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Double Alfvén waves

Published online by Cambridge University Press:  17 October 2011

G. M. WEBB ,
Q. HU ,
B. DASGUPTA  and
G. P. ZANK
G. M. WEBB
Affiliation:
CSPAR, The University of Alabama in Huntsville, Huntsville AL 35805, USA ([email protected])
Q. HU
Affiliation:
CSPAR, The University of Alabama in Huntsville, Huntsville AL 35805, USA ([email protected])
B. DASGUPTA
Affiliation:
CSPAR, The University of Alabama in Huntsville, Huntsville AL 35805, USA ([email protected])
G. P. ZANK
Affiliation:
CSPAR, The University of Alabama in Huntsville, Huntsville AL 35805, USA ([email protected]) Department of Physics, The University of Alabama in Huntsville, Huntsville AL 35899, USA
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Abstract

Double Alfvén wave solutions of the magnetohydrodynamic equations in which the physical variables (the gas density ρ, fluid velocity u, gas pressure p, and magnetic field induction B) depend only on two independent wave phases ϕ1(x,t) and ϕ2(x,t) are obtained. The integrals for the double Alfvén wave are the same as for simple waves, namely, the gas pressure, magnetic pressure, and group velocity of the wave are constant. Compatibility conditions on the evolution of the magnetic field B due to changes in ϕ1 and ϕ2, as well as constraints due to Gauss's law ∇ · B = 0 are discussed. The magnetic field lines and hodographs of B in which the tip of the magnetic field B moves on the sphere |B| = B = const. are used to delineate the physical characteristics of the wave. Hamilton's equations for the simple Alfvén wave with wave normal n(ϕ), and with magnetic induction B(ϕ) in which ϕ is the wave phase, are obtained by using the Frenet–Serret equations for curves x=X(ϕ) in differential geometry. The use of differential geometry of 2D surfaces in a 3D Euclidean space to describe double Alfvén waves is briefly discussed.

Type
Papers
Information
Journal of Plasma Physics , Volume 78 , Issue 1 , February 2012 , pp. 71 - 85
Copyright
Copyright © Cambridge University Press 2011

