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The Model

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Magnetohydrodynamics and Fluid Dynamics: Action Principles and Conservation Laws

Part of the book series: Lecture Notes in Physics ((LNP,volume 946))

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Abstract

The magnetohydrodynamic equations are:

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References

  • Anco, S.C., Bluman, G.: Direct Construction of Conservation Laws from Field Equations. Phys. Rev. Lett. 78(15), 2869–2873 (1997)

    Google Scholar 

  • Berger, M.A., Prior, C.: The Writhe of Open and Closed Curves. J. Phys. A Math. Gen. 39, 8321–8348 (2006)

    Google Scholar 

  • Bieber, J.W., Evenson, P.A., Matthaeus, W.H.: Magnetic Helicity of the Parker Field. Astrophys. J. 315, 700 (1987)

    Google Scholar 

  • Bluman, G.W., Cheviakov, A.F., Anco, S.C.: Applications of Symmetry Methods to Partial Differential Equations. Applied Mathematical Sciences Series 168. Springer, New York (2010)

    Google Scholar 

  • Bruno, R., Carbone, V., Veltri, P., Pieropaolo, E., Bavasonno, B.: Identifying Intermittency Effects in the Solar Wind. Planet. Space Sci. 49, 1201 (2001)

    Article  ADS  Google Scholar 

  • Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. Oxford University Press/Clarendon Press, Oxford (1961)

    MATH  Google Scholar 

  • Chandre, C., Morrison, P.J., Tassi, E.: On the Hamiltonian Formulation of Incompressible Ideal Fluids and Magnetohydrodynamics via Dirac’s Theory of Constraints. Phys. Lett. A 376, 737–743 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Gosling, J.T., McComas, D.J., Roberts, D.A., Skoug, R.M.: A One Sided Aspect of Alfvénic Fluctuations in the Solar Wind. Astrophys. J. 695, L213–L216 (2009)

    Article  ADS  Google Scholar 

  • Holm, D.D.: Geometric Mechanics, Part I: Dynamics and Symmetry. Imperial College Press, London (2008a). Distributed by World Scientific

    Google Scholar 

  • Holm, D.D.: Geometric Mechanics, Part II: Rotating, Translating and Rolling. Imperial College Press, London (2008b). Distributed by World Scientific

    Google Scholar 

  • Janhunen, P.: A Positive Conservative Method for Magnetohydrodynamics Based on HLL and Roe Methods. J. Comput. Phys. 160, 649–661 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Kadomtsev, B.B., Pogutse, O.P.: Nonlinear Helical Perturbations of a Plasma in the Tokamak. J. Exp. Theor. Phys. 38, 283–290 (1974)

    ADS  Google Scholar 

  • Kamchatnov, I.V.: Topological Soliton in Magnetohydrodynamics. Sov. Phys. 55(1), 69–73 (1982)

    MathSciNet  Google Scholar 

  • Kuznetsov, E.A.: Vortex Line Representation for Hydrodynamic Type Equations. J. Nonlinear Math. Phys. 13(1), 64–80 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Kuznetsov, E.A., Ruban, V.P.: Hamiltonian Dynamics of Vortex Lines for Systems of Hydrodynamic Type. J. Exp. Theor. Phys. Lett. 67, 1076 (1998)

    Article  Google Scholar 

  • Kuznetsov, E.A., Ruban, V.P.: Hamiltonian Dynamics of Vortex and Magnetic Field Lines in the Hydrodynamic Type Models. Phys. Rev. E 61, 831–841 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  • Kuznetsov, E.A., Passot, T., Sulem, P.L.: Compressible Dynamics of Magnetic Field Lines for Incompressible MHD Flows. Phys. Plasmas 11, 1410 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  • Low, B.C.: Three-Dimensional Structures of Magnetostatic Atmospheres. Astrophys. J. 293, 31–43 (1985)

    Article  ADS  Google Scholar 

  • Matteini, L., Horbury, T.S., Pantellini, F., Velli, M., Schwartz, S.J.: Ion Kinetic Energy Conservation and Magnetic Field Strength Constancy in Multi-Fluid Solar Wind and Alfvénic Turbulence. Astrophys. J. 802(11), 4 pp. (2015)

    Google Scholar 

  • Morrison, P.J., Greene, J.M.: Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics (Errata). Phys. Rev. Lett. 48, 569 (1982)

    Article  ADS  Google Scholar 

  • Morrison, P.J., Hazeltine, R.D.: Hamiltonian Formulation of Reduced Magnetohydrodynamics. Phys. Fluids 27(4), 886–897 (1984)

    Article  ADS  MATH  Google Scholar 

  • Newcomb, W.A.: Motion of Magnetic Lines of Force. Ann. Phys. N. Y. 3, 347 (1958)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Oughton, S., Matthaeus, W.H., Dmitruk, P.: Reduced MHD in Astrophysical Applications: Two Dimensional or Three Dimensional? Astrophys. J. 839(2), 13 pp. (2017)

    Google Scholar 

  • Panofsky, W.K.H., Phillips, M.: Classical Electricity and Electromagnetism, sect. 9.4, 2nd edn., p. 164. Wesley, Reading (1964)

    Google Scholar 

  • Parker, E.N.: Cosmic Magnetic Fields. Oxford University Press, New York (1979)

    Google Scholar 

  • Pedlosky, J.: Geophysical Fluid Dynamics, 2nd edn., p. 710. Springer, New York (1987)

