Parametric Instability: An Evolutive Mechanism for the Alfvénic Turbulence in the Solar Wind

Abstract

Fluctuations in fast streams or in slow Alfvénic streams of the solar wind, and in the high-latitude wind, are characterized by high cross-helicity and a low level of compressions. Such properties, which are typical of Alfvénic fluctuations, tend to decline with increasing heliocentric distance. Parametric decay, where the energy of an initial Alfvén wave is progressively transferred to both backward-propagating Alfvén and compressive modes, has been proposed as a mechanism responsible for such a behavior. Over the years, the parametric process has been studied, both analytically and numerically, in many configurations, from monochromatic waves to increasingly complex situations which include broad-band turbulent configurations with one- and two-dimensional spectra. In this paper, we give a brief review of this theoretical development, discussing its relevance in the context the evolution of Alfvénic turbulence in the solar wind.

1. Introduction

Alfvén waves play a prominent role in many astrophysical contexts, such as the solar corona and the solar wind, where they have been detected by in situ measurements for several decades (see [1,2] and references therein). Alfvén waves are characterized by a correlation between magnetic field and velocity fluctuations, whose sign can be negative or positive according to the sense of propagation (parallel or anti-parallel to the background magnetic field in the plasma reference frame, respectively), and by a low level of compressive fluctuations (density and pressure remain constant for pure Alfvénic fluctuations). A remarkable property of such fluctuations lies in the fact that they also represent exact solutions of ideal (non-dissipative) magnetohydrodynamics (MHD) equations, regardless of the wave amplitude, provided that the magnitude of the total magnetic field $\mathbf{B}$ is uniform in space. This is verified also for multidimensional fluctuations. Indeed, fluctuations with a high-velocity magnetic field correlation and low levels of density and magnetic field intensity variations are routinely observed in solar wind high-speed streams (HSS) [2], and in the so-called Alfvénic slow-speed streams (ASSS) [3,4,5].

A convenient way to describe the presence of Alfvénic fluctuations in a plasma is the use of the so-called Elsässer variables, ${\mathbf{Z}}^{\pm }$, defined as:

$${\mathbf{Z}}^{\pm }=\mathbf{v}\pm \frac{\mathbf{B}}{\sqrt{4\pi \rho }},$$

where $\mathbf{v}$ and $\rho$ are the plasma velocity and density, respectively. Since for a pure Alfvénic fluctuation it can be either $\delta \mathbf{v}=+\delta \mathbf{B}/\sqrt{4\pi \rho }$ or $\delta \mathbf{v}=-\delta \mathbf{B}/\sqrt{4\pi \rho }$ (where $\delta f$ indicates the fluctuating part of a quantity f) according to the propagation direction in the plasma rest frame with respect to the magnetic field, a situation in which $\delta {\mathbf{Z}}^{+}=0$ and $\delta {\mathbf{Z}}^{-}\ne 0$ mean a fluctuation with a group velocity parallel to the magnetic field and $\delta {\mathbf{Z}}^{-}=0$, and $\delta {\mathbf{Z}}^{+}\ne 0$ a fluctuation with a group velocity opposite to the magnetic field. Such a property can be also summarized by introducing the normalized cross-helicity ${\sigma }_c$, defined as:

$${\sigma }_c=\frac{e^{+}-e^{-}}{e^{+}+e^{-}},$$

where $e^{\pm }={\left(\delta {\mathbf{Z}}^{\pm }\right)}^2/2$ are the pseudo-energy densities associated with ${\mathbf{Z}}^{\pm }$. Therefore, ${\sigma }_c=1$ represents an Alfvénic fluctuation propagating opposite to the magnetic field, which is a pure $\delta {\mathbf{Z}}^{+}$ Elsässer mode, whilst ${\sigma }_c=-1$ represents an Alfvénic fluctuation propagating along the magnetic field, namely, a pure $\delta {\mathbf{Z}}^{-}$ Elsässer mode. We outline that this formalism applies not only for strictly parallel propagating Alfvén fluctuations but also for obliquely propagating ones.

Small-amplitude Alfvén waves are transverse waves that are polarized in a direction perpendicular to both the background magnetic field ${\mathbf{B}}_0$ and the wave vector $\mathbf{k}$ direction. In this case, the magnetic field intensity fluctuation $\delta \left|\mathbf{B}\right|$ is proportional to the squared perturbation amplitude and can be neglected in the small-amplitude limit. This is no longer considered verified for large-amplitude waves; therefore, large-amplitude, obliquely propagating Alfvén waves are not exact solutions of MHD equations and evolve in time, forming magnetosonic shocks and rotational discontinuities [6,7]. However, for a circularly polarized Alfvén wave in parallel propagation ($\mathbf{k}$ parallel to ${\mathbf{B}}_0$), it is $\delta \left|\mathbf{B}\right|=0$ regardless of the wave amplitude; therefore, a large-amplitude circularly polarized Alfvén wave is still an exact solution of MHD equations [8,9]. Other examples of exact solution are given by arc-polarized and spherical-polarized Alfvén waves, where the tip of the $\mathbf{B}$ vector describes a planar arc or moves on a spherical surface with a radius equal to $\left|\mathbf{B}\right|$, so keeping the condition $\delta \left|\mathbf{B}\right|=0$ verified.

The observations made in the solar wind [2,3,4,5,10] indicate that outward propagating Alfvénic fluctuations (let us assume an Elsässer ${\mathbf{Z}}^{+}$ mode) are the dominating component in HSS and ASSS:

$$e^{+}\gg e^{-}.$$

for frequency f in the range ${10}^{-5}\lesssim f\lesssim {10}^{-2}$ Hz, in the spacecraft reference frame [10]. Moreover, such fluctuations are turbulent, with a power-law spectrum covering a broad range of frequencies, and possess low density $\delta \rho$ and pressure $\delta p$ fluctuations; that is, they are almost incompressible:

$$\frac{\delta \rho }{\rho }\simeq \frac{\delta p}{p}\ll 1.$$

The spectrum is dominated by nearly-perpendicular wave vectors [11]. Finally, they appear to be almost circularly polarized, arc polarized or spherical polarized, meaning that such fluctuations have nearly constant magnetic field intensity:

$$\frac{\delta \left|\mathbf{B}\right|}{B_0}\ll 1,$$

where $B_0$ is the background magnetic field intensity.

However, observations in the solar wind also show a progressive decay of the Alfvénic correlation during the expansion of the wind across the heliosphere. Figure 1a reproduces Figure 6 by Bavassano et al. (2000) [12], who showed a composite view of hourly averages from different spacecraft (Helios 1 and 2 and Ulysses) of the radial evolution of $e^{\pm }$ in the polar wind at different heliocentric distances of up to 2.6 AU. It is evident from that plot that both $e^{+}$ and $e^{-}$ decrease due to the expansion of the solar wind, which involves a decay of the amplitude of the magnetic field because of the magnetic flux conservation. However, the rates of decrease in the energies of the Elsässer modes are different; $e^{+}$ decreases at a faster rate (∼$r^{-1.48}$) with respect to $e^{-}$ (∼$r^{-0.42}$). Bavassano et al. observed as this decrease in $e^{+}$ is much faster than the expected value of ∼$r^{-1}$ coming from the WKB approximation for a flow regime such as the one of the polar wind. This behavior is visible also in Figure 3 in the same paper, here reproduced as Figure 1b, where a plot of the so-called Elsässer ratio, namely, the ratio between $e^{-}$ and $e^{+}$, is shown. It can be observed that this ratio increases to between about 1 and 3 AU; then it remains more or less constant after 3 AU, as can be observed also in Figure 6 of the original paper, or in Figure 1b.

A similar result was obtained by Goldstein et al. [13] by analyzing data from the Ulysses spacecraft in the polar wind at two different heliocentric distances: $r\sim 2$ and $r\sim 4$ AU). In Figure 2 (Figure 1 of Paper [13]), panels (b) and (c), the spectra of ${\mathbf{Z}}^{+}$ and ${\mathbf{Z}}^{-}$ are plotted at those two distances, and it is evident that at smaller scales ($\omega >{10}^{-4}$ Hz), the two spectra that were more separated at $r\sim 2$ AU are closer to each other at $r\sim 4$ AU. Additionally, in Figure 3 (Figure 2 of Paper [13]) it is seen that the normalized cross-helicity ${\sigma }_c$ decreases with increasing distance from the Sun.

Such a depletion of Alfvénicity can be expected when the Alfvénic fluctuations propagate across magnetic field inhomogeneities. In fact, several authors (see, for instance, [14] for the incompressible case and [15] for a compressible magnetofluid) have shown that magnetic field inhomogeneities can destroy the correlation of a train of Alfvénic fluctuations, by producing both backward propagating Alfvénic and compressible fluctuations. However, such a mechanism should be much less effective in situations where the magnetic field is more homogeneous, such as in the polar wind and in ASSS.

