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

Andreussi, T.; Morrison, P. J.; Pegoraro, F.

doi:10.1063/1.4964900

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

Journal Article · Thu Oct 20 00:00:00 EDT 2016 · Physics of Plasmas

DOI:https://doi.org/10.1063/1.4964900· OSTI ID:1535285

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SITAEL S.p.A., Pisa (Italy); DOE/OSTI
Univ. of Texas, Austin, TX (United States). Inst. for Fusion Studies and Dept. of Physics
Dipt. di Fisica E. Fermi, Pisa (Italy)

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.

View Accepted Manuscript (DOE)

Research Organization:: Univ. of Texas, Austin, TX (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: FG02-04ER54742

OSTI ID:: 1535285

Alternate ID(s):: OSTI ID: 1329491

Journal Information:: Physics of Plasmas, Journal Name: Physics of Plasmas Journal Issue: 10 Vol. 23; ISSN PHPAEN; ISSN 1070-664X

Publisher:: American Institute of Physics (AIP)Copyright Statement

Country of Publication:: United States

Language:: English

References (61)

Non-linear stability for the Vlasov-Poisson system—the energy-Casimir method Rein, Gerhard Mathematical Methods in the Applied Sciences, Vol. 17, Issue 14 https://doi.org/10.1002/mma.1670171404	journal	November 1994
Singular Casimir Elements of the Euler Equation and Equilibrium Points Yoshida, Zensho; Morrison, Philip J.; Dobarro, Fernando Journal of Mathematical Fluid Mechanics, Vol. 16, Issue 1 https://doi.org/10.1007/s00021-013-0143-4	journal	April 2013
On the nonlinear stability of the rotating Bénard problem via the Lyapunov direct method Mulone, G.; Rionero, S. Journal of Mathematical Analysis and Applications, Vol. 144, Issue 1 https://doi.org/10.1016/0022-247X(89)90362-4	journal	November 1989
Fluid element relabeling symmetry Padhye, Nikhil; Morrison, P. J. Physics Letters A, Vol. 219, Issue 5-6 https://doi.org/10.1016/0375-9601(96)00472-0	journal	August 1996
Inertial magnetohydrodynamics Lingam, M.; Morrison, P. J.; Tassi, E. Physics Letters A, Vol. 379, Issue 6 https://doi.org/10.1016/j.physleta.2014.12.008	journal	March 2015
Hamiltonian gyrokinetic Vlasov–Maxwell system Burby, J. W.; Brizard, A. J.; Morrison, P. J. Physics Letters A, Vol. 379, Issue 36 https://doi.org/10.1016/j.physleta.2015.06.051	journal	September 2015
Concomitant Hamiltonian and topological structures of extended magnetohydrodynamics Lingam, Manasvi; Miloshevich, George; Morrison, Philip J. Physics Letters A, Vol. 380, Issue 31-32 https://doi.org/10.1016/j.physleta.2016.05.024	journal	July 2016
New method of deriving local energy- and momentum-conserving Maxwell-collisionless drift-kinetic and gyrokinetic theories: basic theory Pfirsch, Dieter; Correa-Restrepo, DarÍO Journal of Plasma Physics, Vol. 70, Issue 6 https://doi.org/10.1017/S0022377804002995	journal	January 1999
New method of deriving local energy- and momentum-conserving Maxwell-collisionless drift-kinetic and gyrokinetic theories: conservation laws and their structures Correa-Restrepo, DarÍO; Pfirsch, Dieter Journal of Plasma Physics, Vol. 70, Issue 6 https://doi.org/10.1017/S0022377804003034	journal	January 1999
Linear and nonlinear stability criteria for compressible MHD flows in a gravitational field Moawad, S. M. Journal of Plasma Physics, Vol. 79, Issue 5 https://doi.org/10.1017/S0022377813000627	journal	June 2013
Dynamically accessible perturbations and magnetohydrodynamic stability Hameiri, Eliezer Physics of Plasmas, Vol. 10, Issue 7 https://doi.org/10.1063/1.1577122	journal	July 2003
Bounds on dissipation in magnetohydrodynamic Couette and Hartmann shear flows Alexakis, A.; Pétrélis, F.; Morrison, P. J. Physics of Plasmas, Vol. 10, Issue 11 https://doi.org/10.1063/1.1613962	journal	November 2003
Convective Instability Induced by Gravity in a Plasma with a Frozen-In Magnetic Field Newcomb, William A. Physics of Fluids, Vol. 4, Issue 4 https://doi.org/10.1063/1.1706342	journal	January 1961
Convective Stability in a Plasma with Horizontal Magnetic Fields Yu, C. P. Physics of Fluids, Vol. 9, Issue 2 https://doi.org/10.1063/1.1761688	journal	January 1966
Hamiltonian and action principle formulations of plasma physics Morrison, P. J. Physics of Plasmas, Vol. 12, Issue 5 https://doi.org/10.1063/1.1882353	journal	May 2005
Poisson brackets for fluids and plasmas Morrison, Philip J. AIP Conference Proceedings Volume 88 https://doi.org/10.1063/1.33633	conference	January 1982
Hamiltonian magnetohydrodynamics: Helically symmetric formulation, Casimir invariants, and equilibrium variational principles Andreussi, T.; Morrison, P. J.; Pegoraro, F. Physics of Plasmas, Vol. 19, Issue 5 https://doi.org/10.1063/1.4714761	journal	May 2012
