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# New Observations of Solar Wind 1/f Turbulence Spectrum from Parker Solar Probe

Zesen Huang, Nikos Sioulas, Chen Shi, Marco Velli, Trevor Bowen, Nooshin Davis, B. D. G. Chandran, Lorenzo Matteini, Ning Kang, Xiaofei Shi, et al.

► To cite this version:

Zesen Huang, Nikos Sioulas, Chen Shi, Marco Velli, Trevor Bowen, et al. New Observations of Solar Wind 1/f Turbulence Spectrum from Parker Solar Probe. The Astrophysical journal letters, 2023, 950, 10.3847/2041-8213/acd7f2. insu-04151711

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Submitted on 5 Jul 2023

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Distributed under a Creative Commons Attribution 4.0 International License
# New Observations of Solar Wind 1/f Turbulence Spectrum from Parker Solar Probe

Zesen Huang (黃泽淼)<fnref n="1" /> ID, Nikos Sioulas<fnref n="1" /> ID, Chen Shi (时辰)<fnref n="1" /> ID, Marco Velli<fnref n="1" />,<fnref n="2" /> ID, Trevor Bowen<fnref n="3" /> ID, Nooshin Davis<fnref n="4" /> ID, B. D. G. Chandran<fnref n="4" /> ID, Lorenzo Matteini<fnref n="5" /> ID, Ning Kang (康宁)<fnref n="6" /> ID, Xiaofei Shi (石晓霏)<fnref n="1" /> ID, Jia Huang (黃佳)<fnref n="3" /> ID, Stuart D. Bale<fnref n="3" />,<fnref n="7" /> ID, J. C. Kasper<fnref n="8" />,<fnref n="9" /> ID, Davin E. Larson<fnref n="10" /> ID, Roberto Livi<fnref n="10" /> ID, P. L. Whittlesey<fnref n="10" /> ID, Ali Rahmati<fnref n="10" /> ID, Kristoff Paulson<fnref n="11" /> ID, M. Stevens<fnref n="11" /> ID, A. W. Case<fnref n="11" /> ID, Thierry Dudok de Wit<fnref n="12" /> ID, David M. Malaspina<fnref n="13" />,<fnref n="14" /> ID, J. W. Bonnell<fnref n="3" /> ID, Keith Goetz<fnref n="15" /> ID, Peter R. Harvey<fnref n="3" /> ID, and Robert J. MacDowall<fnref n="16" /> ID

Received 2023 February 27; revised 2023 May 18; accepted 2023 May 23; published 2023 June 12

## Abstract

The trace magnetic power spectrum in the solar wind is known to be characterized by a double power law at scales much larger than the proton gyro-radius, with flatter spectral exponents close to -1 found at the lower frequencies below an inertial range with indices closer to [-1.5, -1.67]. The origin of the 1/f range is still under debate. In this study, we selected 109 magnetically incompressible solar wind intervals ($\delta|\mathbf{B}|/|\mathbf{B}| \ll 1$) from Parker Solar Probe encounters 1-13 that display such double power laws, with the aim of understanding the statistics and radial evolution of the low-frequency power spectral exponents from Alfvén point up to 0.3 au. New observations from closer to the Sun show that in the low-frequency range solar wind, turbulence can display spectra much shallower than 1/f, evolving asymptotically to 1/f as advection time increases, indicating a dynamic origin for the 1/f range formation. We discuss the implications of this result on the Matteini et al. conjecture for the 1/f origin as well as example spectra displaying a triple power law consistent with the model proposed by Chandran et al., supporting the dynamic role of parametric decay in the young solar wind. Our results provide new constraints on the origin of the 1/f spectrum and further show the possibility of the coexistence of multiple formation mechanisms.

*Unified Astronomy Thesaurus concepts: Solar wind (1534); Interplanetary turbulence (830); Magnetohydrodynamics (1964); Space plasmas (1544); Heliosphere (711); Alfvén waves (23)*

## 1. Introduction

The trace magnetic power spectrum (power spectrum density, PSD) in the solar wind is often characterized by a double power law from intermediate to large scales, with power spectral exponents in the inertial range around -5/3 (or closer to -3/2 in the inner heliosphere) and -1 at larger scales, respectively. This double power law is usually found in the fast wind (Bavassano et al. 1982; Denskat & Neubauer 1982; Burlaga & Goldstein 1984) and recently in very long intervals of slow wind (Bruno et al. 2019) and also in extremely long intervals without regard to wind speed (Matthaeus & Goldstein 1986). The low frequency (or large scale in configuration space, converted with modified the Taylor hypothesis; Taylor 1938; Perez et al. 2021a) range of the spectrum has been considered as the energy reservoir that facilitates the

turbulence cascade in the solar wind. The origin and formation mechanism of the 1/f range of the PSD is still not well understood and is under active debate (see, e.g., Matthaeus & Goldstein 1986; Velli et al. 1989; Dmitruk & Matthaeus 2007; Bemporad et al. 2008; Verdini et al. 2012; Matteini et al. 2018; Chandran 2018; Magyar & Doorsselaere 2022). It is worth noting that the 1/f range of the spectrum is indicative of a scale-independent fluctuation energy distribution, i.e., equipartition of energy over all scales (Keshner 1982). Moreover, the 1/f spectrum is also seen in plasma density fluctuations from Ulysses at similar frequencies, especially in the high-latitude intervals (Matthaeus et al. 2007).

