arXiv subjects

Eero Hirvijoki

Publications and source records attributed to Eero Hirvijoki.

At least 19 recordsLinked to original sources

arXiv2022

Multispecies structure-preserving particle discretization of the Landau collision operator

This paper proposes a novel numerical integrator for modeling multispecies Coulomb collisions in kinetic plasmas. The proposed scheme provides an energy-, momentum-, and positivity-preserving particle discretization of the nonlinear Landau collision operator, extending the works of J.A. Carrillo et al., Journal of Computational Physics, 7, 100066 (2020) and E. Hirvijoki, Plasma Physics and Controlled Fusion, 63, 044003 (2021). The discrete-time conservation properties are analyzed both algebraically and numerically, and an efficient, GPU-parallelized implementation is validated against inhomogeneous temperature relaxation, isotropization and thermalization examples. The results agree with analytical estimates, confirming the method capable of reproducing physics.

physics.plasm-ph↗

arXiv2022

Bayesian approach for validation of runaway electron simulations

Plasma-terminating disruptions in future fusion reactors may result in conversion of the initial current to a relativistic runaway electron beam. Validated predictive tools are required to optimize the scenarios and mitigation actuators to avoid the excessive damage that can be caused by such events. Many of the simulation tools applied in fusion energy research require the user to specify several input parameters that are not constrained by the available experimental information. Hence, a typical validation exercise requires multiparameter optimization to calibrate the uncertain input parameters for the best possible representation of the investigated physical system. The conventional approach, where an expert modeler conducts the parameter calibration based on domain knowledge, is prone to lead to an intractable validation challenge. For a typical simulation, conducting exhaustive multiparameter investigations manually to ensure a globally optimal solution and to rigorously quantify the uncertainties is an unattainable task, typically covered only partially and unsystematically. Bayesian inference algorithms offer a promising alternative approach that naturally includes uncertainty quantification and is less subjective to user bias in choosing the input parameters. The main challenge in using these methods is the computational cost of simulating enough samples to construct the posterior distributions for the uncertain input parameters. This challenge can be overcome by combining probabilistic surrogate modelling, such as Gaussian Process regression, with Bayesian optimization, which can reduce the number of required simulations by several orders of magnitude. Here, we implement this type of Bayesian optimization framework for a model for analysis of disruption runaway electrons, and explore for simulations of current quench in a JET plasma discharge with an argon induced disruption.

physics.plasm-ph↗

arXiv2022

Metriplectic foundations of gyrokinetic Vlasov-Maxwell-Landau theory

This letter reports on a metriplectic formulation of collisional, nonlinear full-$f$ electromagnetic gyrokinetic theory compliant with energy conservation and monotonic entropy production. In an axisymmetric background magnetic field, the toroidal angular momentum is also conserved. Notably, a new collisional current, contributing to the gyrokinetic Maxwell-Ampère equation and the gyrokinetic charge conservation law, is discovered.

physics.plasm-ph↗

arXiv2021

Structure-preserving marker-particle discretizations of Coulomb collisions for particle-in-cell codes

This paper contributes new insights into discretizing Coulomb collisions in kinetic plasma models. Building on the previous works [Carrillo et al. J. Comp. Phys. X 7:100066 (2020), Hirvijoki and Burby Phys. Plasmas 27(8):082307 (2020)], I propose deterministic discrete-time energy- and positivity-preserving, entropy-dissipating marker-particle schemes for the standard Landau collision operator and the electrostatic gyrokinetic Landau operator. In case of the standard Landau operator, the scheme preserves also the discrete-time kinetic momentum. The improvements, the extensions of the structure-preserving discretizations in [Carrillo et al. J. Comp. Phys. X 7:100066 (2020), Hirvijoki and Burby Phys. Plasmas 27(8):082307 (2020)] to discrete time, are made possible by exploiting the underlying metriplectic structure of the collision operators involved and the so-called discrete-gradient integrators.

physics.plasm-ph↗

arXiv2021

On the accuracy of the binary-collision algorithm in particle-in-cell simulations of magnetically confined fusion plasmas

Ideally, binary-collision algorithms conserve kinetic momentum and energy. In practice, the finite size of collision cells and the finite difference in the particle locations affect the conservation properties. In the present work, we investigate numerically how the accuracy of these algorithms is affected when the size of collision cells is large compared with gradient scale length of the background plasma, a parameter essential in full-f fusion plasma simulations. Additionally, we discuss implications for the conserved quantities in drift-kinetic formulations when fluctuating magnetic and electric fields are present: we suggest how the accuracy of the algorithms could potentially be improved with minor modifications.

