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

Kaltsas, D. A.; Throumoulopoulos, G. N.; Morrison, P. J.

doi:10.1063/1.5125573

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

Journal Article · Thu Jan 02 04:00:00 EST 2020 · Physics of Plasmas

DOI:https://doi.org/10.1063/1.5125573· OSTI ID:1801010

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Univ. of Ioannina (Greece); University of Ioannina
Univ. of Ioannina (Greece)
Univ. of Texas, Austin, TX (United States)

The formal stability analysis of Eulerian extended magnetohydrodynamics (XMHD) equilibria is considered within the noncanonical Hamiltonian framework by means of the energy-Casimir variational principle and the dynamically accessible stability method. Specifically, we find explicit sufficient stability conditions for axisymmetric XMHD and Hall MHD (HMHD) equilibria with toroidal flow and for equilibria with arbitrary flow under constrained perturbations. The dynamically accessible, second-order variation of the Hamiltonian, which can potentially provide explicit stability criteria for generic equilibria, is also obtained. Moreover, we examine the Lagrangian stability of the general quasineutral two-fluid model written in terms of MHD-like variables, by finding the action and the Hamiltonian functionals of the linearized dynamics, working within a mixed Lagrangian-Eulerian framework. Upon neglecting electron mass, we derive a HMHD energy principle, and in addition, the perturbed induction equation arises from Hamilton’s equations of motion in view of a consistency condition for the relation between the perturbed magnetic potential and the canonical variables.

View Accepted Manuscript (DOE)

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

Sponsoring Organization:: USDOE

Grant/Contract Number:: FG05-80ET53088

OSTI ID:: 1801010

Alternate ID(s):: OSTI ID: 1581013

Journal Information:: Physics of Plasmas, Journal Name: Physics of Plasmas Journal Issue: 1 Vol. 27; ISSN 1070-664X

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

Country of Publication:: United States

Language:: English

References (63)