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References

Barnes, A. 1976 On the non-existence of plane polarized large amplitude Alfvén waves. J. Geophys. Res. 81 (1), 281–282.CrossRefGoogle Scholar
Barnes, A. 1981 Interplanetary Alfvénic fluctuations: a stochastic model. J. Geophys. Res. 86 (A9), 7498–7506.CrossRefGoogle Scholar
Barnes, A. and Hollweg, J. V. 1974 Large amplitude hydromagnetic waves. J. Geophys. Res. 79, 2302–2318.CrossRefGoogle Scholar
Breech, B., Matthaeus, W. H., Minnie, J., Bieber, J. W., Oughton, S., Smith, C. W. and Isenberg, P. A. 2008 Turbulence transport throughout the heliosphere. J. Geophys. Res. 113, A08105. doi:10.1029/2007JA012711.Google Scholar
Bruno, R. and Carbone, V. 2005 The solar wind as a turbulence laboratory, In Living Reviews in Space Physics. http:/www.livingreviews.org/lrsp-2005-4CrossRefGoogle Scholar
Bruno, R., Carbone, V., Veltri, P., Pietropaolo, E. and Bavassano, B. 2001 Identifying intermittency events in the solar wind. Planet. Space Sci. 49, 1201–1210.CrossRefGoogle Scholar
Chen, L. and Hasegawa, A. 1974 Plasma heating by spatial resonance of Alfvén waves. Phys. Fluids 17 (7), 1399–1403.CrossRefGoogle Scholar
Eisenhart, L. P. 1960 A Treatise on the Differential Geometry of Curves and Surfaces. New York: Dover (re-publication of 1909 edition by Ginn and Co.).Google Scholar
Flanders, H. 1963 Differential Forms with Applications to the Physical Sciences. New York: Academic.Google Scholar
Frankel, T. 1997 The Geometry of Physics, An Introduction. Cambridge, UK: Cambridge University Press, p. 161, formula (5.6).Google Scholar
Gekelman, W., Vicenza, S., VanCompernolle, B. Compernolle, B., Morales, G. J., Maggs, J. E., Prybyl, P. and Carter, T. A. 2011 The many faces of shear Alfvén waves. Phys. Plasmas 18, 08511, doi:10.1063/1.3592210.CrossRefGoogle Scholar
Gosling, J. T., McComas, D. J., Roberts, D. A. and Skoug, R. M. 2009 A one-sided aspect of Alfvénic fluctuations in the solar wind. Astrophys. J. 695, L213–L216.CrossRefGoogle Scholar
Gosling, J. T., Teh, W. L. and Eriksson, S. 2010(August 10) A torsional Alfvén wave embedded within a small magnetic flux rope in the solar wind. Ap. J. Lett. 719, L36–L40, doi:10.1088/2041-8205/719/1/L36.Google Scholar
Grundland, A. M. and Huard, B. 2007 Conditional symmetries and Riemann invariants for hyperbolic systems of PDEs. J. Phys. A: Math. Theor. 40, 4093–4123, doi:10.1088/1751-8113/40/15/004.CrossRefGoogle Scholar
Grundland, A. M. and Picard, P. 2004 On conditionally invariant solutions of magnetohydrodynamic equations multiple waves. J. Nonlinear Math. Phys. 11 (1), 47–74.CrossRefGoogle Scholar
Hasegawa, A. and Chen, L. 1975 Kinetic process of plasma heating due to Alfvén wave excitation. Phys. Rev. Lett. 35 (6), 370–373.CrossRefGoogle Scholar
Heinemann, M. and Olbert, S. 1980 Non-WKB Alfvén waves in the solar wind. J. Geophys. Res. 85, 1311.CrossRefGoogle Scholar
Lipschutz, M. 1969 Theory and Problems of Differential Geometry (Schaum Outline Series). New York: McGraw Hill.Google Scholar
MacGregor, K. B. and Charbonneau, P. 1994 Stellar winds with non-WKB Alfvén waves I: wind models for solar coronal conditions. Astrophys. J. 430, 387–398.CrossRefGoogle Scholar
Matthaeus, W. H. and Goldstein, M. L. 1982 Measurement of the rugged invariants of magnetohydrodynamic turbulence in the solar wind. J. Geophys. Res. 87 (A8), 6011–6028.CrossRefGoogle Scholar
Matthaeus, W. H., Oughton, S., Pontius, D. H. Jr and Zhou, Y. 1994 Evolution of energy containing eddies in the solar wind. J. Geophys. Res. A10, 19267–19287.CrossRefGoogle Scholar
Morrison, P. J. 1998 Poisson brackets for fluids and plasmas. Rev. Mod. Phys. 70, 467–521.CrossRefGoogle Scholar
Padhye, N. S. 1998 Topics in Lagrangian and Hamiltonian fluid dynamics: relabeling symmetry and ion acoustic wave stability. PhD Dissertation, University of Texas at Austin.Google Scholar
Roberts, D. A., Klein, L. W., Goldstein, M. L. and Matthaeus, W. H. 1987 The nature and evolution of magnetohydrodynamic fluctuations in the solar wind: Voyager observations. J. Geophy. Res. 92, 11021.CrossRefGoogle Scholar
Rogers, C. and Schief, W. K. 2002 Bäcklund and Darboux Transformations, Geometry and Modern Apllications in Soliton Theory. Cambridge UK: Cambridge University Press.CrossRefGoogle Scholar
Struick, D. J. 1961 Lectures on Classical. Differential Geometry, 2nd edn. Reading, MA: Addison-Wesley.Google Scholar
Webb, G. M., Ratkiewicz, R., Brio, M. and Zank, G. P. 1996 Multi-dimensional MHD simple waves. In: Solar Wind 8, AIP Conf. Proc., Vol. 382 (ed. Winterhalter, D., Gosling, J. T., Habbal, S. R., Kurth, W. S. and Neugebauer, M.). New York: AIP, pp. 335–338.CrossRefGoogle 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.CrossRefGoogle Scholar
Webb, G. M., Hu, Q., Dasgupta, B., Roberts, D. A. and Zank, G. P. 2010 Alfven simple waves: Euler potentials and magnetic helicity. Astrophys. J. 725, 2128–2151, doi:10.1088/0004-637X/725/2/2128.CrossRefGoogle Scholar
Webb, G. M., Zank, G. P., Burrows, R. H. and Ratkiewicz, R. E. 2011 Simple Alfvén waves. J. Plasma Phys. 77 (part 1), 51–93, doi:10.101/S00233377809990596.CrossRefGoogle Scholar
Zank, G. P., Hunana, P., Shaikh, D., Florinski, V. and Webb, G. M. 2010 The transport of low-frequency turbulence throughout the heliosphere. AIP Proc. Conf. 1302, 167–173.CrossRefGoogle Scholar
Zank, G. P., Matthaeus, W. H. and Smith, C. W. 1996 Evolution of turbulent magnetic fluctuation power with heliocentric distance. J. Geophys. Res. 101, 17801.Google Scholar
Zhou, Y. and Matthaeus, W. H. 1990a Transport and turbulence modeling of solar wind fluctuations. J. Geophys. Res. 95, 10291.CrossRefGoogle Scholar
Zhou, Y. and Matthaeus, W. H. 1990b Models of inertial range spectra of interplanetary magnetohydromagnetic turbulence. J. Geophys. Res. 95 (A9), 14881–14892.CrossRefGoogle Scholar