# Dominance of 2 Minute Oscillations near the Alfvén Surface

Zesen Huang, Marco Velli, Chen Shi, Yingjie Zhu, B. D. G. Chandran, Trevor Bowen, Victor Réville, Jia Huang, Chuanpeng Hou, Nikos Sioulas, et al.

► To cite this version:

Zesen Huang, Marco Velli, Chen Shi, Yingjie Zhu, B. D. G. Chandran, et al. Dominance of 2 Minute Oscillations near the Alfvén Surface. The Astrophysical Journal Letters, 2024, 977, 10.3847/2041-8213/ad9271. insu-04833918

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# Dominance of 2 Minute Oscillations near the Alfvén Surface

Zesen Huang (黄泽森)¹ ID, Marco Velli¹ ID, Chen Shi (时辰)¹ ID, Yingjie Zhu (朱英杰)² ID, B. D. G. Chandran³ ID,
Trevor Bowen⁴ ID, Victor Réville⁵ ID, Jia Huang (黄佳)⁴ ID, Chuanpeng Hou (侯传鹏)⁶ ID, Nikos Sioulas⁴ ID,
Mingzhe Liu (刘明哲)⁴ ID, Marc Pulupa⁴ ID, Sheng Huang (黄胜)⁷ ID, and Stuart D. Bale⁸,⁴ ID

¹Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA, USA

²Physikalisch Meteorologische Observatorium, Davos, World Radiation Center (PMOD/WRC), 7260 Davos, Switzerland

³Space Science Center and Department of Physics, University of New Hampshire, Durham, NH 03824, USA

⁴Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA

⁵IRAP, Université Toulouse III-Paul Sabatier, CNRS, Toulouse, France

⁶Institut für Physik und Astronomie, Universität Potsdam, Potsdam, Germany

⁷Center for Space Physics, Boston University, Boston, MA, USA

⁸Physics Department, University of California, Berkeley, CA 94720-7300, USA

Received 2024 October 04; revised 2024 November 05; accepted 2024 November 14; published 2024 December 3

## Abstract

Alfvén waves, considered one of the primary candidates for heating and accelerating the fast solar wind, are ubiquitous in spacecraft observations, yet their origin remains elusive. In this study, we analyze data from the first 19 encounters of the Parker Solar Probe and report the dominance of 2 minute oscillations near the Alfvén surface. The frequency-rectified trace magnetic power spectral density (PSD) of these oscillations indicates that the fluctuation energy is concentrated around 2 minutes for the “youngest” solar wind. Further analysis using wavelet spectrograms reveals that these oscillations primarily consist of outward-propagating, spherically polarized Alfvén wave bursts. Through Doppler analysis, we show that the wave frequency observed in the spacecraft frame can be mapped directly to the launch frequency at the base of the corona, where previous studies have identified a distinct peak around 2 minutes (~8 mHz) in the spectrum of swaying motions of coronal structures observed by the Solar Dynamics Observatory Atmospheric Imaging Assembly. These findings strongly suggest that the Alfvén waves originate from the solar atmosphere. Furthermore, statistical analysis of the PSD deformation beyond the Alfvén surface supports the idea of dynamic formation of the otherwise absent 1/f range in the solar wind turbulence spectrum.

*Unified Astronomy Thesaurus concepts: Solar wind (1534); Fast solar wind (1872); Alfvén waves (23); Interplanetary turbulence (830); Solar corona (1483); Solar coronal heating (1989); Solar coronal holes (1484)*

## 1. Introduction

The heliosphere is formed by the supersonic plasma flow known as the solar wind, originating from the Sun. E. N. Parker (1958) successfully predicted the existence of this supersonic flow, which was later confirmed by in situ observations from the Luna 2 (K. I. Gringauz et al. 1962) and Mariner II spacecraft (M. Neugebauer & C. W. Snyder 1966). However, it was soon recognized (E. N. Parker 1965) that thermal conduction alone cannot accelerate the solar wind to the observed high speeds (≥700 km s⁻¹), implying the need for an additional source of momentum and energy that persists into the supersonic regions of the wind (E. Leer & T. E. Holzer 1980).

Large-amplitude spherically polarized Alfvénic fluctuations are ubiquitous in the solar wind (T. W. J. Unti & M. Neugebauer 1968; J. W. Belcher & L. Davis 1971; S. D. Bale et al. 2019; J. C. Kasper et al. 2019; J. F. Drake et al. 2021; A. Larosa et al. 2021, 2023; A. Tenerani et al. 2021; O. Shi et al. 2022b; D. Telloni et al. 2022; J. Huang et al. 2023a; V. K. Jagarlamudi et al. 2023; Y. Jiao et al. 2023; G. Toth et al. 2023; J. Lee et al. 2024; M. Velli et al. 2024). As the name says, they are characterized by a quasi-constant magnetic magnitude |B|, and the tip of the magnetic field vector undergoes a random walk on a sphere (A. Barnes & J. V. Hollweg 1974; B. T. Tsurutani

et al. 1997; L. Matteini et al. 2014, 2024). In addition, the very high velocity–magnetic field correlation corresponds to that of waves propagating away from the Sun. The constant |B| state is an exact solution of the magnetohydrodynamics equations (L. Matteini et al. 2014; Z. Huang et al. 2024), which can propagate without much nonlinear interaction. Thus, the Alfvén wave is considered to be a prime candidate for the heating and acceleration of the solar wind (G. Alazraki & P. Couturier 1971; J. W. Belcher 1971; J. W. Belcher & S. Olbert 1975; V. H. Hansteen & M. Velli 2012; D. Telloni et al. 2021; C. Shi et al. 2022a; Y. J. Rivera et al. 2024).

