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Carbon sequestration (Cseq) in soils and plant biomass is viewed as an important means of mitigating climate change. Recent global assessments have estimated considerable potential for terrestrial Cseq, but generally lack sensitivity to climate warming, nutrient limitations and perspective on local land use. These are important factors since higher temperatures can accelerate the decomposition of soil organic matter, nutrient availability affects plant productivity, while land use pressures put broader constraints on terrestrial organic matter inputs and storage. Here, we explore the potential for Cseq under changing land use, climate and nutrient conditions in a UK-based national scale case study. We apply an integrated terrestrial C–N–P cycle model with representative ranges of high-resolution climate and land use scenarios to estimate Cseq potential across the UK. If realistic UK targets for grassland restoration and afforestation over the next 30 years are met, we estimate that an additional 120 TgC could be sequestered by 2100 (similar to current annual UK greenhouse gas emissions or roughly 7% of net emission cuts needed in meeting net zero), conditional on climate change of <2 °C. Conversely, we estimate that UK arable expansion would reduce terrestrial carbon storage by a similar magnitude. The most pessimistic climate trajectories are predicted to cause net losses in UK soil carbon storage under all land use scenarios. Warmer climates substantially reduce the potential total terrestrial carbon storage gains offered by afforestation and grassland restoration. We conclude that although concerted land use change could make an important moderate contribution to national level Cseq for countries like the UK, soil Cseq only provides a contribution if we are on a low emission pathway, and is therefore conditional on deep global cuts to emissions from fossil fuels, deforestation and soil degradation.

Carbon sequestration (Cseq) in soils and plant biomass is viewed as an important means of mitigating climate change. Recent global assessments have estimated considerable potential for terrestrial Cseq, but generally lack sensitivity to climate warming, nutrient limitations and perspective on local land use. These are important factors since higher temperatures can accelerate the decomposition of soil organic matter, nutrient availability affects plant productivity, while land use pressures put broader constraints on terrestrial organic matter inputs and storage. Here, we explore the potential for Cseq under changing land use, climate and nutrient conditions in a UK-based national scale case study. We apply an integrated terrestrial C–N–P cycle model with representative ranges of high-resolution climate and land use scenarios to estimate Cseq potential across the UK. If realistic UK targets for grassland restoration and afforestation over the next 30 years are met, we estimate that an additional 120 TgC could be sequestered by 2100 (similar to current annual UK greenhouse gas emissions or roughly 7% of net emission cuts needed in meeting net zero), conditional on climate change of <2 °C. Conversely, we estimate that UK arable expansion would reduce terrestrial carbon storage by a similar magnitude. The most pessimistic climate trajectories are predicted to cause net losses in UK soil carbon storage under all land use scenarios. Warmer climates substantially reduce the potential total terrestrial carbon storage gains offered by afforestation and grassland restoration. We conclude that although concerted land use change could make an important moderate contribution to national level Cseq for countries like the UK, soil Cseq only provides a contribution if we are on a low emission pathway, and is therefore conditional on deep global cuts to emissions from fossil fuels, deforestation and soil degradation.

Carbon stores in the terrestrial biosphere globally represent over 50% of present-day organic carbon reservoirs and have significantly altered over the last three centuries owing to anthropogenic disturbances. Conversion of natural land to agricultural uses often results in a loss of soil carbon, whilst atmospheric deposition of pollutants such as nitrogen has increased carbon storage in both soil and biomass. Terrestrial carbon storage underpins a range of ecosystem services, including climate regulation, food production, and water services. This storage is crucial for sustainable land management. Quantification of terrestrial carbon cycling at regional and national scales, and understanding how human-induced drivers have impacted present-day carbon stores is therefore required to inform sustainable land use policy. This study applies the N14CP model, an integrated soil-plant biogeochemistry carbon-nitrogen-phosphorus model, across the United Kingdom to simulate changes in terrestrial carbon storage from 1700 to 2020. The analysis shows that change in anthropogenic terrestrial carbon storage is a complex picture comprising of gains in natural areas due to nitrogen deposition and afforestation, and losses in arable areas. We observed an overall net increase in total terrestrial carbon storage of 6.9%. We note, however, that continued increases in carbon storage cannot be assumed due to (i) reduced influence of future nitrogen deposition as these systems become limited by other nutrients, (ii) the need to continue enhanced nitrogen inputs to maintain carbon sequestered, and (iii) carbon declines in arable areas continuing alongside diminishing gains in other land use types. This research provides a full picture of anthropogenic impacts on terrestrial organic carbon storage, accounting for changing nutrient cycles at a national scale.

