Cascade and Damping of Alfv\'{e}n-Cyclotron Fluctuations: Application to   Solar Wind Turbulence Spectrum
  With the diffusion approximation, we study the cascade and damping of Alfv\'{e}n-cyclotron fluctuations in solar plasmas numerically. Motivated by wave-wave couplings and nonlinear effects, we test several forms of the diffusion tensor. For a general locally anisotropic and inhomogeneous diffusion tensor in the wave vector space, the turbulence spectrum in the inertial range can be fitted with power-laws with the power-law index varying with the wave propagation direction. For several locally isotropic but inhomogeneous diffusion coefficients, the steady-state turbulence spectra are nearly isotropic in the absence of damping and can be fitted by a single power-law function. However, the energy flux is strongly polarized due to the inhomogeneity that leads to an anisotropic cascade. Including the anisotropic thermal damping, the turbulence spectrum cuts off at the wave numbers, where the damping rates become comparable to the cascade rates. The combined anisotropic effects of cascade and damping make this cutoff wave number dependent on the wave propagation direction, and the propagation direction integrated turbulence spectrum resembles a broken power-law, which cuts off at the maximum of the cutoff wave numbers or the $^4$He cyclotron frequency. Taking into account the Doppler effects, the model can naturally reproduce the broken power-law wave spectra observed in the solar wind and predicts that a higher break frequency is aways accompanied with a greater spectral index change that may be caused by the increase of the Alfv\'{e}n Mach number, the reciprocal of the plasma beta, and/or the angle between the solar wind velocity and the mean magnetic field. These predictions can be tested by future observations. 
Yan Wei Jiang, Siming Liu, Vah\'e Petrosian, Christopher L. Fryer