The damping of the oblique daughter waves observed in Fig. 2 can be explained in terms of the wave-particle processes (e.g., pitch-angle scattering, cyclotron resonance, or Landau damping) in competition with the plasma turbulence evolution. We have computed the time evolution of the spectrum of the magnetic field and density fluctuations. In the late stages of the fluctuation evolution, the one-dimensional reduced spectra (not shown) along the perpendicular wavenumber reveal turbulence cascades (by the fact that a power-law spectrum is being formed) starting from maxima at $k_{⟂}=\mathrm{0}$ specifically for low-beta plasma turbulence. The spectral slope along the parallel wavenumber is flatter in the low wavenumber range and becomes steeper at larger values beyond the pump wavenumber. This spectral anisotropy at low and moderate wavenumbers favors the parallel-propagating modes which remain persistent at the latest stage of the decay process.

The oblique electromagnetic daughter waves in Fig. 3 are followed by corresponding fluctuations in the wavenumber–frequency density spectrum (not shown). Viñas and Goldstein (1991a, b) in their analytical study found that the coupling between the electrostatic and electromagnetic solutions increases with the increase in the propagation angle of the daughter wave. On the basis of this remark, we naively interpret this result by interchanging the density daughter modes with the sideband electromagnetic waves. At large oblique propagation angles, the density daughter modes are electromagnetic waves with fluctuation in both density and electromagnetic fields, while the sideband waves become electrostatic waves. Thus, the three-wave coupling rule is fulfilled. The obliquely propagating modes develop into a turbulent cascade which saturates the instability at a stage when the Alfvén pump wave still dominates the spectrum. A modulation-like process is observed at early times close to the pump wavenumber and saturates before the saturation of the decay instability. The modulation is exclusively oblique and can be reminiscent of the beat instability. This additional parametric instability and the consequences for the decay instability and its early saturation are still under study.

Matteini et al. (2010a) first noticed the occurrence of a perpendicular spectrum of daughter waves developed at the decay of a linearly polarized Alfvén pump wave with oblique direction of propagation to the background magnetic field. The authors concluded that the generation of obliquely propagating daughter waves is a consequence of the obliqueness of the Alfvén pump wave, i.e., its finite perpendicular wavenumber, in the framework of the three-wave coupling process. Our three-dimensional treatment is simplified with respect to that former study in the sense that the pump wave is circularly polarized and propagates parallel to the mean magnetic field. Thus we can make useful the predictions of the MHD analytical analysis for the dispersion relations and the growth rates of the decay instability at different propagation angles. The dispersion relation for left-handed circular polarization, $(k{V}_{\mathrm{A}}/{\mathrm{\Omega }}_{\mathrm{p}}{)}^{\mathrm{2}}=(\mathit{\omega }/{\mathrm{\Omega }}_{\mathrm{p}}{)}^{\mathrm{2}}/(\mathrm{1}-\mathit{\omega }/{\mathrm{\Omega }}_{\mathrm{p}})$, is satisfied for both the Alfvén pump wave (as an imposed initial condition) and Alfvén daughter modes. The value of the pump wavenumber, $k_{\mathrm{0}}{V}_{\mathrm{A}}/{\mathrm{\Omega }}_{\mathrm{p}}=\mathrm{0.21}$, is fairly high with respect to typical MHD scales, but still it can be assumed to be in the dispersionless range on the first order. The analytical dispersion analysis at weak and moderate oblique propagation angles provides the largest growth rates at approximately the same parallel wavenumber as that one predicted for the field-aligned decay. This result can be easily understood by introducing the three-wave coupling equations and the dispersion relations:

where $(k_{\mathrm{0}},{\mathit{\omega }}_{\mathrm{0}}=k_{\mathrm{0}}{V}_{\mathrm{A}})$, $(k_{\parallel }^{-},k_{⟂}^{-},{\mathit{\omega }}^{-}=-k_{\parallel }^{-}{V}_{\mathrm{A}})$, and ($k_{\parallel },k_{⟂},\mathit{\omega }$) are the wavenumbers, frequencies, and dispersion relations characterizing the Alfvén pump, Alfvén daughter, and ion acoustic waves, respectively. The MHD slow mode still preserves the field-aligned dispersion relation at small and moderate propagation angles in the limit of low-beta plasmas, namely, ${\mathit{\omega }}_{\mathrm{slow}}=k_{\parallel }{V}_{\mathrm{A}}\sqrt{\mathit{\beta }}$ (here, $\mathit{\beta }=c_s^{\mathrm{2}}/V_{\mathrm{A}}^{\mathrm{2}}$). Therefore, the sound daughter modes with oblique propagation are normal modes of plasma like their counterpart oblique Alfvén modes, and they fulfill similar dispersion relations, i.e., $\mathit{\omega }=k_{\parallel }{c}_s$. Equations (3)–(5), the dispersion relations, and some elementary algebra provide the parallel wavenumber of the oblique ion acoustic wave,

where k₀ is the wavenumber of the sound daughter driven by the field-aligned decay (see, e.g., Spangler et al., 1997). The overall decay process is not controlled by the field-aligned decay but by the dispersion relation of the participating waves which drives the oblique decay to share identical parallel wavenumbers to those attained in the parallel decay. The above rough evaluation is validated by both the analytical analysis in the MHD framework and by the hybrid simulation applied in the present study. In the analytical analysis, the evolution of the daughter waves is determined by the interaction with the field-aligned pump wave by constructing a system of quasi-linear equations which exhibits wave–wave couplings generating obliquely propagating waves as the daughter component. The growth rate of the oblique modes generally has a maximum value at the parallel wavenumber prescribed by the maximum growth rate of the field-aligned decay instability and depends on the propagation angle.