However, the origin of solar wind Alfvénic fluctuations remains a subject of debate (D. Banerjee et al. 2021). Some suggest they are generated by magnetic reconnection in the lower solar corona (S. D. Bale et al. 2021, 2023; J. F. Drake et al. 2021), while others argue that Alfvén waves originate deeper in the solar atmosphere (D. B. Jess et al. 2009, 2015; R. J. Morton et al. 2023). These deeper sources may include the transition region and chromosphere (B. De Pontieu et al. 2007; D. Kuridze & T. Zaqarashvili 2008; H. Tian et al. 2014; C. Hou et al. 2024) or even the photosphere, where they could be generated by p-mode oscillations (R. K. Ulrich 1970) and reach the corona through double mode conversion (P. S. Cally 2012, 2017; S. C. Hansen & P. S. Cally 2012; R. J. Morton et al. 2013, 2015, 2019; H. Kuniyoshi et al. 2024) or through buffeting of magnetic flux tubes in “G-band bright points” by convective motions (S. R. Cranmer & A. A. Ballegooijen 2005; A. A. V. Ballegooijen &

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Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
M. Asgari-Targhi 2016). Therefore, leveraging state-of-the-art in situ observations is crucial for constraining existing theories.

The Parker Solar Probe (PSP) was launched in late 2018 (N. J. Fox et al. 2016). Since then, it has been providing in situ observations from the magnetically dominated solar corona (J. C. Kasper et al. 2021; R. Chhiber et al. 2024). The unprecedented data from distances to the Sun smaller than 0.25 au have ushered in a new era of solar wind turbulence studies (see, e.g., C. H. K. Chen et al. 2020; W. H. Matthaeus 2021; C. Shi et al. 2021; S. Q. Zhao et al. 2021; N. Sioulas et al. 2022, 2023a, 2023b; G. P. Zank et al. 2022; Zhang et al. 2022a; C. Dunn et al. 2023; A. Larosa et al. 2023; J. R. McIntyre et al. 2023; N. E. Raouafi et al. 2023 for a recent review). In the standard solar wind turbulence model for the fast wind (R. Bruno & V. Carbone 2013), the trace magnetic power spectral density (PSD) is characterized by a “1/f-inertial” double power law, where the 1/f range at low frequencies is known as the energy-containing range. However, one of the surprising discoveries of PSP is the systematic absence of the low-frequency 1/f range near the Alfvén surface (N. Davis et al. 2023; Z. Huang et al. 2023b), i.e., the region where the solar wind flow overtakes the Alfvén speed as it accelerates outward (J. C. Kasper et al. 2021; R. Chhiber et al. 2024). Instead, the turbulence spectrum is predominantly characterized by a “shallow-inertial” double power law, where the shallow part scales like $f^{-0.5}$. Beyond, in the super-Alfvénic wind, the 1/f range gradually forms between the shallow and inertial part of the spectrum, creating a “shallow-1/f-inertial” triple power law. As the solar wind travels outward, the spectrum will eventually transforms into the traditional “1/f-inertial” scaling. As will be elaborated in this Letter, the “shallow-inertial” scaling and its deformation have significant implications on the distribution of fluctuation energy in the spectrum.

In this Letter, we report the dominance of 2 minute oscillations in the trace magnetic PSD near the Alfvén surface. The Letter is organized as follows. In Section 2, we introduce the data from PSP and examine the implications of the dominant “shallow-inertial” double power law near the Alfvén surface. We also define a statistical factor, $f_{mid}$, to capture the radial deformation of the spectrum. In Section 3, we establish a connection between the temporal signals measured in the spacecraft frame and the waves launched from the base of the corona. In Section 4, we show the radial evolution of $1/f_{mid}$ in the solar wind. In Section 5, we discuss the implications of our results and conclude our study.

# 2. Data and Methodology

## 2.1. Data

We use magnetic field data from the fluxgate magnetometer in the FIELDS instrument suite (S. D. Bale et al. 2016; T. A. Bowen et al. 2020) and plasma moments from SPAN-i in the SWEAP instrument suite (J. C. Kasper et al. 2016). In this study, we treat the radial component of the first-order proton moment as the solar wind speed. On PSP, electron number density data from quasi-thermal noise (QTN) spectroscopy serve as a critical calibration standard for scientific analysis (e.g., J. C. Kasper et al. 2021; M. Liu et al. 2021a, 2023; Y. D. Liu et al. 2021b; S. Q. Zhao et al. 2021; X. Zheng et al. 2024), especially given that the alpha-particle abundance is

often negligible (2%–6%; P. Mostafavi et al. 2022). Therefore, we use electron number density data derived from the QTN technique (e.g., M. Moncuquet et al. 2020; O. Kruparova et al. 2023) as a proxy for proton number density in the solar wind.

## 2.2. Example Interval and Methodology

Figure 1 shows an example interval from PSP's E17 perihelion. The interval of interest is highlighted with the pink shade from panels (c) to (h). In panel (c), we show the RTN components and magnitude of the magnetic field vector. In panel (d), we show the time-periodic trace magnetic spectrogram compiled using wavelet transformation. To visually convey the contributions to the total fluctuation energy from different ranges of $\ln(f)$, the PSD $P(f)$ is rectified with $f$ because $P(f)$ $df = fP(f) d\ln f$ (see also Y. Liu et al. 2007). The black dashed curve marking the boundary of the shaded area is the cone of influence (CoI), which indicates the region where the wavelet falls out of the boundary for a given frequency. In panel (e), we show both the radial component of the bulk proton velocity $V_r$ and the local Alfvén speed $V_A = |B|/\sqrt{\mu_0 m_p n_p}$ (where $n_p$ is assumed to be the same as $n_e$ estimated from QTN). The QTN data are interpolated to fill the data gaps, as shown in panel (g). For this specific interval, $V_r/V_A \simeq 1$, indicating that PSP is situated in the neighborhood of the Alfvén surface (J. C. Kasper et al. 2021; R. Chhiber et al. 2024). From panel (f) to (h), we show various plasma parameters, including cross-helicity $\sigma_c$ ($\sigma_c = \frac{z^{+2} - z^{-2}}{z^{+2} + z^{-2}}$, where $z^{\pm} = \delta\vec{V} \mp \delta\vec{B}/\sqrt{\mu_0 m_p n_p}$ are the Elsässer variables), proton plasma $\beta$ ($\beta = 2\mu_0 P/B^2$), proton number density $n_p$, Carrington longitude, PSP's heliocentric distance $R$, and advection time $t_{adv} = (R - R_\odot)/V_r$.

