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Title: Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Examples with translation symmetry

Abstract

Because different constraints are imposed, stability conditions for dissipationless fluids and magnetofluids may take different forms when derived within the Lagrangian, Eulerian (energy-Casimir), or dynamically accessible frameworks. This is in particular the case when flows are present. These differences are explored explicitly here by working out in detail two magnetohydrodynamic examples: convection against gravity in a stratified fluid and translationally invariant perturbations of a rotating magnetized plasma pinch. In this second example, we show in explicit form how to perform the time-dependent relabeling introduced in Andreussi et al. [Phys. Plasmas 20, 092104 (2013)] that makes it possible to reformulate Eulerian equilibria with flows as Lagrangian equilibria in the relabeled variables. The procedures detailed in the present article provide a paradigm that can be applied to more general plasma configurations and in addition extended to more general plasma descriptions where dissipation is absent.

Authors:
ORCiD logo [1]; ORCiD logo [2]; ORCiD logo [3]
  1. SITAEL S.p.A., Pisa (Italy)
  2. Univ. of Texas, Austin, TX (United States). Inst. for Fusion Studies and Dept. of Physics
  3. Dipt. di Fisica E. Fermi, Pisa (Italy)
Publication Date:
Research Org.:
Univ. of Texas, Austin, TX (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1535285
Alternate Identifier(s):
OSTI ID: 1329491
Grant/Contract Number:  
FG02-04ER54742; FG02-04ER-54742
Resource Type:
Accepted Manuscript
Journal Name:
Physics of Plasmas
Additional Journal Information:
Journal Volume: 23; Journal Issue: 10; Journal ID: ISSN 1070-664X
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
70 PLASMA PHYSICS AND FUSION TECHNOLOGY; physics

Citation Formats

Andreussi, T., Morrison, P. J., and Pegoraro, F. Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Examples with translation symmetry. United States: N. p., 2016. Web. doi:10.1063/1.4964900.
Andreussi, T., Morrison, P. J., & Pegoraro, F. Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Examples with translation symmetry. United States. https://doi.org/10.1063/1.4964900
Andreussi, T., Morrison, P. J., and Pegoraro, F. Thu . "Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Examples with translation symmetry". United States. https://doi.org/10.1063/1.4964900. https://www.osti.gov/servlets/purl/1535285.
@article{osti_1535285,
title = {Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Examples with translation symmetry},
author = {Andreussi, T. and Morrison, P. J. and Pegoraro, F.},
abstractNote = {Because different constraints are imposed, stability conditions for dissipationless fluids and magnetofluids may take different forms when derived within the Lagrangian, Eulerian (energy-Casimir), or dynamically accessible frameworks. This is in particular the case when flows are present. These differences are explored explicitly here by working out in detail two magnetohydrodynamic examples: convection against gravity in a stratified fluid and translationally invariant perturbations of a rotating magnetized plasma pinch. In this second example, we show in explicit form how to perform the time-dependent relabeling introduced in Andreussi et al. [Phys. Plasmas 20, 092104 (2013)] that makes it possible to reformulate Eulerian equilibria with flows as Lagrangian equilibria in the relabeled variables. The procedures detailed in the present article provide a paradigm that can be applied to more general plasma configurations and in addition extended to more general plasma descriptions where dissipation is absent.},
doi = {10.1063/1.4964900},
journal = {Physics of Plasmas},
number = 10,
volume = 23,
place = {United States},
year = {Thu Oct 20 00:00:00 EDT 2016},
month = {Thu Oct 20 00:00:00 EDT 2016}
}

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Cited by: 12 works
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Works referenced in this record:

Singular Casimir Elements of the Euler Equation and Equilibrium Points
journal, April 2013

  • Yoshida, Zensho; Morrison, Philip J.; Dobarro, Fernando
  • Journal of Mathematical Fluid Mechanics, Vol. 16, Issue 1
  • DOI: 10.1007/s00021-013-0143-4

Relaxation and magnetic reconnection in plasmas
journal, July 1986


CXLIII. Thermal convection in a magnetic field
journal, December 1951

  • Thompson, W. B.
  • The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 42, Issue 335
  • DOI: 10.1080/14786445108560961

