On the Evolution of the Anisotropic Scaling of Magnetohydrodynamic Turbulence in the Inner Heliosphere

Anisotropy in turbulence represents a local property that relies on both the position and scale. The turbulent fluctuations at a given scale ℓ are greatly influenced by the local mean magnetic field of a size that ranges between 3ℓ and 5ℓ (Gerick et al. 2017). To analyze anisotropy, wavelet analysis has proven to be a useful technique, as it allows signal decomposition into components that are localized in both time and wavelet scale. Recently, the continuous wavelet transform (CWT) has been extensively utilized to estimate the power of magnetic field fluctuations as a function of the direction of the local mean magnetic field (Podesta 2009; Wicks et al. 2010). For a discrete set of measurements such as the time series of the ith component of the magnetic field B_i , where i = R, T, N, and resolution δ t, the wavelet transform is defined as

$$\begin{eqnarray}&&{\omega }_{i}({\ell },\,{t}_{n})=\displaystyle \sum _{j=0}^{N-1}{B}_{i}({t}_{j})\psi * \left(\displaystyle \frac{{t}_{j}-{t}_{n}}{{\ell }}\right),\end{eqnarray} \tag{ 1 }$$

where ψ* denotes the conjugate of the Morlet mother wavelet and $\psi (t)={\pi }^{-1/4}\left[{e}^{i{\omega }_{0}t}-{e}^{-\tfrac{{\omega }_{0}^{2}}{2}}\right]{e}^{-\tfrac{{t}^{2}}{2}}$. The parameter w₀, representing the frequency of the wavelet, is set equal to w₀ = 6. The transformation from the dilation scale, ℓ, to the physical spacecraft frequency, f_sc , is given by

$$\begin{eqnarray}&&{f}_{{sc}}=\displaystyle \frac{{w}_{0}}{2\pi {\ell }{\rm{\Delta }}t},\end{eqnarray} \tag{ 2 }$$

where Δt represents the time interval between successive measurements. The power spectral density (PSD) of the ith component as a function of spacecraft frequency f_sc and the local, scale-dependent field/flow angle θ_BV can be estimated as

$$\begin{eqnarray}&&{F}_{{ii}}({f}_{{sc}},\,{\theta }_{{BV}})=\displaystyle \frac{2\delta t}{N}\displaystyle \sum _{n=0}^{N-1}| {\omega }_{i}({\ell },{t}_{n},{\theta }_{{BV}}){| }^{2},\end{eqnarray} \tag{ 3 }$$

Here N is the number of samples within the range θ_j−1 ≤ θ_BV ≤ θ_j , θ_j = 5° · j, j = 0, 1, ..., 18. At time t_n and wavelet scale ℓ, we estimate the angle θ_BV using the scale-dependent local mean magnetic field B _ℓ and velocity field V _ℓ , where V represents the solar wind velocity in the spacecraft frame (Duan et al. 2021; Cuesta et al. 2022). To calculate the scale-dependent local mean of a field, q , we use a Gaussian weighting scheme centered at t_n :

$$\begin{eqnarray}&&{{\boldsymbol{q}}}_{{\ell }}({t}_{n},{\ell })=\displaystyle \sum _{m=0}^{N-1}{{\boldsymbol{q}}}_{m}\exp \left(-\displaystyle \frac{{\left({t}_{n}-{t}_{m}\right)}^{2}}{2{\lambda }^{2}{{\ell }}^{2}}\right),\end{eqnarray} \tag{ 4 }$$

