Solar wind predominantly permeates the whole heliosphere, providing an indispensable medium to study collisionless plasma. It develops a strong turbulent character on expansion, and for decades, intensive efforts have been directed to understand solar wind and its turbulent nature through various space missions like SOHO (Solar Heliospheric Observatory), Ulysses, Voyager, Helios, Yohkoh, FREJA, POLAR, FAST, Cluster, and TRACE (Transition Region and Coronal Explorer). Emanating from the solar corona and expanding outward into space (Parker, 1958), solar wind acts as a natural laboratory (Bruno and Carbone, 2013) for in situ spacecraft measurements to investigate solar wind turbulence. To understand the turbulent system and perform multipoint measurements, the NASA MIDEX mission HelioSwarm is in a Phase-A study (Klein, 2019; Spence, 2019; Hautaluoma and Fox, 2020). The latest NASA mission, the Parker Solar Probe, is designed to probe the mechanisms leading to acceleration of solar wind. At the closest approach to the Sun, the Parker Solar Probe has progressed one step toward reaching the Sun to explore the mysteries of the evolution of the Sun and provide deeper insights into the flow of energy from the solar corona to the accelerating solar wind. The evolution of astrophysical environments is significantly affected by the turbulent cascade of energy and, so to identify with the dynamics of energy dissipation, is therefore of cardinal importance to the astrophysics communities.

In a turbulent medium like solar wind, the magnetic field lines constantly break and reconnect at some scales (Franci et al., 2017; Mallet et al., 2017; Vech et al., 2018), making magnetic reconnection an inherent part of a turbulent cascade. This process is associated with the release of large amount of energy and is responsible for the transfer of energy between different length scales, thereby leading to the acceleration of particles in solar wind (Lazarian et al., 2020). This review presents an outline of the solar wind plasma portraying the influence of the nonlinear wave–wave interactions (Kraichnan, 1965; Howes & Nielson, 2013; Iroshnikov P.S., 1964; Pezzi et al., 2017a, 2017b; Roberts et al., 2017; Narita, 2018; Roberts et al., 2022) and their role in understanding the nature of turbulence through the transfer of energy from large scales to smaller scales. The understanding of mechanisms which play a decisive part in creation of astrophysical environments through the dissipation of turbulent energy to heat has always been an issue of debate. Different schools of thought have structured this debate. In the first, Alfvénic turbulence is considered one of the processes responsible for the transfer of energy (Coleman, 1968; Isenberg and Hollweg, 1982; Tu et al., 1984; Hu et al., 2000; Isenberg, 2004). Being low-frequency magnetohydrodynamic (MHD) modes and difficult to dissipate, they are able to permeate the whole solar atmosphere and are ubiquitously present in the solar wind. An illustrative characteristic of magnetized turbulence is the propensity to form sheets of current density that are liable to magnetic reconnection (Matthaeus & Lamkin, 1986; Biskamp & Welter, 1989; Dmitruk & Matthaeus, 2006; Retinò et al., 2007; Servidio et al., 2009; Comisso and Sironi, 2019). These reconnecting current sheets are typical sites of particle acceleration and magnetic energy dissipation (Dmitruk et al., 2004). Concurrently, it has long been known that particles can gain energy through random scattering by turbulence fluctuations (Kulsrud & Ferrari, 1971). Therefore, turbulence fluctuations and magnetic reconnection work in alliance, and for a comprehensive understanding of the physics in a turbulent surrounding we need to have a detailed examination of their interplay.

The solar wind is inhabited by Alfvénic fluctuations spanning from fraction of a second to several hours. Their presence in the astrophysical plasma is endorsed by various observational evidences (Belcher and Davis, 1971; Cirtain et al., 2007; He et al., 2009; Okamoto et al., 2007). The restoring force and inertia for the Alfvén wave are provided by the magnetic tension and the ion mass, respectively. Ideal MHD equations administer the propagation of this non-dispersive Alfvén wave which is now called the shear Alfvén wave. The Alfvénic wave disturbance moves with no attenuation with distance along the background magnetic field, and this remarkable property has important ramifications for the transport of energy in plasma fluids. The nonlinear phenomena are of prominent interest to properly assess the acceleration and transportation of the particles in the solar wind. Alfven waves may come across instabilities and may convert into other dispersive modes which may provide a pathway to carry large amounts of energy and then dissipate it as heat (Bogdan et al., 2003).