A key statistic describing climate change impacts is the ‘social cost of carbon dioxide’ (SCCO2), the projected cost to society of releasing an additional tonne of CO2. Cost-benefit integrated assessment models that estimate the SCCO2 lack robust representations of climate feedbacks, economy feedbacks, and climate extremes. We compare the PAGE-ICE model with the decade older PAGE09 and find that PAGE-ICE yields SCCO2 values about two times higher, because of its climate and economic updates. Climate feedbacks only account for a relatively minor increase compared to other updates. Extending PAGE-ICE with economy feedbacks demonstrates a manifold increase in the SCCO2 resulting from an empirically derived estimate of partially persistent economic damages. Both the economy feedbacks and other increases since PAGE09 are almost entirely due to higher damages in the Global South. Including an estimate of interannual temperature variability increases the width of the SCCO2 distribution, with particularly strong effects in the tails and a slight increase in the mean SCCO2. Our results highlight the large impacts of climate change if future adaptation does not exceed historical trends. Robust quantification of climate-economy feedbacks and climate extremes are demonstrated to be essential for estimating the SCCO2 and its uncertainty.

As a result of growing food demands, the area of land used globally for agriculture has rapidly increased over the last 300 years. Clearance of natural vegetation and conversion of land to agriculture is often associated with terrestrial carbon loss, from both vegetation and soil stores. Changes in terrestrial carbon storage has implications for food production, climate and water regulation. Quantifying these changes is therefore vital to understand the risks to and resilience of these benefits. Land use in the East of England has significantly changed during this period and is now predominantly used for agriculture, specifically arable use. In order to map changes to terrestrial carbon storage in this region since 1700, we apply a plant–soil system biogeochemistry model, N14CP. The model indicates carbon storage in the East of England has decreased by 109 Mt (−35.7%) during the study period, and whilst losses are observed in both soil and vegetation stores, vegetation losses as a result of forest clearance dominate. These findings have implications for carbon sequestration strategies; the largest carbon storage gains within the region are likely to be achieved through land-use transitions such as afforestation, rather than soil sequestration through changing arable management practices.

Relatively few people have visited the Earth’s icy cap, the Arctic. Cold and inhospitable, and dark for several months of the year, the region has been home to Inuit, Saami and other indigenous peoples for thousands of years. Adapting to the harsh conditions required plenty of ingenuity and persistence. Yet, the Arctic region is changing, and changing rapidly. The sea ice covering most of the Arctic Ocean, the vast Greenland ice sheet, the snow cover on land and the large area of frozen ground called permafrost, are all melting away. Why? Scientists are unanimous in the verdict: climate change caused by man-made emissions of greenhouse gases (Pachauri et al., 2014). The Arctic has been warming twice faster that the global average as a result (Overland et al., 2015), causing the extensive melting documented by several decades of satellite records and measurements on the ground (Stroeve et al., 2012; Mouginot et al., 2019; Chadburn et al., 2017). It is no wonder that the Arctic is sometimes called the barometer of global risk from climate change

Carbon stores in the terrestrial biosphere globally represent over 50% of present-day organic carbon reservoirs and have significantly altered over the last three centuries owing to anthropogenic disturbances. Conversion of natural land to agricultural uses often results in a loss of soil carbon, whilst atmospheric deposition of pollutants such as nitrogen has increased carbon storage in both soil and biomass. Terrestrial carbon storage underpins a range of ecosystem services, including climate regulation, food production, and water services. This storage is crucial for sustainable land management. Quantification of terrestrial carbon cycling at regional and national scales, and understanding how human-induced drivers have impacted present-day carbon stores is therefore required to inform sustainable land use policy. This study applies the N14CP model, an integrated soil-plant biogeochemistry carbon-nitrogen-phosphorus model, across the United Kingdom to simulate changes in terrestrial carbon storage from 1700 to 2020. The analysis shows that change in anthropogenic terrestrial carbon storage is a complex picture comprising of gains in natural areas due to nitrogen deposition and afforestation, and losses in arable areas. We observed an overall net increase in total terrestrial carbon storage of 6.9%. We note, however, that continued increases in carbon storage cannot be assumed due to (i) reduced influence of future nitrogen deposition as these systems become limited by other nutrients, (ii) the need to continue enhanced nitrogen inputs to maintain carbon sequestered, and (iii) carbon declines in arable areas continuing alongside diminishing gains in other land use types. This research provides a full picture of anthropogenic impacts on terrestrial organic carbon storage, accounting for changing nutrient cycles at a national scale.

