Four-dimensional Frequency–Wavenumber Power Spectrum of a Strong Turbulence Obtained from Hybrid Kinetic Simulations

We carry out three-dimensional hybrid kinetic simulations of a strong decaying turbulence. The turbulence is initiated with a seed spectrum that includes Alfvén wave modes at low wavenumbers. From the temporal and spatial distribution of the simulation output in the quasi-steady phase, we calculate a four-dimensional frequency–wavenumber spectrum of the turbulence. Our analysis shows that kinetic Alfvén waves can be identified in the wavenumber–frequency space in the vicinity of the seed turbulence spectrum. They produce a power peak consistent with a linear dispersion relation. However, further away from the seed spectrum, where most of the wave–particle interaction takes place, the signature of the Alfvén modes disappears among other fluctuations that are not described by any dispersion relations. Furthermore, at higher wavenumbers at which the signature is still identifiable, its frequency broadening becomes comparable to the frequency itself. Therefore, the use of linear waves based on the conventional dispersion relation is not necessarily justified to describe the turbulence if their signature is still present and even more so when it disappears. We find that at larger rms amplitudes of the turbulence, the signature of the dispersion relation is confined to lower parallel wavenumbers. In the range where it is visible, the frequency broadening becomes greater at larger amplitudes. This suggests that stronger nonlinearity makes the fluctuations behave less like wave modes until the conventional wave-mode approach is no longer valid.