Inertial magnetohydrodynamics
journal, March 2015


Hamiltonian gyrokinetic Vlasov–Maxwell system
journal, September 2015


Invariants and Geometric Structures in Nonlinear Hamiltonian Magnetic Reconnection
journal, May 1998


A hierarchy of noncanonical Hamiltonian systems: circulation laws in an extended phase space
journal, May 2014


Bounds on dissipation in magnetohydrodynamic Couette and Hartmann shear flows
journal, November 2003

  • Alexakis, A.; Pétrélis, F.; Morrison, P. J.
  • Physics of Plasmas, Vol. 10, Issue 11
  • DOI: 10.1063/1.1613962

New method of deriving local energy- and momentum-conserving Maxwell-collisionless drift-kinetic and gyrokinetic theories: basic theory
journal, January 1999


Negative-energy perturbations in circularly cylindrical equilibria within the framework of Maxwell-drift kinetic theory
journal, March 1996


Fluid element relabeling symmetry
journal, August 1996


On the nonlinear stability of the rotating Bénard problem via the Lyapunov direct method
journal, November 1989


The free energy of Maxwell–Vlasov equilibria
journal, June 1990

  • Morrison, P. J.; Pfirsch, D.
  • Physics of Fluids B: Plasma Physics, Vol. 2, Issue 6
  • DOI: 10.1063/1.859246

Stability of compressible reduced magnetohydrodynamic equilibria—Analogy with magnetorotational instability
journal, April 2013

  • Morrison, P. J.; Tassi, E.; Tronko, N.
  • Physics of Plasmas, Vol. 20, Issue 4
  • DOI: 10.1063/1.4801027

Hamiltonian and action principle formulations of plasma physics
journal, May 2005


Hydromagnetic Equilibrium ii. Stability in the Variational Formulation
journal, June 1959

  • Woltjer, L.
  • Proceedings of the National Academy of Sciences, Vol. 45, Issue 6
  • DOI: 10.1073/pnas.45.6.769

Concomitant Hamiltonian and topological structures of extended magnetohydrodynamics
journal, July 2016


Hamiltonian magnetohydrodynamics: Helically symmetric formulation, Casimir invariants, and equilibrium variational principles
journal, May 2012

  • Andreussi, T.; Morrison, P. J.; Pegoraro, F.
  • Physics of Plasmas, Vol. 19, Issue 5
  • DOI: 10.1063/1.4714761

Relaxation of Toroidal Plasma and Generation of Reverse Magnetic Fields
journal, November 1974


Convective Instability Induced by Gravity in a Plasma with a Frozen-In Magnetic Field
journal, January 1961


Non-linear stability for the Vlasov-Poisson system—the energy-Casimir method
journal, November 1994


Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics
journal, September 1980


Remarkable connections between extended magnetohydrodynamics models
journal, July 2015

  • Lingam, M.; Morrison, P. J.; Miloshevich, G.
  • Physics of Plasmas, Vol. 22, Issue 7
  • DOI: 10.1063/1.4926821

Action principle for relativistic magnetohydrodynamics
journal, April 2015


Poisson brackets for fluids and plasmas
conference, January 1982

  • Morrison, Philip J.
  • AIP Conference Proceedings Volume 88
  • DOI: 10.1063/1.33633

Gyro-induced acceleration of magnetic reconnection
journal, September 2013

  • Comisso, L.; Grasso, D.; Waelbroeck, F. L.
  • Physics of Plasmas, Vol. 20, Issue 9
  • DOI: 10.1063/1.4821840

Hamiltonian approach to hybrid plasma models
journal, July 2010


Hydromagnetic Equilibrium. III. Axisymmetric Incompressible Media.
journal, July 1959


Hybrid Vlasov-MHD models: Hamiltonian vs. non-Hamiltonian
journal, July 2014


Hamiltonian description of the ideal fluid
journal, April 1998


Hamiltonian formalism of extended magnetohydrodynamics
journal, May 2015

  • Abdelhamid, H. M.; Kawazura, Y.; Yoshida, Z.
  • Journal of Physics A: Mathematical and Theoretical, Vol. 48, Issue 23
  • DOI: 10.1088/1751-8113/48/23/235502

