Stability-beta limit of helical equilibria

Investigation of helical, net longitudinal currentless equilibria with the help of the high-n number ballooning criterion and with the 2D spectral code HERA of m = 1, 2, 3 (2m poloidal node number) internal and external modes suggests that the MHD stability limit of helical equilibria exceeds 〈β〉 = 0.1.     

Transformation of ERATO into a δW code

The spectral code ERATO has been transformed into a stability code based on the energy principle δW. It only gives the stability index, i.e. the “yes or no” answer to the stability of an axisymmetric toroidal plasma equilibrium. Compared to the full ERATO code, the computing time is reduced by a factor of 4. Storage requirements and input/output operations reduced by a factor of 5.     

HCO-TERNARY: A FORTRAN code for calculating P–V–T–X properties and liquid vapor equilibria of fluids in the system H2O–CO2–CH4

The equation of state (EOS) for the system H₂O–CO₂–CH₄ was programmed in a FORTRAN code which allows for its utilization in several modes. In one mode, specifically designed for mathematical modeling of two-phase, two-component flow, the code accepts as independent variables (1) pressure and (2) the composition of the gas phase. Other modes allow for the calculation of phase equilibria and/or the molar volumes of H₂O and binary mixtures, with pressure and temperature as the input variables (just pressure in the case of H₂O). Another mode is used to calculate phase equilibria for ternary mixtures, with pressure, temperature and the mole fraction of water in the gas phase as input variables. The algorithms for automatic convergence utilized in each mode are described.The code was tested extensively against experimental data from the literature. Some of these data were applied in the development of the EOS, and others were published subsequently. Analyses of the performance of the code and EOS for the modes described above, in the range 50–1000°C, 0–1000 bar, are presented. P–T–X regions of best applicability of the code and EOS are also identified.     

Tearing mode stability in 1D and 2D

A stability code for tearing modes in 1D and 2D straight equilibria in the tokamak scaling has been developed. It finds the lowest eigenvalues of a Hermitian problem which is obtained analytically by a reduction of the full problem. The main advantage is the powerful handling of equilibria with several resonant surfaces and displaying poloidal and radial mode couplings. The code has been successfully tested by comparing it with explicitly known analytical results for external kinks.     

On the numerical determination of ideal MHD limits of stability of axisymmetric toroidal configurations

The various problems that are encountered in searching with ERATO the limits of stability of axisymmetric toroidal equilibria are described and illustrated with specific examples.     

GATO: An MHD stability code for axisymmetric plasmas with internal separatrices

The GATO code computes the growth rate of ideal magnetohydrodynamic instabilities in axisymmetric geometries with internal separatrices such as doublet and expanded spheromak. The basic method, which uses a variational principle and a Galerkin procedure to obtain a matrix eigenvalue problem, is common to the ERATO and PEST codes. A new coordinate system has been developed to handle the internal separatrix. Efficient algorithms have been developed to solve the matrix eigenvalue problem for matrices of rank as large as 40 000. Further improvement is expected using graph theoretical techniques to reorder the matrices. Using judicious mesh repartition, the marginal point can be determined with great precision. The code has been extensively used to optimize doublet and general tokamak plasmas.     

MOMCON: A spectral code for obtaining three-dimensional magnetohydrodynamic equilibria

A new code, MOMCON (spectral moments code with constraints), is described that computes three-dimensional ideal magnetohydrodynamic (MHD) equilibria in a fixed toroidal domain using a Fourier expansion for the inverse coordinates (R, Z) representing nested magnetic surfaces. A set of nonlinear coupled ordinary differential equations for the spectral coefficients of (R, Z) is solved using an accelerated steepest descent method. A stream function, λ, is introduced to improve the mode convergence properties of the Fourier series for R and Z. The convergence rate of the R - Z spectra is optimized on each flux surface by solving nonlinear constraint equations relating the m ≥ 2 spectral coefficients of R and Z.     

Conservative global gyrokinetic toroidal full-f five-dimensional Vlasov simulation

A new conservative global gyrokinetic toroidal full-f five-dimensional Vlasov simulation (GT5D) is developed using a novel non-dissipative conservative finite difference scheme. The scheme guarantees numerical stability by satisfying relevant first principles in the modern gyrokinetic theory, and enables robust and accurate simulations of tokamak micro-turbulence. GT5D is verified through comparisons of zonal flow damping tests, linear analyses of ion temperature gradient driven (ITG) modes, and nonlinear ITG turbulence simulations against a global gyrokinetic toroidal δf particle code. In the comparison, global solutions of the ITG turbulence are identified quantitatively by using two gyrokinetic codes based on particle and mesh approaches.     

Linear and nonlinear resistive instability studies

The linear stability properties of the resistive ‘g’ mode are examined. The effects of including a full stress tensor have been examined for this mode. The nonlinear ‘g’ mode has also been examined and a potential saturation mechanism identified. The results of 2D calculations for the m = 0 and m = 1 ‘g’ modes in the reverse field pinch (RFP) are presented. Ergodic field line behaviour is found as a result of the interaction of mixed helicity ‘g’ modes.     