Within the interval, the amplitudes of the magnetic vector fluctuations are very large, yet the magnetic field magnitude remains relatively constant. Meanwhile, $\sigma_c$ remains close to unity. The strong correlation between $B_r$ (blue curve; panel (c)) and $V_r$ (green curve; panel (e)) indicates that the fluctuations are unidirectional, outward-propagating, large-amplitude, spherically polarized Alfvén waves. To further substantiate the spherical polarization, Figure 2 presents the hodograms of the three components of the magnetic field and proton fluctuations. It is evident that the fluctuation amplitudes are comparable to the averaged quantities while remaining constrained on a constant sphere, symbolizing spherical polarization. It should be noted that, due to the blocked field of view of SPAN-i by the heat shield, the tangential and normal components of the partial moment proton velocity are not as good as the radial component.

The trajectory of PSP is plotted in the Carrington corotating frame (with the 27 day Carrington rotation subtracted) in Figure 1(a), where the interval of interest is highlighted with the cyan bar (“selected stream”). The local radial solar wind speed ($V_r$ in panel (e)) is shown by the color of the radial lines. To indicate the location of the Alfvén surface, the Alfvén Mach number ($M_A = V_r/V_A$) is visualized as the red curve. Whenever the red curve dips below the black curve, PSP is inside the magnetically dominant solar corona. To aid the reader, the black dots are plotted every 8 hr, the entering direction is indicated with the green dot, the black dashed circles are plotted every 10 $R_\odot$, and the radial dotted lines are plotted at 5° intervals in Carrington longitude.
<figure><img src="image_4.png" /><figcaption>**Figure 1.** (a) Trajectory of PSP in the solar corotating frame. Black dots are plotted every 8 hr, and the green dot indicates the entering direction of PSP. (b) Trace magnetic PSD of the selected interval. (c) RTN components and magnitude of the magnetic field. The selected interval is highlighted with the vertical pink area. (d) Trace wavelet scalogram of the magnetic field from the selected interval. (e) Radial solar wind $V_r$ and Alfvén speed $V_A$. (f) Cross-helicity $\sigma_c$ (left; blue) and plasma $\beta = 2\mu_0P/B^2$ (right; black). (g) Proton density $n_p$ (left; black) and Carrington longitude (right; orange). (h) Advection time $\tau_{adv} = (R - R_c)/V_r$ (left; red) and helioradial distance $R$ of the spacecraft (right; black).</figcaption></figure>

Following previous work (Z. Huang et al. 2023b), in this study, we primarily focus on the trace PSD of the magnetic field vector, shown in panel (b). The PSD<sub>FFT</sub> (blue curve) is estimated using fast Fourier transformation (FFT). Due to the mathematical properties of FFT, PSD<sub>FFT</sub> is naturally noisy. Following the procedure used in previous works (Z. Huang et al. 2023b; N. Sioulas et al. 2023a, 2023b), we smooth PSD<sub>FFT</sub> with a sliding window with a frequency factor of 2, and the smoothed spectrum is PSD<sub>FFT,smoothed</sub> (green dashed line). However, the trajectory in panel (a) shows that PSP travels very quickly during perihelion. For the interval of interest (cyan bar), PSP traversed about 14° over 4 hr. Thus, the time series measured by PSP around perihelion is highly nonstationary, violating one of the fundamental assumptions of Fourier transformation. The nonstationary effects primarily influence the low-frequency part of the spectrum. To assess the extent of these effects, we plot the averaged wavelet PSD (PSD<sub>WL</sub>), shown as the orange line (averaged over the data within the CoI in panel (d)). Clearly, PSD<sub>WL</sub> and PSD<sub>FFT,smoothed</sub> overlap perfectly in the high-frequency range and begin to diverge around $10^{-3.5}$ Hz, where a significant portion of the data fall outside the CoI. Consequently, we rely on the spectrum down to the frequency where more than 50% of the data fall outside the CoI ($f_{CoI,50\%}$), indicated by the gray shading in panel (b). For the remainder of this work, we will

use PSD<sub>WL</sub> as the "primary" PSD of the magnetic field time series (henceforth referred to as PSD<sub>WL</sub> or $P(f)$ interchangeably), and we will extract factors from PSD<sub>WL</sub>.

PSD<sub>WL</sub>, as mentioned previously, is characterized by a "shallow-inertial" double power-law scaling. In the low-frequency range, the spectrum is shallower than $1/f$, while in the high-frequency range, the scaling aligns with the spectral index range predicted by various turbulence phenomenology theories (e.g., P. S. Iroshnikov 1964; R. Kraichnan 1965; P. Goldreich & S. Sridhar 1995, 1997; S. Boldyrev 2005, 2006; B. D. G. Chandran et al. 2015; G. P. Zank et al. 2020) and is consistent with recent observations (e.g., C. H. K. Chen et al. 2020; L.-L. Zhao et al. 2020, 2022; C. Shi et al. 2021; J. R. McIntyre et al. 2023; N. Sioulas et al. 2023a, 2023b). Therefore, this range is identified as the turbulence inertial range. One interval with a spectrum shallower than $1/f$ was seen in Helios observations of fast solar wind at 0.3 au (C. Y. Tu & E. Marsch 1995; B. D. G. Chandran 2018). Recently, based on the new observations from PSP at much closer distances to the Sun ($R < 0.3$ au), this type of spectrum unexpectedly dominates, and it gradually develops into the typical "$1/f$-inertial" spectrum as $R$ increases (N. Davis et al. 2023; Z. Huang et al. 2023b).

