# Observations of Transition from Imbalanced to Balanced Kinetic Alfvénic Turbulence

Jinsong Zhao, Trevor A. Bowen, Stuart D. Bale, Chen Shi, Thierry Dudok de Wit, Nikos Sioulas

► To cite this version:

Jinsong Zhao, Trevor A. Bowen, Stuart D. Bale, Chen Shi, Thierry Dudok de Wit, et al. Observations of Transition from Imbalanced to Balanced Kinetic Alfvénic Turbulence. The Astrophysical Journal Letters, 2026, 999, 10.3847/2041-8213/ae4515. insu-05537458

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# Observations of Transition from Imbalanced to Balanced Kinetic Alfvénic Turbulence

Jinsong Zhao¹,*, Trevor A. Bowen²,†, Stuart D. Bale²,³,†, Chen Shi⁴,†, Thierry Dudok de Wit⁵,⁶,†, and Nikos Sioulas²,†

¹Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, People's Republic of China; zhaojs82@gmail.com

²Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA

³Physics Department, University of California, Berkeley, CA 94720-7300, USA

⁴School of Physics and Electronic Sciences, Changsha University of Science and Technology, Changsha, People's Republic of China

⁵LPC2E, CNRS, CNES, University of Orléans, Orléans, France

⁶International Space Science Institute (ISSI), Bern, Switzerland

Received 2025 November 28; revised 2026 January 26; accepted 2026 February 2; published 2026 February 27

## Abstract

We report observations of solar wind turbulence derived from measurements by the Parker Solar Probe. Our findings reveal the emergence of finite magnetic helicity within the transition range of the turbulence, aligning with signatures of kinetic Alfvén waves (KAWs). Notably, as the wave scale transitions from superion to subion scales, the ratio between KAWs with opposite signs of magnetic helicity initially increases from approximately one to 6.5, before decreasing back to one. This observation provides, for the first time, compelling evidence for the transition from imbalanced to balanced kinetic Alfvénic turbulence.

*Unified Astronomy Thesaurus concepts: Solar wind (1534); Plasma physics (2089); Space plasmas (1544)*

## 1. Introduction

Alfvénic turbulence is prevalent in the heliosphere (C. Y. Tu & E. Marsch 1995; R. Bruno & V. Carbone 2005) and is believed to exist in astrophysical environments (J. W. Armstrong et al. 1995; P. Goldreich & S. Sridhar 1995). The dissipation of this turbulence is proposed to be a major contributor to heating in the solar wind, solar corona, and other astrophysical plasmas (E. Quataert & A. Gruzinov 1999; C. W. Smith et al. 2006; S. R. Cranmer et al. 2007; B. D. G. Chandran 2010; B. D. G. Chandran et al. 2010; A. A. van Ballegooijen et al. 2011; L. Arzamasskiy et al. 2019; Y. Kawazura et al. 2019; T. A. Bowen et al. 2020b; H. Wu et al. 2020, 2021). Observations indicate that Alfvénic turbulence encompasses a broad range of scales, evolving from magnetohydrodynamics (MHD) scales down to subion kinetic scales, potentially extending to subelectron kinetic scales (R. J. Leamon et al. 1998; S. D. Bale et al. 2005; O. Alexandrova et al. 2009; F. Sahraoui et al. 2009, 2010; C. S. Salem et al. 2012; C. H. K. Chen et al. 2013b). As the turbulence cascade scale approaches the ion gyroradius, anisotropic turbulence models predict a transition from MHD-scale Alfvénic waves to kinetic Alfvén waves (KAWs; E. Quataert 1998; G. G. Howes et al. 2008; A. A. Schekochihin et al. 2009; Y. Voitenko & J. de Keyser 2011; S. Boldyrev & J. C. Perez 2012; J. S. Zhao et al. 2013, 2016; C. H. K. Chen & S. Boldyrev 2017; A. A. Schekochihin 2022). Despite substantial evidence of KAWs occurring at ion kinetic scales in satellite observations and numerical simulations (S. D. Bale et al. 2005; O. Alexandrova et al. 2009; F. Sahraoui et al. 2009, 2010; G. G. Howes et al. 2011; C. S. Salem et al. 2012; C. H. K. Chen et al. 2013b; L. Franci et al. 2015; D. Grošelj et al. 2018, 2019; J. Squire et al. 2022), the precise evolution of turbulence at ion kinetic scales remains unknown.

Solar wind turbulence is often highly imbalanced at MHD scales in the inner heliosphere, with outward-propagating fluctuations away from the Sun prevailing over inward-propagating ones (C. Y. Tu & E. Marsch 1995; B. D. G. Chandran 2008; C. H. K. Chen et al. 2013a; L. Yang et al. 2023). Based on the signatures of magnetic helicity at ion scales observed in solar wind turbulence (J. He et al. 2011; J. J. Podesta & S. P. Gary 2011; D. Telloni et al. 2015, 2020), it has been suggested that KAWs are predominantly outward propagating, indicating that imbalanced kinetic Alfvénic turbulence arises at ion kinetic scales (J. J. Podesta & S. P. Gary 2011; J. He et al. 2012; K. G. Klein et al. 2014). The evolution of such imbalanced turbulence may be influenced by nonlinear interactions among copropagating KAWs or by mechanisms involving the helicity barrier (Y. Voitenko & J. De Keyser 2016; R. Meyrand et al. 2021; J. Squire et al. 2022; J. R. McIntyre et al. 2025; U. Panchal et al. 2025). Moreover, at kinetic scales, the power spectral density of the magnetic field is often characterized by a transition range, occurring at scales comparable to the ion scale, and a kinetic-inertial range between ion and electron scales (S. D. Bale et al. 2005; O. Alexandrova et al. 2009; F. Sahraoui et al. 2009, 2010; C. S. Salem et al. 2012; C. H. K. Chen et al. 2013b; D. Duan et al. 2020, 2021; S. Y. Huang et al. 2021). These observations prompt a fundamental question concerning the evolution of turbulence at kinetic scales: how does imbalanced Alfvénic turbulence evolve toward smaller scales? This phenomenon remains largely unexplored experimentally, primarily due to the absence of reliable diagnostic techniques capable of quantifying wave propagation direction at ion and subion kinetic scales in single-point spacecraft observations.