However, we still point out that the field-aligned decay remains in any case the fastest one, as shown in Fig. 4. Also, this dynamics is different from the case of an oblique pump wave propagating at a given theta angle with respect to B₀, where the oblique mode's decay rate gamma is found to scale cos θ_kB, so controlled by the $k_{\left|\right|}$ projection of the initial oblique wave vector (Del Zanna, 2001). This suggests then that there are two possible ways of generating oblique modes from the parametric decay: (1) from a purely parallel mother wave, as in this study; and (2) from an oblique pump wave (see, e.g., Matteini et al., 2010a). In both cases the oblique modes grow at a rate that is smaller than the parallel decay. In our study the central role is played by the conservation of frequencies and the dispersion relations of the components involved in the three-wave coupling. Matteini et al. (2010a) show that in their analysis, the central role in driving the transversal spectrum is played by the conservation of the momentum and the non-zero perpendicular projection of the pump wave vector. An obliquely propagating (pump) wave is expected to be more compressive, and it can generate a broadband spectrum of compressive fluctuations. Thus, the configuration with an oblique pump wave should be more efficient in driving oblique modes, even though the daughter waves in our study are driven earlier after a short time of nonlinear evolution. On the other hand, one should not ignore the major differences in the involved setups, e.g., the different polarization of the pump wave and different β_e values (see discussion below). Besides the plasma beta, the wavenumber of the pump wave may play a significant role in the process of driving the transverse broadband modulation. In both studies, the medium is still weakly dispersive ($k_{\mathrm{0}}{V}_{\mathrm{A}}/{\mathrm{\Omega }}_{\mathrm{p}}\sim \mathrm{0.2}$). In a more dispersive medium such as that one used by Verscharen et al. (2012), the dispersion relations probably cannot construct a transversal spectrum of waves for the decay of a field-aligned Alfvén wave.

Unlike MHD simulations, where the instability can saturate only through the steepening of the excited sound waves, in hybrid simulations it saturates via particle trapping and phase-space modulation (see, e.g., Matteini et al., 2010b). A consequence of this dynamics is a significant perturbation of the ion velocity distribution function, leading also to the generation of field-aligned beams (see, e.g., Araneda et al., 2008). In low-beta plasmas, the decay instability dominates over the other parametric instabilities. Plasma beta values for both ions and electrons also play an important role in the dynamics of the parametric decay. For instance, β_e is responsible for activating different saturation mechanisms. Matteini et al. (2010b) reported that for cold fluid electrons (β_e∼0), the MHD saturation mechanism is recovered. At larger values (e.g., β_e∼0.1), the trapping and beam formation is in use. The electron plasma beta β_e in our simulation has an intermediate value of 0.01. The parallel distribution in the phase space $z-v_{\parallel }$ (not shown) at a time still close to the linear phase of the instability (tΩ_p=300) suggests that particles are confined and accelerated in different regions along the parallel axis to the mean magnetic field. The phase-space modulation can be a signature for the instability saturation. There is no evidence of a field-aligned velocity beam at the latest time of the simulation. The accelerated particles are thermalized via pitch-angle diffusion by the developed oblique modes.

A consequence of the oblique parametric decay in three-dimensional plasmas is a more efficient heating of the ions; see, e.g., the pitch angle scattering study by Comişel et al. (2018). Their conclusion based on a hybrid simulation by using similar parameters to the actual run was that the plateau levels observed in the proton velocity distribution functions are driven by the pitch-angle diffusion of the ions by obliquely propagating modes. The spectral analysis performed in Fig. 3 in the frequency–wave-vector domain over a broad range of oblique propagation angles proves that the multi-channel decay of the parallel-propagating Alfvén pump wave is the source of obliquely propagating daughter waves assumed to satisfy the resonance condition with ions. The former and actual results reconfirm in particular the role of oblique compressive daughter waves in heating the protons parallel and perpendicular to the mean magnetic field according to the in situ observation in solar wind plasmas (see, e.g., Marsch and Tu, 2001).

We have studied by three-dimensional hybrid simulations the decay of a large-amplitude Alfvén pump wave expected to achieve the largest growth rate for parallel propagation and circular polarization. Our conclusions are drawn below.

• The parallel-propagating Alfvén wave decays into a transversal spectrum of daughter waves. The oblique decay is controlled by the growth rates of the decay instability which are significantly larger at small and moderate propagation angles in low-beta plasmas, in agreement with the analytic analysis.

• The transversal spectrum of daughter waves is controlled by the dispersion relation under the constraint of the three-wave coupling process. Thus, the perpendicular alignment to the mean magnetic field originates in the dispersion state of the genuine circularly polarized Alfvén wave.

Kinetic effects are assumed to become more important at larger propagation angles. Their role correlated with the value of the plasma β parameter will be the subject of a forthcoming study on the decay instability in three-dimensional plasmas.