Integrated assessment models (IAMs) are cross-disciplinary tools that explore how economic activity interacts with the environment and vice versa. They differ in their design and scope and are frequently used by the Intergovernmental Panel on Climate Change (IPCC), national environmental agencies and others to assess the economic consequences of climate change. One particular group of IAMs goes as far as estimating the total economic effect of climate change, which includes three essential components: costs of emissions abatement, costs of adaptation to changing climate, and residual impacts on the economy. The most prominent models in this group, the Dynamic Integrated Model of Climate and the Economy (DICE), Climate Framework for Uncertainty, Negotiation and Distribution (FUND) and Policy Analysis of the Greenhouse Effect (PAGE), follow a commonly adopted approach to use empirical damage functions as a way of translating climate-driven environmental changes into the economic impacts, which could be either costs or benefits. The chapter explores how damage functions are constructed and calibrated in these three IAMs by examining model documentation, and summarizes the limitations of the current approaches. It then reflects on how damage functions could be improved in order to help build confidence in these IAMs, and ultimately to provide better estimates of the impacts of climate change for decision-makers and academics.

The PAGE-ICE integrated assessment model (IAM) is a new version of the Policy Analysis of Greenhouse Effect (PAGE) IAM. It is a major update of the two previous versions of the model called PAGE09 and PAGE2002 (Hope 2006, 2013). PAGE-ICE includes several updates both to climate science and economics from IPCC AR5 and literature that followed, as well as novel statistical emulators of nonlinear Arctic feedbacks presented in the study "Climate policy implications of nonlinear decline of Arctic land permafrost and other cryosphere elements" (Yumashev et al., 2019). This chapter provides a detailed overview of the updates in PAGE-ICE other than the statistical emulators of the nonlinear Arctic feedbacks, which are described in detail in the above-mentioned study (Yumashev et al., 2019). The updates in PAGE-ICE illustrate possible directions for future developments of all small-scale IAMs focusing on the total economic effect of climate change, a group of models that includes PAGE, DICE and FUND. The overarching goal of any such developments is to reduce the uncertainties in the model parameterisation according to the latest scientific results, and to make the models more transparent. Only with a constant pursuit of this goal will the indicative results from the small-scale IAMs be deemed relevant for advising on climate policy decisions. The PAGE-ICE model is one of the many necessary steps in this direction.

IAMs connect physical and social science models to address cross-disciplinary questions, such as how does climate affect economies. There are different types of IAMs. One category of IAMs, for example, derives scenarios for future population, economies, technology and greenhouse gas (GHG) emissions, and explores how these may influence climate variables and, subsequently, the biosphere. These IAMs tend to be large, complex and computationally intensive. As a result, they are typically deterministic, as there is often insufficient information available to define probability distributions for the thousands of parameters of the model. Even if this could be done, computational power is often not sufficient to run large numbers of simulations with varying parameters. Examples of such IAMs include the Integrated Model to Assess the Global Environment (IMAGE) (van Vuuren et al., 2011, p. 6) and the Global Change Assessment Model (GCAM) (Wise et al., 2009). A different category of IAMs primarily estimates the economic costs and benefits of climate change, and then uses cost-benefit analysis (CBA) to assess the relative desirability of different GHG emissions as well as adaptation policies. These IAMs tend to be much smaller and simpler, which means that uncertainties can be explored via Monte Carlo analysis. IAMs in this group sometimes use GHG and socioeconomic scenarios produced by IAMs in the previous category as exogenous inputs, and then generate their own estimates of temperature, sea-level rise (SLR) and the associated economic impacts. Alternatively they may generate the input scenarios themselves using simple internal models, and then use these in other components of the model. Popular examples include the Dynamic Integrated Climate-Economy model (DICE) (Nordhaus, 2017), the Climate Framework for Uncertainty, Negotiation and Distribution (FUND) (Anthoff and Tol, 2014) and Policy Analysis of the Greenhouse Effect (PAGE) (Hope, 2013). This second group of IAMs will o