where λ is a dimensionless parameter that determines the scaling of the average. To ensure the robustness of our findings, we investigated two distinct values of λ, specifically λ = 1 and λ = 3. Remarkably, the results obtained for both cases were comparable, exhibiting differences in spectral exponents of only 0.01−0.02 (see also Gerick et al. 2017). The parameter θ_BV was determined using two distinct methods: the non-scale-dependent time-to-time velocity field value V (t) and the scale-dependent value V _ℓ . Our results indicates that the outcomes obtained from both techniques are practically indistinguishable, which validates the minimal impact of interpolating V at the time points of B or only considering V (t) (see also Verdini et al. 2018; Wang et al. 2020). Throughout the remainder of the study, we utilize B _ℓ and V _ℓ to estimate the θ_BV parameter considering the case where λ = 3. For intervals that are sampled at heliocentric distances greater than 0.5 au and have a significant lack of plasma data, defined as having more than 10% of solar wind velocity measurements missing, the angle θ_BR is used. This angle represents the angle between B _ℓ and the scale-dependent radial component of the magnetic field, denoted as B_Rℓ . To determine the reliability and consistency of using θ_BR instead of θ_BV , both angles were evaluated for intervals with adequate plasma data. Our findings suggest that the anisotropic spectra remained almost unchanged for the majority of intervals, even when sampled as close as 0.3 au. The subsequent analysis examines the trace of the PSD, denoted as F = ∑F_ii . The range of θ_BV is restricted to be between 0° and 90° based on the symmetry of θ_BV around 90° (Chen et al. 2011).

To transform the PSD derived in the spacecraft-frame frequency F(f_sc , θ_BV ) into a wavenumber spectrum expressed in physical units E(κ*, θ_BV ), we employ Taylor's frozen-in hypothesis (Taylor 1938). This hypothesis assumes that the speeds of MHD wave modes, such as shear Alfvén modes propagating at ${V}_{p}={V}_{{\rm{A}}}\cos (\langle {\boldsymbol{k}},{\boldsymbol{B}}\rangle )$ observed in the solar wind plasma, are negligible compared to the bulk flow of the solar wind. This means that the Alfvén Mach number ${M}_{{\rm{A}}}=\tfrac{{V}_{r}}{{V}_{{\rm{A}}}}\gg 1$. However, as the PSP spacecraft approaches the Sun, both the spacecraft velocity V _sc and the speeds of MHD wave modes start to become comparable to V _sw (Perez et al. 2021). For this reason, a modified version of Taylor's hypothesis that accounts for wave propagation and spacecraft velocity is adopted for heliocentric distances below 0.3 au, i.e., κ* = κ · d_i = 2π f_sc /V_tot · d_i , where d_i represents the ion inertial length and V_tot =∣ V _sw + V a − V _sc∣ (Klein et al. 2015; Zank et al. 2022; Zhao et al. 2022). It is important to note that this method assumes the dominance of outwardly propagating waves, which is the case for the vast majority of the analyzed intervals closer to the Sun (Alberti et al. 2022; Sioulas et al. 2022a).

As a first step, observations of PSP between 2018 January 1 and 2022 October 1 were collected, encompassing the first 13 perihelia (E1 − E13) of the PSP mission. Level 2 magnetic field data from the fluxgate magnetometer (FGM; Bale et al. 2016), as well as Level 3 plasma moment data from the Solar Probe Cup (SPC) for E1−E8 and the Solar Probe Analyzer (SPAN) from the Solar Wind Electron, Alpha and Proton (SWEAP) suite for E9−E13 (Kasper et al. 2016), were analyzed. Data from the SCaM product (Bowen et al. 2020) obtained during E1 have also been analyzed and will be presented as a high-quality case study. The plasma data consist of moments of the distribution function computed on board the spacecraft, including the proton velocity vector V _p , number density n_p , and temperature T_p . When available, electron number density data derived from the quasi-thermal noise from the FIELDS instrument (Moncuquet et al. 2020) were preferred over SPAN or SPC data for estimating proton number density. In order to calculate the proton density from the electron density, charge neutrality must be considered, leading to a ≈4% abundance of alpha particles. Therefore, electron density from quasi-thermal noise was divided by 1.08.

The second step involved obtaining magnetic field and particle measurements from the SO mission between 2018 June 1 and 2022 March 1. Magnetic field measurements from the Magnetometer (MAG) instrument (Horbury et al. 2020) have been analyzed using burst magnetic field data when available. Particle moment measurements for our study are provided by the Proton and Alpha Particle Sensor (SWA-PAS) on board the Solar Wind Analyser (SWA) suite of instruments (Owen et al. 2020).

