Mean--field electrodynamics: Critical analysis of various analytical approaches to the mean electromotive force

There are various analytical approaches to the mean electromotive force $\cal E =$ crucial in mean--field electrodynamics, with $\vec{u}$ and $\vec{b}$ being velocity and magnetic field fluctuations. In most cases the traditional approach, restricted to the second--order correlation approximation, has been used. Its validity is only guaranteed for a range of conditions, which is narrow in view of many applications, e.g., in astrophysics. With the intention to have a wider range of applicability other approaches have been proposed which make use of the so--called $\tau$--approximation, reducing correlations of third order in $\vec{u}$ and $\vec{b}$ to such of second order. After explaining some basic features of the traditional approach a critical analysis of the approaches of that kind is given. It is shown that they lead in some cases to results which are in clear conflict with those of the traditional approach. It is argued that this indicates shortcomings of the $\tau$--approaches and poses serious restrictions to their applicability. These shortcomings do not result from the basic assumption of the $\tau$--approximation. Instead, they seem to originate in some simplifications made in order to derive $\cal E$ without really solving the equations governing $\vec{u}$ and $\vec{b}$. A starting point for a new approach is described which avoids the conflict.Comment: 32 pages, no figures; accepted by Geophys. Astrophys. Fluid Dynam. A quenching formula for \alpha and a section on comparisons with numerical simulations added; references amended; changes in presentation and languag

Multi-dimensional Core-Collapse Supernova Simulations with Neutrino Transport

We present multi-dimensional core-collapse supernova simulations using the Isotropic Diffusion Source Approximation (IDSA) for the neutrino transport and a modified potential for general relativity in two different supernova codes: FLASH and ELEPHANT. Due to the complexity of the core-collapse supernova explosion mechanism, simulations require not only high-performance computers and the exploitation of GPUs, but also sophisticated approximations to capture the essential microphysics. We demonstrate that the IDSA is an elegant and efficient neutrino radiation transfer scheme, which is portable to multiple hydrodynamics codes and fast enough to investigate long-term evolutions in two and three dimensions. Simulations with a 40 solar mass progenitor are presented in both FLASH (1D and 2D) and ELEPHANT (3D) as an extreme test condition. It is found that the black hole formation time is delayed in multiple dimensions and we argue that the strong standing accretion shock instability before black hole formation will lead to strong gravitational waves.Comment: 3 pages, proceedings for Nuclei in the Cosmos XIV, Niigata, Japan (2016

Two-Dimensional Core-Collapse Supernova Simulations with the Isotropic Diffusion Source Approximation for Neutrino Transport

The neutrino mechanism of core-collapse supernova is investigated via non-relativistic, two-dimensional (2D), neutrino radiation-hydrodynamic simulations. For the transport of electron flavor neutrinos, we use the interaction rates defined by Bruenn (1985) and the isotropic diffusion source approximation (IDSA) scheme, which decomposes the transported particles into trapped particle and streaming particle components. Heavy neutrinos are described by a leakage scheme. Unlike the "ray-by-ray" approach in some other multi-dimensional supernova models, we use cylindrical coordinates and solve the trapped particle component in multiple dimensions, improving the proto-neutron star resolution and the neutrino transport in angular and temporal directions. We provide an IDSA verification by performing 1D and 2D simulations with 15 and 20 $M_\odot$ progenitors from Woosley et al.~(2007) and discuss the difference of our IDSA results with those existing in the literature. Additionally, we perform Newtonian 1D and 2D simulations from prebounce core collapse to several hundred milliseconds postbounce with 11, 15, 21, and 27 $M_\odot$ progenitors from Woosley et al.~(2002) with the HS(DD2) equation of state. General relativistic effects are neglected. We obtain robust explosions with diagnostic energies $E_{\rm dig} \gtrsim 0.1- 0.5$~B for all considered 2D models within approximately $100-300$ milliseconds after bounce and find that explosions are mostly dominated by the neutrino-driven convection, although standing accretion shock instabilities are observed as well. We also find that the level of electron deleptonization during collapse dramatically affect the postbounce evolution, e.g.~the ignorance of neutrino-electron scattering during collapse will lead to a stronger explosion.Comment: 23 pages. Accepted for publication in Ap

Stochastic Nonlinear Model Predictive Control with Efficient Sample Approximation of Chance Constraints

This paper presents a stochastic model predictive control approach for nonlinear systems subject to time-invariant probabilistic uncertainties in model parameters and initial conditions. The stochastic optimal control problem entails a cost function in terms of expected values and higher moments of the states, and chance constraints that ensure probabilistic constraint satisfaction. The generalized polynomial chaos framework is used to propagate the time-invariant stochastic uncertainties through the nonlinear system dynamics, and to efficiently sample from the probability densities of the states to approximate the satisfaction probability of the chance constraints. To increase computational efficiency by avoiding excessive sampling, a statistical analysis is proposed to systematically determine a-priori the least conservative constraint tightening required at a given sample size to guarantee a desired feasibility probability of the sample-approximated chance constraint optimization problem. In addition, a method is presented for sample-based approximation of the analytic gradients of the chance constraints, which increases the optimization efficiency significantly. The proposed stochastic nonlinear model predictive control approach is applicable to a broad class of nonlinear systems with the sufficient condition that each term is analytic with respect to the states, and separable with respect to the inputs, states and parameters. The closed-loop performance of the proposed approach is evaluated using the Williams-Otto reactor with seven states, and ten uncertain parameters and initial conditions. The results demonstrate the efficiency of the approach for real-time stochastic model predictive control and its capability to systematically account for probabilistic uncertainties in contrast to a nonlinear model predictive control approaches.Comment: Submitted to Journal of Process Contro

Specifying and Placing Chains of Virtual Network Functions

Network appliances perform different functions on network flows and constitute an important part of an operator's network. Normally, a set of chained network functions process network flows. Following the trend of virtualization of networks, virtualization of the network functions has also become a topic of interest. We define a model for formalizing the chaining of network functions using a context-free language. We process deployment requests and construct virtual network function graphs that can be mapped to the network. We describe the mapping as a Mixed Integer Quadratically Constrained Program (MIQCP) for finding the placement of the network functions and chaining them together considering the limited network resources and requirements of the functions. We have performed a Pareto set analysis to investigate the possible trade-offs between different optimization objectives

Towards generating a new supernova equation of state: A systematic analysis of cold hybrid stars

The hadron-quark phase transition in core-collapse supernovae (CCSNe) has the potential to trigger explosions in otherwise nonexploding models. However, those hybrid supernova equations of state (EOS) shown to trigger an explosion do not support the observational 2 M$_\odot$ neutron star maximum mass constraint. In this work, we analyze cold hybrid stars by the means of a systematic parameter scan for the phase transition properties, with the aim to develop a new hybrid supernova EOS. The hadronic phase is described with the state-of-the-art supernova EOS HS(DD2), and quark matter by an EOS with a constant speed of sound (CSS) of $c_{QM}^2=1/3$. We find promising cases which meet the 2 M$_\odot$ criterion and are interesting for CCSN explosions. We show that the very simple CSS EOS is transferable into the well-known thermodynamic bag model, important for future application in CCSN simulations. In the second part, the occurrence of reconfinement and multiple phase transitions is discussed. In the last part, the influence of hyperons in our parameter scan is studied. Including hyperons no change in the general behavior is found, except for overall lower maximum masses. In both cases (with and without hyperons) we find that quark matter with $c_{QM}^2=1/3$ can increase the maximum mass only if reconfinement is suppressed or if quark matter is absolutely stable.Comment: 14 pages, 11 figures, v2: matches published versio