The 2020 Nobel prize in Physics has revived the interest in the singularity theorems and, in particular, in the Penrose theorem published in 1965. In this short paper, I briefly review the main ideas behind the theorems and then proceed to an evaluation of their hypotheses and implications. I will try to dispel some common misconceptions about the theorems and their conclusions, as well as to convey some of their rarely mentioned consequences. In particular, a discussion of space–time extensions in relation to the theorems is provided. The nature of the singularity inside black holes is also analysed. This article is part of the theme issue ‘The future of mathematical cosmology, Volume 1’.
{"title":"A critical appraisal of the singularity theorems","authors":"J. Senovilla","doi":"10.1098/rsta.2021.0174","DOIUrl":"https://doi.org/10.1098/rsta.2021.0174","url":null,"abstract":"The 2020 Nobel prize in Physics has revived the interest in the singularity theorems and, in particular, in the Penrose theorem published in 1965. In this short paper, I briefly review the main ideas behind the theorems and then proceed to an evaluation of their hypotheses and implications. I will try to dispel some common misconceptions about the theorems and their conclusions, as well as to convey some of their rarely mentioned consequences. In particular, a discussion of space–time extensions in relation to the theorems is provided. The nature of the singularity inside black holes is also analysed. 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":"108 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75539178","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, we propose a Boltzmann-type kinetic description of opinion formation on social networks, which takes into account a general connectivity distribution of the individuals. We consider opinion exchange processes inspired by the Sznajd model and related simplifications but we do not assume that individuals interact on a regular lattice. Instead, we describe the structure of the social network statistically, assuming that the number of contacts of a given individual determines the probability that their opinion reaches and influences the opinion of another individual. From the kinetic description of the system, we study the evolution of the mean opinion, whence we find precise analytical conditions under which a polarization switch of the opinions, i.e. a change of sign between the initial and the asymptotic mean opinions, occurs. In particular, we show that a non-zero correlation between the initial opinions and the connectivity of the individuals is necessary to observe polarization switch. Finally, we validate our analytical results through Monte Carlo simulations of the stochastic opinion exchange processes on the social network. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.
{"title":"Opinion polarization in social networks","authors":"N. Loy, Matteo Raviola, A. Tosin","doi":"10.1098/rsta.2021.0158","DOIUrl":"https://doi.org/10.1098/rsta.2021.0158","url":null,"abstract":"In this paper, we propose a Boltzmann-type kinetic description of opinion formation on social networks, which takes into account a general connectivity distribution of the individuals. We consider opinion exchange processes inspired by the Sznajd model and related simplifications but we do not assume that individuals interact on a regular lattice. Instead, we describe the structure of the social network statistically, assuming that the number of contacts of a given individual determines the probability that their opinion reaches and influences the opinion of another individual. From the kinetic description of the system, we study the evolution of the mean opinion, whence we find precise analytical conditions under which a polarization switch of the opinions, i.e. a change of sign between the initial and the asymptotic mean opinions, occurs. In particular, we show that a non-zero correlation between the initial opinions and the connectivity of the individuals is necessary to observe polarization switch. Finally, we validate our analytical results through Monte Carlo simulations of the stochastic opinion exchange processes on the social network. 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":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88195085","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 prove the nonlinear stability of the asymptotic behaviour of perturbations of subfamilies of Kasner solutions in the contracting time direction within the class of polarized T2-symmetric solutions of the vacuum Einstein equations with arbitrary cosmological constant Λ. This stability result generalizes the results proven in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized T2-symmetric vacuum spacetimes. Ann. Henri Poincaré. (doi:10.1007/s00023-021-01142-0)), which focus on the Λ=0 case, and as in that article, the proof relies on an areal time foliation and Fuchsian techniques. Even for Λ=0, the results established here apply to a wider class of perturbations of Kasner solutions within the family of polarized T2-symmetric vacuum solutions than those considered in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized T2-symmetric vacuum spacetimes. Ann. Henri Poincaré. (doi:10.1007/s00023-021-01142-0)) and Fournodavlos G et al. (2020 Stable Big Bang formation for Einstein’s equations: the complete sub-critical regime. Preprint. (http://arxiv.org/abs/2012.05888)). Our results establish that the areal time coordinate takes all values in (0,T0] for some T0>0, for certain families of polarized T2-symmetric solutions with cosmological constant. This article is part of the theme issue ‘The future of mathematical cosmology, Volume 1’.
