Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles

doi:https://doi.org/10.1017/s0022377818000338

Title: Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles

Abstract

Hamiltonian extended magnetohydrodynamics (XMHD) is restricted to respect helical symmetry by reducing the Poisson bracket for the three-dimensional dynamics to a helically symmetric one, as an extension of the previous study for translationally symmetric XMHD (Kaltsaset al.,Phys. Plasmas, vol. 24, 2017, 092504). Four families of Casimir invariants are obtained directly from the symmetric Poisson bracket and they are used to construct Energy–Casimir variational principles for deriving generalized XMHD equilibrium equations with arbitrary macroscopic flows. The system is then cast into the form of Grad–Shafranov–Bernoulli equilibrium equations. The axisymmetric and the translationally symmetric formulations can be retrieved as geometric reductions of the helically symmetric one. As special cases, the derivation of the corresponding equilibrium equations for incompressible plasmas is discussed and the helically symmetric equilibrium equations for the Hall MHD system are obtained upon neglecting electron inertia. An example of an incompressible double-Beltrami equilibrium is presented in connection with a magnetic configuration having non-planar helical magnetic axis.

Authors:

;

Publication Date:: 2018-05-24

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

Sponsoring Org.:: USDOE

OSTI Identifier:: 1538937

DOE Contract Number:: FG05-80ET53088

Resource Type:: Journal Article

Journal Name:: Journal of Plasma Physics

Additional Journal Information:: Journal Volume: 84; Journal Issue: 3; Journal ID: ISSN 0022-3778

Publisher:: Cambridge University Press

Country of Publication:: United States

Language:: English

Subject:: Physics

Citation Formats


                    Kaltsas, D. A., Throumoulopoulos, G. N., and Morrison, P. J. Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles.  United States: N. p., 2018. 
        Web.  doi:10.1017/s0022377818000338.

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                    Kaltsas, D. A., Throumoulopoulos, G. N., & Morrison, P. J. Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles.  United States.  https://doi.org/10.1017/s0022377818000338

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                    Kaltsas, D. A., Throumoulopoulos, G. N., and Morrison, P. J. Thu .  
        "Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles".  United States.  https://doi.org/10.1017/s0022377818000338.

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@article{osti_1538937,

  title        = {Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles},

  author       = {Kaltsas, D. A. and Throumoulopoulos, G. N. and Morrison, P. J.},

  abstractNote = {Hamiltonian extended magnetohydrodynamics (XMHD) is restricted to respect helical symmetry by reducing the Poisson bracket for the three-dimensional dynamics to a helically symmetric one, as an extension of the previous study for translationally symmetric XMHD (Kaltsaset al.,Phys. Plasmas, vol. 24, 2017, 092504). Four families of Casimir invariants are obtained directly from the symmetric Poisson bracket and they are used to construct Energy–Casimir variational principles for deriving generalized XMHD equilibrium equations with arbitrary macroscopic flows. The system is then cast into the form of Grad–Shafranov–Bernoulli equilibrium equations. The axisymmetric and the translationally symmetric formulations can be retrieved as geometric reductions of the helically symmetric one. As special cases, the derivation of the corresponding equilibrium equations for incompressible plasmas is discussed and the helically symmetric equilibrium equations for the Hall MHD system are obtained upon neglecting electron inertia. An example of an incompressible double-Beltrami equilibrium is presented in connection with a magnetic configuration having non-planar helical magnetic axis.},

  doi          = {10.1017/s0022377818000338},

  url          = {https://www.osti.gov/biblio/1538937},
  journal      = {Journal of Plasma Physics},

  issn         = {0022-3778},

  number       = 3,

  volume       = 84,

  place        = {United States},

  year         = {2018},

  month        = {5}

}

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Works referenced in this record:

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Title: Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles

Abstract

Citation Formats

On Force-Free Magnetic Fields. journal, July 1957

Structure and computation of two-dimensional incompressible extended MHD journal, January 2017

Ideal magnetohydrodynamic equilibria with helical symmetry and incompressible flows journal, October 1999

Double Curl Beltrami Flow: Diamagnetic Structures journal, November 1998

Persistent density perturbations at rational- q surfaces following pellet injection in the Joint European Torus journal, November 1987

Hamiltonian four-field model for nonlinear tokamak dynamics journal, January 1987

Studies on Magneto-Hydrodynamic Waves and other Anisotropic Wave Motions journal, March 1960

Lyapunov stability analysis of magnetohydrodynamic plasma equilibria with axisymmetric toroidal flow journal, January 1988

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

Action principles for relativistic extended magnetohydrodynamics: A unified theory of magnetofluid models journal, February 2017

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

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

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

On the variational approach to axisymmetric magnetohydrodynamic equilibria journal, September 2008

Hamiltonian description of the ideal fluid journal, April 1998

On the Stability of Plasma in Static Equilibrium journal, January 1958

MHD equilibrium variational principles with symmetry journal, March 2010

Helical equilibria and magnetic structures in the reversed field pinch and analogies to the tokamak and stellarator journal, November 2009

Astrophysical jets and outflows journal, January 2005

Inertial magnetohydrodynamics journal, March 2015

Theory and observations of magnetic islands journal, September 2009

Force-free helical equilibria journal, November 1973

Variational Principles and Self-Organization in Two-Fluid Plasmas journal, February 2002

Hall magnetohydrodynamics: Conservation laws and Lyapunov stability journal, January 1987

Helically symmetric equilibria with pressure anisotropy and incompressible plasma flow journal, December 2017

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

JET snake magnetohydrodynamic equilibria journal, June 2011

Nonlinear stability of fluid and plasma equilibria journal, July 1985

On Hall magnetohydrodynamics equilibria journal, October 2006

Stellarator and tokamak plasmas: a comparison journal, November 2012

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

Tokamak Magnetohydrodynamic Equilibrium States with Axisymmetric Boundary and a 3D Helical Core journal, July 2010