The "shallow-inertial" double power law has an interesting property. It is well known that $1/f$ scaling indicates that energy
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Illustration of the spherical polarization of the Alfvén wave burst. (a) Components RTN and the magnitude of the magnetic field. The selected interval is highlighted by the vertical pink area, while the Alfvén wave burst event is indicated by the red dashed lines. (b) Trace wavelet scalogram of the magnetic field for the selected interval. (c1)–(c3) Hodograms of $B_r - B_n$, $B_r - B_n$, and $B_t - B_n$ for the Alfvén wave burst event. (d1)–(d3) Hodograms of $V_r - V_t$, $V_r - V_n$, and $V_t - V_n$ from SPAN-i moments.</figcaption></figure>

is distributed evenly across different frequencies on a logarithmic scale (M. Keshner 1982). In other words,

$$ \int_{f_1}^{f_2} \frac{1}{f} \, df = \ln(f_2/f_1) \quad (1) $$

is dependent only on the ratio of the integration bounds, i.e., $f_2/f_1$. Given that the spectrum scales as $P(f) = f^{-\alpha}$, if $\alpha > 1$, the fluctuation energy concentrates at the low-frequency end, and when $\alpha < 1$, the energy concentrates at high frequencies. Consequently, the “shallow–inertial” scaling indicates that the energy concentrates around the “bend.” To substantiate our argument, we consider a fixed window in $\ln(f)$. The energy

stored in the moving window is

$$ \int P(f) \, df = \int P(f) f \, d\ln(f). \quad (2) $$

Thus, the rectified spectrum $P(f) \cdot f$ curve is proportional to the “energy density” of frequency $f$ on a logarithmic scale. Additionally, the area under the $P(f) \cdot f$ curve in a linear-log plot represents the relative amount of fluctuation energy for a given frequency range.

In panel (b), $P(f) \cdot f$ is plotted in both logarithmic scale (orange dashed curve) and linear scale (black dotted–dashed curve, right-hand side twin axis). The logarithmic plot clearly shows that the turbulence scaling is shallower than $1/f$ in the
low-frequency range ($f < 10^{-2}$ Hz; the orange curve goes up) and steeper than $1/f$ in the high-frequency inertial range ($f > 10^{-2}$ Hz; the orange curve goes down). Thus, $P(f) \cdot f$ peaks at $1/f_{peak} = 136$ s, indicated by the vertical black dashed line. The area under the black dotted-dashed $P(f) \cdot f$ curve represents the relative amount of total fluctuation energy. Green shading is overlaid to indicate the frequency range containing the middle 50% of the total fluctuation energy, estimated from the highest-frequency $f_0$ down to the right-hand edge of the gray shading $f_{Col,50\%}$. In other words, the area under the $P(f) \cdot f$ curve in the green shading contains 50% of the total area, with the right side up to $f_0$ containing 25%, and the left side down to $f_{Col,50\%}$ containing the remaining 25%. The middle frequency of the green shading is named $f_{mid}$ and is calculated to be $1/f_{mid} = 116$ s (see Appendix A for a formulaic expression of $f_{mid}$). Both $f_{mid}$ and $f_{peak}$ are indicated by red and black dashed lines in panels (b) and (d).

In this case, the values of $f_{peak}$ and $f_{mid}$ are very close, and the majority of the fluctuation energy concentrates around them (50% within the green shading). Thus, both can be considered as a “representative frequency” of this interval. However, $f_{peak}$ is very sensitive to the specific choice of interval. For example, when the spectrum gradually deforms from “shallow-inertial” to “1/f-inertial”, $f_{peak}$ will rapidly migrate to $f_{Col,50\%}$ (the lowest trustable frequency, as $P(f) \cdot f$ monotonically decreases as a function of $f$). On the other hand, $f_{mid}$ will slowly migrate to the left as the spectrum deforms, because it represents the center of the frequency range containing the middle 50% of the total fluctuation energy (see N. Davis et al. 2023 for details of the deformation in E10 and Z. Huang et al. 2023b for a statistical analysis). Thus, $f_{mid}$ can capture the overall deformation of the spectrum. In this study, we will use $f_{mid}$ as the primary factor for statistical analysis.

### 3. Interpretation of Time Series

The frequencies considered in the previous section are measured in the spacecraft frame. Thus, proper interpretation of the spacecraft time series is pivotal to our study. Generally speaking, there are two approaches to interpret the spacecraft time series: (1) directly interpret the time series as temporal signals in the spacecraft frame or (2) invoke the Taylor hypothesis (G. I. Taylor 1938), including its modified versions tailored for PSP due to the comparable solar wind speed and Alfvén speed near the Alfvén surface (K. G. Klein et al. 2015; J. C. Perez et al. 2021; G. P. Zank et al. 2022; L.-L. Zhao et al. 2024), to translate temporal signals into spatial signals. To make meaningful comparisons with anisotropic turbulence phenomenologies (e.g., P. S. Iroshnikov 1964; R. Kraichnan 1965; P. Goldreich & S. Sridhar 1995, 1997; S. Boldyrev 2005, 2006; B. D. G. Chandran et al. 2015; G. P. Zank et al. 2020), one must transform the frequency spectrum into the wavevector spectrum. Therefore, the latter approach is more commonly used in the solar wind turbulence community (e.g., T. S. Horbury et al. 2012; S. Oughton et al. 2015; D. Duan et al. 2021; D. Telloni et al. 2021, 2024; L. Adhikari et al. 2021, 2024; J. Zhang et al. 2022b; R. Bandyopadhyay et al. 2022; N. Andrés et al. 2022; G. P. Zank et al. 2022; N. Sioulas et al. 2023a).

In this study, we take the first approach. We aim to establish a connection between the waves launched from the base of the corona and the waves measured by PSP near the Alfvén surface. In this section, we will show that the Alfvén wave

frequency observed in the spacecraft frame can be considered identical to the launch frequency for the scenarios considered in this study.

#### 3.1. 1D Solar Wind Model

To begin, we consider a simple 1D solar wind model in the solar corotating frame, similar to earlier studies (M. Heinemann & S. Olbert 1980; J. V. Hollweg et al. 1982; M. Velli 1993). In this model, we ignore the perpendicular movement of PSP and consider only its radial speed, $V_{PSP,r}$.

