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Multi-symplectic magnetohydrodynamics

Published online by Cambridge University Press:  09 June 2014

G. M. Webb ,
J. F. McKenzie  and
G. P. Zank
G. M. Webb*
Affiliation:
Center for Space Plasma and Aeronomic Research, The University of Alabama in Huntsville, Huntsville, AL 35805, USA
J. F. McKenzie
Affiliation:
Center for Space Plasma and Aeronomic Research, The University of Alabama in Huntsville, Huntsville, AL 35805, USA Department of Mathematics and Statistics, Durban University of Technology, Steve Biko Campus, Durban, South Africa, and School of Mathematical Sciences, University of Kwa-Zulu Natal, Durban, South Africa
G. P. Zank
Affiliation:
Center for Space Plasma and Aeronomic Research, The University of Alabama in Huntsville, Huntsville, AL 35805, USA Department of Space Science, The University of Alabama in Huntsville, Huntsville, AL 35805, USA
*
†Email address for correspondence: gmw0002@uah.edu
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Abstract

A multi-symplectic formulation of ideal magnetohydrodynamics (MHD) is developed based on the Clebsch variable variational principle in which the Lagrangian consists of the kinetic minus the potential energy of the MHD fluid modified by constraints using Lagrange multipliers that ensure mass conservation, entropy advection with the flow, the Lin constraint, and Faraday's equation (i.e. the magnetic flux is Lie dragged with the flow). The analysis is also carried out using the magnetic vector potential à where α=Ã⋅dx is Lie dragged with the flow, and B=∇×Ã. The multi-symplectic conservation laws give rise to the Eulerian momentum and energy conservation laws. The symplecticity or structural conservation laws for the multi-symplectic system corresponds to the conservation of phase space. It corresponds to taking derivatives of the momentum and energy conservation laws and combining them to produce n(n−1)/2 extra conservation laws, where n is the number of independent variables. Noether's theorem for the multi-symplectic MHD system is derived, including the case of non-Cartesian space coordinates, where the metric plays a role in the equations.

Type
Papers
Information
Journal of Plasma Physics , Volume 80 , Issue 5 , October 2014 , pp. 707 - 743
Copyright
Copyright © Cambridge University Press 2014 