Marsch and Tu [10] carried out a detailed analysis of the spectra of the Elsässer variables in the solar wind at 0.3 AU in different zones, inside slow streams, where the heliospheric current sheet is embedded, and in fast-speed streams, where both the magnetic and velocity fields are more homogeneous. Moreover, they performed a study of the evolution of the density and magnetic field intensity spectra of the fluctuations at 0.4 and 0.8 AU, in both fast and slow streams. In Figure 4, left (Figure 2, of the original paper), spectra of ${\mathbf{Z}}^{+}$ and ${\mathbf{Z}}^{-}$ are shown at different positions in both the fast and slow streams at 0.3 AU. It is visible that in the slow streams, near to the magnetic sector boundary, where the magnetic field is more inhomogeneous, the spectra of ${\mathbf{Z}}^{+}$ and ${\mathbf{Z}}^{-}$ (panels b and c) are similar to each other, and the spectrum of the density fluctuations exhibits a well-developed power-law. On the other hand, in the regions where the magnetic field is more homogeneous, that is, in the high-speed streams, far away from the sector boundary, the spectra of the Elsässer modes are more separated (panels a, d–g), and the energy of the density fluctuations, especially at larger scales, is lower. A peculiar feature of the spectra shown in Figure 4, left, lies in the fact that in the fast-speed streams, at smaller scales, both the spectra of ${\mathbf{Z}}^{-}$ and of the density exhibit flatter tails at high frequencies, suggesting the presence of a common mechanism for the generation of small scale turbulence that makes the two quantities evolve in a similar way. Finally, a comparison of the spectra of density and magnetic field intensity fluctuations at different radial distances (0.4 and 0.8 AU) in the fast speed streams (see Figure 4 right, corresponding to Figures 6 and 7 of reference [10]) show that both quantities are considerably higher in the slow streams, with respect to the high-speed streams (solid and dashed lines); however, the fluctuations of both density and magnetic field intensity increase, in the latter case, for increasing radial distances from the Sun. Finally, in the same figures it is clear that in HSS both the spectra of the density and magnetic field intensity fluctuations have flatter tails at high frequencies, again indicating a common generation mechanism for the fluctuations of ${\mathbf{Z}}^{-}$, density and magnetic field intensity.

We can therefore summarize the previously reported observations in the following way: fluctuations observed in the HSS and ASSS have a high degree of Alfvénic correlation at lower heliocentric distances; that is, they appear to be almost “pure”, outwardly propagating, arc- or spherically polarized Elsässer modes (${\sigma }_c\simeq 1$, $\delta \left|\mathbf{B}\right|/\left|\mathbf{B}\right|\ll 1$), involving low levels of density and magnetic field intensity fluctuations. With increasing heliocentric distances, however, the situation changes considerably: higher levels of ${\mathbf{Z}}^{-}$, density and magnetic field intensity fluctuations are observed, and the energy of both ${\mathbf{Z}}^{+}$ and ${\sigma }_c$ decreases. This phenomenology seems to indicate that some dynamical mechanism is at work to progressively destroy the initial, almost “pure”, Alfvénic correlation while the fluctuations are propagating through the heliosphere, so that the initial Alfvénic fluctuations decay by producing both back-scattered Alfvénic and magnetosonic fluctuations.

A possible mechanism capable to explain such a phenomenon is the parametric instability [16]: in spite of the fact that circularly polarized, Alfvénic fluctuations are an exact solution of the MHD equations, when compressibility is present, a large amplitude Alfvénic fluctuation (called mother wave or pump wave) can decay by producing back-scattered Alfvénic and magnetosonic fluctuations (daughter waves). The produced fluctuations will be nonlinear; thus, they can produce extended spectra. Such decay instability involves strict resonance conditions between both the frequencies and wave vectors of the mother and daughter waves, thereby giving rise to an eventual production of nonlinear fluctuations with frequencies and wave vectors far from that of the pump wave (namely, it involves nonlocal interactions among modes in the Fourier space). The latter remark can point out the role of the parametric instability as the source of the flatter tails observed at high frequencies in the spectra of ${\mathbf{Z}}^{-}$, density and magnetic field intensity fluctuations in the solar wind.

The instability has been studied in past years by many authors, either as an interesting theoretical example of plasma instability, or for its suitability to explaining the above-mentioned observations of the solar wind. Galeev and Oraevskii [17] and Sagdeev and Galeev [16] studied the instability at low-beta values (cold plasma, beta being the ratio between kinetic and magnetic pressure); Goldstein [18] extended those studies to the case of a generic value of beta; Hoshino and Goldstein [19] studied possible nonlinear effects that can take place during the development of the instability; Longtin and Sonnerup [20] included dispersive effects in the development of the instability. In all the previously cited studies, the pump wave was a monochromatic Alfvén perturbation with either small or large amplitude. However, Alfvénic fluctuations in the solar wind have an extended spectrum already formed near the Sun. It is therefore necessary to see whether the properties of the instability change when the decaying fluctuation has an already developed initial spectrum. This problem was previously addressed by Umeki and Terasawa [21], who considered a mother wave as a superposition of circularly polarized waves. Note, however, that this initial condition is not globally a magnetic field with a constant intensity; therefore, it is not an exact solution of the MHD equations. They found that the instability still takes place with a phenomenology analogous to the case of a monochromatic pump wave. The problem was further studied with a semi-analitical approach by Malara and Velli [22], who studied the parametric instability in the case of a nonmonochromatic pump wave with a power-law spectrum, along with the condition that the mother wave was arc-polarized. Additionally, in this case, the instability showed to be a robust mechanism, leading to the formation of backward propagating Alfvén and magnetosonic fluctuations with some dependency of the growth rate of the instability on the width and phases of the initial spectrum.

Stimulated by those positive results, Malara et al. [23,24] and other authors [25,26] studied numerically the development of the parametric instability in a variety of situations (different configurations of the initial spectrum, 1D and multidimensional cases, expanding box models to keep in account the influence of the expansion of the solar wind on the instability), always finding that the parametric instability is a fairly good candidate to account for the observed evolution of the turbulence in the solar wind.

The plan of the paper is as follows: a review of all the above-studied cases is presented in a more detailed way in Section 2, for the case of a monochromatic pump wave, and in Section 3, for the case of a broad-band Alfvénic initial fluctuation. Then, in Section 4, we discuss in deeper detail the different studies applying the instability to the case of the solar wind, eventually showing that the properties of the turbulence generated by the parametric instability can account also for other features of the Alfvénic turbulence observed in the solar wind. Finally, in Section 5, we draw some conclusions concerning possible future outlooks for making the initial conditions of the numerical simulations increasingly similar to the ones of the real solar wind, thereby improving understanding of this mechanism and its real suitability to explaining the observed dynamics of the solar wind Alfvénic turbulence.

2. Simple Configurations for the Parametric Instability

Alfvén waves are exact solutions of MHD equations; however, in some configurations they become unstable, giving origin to other fluctuations: typically, compressive modes and additional Alfvén-like modes. The unperturbed Alfvén wave is generally indicated as the “pump wave” or “mother wave”, and fluctuations produced by the instability are indicated as “daughter waves”. During the linear stage of instability, when daughter waves have small amplitudes compared with that of the pump wave, their amplitudes grow exponentially in time with a growth rate $\gamma$, which has the same value for all daughter perturbations.

The simplest case corresponds to a monochromatic pump wave, i.e., a plane wave with a well-defined wave vector ${\mathbf{k}}^{\left(0\right)}$ and frequency ${\omega }^{\left(0\right)}$, and monochromatic daughter waves, all propagating in the same direction (say, x). This corresponds to a one-dimensional (1D) configuration in space. The monochromatic 1D case was first studied by Galeev and Oraevskii [17] and Sagdeev and Galeev [16] for the case where the amplitude $B^{\left(0\right)}$ of the pump wave is small and $\beta \ll 1$, where $\beta =c_S^2/c_A^2$, with $c_S$ and $c_A$ being the sound speed and the Alfvén speed, respectively. Moreover, Alfvén waves are circularly polarized and propagate parallel to the background magnetic field ${\mathbf{B}}_0$. In this case, two daughter waves are involved: a sound wave with wave number k and frequency $\omega$ and a backward-propagating Alfvén wave with wave number $k_A$ and frequency ${\omega }_A$. These quantities are related by the resonance conditions:

$${\mathbf{k}}^{\left(0\right)}=\mathbf{k}+{\mathbf{k}}_A,{\omega }^{\left(0\right)}=\omega +{\omega }_A$$

(1)

which express momentum and energy conservation during the wave coupling process. Due to the small amplitude assumption, one can use the dispersion relations: ${\omega }^{\left(0\right)}=k^{\left(0\right)}{c}_A$, ${\omega }_A=-k_A{c}_A$ and $\omega =k{c}_S$. The condition $\beta <<1$ allows one to neglect the frequency $\omega$ of the sound wave with respect to ${\omega }^{\left(0\right)}$ and ${\omega }_A$. Then, Equation (1) gives ${\omega }_A\simeq {\omega }^{\left(0\right)}$, implying $k_A\simeq -k^{\left(0\right)}$ and $k\simeq 2k^{\left(0\right)}$. Therefore, the Alfvénic daughter wave has the same wavelength ${\lambda }^{\left(0\right)}$ as the pump wave, and the sound wave is a quasi-static perturbation at wavelength $\lambda ={\lambda }^{\left(0\right)}/2$.