Throughout the Anthropocene, the conversion of land to agriculture and atmospheric deposition of nitrogen have resulted in significant changes to biogeochemical cycling, including soil carbon stocks. Quantifying these changes is complex due to a number of influential factors (including climate, land use management, soil type) and their interactions. As the largest terrestrial store of carbon, soils are a key component in climate regulation. In addition, soil carbon storage contributes to numerous ecosystem services including food provision. It is therefore imperative that we understand changes to soil carbon stocks, and provide effective strategies for their future management. Modelling soil systems provides a means to estimate changes to soil carbon stocks. Due to linkages between the carbon cycle and other major nutrient cycles (notably nitrogen and phosphorus which often limit the productivity of ecosystems), models of integrated nutrient cycling are required to understand the response of the carbon cycle to global pressures. Simulating the impacts of land use changes requires capacity to model both semi-natural and intensive agricultural systems. In this study, we have developed an integrated carbon-nitrogen-phosphorus model of semi-natural systems to include representation of both arable and grassland systems, and a range of agricultural management practices. The model is applicable to large spatial scales, as it uses readily available input data and does not require site-specific calibration. After being validated both spatially and temporally using data from long-term experimental sites across Northern-Europe, the model was applied at a national scale throughout the United Kingdom to assess the impacts of land use change and management practices during the last two centuries. Results indicate a decrease in soil carbon in areas of agricultural expansion, yet in areas of semi-natural land use, atmospheric deposition of nitrogen has resulted in increased net pri

Soils provide us with multiple essential services, such as food production, water flow regulation, and climate regulation. The loss of soil function endangers provision of these services, in turn endangering the local, regional and global societies and economies that rely on these. Soils, therefore, are in effect a critical infrastructure, which can be defined as an asset, system or process, the loss or compromise of which could result in a major detrimental impact on the availability, integrity or delivery of essential services, with significant economic or social impacts. Conceptualising soil as a critical infrastructure changes the way we as a society need to approach its management. For example, government authorities have a responsibility to reduce the vulnerability of critical infrastructure, and strengthen their security and resilience. To meet this responsibility there is a need to assess infrastructure resilience, identify critical vulnerabilities, and identify and implement strategies for increasing resilience. There has been growing interest and research on soil resilience, particularly drawing on ecological resilience concepts. In this contribution, we will consider our current understanding of soil system resilience from a critical infrastructure perspective and discuss where further science is needed.

In this study, we explore plausible future states of soil organic matter, biomass, food production and soil greenhouse gas emissions across the UK under a range of climate, land use and land management scenarios. We use state-of-the-art soil biochemistry model, N14CP-Ag, combined with UKCP18 climate scenarios and ASSET land cover change and crop scenarios mapped onto a UK-wide grid with around 100,000 land parcels. Historic runs cover the period from the start of the Holocene interglacial (-12 kyr BP) to 2015; scenarios run from 2016 out to 2100. The results show variations of soil organic carbon (SOC) of around 10% between 2016 and 2100 relative to the simulated starting value of 1.4 Gton in 2015, with reductions of up to 7% under arable expansion scenarios and increases of up to 3% under grassland restoration scenarios. The effect of changing cropping patterns on UK-wide SOC is comparatively small. As climate scenarios move from lower to higher global emissions, the SOC reductions under arable expansion become more pronounced, while the SOC increases under grassland restoration diminish and eventually turn into losses. UK-wide crop yields show resilience to climate change and are maximised for the arable expansion scenario with protected sites of special scientific interest. Soil CO2 and nitrogen emissions get progressively higher in warmer climates. The results of this study are expected to contribute to a future UK agricultural policy aimed at rewarding farmers for sustainable land management practices.

