Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-14T06:29:41.216Z Has data issue: false hasContentIssue false
  • English
  • Français

A general metriplectic framework with application to dissipative extended magnetohydrodynamics

Part of: Hamiltonian Methods in Plasma Physics

Published online by Cambridge University Press:  19 May 2020

Baptiste Coquinot Open the ORCID record for Baptiste Coquinot [Opens in a new window]  and
Philip J. Morrison Open the ORCID record for Philip J. Morrison [Opens in a new window]
Baptiste Coquinot*
Affiliation:
Département de Physique, École Normale Supérieure, 24 rue Lhomond, 75005, Paris, France
Philip J. Morrison*
Affiliation:
Department of Physics and Institute for Fusion Studies, University of Texas at Austin, 2515 Speedway, Austin, TX 78712, USA
*
†Email addresses for correspondence: baptiste.coquinot@ens.fr, morrison@physics.utexas.edu
†Email addresses for correspondence: baptiste.coquinot@ens.fr, morrison@physics.utexas.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

General equations for conservative yet dissipative (entropy producing) extended magnetohydrodynamics are derived from two-fluid theory. Keeping all terms generates unusual cross-effects, such as thermophoresis and a current viscosity that mixes with the usual velocity viscosity. While the Poisson bracket of the ideal version of this model has already been discovered, we determine its metriplectic counterpart that describes the dissipation. This is done using a new and general thermodynamic point of view to derive dissipative brackets, a means of derivation that is natural for understanding and creating dissipative dynamics without appealing to underlying kinetic theory orderings. Finally, the formalism is used to study dissipation in the Lagrangian variable picture where, in the context of extended magnetohydrodynamics, non-local dissipative brackets naturally emerge.