Bifurcation to 3D Helical Magnetic Equilibrium in an Axisymmetric Toroidal Device journal, December 2011

On energy conservation in extended magnetohydrodynamics journal, August 2014

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

Hamiltonian formulation and analysis of a collisionless fluid reconnection model journal, June 2008

Beltrami fields in plasmas: High-confinement mode boundary layers and high beta equilibria journal, May 2001

A 3D approach to equilibrium, stability and transport studies in RFX-mod improved regimes journal, November 2010

Snake-like phenomena in Tore Supra following pellet injection journal, April 1997

The Confinement of Plasma by the Heliotron Magnetic Field journal, July 1961

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

Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Examples with translation symmetry journal, October 2016

The Stellarator Concept journal, January 1958

Self-organized helical equilibria as a new paradigm for ohmically heated fusion plasmas journal, June 2009

Magnetohydrodynamic equilibrium. III - Helically symmetric fields. IV - Nonequilibrium of nonsymmetric hydrodynamic topologies journal, August 1982

Remarkable connections between extended magnetohydrodynamics models journal, July 2015

Über die Ausbreitung von Wellen in einem Plasma journal, January 1959

Two-fluid equilibrium with flow: FLOW2 journal, September 2015

Linear stability of certain translationally symmetric MHD equilibria with incompressible flow journal, January 2017

Some Stable Hydromagnetic Equilibria journal, January 1958

Ertel's vorticity theorem and new flux surfaces in multi-fluid plasmas journal, September 2013

High β relaxed states with internal conductor plasma configuration journal, May 2001

Magnetic Fields in Astrophysical Jets: From Launch to Termination journal, June 2012

On Force-Free Magnetic Fields.
journal, July 1957

Structure and computation of two-dimensional incompressible extended MHD
journal, January 2017

Ideal magnetohydrodynamic equilibria with helical symmetry and incompressible flows
journal, October 1999

Double Curl Beltrami Flow: Diamagnetic Structures
journal, November 1998

Persistent density perturbations at rational- q surfaces following pellet injection in the Joint European Torus
journal, November 1987

Hamiltonian four-field model for nonlinear tokamak dynamics
journal, January 1987

Studies on Magneto-Hydrodynamic Waves and other Anisotropic Wave Motions
journal, March 1960

Lyapunov stability analysis of magnetohydrodynamic plasma equilibria with axisymmetric toroidal flow
journal, January 1988

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

Action principles for relativistic extended magnetohydrodynamics: A unified theory of magnetofluid models
journal, February 2017

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

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

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

On the variational approach to axisymmetric magnetohydrodynamic equilibria
journal, September 2008

Hamiltonian description of the ideal fluid
journal, April 1998

On the Stability of Plasma in Static Equilibrium
journal, January 1958

MHD equilibrium variational principles with symmetry
journal, March 2010

Helical equilibria and magnetic structures in the reversed field pinch and analogies to the tokamak and stellarator
journal, November 2009

Astrophysical jets and outflows
journal, January 2005

Inertial magnetohydrodynamics
journal, March 2015

Theory and observations of magnetic islands
journal, September 2009

Force-free helical equilibria
journal, November 1973

Variational Principles and Self-Organization in Two-Fluid Plasmas
journal, February 2002

Hall magnetohydrodynamics: Conservation laws and Lyapunov stability
journal, January 1987

Helically symmetric equilibria with pressure anisotropy and incompressible plasma flow
journal, December 2017

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

JET snake magnetohydrodynamic equilibria
journal, June 2011

Nonlinear stability of fluid and plasma equilibria
journal, July 1985

On Hall magnetohydrodynamics equilibria
journal, October 2006

Stellarator and tokamak plasmas: a comparison
journal, November 2012

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

Tokamak Magnetohydrodynamic Equilibrium States with Axisymmetric Boundary and a 3D Helical Core
journal, July 2010

Bifurcation to 3D Helical Magnetic Equilibrium in an Axisymmetric Toroidal Device
journal, December 2011

On energy conservation in extended magnetohydrodynamics
journal, August 2014

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

Hamiltonian formulation and analysis of a collisionless fluid reconnection model
journal, June 2008

Beltrami fields in plasmas: High-confinement mode boundary layers and high beta equilibria
journal, May 2001

A 3D approach to equilibrium, stability and transport studies in RFX-mod improved regimes
journal, November 2010

Snake-like phenomena in Tore Supra following pellet injection
journal, April 1997

The Confinement of Plasma by the Heliotron Magnetic Field
journal, July 1961

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

Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Examples with translation symmetry
journal, October 2016

The Stellarator Concept
journal, January 1958

Self-organized helical equilibria as a new paradigm for ohmically heated fusion plasmas
journal, June 2009

Magnetohydrodynamic equilibrium. III - Helically symmetric fields. IV - Nonequilibrium of nonsymmetric hydrodynamic topologies
journal, August 1982

Remarkable connections between extended magnetohydrodynamics models
journal, July 2015

Über die Ausbreitung von Wellen in einem Plasma
journal, January 1959

Two-fluid equilibrium with flow: FLOW2
journal, September 2015

Linear stability of certain translationally symmetric MHD equilibria with incompressible flow
journal, January 2017

Some Stable Hydromagnetic Equilibria
journal, January 1958

Ertel's vorticity theorem and new flux surfaces in multi-fluid plasmas
journal, September 2013

High β relaxed states with internal conductor plasma configuration
journal, May 2001

Magnetic Fields in Astrophysical Jets: From Launch to Termination
journal, June 2012