Quality flags for the magnetic field and particle time series have been taken into account, and time intervals missing ≥1% and/or ≥10% in the magnetic field and particle time series have been omitted from further analysis. Additionally, the mean value of the cadence between successive measurements δ τ in the magnetic field time series has been estimated for each of the selected intervals, and time intervals that were found to have a mean cadence of δ τ ≥ 250 ms were discarded. Due to poor data quality, all PSP intervals exceeding R ≃ 0.5 au have also been discarded.

Spurious spikes in the plasma time series were eliminated by replacing outliers exceeding three standard deviations within a moving average window covering 200 points with their median values (Davies & Gather 1993).

The high-resolution data from the first perihelion of the PSP (R ≈ 0.17 au) from 2018 November 1 to 11 were analyzed. A total of 33 intervals, each with a duration of 12 hr, were obtained, and the PSD was estimated, with subsequent intervals overlapping by 50%. The analysis covered 18 bins of the θ_BV angle, but in the following we will only focus on those bins closer to the parallel and perpendicular directions, with θ_BV ≤ 5° and θ_BV ≥ 85°, respectively. It is worth noting that the second half of E1 displayed significantly different characteristics compared to the first half, with the solar wind exhibiting higher speeds and a greater number of magnetic switchbacks (Bale et al. 2019). It is well established that power spectra exhibit different characteristics when different solar wind speeds are considered owing to the different types of fluctuations they transport (Borovsky et al. 2019). Specifically, the fast solar wind is highly Alfvénic and characterized by large-amplitude, incompressible fluctuations, while the slow wind is generally populated by smaller-amplitude, less Alfvénic, compressive fluctuations, which include convected coherent structures (Bruno et al. 2003; Matteini et al. 2014; Shi et al. 2021; Sioulas et al. 2022b; Zhao et al. 2020). Consequently, the spectral variation due to the differing plasma parameters of the selected streams was investigated. More specifically, we considered the dependence of the PSD on the solar wind speed, V_sw, the normalized cross-helicity σ_c ,

$$\begin{eqnarray}&&{\sigma }_{c}=\displaystyle \frac{{E}_{o}-{E}_{i}}{{E}_{o}+{E}_{i}},\end{eqnarray} \tag{ 5 }$$

a measure of the relative amplitudes of inwardly and outwardly propagating Alfvén waves (AWs), and the normalized residual energy σ_r ,

$$\begin{eqnarray}&&{\sigma }_{r}=\displaystyle \frac{{E}_{V}-{E}_{b}}{{E}_{V}+{E}_{b}},\end{eqnarray} \tag{ 6 }$$

indicating the balance between kinetic and magnetic energy, where ${E}_{\phi }=\tfrac{1}{2}\langle \delta {{\boldsymbol{\phi }}}^{2}\rangle$ denotes the energy associated with the fluctuations of the field ϕ . In particular, E_o,i can be estimated using Elsässer variables, defining outward- and inward-propagating Alfvénic fluctuations (Velli et al. 1991; Velli 1993)

$$\begin{eqnarray}&&\delta {{\boldsymbol{Z}}}_{o,i}=\delta {\boldsymbol{V}}\mp {sign}({B}_{0}^{R})\delta {\boldsymbol{b}},\end{eqnarray} \tag{ 7 }$$