There are three reference frames involved: the solar surface frame, $K_\odot$; the solar wind (intrinsic) plasma frame, $K_{wind}$, which moves at speed $U$ relative to $K_\odot$; and the PSP frame, $K_{PSP}$, which moves at speed $V_{PSP,r}$ relative to $K_\odot$. Similar to previous 1D models, we launch Alfvén waves at the base of the corona with a frequency $f_0$. The frequency of the waves in $K_{wind}$ is

$$f_{\text{wind}} = f_0 \frac{V_A}{U + V_A}, \qquad (3)$$

where $V_A$ is the local Alfvén speed. Similarly, the Doppler shift from $K_{wind}$ to $K_{PSP}$ is (the phase speed of Alfvén waves in $K_{wind}$ is $V_A$, and $K_{PSP}$ moves at $U - V_{PSP,r}$ relative to $K_{wind}$)

$$ \begin{aligned} f_{\text{PSP}} &= f_{\text{wind}} \frac{V_A + U - V_{\text{PSP},r}}{V_A} \\ &= f_0 \frac{U + V_A - V_{\text{PSP},r}}{U + V_A}, \end{aligned} \qquad (4) $$

where $V_{PSP,r}$ is positive when pointing radially outward. Obviously, $f_{PSP}$ reduces to $f_0$ when $V_{PSP,r} \ll U + V_A$. Due to the spatial gradient of both $U$ and $V_A$ from the base of the corona to PSP, a substantial portion of the Alfvén waves could be reflected. In this study, we are particularly interested in the behavior of the Alfvén waves observed near the Alfvén surface (e.g., Figure 1). Thus, following M. Velli (1993), we solve for the transmission coefficient profile for outward-propagating Alfvén waves from the base of the corona to the Alfvén surface, and the results are shown in Figure 3 (see Appendix B for a detailed derivation).

We may now examine the Doppler shift effects from the radial movements of PSP and the transmission coefficient of the “representative frequency” found in Figure 1. Around the Alfvén surface, $V_{PSP,r} \lesssim 100 \text{ km s}^{-1}$, and $U + V_A \gtrsim 800 \text{ km s}^{-1}$ (Figure 1(e)). Thus, the radial Doppler shift factor

$$ \frac{U + V_A - V_{\text{PSP},r}}{U + V_A} \sim 1 $$

is mostly negligible, and the effect is strongest during the fast radial scan phase. The “representative frequency” in Figure 1 lies between 2 and 3 minutes. This indicates that the launch frequency of these Alfvén waves at the solar surface should be similar. From the right panel of Figure 3, we can see that the transmission coefficient for a 3 minute Alfvén wave from the base of the corona to the Alfvén surface is very close to unity. Thus, based on our 1D solar wind model, the 2–3 minute Alfvén waves could be launched from the base of the corona and reach PSP around the Alfvén surface without significant reflection.
<figure><img src="image_6.png" /><figcaption>Figure 3. Left: solar wind and Alfvén speed profile. Right: transmission coefficient of Alfvén waves as a function of launch frequency.</figcaption></figure>

However, the analysis above has ignored two very important effects: the perpendicular movement of PSP and nonlinear interactions. In the following subsection, we address the first effect, and in Section 5, we will examine the second effect.

## 3.2. Perpendicular Contamination

The perpendicular motion of the PSP causes a Doppler shift that transforms spatial structures into temporal signals, potentially contaminating time series in the spacecraft's frame of reference. This Doppler shift is described by the following relation:

$$2\pi f = k_{\phi} \cdot V_{\text{PSP},\phi}, \qquad (5)$$

where $k_{\phi}$ is the wavenumber of the spatial structures in the perpendicular $\phi$ direction. The PSD of the perpendicular wavenumber $P(k_{\phi})$ is an unknown function. However, it is well established that Alfvén waves are guided by the background magnetic field, and around the Alfvén surface, this field is predominantly radial. Therefore, the perpendicular spatial signals are likely caused by crossings of intermittent flux tubes. Despite this, it is highly improbable that the concentration of 2 minute oscillations results from perpendicular contamination for two primary reasons.

First, Figure 1(d) shows that the oscillations are highly localized in the wavelet spectrogram. In other words, the dominant contributors to the peak in the rectified $P(f) \cdot f$ spectrum are the few wave bursts (“red islands” in panel (d)). This feature of the waves further supports the temporal nature of the signals, rather than contamination from the Doppler shift of spatial structures. To reinforce the universality of this observation, we present four additional example intervals in Figure 4. All of these intervals are located near the Alfvén surface, and the oscillations remain highly localized in the time-period domain.

Second, for a fixed size at the coronal base, the Doppler-shifted frequency depends on the local ratio $V_{\text{PSP},\phi}/R$. This ratio varies significantly throughout each perihelion since $V_{\text{PSP},\phi}$ increases as PSP approaches perihelion. To illustrate this effect, we calculate an example of the perpendicular

contamination frequency based on the size of a solar granule $\lambda_{\text{granule}} = 1700$ km (M. Rieutord et al. 2010), assuming a superradial expansion factor of 3:

$$f_c = \frac{V_{\text{PSP},\phi}}{3\lambda_{\text{granule}} \cdot R/R_{\odot}}. \qquad (6)$$

This frequency is represented by green dashed lines in the wavelet spectrograms of Figures 1 and 4. In Figure 4(b), PSP is nearing the end of its fast radial scan phase, where $R \simeq 17 R_{\odot}$. Here, $V_{\text{PSP},\phi} \simeq 100$ km s$^{-1}$, yielding $1/f_c \sim 1000$ s. In Figure 4(e), as PSP moves closer to perihelion, $R \simeq 12 R_{\odot}$ and $V_{\text{PSP},\phi} \simeq 150$ km s$^{-1}$, leading to $1/f_c \sim 400$ s. Despite these changes, the peaks of the fluctuation energy across all five intervals remain consistent, close to 2 minutes. This suggests that to consistently produce the 2 minute peaks, flux tubes at the coronal base must fine-tune their sizes according to the local ratio of $V_{\text{PSP},\phi}/R$. Additionally, even for locally generated structures, they must adjust their sizes to account for variations in $V_{\text{PSP},\phi}$, which can fluctuate by at least 50% across the intervals with 2 minute peaks. Achieving such consistent 2 minute peaks under these conditions would be quite remarkable.

In summary, in this section, we have established a connection between the Alfvén waves launched from the base of the corona and the Alfvén waves measured by PSP near the Alfvén surface. The bursty nature of the Alfvén waves provides strong evidence that they are temporal signals rather than Doppler-shifted spatial structures. In the following section, we will conduct a statistical analysis of $f_{\text{mid}}$ to understand its radial evolution in the solar wind.