Using in situ measurements from the Parker Solar Probe (PSP; S. D. Bale et al. 2016; J. C. Kasper et al. 2016), this Letter investigates solar wind turbulence at kinetic scales through a novel magnetic helicity-based method capable of quantifying wave propagation imbalance at kinetic scales. Our findings present, for the first time, observational evidence of the transition from imbalanced to increasingly balanced kinetic Alfvénic turbulence in the solar wind, offering new insights

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into the evolution of turbulence imbalance. The observed imbalance development agrees with key predictions of the helicity barrier model (R. Meyrand et al. 2021; J. Squire et al. 2022), linking our measurements directly to a proposed cascade mechanism near ion kinetic scales.

## 2. Methodology and Data

Quantifying the imbalance in kinetic Alfvénic turbulence at ion and subion scales requires determining the propagation direction of turbulent fluctuations relative to the background magnetic field $B_0$. This propagation information can be extracted from the phase coherence between parallel and perpendicular magnetic field perturbations (see Appendix A), characterized by the magnetic helicity $\sigma_{m_{\perp 2}}$ (see Equation (2) for definition). An accurate diagnosis requires a field-aligned coordinate (FAC) system, with axes defined by $\hat{e}_{\perp 1} = k_{\perp}/k$, $\hat{e}_{\perp 2} = \hat{e}_{\parallel} \times \hat{e}_{\perp 1}$, and $e_{\parallel} \equiv B_0/|B_0|$ constructed from $B_0$ and the wavevector $k = k_{\perp} + k_{\parallel}$, where $k_{\perp}$ and $k_{\parallel}$ are the components perpendicular and parallel to $B_0$, respectively. In this Letter, we develop a diagnostic method within this coordinate system to accurately resolve wave propagation properties in kinetic-scale turbulence. Details of the implementation and the improvements over previous methods are provided in Appendix A.

A central feature of our method is determining the $k_{\perp}$ direction by exploiting a characteristic property of KAWs: their magnetic perturbations exhibit highly elliptical or linear polarization within the plane perpendicular to $B_0$ (J. V. Hollweg 1999; J. S. Zhao et al. 2014; J. Zhao et al. 2022). The dominant component of the magnetic perturbation lies perpendicular to both $B_0$ and $k_{\perp}$, allowing $k_{\perp}$ to be inferred by locating the dominant magnetic perturbation (see Appendix A for implementation details). Once $k_{\perp}$ and the FAC system are determined, the wave propagation direction is inferred from the sign of $\sigma_{m_{\perp 2}}$, where positive values indicate propagation along $B_0$ and negative values indicate propagation counter to $B_0$. Notably, this signature remains consistent in both the solar wind and spacecraft frames.

Because subion Alfvénic turbulence in the near-Sun solar wind extends to frequencies exceeding 100 Hz (T. A. Bowen et al. 2020b), we analyze magnetic turbulence imbalance from 09:30 UTC on 2018 November 4 to 21:20 UTC on 2018 November 7, using high-cadence (~293 Hz) magnetic field data merged from the fluxgate and search coil magnetometers on PSP/FIELDS (T. A. Bowen et al. 2020a). Proton plasma parameters are obtained from Solar Probe Cup (SPC) measurements (A. W. Case et al. 2020) on PSP/SWEAP (J. C. Kasper et al. 2016), and electron parameters are derived from fitting the quasi-thermal noise measurements (M. Moncuquet et al. 2020). Because the cadence of PSP/SPC is lower than ~1 Hz, the imbalance of subion Alfvénic turbulence in the interval of interest cannot be tracked using the conventional cross-helicity parameter.

We apply the Morlet wavelet transform to the time-series magnetic field data to obtain the complex wavelet amplitude $W(f, t)$, where the central frequency at each scale $s$ is given by $f = (w_0 + \sqrt{2 + w_0^2})/(4\pi s)$, with a nondimensional frequency parameter $w_0 = 6$ (C. Torrence & G. P. Compo 1998). From these, we calculate the normalized reduced magnetic helicity defined as follows:

$$ \sigma_{m_{\perp 1} \perp 2} = -2 \operatorname{Im}(W_{\perp 1} W_{\perp 2}^*) / |W|^2, \quad (1) $$

<figure><img src="image_4.png" /><figcaption>**Figure 1.** Plasma environment and magnetic helicity during the time interval from 09:30 UTC on 2018 November 4 to 21:20 UTC on 2018 November 7. (a) The magnitude and three components of the magnetic field $B$ in the spacecraft frame. (b) The solar wind speed $V$ in the spacecraft frame. (c) The distribution of $\sigma_{m_{\perp 1} \perp 2}$. (d) The distribution of $\sigma_{m_{\perp 2} \perp}$. (e) The distribution of $\sigma_{m_{\parallel 1}}$. (f) The distribution of $\theta_{VB}$. The gray curves in (c)–(e) denote the proton cyclotron frequency $f_{CP}$. Because the merged magnetic field data in (a) contain gaps, no corresponding magnetic helicity or $\theta_{VB}$ distributions are available in (c)–(f).</figcaption></figure>

$$ \sigma_{m_{\perp 2}} = -2 \operatorname{Im}(W_{\perp 2} W_{\parallel}^*) / |W|^2, \quad (2) $$

$$ \sigma_{m_{\parallel 1}} = -2 \operatorname{Im}(W_{\parallel} W_{\perp 1}^*) / |W|^2, \quad (3) $$

where “*” denotes the complex conjugate and $|W|^2 = |W_{\perp 1}|^2 + |W_{\perp 2}|^2 + |W|^2$.

