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Potential vorticity in magnetohydrodynamics

Published online by Cambridge University Press:  05 September 2014

G. M. Webb  and
R. L. Mace
G. M. Webb*
Affiliation:
Center for Space Plasma and Aeronomic Research, The University of Alabama in Huntsville, Huntsville AL 35805, USA
R. L. Mace
Affiliation:
School of Chemistry and Physics, The University of KwaZulu-Natal, Durban Westville, Durban, Natal 4000, South Africa
*
†Email address for correspondence: gmw0002@uah.edu
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Abstract

A version of Noether's second theorem using Lagrange multipliers is used to investigate fluid relabelling symmetries conservation laws in magnetohydrodynamics (MHD). We obtain a new generalized potential vorticity type conservation equation for MHD which takes into account entropy gradients and the J × B force on the plasma due to the current J and magnetic induction B. This new conservation law for MHD is derived by using Noether's second theorem in conjunction with a class of fluid relabelling symmetries in which the symmetry generator for the Lagrange label transformations is non-parallel to the magnetic field induction in Lagrange label space. This is associated with an Abelian Lie pseudo algebra and a foliated phase space in Lagrange label space. It contains as a special case Ertel's theorem in ideal fluid mechanics. An independent derivation shows that the new conservation law is also valid for more general physical situations.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 1 , January 2015 , 905810115
Copyright
Copyright © Cambridge University Press 2014 

