Coconut palm tree (Cocos nucifera L.) tissues were used as a readily available, low-cost and green adsorbent to desalinate seawater. The tree bark (CB), husk (CH), leaves (CL) and roots (CR) were examined in their fresh (F) and dry (D) forms. The salinity removal (adsorption) efficiency followed the trend: F_CB ≈ F_CR > F_CL > D_CR > F_CL > D_CR. The sorbents from the coastal region desalinated more efficiently than those from a non-coastal region. Also, the fresh tissues were more effective and efficient than the dry parts. The salinity retention ability (desalination : desorption) follows the trend: F_CR (22.2) > F_CB (19.0) ≫ D_CR (12.3) > D_CB (11.0) > D_CL (6.14) ≈ F_CL (6.10) > F_CH (4.3) > D_CH (2.1). Moreover, the desalination fitted the pseudo-second-order kinetics than the pseudo-first-order, suggesting the predominance of chemisorption over physical removal. Overall, water pH, conductivity, total dissolved solids and dissolved oxygen (DO) correlated positively and strongly with desalination. By contrast, the density and redox potential correlated negatively, whereas temperature and DO showed no definite influence. Conclusively, F_CR and F_CB are the most suitable coconut palm tree tissues for desalination. Future studies should include chemical characterization of the tissues and system optimization for upscaling. This article is part of the theme issue ‘Developing resilient energy systems’.
{"title":"On the potential of pristine Cocos nucifera L. tissues for green desalination","authors":"A. Adelodun","doi":"10.1098/rsta.2021.0145","DOIUrl":"https://doi.org/10.1098/rsta.2021.0145","url":null,"abstract":"Coconut palm tree (Cocos nucifera L.) tissues were used as a readily available, low-cost and green adsorbent to desalinate seawater. The tree bark (CB), husk (CH), leaves (CL) and roots (CR) were examined in their fresh (F) and dry (D) forms. The salinity removal (adsorption) efficiency followed the trend: F_CB ≈ F_CR > F_CL > D_CR > F_CL > D_CR. The sorbents from the coastal region desalinated more efficiently than those from a non-coastal region. Also, the fresh tissues were more effective and efficient than the dry parts. The salinity retention ability (desalination : desorption) follows the trend: F_CR (22.2) > F_CB (19.0) ≫ D_CR (12.3) > D_CB (11.0) > D_CL (6.14) ≈ F_CL (6.10) > F_CH (4.3) > D_CH (2.1). Moreover, the desalination fitted the pseudo-second-order kinetics than the pseudo-first-order, suggesting the predominance of chemisorption over physical removal. Overall, water pH, conductivity, total dissolved solids and dissolved oxygen (DO) correlated positively and strongly with desalination. By contrast, the density and redox potential correlated negatively, whereas temperature and DO showed no definite influence. Conclusively, F_CR and F_CB are the most suitable coconut palm tree tissues for desalination. Future studies should include chemical characterization of the tissues and system optimization for upscaling. This article is part of the theme issue ‘Developing resilient energy systems’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"114 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75251794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Mandal, Tannecia S. Stephenson, Jayaka Campbell, Michael Taylor, Shavel Watson, L. Clarke, David Smith, Junior Darsan, Matthew D. Wilson
Climate change models project that, within the Caribbean basin, rainfall intensity is likely to increase toward the end of this century, although the region is projected to be drier overall. This may affect the frequency and severity of floods in Jamaica and the Caribbean Small Island Developing States. We investigate how flood hazards may be affected by increases in global mean surface temperature of 1.5, 2.0 and 2.5°C above pre-industrial levels using a case study of a Jamaican watershed. Rainfall projections from the PRECIS regional climate model for the Caribbean are analysed. Six members from the Quantifying Uncertainty in Model Predictions (AENWH, AEXSA, AEXSC, AEXSK, AEXSL and AEXSM) were used to create 100-year flood inundation maps for the Hope river for different global warming levels using hydrological and hydraulic models. Model runs projected peak discharges at 2.0, 2.5 and 1.5°C warming that were higher than discharges in the historical record of events that damaged sections of the watershed. Projections from the hydraulic model show increased flow area, depth and extent for 1.5 followed by 2.0 and 2.5°C rises in temperature. These results imply continued flood risk for the vulnerable areas of the watershed. This article is part of the theme issue 'Developing resilient energy systems'.