The majority of the models proposed for the double power law were built on a priori assumption of the existence of the 1/f spectrum at low frequency. For example, Velli et al. (1989) proposed that the secondary incoming waves generated by linear coupling of the dominant outgoing waves to the large-scale solar wind inhomogeneity could facilitate a quasistationary self-similar cascade, resulting in a 1/k scaling. Montroll & Shlesinger (1982) and Matthaeus & Goldstein (1986)

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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">¹ Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA, USA</footnote>

<footnote n="2">² International Space Science Institute, Bern 3012, CH, Switzerland</footnote>

<footnote n="3">³ Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA</footnote>

<footnote n="4">⁴ Space Science Center and Department of Physics, University of New Hampshire, Durham, NH 03824, USA</footnote>

<footnote n="5">⁵ Imperial College London, South Kensington Campus, London SW7 2AZ, UK</footnote>

<footnote n="6">⁶ Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, CA, USA</footnote>

<footnote n="7">⁷ Physics Department, University of California, Berkeley, CA 94720-7300, USA</footnote>

<footnote n="8">⁸ BWX Technologies, Inc., Washington DC 20001, USA</footnote>

<footnote n="9">⁹ Climate and Space Sciences and Engineering, University of Michigan, Ann Arbor, MI 48109, USA</footnote>

<footnote n="10">¹⁰ Space Sciences Laboratory, University of California, Berkeley, CA 94720, USA</footnote>

<footnote n="11">¹¹ Smithsonian Astrophysical Observatory, Cambridge, MA 02138 USA</footnote>

<footnote n="12">¹² LPC2E, CNRS and University of Orléans, Orléans, France</footnote>

<footnote n="13">¹³ Astrophysical and Planetary Sciences Department, University of Colorado, Boulder, CO, USA</footnote>

<footnote n="14">¹⁴ Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO, USA</footnote>

<footnote n="15">¹⁵ School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA</footnote>

<footnote n="16">¹⁶ Solar System Exploration Division, NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA</footnote>
suggest that the 1/f spectrum results from the superposition of uncorrelated samples of solar surface turbulence that have log-normal distributions of correlation lengths corresponding to a scale-invariant distribution of correlation times over an appropriate range of parameters. The scale invariance originates in the dynamo and manifests in the photospheric magnetic field.

There are, however, few notable exceptions. For example, it has been pointed out by Chandran (2018, henceforth denoted as C18) that in Figure 2 of Tu & Marsch (1995), the low frequency $z^+$ (Elsässer variables: $z^+ = V \mp B/\sqrt{\mu_0 \rho}$) spectrum was as shallow as $f^{-0.5}$ in the low-frequency range, which C18 referred to as the “infrared” range. C18 used this as an input to the model, and the parametric decay-induced inverse cascade of the outward-propagating Alfvén wave would eventually produce a triple power-law $z^+$ spectrum with 1/f in the middle frequency range between the “infrared” range and the inertial range.

Another example would be the conjecture proposed by Matteini et al. (2018, henceforth denoted as M18), which tried to build a connection between spectral properties and magnetic compressibility. In particular, there is a paradox between an arbitrary power-law index and low magnetic compressibility at all scales. Their conjecture shows that, for turbulence with low magnetic compressibility, if the magnetic fluctuations fully saturate over the scales, i.e., $\langle\delta B\rangle = \langle\delta B\rangle \sim |B| = B$, we have $\langle\delta B/B\rangle \sim \langle\delta B\rangle/B \sim 1$. According to the well-known relation that connects the slope in the power spectrum $P(k) \propto k^\alpha$ and the exponent in the second-order structure function (SF) $\delta B^2 \propto l^{-2\beta}$ via $\alpha = 2\beta - 1$, $-3 < \alpha < -1$ (see, e.g., Monin & Jaglom 1987), and here we translate the frequency $f$ to wavenumber $k$ with the Taylor hypothesis and scale $l \sim 1/k$, we see that when $\langle\delta B/B\rangle \sim 1$ for sufficiently large scales (note that $\langle\delta B/B\rangle$ is a function of temporal increment $\tau$ or scales $l$ in configuration space), we have $\beta \sim 0$, and hence $\alpha \gtrsim -1$, for the same scale range. Consequently, 1/f is the steepest possible realization of the power spectrum when magnetic fluctuations fully saturate to the $\langle\delta B/B\rangle \sim 1$ state. In addition, their result leads to a straightforward connection between the scale $l_0$ at which magnetic field fluctuations fully saturate (i.e., $\langle\delta B/B\rangle \sim 1$) and the spectral break point $f_B$ of the double power law in PSD. M18 argued that closer to the Sun, unless the 1/f range is formed in the solar corona and advected outwards preserving its shape, it should gradually disappear, moving closer to the Sun where $\delta B/B < 1$.

