Ruderman, M.S. and Simpson, D. (2004) The stability of parallelpropagating circularly polarized Alfvén waves revisited. Journal of Plasma Physics, 70 (2). pp. 143153. ISSN 00223778

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Abstract
The parametric instability of parallelpropagating circularly polarized Alfven waves (pump waves) is revisited. The stability of these waves is determined by the linearized system of magnetohydrodynamic equations with periodic coefficients. The variable substitution that reduces this system of equations to a system with constant coefficients is suggested. The system with constant coefficients is used to derive the dispersion equation that was previously derived by many authors with the use of different approaches. The dependences of general stability properties on the dimensionless amplitude of the pump wave a and the ratio of the sound and Alfven speed b are studied analytically. It is shown that, for any a and b, there are such quantities k(1) and k(2) that a perturbation with the dimensionless wavenumber k is unstable if k(1)(2) < k(2) < k(2)(2), and stable otherwise. It is proved that, for any fixed b, k(2) is a monotonically growing function of a. The dependence of k(1) on a is different for different values of b. When b(2) < 1/3, k(1) is a monotonically decreasing function of a. When 1/3 < b(2) < 1, k(1) monotonically decreases when a varies from zero to a(c)(b), takes its minimum value at a = a(c) and then monotonically increases when a increases from a, to infinity. When b > 1, k(1) is a monotonically increasing function of a. For any b, k(1) tends to a limiting value approximately equal to 1.18 as a > infinity.
Item Type:  Article 

Copyright, Publisher and Additional Information:  © 2004 Cambridge University Press. Reproduced in accordance with the publisher's selfarchiving policy. 
Institution:  The University of Sheffield 
Academic Units:  The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) 
Depositing User:  Repository Assistant 
Date Deposited:  25 Oct 2006 
Last Modified:  05 Jun 2014 23:41 
Published Version:  http://dx.doi.org/10.1017/S0022377803002599 
Status:  Published 
Publisher:  Cambridge University Press 
Refereed:  Yes 
Identification Number:  10.1017/S0022377803002599 
URI:  http://eprints.whiterose.ac.uk/id/eprint/1577 