{"title":"An assessment of the impact of 1.5 versus 2 and 2.5°C global temperature increase on flooding in Jamaica: a case study from the Hope watershed","authors":"A. Mandal, Tannecia S. Stephenson, Jayaka Campbell, Michael Taylor, Shavel Watson, L. Clarke, David Smith, Junior Darsan, Matthew D. Wilson","doi":"10.1098/rsta.2021.0141","DOIUrl":"https://doi.org/10.1098/rsta.2021.0141","url":null,"abstract":"Climate change models project that, within the Caribbean basin, rainfall intensity is likely to increase toward the end of this century, although the region is projected to be drier overall. This may affect the frequency and severity of floods in Jamaica and the Caribbean Small Island Developing States. We investigate how flood hazards may be affected by increases in global mean surface temperature of 1.5, 2.0 and 2.5°C above pre-industrial levels using a case study of a Jamaican watershed. Rainfall projections from the PRECIS regional climate model for the Caribbean are analysed. Six members from the Quantifying Uncertainty in Model Predictions (AENWH, AEXSA, AEXSC, AEXSK, AEXSL and AEXSM) were used to create 100-year flood inundation maps for the Hope river for different global warming levels using hydrological and hydraulic models. Model runs projected peak discharges at 2.0, 2.5 and 1.5°C warming that were higher than discharges in the historical record of events that damaged sections of the watershed. Projections from the hydraulic model show increased flow area, depth and extent for 1.5 followed by 2.0 and 2.5°C rises in temperature. These results imply continued flood risk for the vulnerable areas of the watershed. This article is part of the theme issue 'Developing resilient energy systems'.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74383256","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. Koon Koon, S. Lewis, K. Mohammed-Koon Koon, A. Chen, Kalim U. Shah
The Eastern Caribbean chain of islands is commonly known to exhibit high-enthalpy systems for geothermal energy exploitation. The northernmost Caribbean Community member state of Jamaica possesses physical manifestations of 12 hot springs across the island. Previous investigations indicate that of the potential 12 hot springs, Bath, Windsor and Milk River springs have cogent geothermometry of their thermal fluids with estimated temperature ranges of (80–102°C), (128–156°C), and (158–206°C), respectively. The paper provides numerical findings for each geothermal system of interest and performs Monte Carlo simulations to optimize calculated findings. The determined quantitative findings are considered under the context of environmental savings and policy regime conditions for driving geothermal energy development. The three areas of interest are situated within the Rio Minho Basin, the Dry Harbour Mountains and the Blue Mountain South Basin. Through the consideration of a 25-year lifetime for production, a collective total of 94.81 MWe of geothermal power reserves can be absorbed into the national energy mix, displacing an estimated 0.38 million barrels of oil imports, resulting in approximately 0.44 million tonnes of carbon dioxide emissions being avoided per year. This article is part of the theme issue ‘Developing resilient energy systems’.
{"title":"A Monte Carlo based approach to the resource assessment of Jamaica's geothermal energy potential","authors":"R. Koon Koon, S. Lewis, K. Mohammed-Koon Koon, A. Chen, Kalim U. Shah","doi":"10.1098/rsta.2021.0133","DOIUrl":"https://doi.org/10.1098/rsta.2021.0133","url":null,"abstract":"The Eastern Caribbean chain of islands is commonly known to exhibit high-enthalpy systems for geothermal energy exploitation. The northernmost Caribbean Community member state of Jamaica possesses physical manifestations of 12 hot springs across the island. Previous investigations indicate that of the potential 12 hot springs, Bath, Windsor and Milk River springs have cogent geothermometry of their thermal fluids with estimated temperature ranges of (80–102°C), (128–156°C), and (158–206°C), respectively. The paper provides numerical findings for each geothermal system of interest and performs Monte Carlo simulations to optimize calculated findings. The determined quantitative findings are considered under the context of environmental savings and policy regime conditions for driving geothermal energy development. The three areas of interest are situated within the Rio Minho Basin, the Dry Harbour Mountains and the Blue Mountain South Basin. Through the consideration of a 25-year lifetime for production, a collective total of 94.81 MWe of geothermal power reserves can be absorbed into the national energy mix, displacing an estimated 0.38 million barrels of oil imports, resulting in approximately 0.44 million tonnes of carbon dioxide emissions being avoided per year. This article is part of the theme issue ‘Developing resilient energy systems’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83724718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Wood-burning stoves, in Kenya and Mexico, are reviewed. With a Kenyan stove, burning charcoal, only 24% of the energy released reached the cooking pot. Initially, the proportion of CO in the leaving gases was 3%. Indoor concentrations of particulate matter (less than 2.5 µm diameter) can be abnormally high near a stove. Decarbonization, by using H2, is facilitated by a distribution system. Replacement by H2 would ultimately rest upon wind or water power, or it being a by-product in the production of heavier hydrocarbons from CH4. The averaged burning rate in the Kenyan stove was 10 kW, over 20 min, with an initial peak value of about 30 kW. A possible replacement is a hob, composed of an array of small diameter H2 jet flames. As an example, combustion of a 2 mm internal diameter H2 jet flame, with a H2 exit velocity of 27.2 m s−1, would release 0.84 kW. Bearing in mind its improved efficiency, a single compact hob with an array of about 10 jets would suffice. A difficulty is the low mass-specific energy of H2. H2 has a high acoustic velocity, and both high velocity subsonic combustion and blending with natural gas are briefly discussed. This article is part of the theme issue ‘Developing resilient energy systems’.