To illustrate the instability mechanism in this case, let us consider a pump wave with a smaller amplitude backward Alfvén wave and opposite helicity superimposed on it. In the pump wave, magnetic $B^{\left(0\right)}$ and velocity $v^{\left(0\right)}$ perturbations are negatively correlated, corresponding to a forward-propagating wave; and in the Alfvénic daughter wave $B_A$ and $v_A$ are positively correlated, corresponding to a backward-propagating wave. Looking at the scheme in Figure 5, it is clear that, on average, a quasi-static modulation of the magnetic pressure $p_M=B^2/8\pi$ is generated at a wavelength ${\lambda }^{\left(0\right)}/2$, where $\mathbf{B}={\mathbf{B}}_0+{\mathbf{B}}^{\left(0\right)}+{\mathbf{B}}_A$ is the total magnetic field. Therefore, the ponderomotive force generates a quasi-static density perturbation with density minima (maxima) localized at $p_M$ maxima (minima). On the other hand, at low (high) density sites, where the two Alfvén waves are in phase (anti-phase), the plasma inertia is decreased (increased). At the lowest order, the force acting on the plasma is magnetic tension, which has the same form $\simeq B_0(\partial B^{\left(0\right)}/\partial x)$ as that for the pump wave only, implying that the actual transverse momentum profile is the same as that for the pump wave only: $\rho \left(x\right)v_{\perp }\left(x\right)\simeq {\rho }_0{v}^{\left(0\right)}\left(x\right)$. Therefore, the modulation of $\rho$ increases or decreases $v_{\perp }$ with respect to the pump wave velocity perturbation $v^{\left(0\right)}$. This corresponds to a secondary transverse velocity perturbation $\delta v_{\perp }=v_{\perp }\left(x\right)-v^{\left(0\right)}\left(x\right)\simeq \left\{1-\left[{\rho }_0/\rho \left(x\right)\right]\right\}{v}^{\left(0\right)}\left(x\right)$ that has the same phase as the Alfvénic daughter wave (see Figure 5), which is therefore amplified. Summarizing, the superposition of the Alfvénic daughter on the pump wave induces a density modulation, which, in turn, amplifies the Alfvénic daughter wave itself; such positive feedback is at the basis of the instability mechanism. The corresponding growth rate is given by [16]:

$$\gamma \simeq {\beta }^{-1/4}\frac{B^{\left(0\right)}}{B_0}{\omega }^{\left(0\right)}.$$

(2)

The degree of velocity-magnetic field correlation is measured by the cross-helicity ${\sigma }_c$, which here we redefine as:

$${\sigma }_c=\frac{E^{+}-E^{-}}{E^{+}+E^{-}},$$

(3)

where the pseudo-energies associated with the Elsässer variables ${\mathbf{Z}}^{\pm }=\mathbf{v}\pm \mathbf{B}/\rho$ are

$$E^{\pm }=\frac{1}{2}{\int }_V{\left(\delta {\mathbf{Z}}^{\pm }\right)}^{2}dV,$$

(4)

the fluctuation of a quantity f is indicated by $\delta f=f-\langle f\rangle$ and V is a volume with a size much larger than any involved wave-length. The level of density and magnetic field intensity fluctuations are measured by the quantities:

$$r=\sqrt{\langle \frac{\delta \rho }{{\rho }_0}\rangle }\mathrm{and}m=\sqrt{\langle \frac{\delta \left|\mathbf{B}\right|}{\langle \left|\mathbf{B}\right|\rangle }\rangle },$$

(5)

respectively. During the development of the instability, $|{\sigma }_c\left(t\right)|$ decreases in time, and both $r\left(t\right)$ and $m\left(t\right)$ increase. This behavior is reminiscent of what is observed in the evolution of fluctuations in solar wind Alfvénic streams with the radial distance, and is a common feature of the parametric process in different and more complex situations.

A large-amplitude, circularly-polarized, parallel-propagating Alfvén wave is an exact solution of ideal MHD equations. The form of the corresponding magnetic perturbation is:

$${\mathbf{B}}^{\left(0\right)}(x,t)=B^{\left(0\right)}\left[cos(k^{\left(0\right)}x-{\omega }^{\left(0\right)}t)\widehat{\mathbf{y}}+sin(k^{\left(0\right)}x-{\omega }^{\left(0\right)}t)\widehat{\mathbf{z}}\right],$$

(6)

with $\widehat{\mathbf{y}}$ and $\widehat{\mathbf{z}}$ being the unit vectors along y and z directions, respectively. The wave amplitude $B^{\left(0\right)}$ can be of the order of $B_0$. The linear stage of parametric instability of such a wave has been studied by Goldstein [18], and independently, by Derby [27] for an arbitrary value of the plasma $\beta$. In this case, the large-amplitude pump wave (6) introduces a non-negligible space-time modulation in the background where daughter waves propagate. As a result, higher-order wave couplings come into play, and this determines a broadening of the resonance lines. Sharp resonances that characterize the small-amplitude case are replaced by resonance bands both in frequency and in wave number, and have a finite width that increases with increasing the amplitude of the pump wave. To study this case, a small amplitude perturbation is superposed on the pump wave (6), and equations are linearized. Mixed terms of the form pump × perturbation arise that give origin to fluctuations with wave number and frequency given by

$${\mathbf{k}}^{\pm }=\mathbf{k}\pm {\mathbf{k}}^{\left(0\right)},{\omega }^{\pm }=\omega \pm {\omega }^{\left(0\right)},$$

(7)

coupled with a compressive wave with wave number $\mathbf{k}$ and frequency $\omega$. The two “sideband” fluctuations at wave vectors ${\mathbf{k}}^{\pm }$ and frequency ${\omega }^{\pm }$ are forward- and backward-propagating waves belonging to the Alfvén branch. One peculiarity of such a process, a consequence of the conditions (7), is that Alfvénic daughter waves have wave numbers ${\mathbf{k}}^{\pm }$ that can be quite different from that (${\mathbf{k}}^{\left(0\right)}$) of the pump wave. Therefore, the parametric process represents a mechanism of energy transfer that is non-local in the spectral space. This property is different from what happens in turbulence, where nonlinear interactions are assumed to take place among nearby wave vectors, i.e., to be local in the spectral space. When $k>k_0$, the process is indicated as “parametric decay,” whereas when $k<k_0$ it is indicated as “modulational instability.” The dispersion relation associated with such a process has the following form [18]:

$$\left({\Omega }^2-\beta {\kappa }^2\right)\left(\Omega -\kappa \right)\left[{\left(\Omega +\kappa \right)}^2-4\right]=\frac{\eta {\kappa }^2}{2}\left({\Omega }^3+{\Omega }^2\kappa -3\Omega +\kappa \right),$$

(8)

with $\Omega =\omega /{\omega }^{\left(0\right)}$, $\kappa =k/k^{\left(0\right)}$ and $\eta =B^{\left(0\right)}/B_0$. An example of the solution of Equation (8) is illustrated in Figure 6: for a given value of $\beta$ and of the normalized pump wave amplitude $\eta$, there is an instability range in the wave-number domain, where two branches of $\Re (\Omega )$ are coincident and the corresponding growth rate $\Gamma =\Im (\Omega )$ is non-vanishing. Both the width of the instability range and $max\left\{\Gamma \right\}$ increase when increasing the pump wave amplitude $\eta$. The maximum growth rate decreases with increasing $\beta$; such a feature is also found in other configurations.

Other studies have been carried out concerning the linear stage of the parametric instability in the 1D case with different approximations. The modulational instability of an Alfvén wave in the presence of dispersive effects has been considered by Sakai and Sonnerup [28] in a two-fluid approximation, using an amplitude and dual time scale expansion in which temporal changes, observed when moving with the wave, are assumed to be slow. The interplay of modulational instability with dispersive effects giving origin to Alfvén solitons has been investigated by Ovenden et al. [29] also in comparison with solar wind observations. A case with $k\ll k^{\left(0\right)}$ has been examined in the two-fluid approximation by Longtin and Sonnerup [20], who found that left-hand and right-hand polarized Alfvén waves are unstable when their phase speed is smaller or higher than $c_s$, respectively. A systematic study of the dispersion relation including dispersive effects has been carried out by Wong and Goldstein [30], extending the results by Sakai and Sonnerup [28], and finding the conditions where modulational or decay instability preferentially develops in right- or left-hand polarized waves, according to the value of the plasma $\beta$.

A multidimensional geometry has been considered for a parametric process when perturbations propagates at perpendicular wave vectors. This is the case for the filamentation instability where perturbations grow without oscillations ($\Re \left(\omega \right)=0$) at wave vectors strictly perpendicular to both the wave vector of the pump wave and to the background magnetic field ${\mathbf{B}}_0$ (Kuo et al. [31,32]). In this case, the time evolution leads the initial plane large-amplitude Alfvén wave to be modulated in the perpendicular direction. A two-fluid description is adopted to include dispersive effects. The growth rate depends on plasma $\beta$ and pump wave amplitude and wave number, differently for right- and left-hand polarization; in particular, a threshold condition is present [31].

The above results have been generalized to include cases when perturbations propagate obliquely. Vi$\mathrm{ñ}$as and Goldstein [33] have studied the instability of large-amplitude dispersive Alfvén waves, allowing the daughter waves (both compressive and sideband waves) to propagate at an arbitrary angle with respect to that of the pump wave. This allows coupling with both electrostatic and electromagnetic waves (Vi$\mathrm{ñ}$as and Goldstein [34]). The electromagnetic coupling comes from a three-wave interaction when only one electrostatic (ion acoustic) sideband wave is involved, while the other sideband wave is vanishing. Moreover, besides the filamentation instability, a parametric magnetoacoustic instability has been found at oblique wave vectors, characterized by propagating density fluctuations with frequencies much larger than the instability growth rate $\gamma$. At small propagation angles $\theta$, the growth rate is maximum at parallel propagation and decreases with increasing $\theta$, both for the decay ($k>k^{\left(0\right)}$) and for the modulational ($k<k^{\left(0\right)}$) instability. At large propagation angles, the growth rate of filamentation instability is maximum at perpendicular propagation; when $\theta$ decreases from ${90}^{\circ }$, $\gamma$ decreases as well and a real part of the frequency arises. Therefore, though the filamentation instability covers a wide range of wave numbers k, both smaller and larger than $k^{\left(0\right)}$, its non-propagating character is confined into a very narrow angular band around $\theta ={90}^{\circ }$. In contrast, the magnetoacoustic instability develops into a narrow wave number band with a growth rate typically lower than for the filamentation instability, which decreases with decreasing the propagation angle $\theta$. Vi$\mathrm{ñ}$as and Goldstein also studied the nonlinear development of the above instabilities by means of two-dimensional (2D) hybrid simulations [33,34]. In this case, while the real part of the frequency is consistent with the linear theory, lower values of the growth rate have been found due to the damping effect generated by kinetic mechanisms of ions. Therefore, kinetic effects tend to reduce the efficiency of the parametric process by damping daughter ion-acoustic modes. Those simulations have allowed researchers to describe the saturation stage of above instabilities, which is due to nonlinear excitation of higher-order modes, similarly to what was found in a 1D MHD simulations by Hoshino and Goldstein [19]. The saturation level of normalized density and magnetic field fluctuations is in the order of some percent, for the considered values of parameters.