where δ B = B − B ₀, with B ₀ the background magnetic field, $\delta {\boldsymbol{b}}=\delta {\boldsymbol{B}}/\sqrt{{\mu }_{0}{m}_{p}{n}_{p}}$ the magnetic fluctuations in Alfvén units, and B^r ₀ the ensemble average of B_R , utilized to determine the polarity of the radial magnetic field (Shi et al. 2021). The magnetic field power spectra for fluctuations parallel (θ_BV ≤ 5°) and perpendicular (θ_BV ≥ 85°) to the local magnetic field, resulting from averaging all the respective spectra, are presented in Figure 1. Individual spectra are also shown with the color of the curve keyed to σ_c . The fluctuation power in the MHD range shows a positive correlation with σ_c , but this dependence vanishes in the transition region and kinetic scales. A similar trend was observed with σ_r and V_sw, not shown here. The results are consistent with Vasquez et al. (2007), who found higher, MHD range, turbulence amplitudes associated with faster streams, as well as Pi et al. (2020), who showed that such dependence vanishes in the kinetic scales. The trend also vanishes at the large, energy injection scales, κ d_i ≤ 10⁻³, where the power spectrum is clearly dominated by parallel fluctuations. Focusing our attention on MHD scales, we can observe two distinct ranges, roughly 3 × 10⁻³ κ d_i − 5 × 10⁻² κ d_i and 8 × 10⁻² κ d_i − 2 × 10⁻¹ κ d_i , over which the PSD displays a clear power-law scaling. Light-black and red shadings are used to indicate these regions in the figure, and we will thereby refer to them as ${R}_{| | (\perp )}^{1}$ and ${R}_{| | (\perp )}^{2}$. The power-law fitting has been applied to the PSD for the two ranges, and the bottom and top inset figures illustrate α_B as a function of V_sw. Note that the color of the scatter plot is keyed to σ_c . Furthermore, horizontal lines have been added to indicate the average value of α_B . In the direction parallel to the mean field the PSD scales roughly like −5/3 and −2 in ${R}_{| | }^{1}$ and ${R}_{| | }^{2}$, respectively. For perpendicular fluctuations, only a minor difference may be observed between ${R}_{\perp }^{1}$ and ${R}_{\perp }^{2}$, which are characterized by a power-law scaling with index −3/2 and −1.57, respectively. The absence of a definitive correlation between α_B , V_sw, and σ_c , as reported in the study by Sioulas et al. (2022a), could be ascribed to the relatively extended time intervals that were examined or the limited size of the sample, which, in this case, encompassed only 33 intervals.

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When examining the local spectral index, α_B (κ*), a similar pattern emerges. This is achieved by applying a sliding window of size w = 3, 5, 10 in κ* over the spectra and calculating the best-fit linear gradient in log−log space over this window, shown in cyan, orange, and red, respectively, in the bottom panel of Figure 1. At smaller scales where $\kappa \,* \,\geqslant 0.1{d}_{i}^{-1}$, both parallel and perpendicular fluctuations display a steeper spectrum between the inertial and kinetic ranges. The scaling behavior observed in the transition and kinetic ranges is consistent with the findings reported by Duan et al. (2021). Additionally, Duan et al. (2021) report a scaling exponent of −2 for the parallel spectrum in the inertial range spanning from 4 × 10⁻¹ Hz to 2 Hz, which corresponds to region ${R}_{| | }^{2}$ in our analysis. It is worth noting, however, that ${R}_{| | }^{2}$ does not encompass the entire inertial range. Specifically, ${R}_{| | }^{1}$ covers most of the MHD range and is characterized by a shallower scaling exponent, α_B ≈ −5/3. The two different MHD scalings persist in most of the intervals studied, suggesting that this may be a consistent feature of the solar wind power spectrum in the vicinity of the Sun.