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References

REFERENCES

Akhmetiev, P. and Ruzmaikin, A. 1995 A fourth order topological invariant of magnetic or vortex lines. J. Geom. Phys. 15, 95–101.CrossRefGoogle Scholar
Anderson, I. M. 1989 The Variational Bicomplex. Logan, UT: Utah State University. http://www.math.usu.edu/~fgmp/Publications/VB/vb.pdf.Google Scholar
Anderson, I. M. 1992 Introduction to the variational Bi-complex in mathematical aspects of contemporary field theory, Contemp. Math. 132, 51–73.Google Scholar
Arnold, V. I. and Khesin, B. A. 1998 Topological Methods in Hydrodynamics. New York, NY: Springer.Google Scholar
Berger, M. A. 1990 Third-order link integrals. J. Phys. Math. Gen. 23, 2787–2793.Google Scholar
Berger, M. A. and Field, G. B. 1984 The toplological properties of magnetic helicity. J. Fluid Mech. 147, 133–148.Google Scholar
Bluman, G. W., Cheviakov, A. F. and Anco, S. 2010 Applications of Symmetry Methods to Partial Differential Equations. New York, NY: Springer.Google Scholar
Bluman, G. W. and Kumei, S. 1989 Symmetries and Differential Equations. New York, NY: Springer.Google Scholar
Bridges, T. J. 1992. Spatial Hamiltonian structure, energy flux and the water-wave problem. Proc. Roy. Soc. London, 439, 297–315.Google Scholar
Bridges, T. J. 1997a Multi-symplectic structures and wave propagation. Math. Proc. Camb. Philos. Soc. 121, 147–190.Google Scholar
Bridges, T. J. 1997b A geometric formulation of the conservation of wave action and its implications for signature and classification of instabilities. Proc. Roy. Soc. A 453, 1365–1395.Google Scholar
Bridges, T. J. 2006 Canonical multi-symplectic structure on the total exterior algebra bundle. Proc. Roy. Soc. London A 462, 1531–1551.Google Scholar
Bridges, T. J., Hydon, P. E. and Lawson, J. K. 2010, Multi-symplectic structures and the variational bi-complex. Math. Proc. Camb. Phil. Soc., 148, 159–178.CrossRefGoogle Scholar
Bridges, T. J., Hydon, P. E. and Reich, S. 2005, Vorticity and symplecticity in Lagrangian fluid dynamics. J. Phys. Math. Gen. 38, 1403–1418.Google Scholar
Bridges, T. J. and Reich, S. 2006 Numerical methods for Hamiltonian PDEs. J. Phys. Math. Gen. 39, 5287–5320.Google Scholar
Brio, M., Zakharian, A. R. and Webb, G. M. 2010 Numerical time-dependent partial differential equations for scientists and engineers. In: Mathematics in Science and Engineering, Vol. 123, 1st edn. (ed. Chui, C. K.). Philadelphia, PA: Elsevier, pp. 199–204.Google Scholar
Calkin, M. G. 1963 An action principle for magnetohydrodynamics. Canad. J. Phys. 41, 2241–2251.Google Scholar
Chandre, C., de Guillebon, L., Back, A., Tassi, E. and Morrison, P. J. 2013 On the use of projectors for Hamiltonian systems and their relationship with Dirac brackets. J. Phys. Math. Theoret. 46, 125203 (14 pp), doi:10.10.1088/1751-8133/46/12/125203.Google Scholar
Cotter, C. J., Holm, D. D. and Hydon, P. E. 2007 Multi-symplectic formulation of fluid dynamics using the inverse map. Proc. Roy. Soc. Lond. A, 463, 2617–2687.Google Scholar
Finn, J. H. and Antonsen, T. M. 1985 Magnetic helicity: what is it and what is it good for? Comments Plasma Phys. Contr. Fusion 9 (3), 111.Google Scholar
Finn, J. M. and Antonsen, T. M. 1988 Magnetic helicity injection for configurations with field errors. Phys. Fluids 31 (10), 3012–3017.Google Scholar
Gordin, V. A. and Petviashvili, V. I. 1987 The gauge of vector potential and Lyapunov stable MHD equilibrium. Soviet J. Plasma Phys. 13 (7), 509–511 (English).Google Scholar
Harrison, B. K. and Estabrook, F. B. 1971 Geometric approach to invariance groups and solution of partial differential systems. J. Math. Phys. 12, 653–666.CrossRefGoogle Scholar
Holm, D. D. and Kupershmidt, B. A. 1983a Poisson brackets and Clebsch representations for magnetohydrodynamics, multi-fluid plasmas and elasticity. Physica D, 6D, 347–363.Google Scholar
Holm, D. D. and Kupershmidt, B. A. 1983b Non-canonical Hamiltonian formulation of ideal magnetohydrodynamics. Physica D, 7D, 330–333.Google Scholar
Holm, D. D., Marsden, J. E. and Ratiu, T. S. 1998 The Euler-Lagrange equations and semi-products with application to continuum theories. Adv. Math. 137, 1–81.CrossRefGoogle Scholar
Hydon, P. E., 2005 Multi-symplectic conservation laws for differential and differential-difference equations. Proc. Roy. Soc. A 461, 1627–1637.CrossRefGoogle Scholar
Hydon, P. E. and Mansfield, E. L. 2011 Extensions of Noether's second theorem: from continuous to discrete systems Proc. Roy. Soc. A 467, 3206–3221, doi:10.1098/rspa.2011.0158.Google Scholar
Marsden, J. E. and Shkoller, S. 1999 Multi-symplectic geometry, covariant Hamiltonians and water waves. Math. Proc. Camb. Phil. Soc. 125, 553–575.Google Scholar
Moffatt, H. K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid. Mech. 35, 117.Google Scholar