我们证明了具有任意宇宙常数Λ的真空爱因斯坦方程的偏振t2对称解类中Kasner解的子族在收缩时间方向上的扰动的渐近行为的非线性稳定性。这一稳定性结果推广了Ames et al.(2022)中证明的结果。安。亨利·庞加莱。(doi:10.1007/s00023-021-01142-0))，其重点是Λ=0的情况，并且在那篇文章中，证明依赖于实时叶化和Fuchsian技术。即使对于Λ=0，这里建立的结果也适用于极化t2对称真空解族中Kasner解的更广泛的微扰，而不是Ames E等人(2022)中考虑的。安。亨利·庞加莱。(doi:10.1007/s00023-021-01142-0))和Fournodavlos G et al.(2020爱因斯坦方程的稳定大爆炸形成:完整的亚临界状态。预印本。(http://arxiv.org/abs/2012.05888))。我们的结果证明，对于某些具有宇宙学常数的偏振t2对称解族，在某些T0>0的情况下，实时坐标取(0,T0)内的所有值。本文是主题问题“数学宇宙学的未来，第一卷”的一部分。
{"title":"Stability of asymptotic behaviour within polarized T2-symmetric vacuum solutions with cosmological constant","authors":"E. Ames, F. Beyer, J. Isenberg, T. Oliynyk","doi":"10.1098/rsta.2021.0173","DOIUrl":"https://doi.org/10.1098/rsta.2021.0173","url":null,"abstract":"We prove the nonlinear stability of the asymptotic behaviour of perturbations of subfamilies of Kasner solutions in the contracting time direction within the class of polarized T2-symmetric solutions of the vacuum Einstein equations with arbitrary cosmological constant Λ. This stability result generalizes the results proven in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized T2-symmetric vacuum spacetimes. Ann. Henri Poincaré. (doi:10.1007/s00023-021-01142-0)), which focus on the Λ=0 case, and as in that article, the proof relies on an areal time foliation and Fuchsian techniques. Even for Λ=0, the results established here apply to a wider class of perturbations of Kasner solutions within the family of polarized T2-symmetric vacuum solutions than those considered in Ames E et al. (2022 Stability of AVTD Behavior within the Polarized T2-symmetric vacuum spacetimes. Ann. Henri Poincaré. (doi:10.1007/s00023-021-01142-0)) and Fournodavlos G et al. (2020 Stable Big Bang formation for Einstein’s equations: the complete sub-critical regime. Preprint. (http://arxiv.org/abs/2012.05888)). Our results establish that the areal time coordinate takes all values in (0,T0] for some T0>0, for certain families of polarized T2-symmetric solutions with cosmological constant. 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":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78748824","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 a modest attempt to present potentially new paradigms in cosmology, including its inflationary epoch, and initiate discussions, I review in this article some novel, string-inspired cosmological models, which entail a purely geometrical origin of the dark sector of the Universe but also of its observed matter-antimatter asymmetry. The models contain gravitational (string-model independent, Kalb–Ramond (KR)) axion fields coupled to primordial gravitational anomalies via CP-violating interactions. The anomaly terms are four-space–time-dimensional remnants of the Green–Schwarz counterterms appearing in the definition of the field strength of the spin-one antisymmetric tensor field of the (bosonic) massless gravitational string multiplet, which also plays the role of a totally antisymmetric component of torsion. I show how in such cosmologies the presence of primordial gravitational waves can lead to anomaly condensates and dynamical inflation of a ‘running-vacuum-model’ type, without external inflatons, but also to leptogenesis in the radiation era due to anomaly induced Lorentz and CPT violating KR axion backgrounds. I also discuss how the torsion-related KR-axion could acquire a mass during the QCD epoch, thus playing the role of (a component of) dark matter. Phenomenological considerations of the inflationary and post-inflationary (in particular, modern) eras of the model are briefly discussed, including its potential for alleviating the observed tensions in the cosmological data of the current epoch. This article is part of the theme issue ‘The future of mathematical cosmology, Volume 1’.
