Equilibrium Code Research Articles

First demonstration of real-time kinetic equilibrium reconstruction on TCV by coupling LIUQE and RAPTOR

Kinetic equilibrium reconstruction (KER) in tokamaks is the solution of the free-boundary MHD equilibrium that best fits the external magnetic measurements and the internal plasma profiles as imposed from modeling and/or available internal measurements. In this paper, for the first time, the KER has been performed in real-time by coupling the free-boundary equilibrium code LIUQE to the 1.5D transport code RAPTOR, during a TCV discharge. The transport code is used as a dynamic state observer evolving the current diffusion equation, the electron heat and particle diffusion equations and correcting the prediction with the measurements available in real-time, making use of the extended Kalman filter (EKF) method. The simple coupling strategy, based on a matching of the free functions ( and ) on the right-hand-side of the Grad Shafranov equation, is shown to be effective in providing consistency of pressure and current density profiles between the two codes. We also show that, despite this highly non-linear system, this technique does not require significant additional computational time with respect to the running times of the two independent codes and does not require a tight coupling of the two codes allowing therefore their independent development. The KER is compared to standard real-time equilibrium reconstruction, performed routinely in TCV, where only the external magnetic measurements are taken into account. The following cases are considered: flat central ne, current density broadening due to applied electron cyclotron current drive and neoclassical tearing mode causing a temperature drop. The KER reproduces the expected changes of the internal profiles, tracking their dynamic evolution in their physical timescales.

Internal Ballistics of a Boron-Containing Solid Fuel Ramjet

ABSTRACT The solid fuel ramjet is one of the simplest air-breathing engines, characterized by increased specific impulse. This study focuses on a parametric investigation of the internal ballistics of a boron-containing solid fuel ramjet. A 2-D axisymmetric model of a ramjet combustor flying at Mach 2.5 was developed and solved numerically utilizing a commercial computational fluid dynamics code modified to address the specific issues of the problem. The simulation results were compared to the theoretical performance calculated by the chemical equilibrium code. The solid fuel regression rate was found to depend on the port diameter and thus decreases with time. Thrust regulation of ramjet engine was demonstrated using bypass air. The specific impulse was found to decrease with boron content, probably because of low combustion efficiency caused by either insufficient oxygen supply or short residence time of the boron particles.

A new set of atmosphere and evolution models for cool T–Y brown dwarfs and giant exoplanets

We present a new set of solar metallicity atmosphere and evolutionary models for very cool brown dwarfs and self-luminous giant exoplanets, which we term ATMO 2020. Atmosphere models are generated with our state-of-the-art 1D radiative-convective equilibrium code ATMO, and are used as surface boundary conditions to calculate the interior structure and evolution of 0.001–0.075 M⊙ objects. Our models include several key improvements to the input physics used in previous models available in the literature. Most notably, the use of a new H–He equation of state including ab initio quantum molecular dynamics calculations has raised the mass by ~1−2% at the stellar–substellar boundary and has altered the cooling tracks around the hydrogen and deuterium burning minimum masses. A second key improvement concerns updated molecular opacities in our atmosphere model ATMO, which now contains significantly more line transitions required to accurately capture the opacity in these hot atmospheres. This leads to warmer atmospheric temperature structures, further changing the cooling curves and predicted emission spectra of substellar objects. We present significant improvement for the treatment of the collisionally broadened potassium resonance doublet, and highlight the importance of these lines in shaping the red-optical and near-infrared spectrum of brown dwarfs. We generate three different grids of model simulations, one using equilibrium chemistry and two using non-equilibrium chemistry due to vertical mixing, all three computed self-consistently with the pressure-temperature structure of the atmosphere. We show the impact of vertical mixing on emission spectra and in colour-magnitude diagrams, highlighting how the 3.5−5.5 μm flux window can be used to calibrate vertical mixing in cool T–Y spectral type objects.