审查了肯尼亚和墨西哥的燃木炉。使用肯尼亚炉子燃烧木炭时，只有24%的能量释放到烹饪锅中。最初，剩余气体中CO的比例为3%。炉灶附近的室内颗粒物(直径小于2.5µm)浓度会异常高。利用氢气的脱碳过程由一个分配系统促进。氢气的替代最终取决于风能或水力，或者它是由CH4生产较重碳氢化合物的副产品。肯尼亚炉的平均燃烧速率为10千瓦，超过20分钟，初始峰值约为30千瓦。一种可能的替代品是由一系列直径较小的H2喷射火焰组成的滚刀。以内径2mm的H2射流火焰为例，H2出口速度为27.2 m s−1时，燃烧释放的能量为0.84 kW。考虑到其提高的效率，一个紧凑的滚刀与大约10个喷气机阵列就足够了。一个困难是H2的质量比能很低。H2具有较高的声速，本文简要讨论了高速亚音速燃烧和与天然气混合的问题。本文是“发展有弹性的能源系统”主题问题的一部分。
{"title":"Hydrogen and wood-burning stoves","authors":"A. Palacios, D. Bradley","doi":"10.1098/rsta.2021.0139","DOIUrl":"https://doi.org/10.1098/rsta.2021.0139","url":null,"abstract":"Wood-burning stoves, in Kenya and Mexico, are reviewed. With a Kenyan stove, burning charcoal, only 24% of the energy released reached the cooking pot. Initially, the proportion of CO in the leaving gases was 3%. Indoor concentrations of particulate matter (less than 2.5 µm diameter) can be abnormally high near a stove. Decarbonization, by using H2, is facilitated by a distribution system. Replacement by H2 would ultimately rest upon wind or water power, or it being a by-product in the production of heavier hydrocarbons from CH4. The averaged burning rate in the Kenyan stove was 10 kW, over 20 min, with an initial peak value of about 30 kW. A possible replacement is a hob, composed of an array of small diameter H2 jet flames. As an example, combustion of a 2 mm internal diameter H2 jet flame, with a H2 exit velocity of 27.2 m s−1, would release 0.84 kW. Bearing in mind its improved efficiency, a single compact hob with an array of about 10 jets would suffice. A difficulty is the low mass-specific energy of H2. H2 has a high acoustic velocity, and both high velocity subsonic combustion and blending with natural gas are briefly discussed. This article is part of the theme issue ‘Developing resilient energy systems’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81146872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Decisions relating to energy systems resilience must take into account a range of environmental and societal transitions together with an array of future threats. These must be assessed broadly and systematically but also must consider how risks and vulnerabilities are linked, and that small events can cascade across a system and between systems to escalate into large-scale collapse. Developing resilience involves not only identifying such threats and potential points of failure but acting upon them with an appropriate level of future planning. Decision-making in complex systems is often affected by cognitive biases and runs into contestation and obstacles of politics, policy complexities and differing views of potential futures. This paper discusses the challenges to building resilient systems for the future, given the inevitable biases and differing risk perspectives of decision-makers at all levels that often confound expert analysis. This article is part of the theme issue ‘Developing resilient energy systems’.
