# Evolution of Solar Wind Turbulence during Radial Alignment of Parker Solar Probe and Solar Orbiter in 2022 December

Ashok Silwal, Lingling Zhao, Xingyu Zhu, Luca Sorriso-Valvo, Lina Hadid, Gary Zank, Hui Li, Samuel Badman, Yeimy Rivera, Sujan Prasad Gautam, et al.

► To cite this version:

Ashok Silwal, Lingling Zhao, Xingyu Zhu, Luca Sorriso-Valvo, Lina Hadid, et al. Evolution of Solar Wind Turbulence during Radial Alignment of Parker Solar Probe and Solar Orbiter in 2022 December. The Astrophysical Journal Supplement Series, 2025, 278 (2), pp.40. 10.3847/1538-4365/add011. hal-05158210

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# Evolution of Solar Wind Turbulence during Radial Alignment of Parker Solar Probe and Solar Orbiter in 2022 December

Ashok Silwal<fnref n="1" /> ID, Lingling Zhao<fnref n="1" />,<fnref n="2" /> ID, Xingyu Zhu<fnref n="2" /> ID, Luca Sorriso-Valvo<fnref n="3" />,<fnref n="4" /> ID, Lina Z. Hadid<fnref n="5" /> ID, Gary P. Zank<fnref n="1" />,<fnref n="2" /> ID, Hui Li<fnref n="6" /> ID, Samuel T. Badman<fnref n="7" /> ID, Yeimy J. Rivera<fnref n="7" /> ID, Sujan Prasad Gautam<fnref n="1" /> ID, Monika Karki<fnref n="1" /> ID, Juan G. Alonso Guzman<fnref n="1" /> ID, Nibuna S. M. Subashchandar<fnref n="1" /> ID, and Zeping Jin<fnref n="1" /> ID

Received 2025 February 22; revised 2025 April 22; accepted 2025 April 22; published 2025 May 30

## Abstract

We investigate the radial evolution of solar wind turbulence during the radial alignment of Parker Solar Probe (PSP) and Solar Orbiter (SO) on 2022 December 10, with PSP located at approximately 0.11 au and SO near 0.88 au. To identify nearly the same plasma parcel crossing both spacecraft, we apply a ballistic propagation model with time-constant acceleration constrained by in situ solar wind velocity measurements at PSP and SO. We trace the magnetic footpoint of the plasma parcel back to the photosphere using a potential field source surface model based on a Global Oscillations Network Group synoptic magnetogram. Field and plasma measurements from PSP and SO are used to analyze power spectral density (PSD), spectral scaling, magnetic compressibility, and intermittency. Our results show that (1) the trace PSD of magnetic fluctuations steepens in the inertial range and flattens in the dissipation range with increasing radial distance; (2) the spectral break shifts to lower frequencies at SO; and (3) the Castaing model reveals multifractal intermittency in the inertial range, with slightly weaker intermittency at SO. These findings based on the same plasma parcel are consistent with the results of statistical studies on the radial evolution of turbulence and provide a reference for theoretical modeling of turbulence in the inner heliosphere.

*Unified Astronomy Thesaurus concepts: Solar wind (1534); Interplanetary turbulence (830)*

## 1. Introduction

The solar wind is a turbulent flow of weakly collisional charged particles that originates in the outermost layer of the Sun's atmosphere, the solar corona, and expands radially into the heliosphere. With its broad range of scales and plasma parameter conditions, the solar wind is the best laboratory for studying plasma turbulence (R. Bruno & V. Carbone 2013). In situ observations show that the solar wind farther from the Sun exhibits typical features of fully developed turbulence (P. J. Coleman 1968; R. Bruno & V. Carbone 2016; D. Telloni et al. 2021; C. Phillips et al. 2022; M. E. Cuesta et al. 2023; D. Telloni et al. 2024). Fluctuations in the magnetic field, velocity, and density are indeed observed over a wide range of scales, ranging from kinetic to large-scale ones. Within this large variety of scales, the energy introduced into the system at injection/integral scales in the form of large-wavelength fluctuations undergoes a turbulent cascade that allows it to be transferred to smaller scales and ultimately dissipated at the dissipation scale (R. Bruno & V. Carbone 2016; D. Verscharen et al. 2019; S. P. Gautam et al. 2024). The inertial range is present in between these two scales, where nonlinear interactions dominate the dissipative and dispersive processes

(V. Carbone 1993; U. Frisch 1995; D. Biskamp 2003; W. H. Matthaeus et al. 2015; A. Silwal et al. 2024). The dissipation range corresponds to the scale at which the usual MHD approximation breaks down, favoring a more complex description of plasma where kinetic processes occur (R. J. Leamon et al. 1998; O. Alexandrova et al. 2009; K. Kiyani et al. 2009). Recent observations show that the magnetic power spectra at the kinetic scale becomes steeper, with the spectral index varying between -2 and -4 (R. Bruno & V. Carbone 2013; K. H. Kiyani et al. 2015). The steepening of the spectral index possibly implies that cascaded energy at the end of MHD scale may be gradually dissipated or develop into a dispersive kinetic turbulence (D. Duan et al. 2020). As the solar wind plasma propagates outward, the spectral breaks between two adjacent scales are observed statistically to shift toward lower frequencies with increasing distance from the Sun (R. D'Amicis et al. 2010; R. Bruno & L. Trenchi 2014; D. Duan et al. 2020). In addition, other properties of solar wind turbulence, such as anisotropy, Alfvénicity, intermittency, and compressibility, also evolve during its expansion (R. Bruno & V. Carbone 2013).

The radial alignment of two or more spacecraft at different heliospheric distances is a promising way to study the radial evolution of solar wind turbulence properties. During the twin-Helios era, S. J. Schwartz & E. Marsch (1983) first exploited a radial alignment between Helios 1 at ~0.51 au and Helios 2 at ~0.72 au from the Sun. While their study focused on the radial

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Original content from this work may be used under the terms of the [Creative Commons Attribution 4.0 licence](https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹Department of Space Science, University of Alabama in Huntsville, Huntsville, AL, USA; lz0009@uah.edu</footnote>

<footnote n="2">²Center for Space Plasma and Aeronomic Research (CSPAR), University of Alabama in Huntsville, Huntsville, AL, USA</footnote>

<footnote n="3">³Institute for the Science and Technology of Plasmas, National Research Council, Via Amendola 122/D, I-70126 Bari, Italy</footnote>

<footnote n="4">⁴Space and Plasma Physics, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Teknikringen 31, SE-11428 Stockholm, Sweden</footnote>

<footnote n="5">⁵LPP, CNRS, Observatoire de Paris, Sorbonne Université, Université Paris Saclay, École polytechnique, Institut Polytechnique de Paris, Palaiseau, France</footnote>

<footnote n="6">⁶Los Alamos National Laboratory, Los Alamos, NM 87545, USA</footnote>

<footnote n="7">⁷Center for Astrophysics, Harvard and Smithsonian, Cambridge, MA 02138, USA</footnote>
evolution of the adiabatic invariants and energy budget of the
same high-speed plasma parcel, the evolution of solar wind
turbulence was not addressed in their analysis. Nevertheless,
this study motivated several other efforts to characterize the
evolution of turbulence during successive alignments between
two spacecraft in the inner heliosphere (e.g., Messenger and
Wind (R. Bruno & L. Trenchi 2014; R. Bruno et al. 2014;
D. Telloni et al. 2015), PSP and SO (D. Telloni et al. 2021), or
PSP and BepiColombo (T. Alberti et al. 2022a; D. Telloni
et al. 2022)) and the outer heliosphere (e.g., Wind and Ulysses
(R. Bruno et al. 2014) or ACE and Ulysses (R. D'Amicis et al.
2010)). In particular, the recently launched missions Parker
Solar Probe (PSP; N. Fox et al. 2016), Solar Orbiter (SO;
D. Müller et al. 2020), and BepiColombo (BC; J. Benkhoff
et al. 2010) provide unprecedented opportunities to investigate
the evolution of solar wind turbulence in the inner heliosphere.
For example, D. Telloni et al. (2021) investigated the first
PSP-SO alignment, when PSP and SO were situated at
~0.1 au and ~1 au from the Sun, respectively. Their study
provided compelling evidence for the radial evolution of solar
wind turbulence from a weak, less-developed turbulence state
at 0.1 au to a fully developed and intermittent turbulence state
at 1 au. In the companion paper by D. Telloni et al. (2022), the
authors exploited the radial alignment of PSP (0.11 au) and BC
(0.33 au) and found that the solar wind turbulence already
evolves toward a highly turbulent and intermittent state at
distances less than 0.3 au from the Sun. Based on the first
radial alignment of the PSP and BC spacecraft, T. Alberti et al.
(2022a) studied the evolution of solar-wind magnetic field
scaling features and showed that the efficiency of the nonlinear
energy cascade mechanism increases when moving away from
the Sun.