Stepped pressure equilibrium with relaxed flow and applications in reversed-field pinch plasmas

The multi-region relaxed magnetohydrodynamics (MRxMHD) has been successful in the construction of equilibria in three-dimensional (3D) configurations. In MRxMHD, the plasma is sliced into sub-volumes separated by ideal interfaces, each undergoing relaxation, allowing the formation of islands and chaos. The resulting equilibrium has a stepped pressure profile across sub-volumes. The stepped pressure equilibrium code (SPEC) (S R Hudson et al, Phys. Plasmas 19, 112502 (2012)) was developed to calculate MRxMHD equilibria numerically. In this work, we have extended the SPEC code to compute MRxMHD equilibria with field-aligned flow and rotation, following the theoretical development to incorporate cross-helicity and angular momentum constraints. The code has been verified for convergence and compared to a Grad–Shafranov solver in 2D. We apply our new tool to study the flow profile change before and after the sawtooth crash of a reversed-field pinch discharge, in which data of the parallel flow is available. We find the promising result that under the constraints of cross-helicity and angular momentum, the parallel flow profile in post-crash SPEC equilibrium is flat in the plasma core and the amplitude of the flow matches experimental observations. Finally, we provide an example equilibrium with a 3D helical field structure as the favoured lower energy state. This will be the first 3D numerical equilibrium in which the flow effects are self-consistently calculated.

“PROCESS”: Systems studies of spherical tokamaks

With a reduced aspect ratio, spherical tokamaks have a number of attractive features for a fusion power plant. This can be studied using systems codes, which allow for the rapid conceptual study of power plants covering everything, from the plasma through to electricity generation. In this paper, we describe models in the systems code process that have been added specifically for spherical tokamaks. Within process an alternative relation for the plasma current is included which accounts for the increased ratio of Ip/aB. We have tested this against a series of equilibria created with the free boundary equilibrium code fiesta, and additionally performed our own fit. We also outline the engineering changes that can be made to the device and describe a water-cooled copper centrepost model. To test our models we recreate the published designs for the Fusion Nuclear Science Facility (FNSF) and a High Temperature Superconducting Pilot Plant (HTS-PP) and find good agreement. We conclude by highlighting the efficiencies needed to produce net electricity from small fusion devices.

Analysis of the Magnetic Axis of Plasma Discharges in EAST

The magnetic-axis radial position ${R}_{M}$ is one of the most important parameters of the reliable, real-time, plasma control system. The magnetic-axis model suitable for the non-circular cross section plasma is studied based on the magnetic-axis model of the circular cross section plasma. The model is verified by the discharge parameters of Experimental and Advanced Superconducting Tokamak (EAST) ( $k \sim $ 1.4–1.9, i.e., non-circular cross section). The offline magnetic equilibrium code, EAST equilibrium fitting (EFIT), has been selected as the reference calculation for magnetic-axis radial position. The error of ${R}_{M}$ compared to the EAST EFIT code reference value is reported and discussed for the circular and non-circular cross-section plasmas in EAST. The analysis results show that the developed magnetic axis model is available to calculate the magnetic axis radial position value, with the average error $\overline {\Delta R_{M}}\approx 0.015$ and 0.025 m for the ${\beta }_{p}{ and ${\beta }_{p}{>2}$ cases, respectively. The error level depends on the elongation ratio $k $ and the poloidal plasma beta ${\beta }_{p}$ in the high- ${\beta }_{p}$ cases.

Introduction and Validation of the APEC Code for the Simulation of Plasma Equilibrium and Evolution in the Presence of 3-D Passive Structures

Recently, a novel code for the simulation of the plasma equilibrium and its evolution has been developed and fully implemented in ANSYS. We named it ANSYS Plasma Equilibrium and Evolution Code (APEC). It is based on the solution of the Grad–Shafranov equation and exploits the finite-element method using the magnetic potential vector formulation. In this article, we present a further development on the coupling of the axisymmetric plasma equilibrium equations, describing the plasma behavior, with the full 3-D eddy current equations, defining the surrounding 3-D structures. One test case, related to the International Thermonuclear Experimental Reactor (ITER) tokamak, has been fully analyzed and benchmarked with a validated code showing the validity of the proposed approach. In particular, the benchmark has concerned the comparison of the initial equilibrium and plasma evolution.

Jones-Wilkins-Lee Unreacted and Reaction Product Equations of State for Overdriven Detonations in Octogen- and Triaminotrinitrobenzene-Based Plastic-Bonded Explosives.