When the Alfven waves attain wavelengths comparable to the ion gyroradius in the direction perpendicular to the background magnetic field, they are known as kinetic Alfvén waves (KAWs) (Podesta, 2013; Narita et al., 2020). KAWs are believed to play an indispensable role in particle heating and acceleration mechanisms (Wu, 2003; Wu and Chao, 2004). The transfer of energy from MHD scales to kinetic scales is also possible through the interaction of KAWs with large-scale MHD waves (Zhao et al, 2011). The in situ measurements in the solar wind back up the idea of an Alfvén wave turbulent cascade in the perpendicular direction at the ion or electron scales producing the KAWs (Howes et al., 2008a, Howes et al., 2008b, Howes et al., 2011; Alexandrova et al., 2013; Sahraoui et al., 2009; Zhao et al., 2013). KAWs are also dominating at proton kinetic scales (He et al., 2009; Roberts et al., 2017) and also shorter scales (Chen et al., 2010a). In addition, they can also couple with high-frequency modes (Zhao et al., 2013). The shear and kinetic Alfvén waves account for various nonlinear effects like parametric processes such as three-wave decay interactions, modulational instability, and the background plasma number density being modified by the ponderomotive force of the Alfvén wave. The nonlinear interaction of the KAW and shear Alfvén wave has been intensively studied by many authors and energy transfer processes so as to interpret the observations to have a deeper insight into the turbulent cascade of energy (Zhao et al., 2013). With the finite frequency correction, i.e., when the frequency of the Alfven wave becomes comparable to but less than the ion cyclotron frequency, we get circularly polarized dispersive Alfven waves (DAWs). This dispersion of the wave also occurs when the wavelength is around the ion inertial length, which many authors (Meyrand & Galtier, 2012; Ghosh, et al., 1996) consider the Hall term, and this Hall term contributes significantly to the transfer of energy over and across the small scales as well as has an important role in the increased reconnection rate (Shay et al., 2001; Smith et al., 2004). The interaction of these dispersive Alfvén waves undergoing filamentation with the pre-existing chain of magnetic islands may also contribute toward the turbulent cascade of energy in the solar wind.

Nonlinear processes being dominant in the transfer of energy from long wavelength magnetic fluctuations to shorter wavelengths, the study of the frequency spectrum is of utmost importance to unravel the nature of turbulent plasma. The energy spectrum depicting the turbulence scaling comprises the energy injection scales followed by the energy-containing scale, known as the forcing range or injective range, marking the energy source. Next, the “inertial range,” also called the intermediate range, marks the energy transfer process to smaller scales. This range is often described as the “dissipation range,” “dispersion range,” or the “scattering range.” In this range, the fluctuations are converted to thermal energy, thereby causing the heating of particles. For magnetic fluctuations at very large scales (i.e. f ≤ 10–⁴ Hz), the power spectra go as k ⁻¹ . This range is the energy reservoir feeding the turbulent cascade and so called the 1/f range (Matthaeus and Goldstein 1986; Chandran 2018; Matteini et al., 2018). Various measurements account for the Kolmogorov k ^−5/3 spectrum in the inertial range (10 ⁻³  Hz ≤ f ≤ 10 ⁻¹ Hz) followed by steepening at ion kinetic scales. A spectral break appears around kρ _s >1 (where ρ _s is the ion gyroradius), and the spectral index lies between -2 and -5 (Bale et al., 2005; Shaikh and Shukla, 2009; Sahraoui et al., 2009). Between the ion scales and the electron scales, a small scale turbulent cascade is established (Alexandrova 2009). As the turbulent cascade crosses the ion scales and before reaching the electron scales (i.e. 3 ≤ f ≤ 30 Hz), the magnetic spectra follow ∼ k^−2.8 (Alexandrova et al., 2009; Chen et al., 2010a; Sahraoui et al., 2010). The spectral index is conspicuous of the energy transfer phenomena. Many pioneering works have been carried out to understand the dynamics and the different nonlinear processes responsible for this steepening (Leamon et al., 2000; Sahraoui et al. 2010; Rudakov et al., 2011).