We then analyzed the power anisotropy, defined as E_⊥/E_∣∣ (Podesta 2009), as a function of κ*. The results of this analysis are displayed in Figure 2, where individual intervals are plotted as scatter points and binned based on their σ_c value. The median curve for each bin is shown, and the color of each curve is keyed to σ_c . The median curve (black solid line) in Figure 2 is consistent with previous findings at larger heliocentric distances. Specifically, the curve exhibits a region of near isotropy for kd_i ≤ 10⁻³, which roughly corresponds to the rollover to the f⁻¹ range of the magnetic spectrum (see Figure 1). At smaller scales, the anisotropy becomes more noticeable and shows a power-law scaling that closely resembles the 1/3 value suggested by the CB conjecture. Therefore, a line with a scaling exponent of 1/3 was included in the figure as a point of reference. This κ*1/3 scaling is observed within the range of $4\times {10}^{-2}-\,3\times {10}^{-1}\,[{d}_{i}^{-1}]$, which corresponds to region R² in Figure 1. Additionally, while the anisotropy increases at smaller scales until κ d_i ≈ 4 × 10⁻¹, there is a sudden but noticeable local minimum at around κ d_i ≈ 0.7 followed by a local maximum at κ d_i ≈ 1.7. Both the trough and peak are consistently observed across all intervals considered in this study. The local minimum may be caused by the bump observed in E_∣∣ at κ d_i ≈ 0.06, which coincides with the beginning of the transition region in E_⊥ (see Figure 1). This bump may suggest a local enhancement of energy that could be due to ion kinetic instabilities (Wicks et al. 2010). For a more comprehensive discussion of the double-peak structure in Figure 2(a), see Podesta (2009). The results of this study differ from those of Podesta (2009) in that we observe an increase in anisotropy at smaller scales κ d_i > 2. As shown in Figure 7 of Podesta (2009), a rapid decrease in the power ratio is observed beyond the local kinetic scale maximum of approximately 1 Hz, which is attributed to the dissipation of kinetic AWs. However, as the spacecraft moves farther away from the Sun, the amplitude of the fluctuations at kinetic scales is close to the noise floor of the magnetometer. This can lead to an artificial steepening of the PSD caused by instrumental noise (Woodham 2019). The effect is particularly significant for α_B (κ*) parallel, as most of the power in the solar wind is associated with perpendicular fluctuations. As a result, the parallel PSD systematically obtains lower values at MHD and kinetic scales and is therefore more likely to be affected by instrumental noise. On the other hand, the perpendicular PSD can remain intact. This can cause the parallel PSD to flatten out and the power ratio to decrease with decreasing scale. Considering that (1) the aforementioned paper uses magnetic field data from the STEREO mission (Acuña et al. 2008) at Earth orbit, where the turbulence amplitude is lower compared to that observed by PSP's E1, and (2) the SCaM data product merges FGM and search-coil magnetometer measurements, allowing for magnetic field observations up to 1 MHz with an optimal signal-to-noise ratio, we attribute the discrepancy to instrumental noise that may have affected the parallel PSD in Figure 7 of Podesta (2009).

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4.2.1. Spectral Index Anisotropy

In the following, we investigate the evolution of spectral index anisotropy with heliocentric distance. For this analysis we consider intervals sampled by the PSP and SO at distances between 0.06 and 1 au (see Section 3.2). Previous research has shown that the dominant orientation of fluctuation wavevectors in fast solar wind streams tends to be quasi-parallel to the local magnetic field, while in slow solar wind streams the dominant orientation is quasi-perpendicular (Dasso et al. 2005). In order to examine the distinct features of each type of stream and their potential impact on the development of anisotropy in the solar wind, a comprehensive visual analysis was undertaken to categorize the streams into two distinct groups: slow streams characterized by V_sw ≤ 400 km s⁻¹ and fast streams with V_sw ≥ 400 km s⁻¹. A comprehensive record of the chosen intervals can be accessed in MHDTurbPy.

We shall begin by examining the evolution of slow streams, which compose the majority of the samples collected from PSP and SO. To determine the local spectral index for each interval, we perform calculations in the direction parallel (θ_BV ≤ 5°) and perpendicular (θ_BV ≥ 85°) to the locally dominant magnetic field, utilizing a sliding window of size w = 10, following the methodology outlined in Section 4.1.

We then partitioned our intervals into six heliocentric bins and calculated the mean local spectral index for those intervals that fell within each bin. It should be noted that despite the spectra and local spectral indices being calculated at identical frequencies based on the interval duration and sampling frequency, the normalization process results in an irregular shift along the vertical axis. Consequently, we divided the complete range of κ* into 100 bins and computed the mean for all α_B (κ*) values that fell within each bin, as described in Němeček et al. (2021). It is worth noting that the size of the interval under consideration does not have a significant impact on the outcomes. This is true as long as a sizable statistical sample of fluctuations is taken into account at a given scale, in order to ensure the validity of the statistical analysis and produce accurate spectra (Dudok de Wit et al. 2013). Any intervals that exhibited noisy or otherwise unreliable spectra were excluded from subsequent analyses. However, it is important to note that such intervals made up only an inconsequential proportion of the overall data set.

