Propagation of non-linear circularly polarised Alfvén waves in a homogeneous medium

We study the evolution of non-linear circularly polarised Alfvén waves by solving numerically the time-dependent equations of magnetohydrodynamics (MHD) in one dimension. We examine the behaviour of the waves and find that different physical mechanisms are relevant in different ranges of β. In a low β plasma the wave may undergo a parametric decay. This is because the wave excites a density enhancement that travels slower than the wave itself and thus interacts with the wave. When $\beta \ge 1$ the density enhancement does not interact with the wave and no decay takes place, instead the Alfvén wave is reflected against the density enhancement. The reflection zone propagates with the speed $\frac{1}{n} \, v_{\rm A}$. Because of that the magnetic flux is conserved which results in an amplification of the oscillating magnetic field by a factor $\frac{1}{n}$. We find that n depends on β, and that in particular it is ≤1 for values of $\beta \sim 1$ and ≥1 for $\beta \gg 1$. We discuss the relevance of these mechanisms to the acceleration of the solar wind, and the triggering of MHD turbulence in the polar wind region. In particular these simulations can explain the presence of inward propagating Alfvén waves in the solar corona.