Self-similar evolution of Alfven wave turbulence

We study self-similar solutions of the kinetic equation for MHD wave turbulence derived in (Galtier S et al 2000 J. Plasma Phys. 63 447–88). Motivated by finding the asymptotic behaviour of solutions for initial value problems, we formulate a nonlinear eigenvalue problem comprising in finding a number x∗ such that the self-similar shape function f (η) would have a powerlaw asymptotic η−x ∗ at low values of the self-similar variable η and would be the fastest decaying positive solution at η → ∞. We prove that the solution f (η) of this problem has a tail decaying as a power-law, and not exponentially or super-exponentially. We present a relationship between the power-law exponents in the regions η → 0 and η → ∞, and an integral relation for f (η) and x∗. We confirm these relationships by solving numerically the nonlinear eigenvalue problem, and find that x∗ ≈ 3.80.