## 4. Statistical Results

To investigate the evolution of $f_{\text{mid}}$ in the solar wind, we perform a statistical analysis of data from the first 19 encounters of PSP. We consider data from ±7 days around each perihelion, as highlighted in Figure 4(a). A 4 hr sliding window with a fixed 15 minute step is used. For each window, we compute several factors, including $\text{PSD}_{\text{WL}}$, $f_{\text{mid}}$, $f_{\text{peak}}$, and
<figure><img src="image_7.png" /><figcaption>**Figure 4.** (a) Heliocentric distance of PSP as of 2024 July 1. The time range considered in this study is highlighted with the black lines and pink shaded area. (b1)–(e1) Trajectory of PSP (black line) in the Carrington corotating frame. The black dots are plotted every 8 hr, and the green dot indicates the entering direction of PSP. (b2)–(e2) PSD of the selected interval. (b3)–(e3) Wavelet scalogram of the trace magnetic field power spectrum of the selected interval. The elements in (b), (c), and (d) are identical to Figures 1 (a), (b), and (d).</figcaption></figure>

other parameters such as $\sigma_c$ and $t_{adv} = (R - R_\odot)/V_r$. In total, 26,625 intervals were selected. As a first step, we cleaned the data, retaining only intervals where $0.005 < \delta t_{adv} / \langle t_{adv} \rangle < 0.2$, leaving us with 25,643 intervals. (This criterion is chosen to avoid missing values and abrupt changes in solar wind speed; 982 intervals were filtered out.) The values of $1/f_{mid}$ were then sorted by $R$, as shown in Figure 5(a).

The scatter plot in panel (a) appears highly chaotic. This is because, during solar maximum (starting from E10 in late 2021), the ecliptic solar wind consists of a complex mixture of slow and fast wind. The turbulence outer scale in the slow wind is typically observed at very low frequencies, $\lesssim 10^{-4}$ Hz (R. Bruno et al. 2019), whereas in the fast wind, the low-frequency spectral breaks are often found around $10^{-2}-10^{-3}$ Hz (R. Bruno & V. Carbone 2013). As a result, even for the same $R$, the distribution of $1/f_{mid}$ remains highly dispersed. Nevertheless, the binned mean value of $1/f_{mid}$ decreases with decreasing $R$ and drops to about 3 minutes.

A more suitable parameter for sorting $1/f_{mid}$ is the advection time, $t_{adv}$. This parameter can serve as a proxy for the “age” of the solar wind plasma, as fast wind naturally takes less time to reach a given $R$ than slow wind. The scatter plot of $1/f_{mid}$ as a function of $t_{adv}$ is shown in panel (c). When sorted by $t_{adv}$, the

overall trend becomes much clearer: $1/f_{mid}$ steadily decreases from the saturation value (estimated from a Kolmogorov $f^{-5/3}$ spectrum; see Appendix A for details) to around 2–3 minutes.

What is more intriguing is the evolution of the Alfvén waves. In panels (b) and (d), we keep only the intervals where $|\sigma_c| > 0.75$ and with at least 50% QTN density coverage within the 4 hr window ($\eta_{QTN} > 50\%$), leaving 7237 intervals. Surprisingly, for both $R$ and $t_{adv}$, the overall trends remain almost unchanged. To probe the evolution of $1/f_{mid}$ within a single stream, we highlight two long intervals from the two confirmed in situ coronal holes from the inbounds of E10 (S. T. Badman et al. 2023) and E12 (Z. Huang et al. 2024) (the dots are connected according to the value of $R$). Following a single stream originating from a coronal hole, $1/f_{mid}$ gradually drops toward 2 minutes as the “age” of the plasma decreases.

In summary, the trend of $1/f_{mid}$ as a function of $t_{adv}$ is very clear. For the “youngest” solar wind, $1/f_{mid}$ drops to around 2–3 minutes. The trend remains largely unchanged when considering only the Alfvénic intervals, indicating that this is an intrinsic property of Alfvén waves originating from the base of the corona. In the next section, we will provide a detailed discussion of the nonlinear effects and the implications of our results.
<figure><img src="image_8.png" /><figcaption>Figure 5. (a) Scatter plot of $1/f_{mid}$ as a function of heliocentric distance R. The window limit is the $1/f_{mid}$ value estimated from a Kolmogorov $f^{-5/3}$ spectrum. The error bars indicate one standard deviation in logarithmic scale. (b) Intervals with $|\sigma_c| > 0.75$ are selected. The two known in situ coronal holes from E10 and E12 are highlighted. (c) $1/f_{mid}$ is plotted as a function of $t_{adv} = (R - R_{\odot})/V_r$. (d) Intervals with $|\sigma_c| > 0.75$ are selected.</figcaption></figure>

## 5. Discussion and Conclusions

The trend in Figure 5(d) does not seem to stabilize with $t_{adv}$. Indeed, both the E10 coronal hole in Figure 5(d) and the cases in Figure 4 exhibit $1/f_{mid}$ values smaller than the smallest binned mean from the trend, closer to 2 minutes ($f_{mid} \approx 8.3$ mHz) rather than 3 minutes ($f_{mid} \approx 5.6$ mHz). We argue that the real saturation value of $1/f_{mid}$ is closer to 2 minutes from two perspectives: (1) the actual propagation time of the Alfvén waves and (2) the deformation and evolution of the turbulence spectrum.

The actual propagation time, $t_{prop}$, of the Alfvén waves from the base of the corona to the Alfvén surface is significantly shorter than $t_{adv}$,

$$t_{prop} = \int_{R_{\odot}}^{R_c} \frac{dR}{U + V_A}, \qquad (7)$$

where $R_c$ is the heliocentric distance of the Alfvén surface. Using the profile in Figure 3, we estimate $t_{prop} \approx 1.84$ hr, which is much shorter than the smallest $t_{adv} \approx 5.7$ hr. This indicates that the “age” of the Alfvén waves in the intervals located at the

lower left corner of panel (d) is much “younger” than it appears. However, as the “age” of the Alfvén waves increases, the relative difference between $t_{prop}$ and $t_{adv}$ decreases substantially due to the rapid drop of $V_A$ as a function of R. Thus, when $t_{adv}$ is large, the changes in $t_{adv}$ more closely reflect variations in $t_{prop}$. It is clear from the trend that a difference of 1.84 hr has little impact on the value of $1/f_{mid}$.