physics.plasm-ph↗

arXiv2020

Subcycling of particle orbits in variational, geometric electromagnetic particle-in-cell methods

This paper investigates subcycling of particle orbits in variational, geometric particle-in-cell methods addressing the Vlasov--Maxwell system in magnetized plasmas. The purpose of subcycling is to allow different time steps for different particle species and, ideally, time steps longer than the electron gyroperiod for the global field solves while sampling the local cyclotron orbits accurately. The considered algorithms retain the electromagnetic gauge invariance of the discrete action, guaranteeing a local charge conservation law, while the variational approach provides a bounded long-time energy behavior.

physics.comp-ph↗

arXiv2020

Collisional Gyrokinetics Teases the Existence of Metriplectic Reduction

In purely non-dissipative systems, Lagrangian and Hamiltonian reduction have proven to be powerful tools for deriving physical models with exact conservation laws. We have discovered a hint that an analogous reduction method exists also for dissipative systems that respect the First and Second Laws of Thermodynamics. In this paper, we show that modern electrostatic gyrokinetics, a reduced plasma turbulence model, exhibits a serendipitous metriplectic structure. Metriplectic dynamics in general is a well developed formalism for extending the concept of Poisson brackets to dissipative systems. Better yet, our discovery enables an intuitive particle-in-cell discretization of the collision operator that also satisfies the First and Second Laws of thermodynamics. These results suggest that collisional gyrokinetics, and other dissipative physical models that obey the Laws of Thermodynamics, could be obtained using an as-yet undiscovered metriplectic reduction theory and that numerical methods could benefit from such theory significantly. Once uncovered, the theory would generalize Lagrangian and Hamiltonian reduction in a substantial manner.

physics.plasm-ph↗

arXiv2020

Energy and momentum conservation in the Euler-Poincaré formulation of local Vlasov-Maxwell-type systems

The action principle by Low [Proc. R. Soc. Lond. A 248, 282--287] for the classic Vlasov-Maxwell system contains a mix of Eulerian and Lagrangian variables. This renders the Noether analysis of reparametrization symmetries inconvenient, especially since the well-known energy- and momentum-conservation laws for the system are expressed in terms of Eulerian variables only. While an Euler-Poincaré formulation of Vlasov-Maxwell-type systems, effectively starting with Low's action and using constrained variations for the Eulerian description of particle motion, has been known for a while [J. Math. Phys., 39, 6, pp. 3138-3157], it is hard to come by a documented derivation of the related energy- and momentum-conservation laws in the spirit of the Euler-Poincaré machinery. To our knowledge only one such derivation exists in the literature so far, dealing with the so-called guiding-center Vlasov-Darwin system [Phys. Plasmas 25, 102506]. The present exposition discusses a generic class of local Vlasov-Maxwell-type systems, with a conscious choice of adopting the language of differential geometry to exploit the Euler-Poincaré framework to its full extent. After reviewing the transition from a Lagrangian picture to an Eulerian one, we demonstrate how symmetries generated by isometries in space lead to conservation laws for linear- and angular-momentum density and how symmetry by time translation produces a conservation law for energy density. We also discuss what happens if no symmetries exist. Finally, two explicit examples will be given -- the classic Vlasov-Maxwell and the drift-kinetic Vlasov-Maxwell -- and the results expressed in the language of regular vector calculus for familiarity.

physics.plasm-ph↗

arXiv2019

Charge-conserving, variational particle-in-cell method for the drift-kinetic Vlasov-Maxwell system

This paper proposes a charge-conserving, variational, spatio-temporal discretization for the drift-kinetic Vlasov-Maxwell system, utilizing finite-elements for the electromagnetic fields and the particle-in-cell approach for the Vlasov distribution. The proposed scheme is fully electromagnetic, dealing with fields instead of potentials, and includes the effects of polarization and magnetization in the Gauss and Ampère-Maxwell laws, a consequence of reducing the full particle dynamics to drift-center dynamics. There is, however, no need to invert the Gauss law: it is satisfied automatically at every time-step as a result of a discrete Noether symmetry, and the electric field is updated directly from the Ampère-Maxwell equation. The method provides an update for the magnetic field that is fully explicit, involving only local operations. The update for particles is implicit for each particle individually, also leading to local operations only. The update for the electric field is linearly implicit due to the presence of a finite-element mass matrix and polarization and magnetization effects in the Ampère-Maxwell equation, hence involving a sparse matrix inversion once at every time step. Because the scheme deals with the electromagnetic fields and not the potentials, it also provides the first serious attempt at constructing a structure-preserving numerical scheme for the mixed kinetic-ion--drift-kinetic-electron Vlasov-Maxwell model. Consequently, the proposed method could be used to simulate electromagnetic turbulence in fusion experiments or space plasmas that exhibit a strong background magnetic field while retaining all of the ion physics, most of the necessary electron physics, yet eliminating perhaps the biggest obstacle in reaching macroscopic transport time scales in kinetic simulations, namely the electron cyclotron time scale.

