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A graph theoretical approach to the analysis, comparison, and enumeration of crystal structures
(2008)
As an alternative approach to lattices and space groups, this work explores graph theory as a means to model crystal structures. The approach uses quotient graphs and nets - the graph theoretical equivalent of cells and lattices - to represent crystal structures. After a short review of related work, new classes of cycles in nets are introduced and their ability to distinguish between non-isomorphic nets and their computational complexity are evaluated. Then, two methods to estimate a structure’s density from the corresponding net are proposed. The first uses coordination sequences to estimate the number of nodes in a sphere, whereas the second method determines the maximal volume of a unit cell. Based on the quotient graph only, methods are proposed to determine whether nets consist of islands, chains, planes, or penetrating, disconnected sub-nets. An algorithm for the enumeration of crystal structures is revised and extended to a search for structures possessing certain properties. Particular attention is given to the exclusion of redundant nets and those, which, by the nature of their connectivity, cannot correspond to a crystal structure. Nets with four four-coordinated nodes, corresponding to sp3 hybridised carbon polymorphs with four atoms per unit cell, are completely enumerated in order to demonstrate the approach. In order to render quotient graphs and nets independent from crystal structures, they are reintroduced in a purely graph-theoretical way. Based on this, the issue of iso- and automorphism of nets is reexamined. It is shown that the topology of a net (that is the bonds in a crystal) constrains severely the symmetry of the embedding (that is the crystal), and in the case of connected nets the space group except for the setting. Several examples are studied and conclusions on phases are drawn (pseudo-cubic FeS2 versus pyrite; α- versus β- quartz; marcasite- versus rutile-like phases). As the automorphisms of certain quotient graphs stipulate a translational symmetry higher than an arbitrary embedding of the corresponding net would show, they are examined in more detail and a method to reduce the size of such quotient graphs is proposed. Besides two instructional examples with 2-dimensional graphs, the halite, calcite, magnesite, barytocalcite, and a strontium feldspar structures are discussed. For some of the structures it is shown that the quotient graph which is equivalent to a centred cell is reduced to a quotient graph equivalent to the primitive cell. For the partially disordered strontium feldspar, it is shown that even if it could be annealed to an ordered structure, the unit cell would likely remain unchanged. For the calcite and barytocalcite structures it is shown that the equivalent nets are not isomorphic.
Wetlands such as bogs, swamps, or freshwater marshes are hotspots of biodiversity. For 5.1 million km2 of inland wetlands, the dynamics of area and water storage, which strongly impact biodiversity and ecosystem services, were simulated using the global hydrological model WaterGAP. For the first time, the impacts of both human water use and man‐made reservoirs (WUR) and future climate change (CC) on wetlands around the globe were quantified. WUR impacts are concentrated in arid/semiarid regions, where WUR decreased mean wetland water storage by more than 5% on 8.2% of the mean wetland area during 1986–2005 (Am), with highest decreases in groundwater depletion area. Using output of three climate models, CC impacts on wetlands were quantified, distinguishing unavoidable impacts [i.e., at 2 °C global warming (GW)] from avoidable impacts (difference between 3 °C and 2 °C impacts). Even unavoidable CC impacts are projected to be much larger than WUR impacts, also in arid/semiarid regions. On most wetland area with reliable estimates, avoidable CC impacts are more than twice as large as unavoidable impacts. In case of 2 °C GW, half of Am is estimated to be unaffected by mean storage changes of more than 5%, but only one third in case of 3 °C GW. Temporal variability of water storage will increase for most wetlands. Wetlands in dry regions will be affected the most, particularly by water storage decreases in the dry season. Different from wealthier countries, low‐income countries will dominantly suffer from a decrease in wetland water storage due to CC.
