The stability of parallel-propagating circularly polarized Alfvén waves revisited

The parametric instability of parallel-propagating circularly polarized Alfvén 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 Alfvén 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_c 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 -> infty.