One of the key challenges in radial alignment studies, in
particular for exploring the evolution of solar wind turbulence,
is the identification of the same plasma parcel as it crosses two
radially aligned spacecraft during its outward propagation. This
requires a suitable plasma propagation model and a rigorous
selection of plasma parcels at both spacecraft (e.g., D. Telloni
et al. 2021; T. Alberti et al. 2022a; E. Berriot et al. 2024). As a
result, only a handful of studies have been conducted since the
Helios missions. Recent case studies by D. Telloni et al.
(2021, 2022) and T. Alberti et al. (2022a, 2022b) identified the
time intervals corresponding to the same plasma parcel using a
ballistic propagation model that assumes a constant propagation
speed of the solar wind plasma measured at the inner spacecraft.
These identifications suggest negligible acceleration of the solar
wind during its radial expansion. However, due to the
unavailability of plasma measurements at the outer spacecraft,
the authors were unable to verify this result. Nonetheless,
several statistical (R. Schwenn et al. 1981; M. Maksimovic
et al. 2020; J.-B. Dakeyo et al. 2022; J. Halekas et al. 2022;
T. Ervin et al. 2024; Y. Jiao et al. 2024) and conjunction studies
(Y. J. Rivera et al. 2024, 2025) have reported a non-negligible
acceleration of both slow and fast solar wind in the inner
heliosphere. It is also noteworthy that Parker solar-wind
solutions are not mathematically asymptotic (E. N. Parker
1958), since all wind solutions continue to gradual accelerate
throughout the inner heliosphere. Consequently, any accelera-
tions or nonradial flows are potential sources of uncertainty in
such constant-speed models and may lead to inaccuracies in
identifying the time intervals corresponding to the same plasma
parcel. While the effect on spatial localization is relatively

small, the impact on timing can be significant, in particular for
slower wind speeds, and becomes more pronounced when the
separation between the two spacecraft is larger (J.-B. Dakeyo
et al. 2024). In this paper, we adopt the novel approach
introduced by E. Berriot et al. (2024) that allowed the
identification of the same plasma parcel using a ballistic
approach with constant acceleration during the radial alignment
of PSP (0.11 au) and SO (0.88 au) on 2022 December 10. We
evaluate several identification criteria to show that the constant-
acceleration (over time) model provides a more robust and
physically consistent framework for tracing the same plasma
parcel in the inner heliosphere. We then investigate the
evolution of solar wind turbulence properties—such as aniso-
tropy, compressibility, and intermittency—of the same plasma
parcel during its radial expansion from 0.11 au to 0.88 au.

The paper is organized as follows. Section 2 outlines the
data selection and provides an overview of the spacecraft
configuration. Section 3 describes the ballistic propagation
model used to estimate the time intervals corresponding to the
plasma lineup. Section 4 focuses on the validation of these
estimates. Sections 5, 6, and 7 present the diagnosis of
turbulence properties at the two locations. Section 8 sum-
marizes the key findings and concludes.

## 2. Spacecraft Data and Configuration

We study the radial alignment between PSP and SO on
2022 December 10. Figure 1(a) shows the trajectories of PSP
(blue curve) and SO (red curve) in the X–Y plane of the
ECLIPJ2000 coordinate system during the period of 2022
December 3–17. PSP and SO are located at approximately
0.11 au and 0.88 au, respectively. At about 02:00 UT on 2022
December 10, the two spacecraft are roughly radially aligned
as indicated by the black dashed line. The corresponding
latitude and longitude of PSP and SO during this alignment
are depicted in Figures 1(b) and 1(c), where the dashed
vertical lines denote the time $t_0 \sim 02:00$ UT when both
spacecraft are at the same longitude $\phi$. At $t_0$, the latitudinal
$\theta$ separation between PSP and SO is approximately 4.8.
Figure 1(a) also shows the approximate spacecraft crossing
time of the plasma parcel (green circle along each space-
craft’s trajectory) during this radial alignment. Due to its
higher orbital velocity, PSP moves significantly as the plasma
parcel transits to the SO orbit, as reflected by its larger
variation in both longitude and latitude compared to SO
(panels (b) and (c)). We use $t_0$ as a reference time in the
following discussion to identify the same plasma parcel
observed by the two spacecraft.

The identification of the same plasma parcel observed by
PSP and SO during this radial alignment is based on the
ballistic propagation model (D. Telloni et al. 2021; T. Alberti
et al. 2022a; E. Berriot et al. 2024). We use the magnetic field
data measured by the PSP/FIELDS/MAG instrument (S. Bale
et al. 2016) at 293 Hz, and the plasma measurements are from
the PSP/SWEAP/Solar Probe ANalyzer for Ions (SPAN-I;
J. C. Kasper et al. 2016; R. Livi et al. 2022) instrument with a
resolution of 3.5 s. The magnetic field data from SO are
measured by the MAG (T. Horbury et al. 2020) instrument at
64 Hz, and the proton and alpha moments sampled at 4 s
resolution are from the Proton and Alpha particle sensor (PAS)
of the SO/Solar Wind Analyser (SWA) suite (C. Owen et al.
2020). We also use magnetogram data from the Global
Oscillations Network Group (GONG; J. Harvey et al. 1996) as
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Positions of PSP and SO relative to the Sun for the studied radial alignment event. Panel (a) shows the trajectories of PSP (blue curve) and SO (red curve) in the X-Y plane of the ECLIPJ2000 coordinate system during the period of 2022 December 3–17. The black dashed line indicates the radial line from the Sun passing through the PSP and SO at the alignment time $t_0 \sim 02:00$ UT on 2022 December 10. Blue squares and red triangles indicate the positions of PSP and SO at three different instants in time, respectively. The green circle along each spacecraft’s trajectory denotes the approximate crossing time of the same plasma parcel during this alignment. Panels (b) and (c) display the latitude ($\theta$) and longitude ($\phi$) of PSP and SO, respectively. The black vertical dashed line marks the spacecraft alignment time $t_0$.</figcaption></figure>

<figure><img src="image_5.png" /></figure>

inputs to our PFSS model. GONG provides full-disk magnetograms every minute with a resolution of 5", which are publicly available through the NSO website.<fnref n="8" />

### 3. Plasma Propagation Model

We adopt the ballistic propagation model described in E. Berriot et al. (2024), which allows us to track the plasma parcel at each moment ($t$) after it crosses the inner spacecraft, PSP, at time $t_{in}$. The trajectory can be computed as

$$ \mathbf{R}_{\text{parcel}}(t, t_{\text{in}}) = \mathbf{R}_{\text{PSP}}(t_{\text{in}}) + \int_{t_{\text{in}}}^{t} V(t', t_{\text{in}}) dt', \quad (1) $$

where $\mathbf{R}_{\text{parcel}}(t, t_{\text{in}})$ is the position of the plasma parcel at instant $t$ following its crossing of PSP and $V(t', t_{\text{in}})$ is the propagation velocity of the plasma parcel.

For each $\mathbf{R}_{\text{parcel}}(t, t_{\text{in}})$, we can calculate the distance $d(t, t_{\text{in}})$ between the plasma parcel and the outer spacecraft as

$$ d(t, t_{\text{in}}) = |\mathbf{R}_{\text{SO}}(t) - \mathbf{R}_{\text{parcel}}(t, t_{\text{in}})|. \quad (2) $$

Through Equation (2), we can determine the minimum distance between the plasma parcel and the outer spacecraft at the time $t_{\text{out}}(t_{\text{in}}) = t_{\text{in}} + \tau(t_{\text{in}})$, where $\tau(t_{\text{in}})$ represents the plasma transit time from PSP to SO, i.e.,

$$ d_{\min}(t_{\text{in}}) = \min \{|\mathbf{R}_{\text{SO}}(t) - \mathbf{R}_{\text{parcel}}(t, t_{\text{in}})|\}. \quad (3) $$

At time $t_{\text{out}}$, the position of the plasma parcel is $\mathbf{R}_{\text{parcel}}(t = t_{\text{out}}, t_{\text{in}}) \equiv \mathbf{R}_{\text{out}}(t_{\text{in}})$, and this is the point at which the plasma parcel crossed by PSP (at $t_{\text{in}}$) is at its closest distance to SO (at $t_{\text{out}}(t_{\text{in}})$) after the propagation time $\tau(t_{\text{in}})$.