Among these helicity components, $\sigma_{m_{\perp 2}}$ quantifies the phase coherence between $W_{\perp 2}$ and $W_{\parallel}$, serving as a clear signature of KAWs. By comparison, $\sigma_{m_{\perp 1} \perp 2}$ captures the coherence between $W_{\perp 1}$ and $W_{\perp 2}$, and is commonly used for diagnosing quasi-perpendicular ion cyclotron waves (ICWs) and fast magnetosonic whistler waves (FMWs; T. A. Bowen et al. 2020c; J. L. Verniero et al. 2020; W. Liu et al. 2023; C. Shi et al. 2024). However, both quasi-perpendicular KAWs and quasi-parallel ICWs and FMWs generally exhibit weak coherence between $W_{\perp 1}$ and $W_{\parallel}$, so this helicity component is not a reliable diagnostic for any of these wave modes.

## 3. Event Overview

Figure 1 provides an overview of the plasma environment and the normalized magnetic helicity distributions during the chosen time interval.

Figures 1(a) and (b) display the measured magnetic field and solar wind velocity in the spacecraft frame, with $\hat{e}_z$ oriented sunward. The magnetic field strength remains relatively constant, while its direction exhibits considerable variability due to the presence of switchbacks. In contrast, the solar wind velocity experiences only minor changes, predominantly streaming outward from the Sun along the radial direction.
<figure><img src="image_5.png" /><figcaption>**Figure 2.** (a) The $\sigma_{m_{\perp 1}\parallel}$ distribution in $\theta_{VB}-f$ space, where the gray curve denotes the Doppler shift frequency of quasi-perpendicular waves with $\rho_p k = 1$. (b) The $\sigma_{m_{\perp 1}\perp 2}$ distribution in $\theta_{VB}-f$ space, where the black curve denotes the Doppler shift frequency of quasi-parallel waves with $\lambda_p k = 0.6$ and $\theta = 170^\circ$. (c) The power spectral density of the magnetic field $P_B$ in four different $\theta_{VB}$ ranges: 75°–90°, 90°–105°, 105°–120°, and 120°–135°. (d) $P_B$ as a function of the normalized wavenumber $Lk^*$, with $L = \rho_p$ (thick curve) and $L = \lambda_p$ (thin curve). In (c) and (d), the upper panels show $P_B$ with $\sigma_{m_{\perp 2}\parallel}$ overlaid, whereas the lower panels display the corresponding spectral index $\alpha$.</figcaption></figure>

Figure 1(c) shows the distribution of $\sigma_{m_{\perp 1}\perp 2}$, which quantifies the polarization of the waves relative to the magnetic field. Positive (red) and negative (blue) $\sigma_{m_{\perp 1}\perp 2}$ correspond to right- and left-handed polarized waves, respectively. The data show a predominance of left-handed polarized waves with $\sigma_{m_{\perp 1}\perp 2} \to -1$ at $f \gtrsim f_{cp}$, where $f_{cp}$ is the proton cyclotron frequency. Intermittent right-handed polarized waves with $\sigma_{m_{\perp 1}\perp 2} \to 1$ are also present. These polarization characteristics are consistent with the presence of ICWs and FMWs (T. A. Bowen et al. 2020c; J. L. Verniero et al. 2020; W. Liu et al. 2023).

Figure 1(d) presents the distribution of $\sigma_{m_{\perp 2}\parallel}$, which is used to identify KAWs, as described in Appendix C. A clear signature of $\sigma_{m_{\perp 2}\parallel} < 0$ appears at frequencies $f \gtrsim 1$ Hz, indicating the presence of KAWs in the near-Sun solar wind, consistent with previous observations (S. Y. Huang et al. 2020; D. Duan et al. 2021). Furthermore, the signature of negative $\sigma_{m_{\perp 2}\parallel}$ varies over time. This temporal variation is closely linked to the angle $\theta_{VB}$ between the local solar wind magnetic field and velocity direction (Figure 1(f)), as shown by the joint distributions of $\sigma_{m_{\perp 2}\parallel}$ and $\theta_{VB}$ in Figure 2(a).

Figure 1(e) illustrates the distribution of $\sigma_{m_{\parallel}1}$. This quantity is typically distributed around zero, suggesting a lack of coherence between $W_{\perp 1}$ and $W_{\parallel}$. This behavior aligns with theoretical predictions that $\sigma_{m_{\parallel}1} \sim 0$ for linear KAWs and quasi-parallel coherent ICWs and FMWs.