Morrison, P. J. 1982 Poisson brackets for fluids and plasmas. In: Mathematical Methods in Hydrodynamics and Integrability of Dynamical Systems, (eds. Tabor, M. and Treve, Y. M.), AIP Proc. Conf., Vol. 88. New York, NY: American Institute of Physics, pp 13–46.Google Scholar
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70 (2), 467–521, doi:10.1103/RevModPhys.70.467.Google Scholar
Morrison, P. J. and Greene, J. M. 1980 Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics. Phys. Rev. Lett. 45, 790–794.Google Scholar
Morrison, P. J. and Greene, J. M. 1982 Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics (Errata). Phys. Rev. Lett. 48, 569.CrossRefGoogle Scholar
Newcomb, W. A. 1962 Lagrangian and Hamiltonian methods in magnetohydrodynamics. Nucl. Fusion Suppl. Part 2, 451–463.Google Scholar
Padhye, N. S. and Morrison, P. J. 1996a Fluid relabeling symmetry Phys. Lett. A 219, 287–292.Google Scholar
Padhye, N. S. and Morrison, P. J. 1996b Relabeling symmetries in hydrodynamics and magnetohydrodynamics. Plasma Phys. Rep. 22, 869–877.Google Scholar
Powell, K. G., Roe, P. L., Linde, T. J., Gombosi, T. I. and De Zeeuw, D. 1999 A solution adaptive upwind scheme for ideal magnetohydrodynamics. J. Comput. Phys. 154, 284–309.Google Scholar
Reich, S. 2000 Multi-symplectic Runge–Kutta collocation methods for Hamiltonian wave equations. J. Comp. Phys. 57, 473.Google Scholar
Ruzmaikin, A. and Akhmetiev, P. 1994 Topological invariants of magnetic fields and the effect of reconnections. Phys. Plasmas 1 (2), 331–338.CrossRefGoogle Scholar
Sjöberg, A. and Mahomed, F. M. 2004. Non-local symmetries and conservation laws for one-dimensional gas dynamics equations. Appl. Math. Comput. 150, 379–397.Google Scholar
Tur, A. V. and Yanovsky, V. V. 1993. Invariants in dissipationless hydrodynamic media. J. Fluid Mech. 248, 67–106 (Cambridge Univ. Press).Google Scholar
Webb, G. M., Burrows, R. H., Ao, X. and Zank, G. P. 2014a Ion acoustic travelling waves. J. Plasma Phys. 80 (2), 147–171, doi:10.1017/S0022377813001013, preprint at http://arxiv.org/abs/1312.6406.Google Scholar
Webb, G. M., Dasgupta, B., McKenzie, J. F., Hu, Q. and Zank, G. P. 2014b Local and nonlocal advected invariants and helicities in magnetohydrodynamics and gas dynamics I: Lie dragging approach. J. Phys. Math. and Theoret. 47, 095501 (33 pp), doi:10.1088/1751-8113/49/9/095501, preprint available at http://arxiv.org/abs/1307.1105.Google Scholar
Webb, G. M., Dasgupta, B., McKenzie, J. F., Hu, Q. and Zank, G. P. 2014c Local and nonlocal advected invariants and helicities in magnetohydrodynamics and gas dynamics II: Noether's theorems and Casimirs. J. Phys. A., Math. Theoret. 47, 095502 (31 pp), doi:10.1088/1751-8113/47/9/095502, preprint available at http://arxiv.org/abs/1307.1038.Google Scholar
Webb, G. M., Hu, Q., McKenzie, J. F., Dasgupta, B. and Zank, G. P. 2014d Advected invariants in MHD and gas dynamics. In: Outstanding Problems in Heliophysics: from Coronal Heating to the Edge of the Heliosphere (eds. Zank, G. P. and Hu, Q.) 12th Annual International Astrophysics Conference, Astronomical Society of the Pacific Conf. Series, Vol. 484. Orem, UT: Astronomical Socieity of the Pacific, pp. 229–234.Google Scholar
Webb, G. M., Ko, C. M., Mace, R. L., McKenzie, J. F. and Zank, G. P. 2008, Integrable, oblique travelling waves in charge neutral, two-fluid plasmas. Nonlinear Proc. Geophys. 15, 179–208.Google Scholar
Webb, G. M. and Mace, R. L. 2014 Noether's theorems and fluid relabelling symmetries in magnetohydrodynamics and gas dynamics. J. Phys. AMath. Theoret. article No. JPHYSA-101-057, available at http://arxiv.org/abs/1403.3133 (submitted on March 10).Google Scholar
Webb, G. M., McKenzie, J. F., Mace, R. L., Ko, C. M. and Zank, G. P. 2007 Dual variational principles for nonlinear traveling waves in multifluid plasmas. Phys. Plasmas, 4 (8), 082318-082318-17, doi:10.1063/1.2757154.Google Scholar
Webb, G. M., Pogorelov, N. V. and Zank, G. P. 2010 MHD simple waves and the divergence wave. In: Twelfth International Solar Wind Conference, St. Malo, France, AIP Conference Proceedings, Vol. 1216. College Park, MD: AIP, pp. 300–303, doi:10.1063/1.3396300.Google Scholar
Webb, G. M. and Zank, G. P. 2009 Scaling symmetries, conservation laws and action principles in one-dimensional gas dynamics. J. Phys. Math. Theor. 42, 475205 (23 pp).Google Scholar
Webb, G. M., Zank, G. P., Kaghashvili, E. Kh and Ratkiewicz, R. E. 2005 Magnetohydrodynamic waves in non-uniform flows II: stress energy tensors, conservation laws and lie symmetries. J. Plasma Phys. 71, 811–857, doi: 10.1017/S00223778050003740.Google Scholar
Woltjer, L. 1958 On hydromagnetic equilibria. Proc. Nat. Acad. Sci. 44 (9), 833–841.Google Scholar
Zakharov, V. E. and Kuznetsov, E. A. 1997 Hamiltonian formalism for nonlinear waves. Phys.-Usp. 40 (11), 1087–1116.Google Scholar