# 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:

- Max-Planck-Institut für Plasmaphysik, Boltzmannstraße 2, 85748 Garching (Germany)
- (Germany)
- Aix-Marseille Université, Université de Toulon, CNRS, CPT, UMR 7332, 163 avenue de Luminy, case 907, 13288 cedex 9 Marseille (France)
- 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. Thu .
"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 = {Thu Sep 15 00:00:00 EDT 2016},

month = {Thu Sep 15 00:00:00 EDT 2016}

}