As revealed by the helicity distributions in Figures 1(c)–(d), both coherent waves and broadband turbulent fluctuations appear in the observations. To accurately investigate the properties of the turbulence itself, it is essential to isolate

intervals dominated by turbulent fluctuations. To achieve this, we exclude time intervals containing coherent waves, identified by significant $\sigma_{m_{\perp 1}\perp 2}$ values, based on two criteria: (1) the ratio of the upper to lower boundary frequencies of the relevant frequency band exceeds two; and (2) the maximum value of $|\sigma_{m_{\perp 1}\perp 2}|$ exceeds $3\sigma_{\text{std}}$, where $\sigma_{\text{std}}$ denotes the mean standard deviation of $\sigma_{m_{\perp 1}\perp 2}$ within the low-frequency range of 0.08–0.8 Hz. Additionally, we remove adjacent time intervals that are potentially affected by these coherent waves. The resulting occurrence rate of the identified turbulent fluctuations is approximately 40%. Details of the identification procedure are provided in Appendix B.

## 4. Magnetic Helicity and Turbulent Spectra

A crucial parameter influencing the distributions of reduced magnetic helicity and magnetic power spectral density ($P_B$) is the angle $\theta_{VB}$ (J. He et al. 2011; J. J. Podesta & S. P. Gary 2011). The relationships among these parameters for identified turbulent fluctuations are illustrated in Figure 2. To choose a quiet background for analysis, data with an angular deviation greater than 15° between **B** and $\mathbf{B}_0$, referred to as $\theta_{BB0}$, are excluded from consideration (L. D. Woodham et al. 2021). Moreover, some plasma measurements have low sampling rates or are sampled over 20 s intervals, and thus, we discard any data with average plasma sampling rates below 1 Hz (i.e., fewer than 20 samples in a 20 s interval).

Figure 2(a) presents the distribution of $\sigma_{m_{\perp 2}\parallel}$ as a function of $f$ and $\theta_{VB}$. A region of strongly negative helicity (blue) is observed, with its characteristic frequency decreasing as $\theta_{VB}$ deviates from 90°. This frequency dependence provides direct information on the wave normal angle. The observed frequency $f = f_{\text{pl}} + f_D$ consists of the wave frequency $f_{\text{pl}}$ in the solar wind rest frame and the Doppler-shifted frequency $f_D = V \cdot k$. $f_D$ can be further separated into contributions from the parallel and perpendicular wavevectors: $f_{\text{Dk}_\parallel} = V \cdot k_\parallel/(2\pi) = (Vk_\parallel/2\pi)\cos(\theta_{VB})$ and $f_{\text{Dk}_\perp} = V_\perp \cdot k_\perp/(2\pi) = (Vk_\perp/2\pi)\sin(\theta_{VB})(V_\perp \cdot \hat{e}_{\perp\parallel}/V_\perp)$. The observed variation of the characteristic frequency associated with enhanced negative $\sigma_{m_{\perp 2}\parallel}$ is consistent with the $\theta_{VB}$ dependence of $f_{\text{Dk}_\parallel}$ rather than that of $f_{\text{Dk}_\perp}$, as shown by the predicted frequency $f_{\text{Dk}_\parallel} = V_\perp/(2\pi\rho_p)$ for ion-scale waves with $k_\perp\rho_p = 1$ ($\rho_p \simeq 6.4$ km; gray curve in Figure 2(a)). Moreover, the intrinsic wave frequency $f_{\text{pl}}$ depends on the wavevector (e.g., $f_{\text{pl}} \propto k_\parallel$ for Alfvén-mode waves) and thus cannot account for the observed $\theta_{VB}$ dependence. These comparisons imply that waves with enhanced negative $\sigma_{m_{\perp 2}\parallel}$ exhibit highly oblique propagation ($k_\perp \gg k_\parallel$).

The distribution of $\sigma_{m_{\perp 1}\perp 2}$ in Figure 2(b) shows a weak positive helicity signature within the region of enhanced negative $\sigma_{m_{\perp 2}\parallel}$ near $\theta_{VB} \sim 90^\circ$ (Figure 2(a)). These contrasting helicity signatures—enhanced negative $\sigma_{m_{\perp 2}\parallel}$ and weak positive $\sigma_{m_{\perp 1}\perp 2}$—are consistent with the polarization properties predicted for KAWs (S. P. Gary 1986; J. S. Zhao et al. 2014; L. D. Woodham et al. 2021). In addition, a band of weakly negative $\sigma_{m_{\perp 1}\perp 2}$ appears at $\theta_{VB} \gtrsim 90^\circ$, indicative of left-hand wave signatures. The frequencies of these waves increase with $\theta_{VB}$, consistent with Doppler-shifted frequencies $f = V \cos(\theta_{VB} - 170^\circ)k/2\pi$, expected for quasi-parallel waves under the assumption of wave normal angles near 170° and $\lambda_p k \approx 0.6$, where $\lambda_p$ is the proton inertial length. The origin of these observed left-hand waves remains uncertain and may be related either to residual coherent wave activity driven by plasma instabilities (J. L. Verniero et al. 2020; W. Liu et al.
2023) or to turbulent cascades modified by a helicity barrier (R. Meyrand et al. 2021; J. Squire et al. 2022). While weakly coherent waves, potentially ICWs or FMWs, remain in the turbulence dataset, their frequencies are largely confined below 2 Hz, near the transition from the MHD inertial range to the ion transition range (Figure 2(c)). Thus, their contribution to the imbalance analysis at ion and subion scales is likely limited.