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References

REFERENCES

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: Springer.Google Scholar
Bluman, G. W. and Kumei, S. 1989 Symmetries and Differential Equations. New York: Springer.Google Scholar
Cheviakov, A. F. 2014 Conservation properties and potential systems of vorticity-type equations. J. Math. Phys. 55, 033508 (16pp) (0022-2488/2014/55(3)/033508/16).CrossRefGoogle Scholar
Cotter, C. J. and Holm, D.D. 2013 On Noether's theorem for Euler Poincaré equation on the diffeomorphism group with advected quantities. Found. Comput. Math. 13, 457–477, ISSN:1615-3375.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
Flanders, H. 1963 Differential Forms with Applications to the Physical Sciences. New York: Academic.Google Scholar
Fuchs, J. C. 1991 Symmetry groups of similarity solutions of the MHD equations. J. Math. Phys. 32, 1703–1708.CrossRefGoogle Scholar
Grundland, A. M. and Lalague, L. 1995 Lie subgroups of fluid dynamics and magnetohydrodynamics equations. Canad. J. Phys. 73, 463–477.CrossRefGoogle 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.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. and Mansfield, E. L. 2011 Extensions of Noether's second theorem: From continuous to discrete systems. Proc. Roy. Soc. A, 467 (2135), 3206–3221.CrossRefGoogle Scholar
Ibragimov, N. H. 1985 Transformation Groups Applied to Mathematical Physics Reidel: Dordrecht.CrossRefGoogle Scholar
Ilgisnos, V. I. and Pastukhov, V. P. 2000 Variational approaches to problems of plasma stability and of nonlinear plasma dynamics. JETP Lett. 72 (10), 530–540.Google Scholar
Ivey, T. A. and Landsberg, J. M. 2003 Cartan for beginners: Differential geometry via moving frames and exterior differential systems. In AMS, Graduate Studies in Mathematics, Vol. 61 (eds. Craig, W., Ivanov, N., Krantz, S.G. and Saltman, D.) American Mathematical Society, Providence Rhode Island.Google Scholar
Kats, A. V. 2003 Variational principle in canonical variables, Weber transformation and complete set of local integrals of motion for dissipation-free magnetohydrodynamics. JETP Lett. 77 (12), 657–661.Google Scholar
Kelbin, O., Cheviakov, A. F. and Oberlack, M. 2014 New conservation laws of helically symmetric, plane and rotationally symmetric viscous and inviscid flows J. Fluid Mech. 721 (4), 340–366.Google Scholar
Kuvshinov, B. N. and Schep, T. J. 1997 Geometrical approach to fluid models. Phys. Plasmas 4 (3), 537–550, http://dx.doi.org/10.1063/1.872576.Google Scholar
Moffatt, H. K. 1978 Magnetic field Generation in Electrically Conducting Fluids. Cambridge U.K.: Cambridge University Press.Google Scholar
Moffatt, H. K. and Ricca, R. L. 1992 Helicity and the Calugareanu invariant. Proc. Roy. Soc. London, Ser. A 439, 411.Google Scholar
Moiseev, S. S., Sagdeev, R. Z., Tur, A. V. and Yanovskii, V. V. 1982 On the freezing-in integrals and Lagrange invariants in hydrodynamic models. Sov. Phys. JETP 56 (1), 117–123.Google Scholar
Newcomb, W. A. 1962 Lagrangian and Hamiltonian methods in magnetohydrodynamics Nucl. Fusion Suppl. 2, 451–463.Google Scholar
Noether, E. 1918 Invariante variations probleme, Nachr. König. Gessell. Wissen. Göttingen, Mathphys. Kl., 235257 (1918). See Transp. Theory Stat. Phys. 1, 186207 (1971) for an English translation, which is also posted at physics/0503066.Google Scholar
Olver, P. J. 1993 Applications of Lie Groups to Differential Equations, 2nd edn. New York: Springer.CrossRefGoogle Scholar
Padhye, N. S. 1998 Topics in Lagrangian and Hamiltonian Fluid Dynamics: Relabeling Symmetry and Ion Acoustic Wave Stability, Ph. D. Dissertation, University of Texas at Austin.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
Parker, E. N. 1979 Cosmic Magnetic Fields. New York: Oxford University Press.Google Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics, 2nd edn. New York: Springer.Google Scholar
Salmon, R. 1982 Hamilton's principle and Ertel's theorem. AIP Conf. Proc. 88, 127–135.Google Scholar
Salmon, R. 1988 Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech. 20, 225–256.Google Scholar
Stern, D. P. 1966 The motion of magnetic field lines. Space Sci. Rev. 6, 147–173.Google Scholar
Tur, A. V. and Yanovsky, V. V. 1993 Invariants in dissipationless hydrodynamic media. J. Fluid Mech. 248, Cambridge Univ. Press, 67–106.CrossRefGoogle Scholar
Volkov, D. V., Tur, A. V. and Yanovsky, V. V. 1995 Hidden supersymmetry of classical systems (hydrodynamics and conservation laws). Phys. Lett. A 203, 357–361.CrossRefGoogle Scholar
Webb, G. M., Dasgupta, B., McKenzie, J. F., Hu, Q. and Zank, G. P. 2014a Local and non-local advected invariants and helicities in MHD and gas dynamics I: Lie dragging approach. J. Phys. A, Math. Theor. 47 (2014) 095501 (33pp) a preprint version is available on arxiv at http://arxiv.org/abs/1307.1105.Google Scholar
Webb, G. M., Dasgupta, B., McKenzie, J. F., Hu, Q. and Zank, G. P. 2014b Local and nonl-ocal advected invariants and helicities in MHD and gas dynamics II: Noether's theorems and Casimirs, J. Phys. A, Math. Theor. 47 (2014) 095502 (31pp) preprint available at http://arxiv.org/abs/1307.1038 Google Scholar
Webb, G. M., McKenzie, J. F. and Zank, G. P. 2014c Multi-symplectic magnetohydrodynamics. J. Plasma Phys. firstview article at JPP website (since April 24, 2014), preprint available at http://arxiv.org/abs/1312.4890.Google Scholar
Webb, G. M. and Zank, G. P. 2007 Fluid relabelling symmetries, Lie point symmetries and the Lagrangian map in magnetohydrodynamics and gas dynamics. J. Phys. A., Math. Theor. 40, 545–579, doi:10.1088/1751-8113/40/3/013.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