MINERVA: Ideal MHD stability code for toroidally rotating tokamak plasmas

A new linear MHD stability code MINERVA is developed for investigating a toroidal rotation effect on the stability of ideal MHD modes in tokamak plasmas. This code solves the Frieman-Rotenberg equation as not only the generalized eigenvalue problem but also the initial value problem. The parallel computing method used in this code realizes the stability analysis of both long and short wavelength MHD modes in short time. The results of some benchmarking tests show the validity of this MINERVA code. The numerical study with MINERVA about the toroidal rotation effect on the edge MHD stability shows that the rotation shear destabilizes the intermediate wavelength modes but stabilizes the short wavelength edge localized MHD modes, though the rotation frequency destabilizes both the long and the short wavelength MHD modes.     

Vector processing efficiency of plasma MHD codes by use of the FACOM 230-75 APU

In the framework of pipelined vector architecture, the efficiency of vector processing is assessed with respect to plasma MHD codes in nuclear fusion research. By using a vector processor, the FACOM 230-75 APU, the limit of the enhancement factor due to parallelism of current vector machines is examined for three numerical codes based on a fluid model. Reasonable speed-up factors of approximately 6,6 and 4 times faster than the highly optimized scalar version are obtained for ERATO (linear stability code), AEOLUS-R1 (nonlinear stability code) and APOLLO (1-1/2D transport code), respectively. Problems of the pipelined vector processors are discussed from the viewpoint of restructuring, optimization and choice of algorithms. In conclusion, the important concept of “concurrency within pipelined parallelism” is emphasized.     

Efficient Coordination Mechanisms for Unrelated Machine Scheduling

We present three new coordination mechanisms for scheduling n selfish jobs on m unrelated machines. A coordination mechanism aims to mitigate the impact of selfishness of jobs on the efficiency of schedules by defining a local scheduling policy on each machine. The scheduling policies induce a game among the jobs and each job prefers to be scheduled on a machine so that its completion time is minimum given the assignments of the other jobs. We consider the maximum completion time among all jobs as the measure of the efficiency of schedules. The approximation ratio of a coordination mechanism quantifies the efficiency of pure Nash equilibria (price of anarchy) of the induced game. Our mechanisms are deterministic, local, and preemptive in the sense that the scheduling policy does not necessarily process the jobs in an uninterrupted way and may introduce some idle time. Our first coordination mechanism has approximation ratio Θ(logm) and always guarantees that the induced game has pure Nash equilibria to which the system converges in at most n rounds. This result improves a bound of O(log² m) due to Azar, Jain, and Mirrokni and, similarly to their mechanism, our mechanism uses a global ordering of the jobs according to their distinct IDs. Next we study the intriguing scenario where jobs are anonymous, i.e., they have no IDs. In this case, coordination mechanisms can only distinguish between jobs that have different load characteristics. Our second mechanism handles anonymous jobs and has approximation ratio $O(\frac{\log m}{\log\log m})$ although the game induced is not a potential game and, hence, the existence of pure Nash equilibria is not guaranteed by potential function arguments. However, it provides evidence that the known lower bounds for non-preemptive coordination mechanisms could be beaten using preemptive scheduling policies. Our third coordination mechanism also handles anonymous jobs and has a nice “cost-revealing” potential function. We use this potential function in order, not only to prove the existence of equilibria, but also to upper-bound the price of stability of the induced game by O(logm) and the price of anarchy by O(log² m). Our third coordination mechanism is the first that handles anonymous jobs and simultaneously guarantees that the induced game is a potential game and has bounded price of anarchy. In order to obtain the above bounds, our coordination mechanisms use m as a parameter. Slight variations of these mechanisms in which this information is not necessary achieve approximation ratios of O(m ^? ), for any constant ?>0.     

High β and toroidal effects in three dimensions

The resistive MHD equations for toroidal plasma configurations are reduced by expanding to second order in ?, the inverse aspect ratio, allowing for high β = μ₀p/B² of order ?. The result is a closed system of nonlinear, three-dimensional equations where the fast magnetohydrodynamic time scale is eliminated. In particular, the equation for the toroidal velocity remains decoupled. These equations generalize reduced equations derived earlier. They are now solved numerically. As a first step, various types of axisymmetric equilibria are found by relaxing a given initial configuration. Then, three-dimensional perturbations are introduced and followed in time to investigate the linear and nonlinear properties of tearing modes, etc., for high-β plasmas in toroidal geometry.     

Thermodynamic modelling of the Cr-Nb-Sn system

The Cr-Nb-Sn system has been studied experimentally, by first principles calculation and finally modeled with the Calphad method. The experimental study has been carried out on the Cr-Sn binary system to determine the solubility of Sn in the A2 (Cr) solid solution, but also in the Cr-Nb-Sn ternary system in order to determine phase equilibria of the isothermal section at 800 °C and 1100 °C. Besides, the formation enthalpies of all the ordered configurations of the C15 and A15 phases and the stoichiometric Nb₆Sn₅ and NbSn₂ phases have been calculated using the Density Functional Theory (DFT). The mixing enthalpies of the A2 binary solid solutions have been estimated using the Special Quasirandom Structures (SQS). All these new experimental and calculated data have been taken into account for a new thermodynamic assessment of the three binary and the ternary systems.     