Other studies on damping effects have been conducted based both on 1D hybrid simulations (Vasquez [35], Araneda [36]) and on an analytical approach where the effects of kinetic damping of compressive waves are represented by a collisional-like term in the parallel component of the fluid equations (Gomberoff and Araneda [37]). The results showed that, for $\beta$ larger than unity, new instabilities can develop that are not present in the ideal MHD case: a modulational-like instability and another instability occurring in the region where a decay instability could have been expected in the absence of damping.

Multi-dimensional configurations have also been considered by Del Zanna et al. [25], who studied the parametric process in a 1D large-amplitude circularly-polarized pump Alfvén wave perturbed by 1D, 2D or three-dimensional (3D) fluctuations. For this study, Del Zanna et al. [25] performed nonlinear MHD simulations considering different values of the plasma $\beta$. In the 1D case and at low values of $\beta$ ($\beta =0.1$), the system undergoes a complex dynamics where the back-scattered Alfvén wave saturates at a level comparable with that of the initial pump wave. In this case, at saturation, $|{\sigma }_c|$ reaches values comparable with the initial value, but with the opposite sign, indicating that about half of the energy initially contained in the pump wave is transferred to the back-scattered Alfvén wave. Subsequently, the back-scattered Alfvén wave itself undergoes a secondary parametric instability that leads to the restoration of the initial sign of ${\sigma }_c$. Such changes of sign for ${\sigma }_c$ can repeat several times. The remaining part of the initial pump energy is transferred to compressive modes that grow to a high level; as a consequence, shocks are formed which dissipate energy and heat the plasma. For larger values of $\beta$, a simpler time behavior is observed: ${\sigma }_c$ monotonically decreases until reaching a slight sign reversal for $\beta =0.5$, or until reaching ${\sigma }_c\simeq 0$ for $\beta =1.2$, which implies an asymptotic balance between Alfvénic modes propagating in the two directions. When the initial perturbation is 2D or 3D, several daughter modes are observed to grow, with wave vectors oriented in different directions with respect to that of the pump wave. During the linear stage of the instability, the fastest growing modes are those propagating in parallel, in accordance with previous results. Nevertheless, at saturation, modes with a given wave number $k=\left|\mathbf{k}\right|$ reach a comparable level, leading to a roughly isotropic final spectrum for density fluctuations, and a slight presence of perpendicular wave vectors is observed in the asymptotic magnetic energy spectrum.

Del Zanna et al. [25] have also performed numerical simulations of the parametric instability in open-boundary configurations. In this case, a technique based on projected characteristics was adopted to obtain a steady injection of Alfvén waves on one side of the spatial domain boundary, along with the exit without reflection of all wave modes through the opposite side of the boundary. During its travel from one side to the other, the injected pump wave undergoes parametric decay, leading to a reduction in ${\sigma }_c$ and to an increase in compressions. For low $\beta$ values, the long-time evolution generates a gradual increase in transverse wave vectors, leading to the formation of longitudinal density structures, and to the transverse filamentation of Alfvénic modes. It is suggested that such a behavior is reminiscent of density ray-like features observed in the extended solar corona and pressure-balanced structures found in solar wind data. Similar results have also been obtained by Pruneti and Velli [38] with mild gravity added in the parallel direction.

3. Parametric Instability in Non-Monochromatic Configurations

The theoretical work described in the previous section has been devoted to studying the parametric instability in cases of a monochromatic pump wave, i.e., an Alfvén wave characterized by a single well-defined wave vector ${\mathbf{k}}^{\left(0\right)}$. However, from the perspective of applying theoretical results to solar wind Alfvénic fluctuations, the hypothesis of monochromaticity is not realistic. Indeed, solar wind fluctuations are not monochromatic: instead, they are characterized by a spectrum of wave numbers that extends over several tens of MHz in the MHD range, along with one or two different spectral indexes in different spectral subranges [2]. Therefore, efforts have been devoted to studying the parametric instability of non-monochromatic Alfvénic fluctuations, which are more realistic representations of solar wind fluctuations detected during Alfvénic periods.

The stability of Alfvén waves with an extended spectrum against the parametric process has been considered by Cohen and Dewar [39], with the purpose of determining the stability properties of Alfvénic, outward-directed fluctuations in the solar wind. These authors followed a weak turbulence approach assuming that Alfvén waves forming the turbulence have random-distributed phases and interact in an incoherent way. Moreover, they assumed that compressive daughter waves are strongly damped by kinetic effects during the development of the instability. In the limit $\beta \ll 1$, and assuming a power spectral density of the initial Alfvén waves in the form of a power law: $I_{\mathrm{out}}\propto k^{-\alpha }$, they found that back-scattered, inward-propagating Alfvén waves grow if the spectral index is $\alpha <1$. The resulting prediction for the solar wind fluctuations, where $\alpha >1$, would be that they are stable against the parametric process. However, the assumptions made in this theory do not fit with the solar wind turbulence [2]. In fact, the hypothesis of weak turbulence fluctuations can hardly be justified, since the properties of solar wind fluctuations seem to indicate fully developed turbulence. Moreover, the condition $\beta \ll 1$ has not been verified in the solar wind. Therefore, the prediction of stability for the solar wind turbulence is questionable.

Umeki and Terasawa [21] studied the parametric decay of a large-amplitude disturbance, composed of an incoherent superposition of circularly polarized Alfvén waves with the same helicity and propagation direction, but with different wavelengths. Amplitudes are distributed in a way to reproduce a power-law spectrum. For this study, the authors employed fully nonlinear MHD simulations. Since the initial condition was not an exact solution of the MHD equations, nonlinear steepening of perturbations and the consequent formation of shocks occurred simultaneously and overlapped with the parametric decay process. In any case, such simulations have shown that the parametric process of incoherent Alfvén waves actually takes place, contrary to the predictions of Cohen and Dewar [39], regardless of the slope of the initial spectrum and even for small values of $\beta$; in contrast, it is absent for high values of $\beta$ ($\beta =2$). More importantly, it displays features qualitatively similar to those of the monochromatic coherent case, namely, the production of both sound waves and back-scattered Alfvén waves. These results indicate that parametric instability is a robust mechanism which can take place in situations different from the coherent wave interaction, typical of the monochromatic case.

A further step in the direction of investigating parametric instability of large-amplitude Alfvénic fluctuations with a spectrum of wave numbers has been done by Malara and Velli [22], who studied the linear stage of the instability. The configuration considered by these authors was 1D, i.e., all quantities depended only on one space variable x and time t, though vector quantities can have three non-vanishing components. The main improvement given by this study is in the fact that the pump wave, despite its large amplitude and its extended spectrum, is an exact solution of ideal MHD equations; i.e., it propagates without distortions in a homogeneous background. Moreover, its properties are similar to those of Alfvénic fluctuations observed in the solar wind. Namely: (i) velocity ${\mathbf{v}}^{\left(0\right)}$ and magnetic field ${\mathbf{B}}^{\left(0\right)}$ fluctuations are Alfvénically correlated: ${\mathbf{v}}^{\left(0\right)}/c_{A0}=-{\mathbf{B}}^{\left(0\right)}/B_0$, $c_{A0}=B_0/{\left(4\pi {\rho }_0\right)}^{1/2}$, $B_0$ and ${\rho }_0$ are the background Alfvén velocity, magnetic field and density, respectively; (ii) the total magnetic field is uniform: $|{\mathbf{B}}^{\left(0\right)}+{\mathbf{B}}_0|=\mathrm{const}$; and (iii) density and pressure perturbations are vanishing: ${\rho }^{\left(0\right)}=p^{\left(0\right)}=0$. Conditions (i)–(iii) ensure that the pump wave is an exact solution of MHD equations. In order to implement the condition (ii), the following form for a transverse magnetic perturbation has been considered in reference [22]:

$${\mathbf{B}}^{\left(0\right)}(x,t)=B^{\left(0\right)}\left\{cos\left[\varphi \left(x-c_{A0}t\right)\right]\widehat{\mathbf{y}}+sin\left[\varphi \left(x-c_{A0}t\right)\right]\widehat{\mathbf{z}}\right\},$$

(9)

with ${\mathbf{B}}_0=B_0\widehat{\mathbf{x}}$. The function $\varphi (·)$ represents the phase, and its form determines the Alfvénic pump fluctuation. A monochromatic wave with wave number $k^{\left(0\right)}$ corresponds to $\varphi =k^{\left(0\right)}\left(x-c_{A0}t\right)$; in general, the explicit form of $\varphi$ indirectly determines the spectrum of the pump fluctuation. Assuming periodicity on a length L, the phase is determined by the equation $d\varphi /dX=k^{\left(0\right)}+a{\varphi }_1^{\prime }\left(X\right)$, with ${\varphi }_1^{\prime }\left(X\right)=\sum {\widehat{\varphi }}_{1,n}^{\prime }{e}^{inX}=\sum \left|{\widehat{\varphi }}_{1,n}^{\prime }\right|e^{i(nX+{\delta }_n)}$, and $X=(x-c_{A0}t)/L$ is the phase argument. Along with the function ${\varphi }_1^{\prime }\left(X\right)$, the parameter a controls the pump wave spectrum whose width increases with increasing a. The monochromatic case with wave number $k^{\left(0\right)}$ is obtained by setting $a=0$. The equations for the Fourier time-transforms of a small amplitude perturbation are derived in a matrix form [22]:

$$i\Omega Y+A\frac{dY}{dX}+i\left[k^{\left(0\right)}+a{\varphi }_1^{\prime }\left(X\right)\right]BY=0,$$

(10)

where the elements of column vector Y contain suitable linear combinations of the Fourier-transformed perturbation components, $\Omega$ is the normalized frequency, and A and B are two constant matrices (see reference [22] for the details). The limit of a weakly-non-monochromatic pump wave, corresponding to $a\ll 1$, is obtained by a perturbative procedure in the expansion parameter a.