Stability of compressible reduced magnetohydrodynamic equilibria—Analogy with magnetorotational instability Morrison, P. J.; Tassi, E.; Tronko, N. Physics of Plasmas, Vol. 20, Issue 4 https://doi.org/10.1063/1.4801027	journal	April 2013
Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Theory Andreussi, T.; Morrison, P. J.; Pegoraro, F. Physics of Plasmas, Vol. 20, Issue 9 https://doi.org/10.1063/1.4819779	journal	September 2013
Gyro-induced acceleration of magnetic reconnection Comisso, L.; Grasso, D.; Waelbroeck, F. L. Physics of Plasmas, Vol. 20, Issue 9 https://doi.org/10.1063/1.4821840	journal	September 2013
On energy conservation in extended magnetohydrodynamics Kimura, Keiji; Morrison, P. J. Physics of Plasmas, Vol. 21, Issue 8 https://doi.org/10.1063/1.4890955	journal	August 2014
Hamiltonian and action formalisms for two-dimensional gyroviscous magnetohydrodynamics Morrison, P. J.; Lingam, M.; Acevedo, R. Physics of Plasmas, Vol. 21, Issue 8 https://doi.org/10.1063/1.4891321	journal	August 2014
Action principles for extended magnetohydrodynamic models Keramidas Charidakos, I.; Lingam, M.; Morrison, P. J. Physics of Plasmas, Vol. 21, Issue 9 https://doi.org/10.1063/1.4896336	journal	September 2014
Erratum: “Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Theory” [Phys. Plasmas 20, 092104 (2013)] Andreussi, T.; Morrison, P. J.; Pegoraro, F. Physics of Plasmas, Vol. 22, Issue 3 https://doi.org/10.1063/1.4916504	journal	March 2015
Remarkable connections between extended magnetohydrodynamics models Lingam, M.; Morrison, P. J.; Miloshevich, G. Physics of Plasmas, Vol. 22, Issue 7 https://doi.org/10.1063/1.4926821	journal	July 2015
Derivation of the Hall and extended magnetohydrodynamics brackets D'Avignon, Eric C.; Morrison, Philip J.; Lingam, Manasvi Physics of Plasmas, Vol. 23, Issue 6 https://doi.org/10.1063/1.4952641	journal	June 2016
The free energy of Maxwell–Vlasov equilibria Morrison, P. J.; Pfirsch, D. Physics of Fluids B: Plasma Physics, Vol. 2, Issue 6 https://doi.org/10.1063/1.859246	journal	June 1990
On Hydromagnetic Equilibrium Woltjer, L. Proceedings of the National Academy of Sciences, Vol. 44, Issue 9 https://doi.org/10.1073/pnas.44.9.833	journal	September 1958
Hydromagnetic Equilibrium ii. Stability in the Variational Formulation Woltjer, L. Proceedings of the National Academy of Sciences, Vol. 45, Issue 6 https://doi.org/10.1073/pnas.45.6.769	journal	June 1959
CXLIII. Thermal convection in a magnetic field Thompson, W. B. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Vol. 42, Issue 335 https://doi.org/10.1080/14786445108560961	journal	December 1951
Hydromagnetic Equilibrium. III. Axisymmetric Incompressible Media. Woltjer, L. The Astrophysical Journal, Vol. 130 https://doi.org/10.1086/146731	journal	July 1959
Nonlinear variational method for predicting fast collisionless magnetic reconnection Hirota, M.; Morrison, P. J.; Ishii, Y. Nuclear Fusion, Vol. 53, Issue 6 https://doi.org/10.1088/0029-5515/53/6/063024	journal	May 2013
Interchange instabilities in ideal hydromagnetic theory Greene, John M.; Johnson, John L. Plasma Physics, Vol. 10, Issue 8 https://doi.org/10.1088/0032-1028/10/8/301	journal	January 1968
A hierarchy of noncanonical Hamiltonian systems: circulation laws in an extended phase space Yoshida, Z.; J. Morrison, P. Fluid Dynamics Research, Vol. 46, Issue 3 https://doi.org/10.1088/0169-5983/46/3/031412	journal	May 2014
Variational principles and Lyapunov stability analysis for compressible cylindrical plasma flows with arbitrary cross-section Khater, A. H.; Moawad, S. M. Plasma Physics and Controlled Fusion, Vol. 45, Issue 3 https://doi.org/10.1088/0741-3335/45/3/307	journal	February 2003
MHD equilibrium variational principles with symmetry Andreussi, T.; Morrison, P. J.; Pegoraro, F. Plasma Physics and Controlled Fusion, Vol. 52, Issue 5 https://doi.org/10.1088/0741-3335/52/5/055001	journal	March 2010
Hybrid Vlasov-MHD models: Hamiltonian vs. non-Hamiltonian Tronci, Cesare; Tassi, Emanuele; Camporeale, Enrico Plasma Physics and Controlled Fusion, Vol. 56, Issue 9 https://doi.org/10.1088/0741-3335/56/9/095008	journal	July 2014
Hamiltonian approach to hybrid plasma models Tronci, Cesare Journal of Physics A: Mathematical and Theoretical, Vol. 43, Issue 37 https://doi.org/10.1088/1751-8113/43/37/375501	journal	July 2010
Hamiltonian formalism of extended magnetohydrodynamics Abdelhamid, H. M.; Kawazura, Y.; Yoshida, Z. Journal of Physics A: Mathematical and Theoretical, Vol. 48, Issue 23 https://doi.org/10.1088/1751-8113/48/23/235502	journal	May 2015