Previous observations with Ulysses, Helios, and WIND have only explored the heliosphere beyond 0.3 au. It is well-known that due to solar wind expansion, $B$ decreases like $R^{-2}$ radially per conservation of magnetic flux (or Parker spiral; Parker 1958), because the field is mostly radial close to the Sun), and $\delta B$ decreases like $R^{-1.5}$ per Wentzel-Kramers-Brillouin (WKB) theory (see, e.g., Whang 1973; Heinemann & Olbert 1980; Velli et al. 1991; Tu & Marsch 1995; Huang et al. 2022), where $R$ is the heliocentric distance. And hence $\langle\delta B/B\rangle$ is expected to increase radially as $R^{0.5}$ from the Alfvén critical point (where $\delta B^2$, $\delta V^2 \propto V_{sw}V_A/(V_A + V_{sw})^2$ reaches its absolute maximum per WKB theory. However, the peak is expected to locate inside the Alfvén point due to non-WKB effects (see, e.g., Cranmer & Ballegooijen 2005; Verdini & Velli 2007; Chen et al. 2020; Bandyopadhyay et al. 2022) until it reaches its full saturation at 1 around 0.3 au. Therefore, closer to the Sun, based on the WKB prediction, we expect to see

magnetically incompressible intervals with partial saturation $\langle\langle\delta B/B\rangle\rangle \sim \text{const, const} < 1$) or without saturation $\langle\langle\delta B/B\rangle\rangle$ keeps increasing over scales but smaller than 1). It is, hence, important to validate the connection between 1/f spectrum and low magnetic compressibility with solar wind closer to the Sun where full magnetic fluctuation saturation is not achieved.

As a consequence, with new observations much closer to the Sun from the Parker Solar Probe (PSP; Fox et al. 2016), there are two major questions to be investigated: (1) What are the statistics and radial evolution of low-frequency power spectral exponents? (2) With the new evidence from closer to the Sun, what new constraints can be applied to the existing models of 1/f spectrum? To answer these questions we perform a systematic search for magnetically incompressible ($|B|/|B| \ll 1$) solar wind intervals that are characterized by a double power law with PSP data from Encounter 1 to Encounter 13. The rest of the paper is organized as follows: In Section 2, we explain the data selection and analysis procedure. In Section 3, we show the main results; in Section 4, we discuss the implications from new observations on M18 and C18. In Section 5, we conclude and summarize the main results.

## 2. Data and Analysis Procedures

In order to study the low-frequency spectral properties of magnetically incompressible solar wind observed by PSP, we performed a systematic search for magnetically incompressible turbulence intervals from E1 (2018 November) to E12 (2022 June). We mainly use high-resolution vector magnetic field measurements from FIELDS instruments (Bale et al. 2016) to calculate the magnetic turbulence spectrum and proton measurements from SWEAP instruments (Kasper et al. 2016) and quasi-thermal noise electron density measurements (Pulupa et al. 2017; Moncuquet et al. 2020) to support interval selection.

### 2.1. Interval Selection

Our selection of magnetically incompressible solar wind intervals is based on the evaluation of the scale-dependent magnetic compressibility $\eta_B(T)$:

$$ \eta_B(T) = \left\langle \frac{|B| - \langle |B| \rangle_T}{\langle |B| \rangle} \right\rangle, \qquad (1) $$

where $\langle\rangle_T$ is the ensemble average at scale $T$ and $\langle\rangle$ is the ensemble average throughout the interval. For all intervals, we ensure that $\eta_B$ is smaller than 0.1 for any given scale $T$. Moreover, due to the rapid movement of the spacecraft around perihelia, careful selection is needed to ensure that the spacecraft stays within the same type of solar wind over the selected interval. Therefore, secondary parameters, including solar wind speed $V_{sw} = |V|$, thermal speed $V_{th} = \sqrt{2k_B T/m_p}$, normalized cross helicity $\sigma_c = (z^+ + z^-)/(z^+ + z^-)$, plasma $\beta = 2\mu_0 P/B^2$, proton density $n_p$, Carrington longitude, Heliocentric distance $R$, and advection time $t_{adv} = R/V_{sw}$, are examined to differentiate different solar types (see Figure 1 panels (a1)–(a5)). With our selection criteria, we ended up with 109 nonoverlapping magnetically incompressible solar wind intervals observed by PSP from E1 to E13, with a total 1500 hr worth of data (see Figure 2(c) for histogram of interval lengths). Note that the jets in the solar wind speed in Panel (a2),
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Example of a magnetically incompressible solar wind interval from PSP Encounter 10, close to the perihelion. The particle data of this interval are provided by SPAN (Livi et al. 2022). Right: Panels (a1)–(a5) show time series of magnetic field $B_{rm}$ in radial, tangential, and normal coordinates; solar wind speed $V_{SW}$ and solar wind thermal speed $V_{th}$; cross helicity $\sigma_c$ and plasma beta $\beta$; proton density $n_p$ and Carrington longitude; and heliocentric distance $R$ in solar radii $R_\odot$ and advection time $t_{adv} = R/V_{SW}$, respectively. The selected interval is indicated with the red shaded area enclosed by two red dashed lines. Left: (b) Power-law fitting index $\alpha$ as a function of frequency $f$ of trace PSD. (c) Fast Fourier transformation (FFT) is plotted in blue as background; smoothed FFT PSD is the green line; and Morlet wavelet transformation (WL) PSD is the orange line. The two pairs of red crosses accompanied by red and green shaded areas indicate the ranges of frequency over which the power-law fits are applied, and the two fitted lines are shown in dashed and dashed-dotted lines. The gray shaded area is the frequency range where more than 70% of points fall out of the cone of influence (Col) of wavelet transformation. The red dashed line indicates the intersection of two fitted lines, interpreted as the low-frequency break point. (d) Power-law fitting index $\beta$ as a function of frequency $f$ of the normalized SF. (e) Normalized first-order SF $\langle \delta B/B \rangle$ as a function of frequency $f = 1/(2^*\Delta t)$, where $\Delta t$ is the temporal increment. Two fits are applied to the same frequency ranges from (c), which produce a similar low-frequency break point, indicated by the green dashed line.</figcaption></figure>