Furthermore, the shift of $1/f_{mid}$ to higher values with increasing $t_{adv}$ likely results from two key effects in solar wind turbulence. The first is the gradual deformation of the spectrum from “shallow–inertial” to “shallow–1/f–inertial” and eventually to “1/f–inertial” as the solar wind moves away from the Sun (N. Davis et al. 2023; Z. Huang et al. 2023b). This transformation, proposed to be caused by the parametric-decay-induced inverse cascade of Alfvén waves (B. D. G. Chandran 2018; Réville et al. 2018), typically generates a $1/f$ range starting from the “bend” that expands toward lower frequencies, gradually shifting $1/f_{mid}$ to the left. The second is the cascade and dissipation of turbulent fluctuation energy, which consumes energy in the $1/f$ energy-containing range, pushing both the spectral break of the
“1/f-inertial” spectrum and $f_{mid}$ to lower frequencies (C. Y. Tu & E. Marsch 1995; R. Bruno & V. Carbone 2013; H. Wu et al. 2020, 2021).

Both mechanisms drive $1/f_{mid}$ to larger values by “spreading out” the concentration of fluctuation energy toward lower frequencies. The “shallow-inertial” spectrum observed in the most pristine solar wind indicates a concentration of fluctuation energy. To shift $1/f_{mid}$ to even smaller values, the spectrum as a whole would need to move toward higher frequencies, rather than simply deform. To our knowledge, no current turbulence transport theories or models predict such behavior. Therefore, we propose that the true saturation value of $1/f_{mid}$ is likely closer to 2 minutes.

We would also like to emphasize the importance of the rectification of the spectrum: $P(f) \cdot f$. For comparison, P. Kumar et al. (2023) analyzed proton velocity data from the coronal hole outflow in PSP E10 and discovered distinct period peaks around 3, 5, 10, and 20 minutes, consistent with the period peaks in the emission intensity of jetlets at the base of coronal hole plumes and plumelets (V. M. Uritsky et al. 2021; P. Kumar et al. 2022). However, in their wavelet analysis, the PSD was not rectified with the corresponding frequency, as suggested by Y. Liu et al. (2007). Due to the high Alfvénicity of the coronal outflow, if rectified, the PSD from the proton velocity should resemble the trace magnetic PSD compiled in this study.

Our findings show a striking consistency with the remote sensing spectrum derived from the swaying motions of coronal structures observed by the Solar Dynamics Observatory Atmospheric Imaging Assembly (SDO/AIA) within coronal holes. Specifically, in Figure 6(c) of R. J. Morton et al. (2015), the peak of the histogram for Alfvén(ic) wave periods falls between 100 s and 200 s. Furthermore, in Figure 1(d) of R. J. Morton et al. (2019), after rectifying the SDO/AIA coronal hole PSD with $f$, the true peak of the spectrum emerges clearly at around 8 mHz (or approximately 2 minutes). However, this peak is absent in the PSD compiled from the CoMP Doppler velocity time series. The authors noted that due to instrument uncertainties, they were unable to confirm the authenticity of this peak.

Finally, our results establish a strong connection between the Alfvén waves measured by PSP near the Alfvén surface and those launched from the base of the corona. The dominance of 2 minute Alfvén waves near the Alfvén surface suggests that the Alfvén waves in the solar wind could be the result of leakage from p-mode oscillations, as proposed by previous studies (P. S. Cally 2012, 2017; V. H. Hansteen & M. Velli 2012; R. J. Morton et al. 2013, 2015, 2019). However, current evidence (R. J. Morton et al. 2019) is not yet robust enough to confirm this hypothesis. It is also possible that Alfvén waves at the corona's base are generated by convective motions at the photosphere, which shake the footprints of open magnetic field lines (S. R. Cranmer & A. A. Ballegooijen 2005; A. A. V. Ballegooijen & M. Asgari-Targhi 2016). A detailed discussion to differentiate between these models is, unfortunately, beyond the scope of this study.

Future studies on this matter can focus on two avenues. First, PSP will reach its deepest perihelion in 2024 December. The new data from deeper below the Alfvén surface will provide valuable insights into the evolution of the turbulence spectrum. Thus, a

dedicated cross-validation study between in situ observations and remote sensing in the future is of great interest. Second, the origin of the energy-containing $1/f$ range has been debated in the field for decades (M. Keshner 1982; E. W. Montroll & M. F. Shlesinger 1982; W. H. Matthaeus & M. L. Goldstein 1986; P. Bak et al. 1987; M. Velli et al. 1989; W. H. Matthaeus et al. 2007; A. Bemporad et al. 2008; P. Dmitruk et al. 2011; A. Verdini et al. 2012; B. D. G. Chandran 2018; L. Matteini et al. 2018; V. Réville et al. 2018; N. Magyar & T. V. Doorsselaere 2022; R. Meyrand et al. 2023). A more detailed study on the evolution of the spectrum in the young solar wind will likely shed new light on this topic.

## Acknowledgments

The authors acknowledge the following open-source packages: J. D. Hunter (2007), S. K. Lam et al. (2015), V. Angelopoulos et al. (2019), C. R. Harris et al. (2020), and P. Virtanen et al. (2020). Z.H. thanks Pankaj Kumar for valuable discussions. 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, and NASA HTMS 80NSSC20K1275. M.V. acknowledges support from ISSI via the J. Geiss fellowship. B.C. acknowledges support from NASA grant 80NSSC24K0171. C.S. acknowledges support from NSF SHINE #2229566 and NASA ECIP #80NSSC23K1064. J. H. acknowledges the support by NASA grant 80NSSC23K0737. Part of C.-P.H.'s work was completed prior to joining Potsdam University.

