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Magnetohydrodynamic wave mixing in shear flows: Hamiltonian equations and wave action

Published online by Cambridge University Press:  01 February 2007

G. M. WEBB ,
E. Kh. KAGHASHVILI  and
G. P. ZANK
G. M. WEBB
Affiliation:
Institute of Geophysics and Planetary Physics, University of California Riverside, Riverside, CA 92521, USA ([email protected])
E. Kh. KAGHASHVILI
Affiliation:
Institute of Geophysics and Planetary Physics, University of California Riverside, Riverside, CA 92521, USA ([email protected])
G. P. ZANK
Affiliation:
Institute of Geophysics and Planetary Physics, University of California Riverside, Riverside, CA 92521, USA ([email protected])
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Abstract.

Magnetohydrodynamic wave interactions in a linear shear flow are investigated using the Lagrangian fluid displacement ξ and entropy perturbation Δ S, in which a spatial Fourier solution is obtained in the frame of the background shear flow (Kelvin's method). The equations reduce to three coupled oscillator equations, with time-dependent coefficients and with source terms proportional to the entropy perturbation. In the absence of entropy perturbations, the system admits a wave action conservation integral consisting of positive and negative energy waves. Variational and Hamiltonian forms of the equations are obtained. Examples of wave amplification phenomena and sharp resonant-type wave interactions are obtained. Implications for the interaction of magnetohydrodynamic waves in the shear flow between fast, polar coronal-hole solar wind and slow, streamer belt solar wind are discussed.

Type
Papers
Information
Journal of Plasma Physics , Volume 73 , Issue 1 , February 2007 , pp. 15 - 68
Copyright
Copyright © Cambridge University Press 2006