which are accompanied with partial or total reversal of $B_r$ in Panel (a1), are known as switchbacks and have been studied by numerous recent studies (see, e.g., Bale et al. 2019; Bourouaine et al. 2020; Dudok de Wit et al. 2020; Farrell et al. 2020; Hernández et al. 2020; Mozer et al. 2020; Tenerani et al. 2020; Woolley et al. 2020; Laker et al. 2021; Larosa et al. 2021; Martinović et al. 2021; Tenerani et al. 2021; Meng et al. 2022; Shi et al. 2022b; Huang et al. 2023 and references therein). They are known as large-amplitude or spherically polarized Alfvén waves (Mallet & Chandran 2021; Mallet et al. 2021) with almost constant magnetic modulus.

## 2.2. Diagnostics: Magnetic Trace Power Spectrum and Structure Function

For each interval, the following two diagnostics are calculated: trace PSD and normalized first-order SF for vector magnetic field. To produce a reliable low-frequency turbulence spectrum, for each interval, we calculate the trace PSD with both Morlet WL and FFT. For FFT, we smooth the spectrum by averaging over a sliding window of a factor of 2 in the frequency domain (smoothed FFT, sm-FFT). The smooth

spectra calculated with the two methods (WL and sm-FFT) generally overlap with each other perfectly in the high-frequency range but gradually deviate from each other at the low-frequency end. Therefore, we keep the spectrum up to the frequency where more than 70% of points fall out of the cone of influence (Col) of the WL spectrum, and we trust only the frequency range where WL overlaps with sm-FFT. We calculate the power-law fit index $\alpha$ by fitting on the WL PSD unless specified otherwise, and henceforth referred simply as PSD.

We follow the steps described in M18 to calculate the normalized first-order SF for vector magnetic field. We apply 200 logarithmically spaced lags $\Delta t$ as temporal increment of vector magnetic field $\delta B(t, \Delta t) = B(t) - B(t + \Delta t)$ within the range (1s, length of interval/2). The normalized first-order SF is then calculated by

$$SF(\Delta t) = SF_{\text{norm}}(\Delta t) = \left\langle \frac{|\delta \mathbf{B}(t, \Delta t)|}{B(t, \Delta t)} \right\rangle_t, \quad (2)$$

where we denote $|\delta B|$ as $\delta B$ and $|B|$ as $B$, and $\langle \rangle_t$ is average with regard to $t$ for scale $\Delta t$. Note that the intervals we select are magnetically incompressible, and hence $B$ can be
<figure><img src="image_5.png" /><figcaption>**Figure 2.** (a) Histograms of power-law fit index $\alpha$ of low-frequency (blue) and magnetohydrodynamics (MHD) frequency (red) ranges. The averaged index values are indicated with corresponding dashed lines and are shown in the legends. The expected power spectral exponent value range from various phenomenologies is indicated with red shaded area. (b) Statistics of length and mean radial distance of the selected 109 intervals. (c) Histogram of interval lengths. The average length is 13.72 hr (shown as legend accompanied by dashed line), with minimum of 1.08 hr and maximum of 64.70 hr.</figcaption></figure>

considered as scale- independent and can be taken out of the averaging window $\langle\rangle_t$. The resultant $SF(\Delta t)$ in the temporal domain is finally converted to $SF(f)$ in the frequency domain via $f=1/(2 \cdot \Delta t)$ because a half period in the absolute fluctuation corresponds to the full period in the non- absolute one.