{"title":"Geometrical origins of the universe dark sector: string-inspired torsion and anomalies as seeds for inflation and dark matter","authors":"N. Mavromatos","doi":"10.1098/rsta.2021.0188","DOIUrl":"https://doi.org/10.1098/rsta.2021.0188","url":null,"abstract":"In a modest attempt to present potentially new paradigms in cosmology, including its inflationary epoch, and initiate discussions, I review in this article some novel, string-inspired cosmological models, which entail a purely geometrical origin of the dark sector of the Universe but also of its observed matter-antimatter asymmetry. The models contain gravitational (string-model independent, Kalb–Ramond (KR)) axion fields coupled to primordial gravitational anomalies via CP-violating interactions. The anomaly terms are four-space–time-dimensional remnants of the Green–Schwarz counterterms appearing in the definition of the field strength of the spin-one antisymmetric tensor field of the (bosonic) massless gravitational string multiplet, which also plays the role of a totally antisymmetric component of torsion. I show how in such cosmologies the presence of primordial gravitational waves can lead to anomaly condensates and dynamical inflation of a ‘running-vacuum-model’ type, without external inflatons, but also to leptogenesis in the radiation era due to anomaly induced Lorentz and CPT violating KR axion backgrounds. I also discuss how the torsion-related KR-axion could acquire a mass during the QCD epoch, thus playing the role of (a component of) dark matter. Phenomenological considerations of the inflationary and post-inflationary (in particular, modern) eras of the model are briefly discussed, including its potential for alleviating the observed tensions in the cosmological data of the current epoch. 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":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82726529","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 study the evolution of solutions to the two-dimensional Euler equations whose vorticity is sharply concentrated in the Wasserstein sense around a finite number of points. Under the assumption that the vorticity is merely Lp integrable for some p>2, we show that the evolving vortex regions remain concentrated around points, and these points are close to solutions to the Helmholtz–Kirchhoff point vortex system. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.
{"title":"On the dynamics of point vortices for the two-dimensional Euler equation with Lp vorticity","authors":"S. Ceci, Christian Seis","doi":"10.1098/rsta.2021.0046","DOIUrl":"https://doi.org/10.1098/rsta.2021.0046","url":null,"abstract":"We study the evolution of solutions to the two-dimensional Euler equations whose vorticity is sharply concentrated in the Wasserstein sense around a finite number of points. Under the assumption that the vorticity is merely Lp integrable for some p>2, we show that the evolving vortex regions remain concentrated around points, and these points are close to solutions to the Helmholtz–Kirchhoff point vortex system. 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":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77831462","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}
The spread of COVID-19 has been thwarted in most countries through non-pharmaceutical interventions. In particular, the most effective measures in this direction have been the stay-at-home and closure strategies of businesses and schools. However, population-wide lockdowns are far from being optimal, carrying heavy economic consequences. Therefore, there is nowadays a strong interest in designing more efficient restrictions. In this work, starting from a recent kinetic-type model which takes into account the heterogeneity described by the social contact of individuals, we analyse the effects of introducing an optimal control strategy into the system, to limit selectively the mean number of contacts and reduce consequently the number of infected cases. Thanks to a data-driven approach, we show that this new mathematical model permits us to assess the effects of the social limitations. Finally, using the model introduced here and starting from the available data, we show the effectiveness of the proposed selective measures to dampen the epidemic trends. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.
{"title":"Optimal control of epidemic spreading in the presence of social heterogeneity","authors":"G. Dimarco, G. Toscani, M. Zanella","doi":"10.1098/rsta.2021.0160","DOIUrl":"https://doi.org/10.1098/rsta.2021.0160","url":null,"abstract":"The spread of COVID-19 has been thwarted in most countries through non-pharmaceutical interventions. In particular, the most effective measures in this direction have been the stay-at-home and closure strategies of businesses and schools. However, population-wide lockdowns are far from being optimal, carrying heavy economic consequences. Therefore, there is nowadays a strong interest in designing more efficient restrictions. In this work, starting from a recent kinetic-type model which takes into account the heterogeneity described by the social contact of individuals, we analyse the effects of introducing an optimal control strategy into the system, to limit selectively the mean number of contacts and reduce consequently the number of infected cases. Thanks to a data-driven approach, we show that this new mathematical model permits us to assess the effects of the social limitations. Finally, using the model introduced here and starting from the available data, we show the effectiveness of the proposed selective measures to dampen the epidemic trends. 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":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78822117","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}
The background method is a widely used technique to bound mean properties of turbulent flows rigorously. This work reviews recent advances in the theoretical formulation and numerical implementation of the method. First, we describe how the background method can be formulated systematically within a broader ‘auxiliary function’ framework for bounding mean quantities, and explain how symmetries of the flow and constraints such as maximum principles can be exploited. All ideas are presented in a general setting and are illustrated on Rayleigh–Bénard convection between stress-free isothermal plates. Second, we review a semidefinite programming approach and a timestepping approach to optimizing bounds computationally, revealing that they are related to each other through convex duality and low-rank matrix factorization. Open questions and promising directions for further numerical analysis of the background method are also outlined. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.