Some instabilities of a completely ionized plasma Kruskal, Martin DAvid; Schwarzschild, Martin Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Vol. 223, Issue 1154, p. 348-360 https://doi.org/10.1098/rspa.1954.0120	journal	May 1954
An energy principle for hydromagnetic stability problems Bernstein, I. B.; Frieman, E. A.; Kruskal, Martin David Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Vol. 244, Issue 1236, p. 17-40 https://doi.org/10.1098/rspa.1958.0023	journal	February 1958
Free-energy expressions for Vlasov equilibria Morrison, P. J.; Pfirsch, D. Physical Review A, Vol. 40, Issue 7 https://doi.org/10.1103/PhysRevA.40.3898	journal	October 1989
Action principle for relativistic magnetohydrodynamics D’Avignon, Eric; Morrison, P. J.; Pegoraro, F. Physical Review D, Vol. 91, Issue 8 https://doi.org/10.1103/PhysRevD.91.084050	journal	April 2015
Negative-energy modes in a magnetically confined plasma in the framework of Maxwell-drift kinetic theory Throumoulopoulos, G. N.; Pfirsch, D. Physical Review E, Vol. 49, Issue 4 https://doi.org/10.1103/PhysRevE.49.3290	journal	April 1994
Negative-energy perturbations in circularly cylindrical equilibria within the framework of Maxwell-drift kinetic theory Throumoulopoulos, G. N.; Pfirsch, D. Physical Review E, Vol. 53, Issue 3 https://doi.org/10.1103/PhysRevE.53.2767	journal	March 1996
Relaxation of Toroidal Plasma and Generation of Reverse Magnetic Fields Taylor, J. B. Physical Review Letters, Vol. 33, Issue 19 https://doi.org/10.1103/PhysRevLett.33.1139	journal	November 1974
Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics Morrison, Philip J.; Greene, John M. Physical Review Letters, Vol. 45, Issue 10 https://doi.org/10.1103/PhysRevLett.45.790	journal	September 1980
Invariants and Geometric Structures in Nonlinear Hamiltonian Magnetic Reconnection Cafaro, E.; Grasso, D.; Pegoraro, F. Physical Review Letters, Vol. 80, Issue 20 https://doi.org/10.1103/PhysRevLett.80.4430	journal	May 1998
On Hydromagnetic Stability of Stationary Equilibria Frieman, E.; Rotenberg, Manuel Reviews of Modern Physics, Vol. 32, Issue 4 https://doi.org/10.1103/RevModPhys.32.898	journal	October 1960
Relaxation and magnetic reconnection in plasmas Taylor, J. B. Reviews of Modern Physics, Vol. 58, Issue 3 https://doi.org/10.1103/RevModPhys.58.741	journal	July 1986
Hamiltonian description of the ideal fluid Morrison, P. J. Reviews of Modern Physics, Vol. 70, Issue 2 https://doi.org/10.1103/RevModPhys.70.467	journal	April 1998
Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics. Morrison, Philip J.; Greene, John M. Physical Review Letters, Vol. 48, Issue 8 https://doi.org/10.1103/physrevlett.48.569	journal	February 1982
Hamiltonian magnetohydrodynamics: symmetric formulation, Casimir invariants, and equilibrium variational principles Andreussi, Tommaso; Morrison, Philip J.; Pegoraro, Francesco arXiv https://doi.org/10.48550/arxiv.1202.0468	text	January 2012
Stability of compressible reduced magnetohydrodynamic equilibria - analogy with magnetorotational instability Morrison, Philip J.; Tassi, Emanuele; Tronko, Natalia arXiv https://doi.org/10.48550/arxiv.1302.1822	text	January 2013
Hamiltonian Magnetohydrodynamics: Lagrangian, Eulerian, and Dynamically Accessible Stability - Theory Andreussi, T.; Morrison, P. J.; Pegoraro, F. arXiv https://doi.org/10.48550/arxiv.1308.0737	text	January 2013
Gyro-induced acceleration of magnetic reconnection Comisso, Luca; Grasso, Daniela; Waelbroeck, François L. arXiv https://doi.org/10.48550/arxiv.1309.1753	text	January 2013
A hierarchy of noncanonical Hamiltonian systems: circulation laws in an extended phase space Yoshida, Z.; Morrison, P. J. arXiv https://doi.org/10.48550/arxiv.1401.7698	text	January 2014
Hybrid Vlasov-MHD models: Hamiltonian vs. non-Hamiltonian Tronci, Cesare; Tassi, Emanuele; Camporeale, Enrico arXiv https://doi.org/10.48550/arxiv.1403.2773	text	January 2014
On energy conservation in extended magnetohydrodynamics Kimura, Keiji; Morrison, P. J. arXiv https://doi.org/10.48550/arxiv.1406.2745	text	January 2014
Hamiltonian Formalism of Extended Magnetohydrodynamics Abdelhamid, Hamdi M.; Kawazura, Yohei; Yoshida, Zensho arXiv https://doi.org/10.48550/arxiv.1412.6228	text	January 2014
Negative-Energy Perturbations in Circularly Cylindrical Equilibria within the Framework of Maxwell-Drift Kinetic Theory Throumoulopoulos, G. N.; Pfirsch, D. arXiv https://doi.org/10.48550/arxiv.plasm-ph/9602002	text	January 1996

