# Magnetic field spectral evolution in the inner heliosphere

Nikos Sioulas Department of Earth, Planetary, and Space Sciences, University
of California, Los Angeles
Los Angeles, CA 90095, USA  Zesen Huang (黄泽森) Department of Earth, Planetary,
and Space Sciences, University of California, Los Angeles
Los Angeles, CA 90095, USA  Chen Shi (时辰) Department of Earth, Planetary, and
Space Sciences, University of California, Los Angeles
Los Angeles, CA 90095, USA  Marco Velli Department of Earth, Planetary, and
Space Sciences, University of California, Los Angeles
Los Angeles, CA 90095, USA  Anna Tenerani Department of Physics, The
University of Texas at Austin,
TX 78712, USA  Trevor A. Bowen Space Sciences Laboratory, University of
California,
Berkeley, CA 94720-7450, USA  Stuart D. Bale Physics Department, University of
California, Berkeley, CA 94720-7300, USA  Space Sciences Laboratory,
University of California, Berkeley, CA 94720-7450, USA  Jia Huang Climate and
Space Sciences and Engineering, University of Michigan,
Ann Arbor, MI 48109, USA  Loukas Vlahos Deprtment of Physics, Aristotle
University of Thessaloniki
GR-52124 Thessaloniki, Greece  L. D. Woodham Department of Physics, The
Blackett Laboratory, Imperial College London, London, SW7 2AZ, UK  T. S.
Horbury The Blackett Laboratory, Imperial College London, London, UK  Thierry
Dudok de Wit LPC2E, CNRS and University of Orléans, Orléans, France  Davin
Larson Space Sciences Laboratory, University of California, Berkeley, CA
94720-7450, USA  Justin Kasper BWX Technologies, Inc., Washington DC 20002,
USA  Climate and Space Sciences and Engineering, University of Michigan, Ann
Arbor, MI 48109, USA  Christopher J. Owen Mullard Space Science Laboratory,
University College London, Dorking, RH5 6NT, UK  Michael L. Stevens Harvard-
Smithsonian Center for Astrophysics,
Cambridge, MA 02138, USA  Anthony Case Smithsonian Astrophysical Observatory,
Cambridge, MA 02138, US  Marc Pulupa Space Sciences Laboratory, University of
California, Berkeley, CA 94720-7450, USA  David M. Malaspina Astrophysical and
Planetary Sciences Department, University of Colorado, Boulder, CO, USA
Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder,
CO, USA  J.W. Bonnell Space Sciences Laboratory, University of California,
Berkeley, CA 94720-7450, USA  Roberto Livi Space Sciences Laboratory,
University of California, Berkeley, CA 94720-7450, USA  Keith Goetz School of
Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA
Peter R. Harvey Space Sciences Laboratory, University of California, Berkeley,
CA 94720-7450, USA  Robert J. MacDowall Solar System Exploration Division,
NASA/Goddard Space Flight Center, Greenbelt, MD, 20771  Milan Maksimović
LESIA, Observatoire de Paris, Universit e PSL, CNRS, Sorbonne Universit e,
Universit e de Paris, 5 place Jules Janssen, 92195 Meudon, France  P. Louarn
IRAP, Université Toulouse III - Paul Sabatier, CNRS, CNES, Toulouse, France
A. Fedorov IRAP, Université Toulouse III - Paul Sabatier, CNRS, CNES,
Toulouse, France

###### Abstract

Parker Solar Probe and Solar Orbiter data are used to investigate the radial
evolution of magnetic turbulence between $0.06~{}\lesssim R~{}\lesssim 1$ au.
The spectrum is studied as a function of scale, normalized to the ion inertial
scale $d_{i}$. In the vicinity of the Sun, the inertial range is limited to a
narrow range of scales and exhibits a power-law exponent of,
$\alpha_{B}=-3/2$, independent of plasma parameters. The inertial range grows
with distance, progressively extending to larger spatial scales, while
steepening towards a $\alpha_{B}=-5/3$ scaling. It is observed that spectra
for intervals with large magnetic energy excesses and low Alfvénic content
steepen significantly with distance, in contrast to highly Alfvénic intervals
that retain their near-Sun scaling. The occurrence of steeper spectra in
slower wind streams may be attributed to the observed positive correlation
between solar wind speed and Alfvénicity.