Irrigation intensifies land use by increasing crop yield but also impacts water resources. It affects water and energy balances and consequently the microclimate in irrigated regions. Therefore, knowledge of the extent of irrigated land is important for hydrological and crop modelling, global change research, and assessments of resource use and management. Information on the historical evolution of irrigated lands is limited. The new global Historical Irrigation Dataset (HID) provides estimates of the temporal development of the area equipped for irrigation (AEI) between 1900 and 2005 at 5 arc-minute resolution. We collected subnational irrigation statistics from various sources and found that the global extent of AEI increased from 63 million ha (Mha) in 1900 to 112 Mha in 1950 and 306 Mha in 2005. We developed eight gridded versions of time series of AEI by combining subnational irrigation statistics with different data sets on the historical extent of cropland and pasture. Different rules were applied to maximize consistency of the gridded products to subnational irrigation statistics or to historical cropland and pasture data sets. The HID reflects very well the spatial patterns of irrigated land in the western United States as shown on historical maps. Mean aridity on irrigated land increased and river discharge decreased from 1900–1950 whereas aridity decreased from 1950–2005. The dataset and its documentation are made available in an open data repository at https://mygeohub.org/publications/8 (doi:10.13019/M2MW2G).
Irrigation intensifies land use by increasing crop yield but also impacts water resources. It affects water and energy balances and consequently the microclimate in irrigated regions. Therefore, knowledge of the extent of irrigated land is important for hydrological and crop modelling, global change research, and assessments of resource use and management. Information on the historical evolution of irrigated lands is limited. The new global historical irrigation data set (HID) provides estimates of the temporal development of the area equipped for irrigation (AEI) between 1900 and 2005 at 5 arcmin resolution. We collected sub-national irrigation statistics from various sources and found that the global extent of AEI increased from 63 million ha (Mha) in 1900 to 111 Mha in 1950 and 306 Mha in 2005. We developed eight gridded versions of time series of AEI by combining sub-national irrigation statistics with different data sets on the historical extent of cropland and pasture. Different rules were applied to maximize consistency of the gridded products to sub-national irrigation statistics or to historical cropland and pasture data sets. The HID reflects very well the spatial patterns of irrigated land as shown on historical maps for the western United States (around year 1900) and on a global map (around year 1960). Mean aridity on irrigated land increased and mean natural river discharge on irrigated land decreased from 1900 to 1950 whereas aridity decreased and river discharge remained approximately constant from 1950 to 2005. The data set and its documentation are made available in an open-data repository at https://mygeohub.org/publications/8 (doi:10.13019/M20599).
Climate change and its impacts already pose considerable challenges for societies that will further increase with global warming (IPCC, 2014a, b). Uncertainties of the climatic response to greenhouse gas emissions include the potential passing of large-scale tipping points (e.g. Lenton et al., 2008; Levermann et al., 2012; Schellnhuber, 2010) and changes in extreme meteorological events (Field et al., 2012) with complex impacts on societies (Hallegatte et al., 2013). Thus climate change mitigation is considered a necessary societal response for avoiding uncontrollable impacts (Conference of the Parties, 2010). On the other hand, large-scale climate change mitigation itself implies fundamental changes in, for example, the global energy system. The associated challenges come on top of others that derive from equally important ethical imperatives like the fulfilment of increasing food demand that may draw on the same resources. For example, ensuring food security for a growing population may require an expansion of cropland, thereby reducing natural carbon sinks or the area available for bio-energy production. So far, available studies addressing this problem have relied on individual impact models, ignoring uncertainty in crop model and biome model projections. Here, we propose a probabilistic decision framework that allows for an evaluation of agricultural management and mitigation options in a multi-impact-model setting. Based on simulations generated within the Inter-Sectoral Impact Model Intercomparison Project (ISI-MIP), we outline how cross-sectorally consistent multi-model impact simulations could be used to generate the information required for robust decision making.
Using an illustrative future land use pattern, we discuss the trade-off between potential gains in crop production and associated losses in natural carbon sinks in the new multiple crop- and biome-model setting. In addition, crop and water model simulations are combined to explore irrigation increases as one possible measure of agricultural intensification that could limit the expansion of cropland required in response to climate change and growing food demand. This example shows that current impact model uncertainties pose an important challenge to long-term mitigation planning and must not be ignored in long-term strategic decision making.