Keywords

plasma dynamics
Type
Research Article
Information
Journal of Plasma Physics , Volume 86 , Issue 3 , June 2020 , 835860302
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abdelhamid, H. M., Kawazura, Y. & Yoshida, Z. 2015 Hamiltonian formalism of extended magnetohydrodynamics. J. Phys. A 48, 235502.CrossRefGoogle Scholar
Alfven, H. 1950 Cosmical Electrodynamics. Clarenden Press, Oxford.Google Scholar
Bannon, P. R. 2003 Hamiltonian description of idealized binary geophysical fluids. Am. Meteorol. Soc. 60, 2809–2819.Google Scholar
Braginskii, S. I. 1965 Transport processes in a plasma. Rev. Plasma Phys. 1, 205–311.Google Scholar
Bressan, C., Kraus, M., Morrison, P. J. & Maj, O. 2018 Relaxation to magnetohydrodynamics equilibria via collision brackets. J. Phys.: Conf. Ser. 1125, 012002.Google Scholar
Callen, H. 1960 Thermodynamics. Wiley.Google Scholar
D’Avignon, E., Morrison, P. J. & Pegoraro, F. 2015 Action principle for relativistic magnetohydrodynamics. Phys. Rev. E 91, 084050.Google Scholar
D’Avignon, E. C., Morrison, P. J. & Lingam, M. 2016 Derivation of the Hall and extended magnetohydrodynamics brackets. Phys. Plasmas 23, 062101.Google Scholar
Edwards, B. J. 1998 An analysis of single and double generator thermodynamic formalisms for the macroscopic description of complex fluids. J. Non-Equilib. Thermodyn. 23, 301–333.Google Scholar
Edwards, B. J. & Beris, A. N. 1991 Noncanonical poisson bracket for nonlinear elasticity with extensions to viscoelasticity. J. Phys. A 24, 2461–2480.CrossRefGoogle Scholar
Eldred, C. & Gay-Balmaz, F.2018 Single and double generator bracket formulations of geophysical fluids with irreversible processes. Preprint.Google Scholar
Eldred, C. & Gay-Balmaz, F. 2020 Single and double generator bracket formulations of multicomponent fluids with irreversible processes. J. Phys. A (to appear).CrossRefGoogle Scholar
Gay-Balmaz, F. & Yoshimura, H. 2017a A Lagrangian variational formalism for nonequilibrium thermodynamics. Part i. Discrete systems. J. Geom. Phys. 111, 169–193.CrossRefGoogle Scholar
Gay-Balmaz, F. & Yoshimura, H. 2017b A Lagrangian variational formalism for nonequilibrium thermodynamics. Part ii. Continuum systems. J. Geom. Phys. 111, 194–212.CrossRefGoogle Scholar
Grmela, M. 1984 Particle and bracket formulations of kinetic equations. Contemp. Maths 28, 125–132.CrossRefGoogle Scholar
Grmela, M. & Öttinger, H. C. 1997a Dynamics and thermodynamics of complex fluids. Part i. Development of a general formalism. Phys. Rev. E 56, 6620–6632.Google Scholar
Grmela, M. & Öttinger, H. C. 1997b Dynamics and thermodynamics of complex fluids. Part ii. Illustration of a general formalism. Phys. Rev. E 56, 6633–6655.Google Scholar
de Groot, S. R. & Mazur, P. 1984 Non-Equilibrium Thermodynamics. Dover.Google Scholar
Hagstrom, G. I. & Morrison, P. J. 2011 On Krein-like theorems for noncanonical Hamiltonian systems with continuous spectra: application to Vlasov–Poisson. Trans. Theor. Stat. Phys. 39, 466–501.CrossRefGoogle Scholar
Kampen, N. G. V. & Felderhof, B. U. 1967 Theoretical Methods in Plasma Physics. Interscience Publishers.Google Scholar
Kaufman, A. N. 1984 Dissipative Hamiltonian systems: a unifying principle. Phys. Lett. A 100, 419–422.CrossRefGoogle Scholar
Kaufman, A. N. & Morrison, P. J. 1982 Algebraic structure of the plasma quasilinear equations. Phys. Lett. A 88, 405–406.CrossRefGoogle Scholar
Keramidas Charidakos, I., Lingam, M., Morrison, P. J., White, R. L. & Wurm, A. 2014 Action principles for extended magnetohydrodynamic models. Phys. Plasmas 21, 092118.CrossRefGoogle Scholar
Kimura, K. & Morrison, P. J. 2014 On energy conservation in extended magnetohydrodynamics. Phys. Plasmas 21, 082101.CrossRefGoogle Scholar
Kraus, M. & Hirvijoki, E. 2017 Metriplectic integrators for the Landau collision operator. Phys. Plasmas 24, 102311.CrossRefGoogle Scholar
Kraus, M., Kormann, K., Morrison, P. J. & Sonnendrücker, E. 2017 GEMPIC: geometric electromagnetic particle-in-cell methods. J. Plasma Phys. 83, 905830401.CrossRefGoogle Scholar
Kulsrud, R. M. 1983 MHD description of plasma. In Handbook of Plasma Physics: Basic Plasma Physics (ed. Galeev, A. A. & Sudan, R. N.), vol. 1, chap. 1.4, pp. 115–145. North-Holland Publishing Company.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1960 Course of Theoretical Physics: Mechanics. Pergamon Press.Google Scholar
Lingam, M., Miloshevich, G. & Morrison, P. J. 2016 Concomitant Hamiltonian and topological structures of extended magnetohydrodynamics. Phys. Lett. A 380, 2400–2406.CrossRefGoogle Scholar
Lingam, M., Morrison, P. J. & Miloshevich, G. 2015a Remarkable connections between extended magnetohydrodynamics models. Phys. Plasmas 22, 072111.CrossRefGoogle Scholar
Lingam, M., Morrison, P. J. & Tassi, E. 2015b Inertial magnetohydrodynamics. Phys. Lett. A 379, 570–576.CrossRefGoogle Scholar
Lüst, V. R. 1959 Über die Ausbrecitung von Wellen in einem plasma. Fortschr. Phys. 7, 503–558.CrossRefGoogle Scholar
Materassi, M. 2015 Metriplectic algebra for dissipative fluids in Lagrangian formulation. Entropy 17, 1329–1346.CrossRefGoogle Scholar
Materassi, M. & Morrison, P. J. 2018 Metriplectic torque for rotation control of a rigid body. Cybern. Phys. 7, 78–86.CrossRefGoogle Scholar
Materassi, M. & Tassi, E. 2012 Metriplectic framework for dissipative magneto-hydrodynamics. Physica D 241, 729–734.CrossRefGoogle Scholar
Morrison, P. J. 1982 Poisson brackets for fluids and plasmas. AIP Conf. Proc. 88, 13–46.Google Scholar
Morrison, P. J. 1984a Bracket formulation for irreversible classical fields. Phys. Lett. A 100, 423–427.CrossRefGoogle Scholar
Morrison, P. J.1984b Some observations regarding brackets and dissipation. Tech. Rep. PAM–228. University of California at Berkeley.Google Scholar
Morrison, P. J. 1986 A paradigm for joined Hamiltonian and dissipative systems. Physica D 18, 410–419.Google Scholar
Morrison, P. J. 1998a Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70, 467–521.CrossRefGoogle Scholar
Morrison, P. J. 1998b Thoughts on brackets and dissipation: old and new. J. Phys.: Conf. Ser. 169, 012006.Google Scholar
Morrison, P. J. 2009 On Hamiltonian and action principle formulations of plasma dynamics. AIP Conf. Proc. 1188, 329–344.Google Scholar
Morrison, P. J. 2017 Structure and structure-preserving algorithms for plasma physics. Phys. Plasmas 24, 055502.CrossRefGoogle Scholar
Morrison, P. J. & Greene, J. M. 1980 Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics. Phys. Rev. Lett. 45, 790–793.CrossRefGoogle Scholar
Morrison, P. J. & Vanneste, J. 2016 Weakly nonlinear dynamics in noncanonical Hamiltonian systems with applications to fluids and plasmas. Ann. Phys. 368, 117–147.CrossRefGoogle Scholar
Salmon, R. 1983 Practical use of Hamilton’s principle. J. Fluid Mech. 132, 431–444.CrossRefGoogle Scholar