A simple analytical expression for the power spectrum of cascading Alfv\'en waves in the solar wind
R. Vainio, T. Laitinen, H. Fichtner
Alfv\'en wave transport in the solar wind, including non-linear spectral energy transfer, is studied. We present numerical solutions of wave transport using a diffusive flux function previously introduced for spectral energy transfer, and compare it with the analytical solution obtained for a convective flux function. The two models of cascading produce very similar behavior of a power spectrum initially of 1/f-form at the solar surface, provided that the cascading constants are tuned to produce the same spectral flux in the inertial range. We present an analytical expression for the power spectrum of the diffusively-cascading Alfv\'en waves in the solar wind derived from a solution of the wave transport equation and show that it compares well with the exact solutions. Our expression enables (semi) analytical evaluation of the cyclotron heating rate, the wave pressure gradient, and the energetic-particle mean free path related to the Alfv\'en waves in the corona and solar wind.
A&A, 407, 713-723, 2003