Pyridine nucleotides in glucose metabolism and diabetes: a review Wahlberg, Gustaf; Adamson, Ulf; Svensson, Jan Diabetes/Metabolism Research and Reviews, Vol. 16, Issue 1 https://doi.org/10.1002/(sici)1520-7560(200001/02)16:1<33::aid-dmrr79>3.0.co;2-s	journal	January 2000
Nonlinear stability of fluid and plasma equilibria Holm, Darryl D.; Marsden, Jerrold E.; Ratiu, Tudor Physics Reports, Vol. 123, Issue 1-2 https://doi.org/10.1016/0370-1573(85)90028-6	journal	July 1985
Über die Ausbreitung von Wellen in einem Plasma Lüst, Reimar Fortschritte der Physik, Vol. 7, Issue 9 https://doi.org/10.1002/prop.19590070902	journal	January 1959
Kelvin–Helmholtz Instability in the Solar Atmosphere, Solar Wind and Geomagnetosphere Mishin, V. V.; Tomozov, V. M. Solar Physics, Vol. 291, Issue 11 https://doi.org/10.1007/s11207-016-0891-4	journal	April 2016
Hydromagnetic stability of a diffuse linear pinch Newcomb, William A. Annals of Physics, Vol. 10, Issue 2 https://doi.org/10.1016/0003-4916(60)90023-3	journal	June 1960
Lyapunov stability of perfect fluid and multifluid plasma equilibria Elsässer, Klaus; Spieß, Peter Physics Letters A, Vol. 230, Issue 1-2 https://doi.org/10.1016/S0375-9601(97)00220-X	journal	June 1997
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
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
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
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
Geospace Environmental Modeling (GEM) Magnetic Reconnection Challenge Birn, J.; Drake, J. F.; Shay, M. A. Journal of Geophysical Research: Space Physics, Vol. 106, Issue A3 https://doi.org/10.1029/1999JA900449	journal	March 2001
Stability of long field-reversed configurations Barnes, D. C. Physics of Plasmas, Vol. 9, Issue 2 https://doi.org/10.1063/1.1435425	journal	February 2002
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
Linear stability of static equilibrium states in the Hall-magnetohydrodynamics model Hameiri, Eliezer; Torasso, R. Physics of Plasmas, Vol. 11, Issue 11 https://doi.org/10.1063/1.1784453	journal	November 2004
Ballooning mode stability in the Hall-magnetohydrodynamics model Torasso, R.; Hameiri, Eliezer Physics of Plasmas, Vol. 12, Issue 3 https://doi.org/10.1063/1.1851993	journal	March 2005
A theory for the pressure pedestal in high (H) mode tokamak discharges Guzdar, P. N.; Mahajan, S. M.; Yoshida, Z. Physics of Plasmas, Vol. 12, Issue 3 https://doi.org/10.1063/1.1852468	journal	March 2005
Variational principle for linear stability of flowing plasmas in Hall magnetohydrodynamics Hirota, M.; Yoshida, Z.; Hameiri, E. Physics of Plasmas, Vol. 13, Issue 2 https://doi.org/10.1063/1.2169734	journal	February 2006
A sufficient condition for the linear stability of magnetohydrodynamic equilibria with field aligned incompressible flows Throumoulopoulos, G. N.; Tasso, H. Physics of Plasmas, Vol. 14, Issue 12 https://doi.org/10.1063/1.2817957	journal	December 2007
On Hamiltonian and Action Principle Formulations of Plasma Dynamics Morrison, P. J.; Eliasson, Bengt; Shukla, Padma K. NEW DEVELOPMENTS IN NONLINEAR PLASMA PHYSICS: Proceedings of the 2009 ICTP Summer College on Plasma Physics and International Symposium on Cutting Edge Plasma Physics, AIP Conference Proceedings https://doi.org/10.1063/1.3266810	conference	January 2009
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
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
Ertel's vorticity theorem and new flux surfaces in multi-fluid plasmas Hameiri, Eliezer Physics of Plasmas, Vol. 20, Issue 9 https://doi.org/10.1063/1.4820769	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
Influence of the Hall effect and electron inertia in collisionless magnetic reconnection Andrés, Nahuel; Dmitruk, Pablo; Gómez, Daniel Physics of Plasmas, Vol. 23, Issue 2 https://doi.org/10.1063/1.4942418	journal	February 2016
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
Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Examples with translation symmetry Andreussi, T.; Morrison, P. J.; Pegoraro, F. Physics of Plasmas, Vol. 23, Issue 10 https://doi.org/10.1063/1.4964900	journal	October 2016
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
Ellipticity conditions for the extended MHD Grad-Shafranov-Bernoulli equilibrium equations Kaltsas, D. A.; Throumoulopoulos, G. N.; Morrison, P. J. Physics of Plasmas, Vol. 26, Issue 2 https://doi.org/10.1063/1.5080997	journal	February 2019
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