Magnetohydrodynamics (1964), Solar wind (1534), Alfven waves (23)

··

##  1 Introduction

The solar wind flow transports a wide range of magnetic field and plasma
fluctuations (Coleman, 1968; Velli et al., 1989). Because fluctuations are
predominantly Alfvénic (i.e., magnetic field and velocity fluctuations exhibit
the correlations typical of outwardly propagating Alfvén waves) (Bruno &
Carbone, 2013), and relative density fluctuations are very small solar wind
turbulence is usually discussed within the phenomenologies of incompressible
magnetohydrodynamic (MHD).

During the expansion, non-linear interactions result in a cascade of the
energy towards smaller scales (Matthaeus & Velli, 2011). Therefore, the energy
injected into the solar wind at large scales, likely of solar origin, cascades
downwards until it reaches ion scales, at which point the dynamics involve
kinetic processes and structures such as ion cyclotron damping, kinetic Alfvén
waves, kinetic scale current sheets, etc. (Leamon et al., 1999; Cranmer, 2001;
Dmitruk et al., 2004; TenBarge & Howes, 2013; Karimabadi et al., 2013).
Turbulence is thought to be one of the main processes contributing to the non-
adiabatic expansion, as well as the acceleration of the solar wind (SW)
(Matthaeus & Velli, 2011). MHD turbulence phenomenologies predict different
power law exponents depending on prevailing characteristics of turbulence,
such as spatial wave-number anisotropy (Goldreich & Sridhar, 1995, 1997),
intermittency (Mininni & Pouquet, 2009; Chandran et al., 2015), and the scale-
dependent correlation between velocity and magnetic field (Boldyrev, 2006;
Beresnyak & Lazarian, 2010). The variability of solar wind turbulence
properties in the inner heliosphere reflects the diversity of solar coronal
sources, that modulate the density, velocity, temperature, and ion composition
of the plasma. As a result, several factors, including the role played by
large-scale gradients (Velli et al., 1989; Chandran & Perez, 2019); the
proximity to the heliospheric current sheet (Chen et al., 2021; Shi et al.,
2022a); the presence of magnetic field switchbacks (Martinović et al., 2021;
Bourouaine et al., 2020; Shi et al., 2022b); large-scale velocity shear in the
SW (Coleman, 1968), strongly influence the properties of turbulence, resulting
in a wide range of spectral scalings. By means of fitting the power-spectrum
within a constant range in the frequency domain, recent statistical studies of
PSP data, have recovered a non-evolving velocity spectral index close to
$-3/2$, independent of the radial distance from the Sun (Shi et al., 2021),
while the magnetic field spectrum steepens from a -$3/2$ slope at $\sim 0.2\
au$ to a -$5/3$ slope at $\sim 0.6\ au$ (Chen et al., 2020; Shi et al., 2021).

Two scales are crucial to understanding the radial evolution of turbulence in
the solar wind (1) the ion inertial scale $d_{i}={V_{A}}/{\Omega_{i}}$, and
(2) the thermal ion gyroradius, $\rho_{i}={V_{th,i}}/{\Omega_{i}}$, where,
$\Omega_{i}={e|B|}/{m_{p}}$, is the proton gyrofrequency, $e$ is the
elementary charge, $|B|$ is the magnitude of the magnetic field, and $m_{p}$
is the mass of the proton. With increasing heliocentric distance, both
physical scales ($d_{i}$, $\rho_{i}$) increase (Duan et al., 2020; Cuesta et
al., 2022).

It is thus natural to expect that the relative physical scale of fluctuations
of a given frequency decreases as the solar wind expands. Here we aim to
understand the radial evolution of magnetic turbulence and to study the basic
features of scaling laws for solar wind fluctuations in terms of properly
normalized physical scales. High resolution data from Parker Solar Probe
($PSP$) (Fox et al., 2016), and Solar Orbiter ($SO$) (Müller et al., 2020)
covering heliocentric distances $13\ R_{\odot}\lesssim R\lesssim 220$
$R_{\odot}$ are utilized and the radial evolution of the magnetic spectral
index as a function of normalized wavenumber is investigated.