In addition to the dependence of $\sigma_{m\perp 1\perp 2}$ and $\sigma_{m\perp 2\parallel}$ on $\theta_{VB}$, $P_B$ also exhibits a dependence on this angle (D. Duan et al. 2021). To illustrate the spectral properties, Figure 2(c) shows $P_B$, defined as $P_B(f, \theta_{VB}) = (2dt/N)\sum_i|W(t_i, \theta_{VB})|^2$, across four $\theta_{VB}$ regimes: 75°–90°, 90°–105°, 105°–120°, and 120°–135°. Here, $dt$ represents the time resolution of the magnetic field data, and $N$ is the number of data points in each $\theta_{VB}$ range. Assuming a power-law form for $P_B$, i.e., $P_B = Cf^\alpha$, the spectral index $\alpha$ at each frequency $f$ is obtained by fitting $P_B$ over the range from $f/2$ to $2f$. The distribution of $\alpha$, shown in Figure 2(c), reveals a pronounced steepening in the spectral slope between approximately 3 and 15 Hz, corresponding to the ion transition range. Based on the $P_B$ in the two angle regimes of 75°–90° and 90°–105°, the average spectral index within the ion transition range is approximately $-3.37 \pm 0.05$, compared to $-1.57 \pm 0.02$ in the inertial range (0.2–1 Hz) and $-2.63 \pm 0.01$ in the higher-frequency (20–100 Hz) kinetic-inertial range.

To explore the wavenumber corresponding to the transition range, Figure 2(d) presents $P_B$ with $\theta_{VB}$ restricted to the range 75°–105°, calculated using the expression (N. Sioulas et al. 2023) $P_B(Lk^*, \theta_{VB}) = (dt/\pi N)\sum_i[V_{\perp}(t_i)|W(t_i, \theta_{VB})|^2/L(t_i)$, where $L$ presents either $\rho_p$ or $\lambda_p$ and $k^* = 2\pi f/V_{\perp}$. Both $P_B(\rho_p k^*)$ and $P_B(\lambda_p k^*)$ display similar power-law behavior, with a spectral transition occurring at $\rho_p k^* \approx 0.4–2.0$ and $\lambda_p k^* \approx 0.7–3.7$. These results indicate that the transition from the inertial to transition range starts at a scale larger than the characteristic ion scale, consistent with the observations reported by T. A. Bowen et al. (2020b).

## 5. Transition from Imbalanced to Balanced Kinetic Alfvénic Turbulence

Because the sign of $\sigma_{m\perp 2\parallel}$ serves as a diagnostic for the wave propagation direction, we can elucidate the degree of imbalance in kinetic Alfvénic turbulence by statistically analyzing the number of data points $N$ in $\sigma_{m\perp 2\parallel} - f$ space, as shown in Figure 3.

Figure 3(a) depicts the distribution of $\bar{N}(\sigma_{m\perp 2\parallel}, f)$, defined as $N(\sigma_{m\perp 2\parallel}, f)$ normalized by its maximum at each frequency $f$. Two primary features are evident: (1) as $f$ increases, the location of the maximum $\bar{N} = 1$ rapidly shifts toward lower values of $\sigma_{m\perp 2\parallel}$ within $\sim 1–10$ Hz, primarily arising in the transition range; and (2) in the kinetic-inertial range, $\bar{N}$ exhibits a broader distribution of $\sigma_{m\perp 2\parallel}$ compared to that in the transition range, suggesting a tendency toward balanced signatures at smaller scales.

Figure 3(b) shows the PDF of $N(\sigma_{m\perp 2\parallel}, f)$ at four representative frequencies: 0.3, 3.1, 10.5, and 50.1 Hz. In the inertial range ($f = 0.3$ Hz), the PDF roughly follows a Gaussian distribution centered around zero, indicating that $\sigma_{m\perp 2\parallel}$ behaves randomly. In the transition range ($f = 3.1$ and 10.5 Hz), the PDFs are dominated by negative $\sigma_{m\perp 2\parallel}$, reflecting a significant imbalance. At 50.1 Hz, within the kinetic-inertial range, the PDF becomes broad and flat, indicating a weakened

imbalance signature and a trend toward more balanced turbulence.

To quantify the degree of imbalance, we calculate the ratio $R$ of the amount of data with negative to those with positive $\sigma_{m\perp 2\parallel}$:

$$R(f) = \frac{N_t(f, \sigma_{m\perp 2\parallel} < 0)}{N_t(f, \sigma_{m\perp 2\parallel} > 0)},$$

as illustrated in Figure 3(c). The value of $R$ increases with frequency, peaks ($R \sim 6.5$) near $f \sim 5$ Hz, and then decreases toward one at higher frequencies. Since the transition range spans $f = 3–15$ Hz, the position of this maximum roughly coincides with the onset of the transition range. The corresponding frequency $f \sim 5$ Hz translates to spatial scales of $\rho_p k \sim 0.6$ and $\lambda_p k \sim 1.3$, which lie near the characteristic ion scale.

Figure 3(d) presents the ratio $R$ between the amount of data with negative and positive $\sigma_{m\perp 2\parallel}$ in $k_{\perp} - k_{\parallel}$ space. Using the angle $\phi$ between $k_{\perp}$ and $V_{\perp}$, $k_{\perp}$ is determined via $k_{\perp} = 2\pi f \sin(\theta_{VB})/(V \cos(\phi))$, while $k_{\parallel}$ is calculated as $k_{\parallel} = 2\pi f \cos(\theta_{VB})/V$. At scales with $\rho_p k_{\perp} \lesssim 0.2$, the maximum $R$ occurs at $\lambda_p k_{\parallel} \sim 0.3$. As $\rho_p k_{\perp}$ increases beyond 0.2, the region of maximum $R$ closely follows the core of the contour lines of $P_B$ (the gray curves in Figure 3(d)) in wavenumber space. At subion scales with $\rho_p k_{\perp} \gtrsim 10$, the degree of imbalance tends to diminish, consistent with a trend toward more balanced characteristics at smaller scales.