In 2019 the EU Waste Electrical and Electronic Equipment (WEEE) Directive documented a sizable increase in e-waste collection targets alongside a wider scope of electronic and electrical products covered by the legislation. These changes have significant impact for the UK, as for the past two years UK waste collected has failed to meet the newly adopted set of targets. Understanding the flows and fates of products on and off the market becomes of paramount importance, especially for producer-led organisations who have the responsibility to achieve the targets and cover the operational costs. Historic e-waste estimation methods often assume that one product on the market will equate to one product in the waste stream. In this article, we report on a project commissioned by one of the largest UK producer-led organizations – REPIC Ltd, in search of an explanation of the observed drop-in products on the market and WEEE collected, and the relationship between the two. We argue that we should move away from “one product in and one product out” assumption to include wider parameters that are tailored specifically for the UK, including those linked with the state of the market for electronic and electrical products and of the wider economy, examples include inflation-adjusted GDP per capita, consumer confidence index (CCI), inflation indices (CPI or RPI), number of households, wealth distribution etc. We show how this can be achieved by adapting a state-of the-art e-waste estimation model (Waste Over Time) to the UK context and developing it further to include additional drivers.

We study the revenue management (RM) problem encountered in airport car parks, with the primary objective to maximize revenues under a continuous-time framework. The implementation of pre-booking systems for airport car parks has spread rapidly around the world and pre-booking is now available in most major airports. Currently, most RM practises in car parks are simple adjustments of those developed for hotels, exploiting the similarities between the two industries. However, airport car parks have a distinct setting where the price per day of a parking space is heavily discounted by the length of stay (LoS) of the booking. This is because the customer decision tends to be made after the length of the trip is already set, and it becomes a choice between parking or alternative modes of transport. Consequently, the LoS becomes a critical variable for revenue optimization. Since customers are able to book the parking by the minute, the resulting state space is very large, making a conventional network solution intractable . Instead, decomposed single-resource problems need to be considered. Here we develop a bid-price control strategy to manage the bookings and propose novel approaches to define such bid prices depending on the LoS, which could be utilized in real-time RM algorithms. Managing stochastic car park bookings by LoS in the decomposed single-resource approximation allowed us to achieve within 5% of the expected revenues for a multi-resource approximation, with a fraction of the computational effort. When expected demand exceeds the available parking capacity, the method increases the revenues by up to 45% relative to the first come, first served acceptance policy.

REPIC commissioned the Pentland Centre of Sustainability in Business at Lancaster University to independently investigate and report on existing models and available data for post-consumer e-waste forecasting. The objective of the research was to identify the information currently available on electrical and electronic equipment (EEE) placed on the market and waste electrical and electronic equipment (WEEE) generated, recommend how this can be improved and propose a new dynamic EEE and WEEE forecasting model that will ultimately help industry to better plan and help inform future government waste policy. The Pentland Centre report sets out the findings including a quantitative approach to measuring and forecasting WEEE per product category based on econometric analytics. REPIC hopes that by sharing these insights, it will further inform discussions with key stakeholders on the complexity and uncertainties of estimating how much WEEE will be generated. Visitors to the site will be able to download the free 34-page report, watch videos from various industry stakeholders on the value of the research and hear from Lancaster University and REPIC, regarding the research results and challenges, and why this work is important for extended producer responsibility. http://dynamicweeemodel.co.uk

The extensive melting of Arctic sea ice driven by climate change provides opportunities for commercial shipping due to shorter travel distances of up to 40% between Asia and Europe. It has been estimated that around 5% of the world’s trade could be shipped through the Northern Sea Route (NSR) in the Arctic alone under year-round and unhampered navigability, generating additional income for many European and East Asian countries. Our analysis shows that for Arctic sea ice conditions under the RCP8.5 emissions scenario and business restrictions facing shipping companies, NSR traffic will increase steadily from the mid-2030s onwards, although it will take over a century to reach the full capacity expected for ice-free conditions. However, in order to achieve a balanced view of Arctic shipping, it is important to include its detrimental environmental impacts, most notably emissions of short-lived pollutants such as black carbon, as well as CO2 and non-CO2 emissions associated with the additional economic growth enabled by NSR. The total climate feedback of NSR could contribute 0.05% (0.04%) to global mean temperature rise by 2100 under RCP8.5 (RCP4.5), adding $2.15 trillion ($0.44 trillion) to the NPV of total impacts of climate change over the period until 2200 for the SSP2 socio-economic scenario. The climatic losses offset 33% (24.7%) of the total economic gains from NSR under RCP8.5 (RCP4.5), with the biggest losses set to occur in Africa and India. These findings call for policy instruments aimed at reducing emissions from Arctic shipping and providing compensation to the affected regions.