physics.plasm-ph↗

arXiv2019

High-performance orbit-following code ASCOT5 for Monte Carlo simulations in fusion plasmas

We present a novel implementation of a Monte Carlo particle-following code for solving the distribution function of minority species in fusion plasmas, called ASCOT5, and verify it using theoretical results for neoclassical transport. The code has been developed from ground up with an OpenMP-MPI hybrid paradigm to take full advantage of current and next generation many-core CPUs with multithreading and SIMD operations. Up to 6-fold increase in performance is demonstrated compared to a previous version of the code which only utilizes MPI. The physics model of the code is comprehensively validated against existing theoretical work, and it is shown to faithfully reproduce neoclassical diffusion across three different collisionality regimes. In simulations for realistic tokamak plasmas, including complex non-axisymmetric geometry, ASCOT5 is verified to reproduce results from the previous version ASCOT4.

physics.plasm-ph↗

arXiv2019

Eliminating poor statistics in Monte-Carlo simulations of fast-ion losses to plasma-facing components and detectors

With Wendelstein 7-X now up and running, and the construction of ITER proceeding, predicting fast-ion losses to sensitive plasma-facing components and detectors is gaining significant interest. A common recipe to perform such studies is to push a large population of marker particles along their equations of motion, the trajectories randomized with Monte Carlo operators accounting for Coulomb collisions, and to record possible intersections of the marker trajectories with synthetic detectors or areas of interest in the first wall. While straightforward to implement and easy to parallelize, this Forward Monte Carlo (FMC) approach tends to suffer from poor statistics and error estimation as the detector domain is often small: it is difficult to guess how to set up the initial weights and locations of the markers for them to remain representative of the source distribution, yet record enough hits to the detector for good statistics. As an alternative, the FMC method can be replaced with a so-called Backward Monte Carlo (BMC) algorithm. Instead of starting with a given initial marker population, one starts from the end condition at the detector and records how the hit probability evolves backwards in time. The scheme eliminates the statistics issue present in the FMC scheme and may provide more accurate and efficient simulations of fast-ion loss signals. The purpose of this paper is to explain the BMC recipe in the fast-ion setting and to discuss the associated nuances, especially how to negate artificial diffusion. For illustration purposes, our numerical example considers a 1-D stochastic harmonic oscillator as a mock-up of a charged particle.

physics.plasm-ph↗

arXiv2019

Conservative finite-element method for the relativistic Coulomb collision operator

This research note documents new developments regarding finite-element discretizations of the relativistic Beliaev-Budker Coulomb collision operator and the nonrelativistic Landau operator. Where energy conservation in a finite-element approximation of the relativistic collision operator was previously thought to be elusive, it is now achieved even with linear elements. The same result applies to the nonrelativistic Landau operator for which the energy conservation was thought to require at least quadratic elements. In both cases, the momentum and density conservation are guaranteed as previously. The new outcomes benefit from the findings reported in a recent finite-difference-scheme paper [Shiroto & Sentoku, arXiv:1902.07866] which we generalize to the finite-element method. This note focuses solely on the direct discretization of the collision operator, leaving the discretization of the underlying metriplectic formulation of the relativistic collision operator to future publications.

physics.plasm-ph↗

arXiv2018

Energy-, momentum-, density-, and positivity-preserving spatio-temporal discretizations for the nonlinear Landau collision operator with exact H-theorems

This paper explores energy-, momentum-, density-, and positivity-preserving spatio-temporal discretizations for the nonlinear Landau collision operator. We discuss two approaches, namely direct Galerkin formulations and discretizations of the underlying infinite-dimensional metriplectic structure of the collision integral. The spatial discretizations are chosen to reproduce the time-continuous conservation laws that correspond to Casimir invariants and to guarantee the positivity of the distribution function. Both the direct and the metriplectic discretization are demonstrated to have exact H-theorems and unique, physically exact equilibrium states. Most importantly, the two approaches are shown to coincide, given the chosen Galerkin method. A temporal discretization, preserving all of the mentioned properties, is achieved with so-called discrete gradients. Hence the proposed algorithm successfully translates all properties of the infinite-dimensional time-continuous Landau collision operator to time- and space-discrete sparse-matrix equations suitable for numerical simulation.

