The stability of parallel-propagating circularly polarized Alfvén waves revisited

The parametric instability of parallel-propagating 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.