$$Y\left(X\right)=Y_0\left(X\right)+a{Y}_1\left(X\right)+a^2{Y}_2\left(X\right)+\dots ;\Omega ={\Omega }_0+a{\Omega }_1+a^2{\Omega }_2+\dots$$

(11)

At the expansion order $a^0$, Equation (10) contains only constant coefficients and is Fourier-transformed with respect to X. The corresponding Fourier coefficients ${\widehat{Y}}_0\left(k\right)$ satisfy an eigenvalue problem and can be expressed as a linear combination of 5 linear independent eigenvectors: ${\widehat{Y}}_0\left(k\right)={\sum }_{p=1}^5{\widehat{Y}}_0^{\left[p\right]}\left(k\right)$. Such eigenvectors represent the unstable modes of the monochromatic case, each characterized by a complex eigenvalue ${\Omega }_0^{\left[p\right]}$, which is the corresponding frequency and satisfies the monochromatic dispersion relation (8). The function ${\varphi }^{\prime }\left(X\right)$, representing non-monochromaticity of the pump wave, enters at the order $a^1$; it produces a correction $Y_1^{\left[p\right]}(X;k)={\sum }_{m=-\infty }^{+\infty }{\widehat{Y}}_1^{\left[p\right]}(m;k)e^{imX}$ with the $\left[p\right]$th mode, which, in that order, is no longer purely monochromatic with wave number k, but has an entire spectrum of wave numbers m. The order-$a^1$ correction to the mode frequency is vanishing: ${\Omega }_1^{\left[p\right]}=0$, whereas the first non-vanishing frequency correction appears of the order $a^2$. Therefore, the expansion procedure gives the following quadratic expression for the growth rate of the $\left[p\right]$th eigenmode:

$${\gamma }^{\left[p\right]}\left(k\right)\simeq {\gamma }_0^{\left[p\right]}\left(k\right)-a^2{\Gamma }^{\left[p\right]}\left(k\right),$$

(12)

where ${\gamma }_0^{\left[p\right]}\left(k\right)$ is the growth rate in the monochromatic case. The coefficient ${\Gamma }^{\left[p\right]}\left(k\right)$ has the following properties [22]: (a) it is positive, indicating that the growth rate ${\gamma }^{\left[p\right]}\left(k\right)$ of a given eigenmode is maximal in the case of a monochromatic pump wave, whereas it decreases with increasing the width of the pump wave spectrum; (b) ${\Gamma }^{\left[p\right]}\left(k\right)$ depends on the power spectrum $|{\widehat{\varphi }}_{1,n}^{\prime }{|}^2$ of the function ${\varphi }_1^{\prime }\left(X\right)$, but it does not depend on the phases ${\delta }_n$ of the Fourier expansion of ${\varphi }_1^{\prime }\left(X\right)$.

The above results have been confirmed by a numerical approach to the full eigenvalue problem (10), which has allowed researchers to find the solution for arbitrary large values of the non-monochromaticity parameter a [22]. The following form for the function ${\varphi }_1^{\prime }\left(X\right)$ has been assumed by Malara and Velli [22]:

$${\varphi }_1^{\prime }\left(X\right)=k^{\left(0\right)}+a\sum _{n=-N}^N\left(n^{-\alpha }{e}^{i{\delta }_n}\right)e^{inX}$$

(13)

corresponding to a power-law spectrum for the phase function. Correspondingly, the pump wave spectrum roughly follows a power law for $\left|n\right|\le N$, but details of the pump wave profile and spectrum also depend on the choice of the numbers $\left\{{\delta }_n\right\}$. In Figure 7, left panel (Figure 1 of [22]), the largest growth rate corresponding to the case $k_0=4$, $B^{\left(0\right)}/B_0=0.5$ and $\beta =0.444$ is plotted as a function of a; different curves correspond to different choices of the numbers $\left\{{\delta }_n\right\}$. For small values of a, all curves collapse on a single quadratic dependence, in accordance with the results of the analytical approach. For larger values of a, different choices of $\left\{{\delta }_n\right\}$ give different values for the growth rate. As a general result, the growth rate for a non-monochromatic pump wave is smaller than but remains of the same order of magnitude as that of the monochromatic case. Therefore, the presence of a broad-band spectrum in the pump wave does not stabilize the wave itself with respect to the parametric process. As for the pump wave, unstable perturbations present extended spectra: see Figure 7, right panels (a and b) (Figure 3 of [22]).

The nonlinear evolution and saturation of a broad-band 1D Alfvénic pump wave has been studied by Malara et al. [23] by direct MHD numerical simulations. The initial condition corresponds to an Alfvénic pump wave with magnetic perturbation given by Equations (9) and (13) and a small-amplitude density noise superposed with the purpose of initiating the instability. Several cases have been examined corresponding to different values of parameters a, $\left\{{\delta }_n\right\}$ and $\beta$. A stage of exponential growth is present in time evolution of density r and magnetic field intensity m fluctuations (Equation (5)), along with a decrease in the cross-helicity $|{\sigma }_c|$. In accordance with the results of reference [22], the estimated growth rate ${\gamma }_{\max }$ of the most unstable Fourier mode tends to decrease with increasing the non-monochromaticity parameter a, but remains of the same order as in the monochromatic case. Moreover, different choices of the set of phases $\left\{{\delta }_n\right\}$ give different values of ${\gamma }_{\max }$. The exponential growth stage is followed by instability saturation. Saturation levels depend on the value of $\beta$ and become lower when increasing $\beta$. For $\beta \le 0.5$, the final cross-helicity is ${\sigma }_c\simeq 0$, indicating that the initial velocity-magnetic field correlation is completely destroyed, and the final level of density and magnetic field intensity fluctuations are moderate: $r\lesssim 0.2$ and $m\lesssim 0.05$, respectively. For larger values of $\beta$ ($\beta \simeq 1$), the final cross helicity is ${\sigma }_c\simeq 0.5$, indicating that the parametric process is unable to completely destroy the initial ${\mathbf{v}}_{\perp }$-${\mathbf{B}}_{\perp }$ correlation. Saturation levels of r and m about one order of magnitude lower than in the case $\beta =0.1$ are obtained. Therefore, high-$\beta$ values reduce both the instability growth rate and saturation levels. In contrast, saturation levels do not depend on the spectral width of the initial pump wave. Moreover, for sufficiently high saturation levels (i.e., at low $\beta$), the final kinetic and magnetic energy spectra follow a Kolmogorov power law $\propto k^{-5/3}$ at small wavenumbers. Finally, the time evolution of spectra shows that the parametric process is non-local in the spectral space, even in the non-monochromatic case.

Planar arc-polarized Alfvén waves are large-amplitude Alfvénic fluctuations where magnetic field varies, describing arcs in the plane perpendicular to the propagation direction. Since the average ${\mathbf{B}}_{\perp }$ component is non-vanishing, such waves are obliquely propagating. Arc-polarized Alfvén waves are commonly detected in the solar wind (e.g., [40,41,42]). In many cases the tip of the $\mathbf{B}$ vector moves on a sphere with a nearly constant radius $\left|\mathbf{B}\right|$ (spherical-polarized waves). It has been shown that planar arc-polarized waves naturally form from large-amplitude, linearly polarized, obliquely-propagating monochromatic Alfvén waves: their nonlinear evolution leads to the formation of a secondary oscillating component, thereby achieving the arc-type polarization [43], along with a magnetoacoustic mode that propagates away from the Alfvénic perturbation. However, once formed, this kind of perturbation can be parametrically unstable. The spectrum of an arc-polarized Alfvén waves contains a band of wavelengths; therefore, these fluctuations can be considered as non-monochromatic. The parametric instability of planar arc-polarized Alfvén waves has been studied by Del Zanna [44]. Though planar arc-polarized waves can be expressed in the form (9) with a suitable choice of phase function $\varphi (·)$, Del Zanna [44] used a different approach to define the form of the pump wave, which allowed them to impose a well-defined propagation angle $\theta$ and amplitude perturbation $\eta$. These quantities represent tunable parameters of the model. The resulting wave describes an arc of less than ${180}^{\circ }$; the propagation angle has a lower limit ${\theta }_c$ that increases with increasing the perturbation amplitude $\eta$. Del Zanna [44] have studied the instability by means of MHD simulations in different cases, varying the wave amplitude $\eta$, the propagation angle $\theta$ and the plasma $\beta$. Parametric instability arose in all the considered cases, leading to the generation of compressive waves and back-scattered Alfvén waves. At low $\beta$, in the order of $\beta =0.1$, the back-scattered Alfvén wave grows to the point when it becomes dominant with respect to the initial pump wave, and subsequently, it undergoes secondary parametric instability. This leads to an evolution of the cross-helicity ${\sigma }_c$ that periodically changes sign in time, similar to what was found in reference [25]. Such an oscillatory behavior was not observed for larger values of $\beta$, when a monotonic decrease in $|{\sigma }_c|$ was observed. Moreover, the instability growth rate $\gamma$ decreased with increasing the propagation angle $\theta$, $\gamma$ being roughly proportional to $cos\theta$. Del Zanna [44] has also considered the case of an initially linearly polarized Alfvén wave, showing that the parametric process takes place after nonlinear effects have produced the arc-type polarization (see [43]). Therefore, the condition $\left|\mathbf{B}\right|=\mathrm{const}$ that characterizes arc-polarized waves is not critical for the development of parametric instability.