To extract spectral information from PSD, we apply a power-law fit on PSD in both low- and MHD-inertial-frequency ranges where the spectrum stabilize visually (red and green shaded areas in Figure 1(c)), and we obtain two power-law fits with indices $\alpha_{Low}$ and $\alpha_{MHD}$ (black dashed line and dashed-dotted line in Figure 1(c)). The intersection of the two fit lines is interpreted as the low-frequency break point ($f_{B,PSD}$, red dashed line). To compare the spectral properties of PSD and SF, we apply a power-law fit in the same frequency ranges to SF (extended shaded area in Figure 1(e)), and the intersection of the two fit lines is also interpreted as the low-frequency break point ($f_{B,SF}$, green dashed line). Clearly the two break frequencies $f_{B,PSD}$ and $f_{B,SF}$ are very close to each other, consistent with M18. The small deviation may be attributed to spectral leakage of the SF, which leads to a spectrum smoother than the wavelet or PSD result. For completeness, two moving fits $\alpha(f)$ and $\beta(f)$ with window size of 1/3 decade on PSD and SF are shown in Figures 1(b) and (d).

# 3. Results

## 3.1. Statistics of Power Spectral Exponents

For the 109 intervals, power-law fit is applied to both high-frequency (MHD/inertial) and low-frequency ranges with the method described in Section 2.2. The primary statistical results are presented in Figure 2(a), in a style similar to Figure 6 in Bruno et al. (2019). The histogram of power-law fit index in the MHD inertial range $\alpha_{MHD}$ is shown in dark red. Most of the intervals have an inertial-range spectral exponent that falls within the expected value range [-1.67, -1.5], predicted by many existing phenomenologies (Kolmogorov 1941; Iroshnikov 1964; Kraichnan 1965; Sridhar & Goldreich 1994; Goldreich & Sridhar 1997; Boldyrev 2005, 2006), consistent with recent observations (see, e.g., Chen et al. 2020; Kasper et al. 2021; Shi et al. 2021; Telloni et al. 2021; Zank et al. 2022; Raouafi et al. 2023; Sioulas et al. 2023a, 2023b). However, for the low-frequency range fit index ($\alpha_{Low}$, dark blue), the spectral exponents have an unexpectedly wide distribution, and the majority of them are larger than -1, i.e., the corresponding low-frequency spectra are shallower than 1/f.

## 3.2. Radial Evolution and Dependence on Solar Wind Speed

The radial evolution of the power-law fit index of PSD in the low-frequency range $\alpha_{Low}$ is shown in Figure 3, sorted with advection time $t_{adv} = R/V_{sw}$ in 3(a) and heliocentric distance $R$ in 3(b). To display the trend of evolution, a binned average weighted with interval length is plotted on top of the scatters. When we sort the intervals with advection time $t_{adv}$, a clear asymptotic evolution from shallower spectrum toward 1/f spectrum is seen. However, when sorted with heliocentric distance $R$, no clear trend is found, and instead $\alpha_{Low}$ is scattered in a wide range of values below 0.3 au. For completeness, the dependence of $\alpha_{Low}$ on solar wind speed is shown in panel c. For intervals with very low solar wind speed (~200 km s⁻¹), which are typically observed very close to the Sun, $\alpha_{Low}$ are mostly close to -1, i.e., very close to the 1/f spectrum. For intervals with higher solar wind speed, no obvious trend is observed. Therefore, for the very low speed streams, it is possible that the 1/f spectrum originated from the solar corona (see, e.g., Matthaeus & Goldstein 1986).

However, substantial solar wind acceleration has been observed below 0.3 au (see, e.g., Shi et al. 2021, 2022a; Sioulas et al. 2022). In other words, the solar wind speed is a radially varying parameter even for the same stream, and hence, the dependence of $\alpha_{Low}$ on solar wind speed should be taken with caution. Moreover, it should be also noted that the actual advection time is an integrated quantity, and thus, the $t_{adv}$ used here can be merely considered as a proxy to the real advection time. And the interval length systematically grows with advection time and heliocentric distance, and hence, $\alpha_{low}$ is acquired from lowering frequency range with increasing $t_{adv}$ and $R$. The radial evolution of $\alpha_{low}$ could be associated with the change in fitting frequency range.

# 4. Discussions

## 4.1. Implications on M18

M18 tried to build a connection between low magnetic compressibility and 1/f spectrum via the well-known relation: $\alpha = 2\beta - 1, -1 > \alpha > -3$, where $\alpha$ is the spectral exponent of the PSD and $\beta$ is the power-law index of the first-order SF
<figure><img src="image_6.png" /><figcaption>Figure 3. (a) Radial evolution of low-frequency power-law fit index $\alpha_{Low}$, sorted with advection time ($t_{adv} = R/V_{SW}$). The sizes of the circles indicate the interval length (weight), and the line plot is the binned weighted average of $\alpha_{Low}$. The error bars are 1 standard error of $\alpha_{Low}$ from each bin. (b) Radial evolution of $\alpha_{Low}$ sorted with heliocentric distance $R$. (c) Dependence of $\alpha_{Low}$ on solar wind speed $V_{SW}$.</figcaption></figure>

(here expressed in terms of frequency). Due to low magnetic compressibility, $\langle\delta B/B\rangle$ can be considered as a first-order SF $\delta B$ normalized by its own constant modulus. Therefore, the fully saturated magnetic fluctuations, i.e., $\delta B/B \sim 1$, can be translated to $\beta \sim 0$, and therefore, $\alpha \gtrsim -1$. In other words, $1/f$ is the steepest possible realization of the low-frequency spectrum when magnetic field fluctuations fully saturates. It has also been proposed by M18 that “unless the $1/f$ range is formed in the corona and just advected in interplanetary space preserving its shape, it should gradually disappear, moving closer to the Sun where $\delta B/B < 1$.”