Several early studies have assumed constant velocity of the plasma parcel in the propagation model (e.g., D. Telloni et al. 2021; T. Alberti et al. 2022a, 2022b). However, the observations of slower wind populations show an increasingly important acceleration in the inner heliosphere (E. Sanchez-Diaz et al. 2016;

M. Maksimovic et al. 2020; J.-B. Dakeyo et al. 2022, 2024; J. Halekas et al. 2022; E. Berriot et al. 2024; Y. J. Rivera et al. 2025). This suggests that the assumption of constant plasma velocity during propagation may not allow us to accurately identify the same plasma parcel at two radially aligned spacecraft, in particular when the separation between them is large. Our present study is motivated by the work of E. Berriot et al. (2024), which established a ballistic propagation model incorporating constant acceleration. In this model, we consider the simplest scenario of constant acceleration, $\mathbf{a}$, constrained by the proton velocities measured by PSP and SO. However, this acceleration $\mathbf{a}$ cannot be directly derived from the two sets of measurements, as identifying the corresponding time values at PSP and SO is inherently complex. Therefore, we first need to link $t_{\text{in}}$ and $t_{\text{out}}(t_{\text{in}})$ through the minimization of $d(t, t_{\text{in}})$ for each $t_{\text{in}}$, scanning a broad range of $\mathbf{a}$ constrained by the PSP and SO flow velocity observations.

By constraining the acceleration profile of the plasma parcel with an arbitrary constant acceleration $\mathbf{a}$, we can determine the position and velocity of the plasma parcel at any time $t > t_{\text{in}}$ after crossing the inner spacecraft PSP as follows:

$$ \mathbf{R}(t) = \mathbf{R}_{\text{in}} + (t - t_{\text{in}})V_{\text{in}} + \frac{(t - t_{\text{in}})^2}{2}\mathbf{a}, \quad (4) $$

$$ V(t) = V_{\text{in}} + (t - t_{\text{in}})\mathbf{a}. \quad (5) $$

After the plasma parcel propagates a certain time, $\tau(t_{\text{in}})$, $\mathbf{a}$, its corresponding position and speed at $t = t_{\text{out}}$ are given as

$$ \mathbf{R}(t = t_{\text{out}}) \equiv \mathbf{R}_{\text{out}} = \mathbf{R}_{\text{in}} + \tau V_{\text{in}} + \frac{\tau^2}{2} \mathbf{a}, \quad (6) $$

$$ V(t = t_{\text{out}}) \equiv V_{\text{out}} = V_{\text{in}} + \tau \mathbf{a}. \quad (7) $$

For each plasma parcel crossing PSP at $t_{\text{in}}$, we model the plasma parcel propagation using uniformly spaced constant acceleration values in the range $[0, a_t]$, where $a_t$ is the upper threshold of acceleration constrained by measurements from PSP and SO, defined as $a_t = (V_{\text{out,max}}^2 - V_{\text{in,min}}^2)/2(||\mathbf{R}_{\text{out}} - \mathbf{R}_{\text{in}}||)$. We chose $V_{\text{out,max}}$ and $V_{\text{in,min}}$ close to the maximum and

<footnote n="8">⁸ https://gong.nso.edu/</footnote>
<figure><img src="image_6.png" /><figcaption>**Figure 2.** Plasma propagation considering a constant acceleration *a*. Panel (a) shows the color map of $\Delta V = \langle V_{\text{p,SO}} \rangle - V_{\text{out}}$, where the minimum of $|\Delta V|$ is indicated by the black curve. Panel (b) shows values of $V_{\text{in}}$ (blue curve), calculated as averages of $V_{\text{p, PSP}}$ over a 1 hr interval centered on $t_{\text{in}}$, $\langle V_{\text{p, SO}} \rangle$ (orange curve) and $V_{\text{out}}$ (green curve) given by Equation (7). Panel (c) shows the minimum distance $d_{\text{min}}$ corresponding to the radial acceleration for minimum of $|\Delta V|$. Panel (d) shows the corresponding propagation times $\tau$ (in black) and $t_{\text{out}}$ (in green). All quantities are shown as a function of $t_{\text{in}}$.</figcaption></figure>

minimum of the radial plasma speeds measured by SO and PSP, respectively. Then, by minimizing $d(t, t_{\text{in}})$ shown in Equation (3), for each $a \in [0, a_t]$, we can find the parameters $t_{\text{out}}(t_{\text{in}}, a)$, $\tau(t_{\text{in}}, a)$, and $d_{\text{MIN}}(t_{\text{in}}, a)$, where $d_{\text{MIN}}$ is the minimum value of $d_{\text{min}}$. During this process, the most compatible value of $a$ is the one that yields the minimum value of $|\Delta V|$, where $\Delta V = \langle V_{\text{p, SO}} \rangle - V_{\text{out}}$. Here, $V_{\text{out}}$ is the radial projection computed using Equation (7), and $\langle V_{\text{p, SO}} \rangle$ is the average proton velocity measured by SO over a 1 hr interval centered on $t_{\text{out}}$. Minimization of $|\Delta V|$ ensures that the propagation model remains consistent with the observations from both PSP and SO.

Figure 2 shows the results of plasma propagation model with constant acceleration *a*, constrained by the observed flow speeds at PSP and SO. Panel (a) illustrates the difference between the speeds computed using Equation (7) and the observed plasma speed at the SO position, defined as $\Delta V = \langle V_{\text{p, SO}} \rangle - V_{\text{out}}$. The black curve denotes the acceleration point at each $t_{\text{in}}$ that minimizes $|\Delta V|$. This can also be inferred from the difference between the green and orange curves in panel (b). The standard deviation of the differences between these two curves is ~3 km s⁻¹, demonstrating that the acceleration bins considered are sufficient to reproduce the enhanced solar wind speed observed at Solar Orbiter’s location. Additionally, the blue curve ($V_{\text{in}}$) represents the averages of plasma speed measured by PSP ($V_{\text{p, PSP}}$) over a 1 hr interval centered at $t_{\text{in}}$. A comparison between $V_{\text{in}}$ and $V_{\text{out}}$ reveals a clear increase in plasma flow speed during its propagation from PSP to SO. This result is in agreement with earlier studies that have reported non-negligible acceleration of the slow solar wind in the inner heliosphere (e.g., R. Schwenn et al. 1981; L. Abbo et al. 2016; E. Sanchez-Diaz et al. 2016; M. Maksimovic et al. 2020; J.-B. Dakeyo et al. 2022; Y. J. Rivera et al. 2025). We note that the solar wind

acceleration rate $a_t$ is not strictly constant and tends to decrease with increasing heliocentric distance. Nevertheless, in the inner heliosphere, the acceleration of solar wind remains significant, in particular for the slow wind. For example, J. Halekas et al. (2022) demonstrated that the acceleration of slow solar wind is substantially greater than that of the fast solar wind, and that a significant portion of slow wind acceleration occurs above $13.3 R_\odot$ (approximately ~0.06 au). The time interval we studied is within the typical range of the slow solar wind, which increases in speed by about 120 km s⁻¹ from PSP to Solar Orbiter. This suggest that the slow solar wind is still in its main acceleration phase, and a constant acceleration rate over time is a reasonable approximation based on recent observations (e.g., J. Halekas et al. 2022; T. Ervin et al. 2024; Y. Jiao et al. 2024; Y. J. Rivera et al. 2024, 2025). Also, the constant acceleration rate $a_t$ used in this study does not imply a linear increase of velocity with radial distance. Instead, the solar wind velocity increases with distance as $R^{0.5}$ in our model. Although it is faster than the results shown in Y. Jiao et al. (2024), where an $R^{0.25}$ evolution is found for slow wind, it is not entirely unreasonable given the statistical uncertainties.