Most of the studies about parametric instability concern situations where all involved waves are periodic in the spatial domain. This holds both for monochromatic and for non-monochromatic cases, and is intended to represent a statistically homogeneous system, such as in turbulence, where all waves involved in the parametric process are allowed to repeatedly interact. In contrast, in a recent paper, Li et al. [45] considered the parametric instability for a localized wave packet propagating in a non-periodic spatial domain. In such a case, packets belonging to different wave modes, and then propagating at different speeds, interact only for a finite time. This kind of situation can model open systems where the pump wave has a localized source, such as in the Earth’s foreshock. Li et al. [45] studied this problem by 1D numerical hybrid simulations where ions were advanced in time as individual particles, and electron were represented as a massless neutralizing fluid. Open boundaries were modeled by thin absorbing layers localized just before the actual boundaries. A low-$\beta$ case was considered, which allowed the instability to develop before the pump wave left the domain. Results show that even in this case, the parametric instability takes place, though wave packets belonging to the involved wave modes tend to separate. Scaling laws of the residual packet size as a function of growth rate, amplitude and $\beta$ have been found. Moreover, localized density fluctuations, including cases of cavitation, and ion heating have been observed. Moreover, their results suggest that simultaneous observations of density and magnetic signatures of the instability are not always possible, due to the spatial separation of compressive and Alfvénic wave packets.

4. Parametric Instability in the Solar Wind

Now we briefly review the studies that have been performing concerning some attempts to directly compare the numerical results of the nonlinear evolution of the parametric instability with the observations of the real solar wind. One fundamental problem when trying to compare the results of numerical simulations with observational data is to establish well defined characteristic length and time scales. In fact, in MHD there is no characteristic scale fixed a priori, but everything is scale invariant in the sense that the equations do not change form when the length and time scales are multiplied by the same factor.

In Malara et al. (2001) [24], the authors tried to directly compare the results of the simulations obtained with initial conditions similar to those used in [23], in the case $\beta =1$, which is typically observed in the solar wind (generally, the plasma $\beta$ in the solar wind changes with the distance from the Sun and ranges between $0.5÷1.5$), with the radial evolution of the Alfvén ratio $e^{-}/e^{+}$ observed by Bavassano et al. (2000) [12], shown in Figure 1b (the ratio increases with the heliocentric distance in the inner heliosphere, then saturates around a value of about 0.5 at distances larger than 3 AU). In the case of the numerical simulation, the increasing radial distance of observational data is supposed to be equivalent to the time passing during the developing of the parametric instability.

The numerical simulations of the parametric instability in the presence of a turbulent, circularly polarized, pump wave at $\beta =1$ show that the initial Alfvénic state is only partially destroyed (${\sigma }_c\sim 0.5$), since the growth rate of the instability seems to decrease for increasing values od $\beta$, as happens in the monochromatic case. The simulation by Malara et al. (2001) [24] gives results that are qualitatively in good agreement with the behavior observed by Bavassano et al. (2000) [12], Figure 1b, albeit a direct quantitative comparison is difficult due to the fact that there is strong dependence on the length and time scales (we recall that, due to the Taylor hypothesis, the two are equivalent for solar wind data). Indeed, the Alfvén ratio found by Bavassano et al. [12] was obtained by computing hourly averages of the fluctuating energies, and a direct comparison with the numerical simulations requires, as already said, one to establish a direct correspondence between the scales of the solar wind and that of the MHD simulation.

To tackle this problem, the same authors [46] made another comparison of the Alfvén ratio evolution by computing the energies at different length scales in the simulations. Those results brought about the plot in Figure 8, showing the evolution in time (that corresponds, from what was said, to the evolution with distance in the solar wind data) of the Alfvén ratio $e^{-}/e^{+}$ for different length scales (corresponding to different time scales for the solar wind). Once again, in spite of the simplifying assumptions involved by the model, there is a striking qualitative agreement between the simulation and experimental data: the Alfvén ratio grows in time by starting from low values; then it saturates after some time, exactly as observed in the solar wind data. However, the saturation level of the Alfvén ratio is smaller than that reported by Bavassano et al. (2000) [12] (see Figure 1b). This is due mainly to the difficulty in establishing a direct correspondence between scales of the solar wind and of the simulations, but it may also be caused by several oversimplifications of the numerical model (1.5D model, no effects of the solar wind expansion, unrealistic dissipative terms to preserve numerical stability and so on). However, the qualitative agreement of the comparison is nontrivial.

Further direct comparisons between observational data and numerical simulations of the parametric instability have been attempted in several other papers [47,48]. In particular, in Bruno et al. (2004) [47], the authors tried to investigate how the magnetic field fluctuations in the turbulence observed in the solar wind are polarized, by carrying out a detailed analysis of how the tip of the magnetic field vector fluctuates in space, what kind of statistics it follows and how this may change with the distance from the Sun. This was done by separating the fast and slow speed streams of the solar wind during the Helios 2 mission at 0.3 and 0.9 AU, respectively. Quite interestingly, the authors found that the tips of the magnetic field vector in both the fast and slow wind seem to cover more uniformly the surface of a sphere at 0.3 AU, whereas the behavior of the tips is distributed nonuniformly in space at 0.9 AU, covering different zones of the “sphere” for longer times, then “jumping” to different zones. This behavior is shown in Figure 9, left panels (Figure 2 of the original paper). Analogous behavior was found by studying the displacements: $\delta \left|\mathbf{B}\right|/\langle \left|\mathbf{B}\right|\rangle$ in time, namely, how the modulus of the magnetic field changes in time with respect to its average intensity. Closer to the Sun (0.3 AU), the magnetic field intensity fluctuates in time in a manner closer to ordinary turbulence, whereas further from the Sun there are zones where the magnetic intensity exhibits fluctuations around a well defined value, then quite suddenly changes that value in a more “intermittent” fashion.

A similar analysis performed on the results of the parametric instability obtained by Primavera et al. (2003) at different times of the simulation (where, again, the simulation time was thought to mimic the distance from the Sun in the heliosphere) showed striking qualitative similarities among both the plots of the tips of the magnetic fluctuations (see Figure 9, right panels, corresponding to Figure 17 of the original paper), the plot of the magnetic field intensity fluctuations and the probability distribution functions (PDF) obtained from the data in the HSS of the solar wind.

In the same spirit, but much later, Bruno et al. (2014) [48] carried out a comparison of Helios 2 data at 0.3 and 0.9 AU with the results of the parametric instability, this time focusing on the evolution of the intermittency of proton density at different heliocentric distances. It was shown in the paper that in fast streams, the intermittency of density, measured through the flatness of the fluctuations, decreases with increasing heliocentric distance: see Figure 10, left panel (Figure 2d of the original paper). Again, a comparison was made in the article with the behavior of the flatness during both the development and in the saturation phase of the parametric instability from the simulations carried out in Primavera et al. (2003). Once again, the comparison showed remarkable qualitative agreement with the observations (see Figure 10, right panel, corresponding to Figure 7 of the original paper), with the flatness of the density fluctuations generated by the parametric instability decreasing with increasing time.

Starting from the open magnetic field regions of the Corona, the solar wind expands in the interplanetary space. A given plasma parcel occupies a growing volume as its heliocentric distance progressively increases. This phenomenon has an effect on the spectral anisotropy of fluctuations because it increases the length scales in the directions transverse to the radial, at least in regions where the wind expansion velocity is constant. The amplitude fluctuations relative to the mean field tends to increase. Moreover, expansion rotates the mean magnetic field off the radial direction, thereby generating the Parker spiral. To partially take into account the effects of the expansion of the solar wind on the development of the parametric instability, several papers dealing with numerical simulations have studied the problem in both spherical geometry and in expanding computational domains.

Tanaka et al. (2007) [49] set up a numerical simulation of the parametric instability in spherical geometry. They injected a monochromatic, circularly polarized perturbation with Alfvénic correlation (a ${\mathbf{Z}}^{+}$ Elsässer mode) into the simulation domain, made of a spherical shell where the quantities were changing in the radial direction only, and studied how it propagates in the radially expanding solar wind. The study carried out by the authors used different injecting frequencies and found production of both inward propagating (${\mathbf{Z}}^{-}$) fluctuations and density perturbations. However, the study was conducted for very low values of the plasma $\beta <0.2$, since the evolution of the waves was supposed to happen quite close to the Sun (the computational domain ranges between $1R_{\odot }$ and $40R_{\odot }$, where $R_{\odot }$ is the solar radius).

The effects of solar wind expansion on the development of parametric instability have been considered by Del Zanna et al. [50], who performed numerical simulations using an “expanding box model” [51,52]. Such a model follows a (small) solar wind plasma parcel in its journey using a co-moving reference frame and assuming a constant expansion velocity and periodicity boundary conditions in a 3D spatial domain. The effects of wind expansion translate into additional terms that appear in the MHD equations. Del Zanna et al. [50] have considered low-frequency waves ($f=({10}^{-4}$–${10}^{-2})$ Hz, in the spacecraft reference frame) and various possibilities for the initial Alfvénic pump wave—namely, circularly polarized waves in parallel propagation, arc-polarized waves and non-monochromatic waves. Simulations follow the time evolution during both the linear phase and the nonlinear saturation stage. Various effects due to expansion come into play: in general, the instability growth rate $\gamma$ tends to decrease with increasing the plasma $\beta$, and solar wind expansion makes $\beta$ increase with increasing radial distance $r\left(t\right)$; moreover, the relative fluctuation amplitude increases with $R\left(t\right)$. On the basis of the above mechanisms, a decrease in the growth rate as $\gamma \sim {\left[r\left(t\right)/r_0\right]}^{-2/3}$ is expected [26]. On the other hand, if the development of the Parker spiral is taken into account, after a certain distance, the mean field scales as ∼${\left[r\left(t\right)/r_0\right]}^{-1}$, leading to $\gamma \sim {\left[r\left(t\right)/r_0\right]}^{-5/12}$. In general, a decrease in $\gamma$ with the radial distance is expected, which is less pronounced as the propagation angle increases. Simulation results by Del Zanna et al. [50] show that while the expansion parameter increases, both the growth rate and the saturation level tend to decrease; this happens both for the circularly-polarized parallel propagation and for the arc-polarized oblique-propagating pump wave, indicating that, in general, the wind expansion does not favor the development of the parametric process. The stabilizing effect is more pronounced for low-frequency waves, and below a certain frequency the instability can be suppressed; this could explain the persistence of these outgoing waves in the solar wind. Another feature found in the simulation results [50] is the generation of daughter waves transverse to the local background magnetic field. This aspect is more evident for monochromatic pump waves, but it is completely absent in the non-monochromatic case, where no significant transverse modulation is found.