It is therefore interesting to examine the behaviors of $\alpha_{Low}$ and $\beta_{Low}$ closer to the Sun where $\delta B/B < 1$. There are roughly two possible behaviors of $\langle\delta B/B\rangle$ in the low-frequency range: (1) partial saturation, i.e., $\delta B/B \sim \text{const}$, $\beta \sim 0$; and (2) no saturation, i.e., $\beta < 0$. Figure 4(a) shows the histogram of the $\beta_{Low}$ and $\beta_{MHD}$ in both low- and MHD-inertial-frequency ranges. The histogram of $\beta_{Low}$ clearly shows that substantial portions of intervals have no saturation ($\beta \lesssim -0.1$). However, the scatterplot of ($\beta_{Low}$, $\alpha_{Low}$) in Figure 4(b) shows that the low-frequency PSD of these intervals is close to $1/f$. In fact, for those intervals with partial saturation ($\beta_{Low} \gtrsim -0.1$), the scatterplot shows that the corresponding $\alpha_{Low}$ are much greater than $-1$, i.e., the low-frequency PSD are much shallower than $1/f$.

Figure 4(b) further reveals that in the inertial range, the observed ($\beta$, $\alpha$) pairs follow the relation $\alpha = 2\beta - 1$ pretty well (black dashed line), with a slight systematic shift toward shallower $\alpha$ but in the low-frequency range deviate significantly. To understand such deviation, we generate 1D artificial time series with different power spectral exponent $\alpha_0$. For each $\alpha_0$, we generate an artificial time series $A(t; \alpha_0)$ formed with sinusoidal fluctuations with random phases and amplitudes following the power law:

$$A(t; \alpha_0) = \sum_{i}^{N} \omega_i^{(\alpha_0+1)/2} \sin[2\pi(\omega_i t + \phi_i)], \quad (3)$$

where $\omega_i$ is $i$th frequency log-linearly spaced from $10^{-5}$ to 1 Hz, $\phi_i$ is random phase in range of $(0, 2\pi)$, and $N$ is set to 10,000. For each $A(t; \alpha_0)$, we calculate the power-law fitting

<figure><img src="image_7.png" /><figcaption>Figure 4. (a) Histogram of power-law fit index $\beta$ of the normalized first-order SF from both low- (blue) and MHD- (red) frequency ranges. The averages are indicated with corresponding dashed lines. (b) Scatters of power-law fit index pairs ($\alpha$, $\beta$) of trace PSD and SF from both low- (blue) and MHD- (red) frequency ranges. The ($\alpha$, $\beta$) pairs from artificial data are also shown with black crosses, and the theoretical relation $\alpha = 2\beta - 1$ is indicated with black dashed line. (c) Dependence of $\langle\delta B/B\rangle$ from the saturated intervals ($\beta_{Low} > -0.07$) on heliocentric distance $R$. The value ranges of $\langle\delta B/B\rangle$ in the low-frequency range are indicated with error bars. The newly explored heliosphere radial range ($R < 0.3$ au) is indicated with red shaded area.</figcaption></figure>
index for the PSD (α) and SF (β) from the same frequency range ($10^{-2.5}$, $10^{-1}$) Hz.

The scatterplot of (α, β)Artificial pairs are shown in Figure 4(b) as gray crosses. (α, β)Artificial catch well the trend of the observed (α, β) pairs, especially in the MHD-frequency range, and also deviate significantly from the relation $α = 2β - 1$. Such systematic deviation is possibly due to the spectral leakage effect, because contrary to FFT and WL, the SF has a broad-band response. Therefore, cautions should be taken when one is trying to relate the 1/f spectrum with magnetic fluctuation saturation (regardless of full or partial). For completeness, shown in Figure 4(c), for those 55 intervals with partially saturated magnetic fluctuations (here defined as $β > -0.07$), we see the radial evolution of saturation values $\langle\delta B/B\rangle$ follows well with the WKB prediction of $R^{0.5}$.

In summary, one should be careful when relating magnetic fluctuations saturation ($\langle\delta B/B\rangle \sim \text{const}$) to 1/f spectrum because of the spectral leakage effect. And closer to the Sun where $\delta B/B < 1$, we see low-frequency spectra much shallower than 1/f with partially saturated magnetic fluctuations. One good example has already been shown in Figure 1. Therefore, according to M18, it is possible that some of these low-frequency range spectra originated from the solar corona (see also Matthaeus et al. 1982, 2007; Bemporad et al. 2008 for the coronal origin of the 1/f spectrum.)