## 6. Conclusions and Discussion

This Letter investigates kinetic Alfvénic turbulence in the near-Sun solar wind, utilizing magnetic field measurements from the PSP. To effectively identify wave modes, we introduce a magnetic helicity method that employs FAC system defined by the background magnetic field and the wavevector. Our primary focus is on the reduced magnetic helicity $\sigma_{m\perp 2\parallel}$, which quantifies the coherence between fluctuations in the parallel magnetic field and principal perpendicular magnetic field fluctuations. This methodology successfully identifies signatures of KAWs. Our analysis reveals the prevalence of KAWs characterized by negative values of $\sigma_{m\perp 2\parallel}$, particularly when the observed wave frequency exceeds the proton cyclotron frequency.

Our examination of the PDF($\sigma_{m\perp 2\parallel}, f$) and the ratio $R(f)$ reveals two distinct trends as wave frequency increases. First, the PDF($\sigma_{m\perp 2\parallel}, f$) broadens with frequency. Second, $R(f)$ initially rises, reaches a peak, and then decreases. These trends reflect the scale-dependent evolution of KAW propagation: an initial dominance of outward-propagating waves cascading to smaller scales, followed by a relative increase in inward-propagating waves, leading to a more balanced state at smaller scales. This scenario is further supported by the ratio of outward to inward waves in the two-dimensional wavenumber space. This ratio distribution also reveals a notable feature of ion-scale turbulence: the maximum ratio closely aligns with the cascade direction inferred from the $P_B$ distribution in wavenumber space.

These findings prompt two critical questions regarding our understanding of solar wind turbulence at and below the ion scale.

The first question addresses why the imbalance increases toward the transition range. Our observations of imbalance
<figure><img src="image_6.png" /><figcaption>**Figure 3.** (a) The number of $N(\sigma_{m\perp 2}, f)$ data points normalized by the maximum $N(\sigma_{m\perp 2}, f)$ at each $f$, denoted by $\bar{N}(\sigma_{m\perp 2}, f)$. (b) The probability distribution function (PDF) of $N(\sigma_{m\perp 2}, f)$, denoted by $PDF(\sigma_{m\perp 2}, f)$, at four typical frequencies: $f = 0.3, 3.1, 10.5$, and $50.1$ Hz. (c) The ratio $R(f)$ between the total amount of data points with negative and positive $\sigma_{m\perp 2}$, $R(f) = N_-(f, \sigma_{m\perp 2} \ll 0)/N_+(f, \sigma_{m\perp 2} \gg 0)$. (d) The ratio $R$ between the amount of data with negative and positive $\sigma_{m\perp 2}$ in $k_{\perp}-k_{\parallel}$ space, where the gray curves are the contour lines of $P_B$. The data in (a)–(c) are limited to $|\theta_{VB} - 90^{\circ}| < 45^{\circ}$, and the data in (d) are limited to $|\sigma_{m\perp 2}|/|\sigma_{m\parallel 1}| > 2$; all data are subject to the condition $|W_{\parallel}|^2/(|W_{\perp 1}|^2 + |W_{\perp 2}|^2) < 1$.</figcaption></figure>

development are consistent with the helicity barrier model (R. Meyrand et al. 2021; J. Squire et al. 2022). The conservation of generalized helicity selectively suppresses the cascade of outward-propagating waves near and beyond the ion scale, leading to a pileup of waves at scales approaching $\rho_p k_{\perp} \sim 1$ (J. Squire et al. 2022). This pileup and associated imbalance increase are evident in our observations. Moreover, as shown by the $P_B$ distribution in wavenumber space, the turbulent cascade preferentially channels energy toward $\lambda_p k_{\parallel} \sim 1$ and $\rho_p k_{\perp} \sim 1$, consistent with turbulent behavior predicted by numerical simulations of imbalanced Alfvénic turbulence influenced by the helicity barrier (J. Squire et al. 2023; M. F. Zhang et al. 2025). Additionally, although previous studies have suggested that plasma instabilities can generate KAWs in the solar wind (O. Alexandrova et al. 2013; P. Hellinger et al. 2015), direct observational evidence remains limited, and their potential contribution as a complementary source of KAWs near the ion scale warrants further investigation.

The second question involves the evolution of turbulence within the kinetic-inertial range. Our observations reveal that turbulence in this range remains weakly imbalanced down to subion scales at $\rho_p k_{\perp} \sim 10$. This residual imbalance may indicate that energy transfer at kinetic scales is influenced by nonlocal interactions, such as couplings with larger-scale fluctuations or nonlocal spectral transfer processes (Y. Voitenko & M. Goossens 2005; Y. Lin et al. 2012). Moreover, differential damping rates for outward- and inward-propagating KAWs in the near-Sun

solar wind, potentially affected by the presence of ion beam populations, may contribute to sustaining the weak imbalance. Turbulence in the kinetic-inertial range may also be influenced by additional processes, including KAW generation by plasma instabilities and the presence of coherent structures (A. Vinogradov et al. 2024). Our observations further show that the maximum imbalance tends to align with the cone in the $P_B$ distribution in wavenumber space at $\rho_p k_{\perp} \gtrsim 1$, consistent with predictions from numerical simulations (J. Squire et al. 2022, 2023; M. F. Zhang et al. 2025). This alignment may indicate a preferential cascade pathway at subion scales, although the physical mechanisms governing this behavior remain to be fully understood.