Figure 2(c) shows the minimum distance ($d_{\text{min}}$), calculated using Equation (3), based on the constant acceleration bins represented by the black curve in panel (a). The vertical line in panel (c) represents the value of $t_{\text{in}}$ that corresponds to the minimum of $d_{\text{min}}$. This is simply the closest approach between plasma parcel and the outer spacecraft. At this instant, $\min(d_{\text{min}}) \sim 11 \times 10^6$ km, which is close to a length scale of $l_{\Delta\theta} \sim 10 \times 10^6$ km or 0.07 au. This length scale $l_{\Delta\theta}$ corresponds to the difference in latitude of two spacecraft ($\Delta\theta \sim 4.8^\circ$) around their alignment time and is calculated at the distance of SO. In other words, $l_{\Delta\theta}$ imposes a lower bound on the scale of the plasma parcel we can expect to observe at both
spacecraft (E. Berriot et al. 2024). Panel (d) presents the corresponding propagation time $\tau(t_{in}, a)$ and the time at the outer spacecraft $t_{out}(t_{in}, a)$. Based on the closest approach, as indicated by the minimum value of $d_{min}$, the time intervals associated with the same plasma parcel at PSP and SO should be around $t_{in} \sim 3.25$ hr and $t_{out} \sim 121$ hr, respectively. Although not shown here, the propagation model assuming constant propagation speed results in a delayed propagation time of nearly 40 hr compared to the constant-acceleration model. This outcome is expected, as the observed plasma speed at SO exceeds that at PSP by more than approximately 120 km s⁻¹. Moreover, for a 12 hr interval of $t_{in}$, the variations in $t_{out}$ for the constant-acceleration model (30 hr) are reduced by nearly half compared to those for the constant-speed model (60 hr). This suggests that the constant-acceleration model provides a more reliable estimate for tracing the same plasma parcel during a radial alignment event in the inner heliosphere.

# 4. Identification of the Same Plasma Parcel

Using the propagation model described above, we can roughly estimate the time intervals corresponding to the same plasma parcel. With $t_0 = 2022$ December 10 02:00 UT as the reference time, the model predicts that the same plasma parcel crosses PSP at approximately $t_{in} = t_0 + 3.25$ hr (2022 December 10 05:15 UT) and SO at $t_{out} = t_0 + 121$ hr (2022 December 15 03:00 UT). To validate these estimates, we adopt four different methods: (1) searching similar plasma structures that pass through both spacecraft; (2) matching magnetic field polarity; (3) matching helium abundance (He/H number density ratio); and (4) matching the plasma solar source region. The plasma structures are typically density structures transported by the solar wind that remain largely unaffected by turbulence (N. Sheeley et al. 1997; L. Kepko et al. 2024), making them reliable markers of the same plasma parcel. However, it is important to note that the two spacecraft are separated by $\sim 4.8$ in latitude, which may introduce differences in the observed plasma structure. Prior studies have shown that latitudinal variations in solar wind properties are more pronounced during solar minimum but become less significant during solar maximum (M. J. Owens et al. 2020; R. Laker et al. 2021). Given that our observations occur in the ascending phase of solar cycle 25, latitudinal effects may still be present. Therefore, to identify matching structures, we use a combination of visual inspection and cross-correlation of density and magnetic field strength to assess the correspondence between the two spacecraft measurements.

Earlier studies on radial alignment (e.g., D. Telloni et al. 2021, 2022; T. Alberti et al. 2022a) have examined the correspondence of in situ time series between the two spacecraft by evaluating only the cross-correlation of magnetic field magnitude, partly due to the lack of plasma measurements. However, the cross-correlation function is a linear operator, and it may not accurately capture any nonlinear evolution of the plasma. Additionally, G. E. Box et al. (2015) have shown that the cross-correlation method can sometimes be misleading, and thus it should not be relied upon as a standalone approach for robustly identifying the same plasma parcel crossing two spacecraft. This limitation is particularly significant when the radial separation between the two spacecraft is large, as in the present case, since plasma structures are more likely to evolve over such large separation. Therefore, a physical analysis of relevant plasma quantities,

such as density and magnetic field, is essential to improve the precision of the plasma parcel identification. In this study, we scale PSP measurements to the SO position by accounting for the volumetric expansion of the solar wind plasma and perform a localized comparison of possible structures observed at both locations.

## 4.1. Interval Selection from Parker Solar Probe and Solar Orbiter Observations

Figure 3 shows the plasma parameters observed by PSP and SO around the time $t_{in} \sim 3.25$ hr and $t_{out} \sim 121$ hr, as determined in Section 3. However, to account for the uncertainties inherent in the ballistic propagation model, we examine an extended time window around these predictions: $t_{in} \pm 2.25$ hr for PSP and $t_{out} \pm 3$ hr for SO, ensuring that possible observational matches are not overlooked.

The left and right columns show the PSP and SO observations centered at $t_{in}$ and $t_{out}$ during the above time intervals. We further inspect the same plasma parcel in these two intervals by density structure identification and correlation analysis. Each column, from top to bottom, shows the magnitude of the magnetic field, *B*, components of the magnetic field in the radial tangential normal (RTN) reference frame (radial in blue, tangential in green, and normal in red), *B*; the proton density, $N_p$; the radial solar wind speed, $V_r$; the cosine of the angle between *B* and *V*, $\cos\theta_{BV}$; plasma beta, $\beta$; and normalized cross-helicity, $\sigma_c$. From a visual inspection of panels (a) and (b) in Figure 3, a strikingly similar structure can be identified within the shaded regions. The magnetic field magnitude exhibits an increase during the initial phase of both intervals, followed by two notable drops toward the end. A similar feature is also present in the proton density measurements, shown in panels (e) and (f). Both intervals are characterized by an almost constant proton density for most of the time, except for a significant drop toward the end. It is worth mentioning that the fluctuations in proton moments are more pronounced for PSP, which may be attributed to instrumental noise in SPAN-I caused by field-of-view (FOV) limitations (J. C. Kasper et al. 2016). Indeed, by examining the EFLUX_VS_PHI variable in the SPAN-I L3 proton moment data, we found that the peak of the velocity distribution function (VDF) lies largely within the instrument's field of view (R. Livi et al. 2022; D. Ruffolo et al. 2024; L.-L. Zhao et al. 2025) and the measured proton and alpha particle number densities are reliable for the studied interval. Additionally, a comparison between the electron density derived from quasi-thermal noise (QTN) electric field measurements by the FIELDS/RFS/LFR instrument (M. Moncuquet et al. 2020) and the proton density from SPAN-I indicates that the electron density exhibits significantly smaller fluctuations. However, the uncertainty in QTN-derived $N_e$ is very large for this interval. Additionally, SPAN-I provides proton density ($N_p$) at a higher cadence than QTN, which is crucial for local comparisons of structures within plasma parcels. Therefore, SPAN-I-derived proton moments are used in this study despite the noted fluctuations.

The radial component of the solar wind speed, as shown in panels (g) and (h), exhibits characteristics typical of slow solar wind during both intervals, with the flow speed increasing at larger distances observed by SO. However, the degree of field alignment differs significantly on average, consistent with the evident *B* rotation observed in the components shown in
<figure><img src="image_7.png" /><figcaption>**Figure 3.** Characteristics of same plasma parcel, highlighted by the cyan shadows, observed on 2022 December 10 by PSP (left column) and on 2022 December 15 by SO (right column). Panels from top to bottom show the magnetic field magnitude *B*, component of magnetic field vector in RTN coordinates *B*<sub>rtn</sub>, proton density *N*<sub>p</sub>, radial component of solar wind speed *V*, cosine of the angle between the magnetic field and velocity field vectors cos θ<sub>BV</sub>, plasma beta β, and normalized cross-helicity σ<sub>c</sub>. The vertical black dashed line in each column indicates the model-predicted time *t*<sub>in</sub> at PSP and *t*<sub>out</sub> at SO, corresponding to the same plasma parcel.</figcaption></figure>