The radial size of each bin is shown in the legend of Figure 3. The slow wind local spectral indices, as a function of heliocentric distance, are shown in blue for perpendicular fluctuations and in red for parallel fluctuations, along with error bars indicating the standard error of the mean. The standard error is given by ${\sigma }_{i}/\sqrt{N}$, where σ_i is the standard deviation and N is the number of samples inside the bin, as described in Gurland & Tripathi (1971). We focus our attention on MHD scales, κ d_i ≤ 3 × 10⁻¹, here simply because instrumental noise artificially steepens the PSD with increasing distance, as discussed in Section 4.1.

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It is evident from Figure 3 that the spectral index anisotropy of slow wind turbulence diminishes closer to the Sun. Within 0.1 au, both parallel and perpendicular spectra are characterized by a poorly developed inertial range, viz., the range of scales over which the spectral index is constant is limited to 3 × 10⁻³ ≲ κ d_i ≲ 2 × 10⁻², with a scaling exponent −1.47 ± 0.04 and −1.55 ± 0.05 for perpendicular and parallel fluctuations, respectively. At distances 0.1–0.2 au, two subranges (R¹ and R²) emerge within the inertial range. The transition occurs at κ d_i ≈ 6 × 10⁻², and the scaling exponents in these ranges are similar to those shown in Figure 1. However, the steepened region R² is not as well defined in this case. Considering that the PSP was at a distance of 0.17 au during E1, we attribute this discrepancy to the fact that R² actually appears closer to 0.2 au. By shifting the left boundary of the bin toward 0.2 au, we confirmed this expectation, as the steepened region displayed a parallel power-law scaling of −2 when the left boundary was shifted to approximately 0.15 au.

In R¹ both parallel and perpendicular spectra dynamically evolve with increasing distance and steepen toward –5/3, which in the case of the parallel spectrum occurs within 0.1 au. The steepening occurs in a scale-dependent fashion, which results in R² extending to larger scales with distance. As a result, for distances exceeding 0.5 au, R¹ practically vanishes, and the power spectra are characterized by a power-law exponent that changes from −5/3 in the direction perpendicular to −2 in the direction parallel to the locally dominant mean field, in good agreement with the predictions of CB theory. It should be noted that the analysis was iterated over bins of width 10°, yielding consistent outcomes. Notably, the obtained PSDs for θ_BV ≥ 80° or θ_BV ≥ 85° exhibited indistinguishable scaling behavior across all distances. Conversely, a comparison of the PSDs obtained for θ_BV ≤ 10° and θ_BV ≥ 5° revealed marginally steeper scaling behavior in the latter case for the inertial range. Specifically, in the instance of slow solar wind intervals, when distances exceeded 0.5 au, a consistent –2 scaling was observed for θ_BV ≥ 5°, whereas for θ_BV ≥ 10° the scaling behavior obtained was closer to −1.89.

We next examined the evolution of fast streams (V_sw ≥ 400 km s⁻¹). It is important to consider that as the solar wind expands in the heliosphere, the local mean magnetic field vectors become increasingly oriented at larger angles relative to the radial direction. This radial trend causes sampling at 0.06 au to be more quasi-longitudinal and sampling at 1.0 au to be more quasi-perpendicular. As parallel fluctuations decrease with increasing distance, our ability to accurately estimate the low-frequency part of the parallel power spectrum is reduced. This effect makes the determination of the anisotropic scaling laws for high-speed streams in the ecliptic plane challenging, as there is insufficient data to make accurate measurements at low frequencies. While using longer records could resolve this issue, the limited lifetime of the streams restricts the length of the record. In an effort to address this issue, we imposed a minimum interval length that would allow for a large enough interval size but still enable us to gather a sufficient number of intervals for our statistical study. Specifically, for heliographic distances exceeding 0.3 and 0.5 au, we set the minimum interval size to 12 and 20 hr, respectively. This resulted in a total of 274 intervals sampled across the inner heliosphere. The results of this analysis are presented in Figure 4. It is readily seen that the differences between fast and slow intervals are significant. When examining the lower frequencies, we observe that within 0.2 au the energy injection range of the PSD is dominated by parallel fluctuations. In particular, a remarkably extended and relatively shallow range with α_B ≈ − 0.8 is observed within 0.1 au, which steepens toward −1 with distance. This is particularly noteworthy, as previous research has shown that AWs can parametrically decay into slow magnetosonic waves and counterpropagating AWs (Galeev & Oraevskii 1963; Tenerani et al. 2017; Malara et al. 2022). This process may lead to the development of a ${k}_{| | }^{-1}$ spectrum for outward-propagating AWs by the time they reach a heliocentric distance of 0.3 au in the fast solar wind (Chandran 2018). For a more comprehensive investigation of the radial evolution of the lower-frequency part of the spectrum, see Huang et al. (2023). Due to the issues with interval size that were discussed earlier, we do not attempt to interpret the evolution of the lower-frequency part of the spectrum beyond 0.3 au.