In addition, the time interval analyzed contains switchbacks spanning a range of timescales, characterized by changes (including reversals) of the radial magnetic field direction (T. Dudok de Wit et al. 2020). We therefore examine the imbalance distribution during magnetic field reversal intervals, as shown in Appendix C. We observe the same transition from imbalanced to balanced turbulence within the reversal regions. This indicates that such field reversals do not change the wave propagation direction relative to the local background magnetic field, as illustrated by the schematic in Appendix C. However, magnetic field reversals can change the apparent dominance of outward- versus inward-propagating Alfvénic fluctuations in the spacecraft frame, as reflected by the sign change of the cross helicity (M. D. McManus et al. 2020). Moreover, the reduced cross helicity within switchbacks (S. Bourouaine et al. 2020) may lead to an imbalance evolution that quantitatively differs from that in the surrounding solar wind, since high cross helicity can induce a steeper transition range than in a low-cross-helicity environment (J. R. McIntyre et al. 2025). A systematic comparison of turbulence imbalance inside switchbacks and in the ambient solar wind will be conducted in a future study.

In conclusion, the observations presented in this Letter emphasize the need for refined theoretical frameworks to accurately model the solar wind turbulence at kinetic scales.

## Acknowledgments

This study is supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (grant No. XDB0560000) and the National Key R&D Program of China 2022YFF0503000 (2022YFF0503003). Z.J.S. is grateful for the support from NSFC 42374196. T.D.dW. acknowledges support from CNES. The FIELDS and SWEAP instruments were designed and built under NASA Contract No. NNN06AA01C.

## Appendix A
### Three Magnetic Helicity Methods

In this appendix, we introduce three methods for calculating the reduced magnetic helicity: the radial–tangential–normal (RTN) coordinate method (Section A.1), the $B_0-V_0$ FAC method (Section A.2), and the $B_0-k$ FAC method developed in this Letter (Section A.3). We also compare the results from the two FAC methods to demonstrate the advantages of our method (Section A.4).
### A.1. Method I: Radial–Tangential–Normal Coordinate Method

Using the two magnetic field components $B_T$ and $B_N$ in RTN coordinates, W. H. Matthaeus & M. L. Goldstein (1982) first employed magnetic helicity to analyze solar wind turbulence. The reduced magnetic helicity is defined as $\sigma_{mTN}(f) = 2 \operatorname{Im}(W_T W_N^*) / |W|^2$, where $W_T$, $W_N$, and $|W|$ represent the complex amplitudes of $B_T$, $B_N$, and the magnitude of the complex wave amplitude at frequency $f$, respectively.

The distribution of $\sigma_{mTN}$ as a function of the angle $\theta_{VB}$ is widely used to diagnose ICWs and KAWs in the solar wind (J. He et al. 2011, 2012; J. J. Podesta & S. P. Gary 2011; K. G. Klein et al. 2014; S. Y. Huang et al. 2020), where $\theta_{VB}$ is the angle between the solar wind magnetic field $\mathbf{B}$ and the bulk velocity $\mathbf{V}$. Positive (negative) values of $\sigma_{mTN}$ in the outward (inward) magnetic field sectors, particularly near $\theta_{VB} \sim 90^\circ$, are characteristic signatures of KAWs (J. He et al. 2011, 2012; J. J. Podesta & S. P. Gary 2011; K. G. Klein et al. 2014; S. Y. Huang et al. 2020). By contrast, finite $\sigma_{mTN}$ values near $\theta_{VB} \to 0^\circ$ or $180^\circ$ indicate the presence of quasi-parallel coherent ion-scale waves.

### A.2. Method II: $B_0-V_0$ Field-aligned Coordinate Method

To separate the signatures of highly oblique KAWs from quasi-parallel coherent ion-scale waves, L. D. Woodham et al. (2021) developed an identification method based on magnetic helicity defined in an FAC system, constructed using the background magnetic field $B_0$ and solar wind velocity $V_0$. The three unit axes are defined as $\hat{e}_{\perp 1}' \equiv \hat{e}_{\perp 2}' \times \hat{e}_{\parallel}$, $\hat{e}_{\perp 2}' \equiv (B_0 \times V_0)/(|B_0||V_0|)$, and $\hat{e}_{\parallel} \equiv B_0/|B_0|$. The reduced magnetic helicity components are then defined as

$$ \sigma_{m\perp 1'\perp 2'} = -2 \operatorname{Im}(W_{\perp 1}' W_{\perp 2}') / |W|^2, \quad (A1) $$

$$ \sigma_{m\perp 2'\parallel} = -2 \operatorname{Im}(W_{\perp 2}' W_{\parallel}^*) / |W|^2, \quad (A2) $$

$$ \sigma_{m\parallel\perp 1'} = -2 \operatorname{Im}(W_{\parallel} W_{\perp 1}') / |W|^2, \quad (A3) $$

where $|W|^2 = |W_{\perp 1}'|^2 + |W_{\perp 2}'|^2 + |W_{\parallel}|^2$.

Using $\sigma_{m\perp 1'\perp 2'}$ and $\sigma_{m\perp 2'\parallel}$, quasi-parallel coherent ion-scale waves and KAWs can be separately diagnosed (L. D. Woodham et al. 2021). Specifically, $\sigma_{m\perp 2'\parallel}$ is suggested as a reliable indicator for identifying KAWs. In addition, $\sigma_{m\parallel\perp 1'}$ also captures the characteristic signatures of KAWs.