<table><caption>**Table 1**

Solar Wind Parameters, Averaged over the Selected Same-plasma Parcel Interval Observed by PSP and SO, as Highlighted by the Cyan-shaded Region in Figure 3</caption><thead><tr><th>SC</th><th>DOY</th><th>Start (UT)</th><th>End (UT)</th><th><i>R</i> (au)</th><th><i>B</i> (nT)</th><th><i>N</i><sub>p</sub> (cm<sup>-3</sup>)</th><th><i>V</i> (km s<sup>-1</sup>)</th><th>β</th><th><i>M</i><sub>A</sub></th><th>σ<sub>c</sub></th></tr></thead><tbody><tr><td>PSP</td><td>344/2022</td><td>05:03</td><td>06:03</td><td>0.11</td><td>319.4 ± 4.8</td><td>1696 ± 280</td><td>220 ± 29</td><td>0.40 ± 0.11</td><td>1.25 ± 0.16</td><td>0.82</td></tr><tr><td>SO</td><td>349/2022</td><td>02:30</td><td>04:00</td><td>0.88</td><td>8.4 ± 0.6</td><td>14.4 ± 0.8</td><td>342 ± 6</td><td>1.52 ± 0.54</td><td>7.14 ± 0.56</td><td>0.80</td></tr></tbody></table>

panels (c) and (d). At PSP, the magnetic field and flow velocity are relatively well-aligned, as shown in panel (i). On the other hand, the average θ<sub>VB</sub> at SO location is about 100°, as shown in panel (j), indicating that the flow is almost perpendicular to the magnetic field. We also calculate the normalized cross-helicity (σ<sub>c</sub>) in these two intervals (panels (m) and (n)), which is often used as a proxy for the Alfvénicity of the plasma (R. D'Amicis & R. Bruno 2015; T. Ervin et al. 2024). The normalized cross-helicity, σ<sub>c</sub>, is usually calculated as σ<sub>c</sub> = 2⟨δ<i>v</i> · δ<b</b>/⟨δ<i>v</i><sup>2</sup>⟩ + ⟨δ<b>z</b><sup>2</sup>⟩), where δ<b>z</b> = δ<b><i>B</i></b>/√4πn<sub>p</sub>m<sub>p</sub> is the magnetic fluctuation in Alfvén units (W. H. Matthaeus & M. L. Goldstein 1982; G. Zank et al. 2012; L.-L. Zhao et al. 2020a). The fluctuating magnetic field δ<b><i>B</i></b> and velocity δ<i>v</i> are computed as δ<b><i>B</i></b> = <b><i>B</i></b> − <b>B</b><sub>0</sub> and δ<i>v</i> = <i>V</i> − <i>V</i><sub>0</sub>. <b>B</b><sub>0</sub> and <i>V</i><sub>0</sub> represent the magnetic and velocity fields averaged over a

10 minute time window. As listed in Table 1, the averaged |σ<sub>c</sub>| for both intervals is relatively high.

## 4.2. Identification of the Same Plasma Parcel with Cross-correlation and Local Comparison

To further search the same plasma parcel during the time period shown in Figure 3, we assess the correspondence between the measurements at two spacecraft using Pearson's cross-correlation analysis.

We calculate the correlation coefficient between the PSP measurements of proton density *N*<sub>p</sub> and magnetic field magnitude *B* during 2022 December 15 05:03–06:03 UT and intervals of same temporal length (1 hr) at SO for time shifts τ ∈ [112, 132] with a sliding window of 0.1 hr. The choice of PSP interval is based on [t<sub>in</sub> − 0.7 hr, t<sub>in</sub> + 0.3 hr], which
<figure><img src="image_8.png" /><figcaption>**Figure 4.** Pearson's cross-correlation coefficient for proton density $N_p$ (black curve) and the magnetic field magnitude $B$ (red curve) between PSP and SO measurements.</figcaption></figure>

corresponds to the time interval at PSP for which the cross-correlation function for both $B$ and $N_p$ measured at SO peaks. The time interval corresponding to the maximum cross-correlation coefficient is assumed to represent the same plasma parcel observed by both spacecraft. As presented in Figure 4, the correlation coefficients reach their maximum at $\tau \sim 121$ hr, which is consistent with the prediction of $t_{\text{out}}$ from constant-acceleration model. Figure 4 also shows that the correlation coefficient of magnetic field magnitude exhibits local maxima at multiple time shifts, indicating potential ambiguity in parcel identification based solely on cross-correlation of $B$. Further confirmation of parcel identification is required through physical analysis of the measurements. We therefore perform a local comparison of the structures by scaling the PSP measurements of $|B|$ and $N_p$ to the SO position. Figure 5 shows the scaled results together with SO measurements. This comparison involves scaling the PSP measurements of $|B|$ and $N_p$ by $(R/R_0)^{-1.8}$ and $(R/R_0)^{-2}$, respectively, where $R_0 = 0.88$ au represents the radial distance of SO from the Sun. These scaling factors account for systematic radial variations, such as those arising from volumetric expansion. The proton density in the solar wind approximately follows an $R^{-2}$ dependence, consistent with mass conservation in a spherically expanding flow. The magnetic field also varies with heliocentric distance, but not as simply as $R^{-2}$. In the inner heliosphere, the Parker spiral model predicts that the radial component of the magnetic field, $B_r$, scales as $R^{-2}$ due to magnetic flux conservation (analogous to mass conservation) as field lines are distributed over an expanding spherical surface. However, the tangential component, $B_t$, due to the rotation of the Sun, decreases more slowly with distance, typically as $R^{-1}$. The normal component, $B_n$, due to turbulence and fluctuations, also decays more gradually than $R^{-2}$. As a result, the total magnetic field strength, $|B| = \sqrt{B_r^2 + B_t^2 + B_n^2}$, deviates from the ideal $R^{-2}$ scaling. This deviation becomes more pronounced in the outer heliosphere, where the tangential component dominates. In contrast, within the inner heliosphere, where $B_r$ still contributes significantly, the deviation remains small, typically between $R^{-1.8}$ and $R^{-2}$. Observationally, P. Hellinger et al. (2011) and D. Perrone et al. (2019, 2022) reported that the magnetic field strength scales approximately as $R^{-1.6}$ in the inner heliosphere, suggesting that turbulent processes may affect the expansion of the magnetic field. More recently, Z. Huang et al. (2024)

introduced a novel time-series visualization technique and applied to PSP measurements, finding that the magnetic field strength deviates from $R^{-2}$ scaling. Instead, it follows a remarkably stable scaling of $R^{1.87 \pm 0.02}$, regardless of locations and scales (see also S. Bale et al. 2019). It has been suggested that this persistent deviation from the $R^{-2}$ scaling could be evidence of active acceleration of the solar wind in the inner heliosphere.

Figure 5 shows clearly that the two sets of measurements exhibit a relatively good correspondence. Given the uncertainties in the radial dependence of $B$ and $N_p$, the large consistency within these two intervals suggests that they represent the same plasma parcel.

To further corroborate our finding, we compare the mass flux, $g = N_p V R^2$, which is expected to be conserved for the same plasma parcel (Y. J. Rivera et al. 2024, 2025). Using the average values from Table 1, we compute the ratio of mass flux, $g_{\text{PSP}}/g_{\text{SO}}$, as $1.18 \pm 0.26$. Since we obtain a value close to 1 within the limit of error (1$\sigma$ uncertainty), this suggests that the plasma parcel maintains mass flux conservation. Taken together, these results indicate that the plasma parcel observed by PSP on 2022 December 10 between 05:03–06:03 UT and SO on 2022 December 15 02:30–04:00 UT can be reliably assumed to represent the same plasma parcel for further analysis. We note that the interval chosen at SO is half an hour longer than that at PSP, to account for the expansion of the solar wind. Table 1 displays the typical solar wind parameters, averaged within the intervals of interest, corresponding to the same plasma parcel (highlighted by cyan shadows in Figure 3).