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Focusing on MHD scales, we notice that the perpendicular PSD only extends up to κ d_i ≈ 10⁻². This implies that fast streams in proximity to the Sun exhibit a nearly radial magnetic field at low frequencies. Interestingly, within 0.1 au, the scaling of the perpendicular spectrum is consistent with −5/3, but at larger distances a scaling that is roughly consistent with −3/2, fluctuating between −1.49 and −1.55, is observed. This suggests that the MHD range spectral index of the perpendicular spectrum for fast streams may not evolve in a consistent manner with increasing distance in the inner heliosphere. It is worth noting, however, that within 0.1 au only four intervals with V_sw ≥ 400 km s^–1 were sampled by PSP. More data from fast streams near the Sun are needed to statistically confirm these findings. For parallel fluctuations, the inertial range scaling remains remarkably similar across all heliographic bins, with the spectral index progressively steepening toward smaller scales from −5/3 toward −2, where a narrow range of scales over which the local spectral index obtains a constant value appears. In contrast to slow wind streams, the high-frequency point in fast wind streams does not remain anchored in a normalized wavenumber but gradually drifts toward larger scales with distance. This is an interesting finding that suggests that the evolution of the high-frequency point is different between fast and slow wind streams; this is discussed further in Section 5.

4.2.2. Power Anisotropy

In this section we examine the radial evolution of the power anisotropy, represented by the ratio E_⊥/E_∣∣, where E_⊥ and E_∣∣ are the PSDs for θ_BV ≥ 85° and θ_BV ≤ 5°, respectively. To do this, we utilized the method described in Section 4.2.1 and calculated the mean of E_⊥/E_∣∣ in six heliocentric bins. The results of this analysis are presented in Figure 5(a) for slow streams and Figure 5(b) for fast streams. According to theories based on "dynamical alignment," the inertial range scaling index should be 1/2 when considering E_⊥/E_∣∣, while a slope of 1/3 is predicted by CB theories.

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For slow wind streams, the power anisotropy becomes more significant with increasing distance, particularly at smaller scales (see Figure 5(a)). This suggests that the turbulence undergoes an anisotropic cascade, transporting the majority of its magnetic energy toward larger perpendicular wavenumbers. In contrast, fast streams show practically no significant radial trend, especially when taking into account the error bars (not shown here). As a result, even though the power anisotropy is more pronounced for fast winds closer to the Sun, at distances of around 1 au, the situation is reversed, and slow wind exhibits higher values of E_⊥/E_∣∣. In terms of anisotropic scaling, we observe that E_⊥/E_∣∣ evolves in a manner similar to what was described in Section 4.2.1. Specifically, the scaling of E_⊥/E_∣∣ for slow wind streams does not fit the predictions of any of the existing anisotropic theories closer to the Sun, but with increasing distance, it evolves toward a scaling that is consistent with CB theories. The situation is more complex for fast streams. In particular, for the bin closest to the Sun, the scaling of E_⊥/E_∣∣ is closer to that predicted by CB theories (κ*1/3), but for the rest of the bins, the scaling exponent fluctuates in the range between 1/2 and 1/3. Additionally, the double-peak structure discussed in Section 4.1 is also observed for most of the curves in this analysis, especially in fast wind intervals. In contrast to the data presented in Section 4.1, at smaller scales the utilization of FGM data leads to a significant impact of instrumental noise on the resulting curves, ultimately causing a marked decrease in the power ratio.