physics.plasm-ph↗

arXiv2018

A Langevin approach to multi-scale modeling

In plasmas, distribution functions often demonstrate long anisotropic tails or otherwise significant deviations from local Maxwellians. The tails, especially if they are pulled out from the bulk, pose a serious challenge for numerical simulations as resolving both the bulk and the tail on the same mesh is often challenging. A multi-scale approach, providing evolution equations for the bulk and the tail individually, could offer a resolution in the sense that both populations could be treated on separate meshes, or different reduction techniques applied to the bulk and the tail population. In this letter, we propose a multi-scale method which allows us to split a distribution function into a bulk and a tail so that both populations remain genuine, non-negative distribution functions and may carry density, momentum, and energy. The proposed method is based on the observation that the motion of an individual test particle in a plasma obeys a stochastic differential equation, also referred to as a Langevin equation. This allows us to define transition probabilities between the bulk and the tail and to provide evolution equations for both populations separately.

physics.plasm-ph↗

arXiv2018

The role of magnetic moment in the collisionless pitch-angle scattering of runaway electrons

Recently, the validity of the guiding-center approach to model relativistic runaway electrons in tokamaks has been challenged by full-orbit simulations that demonstrate the breakdown of the standard magnetic moment conservation. In this paper, we derive a new expression for the magnetic moment of relativistic runaway electrons, which is conserved significantly better than the standard one. The new result includes one of the second-order corrections in the standard guiding-center theory which, in case of runaway electrons with $p_{\parallel}\gg p_{\perp}$, can peculiarly be of the same order as the lowest-order term. The better conservation of the new magnetic moment also explains the collisionless pitch-angle-scattering effect observed in full-orbit simulations since it allows momentum transfer between the perpendicular and parallel directions when the runaway electron is accelerated by an electric field. While the derivation of the second-order correction to the magnetic moment in general case would require the full extent of the relativistic second-order guiding-center theory, we exploit the Lie-perturbation method at the limit $p_{\parallel}\gg p_{\perp}$ which simplifies the computations significantly. Consequently, we present the corresponding guiding-center equations applicable to the highly relativistic runaway electrons.

physics.plasm-ph↗

arXiv2018

The effects of kinetic instabilities on the electron cyclotron emission from runaway electrons

In this paper we show that the kinetic instabilities associated with runaway electron beams play an essential role for the production of high-level non-thermal electron-cyclotron-emission (ECE) radiation. Most of the non-thermal ECE comes from runaway electrons in the low-energy regime with large pitch angle, which are strongly scattered by the excited whistler waves. The power of ECE from runaway electrons is obtained using a synthetic diagnostic model based on the reciprocity method. The electron distribution function is calculated using a kinetic simulation model including the whistler wave instabilities and the quasilinear diffusion effects. Simulations based on DIII-D low-density discharge reproduces the rapid growth of the ECE signals observed in DIII-D experiments. Unlike the thermal ECE where radiation for a certain frequency is strongly localized inside the resonance region, the non-thermal ECE radiation from runaway electrons is nonlocal, and the emission-absorption ratio is higher than that of thermal electrons. The runaway electron tail is more significant for ECE with higher frequencies, and the ECE spectrum becomes flatter as RE population grows. The nonlinear behavior of the kinetic instabilities is illustrated in the osculations of the ECE waves. The good agreement with the DIII-D experimental observations after including the kinetic instabilities clearly illustrate the significance of the scattering effects from wave-particle interactions, which can also be important for runaway electrons produced in disruptions.

physics.plasm-ph↗

arXiv2018

A fluid-kinetic framework for self-consistent runaway-electron simulations

The problem of self-consistently coupling kinetic runaway-electron physics to the macroscopic evolution of the plasma is addressed by dividing the electron population into a bulk and a tail. A probabilistic closure is adopted to determine the coupling between the bulk and the tail populations, preserving them both as genuine, non-negative distribution functions. Macroscopic one-fluid equations and the kinetic equation for the runaway-electron population are then derived, now displaying sink and source terms due to transfer of electrons between the bulk and the tail.

physics.plasm-ph↗

arXiv2018

Metriplectic particle-in-cell integrators for the Landau collision operator

In this paper, we present a new framework for addressing the nonlinear Landau collision operator in terms of particle-in-cell methods. We employ the underlying metriplectic structure of the collision operator and, using a macro particle discretization for the distribution function, we transform the infinite-dimensional system into a finite-dimensional time-continuous metriplectic system for advancing the macro particle weights. Temporal discretization is accomplished using the concept of discrete gradients. The conservation of density, momentum, and energy, as well as the positive semi-definite production of entropy in both the time-continuous and the fully discrete system is demonstrated algebraically. The new algorithm is fully compatible with the existing particle-in-cell Poisson integrators for the Vlasov-Maxwell system.

physics.comp-ph↗