Zero-frequency means purely spatial structure with no propagation. Since excitation of eddies is forbidden in the present study due to the dimensionality and most of the fluctuation energy is contained along the wave vector perpendicular to the mean magnetic field, filaments are formed as spatial structures in the simulation (Figure 1). This result is in agreement with recent observation showing clear anisotropy preferring the picture of perpendicular cascade, e.g., [1]. The filaments can be recognized in the first appearance in Figure 1 as the elongated fluctuation structure along the mean magnetic field (in the *z*-direction). It is, however, a time snapshot representation, which implies integration of the Fourier component over the frequencies. Therefore, both the zero-frequency mode and the propagating wave modes (e.g., whistler, Bernstein, and cyclotron modes) are present in the time snapshot. To decompose those elongated structures into non-propagating, coherent structure and propagating waves, the two-dimensional Fourier transform is applied in the data analysis.

Time evolution of the fluctuation energy is tracked in the Fourier domain spanned by the perpendicular wavevectors and the frequencies (Figures 2 and 3). The fluctuation amplitude is evolving as the system is about to develop into turbulence. For this reason, we choose to optimize the color bar range separately in Figures 2 and 3, otherwise one cannot identify dispersion relations in the wavenumber-frequency spectrum due to the saturation effect. The large-scale field appears as broad-band frequencies on the fluid scales *k*_{⊥}*V*_{
A
}/*Ω*_{p}<0.2 (e.g., right panel Figure 2) because the mean field was not subtracted when computing the spectra using fast Fourier transform (FFT). The effect of the large-scale field *B*_{0} (assumed to be constant) on the spectral representation can readily be seen by constructing the delta function using Fourier transform (in the one-dimensional configuration, for simplicity) as

\underset{-\infty}{\overset{\infty}{\int}}{B}_{0}\phantom{\rule{0.3em}{0ex}}{\mathrm{e}}^{\mathrm{i\kappa x}}\mathrm{d}\phantom{\rule{0.3em}{0ex}}x={B}_{0}\phantom{\rule{0.3em}{0ex}}\delta \left(\kappa \right),

(1)

that is, a spatially uniform field ends up with the spectrum with the peak at the origin of the wavenumber, *κ*=0. The large-scale inhomogeneity appears as the broadband spectrum at nearly-zero wavenumbers over a wide range frequencies. Several local peaks at the normalized frequencies in the range 4-6 (Figure 2, right panel, large-box setting) or around 5 (left panel, small-box setting) represent physically non-causal fluctuations, that is, the phase speed is so high that the whole system oscillates altogether with very large wavelengths. This effect most likely represents the numerical discretization effect because even ion Bernstein modes that may have small wavenumbers undergo strong damping at higher frequencies and cannot be sustained by the system for a long time [5]. Turbulence evolution is similar between the two box-size setups in that both the zero-frequency mode (which is the coherent structure) and the propagating waves with dispersion relations appear, and they grow together. Yet, the identified dispersion relations are different between the two setups. The zero-frequency mode grows with the ion Bernstein (fundamental and harmonics) and oblique ion cyclotron wave modes for the small-box setting, while the fluctuation energy is already distributed among them and oblique whistlers for the large-box setting. Furthermore, we observe the following properties in the spectral evolution. First, the wavenumber range of the excited zero-frequency mode agrees with that of the ion Bernstein modes in the early-time spectrum for the small-box setting. Second, the existence of oblique ion cyclotron and whistler modes for the small- and large-box settings, respectively, is clearly visible in the late-time spectrum. Third, the zero-frequency mode has more energy at higher wavenumbers for the large-box setting. In the small-box setting the fluctuation energy of the whistler mode is weaker than that of the large-box setup by a factor about ten.