The Jones-Wilkins-Lee (JWL) equation of state (EOS) is used to calculate expansion of detonation reaction products from the chemical equilibrium Chapman-Jouguet (C-J) state to large volumes. Overdriven detonation waves with shock pressures higher than C-J are created by high-velocity impacts or converging detonation waves. Reflection from high-impedance materials, multiple shock impacts, and Mach stem wave interactions creates similar pressures. When overdriven states were first measured experimentally, the original reaction product JWL EOSs predicted excess compression. This problem was resolved by modifying the JWL EOS to produce less compression at high pressures while still correctly calculating expansion from the C-J state. Zeldovich-von Neumann-Doring (ZND) reactive flow models, which include the measured reaction zone momentum, explained experimental observations that lower C-J pressures are required to smoothly connect the C-J state to overdriven states on the product Hugoniot curve. Experimental data on overdriven detonation waves for two octogen (HMX)-based plastic-bonded explosives (PBXs), PBX 9501 and PBX 9404, and for two triaminotrinitrobenzene (TATB)-based PBXs, LX-17 and PBX 9502, are compared to various JWL reaction product EOSs, including ones generated by the CHEETAH chemical equilibrium code. Excellent agreement is obtained using JWL EOSs for overdriven shock pressures and densities up to 130 GPa and 3.8 g/cm3 for both HMX- and TATB-based PBXs.

Optimization design for plasma configuration at the CFETR

Equilibrium is essential in the design of the Chinese Fusion Engineering Test Reactor (CFETR). It is based on the principle underlying various physical and engineering issues, such as vertical instability, core plasma physics, divertor physics and design, etc. Built on the 0-D parameters with elongation κ∼2.0 using the equilibrium code TEQ, series of equilibria with different lower triangularity (range of 0.3–0.8) were constructed. Through iterations with the divertor and blanket structures, we optimized the plasma shape, including, in particular, the position of the X-point, hence obtaining the first acceptable single-null plasma with δ = 0.418. This optimal ITER-like equilibrium has in turn served as the starting point for various other design works. Besides the standard divertor configuration, the quasi-snowflake-plus (QSF+) divertor configurations with different total plasma currents have also been produced, which aimed at controlling excessive heat flux load on the divertor target. The results indicate that the poloidal field (PF) coil system has the capacity to sustain the shape of the QSF+; however, the PF coil system will continue to be optimized with particular attention of the divertor coil.

Constructing stellarators with quasisymmetry to high order

A method is given to rapidly compute quasisymmetric stellarator magnetic fields for plasma confinement, without the need to call a three-dimensional magnetohydrodynamic equilibrium code inside an optimization iteration. The method is based on direct solution of the equations of magnetohydrodynamic equilibrium and quasisymmetry using Garren & Boozer’s expansion about the magnetic axis (Phys Fluids B, vol. 3, 1991, p. 2805), and it is several orders of magnitude faster than the conventional optimization approach. The work here extends the method of Landremanet al. (J. Plasma Phys., vol. 85, 2019, 905850103), which was limited to flux surfaces with elliptical cross-section, to higher order in the aspect-ratio expansion. As a result, configurations can be generated with strong shaping that achieve quasisymmetry to high accuracy. Using this construction, we give the first numerical demonstrations of Garren and Boozer’s ideal scaling of quasisymmetry breaking with the cube of the inverse aspect ratio. We also demonstrate a strongly non-axisymmetric configuration (vacuum rotational transform$\unicode[STIX]{x1D704}>0.4$) in which symmetry-breaking mode amplitudes throughout a finite volume are${<}2\times 10^{-7}$, the smallest ever reported. To generate boundary shapes of finite-minor-radius configurations, a careful analysis is given of the effect of substituting a finite minor radius into the near-axis expansion. The approach here can provide analytic insight into the space of possible quasisymmetric stellarator configurations, and it can be used to generate good initial conditions for conventional stellarator optimization.

Origin of hydrogen fluoride emission in the Orion Bar

Context. The hydrogen fluoride (HF) molecule is seen in absorption in the interstellar medium (ISM) along many lines of sight. Surprisingly, it is observed in emission toward the Orion Bar, which is an interface between the ionized region around the Orion Trapezium stars and the Orion molecular cloud. Aims. We aim to understand the origin of HF emission in the Orion Bar by comparing its spatial distribution with other tracers. We examine three mechanisms to explain the HF emission: thermal excitation, radiative dust pumping, and chemical pumping. Methods. We used a Herschel/HIFI strip map of the HF J = 1 → 0 line, covering 0.5′ by 1.5′ that is oriented perpendicular to the Orion Bar. We used the RADEX non-local thermodynamic equilibrium (non-LTE) code to construct the HF column density map. We use the Meudon PDR code to explain the morphology of HF. Results. The bulk of the HF emission at 10 km s−1 emerges from the CO-dark molecular gas that separates the ionization front from the molecular gas that is deeper in the Orion Bar. The excitation of HF is caused mainly by collisions with H2 at a density of 105 cm−3 together with a small contribution of electrons in the interclump gas of the Orion Bar. Infrared pumping and chemical pumping are not important. Conclusions. We conclude that the HF J = 1 → 0 line traces CO-dark molecular gas. Similarly, bright photodissociation regions associated with massive star formation may be responsible for the HF emission observed toward active galactic nuclei.