{"title":"Policy and political perceptions of risk: the challenges to building resilient energy systems","authors":"P. Gluckman, Anne Bardsley","doi":"10.1098/rsta.2021.0146","DOIUrl":"https://doi.org/10.1098/rsta.2021.0146","url":null,"abstract":"Decisions relating to energy systems resilience must take into account a range of environmental and societal transitions together with an array of future threats. These must be assessed broadly and systematically but also must consider how risks and vulnerabilities are linked, and that small events can cascade across a system and between systems to escalate into large-scale collapse. Developing resilience involves not only identifying such threats and potential points of failure but acting upon them with an appropriate level of future planning. Decision-making in complex systems is often affected by cognitive biases and runs into contestation and obstacles of politics, policy complexities and differing views of potential futures. This paper discusses the challenges to building resilient systems for the future, given the inevitable biases and differing risk perspectives of decision-makers at all levels that often confound expert analysis. This article is part of the theme issue ‘Developing resilient energy systems’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"80 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79317525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sea-level rise is a key component of many climate change scenarios. Such increases are expected to cause greater coastal inundation by seawater, reduce the depth of the freshwater-saline water lens and have a range of consequences for coastal ecosystems. Soil salinity determines the distribution patterns of coastal vegetation across a sea-to-land gradient, and sea-level rise would be expected to alter these distributions. Our study done along the east coast of Sri Lanka shows that soil salinity decreases predictably from sea to land, varying between 21‰ and 30‰ over a distance of 50 m from the mean high water springs level. Soil salinity also showed declines with increasing soil depth (0, 10, 50 cm) although this was marginally non-significant. The emergence of terrestrial vegetation coincided with a cross-shore ecotone where the soil salinity approached 0‰. Our predictions suggest that if the mean sea-level were to increase by 1 m in the Batticaloa region (Sri Lanka), this would cause landward communities to be inundated with brackish water at various depths. The present permanent coastal vegetation front would likely shift landward by 30–45 m under scenarios involving a 1 m increase in sea level. Sea-level rise and its consequences could result in a variety of changes to coastal vegetation such as altered physiognomy and diversity, colonization of new territory, expansion of salt-tolerant species such as mangroves and saltmarsh and will therefore greatly influence adaptive management and future planning. This article is part of the theme issue ‘Developing resilient energy systems’.
{"title":"Sea-level rise, coastal salinity and vegetation changes in Sri Lanka","authors":"T. Mathiventhan, D. Gorman, T. Jayasingam","doi":"10.1098/rsta.2021.0142","DOIUrl":"https://doi.org/10.1098/rsta.2021.0142","url":null,"abstract":"Sea-level rise is a key component of many climate change scenarios. Such increases are expected to cause greater coastal inundation by seawater, reduce the depth of the freshwater-saline water lens and have a range of consequences for coastal ecosystems. Soil salinity determines the distribution patterns of coastal vegetation across a sea-to-land gradient, and sea-level rise would be expected to alter these distributions. Our study done along the east coast of Sri Lanka shows that soil salinity decreases predictably from sea to land, varying between 21‰ and 30‰ over a distance of 50 m from the mean high water springs level. Soil salinity also showed declines with increasing soil depth (0, 10, 50 cm) although this was marginally non-significant. The emergence of terrestrial vegetation coincided with a cross-shore ecotone where the soil salinity approached 0‰. Our predictions suggest that if the mean sea-level were to increase by 1 m in the Batticaloa region (Sri Lanka), this would cause landward communities to be inundated with brackish water at various depths. The present permanent coastal vegetation front would likely shift landward by 30–45 m under scenarios involving a 1 m increase in sea level. Sea-level rise and its consequences could result in a variety of changes to coastal vegetation such as altered physiognomy and diversity, colonization of new territory, expansion of salt-tolerant species such as mangroves and saltmarsh and will therefore greatly influence adaptive management and future planning. This article is part of the theme issue ‘Developing resilient energy systems’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"380 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87522218","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose two different schemes of realizing a virtual walk corresponding to a kinetic exchange model of opinion dynamics. The walks are either Markovian or non-Markovian in nature. The opinion dynamics model is characterized by a parameter p which drives an order disorder transition at a critical value pc. The distribution S(X,t) of the displacements X from the origin of the walkers is computed at different times. Below pc, two time scales associated with a crossover behaviour in time are detected, which diverge in a power law manner at criticality with different exponent values. S(X,t) also carries the signature of the phase transition as it changes its form at pc. The walks show the features of a biased random walk below pc, and above pc, the walks are like unbiased random walks. The bias vanishes in a power law manner at pc and the width of the resulting Gaussian function shows a discontinuity. Some of the features of the walks are argued to be comparable to the critical quantities associated with the mean-field Ising model, to which class the opinion dynamics model belongs. The results for the Markovian and non-Markovian walks are almost identical which is justified by considering the different fluxes. We compare the present results with some earlier similar studies. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.