It is shown that closer to the Sun the magnetic field power-spectrum exhibits
a poorly developed inertial range that is characterized by a $-3/2$ spectral
index. The inertial range extends to larger and larger scales as the solar
wind expands into the interplanetary medium, with the inertial range spectral
index steepening towards a $-5/3$ value. We demonstrate, that the rate at
which the steepening occurs is strongly dependent on magnetic energy excess
and Alfvénicity of the fluctuations.

Figure 1:  Magnetic field power spectrum, $PSD$ at different heliocentric
distances. The power-spectrum is shown, as a function of (a) spacecraft
frequency; (b) wavenumber $k^{∗}=\ell^{-1}$ in units of $d_{i}$

##  2 Radial Evolution of magnetic field spectral index

We considered overlapping intervals of duration $d=24$ hours such that the
beginnings of adjacent intervals are $8$ hours apart. For each interval, the
Fourier trace power spectral density $F(f_{sc})$ was calculated, smoothed by
averaging over a sliding window of a factor of 2, and transformed into a
wavenumber spectrum expressed in physical units $E(\kappa^{\ast})$ by virtue
of the modified TH:

$E(\kappa^{\ast})=\frac{V_{tot}}{2\pi\cdot\xi}F(f_{sc})\ [nT^{2}\cdot\xi],$
(1)

where $\kappa^{\ast}=\kappa\cdot\xi=\frac{2\pi f_{sc}}{V_{tot}}\cdot\xi$, and,
$\xi=d_{i},\rho_{ci}$.

The radial evolution of the PSD ad a function of spacecraft frequency and
normalized by $d_{i}$ is presented In Fig. 1a,b respectively. Due to the
expansion of the solar wind but also in part because of the turbulent cascade,
a decrease of $\sim 4$ orders of magnitude in magnetic power is observed with
increasing heliocentric distance.

The spectral index, $\alpha_{B}$ is obtained by taking a sliding window of one
decade in the spacecraft-frame frequency (wavenumber) domain, over the
smoothed spectra and calculating the best-fit linear gradient in log-log space
over this window. For clarity, ten radial bins have been used, and the median
value of the spectral index as a function of frequency has been estimated for
intervals that fall within the same bin. The color of the curve is keyed to
the mean value of the distance $R$ corresponding to the intervals within each
bin. The results of this analysis are presented in Figure 2a. In the inertial
range, an energy cascade rate that is independent of scale is expected,
reflecting on the power-spectrum in the form of a constant spectral index over
this range of scales. In light of this, it can be seen that close to the Sun
(dark blue line in Figure 2a), the inertial range is limited into a narrow
range of frequencies ($2\times 10^{-2}-2\times 10^{-1}$Hz). As the solar wind
expands in the interplanetary medium (1) a universal steepening (i.e., across
all frequencies) is observed for the spectral index, $\alpha_{B}$, at a
constant $f_{sc}$; (2) The curves shift horizontally to lower and lower
frequencies. As illustrated in Figure 2a, the frequency range over which the
spectral index is constant is migrating to the left while steepening with
increasing distance, from $\alpha_{B}\approx-3/2$ to $\alpha_{B}\approx-5/3$.
Similar behavior is observed at the largest scales. Closer to the Sun for
$f_{sc}\leq 2\times 10^{-2}$ Hz, the spectrum gets progressively shallower at
lower frequencies and obtains a value of $\alpha_{B}\approx-1$ at $f_{SC}=\
3\times 10^{-4}$ Hz. As heliocentric distance increases, this low-frequency
part of the spectrum gradually steepens, with all the frequencies approaching
a $-5/3$ scaling. Therefore, as the solar wind propagates outward, the
inertial range of the spectrum develops gradually, extending from higher
frequencies to progressively lower and lower frequencies. Additionally, in
accordance with (Duan et al., 2020) the ion scale break, separating the
inertial from the kinetic range is observed to migrate to lower frequencies
with distance.

To cast the results in terms of relevant physical scales, we considered the
evolution of $\alpha_{B}$ into the wavenumber domain normalizing by either the
ion inertial length ($d_{i}$) or the ion gyroradius ($\rho_{i}$).