Approximate solutions of polarization state evolution in tokamak plasma polarimetry and their precision

Polarimetry measurement results are used in tokamak plasma equilibrium codes as a constraint, decreasing the measurement uncertainty in determination plasma current, safety factor and density profile. In numerical codes, instead of solving rigorous equation of polarimetric state evolution, approximate analytical expression could be used to increase computing efficiency. In this paper is analyzed the precision of three different models compared with exact solution of polarization state evolution equation. Calculations are made for circular plasma model, with plasma parameters representative for operating domain of tokamak device. The obtained results define the limits of applicability of the separate proposed approximated methods in equilibrium codes for plasma in configuration similar to considered in the paper.

On the identification of the electron temperature profile from polarimetry Stokes vector measurements in Tokamak free-boundary equilibrium reconstruction

This paper reports numerical investigations on the identification of the electron temperature profile Te from interferometry and polarimetry Stokes vector measurements with the equilibrium code Newton direct and inverse computation for equilibrium. This latter enables the consistent resolution of the inverse equilibrium reconstruction problem in the framework of nonlinear free-boundary equilibrium coupled to the Stokes model equation for polarimetry. We find that for ITER plasma with high Ip, Ne and Te the identification from noisy measurements is possible.

The Meshfree Localized Petrov-Galerkin Approach in Slope Stability Analysis

Abstract The article focuses on the use of the meshfree numerical method in the field of slope stability computations. There are many meshfree implementations of numerical methods. The article shows the results obtained using the meshfree localized Petrov-Galerkin method (MLPG) – localized weak-form of the equilibrium equations with an often used elastoplastic material model based on Mohr-Coulomb (MC) yield criterion. The most important aspect of MLPG is that the discretization process uses a set of nodes instead of elements. Node position within the computational domain is not restricted by any prescribed relationship. The shape functions are constructed using just the set of nodes present in the simple shaped domain of influence. The benchmark slope stability numerical model was performed using the developed meshfree computer code and compared with conventional finite element (FEM) and limit equilibrium (LEM) codes. The results showed the ability of the implemented theoretical preliminaries to solve the geotechnical stability problems.

Much‐needed tools to reduce the experimental burden for determining the gas flammability parameters, Pmax and KG

AbstractThis article provides much‐needed analytical tools to reduce the experimental burden for determining the maximum pressure for flammable gas combustion, Pmax, and the deflagration index, KG, as a function of the gas concentration. Both of these parameters are important for risk assessment, process design, and design of preventive and mitigative safeguards. A very large experimental database for hydrogen/oxygen/nitrogen mixtures in a 20 L spherical vessel was used to validate the tools for Pmax and KG. The first tool considered was an equilibrium code to estimate the maximum combustion pressure, Pmax. The calculated equilibrium pressure agreed very well with the experimental pressures in a broad range around the stoichiometric concentration. An increasing difference occurred as the concentration approached the flammable limits. This result is consistent with homogeneous and complete combustion near the stoichiometric concentration and upward partial combustion near the flammable limits. The ratio of these two pressures is likely representative of the total volume combusted. The second tool was the application of a commercial computer program, TableCurve 2D, to fit experimental data for Pmax and KG. A set of simple empirical equations were identified to estimate Pmax and KG using just a few experimental data points. Methods of application are presented. Reliable experimental data is always preferred, but these tools should help reduce the experimental data burden.

