2 emissions necessary for developing countries to achieve particular HDI thresholds are assessed following a Development As Usual approach (DAU). If current demographic and development trends are maintained, we estimate that by 2050 around 85% of the world's population will live in countries with high HDI (above 0.8). In particular, 300Gt of cumulative CO2 emissions between 2000 and 2050 are estimated to be necessary for the development of 104 developing countries in the year 2000. This value represents between 20% to 30% of previously calculated CO2 budgets limiting global warming to 2{\deg}C. These constraints and results are incorporated into a CO2 reduction framework involving four domains of climate action for individual countries. The framework reserves a fair emission path for developing countries to proceed with their development by indexing country-dependent reduction rates proportional to the HDI in order to preserve the 2{\deg}C target after a particular development threshold is reached. Under this approach, global cumulative emissions by 2050 are estimated to range from 850 up to 1100Gt of CO2. These values are within the uncertainty range of emissions to limit global temperatures to 2{\deg}C.Comment: 14 pages, 7 figures, 1 tabl

The Carbon Dioxide Removal Model Intercomparison Project (CDRMIP): rationale and experimental protocol for CMIP6

The recent IPCC reports state that continued anthropogenic greenhouse gas emissions are changing the climate, threatening "severe, pervasive and irreversible" impacts. Slow progress in emissions reduction to mitigate climate change is resulting in increased attention to what is called geoengineering, climate engineering, or climate intervention – deliberate interventions to counter climate change that seek to either modify the Earth's radiation budget or remove greenhouse gases such as CO2 from the atmosphere. When focused on CO2, the latter of these categories is called carbon dioxide removal (CDR). Future emission scenarios that stay well below 2 °C, and all emission scenarios that do not exceed 1.5 °C warming by the year 2100, require some form of CDR. At present, there is little consensus on the climate impacts and atmospheric CO2 reduction efficacy of the different types of proposed CDR. To address this need, the Carbon Dioxide Removal Model Intercomparison Project (or CDRMIP) was initiated. This project brings together models of the Earth system in a common framework to explore the potential, impacts, and challenges of CDR. Here, we describe the first set of CDRMIP experiments, which are formally part of the 6th Coupled Model Intercomparison Project (CMIP6). These experiments are designed to address questions concerning CDR-induced climate "reversibility", the response of the Earth system to direct atmospheric CO2 removal (direct air capture and storage), and the CDR potential and impacts of afforestation and reforestation, as well as ocean alkalinization.

Estimating Carbon Budgets for Ambitious Climate Targets

Carbon budgets, which define the total allowable CO2 emissions associated with a given global climate target, are a useful way of framing the climate mitigation challenge. In this paper, we review the geophysical basis for the idea of a carbon budget, showing how this concept emerges from a linear climate response to cumulative CO2 emissions. We then discuss the difference between a “CO2-only carbon budget” associated with a given level of CO2-induced warming and an “effective carbon budget” associated with a given level of warming caused by all human emissions. We present estimates for the CO2-only and effective carbon budgets for 1.5 and 2 °C, based on both model simulations and updated observational data. Finally, we discuss the key contributors to uncertainty in carbon budget estimates and suggest some implications of this uncertainty for decision-making. Based on the analysis presented here, we argue that while the CO2-only carbon budget is a robust upper bound on allowable emissions for a given climate target, the size of the effective carbon budget is dependent on the how quickly we are able to mitigate non-CO2 greenhouse gas and aerosol emissions. This suggests that climate mitigation efforts could benefit from being responsive to a changing effective carbon budget over time, as well as to potential new information that could narrow uncertainty associated with the climate response to CO2 emissions