skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Variational integrators for reduced magnetohydrodynamics

Abstract

Reduced magnetohydrodynamics is a simplified set of magnetohydrodynamics equations with applications to both fusion and astrophysical plasmas, possessing a noncanonical Hamiltonian structure and consequently a number of conserved functionals. We propose a new discretisation strategy for these equations based on a discrete variational principle applied to a formal Lagrangian. The resulting integrator preserves important quantities like the total energy, magnetic helicity and cross helicity exactly (up to machine precision). As the integrator is free of numerical resistivity, spurious reconnection along current sheets is absent in the ideal case. If effects of electron inertia are added, reconnection of magnetic field lines is allowed, although the resulting model still possesses a noncanonical Hamiltonian structure. After reviewing the conservation laws of the model equations, the adopted variational principle with the related conservation laws is described both at the continuous and discrete level. We verify the favourable properties of the variational integrator in particular with respect to the preservation of the invariants of the models under consideration and compare with results from the literature and those of a pseudo-spectral code.

Authors:
 [1];  [2];  [3];  [4]
  1. Max-Planck-Institut für Plasmaphysik, Boltzmannstraße 2, 85748 Garching (Germany)
  2. (Germany)
  3. Aix-Marseille Université, Université de Toulon, CNRS, CPT, UMR 7332, 163 avenue de Luminy, case 907, 13288 cedex 9 Marseille (France)
  4. ISC-CNR and Politecnico di Torino, Dipartimento Energia, C.so Duca degli Abruzzi 24, 10129 Torino (Italy)
Publication Date:
OSTI Identifier:
22572349
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational Physics; Journal Volume: 321; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASTROPHYSICS; CONSERVATION LAWS; EQUATIONS; HAMILTONIANS; LAGRANGIAN FUNCTION; MAGNETIC FIELDS; MAGNETOHYDRODYNAMICS; PLASMA; VARIATIONAL METHODS