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References

Abramowitz, M. and Stegun, I. A. 1965 Handbook of Mathematical Functions. New York: Dover.Google Scholar
Barut, A. O. 1967 The Theory of the Scattering Matrix. New York: MacMillan.Google Scholar
Bender, C. M. and Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. New York: McGraw-Hill.Google Scholar
Bernstein, I., Frieman, E. A., Kruskal, M. D. and Kulsrud, R. M. 1958 An energy principle for hydromagnetic stability problems. Proc. Roy. Soc. A 244, 17–40.Google Scholar
Bluman, G. W. and Kumei, S. 1989 Symmetries and Differential Equations. New York: Springer.CrossRefGoogle Scholar
Bodo, G., Poedts, S., Rogava, A. and Rossi, P. 2001 Spatial aspect of wave transformations in astrophysical flows. Astron. Astrophys. 374, 337–347.CrossRefGoogle Scholar
Breech, B., Matthaeus, W. H., Minnie, J., Oughton, S., Parhi, S., Bieber, J. W. and Bavassano, B. 2005 Radial evolution of cross helicity in the solar wind. Geophys. Res. Lett. 32, L06103, doi:1029/2004GL022321.CrossRefGoogle Scholar
Brio, M. and Wu, C. C. 1988 An upwind differencing scheme for the equations of ideal magnetohydrodynamics. J. Comput. Phys. 75, 400.CrossRefGoogle Scholar
Chagelishvili, G. D., Chanishvili, R. G., Lominadze, J. G. and Tevzadze, A. G. 1997 Magnetohydrodynamic waves linear evolution in parallel shear flows: amplification and mutual transformations. Phys. Plasmas 4 (2), 259–269.CrossRefGoogle Scholar
Dewar, R. L. 1970 Interaction between hydromagnetic waves and a time-dependent inhomogeneous medium. Phys. Fluids 13 (11), 2710–2720.CrossRefGoogle Scholar
Dewar, R. L. 1977 Energy-momentum tensors for dispersive electromagnetic waves. Aust. J. Phys. 30, 533–575.CrossRefGoogle Scholar
Ermakov, V. P. 1880, Univ. Izv. Kiev, 20, no. 9, 1.Google Scholar
Falle, S. A. E. G. and Hartquist, T. W. 2002 Generation of density inhomogeneities by magnetohydrodynamic waves. Mon. Not. Roy. Astron. Soc. 329, 195–203.CrossRefGoogle Scholar
Frieman, E. A. and Rotenberg, M. 1960. On hydromagnetic stability of stationary equilibria. Rev. Mod. Phys. 32 (4), 898–902.CrossRefGoogle Scholar
Gogoberidze, G., Chagelishvili, G. D., Sagdeev, R. Z. and Lominadze, D. G. 2004 Linear coupling and over-reflection phenomena of magnetohydrodynamic waves in smooth shear flows. Phys. Plasmas 11 (10), 4672–4685.CrossRefGoogle Scholar
Holm, D. D., Marsden, J. E., Ratiu, T. and Weinstein, A. 1985 Nonlinear stability of fluid and plasma equilibria. Phys. Reports 123 (1–2), 1–116.CrossRefGoogle Scholar
Kaghashvili, E. Kh. 1999 On the acceleration of the solar wind: role of inhomogeneous flow. Astrophys. J. 512, 969–974.CrossRefGoogle Scholar
Kaghashvili, E. Kh. 2002 Mode conversion and wave particle interaction processes in the solar wind. PhD thesis, University of New Hampshire, Durham, NH, USA.Google Scholar
Kelvin, Lord (Thomson, W.) 1887 Stability of fluid motion: rectilinear motion of viscous fluid between two parallel plates. Phil. Mag. 24 (5), 188–196.Google Scholar
Landau, L. D. and Lifshitz, E. M. 1965 Quantum Mechanics, Non-relativistic Theory. Course of Theoretical Physics, Vol. 3, Oxford: Pergamon Press.Google Scholar
Leach, P. G. L. 1991 Generalized Ermakov systems. Phys. Rev. Lett. 15, 102–106.CrossRefGoogle Scholar
Lundgren, T. S. 1963 Hamilton's variational principle for a perfectly conducting plasma continuum. Phys. Fluids 6, 898–904.CrossRefGoogle Scholar
Lutzky, M. 1978 Noether's theorem and the time-dependent harmonic oscillator. Phys. Lett. 68A, 3–4.CrossRefGoogle Scholar
Matthaeus, W. H., Minnie, J., Breech, B., Parhi, S. and Oughton, S. 2004 Transport of cross helicity and radial evolution of Alfvenicity in the solar wind. Geophys. Res. Lett. 31, L12803, doi:10.1029/2004GL019645.CrossRefGoogle Scholar
McComas, D. J., Riley, P., Gosling, J. T., Balogh, A. and Forsyth, R. 1998 Ulysses' rapid crossing of the polar coronal hole boundary. J. Geophys. Res. 103 (A2), 1955–1968.CrossRefGoogle Scholar
McKenzie, J. F. 1970 Hydromagnetic wave interaction with the magnetopause and the bow shock. Planet. Space Sci. 18, 1–23.CrossRefGoogle Scholar
Milano, L. J., Dasso, S., Matthaeus, W. H. and Smith, C. W. 2004 Spectral distribution of cross helicity in the solar wind. Phys. Rev. Lett. 93 (15), 155 005.CrossRefGoogle ScholarPubMed
Olver, P. J. 1993 Applications of Lie groups to Differential Equations, 2nd edn. New York: Springer.CrossRefGoogle Scholar
Poedts, S., Rogava, A. D. and Mahajan, S. M. 1998 Shear-flow-induced coupling in the solar wind. Astrophys. J. 505, 369–375.CrossRefGoogle Scholar
Rogava, A. D., Bodo, G., Massaglia, S. and Osmanov, Z. 2003 Amplification of MHD waves in swirling astrophysical flows. Astron. Astrophys. 408, 401–408, doi:10.1051/004-6361:20030454.CrossRefGoogle Scholar
Roe, P. L. and Balsara, D. 1996 Notes on the eigensystem of magnetohydrodynamics. SIAM J. Appl. Math. 56 (1), 57–67.CrossRefGoogle Scholar
Sagdeev, R. Z. and Galeev, A. A. 1969 Nonlinear Plasma Theory. New York: Benjamin.Google Scholar
Swanson, D. G. 1989 Plasma Waves. Boston: Academic Press.CrossRefGoogle Scholar
Swanson, D. G. 1998 Theory of Mode Conversion and Tunneling in Inhomogeneous Plasmas. New York: Wiley-Interscience.Google Scholar
Tracy, E. R. and Kaufman, A. N. 1993 Metaplectic formulation of linear mode conversion. Phys. Rev. E 48 (3), 2196–2211.CrossRefGoogle ScholarPubMed
Van der Holst, B., Nijober, R. J. and Goedbloed, J. P. 1999 Magnetohydrodynamic spectrum of gravitating plane plasmas with flow. J. Plasma Phys. 61 (2), 221–240.CrossRefGoogle Scholar
Walker, A. D. M. 2000 Reflection and transmission at the boundary between two counter-streaming MHD plasmas—active boundaries or negative-energy waves? J. Plasma Phys. 63 (3), 203–219.CrossRefGoogle Scholar
Webb, G. M., Kaghashvili, E. Kh. and Zank, G. P. 2005a MHD waves in solar wind shear flows. Proc. Solar Wind 11/Soho 16, Whistler, Canada, June 12–17, 2005.Google Scholar
Webb, G. M., Zank, G. P., Kaghashvili, E. Kh. and Ratkiewicz, R. E. 2005b Magneto-hydrodynamic waves in non-uniform flows I: a variational approach. J. Plasma Phys. 71 (6), 785–809, doi:10.1017/S00223778050003739.CrossRefGoogle Scholar
Webb, G. M., Zank, G. P., Kaghashvili, E. Kh. and Ratkiewicz, R. E. 2005c Magneto-hydrodynamic waves in non-uniform flows II: stress energy tensors, conservation laws and Lie symmetries. J. Plasma Phys. 71 (6), 811–857, doi:10.1017/S00223778050003740.CrossRefGoogle Scholar
Weinberg, S. 1995 The Quantum Theory of Fields 1. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Zank, G. P., Matthaeus, W. H. and Smith, C. W. 1996 Evolution of magnetic fluctuation power with heliospheric distance. J. Geophys. Res. 101, 17 081.Google Scholar