The prime candidates contributing to the spectral properties and the steepening of the spectra are the various nonlinear processes like transverse collapse or filamentation due to KAW, its interaction with other wave modes, and/or the formation of coherent (magnetic) structures resulting from the current sheets. In the literature, the analysis of the field fluctuations from the proton to the electron scales shows the presence of current sheets or possible coherent structures of sub-proton scales which are possible sites of magnetic reconnection and energy dissipation (Perri et al., 2012). The current sheets may be formed self-consistently from the Alfven wave-driven turbulence as discussed by Tanbarge & Howes, (2013). These coherent structures play an important role in driving the nonlinear transfer of energy to the smaller scales as they make some additional self-consistent energy injection available (Ma et al., 1995; Sturrock et al., 1999) and thus support the continuation of the turbulence over the smaller scales (Cerri & Califano, 2017). Therefore, it is required to further examine if this turbulent energy transfer across and below the ion scales takes place as a result of instabilities or some other mechanism such as magnetic reconnection that causes the formation of coherent/localized structures or if it is both. A plethora of studies show that instabilities such as those driven by the temperature anisotropy (Gary and Lee, 1994; Verscharen et al., 2014), stream-shear-driven instabilities (Roberts et al., 1992), and parametric instabilities (Longtin and Sonnerup, 1986; Brodin and Stenflo, 1988; Viñas and Goldstein, 1991; Stenflo and Shukla, 2007; Primavera et al., 2019) such as filamentation, all may possibly lead to the dissipation of solar wind turbulence. At the same time, observations also show that reconnection can typically cause turbulence in the solar wind (Vörös et al., 2014). The spectral scaling of these fluctuations driven by the reconnection flow (obtained using the WIND spacecraft data) bears resemblance with the observations in the inertial and dispersive regime of the solar wind. Thus, these two processes, that is, turbulence and magnetic reconnection, work in alliance, and an associative study is required to be carried out.

Here, we revisit the nonlinear effects caused by dispersive Alfvén waves and KAWs to comprehend the dynamics of solar wind turbulence. Part A discusses the nonlinear effects and the turbulence due to the KAW presenting the effect of initial conditions on the spectra and the detailed explanation governing the evolution pattern through a simplified model. In part B, we study the nonlinear evolution of dispersive Alfvén waves in the vicinity of pre-existing chains of magnetic islands. The wave becomes dispersive due to the wave frequency which is finite but less than the ion cyclotron frequency.

Many simulations backed up with observations have led to a deeper understanding of turbulence at 1 AU (Howes and Quataert, 2010; Sahraoui et al., 2010; Narita et al., 2020) unwinding the energy transfer processes. This section predominantly focuses on the role of KAWs in understanding the dissipation of energy through different nonlinear processes. One of the processes leading to energy transfer is filamentation or transverse collapse which has been widely investigated. The sensitivity of the nonlinear coupling of KAWs/IAWs resulting in the transverse collapse on different initial conditions is addressed. Considering the wave propagation in the x–z plane and background field   B 0 z ^ and using a two-fluid model, the following system of equations is derived in the dimensionless form (Gaur and Sharma, 2015): i ∂ B y ∂ t + i ∂ B y ∂ z + 2 i c 1 ∂ B y ∂ x + ∂ 2 B y ∂ x 2 + c 2 ∂ 2 B y ∂ z 2 + n B y = 0 , ∂ 2 n ∂ t 2 − ∂ 2 ∂ t 2 ∂ 2 n ∂ x 2 − β ∂ 2 n ∂ z 2 = − ∂ 2 ∂ t 2 ∂ 2 ∂ x 2 | B y | 2 − β ∂ 2 ∂ z 2 | B y | 2 . (1)