{"title":"The background method: theory and computations","authors":"Giovanni Fantuzzi, A. Arslan, A. Wynn","doi":"10.1098/rsta.2021.0038","DOIUrl":"https://doi.org/10.1098/rsta.2021.0038","url":null,"abstract":"The background method is a widely used technique to bound mean properties of turbulent flows rigorously. This work reviews recent advances in the theoretical formulation and numerical implementation of the method. First, we describe how the background method can be formulated systematically within a broader ‘auxiliary function’ framework for bounding mean quantities, and explain how symmetries of the flow and constraints such as maximum principles can be exploited. All ideas are presented in a general setting and are illustrated on Rayleigh–Bénard convection between stress-free isothermal plates. Second, we review a semidefinite programming approach and a timestepping approach to optimizing bounds computationally, revealing that they are related to each other through convex duality and low-rank matrix factorization. Open questions and promising directions for further numerical analysis of the background method are also outlined. 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":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83089452","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}
Motivated by recent successes in model-based pre-election polling, we propose a kinetic model for opinion formation which includes voter demographics and socio-economic factors like age, sex, ethnicity, education level, income and other measurable factors like behaviour in previous elections or referenda as a key driver in the opinion formation dynamics. The model is based on Toscani’s kinetic opinion formation model (Toscani G. 2006 Kinetic models of opinion formation. Commun. Math. Sci. 4, 481–496.) and the leader–follower model of Düring et al. (Düring B. et al. 2009 Boltzmann and Fokker–Planck equations modelling opinion formation in the presence of strong leaders. Proc. R. Soc. A 465, 3687–3708.), and leads to a system of coupled Boltzmann-type equations and associated, approximate Fokker–Planck-type systems. Numerical examples using data from general elections in the UK show the effect different demographics have on the opinion formation process and the outcome of elections. This article is part of the theme issue ‘Kinetic exchange models of societies and economies’.
受最近基于模型的选前民意调查成功的启发，我们提出了一个民意形成的动态模型，其中包括选民人口统计学和社会经济因素，如年龄、性别、种族、教育水平、收入和其他可测量因素，如以前选举或公投中的行为，作为民意形成动态的关键驱动因素。该模型基于Toscani的动态意见形成模型(Toscani G. 2006)。Commun。数学。科学学报，4,481-496 .)和d ring等人的领导-追随者模型(d ring B. et al. 2009)。Boltzmann和Fokker-Planck方程对强领导存在下意见形成的建模。程序R. Soc(A 465, 3687-3708 .)，并导致耦合boltzmann型方程和相关的近似fokker - planck型系统的系统。使用英国大选数据的数值示例显示了不同人口统计对意见形成过程和选举结果的影响。本文是“社会和经济的动态交换模型”主题的一部分。
{"title":"On a kinetic opinion formation model for pre-election polling","authors":"Bertram Düring, Oliver Wright","doi":"10.1098/rsta.2021.0154","DOIUrl":"https://doi.org/10.1098/rsta.2021.0154","url":null,"abstract":"Motivated by recent successes in model-based pre-election polling, we propose a kinetic model for opinion formation which includes voter demographics and socio-economic factors like age, sex, ethnicity, education level, income and other measurable factors like behaviour in previous elections or referenda as a key driver in the opinion formation dynamics. The model is based on Toscani’s kinetic opinion formation model (Toscani G. 2006 Kinetic models of opinion formation. Commun. Math. Sci. 4, 481–496.) and the leader–follower model of Düring et al. (Düring B. et al. 2009 Boltzmann and Fokker–Planck equations modelling opinion formation in the presence of strong leaders. Proc. R. Soc. A 465, 3687–3708.), and leads to a system of coupled Boltzmann-type equations and associated, approximate Fokker–Planck-type systems. Numerical examples using data from general elections in the UK show the effect different demographics have on the opinion formation process and the outcome of elections. 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":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88905331","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 prove an estimate of total (viscous plus modelled turbulent) energy dissipation in general eddy viscosity models for shear flows. The ratio of the near wall average viscosity to the effective global viscosity is the key parameter in the estimate. This result is then applied to the 1-equation, URANS model of turbulence for which this ratio depends on the specification of the turbulence length scale. The model, which was derived by Prandtl in 1945, is a component of a 2-equation model derived by Kolmogorov in 1942 and is the core of many unsteady, Reynolds averaged models for prediction of turbulent flows. Let τ denote a selected time scale. Away from walls, interpreting an early suggestion of Prandtl, we set l=2k1/2τ.In the near-wall region analysis suggests replacing the traditional l=0.41d (d= wall normal distance) with l=0.41dd/L giving l=min{2k 1/2τ, 0.41ddL}.This specification of l results in a simpler model with correct near wall asymptotics. Its energy dissipation rate scales no larger than the physically correct O(U3/L), balancing energy input with energy dissipation. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.