The extensive melting of Arctic sea ice driven by climate change provides opportunities for commercial shipping due to shorter travel distances of up to 40% between Asia and Europe. It has been estimated that around 5% of the world’s trade could be shipped through the Northern Sea Route (NSR) in the Arctic alone under year-round and unhampered navigability, generating additional income for many European and East Asian countries. Our analysis shows that for Arctic sea ice conditions under the RCP8.5 emissions scenario and business restrictions facing shipping companies, NSR traffic will increase steadily from the mid-2030s onwards, although it will take over a century to reach the full capacity expected for ice-free conditions. However, in order to achieve a balanced view of Arctic shipping, it is important to include its detrimental environmental impacts, most notably emissions of short-lived pollutants such as black carbon, as well as CO2 and non-CO2 emissions associated with the additional economic growth enabled by NSR. The total climate feedback of NSR could contribute 0.05% (0.04%) to global mean temperature rise by 2100 under RCP8.5 (RCP4.5), adding $2.15 trillion ($0.44 trillion) to the NPV of total impacts of climate change over the period until 2200 for the SSP2 socio-economic scenario. The climatic losses offset 33% (24.7%) of the total economic gains from NSR under RCP8.5 (RCP4.5), with the biggest losses set to occur in Africa and India. These findings call for policy instruments aimed at reducing emissions from Arctic shipping and providing compensation to the affected regions.

In recent years the Arctic has been warming nearly twice as fast as the global average. Such change leads to various economic opportunities in the Arctic, but also negative impacts on the climate, ecosystems and communities in the region itself and also worldwide. It is therefore essential to accurately quantify the global costs associated with Arctic change. To do this properly we need to bring together up-to-date, transdisciplinary knowledge on both the climate and economic systems. We assess if and how leading Integrated Assessment Models, including PAGE09, can be used to compute the economic valuation of potential long-term global impacts of the changing Arctic.

In this paper, we study optimal revenue management applied to carparks, with primary objective to maximize revenues under a continuous-time framework. We develop a stochastic discrete-time model and propose a rejection algorithm that makes optimal decisions (accept/reject) according to the future expected revenues generated and on the opportunity cost that arises before each sale. For this aspect of the problem, a Monte Carlo approach is used to derive optimal rejection policies. We then extend this approach to show that there exists an equivalent continuous-time methodology that yields to a partial differential equation (PDE). The nature of the PDE, as opposed to the Monte Carlo approach, generates the rejection policies quicker and causes the optimal surfaces to be significantly smoother. However, because the solution to the PDE is considered not to solve the ‘full’ problem, we propose an approach to generate optimal revenues using the discrete-time model by exploiting the information coming from the PDE. We give a worked example of how to generate near-optimal revenues with an order of magnitude decrease in computation speed.

The possibility of controlling the laminar–turbulent transition in hypersonic shock layers by means of porous coatings is considered. The linear stability of the shock layer to acoustic disturbances is analyzed. A dispersion relation is derived in an analytical form and analyzed for different characteristic values of porosity of the wall, which allows one to study the spectrum of acoustic disturbances in the shock layer. Analytical expressions for the growth rate of instability of acoustic disturbances are presented as functions of the reflection factor. Their structure indicates that the porous coating effectively decreases acoustic instability of the shock layer.

In 2019 the EU Waste Electrical and Electronic Equipment (WEEE) Directive documented a sizable increase in e-waste collection targets alongside widening the scope of electrical and electronic products covered by the legislation. These changes have significant impact for the UK, where e-waste collection has been below the levels necessary to meet the targets. Understanding the flows and fates of products on and off the market becomes of paramount importance, especially for producer-led organisations who have the responsibility to achieve the targets and cover the operational costs. Historic e-waste estimation methods often assume that one product on the market will equate to one product in the waste stream. In this article, we introduce our research commissioned by one of the largest UK producer-led organisations, REPIC Ltd, to explain the gap in products on the market and WEEE collected, and the relationship between the two. We argue that we should move away from the “one-in-one-out” estimation to include a wider set of parameters that are tailored specifically for the UK, including those linked with the state of the market for electronic and electrical products and a broader range of socioeconomic indicators. We show how this can be achieved by adapting a state-of-the-art e-waste estimation model, Waste Over Time, to the UK context and developing it further to include additional drivers.