A sufficient condition for the ideal instability of shear flow with parallel magnetic field Chen, X. L.; Morrison, P. J. Physics of Fluids B: Plasma Physics, Vol. 3, Issue 4 https://doi.org/10.1063/1.859841	journal	April 1991
Hall magnetohydrodynamics: Conservation laws and Lyapunov stability Holm, Darryl D. Physics of Fluids, Vol. 30, Issue 5 https://doi.org/10.1063/1.866246	journal	January 1987
Lyapunov stability analysis of magnetohydrodynamic plasma equilibria with axisymmetric toroidal flow Almaguer, Juan Antonio; Hameiri, Eliezer; Herrera, Julio Physics of Fluids, Vol. 31, Issue 7 https://doi.org/10.1063/1.866640	journal	January 1988
Variational formulation for a multifluid flowing plasma with application to the internal tilt mode of a field-reversed configuration Ishida, A.; Momota, H.; Steinhauer, L. C. Physics of Fluids, Vol. 31, Issue 10 https://doi.org/10.1063/1.866959	journal	January 1988
Stabilization of ballooning modes with sheared toroidal rotation Miller, R. L.; Waelbroeck, F. L.; Hassam, A. B. Physics of Plasmas, Vol. 2, Issue 10 https://doi.org/10.1063/1.871067	journal	October 1995
Shear flow destabilization of a slowly rotating tokamak Chu, M. S. Physics of Plasmas, Vol. 5, Issue 1 https://doi.org/10.1063/1.872687	journal	January 1998
Variational principles for equilibrium states with plasma flow Hameiri, Eliezer Physics of Plasmas, Vol. 5, Issue 9 https://doi.org/10.1063/1.872995	journal	September 1998
Two variational principles for incompressible plasmas Spiess, Peter; Elsässer, Klaus Physics of Plasmas, Vol. 6, Issue 11 https://doi.org/10.1063/1.873687	journal	November 1999
Stabilization of the internal kink mode in a tokamak by toroidal plasma rotation Wahlberg, C.; Bondeson, A. Physics of Plasmas, Vol. 7, Issue 3 https://doi.org/10.1063/1.873889	journal	March 2000
A powerful local shear instability in weakly magnetized disks. I - Linear analysis. II - Nonlinear evolution Balbus, Steven A.; Hawley, John F. The Astrophysical Journal, Vol. 376 https://doi.org/10.1086/170270	journal	July 1991
A quarter-century of H-mode studies Wagner, F. Plasma Physics and Controlled Fusion, Vol. 49, Issue 12B https://doi.org/10.1088/0741-3335/49/12B/S01	journal	November 2007
Hamiltonian formulation and analysis of a collisionless fluid reconnection model Tassi, E.; Morrison, P. J.; Waelbroeck, F. L. Plasma Physics and Controlled Fusion, Vol. 50, Issue 8 https://doi.org/10.1088/0741-3335/50/8/085014	journal	June 2008
The effects of sheared toroidal rotation on stability limits in tokamak plasmas Chapman, I. T.; Walkden, N. R.; Graves, J. P. Plasma Physics and Controlled Fusion, Vol. 53, Issue 12 https://doi.org/10.1088/0741-3335/53/12/125002	journal	October 2011
Ideal and resistive magnetohydrodynamic instabilities in cylindrical geometry with a sheared flow along the axis Brunetti, D.; Lazzaro, E.; Nowak, S. Plasma Physics and Controlled Fusion, Vol. 59, Issue 5 https://doi.org/10.1088/1361-6587/aa6423	journal	March 2017
Hamiltonian structure and stability analysis of a reduced four-field model for plasmas in the presence of a strong guide field Tassi, E.; Ratiu, T. S.; Lazzaro, E. Journal of Physics: Conference Series, Vol. 401 https://doi.org/10.1088/1742-6596/401/1/012023	journal	December 2012
Canonical Hamiltonian mechanics of Hall magnetohydrodynamics and its limit to ideal magnetohydrodynamics Yoshida, Z.; Hameiri, E. Journal of Physics A: Mathematical and Theoretical, Vol. 46, Issue 33 https://doi.org/10.1088/1751-8113/46/33/335502	journal	July 2013
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
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
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
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
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
Suppression of turbulence and transport by sheared flow Terry, P. W. Reviews of Modern Physics, Vol. 72, Issue 1 https://doi.org/10.1103/RevModPhys.72.109	journal	January 2000
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
Variational Stability Problem for MHD Model with Hall Effect Dgisonis, V. I. Fusion Technology, Vol. 35, Issue 1T https://doi.org/10.13182/FST99-A11963845	journal	January 1999
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
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
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
Influence of the Hall effect and electron inertia in collisionless magnetic reconnection Andrés, Nahuel; Dmitruk, Pablo; Gómez, Daniel arXiv https://doi.org/10.48550/arxiv.1509.01169	text	January 2015

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