### 4.3. Composition Analysis: Helium Abundance

To make our identification more robust, we also compared the helium abundance ($N_{\text{He}}/N_p$) of the plasma parcels observed at the two spacecraft, as shown in Figure 6. To compute the helium composition of the solar wind from SO/PAS, we used velocity distribution functions (VDFs) of the thermal component of the solar wind, derived from the differential energy flux. For the SO interval, the VDFs exhibit a clear separation between protons and alpha particles ($\text{He}^{2+}$), which allows us to fit the alpha particle distribution separately with a Maxwellian function. These fits are then used to determine the density of alpha particles, following an approach similar to that implemented by M. Johnson et al. (2024).
<figure><img src="image_9.png" /><figcaption>**Figure 5.** Local comparison of the plasma parcel observed by both PSP and SO by scaling PSP-measured proton density $N_p$ and magnetic field magnitude $B$ with a factor $R^{-2}$ and $R^{-1.8}$, respectively, to take the adiabatic expansion from PSP (blue) to SO (orange) into account.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 6.** Helium abundance as the He/H number density ratio for PSP (left) and SO (right). The black points show the spacecraft-observed values, while red solid curves represent the 50 point median filter. Large circles with error bars represent mean values with $1\sigma$ uncertainty within the shaded interval.</figcaption></figure>

During the expansion of a plasma parcel, the helium abundance is typically expected to remain consistent once leaving the corona (Y. J. Rivera et al. 2022). As seen in Figure 6, the helium abundance remains consistently around 3% in both plasma parcels, though some substructure develops. Since plasma composition is expected to remain unchanged throughout the inner heliosphere, this agreement further supports the idea that the two intervals correspond to the same plasma parcel.

### 4.4. Connection to the Solar Source Regions

To check whether the plasma parcel observed by PSP and SO in the selected intervals comes from the same source regions at the Sun, we use the potential field source surface (PFSS) model (J. M. Wilcox & N. F. Ness 1967; M. D. Altsc'huler & G. Newkirk 1969) along with an ideal Parker spiral to estimate the photospheric footpoints of the plasma parcel of interest. From the locations of PSP and SO, the solar wind is ballistically backmapped using the average solar wind speed.

The PFSS model is then used to trace the magnetic field lines from the photosphere out to the radial source surface. To account for the uncertainties associated with constant flow speed in ballistically backmapping, we vary the solar wind speed by $\pm40 \text{ km s}^{-1}$ and find that it does not change the resulting source regions of the observed wind. As inputs to the PFSS model, we use Global Oscillations Network Group (GONG) synoptic magnetic field maps and a value for the source surface height ($R_{ss}$) at which the field is assumed to be fully radial. We chose $R_{ss} = 3.0 R_\odot$ to match the modeled and observed HCS (S. T. Badman et al. 2022), specifically noting that lower source surface heights led to a prediction of a separate positive sector around 270°–280° longitude, which was not observed in situ by either spacecraft.

Figure 7 shows the predicted footpoints of the plasma parcel measured in situ by PSP on 2022 December 10 00:00 and SO on 2022 December 15 05:00. The blue/red loop represents PSP's trajectory, and the horizontal blue/red stripe represents SO's trajectory, both ballistically backmapped to $3.0 R_{ss}$. The
<figure><img src="image_11.png" /><figcaption>**Figure 7.** PFSS model for PSP on 2022 December 10 05:00 and SO on 2022 December 15 02:30 using an input ADAPT-GONG magnetogram.</figcaption></figure>

black square and triangle mark the ballistically mapped locations of the observed plasma parcel for the identified intervals corresponding to PSP and SO, respectively. The PFSS-modeled HCS aligns well with the measured radial magnetic field polarity from both spacecraft, suggesting that the ballistic propagation method is effective and the coronal model is accurate. This implies that our magnetic footpoints are well estimated. Both PSP and SO are connected to the near-equatorial coronal hole extension around 290° Carrington longitude, providing evidence that the plasma measurements studied here come from the same solar source. We note that the minor separation between the black square and triangle reflects the difference in localization when using a constant-speed model versus a constant-acceleration model. The fact that the source mapping result remains unchanged despite this small separation suggests that the inference is robust. Furthermore, the mapping is independent of the spacecraft latitudinal separation. The confirmation that both plasma parcels originate from the same solar source enables a robust comparison of their turbulence properties during outward radial expansion in the inner heliosphere, as it minimizes variations caused by different source regions.

## 5. Turbulence Anisotropy

We analyze the turbulence properties of the plasma parcel to characterize the evolution of turbulence with radial distance. We first calculate the power spectral densities (PSDs) of the magnetic field fluctuations, $P$, and magnetic compressibility ($C_B = P_{B}/P_{Tr}$) for the two intervals, respectively. Figure 8 shows the PSDs of the magnetic field fluctuations measured by PSP at 0.11 au (left panel) and by SO at 0.88 au. The PSDs are computed in the mean magnetic field coordinate system (J. W. Belcher & L. Davis 1971), $\hat{z} = B_0/|B_0|$, $\hat{y} = -\hat{z} \times \hat{R}/|\hat{z} \times \hat{R}|$, and $\hat{x} = \hat{y} \times \hat{z}$, where $B_0$ is the mean magnetic field vector averaged over the interval and $\hat{R}$ is the radial unit vector. Accordingly, the parallel compressible spectrum $P_\parallel$ (pink curve) represents the PSD along the mean magnetic field $B_0$ direction, and the transverse spectrum $P_\perp$ (green curve) is the sum of the PSDs in the two perpendicular directions. Earlier studies on radial alignment (e.g., D. Telloni et al. 2021, 2022; T. Alberti et al. 2022a) focused on the

variation of the inertial-range turbulence spectra by analyzing the trace spectra of magnetic field fluctuations. In contrast, this study examines the radial evolution of the spectral slope in both the inertial range and dissipation ranges.

As expected, the spectral power density decreases as the radial distance increases. At 0.11 au measured by PSP, the total magnetic field fluctuation power is dominated by the transverse fluctuation. In contrast, the parallel fluctuations are almost comparable to the transverse fluctuations at 0.88 au. This feature is also reflected in the magnetic compressibility $C_B$, shown in the bottom panels. It is evident that the plasma parcel observed by the PSP is less compressible than that observed at the SO location as it propagates outward. This is consistent with statistical studies showing that the magnetic compressibility of solar wind turbulence increases with radial distance (e.g., C. Chen et al. 2020; D. Telloni et al. 2021, 2022; X. Zhu et al. 2024)). We also perform the power-law fits in both the inertial range and the dissipation range. For the PSP observation at 0.11 au, the power-law fits are performed between $8 \times 10^{-3}$ and $5 \times 10^{-1}$ Hz in the inertial range and between 7 and 20 Hz in the dissipation range. For the SO observation at 0.88 au, the fitting is done between $3 \times 10^{-3}$ and $3 \times 10^{-1}$ Hz in the inertial range and between 0.7 and 2.5 Hz in the dissipation range. The observed 1D reduced perpendicular spectrum $P_\perp$ index for PSP in the inertial range is close to the Iroshnikov-Kraichnan (IK) scaling of $-3/2$ (R. H. Kraichnan 1965), and for SO, the spectral index is consistent with Kolmogorov $-5/3$ scaling. This is consistent with statistical results from C. Chen et al. (2020), L.-L. Zhao et al. (2022), and N. Sioulas et al. (2023). The parallel spectrum $P_\parallel$ shows a similar trend, although it becomes even steeper at the SO location (e.g., close to $-2$ scaling). One possible explanation is that, at PSP, the magnetic field $B$ is aligned with the wavevector $k$, while at SO, they are perpendicular, as indicated by the $\theta_{BV}$ angle (and Taylor hypothesis) in panels (i) and (j) of Figure 3. Consequently, at PSP, both the parallel and perpendicular spectra are associated with $k_\parallel$, whereas at SO, they correspond to $k_\perp$, which is consistent with the observed differences in scaling.