The localized structures of KAWs with the magnetic field B y in the transverse direction are studied at 1 AU under different initial conditions by performing numerical simulation of this set of equations. Here, c 1 = k 0 x ρ s , c 2 = k 0 x 2 ρ s 2 / 4 , and β = c s 2 / V A 2 . The equation is normalized using the parameters x n = ρ s , z n = 2 / k 0 z , t n = ( 2 ω / V A 2 k 0 z 2 ) , n n = n 0 , and B n = [ { 1 − η ( 1 + δ ) } V A 2 k 0 z 2 / 16 π n 0 T ω 2 ] − 1 2 . Here, η = ω 2 ω c i 2 and δ = m e k 0 x 2 m i k 0 z 2 , and T e + T i = T , k 0 x is the perpendicular wave vector component and   k 0 z is the parallel wave vector component to   z ^ B 0 , where ω refers to the KAW frequency, c s   is the acoustic speed, V A is the Alfvén speed, and ρ s   is the ion acoustic gyroradius. Three different perturbations imposed on a uniform plane KAW are periodic perturbation (IC-1), Gaussian perturbation (IC-2), and the third random perturbation (IC-3): IC - 1         B y ( x , z , 0 ) = B y 0 ( 1 + ε ⁡ cos ( α x x ) ) ( 1 + ε ⁡ cos ( α z z ) )   and   n ( x , z , 0 ) = | B y ( x , z , 0 ) | 2 IC - 2         B y ( x , z , 0 ) = B y 0 ( 1 + ε ⁡ exp ( − x 2 r 01 2 ) ) ( 1 + ε ⁡ exp ( − z 2 r 02 2 ) ) IC - 3         B y ( x , z , 0 ) = B y 0 ( 1 + ε ⁡ exp ( 2 π i θ ( x ) ) ) ( 1 + ε ⁡ exp ( 2 π i θ ( z ) ) )

Here, the initial amplitude of the main KAW is B y 0 = 1 , ε = 0.1 is the magnitude of the perturbation, r 01 (normalized by x n ) is the transverse scale size of the perturbation, r 02 (normalized by z n ) is the longitudinal scale size of the perturbation, and θ ( x ) and θ ( z ) are the random variables uniformly distributed on [0,1] (Sharma et al., 1996). The random value for theta is attributed at each grid point in the x–z plane. A seed value was initially provided, and random variables were generated and uniformly distributed on [0,1] in the x–z plane.

Figures 1A–3A,C present the simulation results indicating the splitting of pump KAW at different instances of time with periodic, Gaussian, and random perturbation forms, respectively, in real space alongside contour plots in the Fourier space, i.e., Figures 1B–3B,D. The figures demonstrate that there is no regularity in the filament formation with the formation of the most intense structures at the early time and an increase in complexity at later times. The contour plot shows the dependence of | B y k | 2 on Fourier modes. As is noticeable in these figures, the filamentary structures obtained have different intensities and patterns. Also, there is a varied scale size of the structures under the three initial conditions ( ≈ 0.3 ρ s with periodic, ≈ 1.6 ρ s with Gaussian, and ≈ 0.5 ρ s with random perturbation). For β   > m e m i , the transverse size of Alfvén vortex tubes is of the order of ρ s (Chmyrev et al., 1988).

Figure 4 illustrates the effect of formation of localized structures on the wave number spectrum. In the power spectra obtained by plotting   | B k | 2 against k ,   k x = 0 , the scaling law approaches k − 5 / 3 for   k < 1 followed by a spectral break at around k ≈ 1 . The scaling index for k > 1 shows dependence on the initial conditions being k − 2.6 for periodic, k − 2.2 for Gaussian, and k − 2.6 for the random forms of perturbation.

In this review, we have discussed the role of kinetic Alfvén waves and dispersive Alfvén waves interacting with the surrounding, nonlinearities, other low-frequency modes, and pre-existing chain of magnetic islands in the evolution of solar wind turbulence. Amongst the different validated processes for explaining the small-scale physics, emphasis is given on the en route generated coherent or localized structures which lead to the turbulent behavior and transfer of energy along and across the ion scales. The background vicinity and initial conditions affect the evolution of these structures, but the energy transfer continues to occur from larger length scales to smaller length scales. Although some variation in the spectral index is observed due to these effects, it lies within the observed spectral range of the solar wind turbulence, that is, ∼1.41.7 for large scales and ∼2.62.8 for small scales. As indicated by the observations, a reconnection flow can also generate turbulence in the solar wind, and the magnetic reconnection (magnetic island sites) may customize the plasma conditions such that several dissipation mechanisms may contribute to the evolution of the solar wind (Voros et al., 2014). Thus, solar wind turbulence is a result of various nonlinear processes like transverse collapse or localization due to KAW, its interaction with other wave modes, and/or the formation of coherent (magnetic) structures resulting from the current sheets, and all these processes gives solar wind turbulence the structure it is found in.