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A class of three-dimensional gyroviscous magnetohydrodynamic models Lingam, Manasvi; Morrison, Philip J.; Wurm, Alexander arXiv https://doi.org/10.48550/arxiv.2002.11272	text	January 2020
Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles Kaltsas, D. A.; Throumoulopoulos, G. N.; Morrison, P. J. Journal of Plasma Physics, Vol. 84, Issue 3 https://doi.org/10.1017/s0022377818000338	journal	May 2018
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Translationally symmetric extended MHD via Hamiltonian reduction: Energy-Casimir equilibria Kaltsas, D. A.; Throumoulopoulos, G. N.; Morrison, P. J. Physics of Plasmas, Vol. 24, Issue 9 https://doi.org/10.1063/1.4986013	journal	September 2017
Energy-Casimir, dynamically accessible, and Lagrangian stability of extended magnetohydrodynamic equilibria Kaltsas, D. A.; Throumoulopoulos, G. N.; Morrison, P. J. Physics of Plasmas, Vol. 27, Issue 1 https://doi.org/10.1063/1.5125573	journal	January 2020
Energy-Casimir, dynamically accessible, and Lagrangian stability of extended magnetohydrodynamic equilibria Kaltsas, D. A.; Throumoulopoulos, G. N.; Morrison, P. J. arXiv https://doi.org/10.48550/arxiv.1908.08821	text	January 2019

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