In the dissipative range, the PSD observed by PSP at 0.11 au appears to be slightly steeper than the typically observed $f^{-3}$ scaling at 1 au (R. Bruno & L. Trenchi 2014). Nevertheless, the overall evolution appears to be different from the inertial
<figure><img src="image_12.png" /><figcaption>**Figure 8.** Power spectral density (PSD) of the total magnetic field fluctuations, separated into compressible parallel ($P_‖$, pink solid lines) and transverse fluctuations ($P_⊥$, green solid lines) (panels (a) and (b)), and magnetic compressibility $C_B$ (panels (c) and (d)) for the intervals observed by PSP (left column) at 0.11 au and SO (right column) at 0.88 au. The best power-law fits are represented by dashed lines in the respective colors, and the corresponding spectral break frequency $f_b$ is indicated by a black arrow. The vertical dashed lines indicate the proton gyrofrequency $f_p$ in the plasma frame.</figcaption></figure>

range, with steeper dissipation range spectra observed at closer distance. This is also found in D. Duan et al. (2020) and D. Perrone et al. (2022). At sub-ion scales, SO exhibits ~$f^{-3}$ scaling. The dissipation range spectrum at the PSP is steeper, indicating that stronger dissipation is observed at PSP location, which is partially due to the parallel wavevector $k_‖$ it samples. Since Landau and cyclotron damping are most efficient for waves with higher $k_‖$ (e.g., T. A. Bowen et al. 2020), the presence of more parallel-propagating fluctuations at PSP results in stronger dissipation. On the other hand, as the solar wind expands, the Parker spiral bends more azimuthally, reducing the sampled parallel wavevector components, leading to a decrease in the Landau damping efficiency and thus a shallower dissipation range spectrum observed at SO. However, other possible causes, e.g., expansion effects may also play a role in shaping these spectra. This dissipation range spectral evolution has not been identified in previous radial alignment studies (e.g., R. D'Amicis et al. 2010; D. Telloni et al. 2021, 2022). Additionally, the spectral break frequency $f_b$ between the two scales shifts toward lower frequencies with increasing radial distance, which is in agreement with the previous findings of R. Bruno & L. Trenchi (2014), D. Duan et al. (2020), M. Terres & G. Li (2022), and S. Lotz et al. (2023).

## 6. Intermittency Analysis

Here, we analyze the intermittency of the magnetic field fluctuations at different scales. We perform a standard

analysis, starting from the comparison of probability density functions (PDFs) of the standardized variable of increments at different timescales with the Gaussian distribution, namely

$$ \delta X_i(t, \tau) = X_i(t + \tau) - X_i(t), \quad (8) $$

where $X_i$ are the components of the magnetic field vector and $\tau$ is the time lag scale.

The degree of deviation of the standardized variables of the increments from a Gaussian distribution quantifies the intermittency level, which can be assessed via the fourth-order moments of the distribution of the increments, the so-called kurtosis or flatness (U. Frisch 1995; L.-L. Zhao et al. 2020b), defined as

$$ F_i(\tau) = \frac{\langle \delta X_i(t, \tau)^4 \rangle}{\langle \delta X_i(t, \tau)^2 \rangle^2}, \quad (9) $$

where $\langle \dots \rangle$ represents the time average over a suitable interval. For reference, a standard Gaussian distribution has a flatness of $F(\tau) = 3$.

Figure 9 shows the PDFs of the magnetic field increments for the plasma parcel observed by PSP at 0.11 au (left column) and SO at 0.88 au (right column) for three different time lags, $\tau$. For PSP, we have chosen $\tau_1 = 10$ s, $\tau_2 = 0.5$ s, and $\tau_3 = 0.1$ s. For SO, we have chosen $\tau_1 = 100$ s, $\tau_2 = 5$ s, and $\tau_3 = 1$ s. The three different time lags ($\tau_1$, $\tau_2$, and $\tau_3$) chosen in our study are representatives of three characteristic scales: (1) the scale well within the MHD inertial range; (2) the scale near the end of the MHD regime but not yet reaching the spectral
<figure><img src="image_13.png" /><figcaption>Figure 9. PDFs of magnetic field increments at three different timescales (τ; see legends) for the radial (δB_r; top row), tangential (δB_t; middle row), and normal (δB_n; bottom row) components. A Gaussian distribution is shown as a reference (dashed black line).</figcaption></figure>

break; and (3) the scale beyond the break and near the onset of the kinetic range. Different τ were used for PSP and SO, since the inertial range shifts from higher (smaller) to lower (larger) frequencies (timescales), as shown in Figure 8 (panels (a) and (b)) from 0.11 to 0.88 au. In Figure 9, the magnetic field fluctuations show clear heavy-tail distributions for both PSP and SO, indicating a significant departure from the Gaussian distribution. The presence of non-Gaussian tails is a signature of strong intermittency (J. He et al. 2019; L.-L. Zhao et al. 2020b; R. Chhiber et al. 2021). It is noteworthy that the heavier tail of PSP appears in the radial direction, while it appears in the tangential direction for SO observation.

## 7. Castaing Multifractal Model for Intermittent PDFs

Intermittency in turbulence arises from the multifractal, inhomogeneous nature of the energy cascade (L. Sorriso-Vallvo et al. 2001; V. Carbone et al. 2004). To quantitatively describe the statistical characteristics of intermittency, we further use the Castaing distribution (B. Castaing et al. 1990) to characterize the behavior of the intermittent PDFs in the

framework of a multiplicative cascade model, which is represented by a Gaussian convolution whose variance follows a log-normal distribution. We begin with the PDF of a Castaing distribution:

$$ \text{PDF}_{\text{Castaing}}(\delta B(\tau)) = \frac{1}{2\pi\lambda} \int_{0}^{\infty} \exp\left(-\frac{\delta B^2}{2\sigma^2}\right) \times \exp\left(-\frac{(\ln\sigma - \mu)^2}{2\lambda^2}\right) \frac{d\sigma}{\sigma^2}, \quad (10) $$

where σ(τ) is the standard deviation in the Gaussian distribution of the magnetic field increment, δB(τ). We used the magnetic field increments of the normal component B_n only because B_n is always perpendicular to the mean magnetic field, and thus it minimizes the influence of the Parker spiral on our analysis. The Castaing PDF shown in Equation (10) is simply the integral of a joint PDF, formed by combining a normal distribution of δB(τ) and a log-normal distribution of σ(τ), i.e., PDF(δB(τ), ln(σ(τ))), over the range of the variable ln(σ). By invoking the Castaing
<figure><img src="image_14.png" /><figcaption>**Figure 10.** Top panels: Comparison of reconstructed structure functions for PSP (panel (a)) and SO (panel (b)) intervals, based on the fitted parameters $\mu(\tau)$ and $\lambda^2(\tau)$ of the Castaing function (colored lines, corresponding to different orders, $q$), with observational results (black circles). The structure functions are shown for orders $q = 0.5, 1.0, 1.5, 2.5, 3.0$, and 3.5. The vertical blue dashed lines indicate the fitting range of the structure functions used to determine the scaling exponent. Bottom panels: Scale dependence of flatness for PSP (panel (c)) and SO (panel (d)) intervals. Black lines represent flatness from observational data, while red lines correspond to values derived from the Castaing parameter $\lambda^2(\tau)$.</figcaption></figure>

function, the intermittent turbulence can be approximately represented with two scale-dependent parameters ($\mu(\tau)$, $\lambda^2(\tau)$), which are obtained through a nonlinear fit of the PDFs to Equation (10). The scaling of the parameter $\lambda^2(\tau)$ controls the shape of the PDF tails and describes the deviation from the parent distribution, characterizing the intermittency in the inertial range (B. Castaing et al. 1990; L. Sorriso-Valvo et al. 1999).

Based on the framework of the Castaing PDF, one can derive the following expression for a qth-order structure function:

$$S^q(\tau) = C_q \exp\left[\frac{q^2}{2}\lambda^2(\tau) + q\mu(\tau)\right], \quad (11)$$

where $C_q = \frac{\sqrt{2^q}}{\sqrt{\pi}}\Gamma(\frac{1+q}{2})$. The scaling exponent is therefore

$$\zeta(q) = \frac{d(\ln S^q)}{d \ln \tau} = \frac{q^2}{2} \frac{d\lambda^2}{d \ln \tau} + q \frac{d\mu}{d \ln \tau}. \quad (12)$$

The scaling of the flatness in terms of $\lambda^2(\tau)$ can be obtained from the structure function $S^q(\tau)$ (L. Sorriso-Valvo et al. 2015) as

$$F(\tau) = \frac{S^4(\tau)}{(S^2(\tau))^2} = 3e^{4\lambda^2(\tau)}. \quad (13)$$