## 4.2. Implications on C18

C18 proposed that the 1/f spectrum is formed by the inverse cascade of Alfvén waves facilitated by parametric decay within the context of weak-turbulence theory. The author initializes the simulation with primarily outward-propagating Alfvén waves, i.e., $e^+ \gg e^-$, where $e^±$ are the frequency (f) spectra of Alfvén waves propagating in opposite directions along the magnetic field lines. If the initial $e^+$ has a peak frequency $f_0$ (where $f e^+ = \text{maximized}$), and an “infrared” scaling $f_p$ at smaller f with $-1<p<1$, then $e^+$ acquires an $f^{-1}$ scaling throughout a range of frequencies that spreads out in both directions from $f_0$. The final state of their model evolution is a triple power law with “infrared” scaling in the low frequency, 1/f in the intermediate-frequency range, and $f^{-2}$ scaling in the inertial range (the inertial range is initialized with the critical-balanced parallel spectrum of $k_{\parallel}^{-2}$ (see, e.g., Goldreich & Sridhar 1995; Podesta 2009; Forman et al. 2011).

Therefore, it is interesting to explore the magnetically incompressible (yet weakly compressible plasma $β \ll 1$ and $\delta\vec{B} \sim |B|$ to allow parametric decay; Fu et al. 2018) and Alfvénic turbulence (primarily outward-propagating Alfvén waves, i.e., $|\sigma_c| \sim 1$) intervals close to the Sun to see if there exists the triple power law $z^+$ spectrum. Among the 109 intervals, there are four intervals displaying clear triple power law in trace $z^+$ PSD with stable intermediate-frequency range. One of the best examples is shown in Figure 5, which is an interval from E12, with $\langle\sigma_c\rangle = 0.82$. Note that due to the low-resolution particle data, the time series is resampled to 5 s, and hence, the WL spectrum is very wavy. To acquire a better-moving fit $α(f)$ profile, the moving power-law fit with window size of 1/3 decade is applied to the smoothed FFT PSD (green line).

In the low-frequency range, we see a stable “infrared” spectrum with $f^{-0.627}$. In the intermediate range, the spectrum is very close to 1/f. And in the inertial range, the scaling (which can be converted to configuration space with the Taylor

<figure><img src="image_8.png" /><figcaption>**Figure 5.** Example of C18 type of trace $z^+$ spectrum. The FFT spectrum is shown with blue line as background, the smoothed FFT is the green line, and the trace Morlet Wavelet (WL) PSD is the orange line. The gray shaded area indicates that more than 70% of points fall out of the Cone of Influence (CoI) of WL. The power-law fit is conducted on the smoothed FFT, and the moving window fit $α(f)$ (window sized 1/3 decade) is shown in the overhead panel. The trace PSD is separated into three distinct frequency ranges (low, mid, MHD); each possesses a quasi-steady power-law range, indicated with red (low), blue (mid), and green (MHD) shaded areas. Power-law fits are applied to each of the ranges, and the fit indices $α_{Low/Mid/MHD}$ are shown in the legends. A “low-frequency break” is found as the intersection between the power-law fits from low-frequency and MHD-inertial ranges, indicated with orange dashed line.</figcaption></figure>

hypothesis; Taylor 1986; Perez et al. 2021b) is agreeable with the perpendicular spectrum of $k_{\perp}^{-5/3}$ from the critical balance theory, which would mask the steeper parallel spectrum because we are observing the trace $z^+$ PSD. This is also supported by Cuesta et al. (2022) and Sioulas et al. (2023b), which has shown that when considering turbulence anisotropy, for fast streams ($V_{sw} \gtrsim 400\text{km s}^{-1}$) close to the Sun, the low-frequency spectrum is dominated by parallel fluctuations, and in the inertial range, the perpendicular spectrum has an $f^{-5/3}$ scaling. This is also consistent with the simulation by Verdini et al. (2012), where at higher frequencies the steep parallel spectral slope ($-2$) is masked by the more energetic perpendicular spectrum (slope $-5/3$). In summary, this triple power law is consistent with the simulation result from C18 and further provides evidence for the presence of parametric decay instability in the solar wind (see, e.g., Matteini et al. 2010; Tenerani & Velli 2013; Zanna et al. 2015; Tenerani et al. 2017; Shi et al. 2017; Bowen et al. 2018; Réville et al. 2018).

It should also also noted that due to the existence of the triple power law, the concept of “low-frequency break point” is now questionable. For example the orange dashed line in Figure 5 is obtained as the intersection between the low-frequency “infrared” spectrum and the MHD spectrum and interpreted
as the low-frequency break point $f_{B,PSD}$. However, any frequencies in between these two ranges should be considered as the “break frequency.” Therefore, the low-frequency break point obtained with this method would have an unacceptably large uncertainty. For this reason, we have not shown any statistics of $f_{B,PSD}$ in this study.

## 5. Conclusions and Summary

In this Letter, we have selected 109 magnetically incompressible intervals with a total 1500 hr worth of data from PSP encounter 1–13. All of the intervals display double power law in the intermediate to large scales in their trace magnetic PSD. Traditionally, the double power law in PSD is characterized by power indices $-5/3$ and $-1$ at the two scale ranges, respectively, and there are many models in the literatures to explain the origin of the $1/f$ range (e.g., Matthaeus & Goldstein 1986; Velli et al. 1989; Dmitruk & Matthaeus 2007; Verdini et al. 2012; C18; M18; Magyar & Doorsselaere 2022). Previous observations from Ulysses, Helios, and WIND have only explored the heliosphere beyond 0.3 au. In this study, we aim to use the latest observations from PSP to provide constraints on the origin of the $1/f$ range.

