Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-19T00:38:48.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Alfvén simple waves

Published online by Cambridge University Press:  04 February 2010

G. M. WEBB ,
G. P. ZANK ,
R. H. BURROWS  and
R. E. RATKIEWICZ
G. M. WEBB
Affiliation:
CSPAR, The University of Alabama in Huntsville, Huntsville, AL 35805, USA (gary.webb@uah.edu)
G. P. ZANK
Affiliation:
CSPAR, The University of Alabama in Huntsville, Huntsville, AL 35805, USA (gary.webb@uah.edu) Department of Physics, The University of Alabama in Huntsville, Huntsville, AL 35899, USA
R. H. BURROWS
Affiliation:
CSPAR, The University of Alabama in Huntsville, Huntsville, AL 35805, USA (gary.webb@uah.edu)
R. E. RATKIEWICZ
Affiliation:
Space Research Center, Bartycka 18A, 00-716 Warsaw, Poland
Get access Rights & Permissions [Opens in a new window]

Abstract

Multi-dimensional Alfvén simple waves in magnetohydrodynamics (MHD) are investigated using Boillat's formalism. For simple wave solutions, all physical variables (the gas density, pressure, fluid velocity, entropy, and magnetic field induction in the MHD case) depend on a single phase function ϕ, which is a function of the space and time variables. The simple wave ansatz requires that the wave normal and the normal speed of the wave front depend only on the phase function ϕ. This leads to an implicit equation for the phase function and a generalization of the concept of a plane wave. We obtain examples of Alfvén simple waves, based on the right eigenvector solutions for the Alfvén mode. The Alfvén mode solutions have six integrals, namely that the entropy, density, magnetic pressure, and the group velocity (the sum of the Alfvén and fluid velocity) are constant throughout the wave. The eigenequations require that the rate of change of the magnetic induction B with ϕ throughout the wave is perpendicular to both the wave normal n and B. Methods to construct simple wave solutions based on specifying either a solution ansatz for n(ϕ) or B(ϕ) are developed.

Type
Papers
Information
Journal of Plasma Physics , Volume 77 , Issue 1 , February 2011 , pp. 51 - 93
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. and Stegun, I. A. 1965 Handbook of Mathematical Functions. New York: Dover.Google Scholar
Barnes, A. 1976 On the non-existence of plane polarized large amplitude Alfvén waves. J. Geophys. Res. 81 (1), 281–282.CrossRefGoogle Scholar
Berger, M. and Field, G. B. 1984 The topological properties of magnetic helicity. J. Fluid Mech. 147, 133–148.CrossRefGoogle Scholar
Berger, M. A. and Prior, C. 2006 The writhe of open and closed curves. J. Phys. A.: Math. Gen. 39, 8321–8348.CrossRefGoogle Scholar
Bishop, R. L. 1975 There is more than one way to frame a curve. Am. Math. Mon. 82, 246–251.CrossRefGoogle Scholar
Boillat, G. 1970 Simple waves in N-dimensional propagation. J. Math. Phys. 11, 1482–1483.CrossRefGoogle Scholar
Brio, M. and Wu, C. C. 1988 An upwind differencing scheme for the equations of magnetohydrodynamics. J. Comput. Phys. 75, 400–422.CrossRefGoogle Scholar
Cabannes, H. 1970 Theoretical Magnetohydrodynamics. New York: Academic Press.Google 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
Goriely, A. and Tabor, M. 1997 Nonlinear dynamics of filaments I: dynamical instabilities. Physica D 105, 45–61.CrossRefGoogle Scholar
Holm, D. D., Marsden, J. E. and Ratiu, T. S. 1998 The Euler–Lagrange equations and semiproducts with application to continuum theories. Adv. Math. 137, 1–81.CrossRefGoogle Scholar
Janhunen, P. 2000 A positive conservative method for magnetohydrodynamics based on HLL and Roe methods. J. Comput. Phys. 160, 649–661.CrossRefGoogle Scholar
Jeffrey, A. and Taniuti, T. 1964 Nonlinear Wave Propagation with Application to Physics, and Magnetohydrodynamics. New York: Academic Press.Google Scholar
Lipschutz, M. M. 1969 Theory and Problems of Differential Geometry. Schaum's Outline Series. New York: McGraw-Hill.Google Scholar
Marsden, J. E. and Ratiu, T. S. 1994 Introduction to Mechanics and Symmetry (Texts in Applied Mathematics, 17). New York: Springer.Google Scholar
Nadjafikhah, M. and Mahdipour-Shirayeh, A. 2009 Symmetry analysis for a new form of the vortex mode equation. arXiv:0905.0175v1 [math.DG]. Differential Geometry Dynamical Systems, 11, 144–154.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.CrossRefGoogle Scholar
Rajee, L., Eshraghi, H. and Popovych, R. O. 2008, Multi-dimensional quasi-simple waves in weakly dissipative flows. Physica D 237, 405–419.CrossRefGoogle Scholar
Rajee, L. and Eshraghi, H. 2009 Multi-dimensional vortex quasi-simple waves. Physica D 238, 477–489.CrossRefGoogle Scholar
Roe, P. L. and Balsara, D. 1996 Notes on the eigensystem of magnetohydrodynamics. SIAM J. Appl. Math. 56, 57–67.CrossRefGoogle Scholar
Sahihi, T., Eshraghi, H. and Mahdipour-Shirayeh, A. 2008 Multi-dimensional simple waves in fully relativistic fluids. arXiv:0811.2307v1 [physics.flu-dyn].Google Scholar
Sneddon, I. N. 1957 Elements of Partial Differential Equations. New York: McGraw-Hill.Google Scholar
Webb, G. M., Ratkiewicz, R., Brio, M. and Zank, G. P. 1995 Multi-dimensional MHD simple waves. In Solar Wind, Vol. 8 (ed. Winterhalter, D., Gosling, J. T., Habbal, S. R., Kurth, W. S. and Neugebauer, M.), pp. 335–338; AIP Conference Proceedings, Vol. 382. New York: AIP.Google Scholar
Webb, G. M., Ratkiewicz, R., Brio, M. and Zank, G. P. 1998 Multidimensional simple waves in gas dynamics. J. Plasma Phys. 59, 417–460.CrossRefGoogle Scholar
Webb, G. M., Pogorelov, N. V. and Zank, G. P. 2009 MHD simple waves and the divergence wave. Solar Wind 12, 4, held in St. Malo France, June 21–26, 2009.Google Scholar