The evolution of the spectral index as a function of distance ($R$) in the
wavenumber domain normalized by $d_{i}$, is illustrated in Figure 2b. It is
readily seen, that the vertical shifting of the curves to lower frequencies,
observed in Figure 2a, has vanished: all the curves roll over at $\kappa
d_{i}\approx~{}0.1$ and overlap at smaller scales. The normalization does not
appear to have a substantial impact on the radial development of the spectral
index at large scales, $\kappa^{\ast}\lesssim 8\times 10^{-2}$, since a
steepening closely resembles Figure 1a is obtained. On the other hand, as
shown in Figure 1b, the small scale break, demarcating the beginning of the
transition region, $\kappa^{\ast}\approx 9\times 10^{-2}\ (\rho^{-1}_{i})$,
does not show any remarkable evolution with distance and stays constant in
physical space. We do not show plots using $\rho_{i}$ as normalization because
the spectra do not collapse as clearly into one curve at small scales,
demonstrating that $d_{i}$ is the more appropriate scale for such a
normalization.

Figure 2:  Evolution of magnetic field spectral index ($\alpha_{B}$) as a
function of distance, & (a) frequency in units of $Hz$ (b) normalized
wavenumber $\kappa^{\ast}$ in units of $d_{i}$.

###  2.1 Dependence of $\alpha_{B}$ on plasma parameters

To disentangle the spectral variation with distance from changes due to the
differing plasma parameters of different solar wind streams the dependence of
$\alpha_{B}$ on the normalized cross helicity $\sigma_{c}$

$\sigma_{c}=\frac{E_{o}-E_{i}}{E_{o}+E_{i}},$ (2)

a measure of the relative amplitudes of inwardly and outwardly propagating
Alfvén waves, and the normalized residual energy $\sigma_{r}$

$\sigma_{r}=\frac{E_{V}-E_{b}}{E_{V}+E_{b}},$ (3)

indicating the balance between kinetic and magnetic energy is examined.
$E_{q}=\frac{1}{2}\langle\delta\bm{q}^{2}\rangle$ denotes the energy
associated with the fluctuations of the field $\bm{q}$. In particular,
$E_{o,~{}i}$ can be estimated using Elsasser variables, defining outward and
inward propagating Alfvénic fluctuations (Velli et al., 1991; Velli, 1993)

$\delta\bm{Z}_{o,i}=\delta\bm{V}\mp sign(B^{R}_{0})\delta\bm{b},$ (4)

$\delta\bm{B}=\bm{B}-\bm{B_{0}}$, $\bm{B_{0}}$ the background magnetic field,
$\delta\bm{b}={\delta\bm{B}}/{\sqrt{\mu_{0}m_{p}n_{p}}}$ 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 variation of $\alpha_{B}$ with $V_{sw}$, the ratio of magnetic to
thermal pressure, $\beta\equiv n_{p}K_{B}T/(B^{2}/2\mu_{0})\ll 1$, and the
field/flow angle $\Theta_{BV}$ was also examined. Though we do not focus on
$\beta$, and $\Theta_{BV}$ here, we will comment on these in Section 3. The
evolution of $\alpha_{B}$ is investigated by fitting the magnetic spectrum
over a constant range ($10^{-3}-3\times 10^{-2}\ d_{i}^{-1}$). To ensure that
the plasma parameters under study do not vary significantly within the
interval the duration of intervals has been reduced to $d=1$ hr.

Figure 3:  The joint distribution of normalized cross-helicity $\sigma_{c}$
and normalized residual energy $\sigma_{r}$ at different heliocentric
distances.

###  2.2 Solar Wind Speed, $V_{SW}$

As shown in Figure 4a, Within 30 $R_{s}$, no significant differences in
spectral index with solar wind speed are found with an inertial range scaling
, $\alpha_{B}\approx-3/2$. As the solar wind expands, the dependence on solar
wind speed becomes more evident: steepening occurs regardless of solar wind
speed, but it is more efficient for slower solar wind streams. As a result, at
$R\approx 1$ au, the dependence of the spectral index on speed is clear, with
the spectral index being consistent with a K41 scaling in the fast wind and a
steeper scaling of $\approx-1.8$ for the slowest winds. Categorizing the
spectral index as a function of $\tau_{adv}$, Figure 4b, instead of radial
distance one finds that for $\tau_{adv}<40Hrs$ no clear trend is observed for
the spectral index as a function of wind speed. Beyond, this point, though
steepening is monotonic with $\tau_{adv}$ at all wind speeds. Overall, lower
speed intervals display a significant radial steepening as compared to faster
winds that only display a slight steepening. Closer to the Sun, however, there
seems to be no dependence on wind speed on the spectral index, suggesting that
the spectra are initially similar regardless of the wind speed.