Tenerani and Velli (2013) [26] in another paper used instead the accelerating expanding box model, which mimics the expansion of the solar wind by taking into account the acceleration of the solar wind. The simulations have been initialized with a condition similar to Equation (9), with a circularly polarized, one dimensional, initial perturbation, both monochromatic and with an initial spectrum. However, the study was performed in a low-$\beta$ plasma ($\beta =0.1$). The results of those numerical simulations indicate that the parametric instability is indeed affected by the expansion of the wind, which can even lead to stabilization of the instability. However, as the authors themselves pointed out, the phenomena can have a different importance in multi-dimensional simulations; therefore, further investigations are needed in order to clarify how the expansion really affects the evolution of the instability.

Several attempts have further been made to study several by-products of the development of the parametric instability, for instance, to study the reflection properties (that is, the production of inward propagating Elsässer modes starting from an outward propagating one) of an initial turbulent perturbation with Alfvénic correlation, to study the decay of Alfvénicity with the distance from the Sun. For instance, Shoda and Yokoyama (2016) [53] set up a one-dimensional numerical simulation starting with a linearly polarized Alfvén wave. The authors claim to have carried out this study to explain the decay of Alfvén waves in both the solar corona and the solar wind. The initial conditions the authors considered were a red-noise ($f^{-1}$) initial Alfvénic turbulent state, with linearly polarized fluctuations, and the plasma was supposed to be isothermal. However, this initial state is not an exact solution of the MHD equations: in fact, a linearly polarized large amplitude wave, like the one used by the authors, immediately decays due to the pressure unbalance induced by the large amplitude fluctuations, and this decay process is in no way related to the parametric instability. Furthermore, the fluctuations observed in the solar wind appear to be mainly arc-polarized, not linearly polarized. Finally, the use of a spectrum in $f^{-1}$ is debatable: such a spectrum is typical of the large scale fluctuations in the solar wind, and according to some authors, it is caused by the expansion of the wind; some others think that it may be caused by a turbulent evolution. In summary, although the study is interesting in itself, some of the choices made by the authors of the initial conditions are questionable.

To improve the above mentioned study, the same authors (Shoda et al., 2018 [54]) numerically investigated the effects of the expansion of the solar wind by considering a simulation in spherical geometry in which the only variations were along the radial direction. The authors injected from the left boundary (which is supposed to be the one closer to the Sun) of the computational domain either a monochromatic or a broadband fluctuation under the form of a circularly polarized ${\mathbf{Z}}^{+}$ Elsässer mode and investigated numerically the development of the instability and its evolution, by finding that, as in Tenerani and Velli (2013) [26], the instability can even be suppressed by the expansion of the wind. Though the study in itself is interesting, it suffers from some drawbacks: the initial wave is considered either as monochromatic or with a $f^{-1}$ spectrum, the latter being consistent only with the low-frequency part of the spectrum of the fluctuations in the solar wind. Moreover, the finite volume scheme used by the authors in the simulations can heavily dump high frequency fluctuations and destroy the non-local interactions typical of the parametric instability. Finally, as already mentioned by Tenerani and Velli (2013) [26], the effects of considering a multi-dimensional domain can change drastically the suppression of the instability induced by the solar wind expansion.

As said above, observations of the solar wind show the presence of transverse correlated magnetic-velocity field fluctuations, and an anti-correlation between density and magnetic field intensity fluctuations, attributable either to the existence of slow-magnetosonic fluctuations or pressure-balanced structures or kinetic slow waves. However, such modes can be heavily damped because of kinetic effects (Landau damping, etc.), thereby requiring a local mechanism that can produce such anti-correlated fluctuations.

Shi and Li (2017) [55], in an attempt to phase out the importance of the parametric decay instability in justifying the presence of anti-correlated density and magnetic field intensity fluctuations observed in the solar wind, performed some numerical simulations using the Pluto code ([56]) of a 3D MHD turbulent state, initially generated by studying the decay of a circularly polarized, monochromatic Alfvén wave propagating in a turbulent state made of the steady evolution of a set of six waves. The authors performed a detailed study of the development of the instability by changing the amplitude of the initial Alfvénic fluctuations (with respect to the average magnetic field) between 0.1 and 0.5 and a value of $\beta =0.5$. Although the initial conditions used by the authors are not really close to those of the solar wind (especially given the fact that no condition on the constant intensity of the global magnetic field was imposed) and the values of $\beta$ that are only typical for distances closer to the Sun, along with the fact that Pluto is a finite volume code, which generally involves enhanced dissipation of turbulent states due to the numerical method itself (which could justify the reduced growth rates found by the authors with respect to the theoretical ones), the simulations showed that compressible slow-magnetosonic fluctuations are indeed produced by the parametric instability, thereby pointing out the importance of the instability in explaining the observations of the solar wind.

In the same spirit, Bowen et al. (2018) [57] investigated further the possibility of generating such fluctuations as a sub-product of the parametric decay instability in an observational framework. They used data from Wind spacecraft from 1 January 1996 to 31 December 2005 and investigated the cross correlations between the fluctuations of the Elsässer variable ${\mathbf{Z}}^{-}$, the magnetic field intensity and the density fluctuations, along with an assessment of the theoretical growth rates and kinetic damping rates predicted theoretically [18]. The authors showed that the parametric decay instability can heavily influence the behavior of the turbulence in the solar wind and actively produce slow modes with anti-correlated density and magnetic field intensity fluctuations, thereby pointing out the parametric instability as a fundamental process in explaining the dynamics of the turbulence observed in the solar wind.

A study of parametric instability in a turbulent medium based on weak-turbulence theory has been performed by Chandran [58,59]. In this kind of approach, it is assumed that the time scale of nonlinear interactions is much longer than the inverse frequency of wave modes. This allows one to preserve some properties of linear modes, such as dispersion relation, at least to a certain degree of approximation. This leads to a hierarchy of equations for the moments, defined as the structure functions of fluctuations, where the derivative of the nth-order moment is expressed in terms of the $(n+1)$th moment (Galtier et al. [60]). This hierarchy is closed by a random phase assumption, which allows the fourth-order correlation functions to be expressed as products of second-order correlation functions. In the approach followed by Chandran [58,59] a collection of interacting quanta was considered belonging to different propagating modes, where the resonance conditions translated into the conservation of momentum and energy during quanta interactions. In particular, a 3D configuration was considered at low $\beta$ values. At each interaction, an Alfvénic quantum decayed to form another oppositely-propagating Alfvénic quantum plus a slow magnetosonic quantum, all propagating along the mean magnetic field. An equation for the time derivative of the both Alfvénic spectra was derived (wave kinetic equation) that describes both an exponential growth of daughter waves in the linear regime and an inverse cascade in the general case. In reference [59], numerical solutions of wave kinetic equation were described: starting from an initial condition, intended as a representative of the spectrum of Alfvén waves launched from the Sun, the spectrum $e^{+}\left(f\right)$ of outward-propagating Alfvén waves evolved to form a broad $f^{-1}$ range (f being the frequency in the spacecraft reference frame), while the spectrum $e^{-}\left(f\right)$ of inward-propagating Alfvén waves grew until reaching a $f^{-2}$ power law. Such features positively compare with what has been observed in fast-speed streams of the solar wind. While the conditions for the applicability of weak-turbulence theory are at least marginally satisfied at heliocentric distances $R<0.4$ AU, the low-$\beta$ assumption does not hold in the solar wind, where typically it is $\beta \sim 1$. This represents a limitation for the direct applicability of this theory to the solar wind case; however, it can at least give indications on the role that the parametric process can have in determining the spectra of Alfvénic fluctuations in solar wind fast-speed streams.

More recently, Primavera et al. (2019, 2021) [61,62,63] carried out a series of numerical simulations, where they built up initial conditions more similar to those of the solar wind after the Alfvénic point, where only one of the two possible Elsässer modes is dominant, with an extended, anisotropic spectrum of Alfvénic fluctuations. The initial conditions satisfied the constraint that the global magnetic field (background field plus fluctuations) has constant intensity, to mimic spherically polarized, large-amplitude Alfvénic fluctuations. The simulations were carried out in a 2.5-dimensional periodic box (all the vector quantities retained three components, but they only depended on two spatial coordinates). At odds with the 1D case previously studied, it is not easy to construct such an initial condition in a multi-dimensional space by respecting the constraint on the constancy of the global magnetic field.

The magnetic field was constructed as a constant average field $B_{0,x}$ directed along the x axis, plus a fluctuation $\Delta \mathbf{B}$ having three components:

$$\mathbf{B}=(B_{0,x}+\Delta B_x){\widehat{\mathbf{e}}}_x+\Delta B_y{\widehat{\mathbf{e}}}_y+B_z{\widehat{\mathbf{e}}}_z.$$

The components of the magnetic field $\Delta B_{x,y}$ were chosen, respectively, as obtained from the z component of a 2D potential vector $\mathbf{A}=A(x,y){\widehat{\mathbf{e}}}_z$:

$$\begin{array}{cc}\hfill \Delta B_x& =\frac{\partial A(x,y)}{\partial y}-\frac{\partial \langle A(x,y)\rangle }{\partial y},\hfill \\ \hfill \Delta B_y& =-\frac{\partial A(x,y)}{\partial x}+\frac{\partial \langle A(x,y)\rangle }{\partial x}.\hfill \end{array}$$

The condition that $\left|\mathbf{B}\right|$ is constant requires that:

$$B_z=\sqrt{B_T-{(B_{0,x}+\Delta B_x)}^2-\Delta B_y^2},$$

where $B_T$ is the total constant intensity. Notice that such a definition for $B_z$ implies that it is positive definite (as is the case for an arc-polarized fluctuation), and it has an average part $B_{0,z}$ and a fluctuating part $\Delta B_z$. Therefore, the resulting total magnetic field will have an average part along x and z directions: ${\mathbf{B}}_0=B_{0,x}{\widehat{\mathbf{e}}}_x+B_{0,z}{\widehat{\mathbf{e}}}_z$, and a fluctuating part $\Delta \mathbf{B}$ along the three directions x, y and z.