{"title":"Virtual walks inspired by a mean-field kinetic exchange model of opinion dynamics","authors":"S. Saha, P. Sen","doi":"10.1098/rsta.2021.0168","DOIUrl":"https://doi.org/10.1098/rsta.2021.0168","url":null,"abstract":"We propose two different schemes of realizing a virtual walk corresponding to a kinetic exchange model of opinion dynamics. The walks are either Markovian or non-Markovian in nature. The opinion dynamics model is characterized by a parameter p which drives an order disorder transition at a critical value pc. The distribution S(X,t) of the displacements X from the origin of the walkers is computed at different times. Below pc, two time scales associated with a crossover behaviour in time are detected, which diverge in a power law manner at criticality with different exponent values. S(X,t) also carries the signature of the phase transition as it changes its form at pc. The walks show the features of a biased random walk below pc, and above pc, the walks are like unbiased random walks. The bias vanishes in a power law manner at pc and the width of the resulting Gaussian function shows a discontinuity. Some of the features of the walks are argued to be comparable to the critical quantities associated with the mean-field Ising model, to which class the opinion dynamics model belongs. The results for the Markovian and non-Markovian walks are almost identical which is justified by considering the different fluxes. We compare the present results with some earlier similar studies. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80680473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, I shall show how the notions of Finsler geometry can be used to construct a similar geometry using a scalar field, f, on the cotangent bundle of a differentiable manifold M. This will enable me to use the second vertical derivatives of f, along with the differential of a scalar field ϕ on M, to construct a Lorentzian metric on M that depends upon ϕ. I refer to a field theory based upon a manifold with such a Lorentzian structure as a scalar–scalar field theory. We shall study such a theory when f is chosen so that the resultant metric on M has the form of a Friedmann–Lemaître–Robertson–Walker metric, and the Lagrangian has a particularly simple form. It will be shown that the scalar–scalar theory determined by the Lagrangian can generate self-inflating universes, which can be pieced together to form multiverses with non-Hausdorff topologies, in which the global time function multifurcates at t = 0. Some of the universes in these multiverses begin explosively, and then settle down to a period of much quieter accelerated expansion, which can be followed by a collapse to its original, pre-expansion state. This article is part of the theme issue ‘The future of mathematical cosmology, Volume 1’.
{"title":"Reformulating scalar–tensor field theories as scalar–scalar field theories using a novel geometry","authors":"G. W. Horndeski","doi":"10.1098/rsta.2021.0183","DOIUrl":"https://doi.org/10.1098/rsta.2021.0183","url":null,"abstract":"In this paper, I shall show how the notions of Finsler geometry can be used to construct a similar geometry using a scalar field, f, on the cotangent bundle of a differentiable manifold M. This will enable me to use the second vertical derivatives of f, along with the differential of a scalar field ϕ on M, to construct a Lorentzian metric on M that depends upon ϕ. I refer to a field theory based upon a manifold with such a Lorentzian structure as a scalar–scalar field theory. We shall study such a theory when f is chosen so that the resultant metric on M has the form of a Friedmann–Lemaître–Robertson–Walker metric, and the Lagrangian has a particularly simple form. It will be shown that the scalar–scalar theory determined by the Lagrangian can generate self-inflating universes, which can be pieced together to form multiverses with non-Hausdorff topologies, in which the global time function multifurcates at t = 0. Some of the universes in these multiverses begin explosively, and then settle down to a period of much quieter accelerated expansion, which can be followed by a collapse to its original, pre-expansion state. This article is part of the theme issue ‘The future of mathematical cosmology, Volume 1’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"160 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82646961","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
First, we discuss the non-Gaussian type of self-similar solutions to the Navier–Stokes equations. We revisit a class of self-similar solutions which was studied in Canonne et al. (1996 Commun. Partial. Differ. Equ. 21, 179–193). In order to shed some light on it, we study self-similar solutions to the one-dimensional Burgers equation in detail, completing the most general form of similarity profiles that it can possibly possess. In particular, on top of the well-known source-type solution, we identify a kink-type solution. It is represented by one of the confluent hypergeometric functions, viz. Kummer’s function M. For the two-dimensional Navier–Stokes equations, on top of the celebrated Burgers vortex, we derive yet another solution to the associated Fokker–Planck equation. This can be regarded as a ‘conjugate’ to the Burgers vortex, just like the kink-type solution above. Some asymptotic properties of this kind of solution have been worked out. Implications for the three-dimensional (3D) Navier–Stokes equations are suggested. Second, we address an application of self-similar solutions to explore more general kind of solutions. In particular, based on the source-type self-similar solution to the 3D Navier–Stokes equations, we consider what we could tell about more general solutions. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.
