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Helically symmetric extended magnetohydrodynamics: Hamiltonian formulation and equilibrium variational principles

  • D. A. Kaltsas (a1), G. N. Throumoulopoulos (a1) and P. J. Morrison (a2)

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 (Kaltsas et 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.

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Corresponding author

Email address for correspondence: gthroum@cc.uoi.gr

References

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Abdelhamid, H. M., Kawazura, Y. & Yoshida, Z. 2015 Hamiltonian formalism of extended magnetohydrodynamics. J. Phys. A 48 (23), 235502.
Almaguer, J. A., Hameiri, E., Herrera, J. & Holm, D. D. 1988 Lyapunov stability analysis of magnetohydrodynamic plasma equilibria with axisymmetric toroidal flow. Phys. Fluids 31 (7), 19301939.
Andreussi, T., Morrison, P. J. & Pegoraro, F. 2010 MHD equilibrium variational principles with symmetry. Plasma Phys. Control. Fusion 52 (5), 055001.
Andreussi, T., Morrison, P. J. & Pegoraro, F. 2012 Hamiltonian magnetohydrodynamics: helically symmetric formulation, casimir invariants, and equilibrium variational principles. Phys. Plasmas 19 (5), 052102.
Andreussi, T., Morrison, P. J. & Pegoraro, F. 2013 Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability-theory. Phys. Plasmas 20 (9), 092104.
Andreussi, T., Morrison, P. J. & Pegoraro, F. 2016 Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability-examples with translation symmetry. Phys. Plasmas 23 (10), 102112.
Andreussi, T. & Pegoraro, F. 2008 On the variational approach to axisymmetric magnetohydrodynamic equilibria. Phys. Plasmas 15 (9), 092108.
Barberio-Corsetti, P. 1973 Force-free helical equilibria. Plasma Phys. 15 (11), 1131.
Bergerson, W. F., Auriemma, F., Chapman, B., Ding, W. X., Zanca, P., Brower, D. L., Innocente, P., Lin, L., Lorenzini, R. & Martines, E. 2011 Bifurcation to 3D helical magnetic equilibrium in an axisymmetric toroidal device. Phys. Rev. Lett. 107, 255001.
Bogoyavlenskij, O. I. 2000 Exact helically symmetric plasma equilibria. Lett. Math. Phys. 51 (4), 235247.
Chandrasekhar, S. & Kendall, P. C. 1957 On force-free magnetic fields. Astrophys. J. 126, 457.
Cooper, W., Graves, J. & Sauter, O. 2011 Jet snake magnetohydrodynamic equilibria. Nucl. Fusion 51 (7), 072002.
Cooper, W. A., Graves, J. P., Pochelon, A., Sauter, O. & Villard, L. 2010 Tokamak magnetohydrodynamic equilibrium states with axisymmetric boundary and a 3D helical core. Phys. Rev. Lett. 105, 035003.
Evangelias, A., Kuiroukidis, A. & Throumoulopoulos, G. N. 2018 Helically symmetric equilibria with pressure anisotropy and incompressible plasma flow. Plasma Phys. Control. Fusion 60 (2), 025005.
de Gouveia Dal Pino, E. M. 2005 Astrophysical jets and outflows. Adv. Space Res. 35 (5), 908924.
Grasso, D., Tassi, E., Abdelhamid, H. M. & Morrison, P. J. 2017 Structure and computation of two-dimensional incompressible extended MHD. Phys. Plasmas 24 (1), 012110.
Guazzotto, L. & Betti, R. 2015 Two-fluid equilibrium with flow: Flow2. Phys. Plasmas 22 (9), 092503.
Hameiri, E. 2013 Ertel’s vorticity theorem and new flux surfaces in multi-fluid plasmas. Phys. Plasmas 20 (9), 092503.
Hazeltine, R. D., Hsu, C. T. & Morrison, P. J. 1987 Hamiltonian four-field model for nonlinear tokamak dynamics. Phys. Fluids 30 (10), 32043211.
Helander, P., Beidler, C. D., Bird, T. M., Drevlak, M., Feng, Y., Hatzky, R., Jenko, F., Kleiber, R., Proll, J. H. E., Turkin, Y. et al. 2012 Stellarator and tokamak plasmas: a comparison. Plasma Phys. Control. Fusion 54 (12), 124009.
Holm, D. D. 1987 Hall magnetohydrodynamics: conservation laws and Lyapunov stability. Phys. Fluids 30 (5), 13101322.
Holm, D. D., Marsden, J. E., Ratiu, T. & Weinstein, A. 1985 Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123 (1), 1116.
Iqbal, M., Mirza, A. M., Murtaza, G. & Yoshida, Z. 2001 High $\unicode[STIX]{x1D6FD}$ relaxed states with internal conductor plasma configuration. Phys. Plasmas 8 (5), 15591564.
Johnson, J. L., Oberman, C. R., Kulsrud, R. M. & Frieman, E. A. 1958 Some stable hydromagnetic equilibria. Phys. Fluids 1 (4), 281296.
Kaltsas, D. A., Throumoulopoulos, G. N. & Morrison, P. J. 2017 Translationally symmetric extended mhd via Hamiltonian reduction: Energy–Casimir equilibria. Phys. Plasmas 24 (9), 092504.
Kawazura, Y., Miloshevich, G. & Morrison, P. J. 2017 Action principles for relativistic extended magnetohydrodynamics: a unified theory of magnetofluid models. Phys. Plasmas 24 (2), 022103.
Kimura, K. & Morrison, P. J. 2014 On energy conservation in extended magnetohydrodynamics. Phys. Plasmas 21 (8), 082101.