Figure 10 presents the fitting results for the structure functions (panels (a) and (b)) and the scale dependence of flatness (panels (c) and (d)), obtained using the Castaing parameters in Equations (11) and (13), for the PSP (left column) and SO (right column) intervals. In panels (a) and (b), the black circles represent the structure functions (SFs) obtained directly from observation for orders $q = 0.5, 1.0, 1.5, 2.5, 3.0$, and 3.5, plotted against the second-order structure function $S^2(\tau)$. This is done using the extended self-similarity (ESS) method (R. Benzi et al. 1993) to estimate scaling exponents more reliably. With approximately 43,680
<figure><img src="image_15.png" /><figcaption>**Figure 11.** Scaling exponent of the structure function vs. order q in the MHD inertial range for PSP (panel (a)) and SO (panel (b)) intervals. The scaling exponents $\zeta(q)$, shown in black curves, are obtained from the power-law fit of data with $S^q(\tau) \sim (S^2(\tau))^{-\zeta(q)/\zeta(2)}$, while the red curves represent the theoretical scaling exponent $\zeta(q)$ obtained on the basis of Castaing function using Equation (12).</figcaption></figure>

data points in both the PSP and SO intervals, the maximum order of convergence of structure functions is about 3.5 (T. Dudok de Wit 2004). It can be seen that the SF based on the Castaing distribution fits well with the observed SF at both PSP and SO locations. Panels (c) and (d) shows the comparison between observed (black curve) and fitting-derived (red curve) flatness profile in the log-log scale for PSP at 0.11 au and SO at 0.88 au, respectively. At PSP, the flatness shows an increasing trend at larger scales (power-law increase), saturation near intermediate range and finally a weak decrease at small scales. On the other hand, the SO interval also shows an increase in flatness at larger scales, although it is not a clear power law. Additionally, the fitting-derived flatness is slightly larger than the observed counterpart, which may be related to the fitting defect on the PDF's tail (J. He et al. 2019). Overall, the fitting results are in good agreement with observations of PSP and SO.

Figure 11 shows the comparison between the observed scaling exponents, $\zeta(q)$, of the qth-order structure function (black curve) and the scaling exponents (red curve) calculated from the fitting parameters $\mu(\tau)$ and $\lambda^2(\tau)$ of the Castaing function, according to Equation (12). As mentioned above, the observed scaling exponents $\zeta(q)$ are obtained using the ESS method through power-law fits of $S^q(\tau)$ versus $S^2(\tau)$ and are normalized using the spectral index $\zeta(2) = \alpha - 1$, where $\alpha$ is the inertial-range spectral index, shown in Figure 8. For both PSP and SO, $\zeta(q)$ shows a concave nonlinear trend, a signature of multifractal scaling of intermittency, which can be mathematically explained by and attributed to the small ratio of $\frac{d\lambda^2}{d\ln\tau}$ to $\frac{d\mu}{d\ln\tau}$ ($\left|\frac{d\lambda^2}{d\ln\tau} \cdot \left(\frac{d\mu}{d\ln\tau}\right)^{-1}\right| \ll 1$) and $\frac{d\lambda^2}{d\ln\tau} < 0$. In comparison to PSP, the shape of the scaling exponent at SO shows less curvature in the concave profile, indicating that the intermittency has weakened in the plasma parcel during its propagation. This is also supported by the flatness profile in the bottom panel of Figure 10, which shows a notably shallower scaling at SO compared to PSP.

The decrease in intermittency with distance is consistent with previous statistical studies on the evolution of turbulent intermittency (T. N. Parashar et al. 2019; D. Perrone et al. 2022; L. Sorriso-Valvo et al. 2023). However, earlier studies on radial alignment (e.g., D. Telloni et al. 2021, 2022; T. Alberti et al. 2022a) reported an increase in intermittency with radial distance. One possible reason for this discrepancy is that, in our case, the plasma parcel observed by PSP may be associated with large number of coherent advected structures, which can enhance intermittency. By the time the parcel reaches SO, the fluctuations may have become more stochastic in nature, leading to a reduction in intermittency. It is noteworthy that this behavior of scaling exponent is well captured by the Castaing model, making it an effective tool for analyzing intermittency in solar wind turbulence.

## 8. Conclusions

In this work, we have investigated the evolution of solar wind turbulence properties, such as anisotropy in the magnetic field fluctuation power and spectral index, compressibility, and intermittency for the same plasma parcel observed by PSP at 0.11 au and SO at 0.88 au during the radial alignment in 2022 December 10. Our main conclusions are as follows:

1. The plasma propagation model with constant acceleration rate, constrained by velocity measurements from both spacecraft, provides a more reliable estimate of the “plasma-lineup” time during radial alignment of two spacecraft. This can be attributed to the non-negligible acceleration of slow solar wind in the inner heliosphere. The same plasma parcel can be further verified by matching magnetic field polarity, mass flux, helium abundance, and solar source region, and then identifying similar magnetic field and density variations through cross-correlation and local comparison of large-scale and fine-scale structures. A limitation of our study is that we did not account for interactions between the plasma
parcel and adjacent streams, which could play a role in the field-line modification and plasma travel time, potentially introducing additional deviations in the ballistic estimates (E. Sanchez-Diaz et al. 2016).

2. The magnetic field fluctuations decrease with distance, as expected. The magnetic fluctuations observed at the Solar Orbiter location are more compressible, although transverse fluctuations still dominate. The increase in compressibility is also observed in D. Telloni et al. (2021, 2022). In addition, the inertial-range turbulence spectrum is steeper at the Solar Orbiter location, while the dissipation range spectrum becomes flatter. The distinct spectral evolutions in the two ranges are consistent with earlier statistical results (e.g., D. Duan et al. 2020).

3. By invoking the Castaing model, we find that the intermittent nature of the plasma parcel changes significantly during its radial evolution. The scaling exponents of the structure functions in the MHD inertial range observed by PSP and SO both follow a concave nonlinear trend, which is a characteristic of intermittent multifractal scaling (L. Sorriso-Valvo et al. 2001). However, the reduced curvature of the multifractal scaling at SO location suggests that the plasma parcel becomes less intermittent with increasing radial distance, which is consistent with earlier studies (e.g., D. Perrone et al. 2022; L. Sorriso-Valvo et al. 2023). This may be because the plasma parcel observed at PSP location is more structured, and as the solar wind propagates outward, these sharp structures are smoothed out by nonlinear interactions and expansion effects.

Further studies utilizing coordinated multi-spacecraft observations will be essential to comprehensively characterize the evolution of heliospheric solar wind plasma, not only as a function of radial distance but also across different phases of the solar cycle and under varying solar wind conditions, thereby advancing our understanding of plasma turbulence and its role in heliospheric dynamics.

## Acknowledgments

The PSP and Solar Orbiter data used in this study are available at the NASA Space Physics Data Facility (SPDF): https://spdf.gsfc.nasa.gov. GONG data are obtained from the NSO Integrated Synoptic Program, managed by the National Solar Observatory, which is operated by AURA, Inc., under a cooperative agreement with the National Science Foundation and with contribution from the National Oceanic and Atmospheric Administration. We acknowledge the partial support of the NSF EPSCoR RII-Track-1 Cooperative Agreement OIA-2148653, NASA awards 80NSSC20K1783, 80NSSC23K0415, NASA FINESST award 80NSSC24K1867, NSF EPSCoR GRSP award 735A1W, and a NASA IMAP subaward under NASA contract 80GSFC19C0027. L.Z., L.Z.H., and L.S.-V. were partially supported by the International Space Science Institute (ISSI) in Bern, through ISSI International Team project #23-591 (Evolution of Turbulence in the Expanding Solar Wind). L.S.-V. was supported by the Swedish Research Council (VR) Research grant N. 2022-03352. Y.J.R. and S.T.B. are partially supported by the Parker Solar Probe project through the SAO/SWEAP subcontract 975569. J.G.A.G is partially supported by NASA grant 80NSSC24K0568.

## ORCID iDs

Ashok Silwal https://orcid.org/0000-0001-6286-2106

Lingling Zhao https://orcid.org/0000-0002-4299-0490

Xingyu Zhu https://orcid.org/0000-0002-1541-6397

Luca Sorriso-Valvo https://orcid.org/0000-0002-5981-7758

Lina Z. Hadid https://orcid.org/0000-0002-8587-0202

Gary P. Zank https://orcid.org/0000-0002-4642-6192

Hui Li https://orcid.org/0000-0003-3556-6568

Samuel T. Badman https://orcid.org/0000-0002-6145-436X

Yeimy J. Rivera https://orcid.org/0000-0002-8748-2123

Sujan Prasad Gautam https://orcid.org/0000-0001-7379-4268

Monika Karki https://orcid.org/0000-0001-5356-2267

Juan G. Alonso Guzman https://orcid.org/0000-0001-9581-3167

Nibuna S. M. Subashchandar https://orcid.org/0009-0008-2509-9330

Zeping Jin https://orcid.org/0000-0001-8747-7407

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