Development of a Gyrokinetic Particle-in-Cell Code for Whole-Volume Modeling of Stellarators

We present initial results in the development of a gyrokinetic particle-in-cell code for the whole-volume modeling of stellarators. This is achieved through two modifications to the X-point Gyrokinetic Code (XGC), originally developed for tokamaks. One is an extension to three-dimensional geometries with an interface to Variational Moments Equilibrium Code (VMEC) data. The other is a connection between core and edge regions that have quite different field-line structures. The VMEC equilibrium is smoothly extended to the edge region by using a virtual casing method. Non-axisymmetric triangular meshes in which triangle nodes follow magnetic field lines in the toroidal direction are generated for field calculation using a finite-element method in the entire region of the extended VMEC equilibrium. These schemes are validated by basic benchmark tests relevant to each part of the calculation cycle, that is, particle push, particle-mesh interpolation, and field solver in a magnetic field equilibrium of Large Helical Device including the edge region. The developed code also demonstrates collisionless damping of geodesic acoustic modes and steady states with residual zonal flow in the core region.

Elimination of current sheets at resonances in three-dimensional toroidal ideal-magnetohydrodynamic equilibria

On the basis of the ideal-magnetohydrodynamic equilibrium code VMEC (Hirshman et al 1986 Comput. Phys. Commun. 43 143) supplemented by transformation of its results into magnetic coordinates, it is shown how current sheets, radially widened by the finite grid size, at rational values of rotational transform in 3d toroidal equilibria can be removed by variation of the equilibrium geometry.

ASDEX Upgrade flight simulator development

Fenix, the ASDEX Upgrade (AUG) flight simulator under development, is based on the Plasma Control System Simulation Platform (PCSSP) developed for ITER, the ASTRA transport code and the SPIDER equilibrium code. Fenix will give a session leader the possibility to check whether the discharge will meet experimental goals prior to execution. It is also designed to facilitate the development of control system features and the validation of physical models. It reads the AUG discharge program and checks if all the parameters and reference waveforms are reasonable during the discharge simulation.ASTRA serves as a plant model (physical model of AUG tokamak) which outputs idealised diagnostic signals (temperature, density, etc.). ASTRA calculates these data from the particle and energy transport in the plasma core, plasma edge, Scrape of Layer (SOL) and divertor. It also includes particle balance, L-H transition and sawtooth models, and it is equipped with the 2D equilibrium reconstruction code SPIDER.The second component of Fenix is a model of the Discharge Control System (DCS) and AUG actuators. The DCS model processes diagnostic signals from ASTRA, computes commands for actuators and sends them back to ASTRA closing the feedback loop. Controllers for coil currents to control plasma current, position and shape are implemented in the model as well as gas puff valves (divertor and midplane) and the pellet injector to control electron density. The DCS model also simulates external heating actuators such as NBI, ECRH and ICRH.Fenix has methods for reading configuration files and archiving. This article presents modelling of the ASTRA and DCS components, and the first results from the simulations of the flattop phase.

Approximate analytic expressions using Stokes model for tokamak polarimetry and their range of validity

The analysis of the polarimetry measurements has the aim of validating models (De Marco and Segre 1972 Plasma Phys. 14 245), with a careful attention to the clarification of their limits of application. In this paper a new approximation method is introduced, the so-called special constant Omega direction (SCOD), which gives an analytical solution to the polarimetry exact Stokes model equations. The available approximate solutions (including SCOD) of the polarimetry propagation equations are presented, compared and their application limits determined, using a reference tokamak configuration, which is a simplified equilibrium for a circular tokamak. The SCOD approximation is compared successfully to the Stokes model in the context also of equilibria evaluated for two JET discharges. The approximation methods are analytical or very simple mathematical expressions which can also be used in equilibrium codes for their optimization.

Comparative collisionless alpha particle confinement in stellarator reactors with the XGC gyrokinetic code

Whole volume simulations of stellarators are necessary to address a number of important physics and engineering issues, including turbulent transport prediction and optimising divertor operation. In this work, advances in the development of a whole volume stellarator capability for the global gyrokinetic particle-in-cell code XGC are detailed. A 3D interpolation of the equilibrium magnetic field to the last close flux surface, calculated using the VMEC ideal magnetohydrodynamic equilibrium code, has been implemented. Orbit tracing with the XGC code is verified for the Wendelstein 7-X stellarator by comparison with the BEAMS3D and EUTERPE codes. The XGC stellarator version has then been used to consider collisionless alpha particle confinement in representative quasi-isodynamic and quasi-axisymmetric designs. It has been shown that, at least without further optimisation for energetic particle confinement, the Wendelstein 7-X-like quasi-isodynamic reactor design exhibits better alpha particle confinement.