    Google Scholar 

  • Powell, K.G., Roe, P.L., Linde, T.J., Gombosi, T.I., De Zeeuw, D.: A Solution Adaptive Upwind Scheme for Ideal Magnetohydrodynamics. J. Comput. Phys. 154, 284–309 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Prim, R., Truesdell, C.: A Derivation of Zorawski’s Criterion for Permanent Vector Lines. Proc. Am. Math. Soc. 1, 32–34 (1950)

    MathSciNet  MATH  Google Scholar 

  • Pshenitsin, D.: Conservation laws of magnetohydrodynamics and their symmetry transformation properties. Ph.D. Thesis, Department of Physics, Brock University, Saint Catharines (2016). Available at https://dr.library.ca.handle/10464/9801

  • Sagdeev, R.Z., Moiseev, S.S., Tur, A.V., Yanovsky, V.: Problems of the Theory of Strong Turbulence and Topological Solitons. In: Sagdeev, R.Z. (ed.) Nonlinear Phenomena in Plasma Physics and Hydrodynamics, pp. 137–182. Mir, Moscow (1986)

    Google Scholar 

  • Semenov, V.S., Korvinski, D.B., Biernat, H.K.: Euler Potentials for the MHD Kamchatnov-Hopf Soliton Solution. Nonlinear Process. Geophys. 9, 347–354 (2002)

    Article  ADS  Google Scholar 

  • Stern, D.P.: The Motion of Magnetic Field Lines. Space Sci. Rev. 6, 143–173 (1966)

    Article  ADS  Google Scholar 

  • Strauss, H.R.: Nonlinear Three Dimensional Magnetohydrodynamics of Non-circular Tokamaks. Phys. fluids 19, 134–140 (1976)

    Article  ADS  Google Scholar 

  • Taylor, J.B.: Relaxation of Toroidal Plasma and Generation of Reverse Magnetic Fields. Phys. Rev. Lett. 33, 1139–1141 (1974)

    Article  ADS  Google Scholar 

  • Taylor, J.B.: Relaxation and Magnetic Reconnection in Plasmas. Rev. Mod. Phys. 58, 741–763 (1986)

    Article  ADS  Google Scholar 

  • Thompson, A., Sweargin, J., Wickes, A., Bouwmeester, D.: Constructing a Class of Topological Solitons in Magnetohydrodynamics. Phys. Rev. E 89, 043104 (2014)

    Article  ADS  Google Scholar 

  • Torok, T., Kliem, B., Berger, M.A., Linton, M.G., Demoulin, P., Van Driel-Gesztelyi, L.: The Evolution of Writhe in Kink-Unstable Flux Ropes and Erupting Filaments. Plasma Phys. Control. Fusion 56, 064012 (7 pp.) (2014)

    Google Scholar 

  • Truesdell, C.: The Kinematics of Vorticity. Indiana University Press, Bloomington (1954)

    MATH  Google Scholar 

  • Truesdell, C., Toupin, R.A.: In: Flugge, S. (ed.) The Classical Field Theories. Handbuch der Physik III/I, pp. 226–793. Springer, Heidelberg (1960)

    Google Scholar 

  • Webb, G.M., Axford, W.I., Terasawa, T.: On the Drift Mechanism for Energetic Charged Particles at Shocks. Astrophys. J. 270, 537–553 (1983)

    Article  ADS  Google Scholar 

  • Webb, G.M., Woodward, T.I., Brio, M., Zank, G.P.: Linear Magnetosonic N-waves and Green’s Functions. J. Plasma Phys. 49(Part 3), 465–513 (1993)

    Google Scholar 

  • Webb, G.M., Pogorelov, N.V., Zank, G.P.: MHD Simple Waves and the Divergence Wave. In: Twelfth International Solar Wind Conference, St. Malo. AIP Conference Proceedings 1216, pp. 300–303 (2009). https://doi.org/10.1063/1.3396300

    ADS  Google Scholar 

  • Webb, G.M., Hu, Q., Dasgupta, B., Zank, G.P.: Homotopy Formulas for the Magnetic Vector Potential and Magnetic Helicity: The Parker Spiral Interplanetary Magnetic Field and Magnetic Flux Ropes. J. Geophys. Res. (Space Phys.) 115, A10112 (2010a). https://doi.org/10.1029/2010JA015513. Corrections: J. Geophys. Res. 116, A11102 (2011). https://doi.org/10.1029/2011JA017286

  • Webb, G.M., Hu, Q., Dasgupta, B., Roberts, D.A., Zank, G.P.: Alfven Simple Waves: Euler Potentials and Magnetic Helicity. Astrophys. J. 725, 2128–2151 (2010b). https://doi.org/10.1088/0004-637X/725/2/2128

    Article  ADS  Google Scholar 

  • Whitham, G.B.: Linear and Nonlinear Waves. New York, Wiley (1974)

    MATH  Google Scholar 

  • Zank, G.P., Matthaeus, W.H.: The Equations of Reduced Magnetohydrodynamics. J. Plasma Phys. 48(1), 85–100 (1992)

    Article  ADS  Google Scholar 

  • Zorawski, K.: Uber die Erhaltung der Wirbelbewegung. Bull. Acad. Sci. Cracov. C. R. 335 (1900)

    Google Scholar 

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Authors and Affiliations

  1. CSPAR, The University of Alabama in Huntsville, Huntsville, Alabama, USA

    Gary Webb

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Webb, G. (2018). The Model. In: Magnetohydrodynamics and Fluid Dynamics: Action Principles and Conservation Laws. Lecture Notes in Physics, vol 946. Springer, Cham. https://doi.org/10.1007/978-3-319-72511-6_2

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