The work provides an extensive theoretical study of an inviscid transonic flow in the vicinity of a wall curvature discontinuity. Depending on the ratio of the curvatures upstream and downstream of the break, several physically different regimes can exist, including a special type of supersonic flows which decelerate to subsonic speeds without a shock wave, transonic Prandtl--Meyer flow and supersonic flows with a weak shock. Using a new numerical technique of solving the Karman--Guderley equation in the ODE form, we perform computations and then employ the \emph{hodograph method} along with the \emph{phase portrait technique} to obtain a complete theoretical description of the flow. It appears that if the flow can be extended beyond the \emph{limiting characteristic}, it subsequently develops a shock wave. As a consequence, a fundamental link between the local and the global flow patterns is observed in our problem (a detailed description of this is given in Part 2). The curvature discontinuity leads to singular pressure gradients $\partial p/\partial x \sim G_{\mp}\, (\mp x)^{-1/3}$ upstream and downstream of the break point, respectively. Analytical expressions for the amplitude coefficients $G_{\mp}$ are derived as functions of the ratio of the curvatures. These results are important for a subsequent study of the boundary layer separation.

The textbook is based on a selection of problems with detailed solutions, which have been taught over a number of years as part of the Theoretical Physics module to students at the Moscow Institute of Physics and Technology. All the problems are related to various electromagnetic phenomena in the Earth's ionosphere, including the scattering of the sunlight, transmission of EM waves and interaction of the solar wind with the Earth's magnetosphere.

This work is Part 2 of the extensive theoretical study of an inviscid transonic flow near a discontinuity in wall curvature, focusing on the flow regimes with the \emph{limiting characteristic}. The closed-form solutions obtained in Part 1 using the \emph{hodograph method} are extended beyond the limiting characteristic by means of analytical continuation, and the results are analysed using the \emph{phase portrait technique}. It is shown that if the flow can be extended beyond the limiting characteristic, it subsequently develops a shock wave. This leads to multiple solutions and a wider variety of flow regimes, including the ones with a concave downstream wall. As a consequence, a fundamental link between the local and the global flow patterns is observed in our problem.

A brief review of the key results in the analytic number theory, based on Marcus du Sautoy's bestseller "The music of the primes" (2003). Provides a concise and clear description of the key formulas which link the prime counting function with Riemann's zeta function, focusing both on their mathematical and philosophical interpretation.

Theoretical study of receptivity of a thin weakly non-parallel shock layer to acoustic disturbances is performed. We apply the general theory of boundary layer receptivity to the case of shock layer eigen modes excited by the impinging acoustic wave. The spectrum of eigen modes is obtained in analytical form in the thin shock layer approximation, assuming the flow inside the shock layer is uniform. This allows to predict the effective directions of an impinging acoustic wave that cause resonance with the eigen modes, as well as locate the regions of the shock layer where the resonance is taking place.

In 2019 the EU Waste Electrical and Electronic Equipment (WEEE) Directive documented a sizable increase in e-waste collection targets alongside widening the scope of electrical and electronic products covered by the legislation. These changes have significant impact for the UK, where e-waste collection has been below the levels necessary to meet the targets. Understanding the flows and fates of products on and off the market becomes of paramount importance, especially for producer-led organisations who have the responsibility to achieve the targets and cover the operational costs. Historic e-waste estimation methods often assume that one product on the market will equate to one product in the waste stream. In this article, we introduce our research commissioned by one of the largest UK producer-led organisations, REPIC Ltd, to explain the gap in products on the market and WEEE collected, and the relationship between the two. We argue that we should move away from the “one-in-one-out” estimation to include a wider set of parameters that are tailored specifically for the UK, including those linked with the state of the market for electronic and electrical products and a broader range of socioeconomic indicators. We show how this can be achieved by adapting a state-of-the-art e-waste estimation model, Waste Over Time, to the UK context and developing it further to include additional drivers.