Vegetation responds to drought through a complex interplay of plant hydraulic mechanisms, posing challenges for model development and parameterization. We present a mathematical model that describes the dynamics of leaf water-potential over time while considering different strategies by which plant species regulate their water-potentials. The model has two parameters: the parameter λ describing the adjustment of the leaf water potential to changes in soil water potential, and the parameter Δψww describing the typical ‘well-watered’ leaf water potentials at non-stressed (near-zero) levels of soil water potential. Our model was tested and calibrated on 110 time-series datasets containing the leaf- and soil water potentials of 66 species under drought and non-drought conditions. Our model successfully reproduces the measured leaf water potentials over time based on three different regulation strategies under drought. We found that three parameter sets derived from the measurement data reproduced the dynamics of 53% of an drought dataset, and 52% of a control dataset [root mean square error (RMSE) < 0.5 MPa)]. We conclude that, instead of quantifying water-potential-regulation of different plant species by complex modeling approaches, a small set of parameters may be sufficient to describe the water potential regulation behavior for large-scale modeling. Thus, our approach paves the way for a parsimonious representation of the full spectrum of plant hydraulic responses to drought in dynamic vegetation models.
Artificial drainage of agricultural land, for example with ditches or drainage tubes, is used to avoid water logging and to manage high groundwater tables. Among other impacts it influences the nutrient balances by increasing leaching losses and by decreasing denitrification. To simulate terrestrial transport of nitrogen on the global scale, a digital global map of artificially drained agricultural areas was developed. The map depicts the percentage of each 5’ by 5’ grid cell that is equipped for artificial drainage. Information on artificial drainage in countries or sub-national units was mainly derived from international inventories. Distribution to grid cells was based, for most countries, on the "Global Croplands Dataset" of Ramankutty et al. (1998) and the "Digital Global Map of Irrigation Areas" of Siebert et al. (2005). For some European countries the CORINE land cover dataset was used instead of the both datasets mentioned above. Maps with outlines of artificially drained areas were available for 6 countries. The global drainage area on the map is 167 Mio hectares. For only 11 out of the 116 countries with information on artificial drainage areas, sub-national information could be taken into account. Due to this coarse spatial resolution of the data sources, we recommended to use the map of artificially drained areas only for continental to global scale assessments. This documentation describes the dataset, the data sources and the map generation, and it discusses the data uncertainty.
An easy-to-use model to evaluate conductivities at high and middle latitudes in the height range 70–100 km is presented. It is based on electron density profiles obtained with the EISCAT VHF radar during 11 years and on the neutral atmospheric model MSIS95. The model uses solar zenith angle, geomagnetic activity and season as input parameters. It was mainly constructed to study the properties of Schumann resonances that depend on such conductivity profiles.
Immersion freezing is the most relevant heterogeneous ice nucleation mechanism through which ice crystals are formed in mixed-phase clouds. In recent years, an increasing number of laboratory experiments utilizing a variety of instruments have examined immersion freezing activity of atmospherically relevant ice nucleating particles (INPs). However, an inter-comparison of these laboratory results is a difficult task because investigators have used different ice nucleation (IN) measurement methods to produce these results. A remaining challenge is to explore the sensitivity and accuracy of these techniques and to understand how the IN results are potentially influenced or biased by experimental parameters associated with these techniques.
Within the framework of INUIT (Ice Nucleation research UnIT), we distributed an illite rich sample (illite NX) as a representative surrogate for atmospheric mineral dust particles to investigators to perform immersion freezing experiments using different IN measurement methods and to obtain IN data as a function of particle concentration, temperature (T), cooling rate and nucleation time. Seventeen measurement methods were involved in the data inter-comparison. Experiments with seven instruments started with the test sample pre-suspended in water before cooling, while ten other instruments employed water vapor condensation onto dry-dispersed particles followed by immersion freezing. The resulting comprehensive immersion freezing dataset was evaluated using the ice nucleation active surface-site density (ns) to develop a representative ns(T) spectrum that spans a wide temperature range (−37 °C < T < −11 °C) and covers nine orders of magnitude in ns.