Kruskal, M. D. & Oberman, C. R. 1958 On the stability of plasma in static equilibrium. Phys. Fluids 1 (4), 275280.
Lighthill, M. J. 1960 Studies on magneto-hydrodynamic waves and other anisotropic wave motions. Phil. Trans. R. Soc. Lond. A 252 (1014), 397430.
Lingam, M., Morrison, P. & Tassi, E. 2015a Inertial magnetohydrodynamics. Phys. Lett. A 379 (6), 570576.
Lingam, M., Morrison, P. J. & Miloshevich, G. 2015b Remarkable connections between extended magnetohydrodynamics models. Phys. Plasmas 22 (7), 072111.
Lorenzini, R., Martines, E., Piovesan, P., Terranova, D., Zanca, P., Zuin, M., Alfier, A., Bonfiglio, D., Bonomo, F., Canton, A. et al. 2009 Self-organized helical equilibria as a new paradigm for ohmically heated fusion plasmas. Nat. Phys. 5, 570574.
Lüst, R. 1959 Über die ausbreitung von wellen in einem plasma. Fortschr. Phys. 7 (9), 503558.
Mahajan, S. M. & Yoshida, Z. 1998 Double curl beltrami flow: diamagnetic structures. Phys. Rev. Lett. 81, 48634866.
Moawad, S., Ramadan, A., Ibrahim, D., El-Kalaawy, O. & Hussain, E. 2017 Linear stability of certain translationally symmetric mhd equilibria with incompressible flow. Results in Physics 7, 21592171.
Moawad, S. M. 2013 Linear and nonlinear stability criteria for compressible MHD flows in a gravitational field. J. Plasma Phys. 79 (5), 873883.
Morrison, P. J. 1982 Poisson brackets for fluids and plasmas. AIP Conf. Proc. 88 (1), 1346.
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70, 467521.
Morrison, P. J. & Greene, J. M. 1980 Noncanonical hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics. Phys. Rev. Lett. 45, 790794.
Morrison, P. J., Lingam, M. & Acevedo, R. 2014 Hamiltonian and action formalisms for two-dimensional gyroviscous magnetohydrodynamics. Phys. Plasmas 21 (8), 082102.
Morrison, P. J., Tassi, E. & Tronko, N. 2013 Stability of compressible reduced magnetohydrodynamic equilibria-analogy with magnetorotational instability. Phys. Plasmas 20 (4), 042109.
Pecquet, A. L., Cristofani, P., Mattioli, M., Garbet, X., Laurent, L., Geraud, A., Gil, C., Joffrin, E. & Sabot, R. 1997 Snake-like phenomena in Tore Supra following pellet injection. Nucl. Fusion 37 (4), 451.
Pudritz, R., Hardcastle, M. & Gabuzda, D. 2012 Magnetic fields in astrophysical jets: from launch to termination. Space Sci. Rev. 169 (1–4), 2772.
Puiatti, M. E., Alfier, A., Auriemma, F., Capello, S., Carraro, L., Cavazzana, R., Dal Bello, S., Fassina, A., Escande, D. F., Franz, P. et al. 2009 Helical equilibria and magnetic structures in the reversed field pinch and analogies to the tokamak and stellarator. Plasma Phys. Control. Fusion 51 (12), 124031.
Spitzer, L. 1958 The stellarator concept. Phys. Fluids 1 (4), 253264.
Tassi, E., Morrison, P. J., Waelbroeck, F. L. & Grasso, D. 2008 Hamiltonian formulation and analysis of a collisionless fluid reconnection model. Plasma Phys. Control. Fusion 50 (8), 085014.
Terranova, D., Bonfiglio, D., Boozer, A. H., Cooper, A. W., Gobbin, M., Hirshman, S. P., Lorenzini, R., Marrelli, L., Martines, E., Momo, B. et al. 2010 A 3D approach to equilibrium, stability and transport studies in RFX-mod improved regimes. Plasma Phys. Control. Fusion 52 (12), 124023.
Throumoulopoulos, G. N. & Tasso, H. 1999 Ideal magnetohydrodynamic equilibria with helical symmetry and incompressible flows. J. Plasma Phys. 62 (4), 449459.
Throumoulopoulos, G. N. & Tasso, H. 2006 On hall magnetohydrodynamics equilibria. Phys. Plasmas 13 (10), 102504.
Tsinganos, K. C. 1982 Magnetohydrodynamic equilibrium. III – helically symmetric fields. Astrophys. J. 259, 820831.
Uo, K. 1961 The confinement of plasma by the heliotron magnetic field. J. Phys. Soc. Japan 16 (7), 13801395.
Waelbroeck, F. 2009 Theory and observations of magnetic islands. Nucl. Fusion 49 (10), 104025.
Weller, A., Cheetham, A. D., Edwards, A. W., Gill, R. D., Gondhalekar, A., Granetz, R. S., Snipes, J. & Wesson, J. A. 1987 Persistent density perturbations at rational-q surfaces following pellet injection in the joint European torus. Phys. Rev. Lett. 59, 23032306.
Yoshida, Z. & Hameiri, E. 2013 Canonical Hamiltonian mechanics of hall magnetohydrodynamics and its limit to ideal magnetohydrodynamics. J. Phys. A 46 (33), 335502.
Yoshida, Z. & Mahajan, S. M. 2002 Variational principles and self-organization in two-fluid plasmas. Phys. Rev. Lett. 88, 095001.
Yoshida, Z., Mahajan, S. M., Ohsaki, S., Iqbal, M. & Shatashvili, N. 2001 Beltrami fields in plasmas: high-confinement mode boundary layers and high beta equilibria. Phys. Plasmas 8 (5), 21252131.
Yoshida, Z., Morrison, P. & Dobarro, F. J. 2014 Singular casimir elements of the Euler equation and equilibrium points. J. Math. Fluid Mech. 16 (1), 4157.
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