Citation Formats

Kraus, Michael, E-mail: michael.kraus@ipp.mpg.de, Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, 85748 Garching, Tassi, Emanuele, E-mail: tassi@cpt.univ-mrs.fr, and Grasso, Daniela, E-mail: daniela.grasso@infm.polito.it. Variational integrators for reduced magnetohydrodynamics. United States: N. p., 2016. Web. doi:10.1016/J.JCP.2016.05.047.
Kraus, Michael, E-mail: michael.kraus@ipp.mpg.de, Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, 85748 Garching, Tassi, Emanuele, E-mail: tassi@cpt.univ-mrs.fr, & Grasso, Daniela, E-mail: daniela.grasso@infm.polito.it. Variational integrators for reduced magnetohydrodynamics. United States. doi:10.1016/J.JCP.2016.05.047.
Kraus, Michael, E-mail: michael.kraus@ipp.mpg.de, Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, 85748 Garching, Tassi, Emanuele, E-mail: tassi@cpt.univ-mrs.fr, and Grasso, Daniela, E-mail: daniela.grasso@infm.polito.it. 2016. "Variational integrators for reduced magnetohydrodynamics". United States. doi:10.1016/J.JCP.2016.05.047.
@article{osti_22572349,
title = {Variational integrators for reduced magnetohydrodynamics},
author = {Kraus, Michael, E-mail: michael.kraus@ipp.mpg.de and Technische Universität München, Zentrum Mathematik, Boltzmannstraße 3, 85748 Garching and Tassi, Emanuele, E-mail: tassi@cpt.univ-mrs.fr and Grasso, Daniela, E-mail: daniela.grasso@infm.polito.it},
abstractNote = {Reduced magnetohydrodynamics is a simplified set of magnetohydrodynamics equations with applications to both fusion and astrophysical plasmas, possessing a noncanonical Hamiltonian structure and consequently a number of conserved functionals. We propose a new discretisation strategy for these equations based on a discrete variational principle applied to a formal Lagrangian. The resulting integrator preserves important quantities like the total energy, magnetic helicity and cross helicity exactly (up to machine precision). As the integrator is free of numerical resistivity, spurious reconnection along current sheets is absent in the ideal case. If effects of electron inertia are added, reconnection of magnetic field lines is allowed, although the resulting model still possesses a noncanonical Hamiltonian structure. After reviewing the conservation laws of the model equations, the adopted variational principle with the related conservation laws is described both at the continuous and discrete level. We verify the favourable properties of the variational integrator in particular with respect to the preservation of the invariants of the models under consideration and compare with results from the literature and those of a pseudo-spectral code.},
doi = {10.1016/J.JCP.2016.05.047},
journal = {Journal of Computational Physics},
number = ,
volume = 321,
place = {United States},
year = 2016,
month = 9
}
  • Symplectic integrators are widely used for long-term integration of conservative astrophysical problems due to their ability to preserve the constants of motion; however, they cannot in general be applied in the presence of nonconservative interactions. In this Letter, we develop the “slimplectic” integrator, a new type of numerical integrator that shares many of the benefits of traditional symplectic integrators yet is applicable to general nonconservative systems. We utilize a fixed-time-step variational integrator formalism applied to the principle of stationary nonconservative action developed in Galley et al. As a result, the generalized momenta and energy (Noether current) evolutions are well-tracked. Wemore » discuss several example systems, including damped harmonic oscillators, Poynting–Robertson drag, and gravitational radiation reaction, by utilizing our new publicly available code to demonstrate the slimplectic integrator algorithm. Slimplectic integrators are well-suited for integrations of systems where nonconservative effects play an important role in the long-term dynamical evolution. As such they are particularly appropriate for cosmological or celestial N-body dynamics problems where nonconservative interactions, e.g., gas interactions or dissipative tides, can play an important role.« less
  • This paper formulates variational integrators for finite element discretizations of deformable bodies with heat conduction in the form of finite speed thermal waves. The cornerstone of the construction consists in taking advantage of the fact that the Green–Naghdi theory of type II for thermo-elastic solids has a Hamiltonian structure. Thus, standard techniques to construct variational integrators can be applied to finite element discretizations of the problem. The resulting discrete-in-time trajectories are then consistent with the laws of thermodynamics for these systems: for an isolated system, they exactly conserve the total entropy, and nearly exactly conserve the total energy over exponentiallymore » long periods of time. Moreover, linear and angular momenta are also exactly conserved whenever the exact system does. For definiteness, we construct an explicit second-order accurate algorithm for affine tetrahedral elements in two and three dimensions, and demonstrate its performance with numerical examples.« less
  • In this contribution, we develop a variational integrator for the simulation of (stochastic and multiscale) electric circuits. When considering the dynamics of an electric circuit, one is faced with three special situations: 1. The system involves external (control) forcing through external (controlled) voltage sources and resistors. 2. The system is constrained via the Kirchhoff current (KCL) and voltage laws (KVL). 3. The Lagrangian is degenerate. Based on a geometric setting, an appropriate variational formulation is presented to model the circuit from which the equations of motion are derived. A time-discrete variational formulation provides an iteration scheme for the simulation ofmore » the electric circuit. Dependent on the discretization, the intrinsic degeneracy of the system can be canceled for the discrete variational scheme. In this way, a variational integrator is constructed that gains several advantages compared to standard integration tools for circuits; in particular, a comparison to BDF methods (which are usually the method of choice for the simulation of electric circuits) shows that even for simple LCR circuits, a better energy behavior and frequency spectrum preservation can be observed using the developed variational integrator.« less
  • Direct numerical simulations of low Mach number compressible three-dimensional magnetohydrodynamic (CMHD3D) turbulence in the presence of a strong mean magnetic field are compared with simulations of reduced magnetohydrodynamics (RMHD). Periodic boundary conditions in the three spatial coordinates are considered. Different sets of initial conditions are chosen to explore the applicability of RMHD and to study how close the solution remains to the full compressible MHD solution as both freely evolve in time. In a first set, the initial state is prepared to satisfy the conditions assumed in the derivation of RMHD, namely, a strong mean magnetic field and plane-polarized fluctuations,more » varying weakly along the mean magnetic field. In those circumstances, simulations show that RMHD and CMHD3D evolve almost indistinguishably from one another. When some of the conditions are relaxed the agreement worsens but RMHD remains fairly close to CMHD3D, especially when the mean magnetic field is large enough. Moreover, the well-known spectral anisotropy effect promotes the dynamical attainment of the conditions for RMHD applicability. Global quantities (mean energies, mean-square current, and vorticity) and energy spectra from the two solutions are compared and point-to-point separation estimations are computed. The specific results shown here give support to the use of RMHD as a valid approximation of compressible MHD with a mean magnetic field under certain but quite practical conditions.« less