###  2.3 Normalized Cross Helicity, $\sigma_{c}$, $\&$ Normalized Residual
Energy $\sigma_{r}$

The joint $\sigma_{c}$ \- $\sigma_{r}$ distribution, estimated using 1 minute-
long moving averages of the respective timeseries is presented in Figure 3.
The median and mean value of $\sigma_{c}$ and $\sigma_{r}$ for each bin are
also shown as red and black crosses respectively. The gray circle defines
fluctuations with perfect alignment between velocity and magnetic field, given
by $\sigma_{c}^{2}+\sigma_{r}^{2}=1$. Closer to the sun (0.06-0.1 au)
turbulence is highly Alfvénic, dominated by outwardly propagating waves
($\sigma_{c}\approx 0.85$), and in slight excess of magnetic energy
($\sigma_{r}\approx-0.15$). A small population of strongly magnetically
dominated intervals characterised by very low alfvénic content (i.e.,
$\sigma_{r}\approx-1$, and $\sigma_{c}\approx 0$, mostly associated with
heliospheric current sheet (HCS) crossings is also observed (see (Shi et al.,
2022a)). At larger heliocentric distances the mean/median value of
$\sigma_{c}$ progressively decreases (Chen et al., 2020; Shi et al., 2021).
Several mechanisms have been proposed to explain the diminishing dominance of
outwardly propagating waves with increasing heliocentric distance due to wave
reflection, including velocity shears (Bavassano et al., 1982) and the
parametric decay instability (Tenerani & Velli, 2013; Shoda et al., 2019). At
1au, $\sigma_{r}$ is clearly more negative than in the near-Sun environment,
but it does not show a clear trend with radial distance. In the distance range
of 0.6-1 au, most of the data points are concentrated in the lower half, with
a few intervals having slightly positive $\sigma_{r}$ values. In addition,
datapoints located in the bottom left quadrant are increasing with distance,
indicating a radially decreasing dominance of waves propagating outward

The power-spectra of the fluctuating fields
$\delta\bm{b},\delta\bm{V},\delta\bm{Z}_{o,i}$ have been obtained and both
$\sigma_{c}$, and $\sigma_{r}$ have been estimated by integrating the
resulting spectra over a constant range ($10^{-3}-5\times 10^{-2}\ d_{i}^{-1}$
) in the wavenumber domain normalized by the ion inertial length. The
dependence of the spectral index on $|\sigma_{c}|$ and $\sigma_{r}$ as well as
the radial distance (R) is presented in Figure 4c,d for $\sigma_{c}$ and
$\sigma_{r}$ respectively. These show how highly alfvénic
($|\sigma_{c}|\approx 1$) and energetically equipartitioned intervals display
little spectral evolution, while evolution to significantly steeper spectra is
associated with low $|\sigma_{c}|$ and/or large magnetic energy excess, with
the data at large distances consistent with 1 AU results (Podesta & Borovsky,
2010; Chen et al., 2013; Bowen et al., 2018).

Figure 4: Magnetic field spectral index $a_{B}$ as a function of $V_{sw}$ and
(a) heliocentric distance, (b) advection time $\tau_{adv},$ as well as, a
function of distance and (c) normalized cross helicity ($\sigma_{c}$), (d)
normalized residual energy ($\sigma_{r}$)

##  3 Conclusions

Using $PSP$ and $SO$ data from the inner heliosphere we have analyzed 1) how
the statistical signatures of turbulence evolve with heliocentric distance and
(2) the plasma parameters driving the evolution.