To simulate the Alfvénic correlation of the fluctuations, the velocity field was chosen as:

$$\Delta \mathbf{v}=-\frac{1}{\sqrt{{\rho }_0}}\Delta \mathbf{B},$$

where ${\rho }_0$ is the (constant) initial density of the plasma. In order to have a spectrum of the fluctuations, the potential vector $A(x,y)$ was built in the Fourier space as:

$$\widehat{A}(k_x,k_y)=\left\{\begin{array}{cc}{A}_0\frac{exp[-6{(k_y/k_c)}^6+i{\varphi }_{k_x,k_y}]}{{[1+{(k_y/k_{y0})}^4]}^{1/2}}\hfill & \mathrm{for}|k_x|\le k_{x,\max }\hfill \\ 0\hfill & \mathrm{for}|k_x|>k_{x,\max }\hfill \end{array}\right.$$

where $k_{x,\max }=k_{0y}=4$ and $k_c=50$. This choice of the potential vector yielded an anisotropic spectrum in the $k_x$-$k_y$ space, with perpendicular (to the mean magnetic field) wave vectors $k_y$ dominating over the longitudinal component $k_x$, with a spectrum of $k_y\propto k^{-5/3}$ approximately. This situation is similar to the one observed in the true solar wind.

In this (supposedly) more “realistic” initial situation, Primavera et al. [62,63] showed that the parametric instability still takes place, and even, remarkably, that the growth rates of the instability are slightly larger for larger values of the plasma $\beta$, which is the opposite situation to the theoretical linear prediction by Sagdeev and Galeev (1969) [16] and to what was found in the 1D simulations. This points out that the role of the parametric instability can be even more fundamental for the solar wind, since one of the main criticisms against the applicability of this mechanism to the solar wind was just the fact that the growth rates can be too small to be effective at the typical values of $\beta$ observed in the solar wind.

Another striking feature of the results obtained in those simulations [61,62,63] is the fact that the fluctuations produced by the instability seem to localize in some special zones of the computational domain, with the formation of large-amplitude, solitary fluctuations (resembling solitons in fluids) which continue to propagate mostly along the average magnetic field direction. Moreover, such fluctuations produce nearly pressure balanced structures at almost all scales, with strong anti-correlation between the density and magnetic field intensity fluctuations, and some positive correlation only at large scales.

Some of the results obtained in the 2.5D simulations of Primavera et al. [61,62,63], in particular, the fact that the instabilities have larger growth rates at higher values of $\beta$ and the fact that the fluctuations appear to be mostly localized in small-scale structures propagating on a turbulent background, have been confirmed by Gonzales et al. (2020) [64], who used a completely different hybrid Vlasov–Maxwell numerical application (CAMELIA), albeit starting with an initial condition made of a monochromatic wave.

In situ measurements in the solar wind have revealed the presence of structures denoted as “switchbacks” characterized by a local inversion of the magnetic polarity, embedded in the turbulent flow emanating from the Sun. Switchbacks are characterized by a pronounced Alfvénic-like correlation between magnetic field and velocity fluctuations, by a nearly constant $\left|\mathbf{B}\right|$ and by a low level of density fluctuations. Therefore, they can be considered as Alfvénic disturbances with very large amplitudes, comparable with those of the background magnetic field, so then one is able to locally change sign to the dominant magnetic field component. In a switchback, magnetic lines follow zigzagging paths, as revealed by the analysis of strahl electrons which come from the Sun but seem to locally revert their motion inside a switchback. Switchbacks have been observed in fast streams at heliocentric distances comprising between $R\sim 0.3$ AU [65] and $R\ge 1.3$ AU [66,67]; recently, a vast literature has been published on switchbacks detected at smaller distances by the Parker Solar Probe [68]. Such observations show that the switchback occurrence frequency decreases with heliocentric distances, and that these structures occur in patches. Since a switchback can be considered as an Alfvénic large-amplitude fluctuation, the question of its stability against the parametric process is apt. Such a problem was studied by Tenerani et al. [69], who built-up a 2.5D model for a localized switchback, using a technique similar to that in reference [62] to obtain an initial magnetic fluctuation that satisfies both constraints, $\nabla \times \mathbf{B}=0$ and $\left|\mathbf{B}\right|=\mathrm{const}$. The magnetic perturbation $\delta \mathbf{B}$ was in the order of the background magnetic field, so that the polarity of the total magnetic field $\mathbf{B}={\mathbf{B}}_0+\delta \mathbf{B}$ would locally invert. Due to the additional conditions $\mathbf{v}=\delta \mathbf{B}/\sqrt{4\pi {\rho }_0}$ and $\rho ={\rho }_0=\mathrm{const}$, the structure propagated along ${\mathbf{B}}_0$ at a constant speed without distortions. The spatial domain was periodic so that perturbations were allowed to re-enter the domain and interact among each other, even if their propagation velocities were different. Various runs were performed with different values of $\beta$ and of the periodicity length. Adding a small amplitude perturbation the instability was activated. Density perturbations were observed to exponentially grow until saturation, with a growth rate that was larger for low values of $\beta$. At saturation, the level of the cross-helicity was moderately reduced with respect to the initial value, but ${\sigma }_c$ can locally revert its sign, indicating that high-level counter-propagating Alfvénic fluctuations have locally been generated. Such results give information about the lifetimes of switchbacks in the solar wind and indicate that if switchbacks are originated in the lower corona, they probably survive at least until they are detected by the Parker Solar Probe. The observed decrease in the occurrence frequency of switchbacks with heliocentric distance could be related with the parametric instability, since such a process tends to destroy the Alfvénic correlation that characterizes these structures. Some caveats about such conclusions is that the periodicity condition probably leads to underestimation of the life time; on the other hand, large-scale inhomogeneities encountered by a switchback will shorten its life by destroying its typical features, such as the uniformity of magnetic field intensity and/or the Alfvénic correlation.

5. Conclusions

The parametric instability of an Alfvénic fluctuation is a mechanism that tends to reduce the velocity–magnetic field correlation typical of fluctuations of this kind, while at the same time increasing the level of density and magnetic field intensity fluctuations. The parametric process takes place in consequence of a coupling among different propagation modes, belonging both to Alfvénic and to compressive branches. This coupling involves different wave vectors; therefore, the parametric process is considered to be non-local in the spectral space. These properties characterize the parametric process in many different configurations, from simpler to more complex ones. In this review, we have illustrated the main results concerning parametric instability in many different contexts and configurations, starting from the earliest results dating several decades ago, and getting all the way to the most recent ones.

The theory of parametric instability is relevant in the context of the evolution of MHD turbulence in solar wind. In fact, Alfvénic fluctuations, characterized by a high-velocity magnetic field correlation and by a low level of compressive fluctuations, are dominant in most of the solar wind, namely, in low-latitude high-speed streams, in the high-latitude wind and in the recently-detected Alfvénic slow streams. A peculiar aspect of such fluctuations is that they represent an exact solution of MHD equations, at least if the conditions ${\sigma }_c=\pm 1$, $\rho =\mathrm{const}$ and $\left|\mathbf{B}\right|=\mathrm{const}$ are exactly verified, regardless of the amplitude and of the dimensionality of the fluctuations. This means that fluctuations of such kind do not evolve in time but propagate at a constant speed without distortions. Therefore, investigating the stability of such fluctuations is crucial to understanding mechanisms which can affect the observed evolution of fluctuation properties in the heliosphere. In particular, observations show that the cross-helicity $|{\sigma }_c|$ of solar wind Alfvénic fluctuations tends to decrease as the heliocentric distance increases, and the levels of density and magnetic field intensity tend to increase. This evolution is similar to that observed during the evolution of parametric instability, which therefore has been considered as a possible mechanism acting in the evolution of solar wind turbulence.

The above considerations have motivated, at least partially, the investigation of theoretical properties of the parametric process in various configurations that have progressively approached better modeling of solar wind fluctuations. Other motivations come from the domain of laboratory plasmas, but those are outside of the scope of the present review. Starting from the linear stage in the 1D monochromatic case at a small or arbitrary amplitude, linearly or circularly polarized, other situations have been examined–namely, a 1D monochromatic pump wave with multi-dimensional perturbation and 1D non-monochromatic cases, in both linear and nonlinear stages; dispersive effects on the instability dispersion relation have been studied, and those of kinetic effects on the damping of compressive daughter fluctuations; the effects of the solar wind expansion on parallel and oblique-propagating waves have also been included; the parametric instability of localized wavepackets in non-periodic configurations has been studied; the role of the parametric process in the proton heating has been investigated; finally, a case of an Alfvénic pump wave with a 2D spectrum of wave vectors has been considered. In most of these cases, despite the complex geometry or other factors, the parametric process is active and determines both cross-helicity reduction and compressive fluctuation increases. This proves that the parametric process is a very robust mechanism that is very likely to play a role in the evolution of the solar wind turbulence. Moreover, several studied have been devoted to investigating the application of theoretical results to solar wind fluctuations.

As a final remark, we mention a recent paper [70] where a fully 3D, constant-magnitude magnetic field with an extended spectrum was explicitly obtained by a numerical procedure. This solution can be used to build-up a 3D Alfvénic state that closely mimics the Alfvénic turbulence in the solar wind, and subsequently, they investigated its possible parametric instability. We plan to perform this study in the near future.