From the statistics of the 109 intervals, we found that within 0.3 au, the majority the intervals display spectra that are much shallower than $1/f$ in the low-frequency range. And as advection time $t_{adv} = R/V_{SW}$ increases, the low-frequency power-law fit index $\alpha_{Low}$ asymptotically approach $-1$. This suggests a dynamical formation of such range from the Alfvén point up to 0.3 au. However, when sorted with heliocentric distance, no obvious evolution of $\alpha_{Low}$ is observed. Moreover, for those extremely slow solar wind streams that were observed very close to Sun, the low-frequency spectra scaling are very close to $1/f$. Therefore, we cannot rule out the possibility that the $1/f$ originated from the corona and are advected out (Matthaeus & Goldstein 1986). Moreover, it should be noted that such $1/f$ scaling is also present at the density spectrum from Ulysses observations (Matthaeus et al. 2007), which appeared to have latitudinal dependence at 1 au. At solar midlatitudes, such scaling matches reasonably well with the longitudinal spectra of the photospheric magnetic field (Nakagawa & Levine 1974).

Unlike other models, M18 and C18 allow the low-frequency spectrum to be shallower than $1/f$. In fact, as has been pointed out by C18, it has been reported by Tu & Marsch (1995) that in the low-frequency range, $z^+$ has spectrum as shallow as $f^{-0.5}$. The new observations provide evidence inconsistent with the conjecture by M18 because closer to the Sun where $\delta B/B < 1$, the low-frequency spectrum does not gradually disappear, but instead they are omnipresent. On the other hand, some Alfvénic intervals display clear triple power law in the $z^+$ spectra with $1/f$ scaling in the intermediate-frequency range and therefore provide some evidence for the model from C18.

The new observations from PSP encounter 1–13 provide abundant new evidence for the origin of the low-frequency trace magnetic $1/f$ spectrum. Unfortunately, up to this point, the existing models have had difficulty covering all of the observed behaviors of the $1/f$ scaling in trace magnetic PSD. In fact, it is not even appropriate to call this part of the spectrum “$1/f$” spectrum because of the omnipresent shallower spectra observed closer to the Sun. However, based on this study, the new evidence indicates that for different intervals from different solar wind conditions, the low-frequency spectrum

might have different formation mechanisms. Therefore, we need to accumulate more observations from future PSP orbits to obtain a clearer picture of the low-frequency turbulence spectrum.

## Acknowledgements

The authors acknowledge the following open-source packages: Harris et al. (2020), Virtanen et al. (2020), Hunter (2007), Lam et al. (2015), The Pandas Development Team (2023), and Angelopoulos et al. (2019). This research was funded in part by the FIELDS experiment on the Parker Solar Probe spacecraft, designed and developed under NASA contract UCB #00010350/NASA NNN06AA01C, and the NASA Parker Solar Probe Observatory Scientist grant NASA NNX15AF34G. M.V. acknowledges support from ISSI via the J. Geiss fellowship.

## ORCID iDs

Zesen Huang

(黄泽淼) https://orcid.org/0000-0001-9570-5975

Nikos Sioulas https://orcid.org/0000-0002-1128-9685

Chen Shi (时辰) https://orcid.org/0000-0002-2582-7085

Marco Velli https://orcid.org/0000-0002-2381-3106

Trevor Bowen https://orcid.org/0000-0002-4625-3332

Nooshin Davis https://orcid.org/0000-0001-7222-3869

B. D. G. Chandran https://orcid.org/0000-0003-4177-3328

Lorenzo Matteini https://orcid.org/0000-0002-6276-7771

Ning Kang (康宁) https://orcid.org/0000-0002-7317-8665

Xiaofei Shi

(石晓霏) https://orcid.org/0000-0003-3367-5074

Jia Huang (黄佳) https://orcid.org/0000-0002-9954-4707

Stuart D. Bale https://orcid.org/0000-0002-1989-3596

J. C. Kasper https://orcid.org/0000-0002-7077-930X

Davin E. Larson https://orcid.org/0000-0001-5030-6030

Roberto Livi https://orcid.org/0000-0002-0396-0547

P. L. Whittlesey https://orcid.org/0000-0002-7287-5098

Ali Rahmati https://orcid.org/0000-0003-0519-6498

Kristoff Paulson https://orcid.org/0000-0002-5699-090X

M. Stevens https://orcid.org/0000-0002-7728-0085

A. W. Case https://orcid.org/0000-0002-3520-4041

Thierry Dudok de Wit https://orcid.org/0000-0002-4401-0943

David M. Malaspina https://orcid.org/0000-0003-1191-1558

J. W. Bonnell https://orcid.org/0000-0002-0675-7907

Keith Goetz https://orcid.org/0000-0003-0420-3633

Peter R. Harvey https://orcid.org/0000-0002-6938-0166

Robert J. MacDowall https://orcid.org/0000-0003-3112-4201

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