In our study, using data from PSP and SO, we estimated perpendicular inertial range spectral indices for the fast wind with values in the range of [−1.49, − 1.55]. These values are slightly shallower than those reported in previous studies (Horbury et al. 2008; Podesta 2009; Wicks et al. 2010), which estimate values in the range of [−1.55, − 1.67]. One possible reason for this discrepancy could be that the PSP and SO data were only collected in the ecliptic plane during the minimum and early rising phase of the solar cycle. It is known that solar wind conditions can vary significantly over the course of the solar cycle, and it is possible that these variations could affect the observed scalings of the perpendicular spectra. In addition, due to the phase of the solar cycle, only a limited number of extended fast wind streams were collected. For example, PSP only sampled four intervals with V_sw ≥ 400 km s⁻¹ within 0.1 au. This limitation may affect the statistical significance of the results and make it difficult to accurately measure the anisotropic scaling laws for these streams at lower frequencies. As a result, it may be premature to draw firm conclusions about the agreement of the scaling in the fast solar wind sampled in the ecliptic plane by PSP and SO with any particular theory of anisotropic MHD turbulence.

Our results indicate that, when analyzing slow wind streams, normalizing the PSD with d_i allows us to fix the high-frequency break point, f_b , in normalized wavenumber space, as previously reported in Sioulas et al. (2022a). However, for fast solar wind streams, f_b tends to shift toward larger κ d_i as the distance increases. This phenomenon can be attributed to the fact that fast solar wind streams are characterized by higher proton temperatures (T_p ; Maksimovic et al. 2020; Shi et al. 2021, 2023), which lead to higher plasma pressure and, consequently, higher plasma β values. The plasma β is defined as the ratio of thermal to magnetic pressure, β ≡ n_p K_B T_p /(B²/2μ₀), and it is comparatively higher in fast streams than in slow ones. It should be noted that the V_sw − β correlation was verified, although it is not presented in this report. Chen et al. (2014) and Vech et al. (2018) have shown that the f_b of the magnetic PSD between inertial and kinetic scales correlates better with d_i when the intervals are characterized by β < 1 values, while high β intervals are characterized by a small-scale break at the thermal ion gyroradius (ρ_i ). In line with this, our analysis confirms the findings of Wicks et al. (2010), who used five fast solar wind streams with β > 1 between 1.5 and 2.8 au and found that the small-scale end of the inertial range seems to naturally scale with the ion gyroradius when normalized with ρ_i . Given that ρ_i grows radially as ∝R^1.48±0.02 (Sioulas et al. 2022a), we expect that f_b will display a similar radial trend for fast solar wind streams.

The use of high-resolution data from E1 of PSP allowed us to confirm the existence of two subranges (Telloni 2022; Wu et al. 2022) within the inertial range. The transition occurs at κ d_i ≈ 6 × 10⁻² and signifies a shift from −5/3 to −2 scaling in the parallel spectra and from −3/2 to −1.57 scaling in the perpendicular spectra. The difference between the two ranges (R¹, R²) is most apparent in the parallel spectrum and could signify a transition from weak to strong turbulence (Sridhar & Goldreich 1994; Meyrand et al. 2016; Zank et al. 2020). It is important to note that the parallel spectral index we report here for ${R}_{| | }^{2}$, α_B ≈ − 2, is steeper than the one reported by Wu et al. (2022). It is unlikely that the variations seen in the outcomes are due to the utilization of structure functions in the analysis carried out by Wu et al. (2022). This is because we used second-order structure functions to confirm the anisotropic scaling. Nonetheless, it is feasible that the differences could be linked to the usage of better-quality SCaM data in our research. In particular, Wu et al.'s parallel structure function in Figure 4(b) seems to become steeper at shorter timescales, but the limited cadence of around 1 Hz might prevent the clear detection of such scaling.

Moreover, it has been observed that there exist noteworthy differences between fast and slow streams in terms of their anisotropic properties and dynamic evolution. Besides the distinctions noted in the evolution of the inertial range scaling, the high-frequency break point displays a more rapid shift toward lower frequencies in the analysis of fast wind streams. These findings are at odds with the assertions put forth by Wu et al. (2022) and emphasize the necessity of analyzing wind streams of comparable speeds for making meaningful comparisons.