Our inter-comparison results revealed a discrepancy between suspension and dry-dispersed particle measurements for this mineral dust. While the agreement was good below ~ −26 °C, the ice nucleation activity, expressed in ns, was smaller for the wet suspended samples and higher for the dry-dispersed aerosol samples between about −26 and −18 °C. Only instruments making measurement techniques with wet suspended samples were able to measure ice nucleation above −18 °C. A possible explanation for the deviation between −26 and −18 °C is discussed. In general, the seventeen immersion freezing measurement techniques deviate, within the range of about 7 °C in terms of temperature, by three orders of magnitude with respect to ns. In addition, we show evidence that the immersion freezing efficiency (i.e., ns) of illite NX particles is relatively independent on droplet size, particle mass in suspension, particle size and cooling rate during freezing. A strong temperature-dependence and weak time- and size-dependence of immersion freezing efficiency of illite-rich clay mineral particles enabled the ns parameterization solely as a function of temperature. We also characterized the ns (T) spectra, and identified a section with a steep slope between −20 and −27 °C, where a large fraction of active sites of our test dust may trigger immersion freezing. This slope was followed by a region with a gentler slope at temperatures below −27 °C. A multiple exponential distribution fit is expressed as ns(T) = exp(23.82 × exp(−exp(0.16 × (T + 17.49))) + 1.39) based on the specific surface area and ns(T) = exp(25.75 × exp(−exp(0.13 × (T + 17.17))) + 3.34) based on the geometric area (ns and T in m−2 and °C, respectively). These new fits, constrained by using an identical reference samples, will help to compare IN measurement methods that are not included in the present study and, thereby, IN data from future IN instruments.
Immersion freezing is the most relevant heterogeneous ice nucleation mechanism through which ice crystals are formed in mixed-phase clouds. In recent years, an increasing number of laboratory experiments utilizing a variety of instruments have examined immersion freezing activity of atmospherically relevant ice-nucleating particles. However, an intercomparison of these laboratory results is a difficult task because investigators have used different ice nucleation (IN) measurement methods to produce these results. A remaining challenge is to explore the sensitivity and accuracy of these techniques and to understand how the IN results are potentially influenced or biased by experimental parameters associated with these techniques.
Within the framework of INUIT (Ice Nuclei Research Unit), we distributed an illite-rich sample (illite NX) as a representative surrogate for atmospheric mineral dust particles to investigators to perform immersion freezing experiments using different IN measurement methods and to obtain IN data as a function of particle concentration, temperature (T), cooling rate and nucleation time. A total of 17 measurement methods were involved in the data intercomparison. Experiments with seven instruments started with the test sample pre-suspended in water before cooling, while 10 other instruments employed water vapor condensation onto dry-dispersed particles followed by immersion freezing. The resulting comprehensive immersion freezing data set was evaluated using the ice nucleation active surface-site density, ns, to develop a representative ns(T) spectrum that spans a wide temperature range (−37 °C < T < −11 °C) and covers 9 orders of magnitude in ns.
In general, the 17 immersion freezing measurement techniques deviate, within a range of about 8 °C in terms of temperature, by 3 orders of magnitude with respect to ns. In addition, we show evidence that the immersion freezing efficiency expressed in ns of illite NX particles is relatively independent of droplet size, particle mass in suspension, particle size and cooling rate during freezing. A strong temperature dependence and weak time and size dependence of the immersion freezing efficiency of illite-rich clay mineral particles enabled the ns parameterization solely as a function of temperature. We also characterized the ns(T) spectra and identified a section with a steep slope between −20 and −27 °C, where a large fraction of active sites of our test dust may trigger immersion freezing. This slope was followed by a region with a gentler slope at temperatures below −27 °C. While the agreement between different instruments was reasonable below ~ −27 °C, there seemed to be a different trend in the temperature-dependent ice nucleation activity from the suspension and dry-dispersed particle measurements for this mineral dust, in particular at higher temperatures. For instance, the ice nucleation activity expressed in ns was smaller for the average of the wet suspended samples and higher for the average of the dry-dispersed aerosol samples between about −27 and −18 °C. Only instruments making measurements with wet suspended samples were able to measure ice nucleation above −18 °C. A possible explanation for the deviation between −27 and −18 °C is discussed. Multiple exponential distribution fits in both linear and log space for both specific surface area-based ns(T) and geometric surface area-based ns(T) are provided. These new fits, constrained by using identical reference samples, will help to compare IN measurement methods that are not included in the present study and IN data from future IN instruments.