On Hydromagnetic Equilibrium
journal, September 1958

  • Woltjer, L.
  • Proceedings of the National Academy of Sciences, Vol. 44, Issue 9
  • DOI: 10.1073/pnas.44.9.833

Dynamically accessible perturbations and magnetohydrodynamic stability
journal, July 2003


Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Theory
journal, September 2013

  • Andreussi, T.; Morrison, P. J.; Pegoraro, F.
  • Physics of Plasmas, Vol. 20, Issue 9
  • DOI: 10.1063/1.4819779

Variational principles and Lyapunov stability analysis for compressible cylindrical plasma flows with arbitrary cross-section
journal, February 2003


Hamiltonian and action formalisms for two-dimensional gyroviscous magnetohydrodynamics
journal, August 2014

  • Morrison, P. J.; Lingam, M.; Acevedo, R.
  • Physics of Plasmas, Vol. 21, Issue 8
  • DOI: 10.1063/1.4891321

Derivation of the Hall and extended magnetohydrodynamics brackets
journal, June 2016

  • D'Avignon, Eric C.; Morrison, Philip J.; Lingam, Manasvi
  • Physics of Plasmas, Vol. 23, Issue 6
  • DOI: 10.1063/1.4952641

Interchange instabilities in ideal hydromagnetic theory
journal, January 1968


Free-energy expressions for Vlasov equilibria
journal, October 1989


MHD equilibrium variational principles with symmetry
journal, March 2010


Linear and nonlinear stability criteria for compressible MHD flows in a gravitational field
journal, June 2013


Nonlinear variational method for predicting fast collisionless magnetic reconnection
journal, May 2013


Negative-energy modes in a magnetically confined plasma in the framework of Maxwell-drift kinetic theory
journal, April 1994


On Hydromagnetic Stability of Stationary Equilibria
journal, October 1960


Action principles for extended magnetohydrodynamic models
journal, September 2014

  • Keramidas Charidakos, I.; Lingam, M.; Morrison, P. J.
  • Physics of Plasmas, Vol. 21, Issue 9
  • DOI: 10.1063/1.4896336

Convective Stability in a Plasma with Horizontal Magnetic Fields
journal, January 1966


On energy conservation in extended magnetohydrodynamics
journal, August 2014

  • Kimura, Keiji; Morrison, P. J.
  • Physics of Plasmas, Vol. 21, Issue 8
  • DOI: 10.1063/1.4890955

Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics.
journal, February 1982


Hamiltonian magnetohydrodynamics: symmetric formulation, Casimir invariants, and equilibrium variational principles
text, January 2012


Gyro-induced acceleration of magnetic reconnection
text, January 2013


Hybrid Vlasov-MHD models: Hamiltonian vs. non-Hamiltonian
text, January 2014


On energy conservation in extended magnetohydrodynamics
text, January 2014


Hamiltonian Formalism of Extended Magnetohydrodynamics
text, January 2014


Works referencing / citing this record:

Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles
journal, May 2018

  • Kaltsas, D. A.; Throumoulopoulos, G. N.; Morrison, P. J.
  • Journal of Plasma Physics, Vol. 84, Issue 3
  • DOI: 10.1017/s0022377818000338

Structure and structure-preserving algorithms for plasma physics
journal, April 2017


Translationally symmetric extended MHD via Hamiltonian reduction: Energy-Casimir equilibria
journal, September 2017

  • Kaltsas, D. A.; Throumoulopoulos, G. N.; Morrison, P. J.
  • Physics of Plasmas, Vol. 24, Issue 9
  • DOI: 10.1063/1.4986013

Energy-Casimir, dynamically accessible, and Lagrangian stability of extended magnetohydrodynamic equilibria
journal, January 2020

  • Kaltsas, D. A.; Throumoulopoulos, G. N.; Morrison, P. J.
  • Physics of Plasmas, Vol. 27, Issue 1
  • DOI: 10.1063/1.5125573

Energy-Casimir, dynamically accessible, and Lagrangian stability of extended magnetohydrodynamic equilibria
text, January 2019


A class of three-dimensional gyroviscous magnetohydrodynamic models
text, January 2020