{"title":"Self-similarity in turbulence and its applications","authors":"K. Ohkitani","doi":"10.1098/rsta.2021.0048","DOIUrl":"https://doi.org/10.1098/rsta.2021.0048","url":null,"abstract":"First, we discuss the non-Gaussian type of self-similar solutions to the Navier–Stokes equations. We revisit a class of self-similar solutions which was studied in Canonne et al. (1996 Commun. Partial. Differ. Equ. 21, 179–193). In order to shed some light on it, we study self-similar solutions to the one-dimensional Burgers equation in detail, completing the most general form of similarity profiles that it can possibly possess. In particular, on top of the well-known source-type solution, we identify a kink-type solution. It is represented by one of the confluent hypergeometric functions, viz. Kummer’s function M. For the two-dimensional Navier–Stokes equations, on top of the celebrated Burgers vortex, we derive yet another solution to the associated Fokker–Planck equation. This can be regarded as a ‘conjugate’ to the Burgers vortex, just like the kink-type solution above. Some asymptotic properties of this kind of solution have been worked out. Implications for the three-dimensional (3D) Navier–Stokes equations are suggested. Second, we address an application of self-similar solutions to explore more general kind of solutions. In particular, based on the source-type self-similar solution to the 3D Navier–Stokes equations, we consider what we could tell about more general solutions. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87668658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This review article offers a survey of the research program focused on a systematic computational search for extreme and potentially singular behaviour in hydrodynamic models motivated by open questions concerning the possibility of a finite-time blow-up in the solutions of the Navier–Stokes system. Inspired by the seminal work of Lu & Doering (2008 Ind. Univ. Math. 57, 2693–2727), we sought such extreme behaviour by solving PDE optimization problems with objective functionals chosen based on certain conditional regularity results and a priori estimates available for different models. No evidence for singularity formation was found in extreme Navier–Stokes flows constructed in this manner in three dimensions. We also discuss the results obtained for one-dimensional Burgers and two-dimensional Navier–Stokes systems, and while singularities are ruled out in these flows, the results presented provide interesting insights about sharpness of different energy-type estimates known for these systems. Connections to other bounding techniques are also briefly discussed. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.
{"title":"Systematic search for extreme and singular behaviour in some fundamental models of fluid mechanics","authors":"B. Protas","doi":"10.1098/rsta.2021.0035","DOIUrl":"https://doi.org/10.1098/rsta.2021.0035","url":null,"abstract":"This review article offers a survey of the research program focused on a systematic computational search for extreme and potentially singular behaviour in hydrodynamic models motivated by open questions concerning the possibility of a finite-time blow-up in the solutions of the Navier–Stokes system. Inspired by the seminal work of Lu & Doering (2008 Ind. Univ. Math. 57, 2693–2727), we sought such extreme behaviour by solving PDE optimization problems with objective functionals chosen based on certain conditional regularity results and a priori estimates available for different models. No evidence for singularity formation was found in extreme Navier–Stokes flows constructed in this manner in three dimensions. We also discuss the results obtained for one-dimensional Burgers and two-dimensional Navier–Stokes systems, and while singularities are ruled out in these flows, the results presented provide interesting insights about sharpness of different energy-type estimates known for these systems. Connections to other bounding techniques are also briefly discussed. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89393864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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