## Appendix A
### PSD and Window Limit

Figure 6 is an illustration of the window limit of 13 minutes for the 4 hr window (the dotted-dashed line in Figure 5), and the definitions of the lines are consistent with Figure 1(b). An artificial Kolmogorov spectrum $P(f) = f^{-5/3}$ is plotted with the orange line. The middle frequency $f_{mid}$ is defined as the median value of the frequency range that contains the middle 50% of the total fluctuation energy. The calculation of $f_{mid}$ proceeds as follows. The normalized cumulative fluctuation energy $E(f)$ is defined by

$$E(f) = \frac{\int_{f_0}^{f} P(f') df'}{\int_{f_0}^{f_{\text{col,50\%}}} P(f') df'} \in [0, 1], \quad (A1)$$

where $f_0$ is the highest frequency and $f_{\text{col,50\%}}$ represents the lowest available frequency due to the CoI. The midpoint frequency $f_{mid}$ is then defined as

$$\ln f_{\text{mid}} = \frac{\ln f_{0.75} + \ln f_{0.25}}{2}, \quad (A2)$$

where $E(f_{0.75}) = 0.75$ and $E(f_{0.25}) = 0.25$.
<figure><img src="image_9.png" /><figcaption>**Figure 6.** Illustration of the saturation $1/f_{\text{mid}}$ with a 4 hr window. Orange: Kolmogorov $P(f) = f^{-5/3}$ spectrum. Dotted-dashed: compensated spectrum $P(f) \cdot f$ in linear scale on twin axis. Gray area: frequency range where >50% of the points are out of the COI. Green area: frequency range that contains 50% of the fluctuation energy (from 25% to 75%).</figcaption></figure>

## Appendix B
### Transmission of Alfvén Waves in the Solar Corona

Here we use a classical Alfvén speed and solar wind speed profile (M. Velli et al. 1991),

$$ 
\begin{aligned}
\rho &= \rho_0 \frac{\exp\{-\alpha/2 \cdot [1 - (R_{\odot}/R)]\}}{\{1 + \beta[(R/R_{\odot} - 1)]\}^2} \\
V_A &= V_{A0} \left(\frac{R_{\odot}}{R}\right)^2 \left(\frac{\rho_0}{\rho}\right)^{1/2} \\
U &= \frac{U_{\infty}}{\beta^2} \exp(-\alpha/2) \left(\frac{R}{R_{\odot}}\right)^2 \left(\frac{V_A}{V_{A0}}\right)^2,
\end{aligned}
\quad (B1) $$

where $V_{A0}$ is the initial Alfvén speed at the base of the corona, $U_{\infty}$ is the asymptotic solar wind speed at infinity, and $\alpha$ and $\beta$ are free parameters. As an example, we choose the following parameters: $\alpha = 0$, $\beta = 5.6$, $V_{A0} = 1000 \text{ km s}^{-1}$, and $U_{\infty} = 500 \text{ km s}^{-1}$. Our choice produces an Alfvén surface at $12 R_{\odot}$ with $U = V_A = 434 \text{ km s}^{-1}$, consistent with PSP observations in Figure 1. The profiles of $U$, $V_A$, and $U + V_A$ are shown in Figure 3. To obtain the transmission coefficient profile, we solve the linear propagation equation for the two Elsässer variables,

$$ -i\omega z^{\pm} + (\vec{U} \pm \vec{V}_{\alpha}) \cdot \nabla z^{\pm} + z^{\mp} \cdot \nabla (\vec{U} \mp \vec{V}_{\alpha}) + \frac{1}{2}(z^{-} - z^{+})\nabla \cdot (\vec{V}_{\alpha} \mp \frac{1}{2}\vec{U}) = 0, \quad (B2) $$

where $\omega$ is the angular frequency of the wave in the solar wind frame at distance $R$. The transmission coefficient is defined in terms of the total wave action flux, which is otherwise conserved in the Wentzel-Kramers-Brillouin limit,

$$ S^{\pm} = \frac{1}{2} \rho \frac{(z^{\pm}/2)^2}{\omega_0 \frac{V_A}{U \pm V_A}} \cdot (U \pm V_A) \cdot A, \quad (B3) $$

where $A$ is the cross-sectional area of the flux tube. Note that here the wave action density (see, e.g., F. P. Bretherton 1968; R. L. Dewar 1970) is defined as the energy density divided by

the wave frequency $\omega$ in the solar wind frame,

$$ \frac{1}{2} \rho \frac{(z^{+}/2)^2}{\omega} = \frac{1}{2} \rho \frac{(z^{+}/2)^2}{\omega_0 \frac{V_A}{U + V_A}}, $$

where $\omega$ is Doppler-shifted downward compared to the launch frequency $\omega_0$ due to the presence of the wind. There are two important properties of Equation (B3). First, the net outward-propagating wave action flux is conserved,

$$ S^{+} - S^{-} = S_{\infty} = S_{c}, \quad (B4) $$

where $S_{\infty}$ is the total wave action flux at infinity and $S_c$ is the flux at the Alfvén surface ($S^{-} = 0$ at the Alfvén surface because $U = V_A$). The transmission coefficient for a given launch angular frequency $\omega_0$ is then defined as,

$$ T = \frac{S_c}{S_0^+}, \quad (B5) $$

where $S_0^+$ is the total wave action flux at the base of the corona. Second, the singular point at the Alfvén surface ($U = V_A$) provides a natural boundary condition for the system. At this singular point, Equation (B3) reduces to

$$ -i\omega z^{-} + \frac{1}{2}(z^{-} - z^{+})\nabla \cdot (\vec{V}_{A} + \frac{1}{2}\vec{U}) = 0. \quad (B6) $$

Now, combining Equations (B3)–(B6), we integrate the transmission coefficient from the given $U$ and $V_A$ profiles, and the result is shown in the right panel of Figure 3.

## ORCID iDs

Zesen Huang (黄泽森) https://orcid.org/0000-0001-9570-5975

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

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

Yingjie Zhu (朱英杰) https://orcid.org/0000-0003-3908-1330

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

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

Victor Réville https://orcid.org/0000-0002-2916-3837

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

Chuanpeng Hou (侯传鹏) https://orcid.org/0000-0001-7205-2449

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

Mingzhe Liu (刘明哲) https://orcid.org/0000-0003-2981-0544

Marc Pulupa https://orcid.org/0000-0002-1573-7457

Sheng Huang (黄胜) https://orcid.org/0000-0002-9951-8787

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

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