Extending the notion of rationality of selfish agents: Second Order Nash equilibria

Motivated by the increasing interest of the Computer Science community in the study and understanding of non-cooperative systems, we present a novel model for formalizing the rational behavior of agents with a more farsighted view of the consequences of their actions. This approach yields a framework creating new equilibria, which we call Second Order equilibria, starting from a ground set of traditional ones. By applying our approach to pure Nash equilibria, we define the set of Second Order pure Nash equilibria and present their applications to the Prisoner’s Dilemma game, to an instance of Braess’s Paradox in the Wardrop model and to the KP model with identical machines.     

Extension of the Newcomb equation into the vacuum for the stability analysis of tokamak edge plasmas

The formulation for solving numerically the two-dimensional Newcomb equation has been extended to calculate the vacuum energy integral by using a vector potential method. According to this extension, a stability code MARG2D has been adapted, and coded for parallel computing in order to reduce substantially the CPU time. The MARG2D code enables a fast stability analysis of ideal external MHD modes from low to high toroidal mode numbers on the basis of the single physical model, and then the code works as a powerful tool in an integrated simulation where it is combined with transport codes, and also in the analysis of tokamak edge plasma experiments.     

The CHEASE code for toroidal MHD equilibria

The CHEASE code (Cubic Hermite Element Axisymmetric Static Equilibrium) solves the Grad-Shafranov equation for toroidal MHD equilibria using a Hermite bicubic finite element discretization with pressure, current profiles and plasma boundaries specified by analytical forms or sets of experimental data points. Moreover, CHEASE allows the automatic generation of pressure profiles marginally stable to ballooning modes or with a prescribed fraction of bootstrap current. The code provides equilibrium quantities for several stability and global wave propagation codes.     

On the dynamics of a gyrostat on Lagrangian equilibria in the three body problem

In this paper, we consider the noncanonical Hamiltonian dynamics of a gyrostat in the three-body problem. By means of geometric-mechanics methods, we study the approximate Poisson dynamics that arises when we develop the potential of the system in series of Legendre and truncate this in an arbitrary order k. After reduction of the dynamics by means of the two symmetries of the system, the existence and number of equilibria that we denominate of Lagrangian type in analogy with classic results on the topic, are considered. Necessary and sufficient conditions are established for their existence in an approximate dynamics of order k, and explicit expressions of these equilibria are given; this being useful for the subsequent study of the stability of these equilibria. The number of Lagrangian equilibria is thoroughly studied in approximate dynamics of orders zero and one. The main result of this work indicates that the number of Lagrangian equilibria in an approximate dynamics of order k for k≥1 is independent of the order of truncation of the potential, if the gyrostat S ₀ is almost spherical. In relation to the stability of these equilibria, necessary and sufficient conditions are given for linear stability of Lagrangian equilibria when the gyrostat is almost spherical. The rotational Poisson dynamics of the gyrostat placed at an Lagrangian equilibrium is explored and the study of nonlinear stability conditions is considered. In this way, we generalize the classical results on equilibria of the three-body problem and many results provided by other authors using more classical techniques for the case of rigid bodies.     

Bounded Budget Connection (BBC) games or how to make friends and influence people,on a budget

Motivated by applications in social and peer-to-peer networks, we introduce the Bounded Budget Connection (BBC) game and study its pure Nash equilibria. We have a collection of n   players, each with a budget for purchasing links. Each link has a cost and a length. Each node has a preference weight for each node, and its objective is to purchase outgoing links within its budget to minimize its sum of preference-weighted distances to the nodes. We show that determining if a BBC game has pure Nash equilibria is NP-hard. We study (n,k)$(n,k)$-uniform BBC games, where all link costs, lengths, and preferences are equal and every budget equals k  . We show that pure Nash equilibria exist for all (n,k)$(n,k)$-uniform BBC games and all equilibria are essentially fair. We construct a family of equilibria spanning the spectrum from minimum to maximum social cost. We also analyze best-response walks and alternative node objectives.     

On the Performance of Approximate Equilibria in Congestion Games

We study the performance of approximate Nash equilibria for congestion games with polynomial latency functions. We consider how much the price of anarchy worsens and how much the price of stability improves as a function of the approximation factor ε. We give tight bounds for the price of anarchy of atomic and non-atomic congestion games and for the price of stability of non-atomic congestion games. For the price of stability of atomic congestion games we give non-tight bounds for linear latencies. Our results not only encompass and generalize the existing results of exact equilibria to ε-Nash equilibria, but they also provide a unified approach which reveals the common threads of the atomic and non-atomic price of anarchy results. By expanding the spectrum, we also cast the existing results in a new light.