Identifying a plasma scale that grows radially at the same rate as the high
frequency break point is crucial for this study in order to anchor the
spectrum in normalized wavenumber space and enable meaningful comparison
between intervals sampled at different heliocentric distances. Since the high
frequency break point exhibits a power-law radial dependence with a scaling
exponent $1.08\pm 0.03$ (Duan et al., 2020; Lotz et al., 2022), then $d_{i}$
which grows radially as $\propto R^{1.04\pm 0.01}$ provides a better
normalization than $\rho_{i}$ which is characterized by a $\propto R^{1.48\pm
0.02}$ radial scaling. It is important to emphasize that the goal here is not
to find the quantity that has 1:1 correspondence with the high frequency break
point. In fact, it is well known that intervals characterized by low ion
$\beta$ values exhibit a magnetic power spectral density that breaks at the
ion inertial length ($d_{i}$), while high $\beta$ intervals are characterized
by a small scale break at the thermal ion gyroradius ($\rho_{i}$) (Chen et
al., 2014). Thus, the pinning of the power-spectrum at a constant $\kappa
d_{i}$ scale, with increasing distance may be explained by the fact that
plasma $\beta$ values remain rather low in the inner heliosphere, see Figure
5. Another plasma scale that is correlated with the high frequency break point
and should be considered by future investigations is the proton cyclotron
resonance (Woodham et al., 2018).

Additionally, our analysis indicates that closer to the Sun, the inertial
range of the magnetic field power-spectrum is poorly developed i.e., the range
of scales over which $\alpha_{B}$ remains constants is limited; its value is
closer to $\alpha_{B}=-3/2$. As the solar wind expands into the interplanetary
medium, the inertial range extends to progressively larger scales, while at
the same time the inertial range spectral index steepens towards
$\alpha_{B}=-5/3$.

We demonstrate that the rate at which $\alpha_{B}$ steepens is strongly
dependent on the normalized residual energy and normalized cross helicity of
the intervals under study. In particular, intervals with high alfvénic content
($|\sigma_{c}|\approx~{}1$), and equipartitioned in $E_{V}$-$E_{b}$
($\sigma_{r}\approx~{}0$) seem to retain their near-Sun scaling, and show a
minor steepening with radial distance. In contrast, magnetically dominated and
balanced intervals are observed to strongly steepen, resulting in anomalously
steep inertial range slopes at 1 au, consistent with previous studies (Podesta
& Borovsky, 2010; Chen et al., 2013; Bowen et al., 2018).

While $|\sigma_{c}|\approx$ 1 and $\sigma_{r}\approx 0$ values may be found in
slow wind streams, especially closer to the sun, they are statistically less
relevant than in fast winds (Shi et al., 2021). As a result, the occurrence of
steeper spectral indices in slower wind streams may be attributed to the
observed positive correlation between solar wind speed and
$\sigma_{c},~{}\sigma_{r}$.

Intervals with large magnetic energy excess closer to the Sun do not display
the steep spectra observed at 1 au, attributed by Bowen et al. (2018) to the
correlation between magnetic coherent structures and highly negative
$\sigma_{r}$ values Mininni & Pouquet (2009). Recent studies (Cuesta et al.,
2022; Sioulas et al., 2022), suggest that magnetic field intermittency is
strengthened with increasing heliocentric distance in the inner heliosphere,
but no similar analysis has been conducted for the velocity field. However,
velocity spectra do not display radial evolution (Shi et al., 2021) and
exhibit a scaling of $a_{v}=-3/2$ at 1 au (Chen et al., 2013). Based on our
results, we expect that both the magnetic and velocity field spectra display a
$-3/2$ scaling closer to the Sun, with the evolution of the magnetic spectrum
related to the in-situ generation of magnetic coherent structures during
expansion. A study of the evolution of $\alpha_{B}$ and $a_{v}$ as a function
of radial distance as well as intermittency is ongoing. Turbulence in the
solar wind is anisotropic with respect to the mean magnetic field (see, e.g.,
reviews by Schekochihin et al., 2009; Horbury et al., 2012; Oughton et al.,
2015, and references therein). Horbury et al. (2008); Wicks et al. (2010);
Kiyani et al. (2012) have shown that when the field/flow angle $\Theta_{BV}$
is $\Theta_{BV}=90^{\circ}$, then the inertial range range scales like either
$\alpha_{B}\approx\textminus 5/3$, or sometimes $\approx\textminus 3/2$,
consistent with a critical balance cascade and dynamical alignment models
respectively. In the parallel direction, $\Theta_{BV}=0^{\circ}$, it is nearer
$\alpha_{B}\approx-2$. In contrast, when a global magnetic field is utilized
to estimate $\theta_{BV}$, no anisotropy in the spectral index as a function
of $\Theta_{BV}$ is observed (Tessein et al., 2009; Chen et al., 2011). Though
it is not shown here, we find no correlation between $\Theta_{BV}$ and
$\alpha_{B}$, using a global magnetic field. A similar result was obtained
when considering the dependence of $\alpha_{B}$ on plasma $\beta$, suggesting
that these two parameters are not related to the steepening of the spectrum.
Further work to clarify the debate between a local, scale-dependent and global
background magnetic field and it’s role on the spectral evolution is presented
in a companion paper (Sioulas et al, Submitted).