{"title":"On the Prandtl–Kolmogorov 1-equation model of turbulence","authors":"Kiera Kean, W. Layton, M. Schneier","doi":"10.1098/rsta.2021.0054","DOIUrl":"https://doi.org/10.1098/rsta.2021.0054","url":null,"abstract":"We prove an estimate of total (viscous plus modelled turbulent) energy dissipation in general eddy viscosity models for shear flows. The ratio of the near wall average viscosity to the effective global viscosity is the key parameter in the estimate. This result is then applied to the 1-equation, URANS model of turbulence for which this ratio depends on the specification of the turbulence length scale. The model, which was derived by Prandtl in 1945, is a component of a 2-equation model derived by Kolmogorov in 1942 and is the core of many unsteady, Reynolds averaged models for prediction of turbulent flows. Let τ denote a selected time scale. Away from walls, interpreting an early suggestion of Prandtl, we set l=2k1/2τ.In the near-wall region analysis suggests replacing the traditional l=0.41d (d= wall normal distance) with l=0.41dd/L giving l=min{2k 1/2τ, 0.41ddL}.This specification of l results in a simpler model with correct near wall asymptotics. Its energy dissipation rate scales no larger than the physically correct O(U3/L), balancing energy input with energy dissipation. 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":"81 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83865993","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 paper develops an econometric price model with fundamental impacts for intraday electricity markets of 15-min contracts. A unique dataset of intradaily updated forecasts of renewable power generation is analysed. We use a threshold regression model to examine how 15-min intraday trading depends on the slope of the merit order curve. Our estimation results reveal strong evidence of mean reversion in the price formation mechanism of 15-min contracts. Additionally, prices of neighbouring contracts exhibit strong explanatory power and a positive impact on prices of a given contract. We observe an asymmetric effect of renewable forecast changes on intraday prices depending on the merit-order-curve slope. In general, renewable forecasts have a higher explanatory power at noon than in the morning and evening, but price information is the main driver of 15-min intraday trading. This article is part of the theme issue ‘The mathematics of energy systems’.
{"title":"An econometric model for intraday electricity trading","authors":"Marcel Kremer, Ruediger Kiesel, Florentina Paraschiv","doi":"10.2139/ssrn.3489214","DOIUrl":"https://doi.org/10.2139/ssrn.3489214","url":null,"abstract":"This paper develops an econometric price model with fundamental impacts for intraday electricity markets of 15-min contracts. A unique dataset of intradaily updated forecasts of renewable power generation is analysed. We use a threshold regression model to examine how 15-min intraday trading depends on the slope of the merit order curve. Our estimation results reveal strong evidence of mean reversion in the price formation mechanism of 15-min contracts. Additionally, prices of neighbouring contracts exhibit strong explanatory power and a positive impact on prices of a given contract. We observe an asymmetric effect of renewable forecast changes on intraday prices depending on the merit-order-curve slope. In general, renewable forecasts have a higher explanatory power at noon than in the morning and evening, but price information is the main driver of 15-min intraday trading. This article is part of the theme issue ‘The mathematics of energy systems’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84216826","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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