Our findings will help us gain a better understanding of how solar wind
turbulence is generated and transported and will guide future models of solar
wind turbulence.

This research was funded in part by the FIELDS experiment on the Parker Solar
Probe spacecraft, designed and developed under NASA contract NNN06AA01C; the
NASA Parker Solar Probe Observatory Scientist grant NNX15AF34G and the HERMES
DRIVE NASA Science Center grant No. 80NSSC20K0604. The instruments of PSP were
designed and developed under NASA contract NNN06AA01C.

Figure 5: The radial evolution of (a) the ion inertial length $d_{i}$, (b) the
ion gyroradius$\rho_{i}$, (c) ion plasma $\beta$

##  Appendix A
Data Selection and Processing

We analyze magnetic field data from the Flux Gate Magnetometer (FGM) (Bale et
al., 2016), as well as plasma moment data from the Solar Probe Cup (SPC) and
Solar Probe Analyzer (SPAN) part of the Solar Wind Electron, Alpha and Proton
(SWEAP) suite between January 1, 2018, and June 15, 2022, encompassing the
first twelve perihelia (E1-E12) of the PSP mission. SPC data have been
utilized for E1-E8, while SPAN data for E9-E12. Quasi-thermal noise derived
data (Moncuquet et al., 2020), have been preferred over SPAN or SPC data when
available. Additionally, magnetic field and particle moment measurements from
the Magnetometer (MAG) instrument (Horbury et al., 2020), prioritizing burst
data when available, and the Proton and Alpha Particle Sensor (SWA-PAS) (Owen
et al., 2020) onboard the SO mission between June 1, 2018, to March 1, 2022,
were considered. Following consideration of quality flags, time intervals that
were found to be missing $\geq 1\%$ and/or $\geq 10\%$, in the magnetic field
and particle timeseries have been omitted from further analysis. The remaining
intervals have been resampled linearly to the highest cadence possible, based
on their initial resolution. Finally, in order to eliminate spurious spikes, a
Hampel filter (Davies & Gather, 1993) was applied to the plasma time series.

Converting the spacecraft-frame frequency derived PSD, $F(f_{sc})$ to a
wavenumber PSD, $E(\kappa)$, far from the sun is possible by means of Taylor’s
hypothesis (TH) (Taylor, 1938), $\kappa={2\pi f_{sc}}/{V_{SW}}$, that becomes
questionable when both the Alfvén and spacecraft velocity are comparable to
the velocity of the solar wind. Therefore, a modified version of Taylor’s
hypothesis that accounts for both wave propagation and spacecraft velocity is
adopted (Klein et al., 2015): in the above expression for $\kappa$ $V_{sw}$ is
replaced by $V_{tot}=|\bm{V}_{sw}+\bm{V}_{a}-\bm{V}_{sc}|$ where $\bm{V}_{sc}$
is the spacecraft velocity, where turbulence is assumed to be dominated by
outwardly propagating Alfvén waves. Note that the TH remained either
moderately or highly valid for the majority of time intervals examined. with
only $\sim 1.53\%$ of the intervals under study exhibiting $M_{A}<1.5$,
including a number sub-Alfvénic intervals during PSP $E_{8}-E_{12}$ ($\sim
0.45\%$ of the entire dataset). Figure 5 illustrates the radial evolution of
ion inertial length $d_{i}$, ion gyroradius $\rho_{i}$, and ion plasma
$\beta$, quantities relevant to this study.

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