### A.3. Method III: $B_0-k$ Field-aligned Coordinate Method

Because KAWs are highly elliptical or even linearly polarized in the perpendicular direction (J. V. Hollweg 1999; J. S. Zhao et al. 2014; J. Zhao et al. 2022), their major perpendicular magnetic perturbations do not always coincide with either $W_{\perp 1}'$ or $W_{\perp 2}'$. Consequently, both $\sigma_{m\perp 1'\parallel}$ and $\sigma_{m\perp 2'\parallel}$ tend to underestimate the coherence between the parallel and the major perpendicular magnetic field perturbations of KAWs.

To address this limitation in Method II, we develop a new wave diagnostic method based on an FAC system defined by the wavevector $\mathbf{k}$ and $B_0$. In this system, the three axes are given by $\hat{e}_{\perp 1} \equiv \hat{e}_{\perp 2} \times \hat{e}_{\parallel}$, $\hat{e}_{\perp 2} \equiv (B_0 \times \mathbf{k}) / (|B_0||\mathbf{k}|)$, and $\hat{e}_{\parallel}$. The major magnetic perturbation of KAWs, $W_{\perp}$, resides along the $\hat{e}_{\perp 2}$ direction. Therefore, the coherence between $W_{\perp}$ and $W_{\parallel}$ serves as a reliable parameter for identifying KAWs.

<figure><img src="image_7.png" /><figcaption>**Figure 4.** (a) The FAC system defined by the background magnetic field $B_0$ and solar wind velocity $V_0$, where $\hat{e}_{\parallel} \equiv B_0/|B_0|$, $\hat{e}_{\perp 1}' \equiv \hat{e}_{\perp 2}' \times \hat{e}_{\parallel}$, and $\hat{e}_{\perp 2}' \equiv B_0 \times V_0 / (|B_0||V_0|)$. (b) The rotation from $(\hat{e}_{\perp 1}', \hat{e}_{\perp 2}')$ into $(\hat{e}_{\perp 1}, \hat{e}_{\perp 2})$, where $\phi$ represents the rotation angle between the two sets of coordinates.</figcaption></figure>

In the super-Alfvénic solar wind, the variability of the perpendicular wavevector $\mathbf{k}_{\perp}$ in the spacecraft frame exhibits the following characteristic property. Because low-frequency MHD- and kinetic-scale Alfvén-mode waves are convected by the super-Alfvénic flow, the observed wave propagation in the plane defined by the magnetic field and flow directions is dominated by advection, while the wave propagation perpendicular to this plane is unaffected. As a result, the component of $\mathbf{k}_{\perp}$ along the $\hat{e}_{\perp 1}'$ direction remains fixed in sign, whereas its component along the $\hat{e}_{\perp 2}'$ direction can be either parallel or antiparallel to this axis. This variability implies that the angle $\phi$ between $\mathbf{k}_{\perp}$ and $\hat{e}_{\perp 1}'$ can range from $-\pi/2$ to $\pi/2$ (see Figure 4).

Given the range of $\phi$, we determine $\mathbf{k}_{\perp}$ based on the wave polarization properties in the $\hat{e}_{\perp 1}'-\hat{e}_{\perp 2}'$ plane. Using the relations $W_{\perp 1} = W_{\perp 1}' \cos(\phi) + W_{\perp 2}' \sin(\phi)$ and $W_{\perp 2} = -W_{\perp 1}' \sin(\phi) + W_{\perp 2}' \cos(\phi)$ in this plane, the direction of $\mathbf{k}_{\perp}$ is identified by finding the value of $\phi$ that maximizes $|W_{\perp 2}|$ as $\phi$ varies from $-\pi/2$ to $\pi/2$. Simultaneously, we obtain the two perpendicular magnetic field perturbation components in the $(\hat{e}_{\perp 1}, \hat{e}_{\perp 2}, \hat{e}_{\parallel})$ coordinate system.

This approach may not accurately characterize the perpendicular wavevector for quasi-parallel waves, as the two perpendicular magnetic field components are approximately equal in magnitude. However, their polarization characteristics (left hand or right hand) remain consistent across both sets of FACs.

### A.4. A Comparison of the Magnetic Helicity Distributions from the Two Field-aligned Coordinate Methods

Here we compare the reduced magnetic helicity distributions obtained from the two FAC methods.

To calculate the magnetic field perturbations at each frequency and time, we apply a Morlet wavelet transform to the time-series magnetic field data merged from the fluxgate and search coil magnetometers on PSP/FIELDS (T. A. Bowen et al. 2020a). Narrowband noise induced by PSP's reaction wheels (T. A. Bowen et al. 2020c) is removed using short-time Fourier transform techniques. Because the fast Fourier transform in the wavelet analysis assumes waveforms in the form $e^{i\omega t}$, which differs from the conventional form $e^{-i\omega t}$, we apply a conjugate transformation to the complex wavelet amplitude so that it follows the standard definition of wave evolution as $e^{-i\omega t}$. In addition, we average the wavelet amplitudes over eight periods to compute the coherence at each scale before calculating the magnetic helicity.

We construct the $B_0-V_0$ FAC system $(\hat{e}_{\perp 1}', \hat{e}_{\perp 2}', \hat{e}_{\parallel})$, in which the local background magnetic field $B_0$ and solar wind velocity
<figure><img src="image_8.png" /><figcaption>Figure 5. Overview of the reduced magnetic helicity distributions obtained from the two FAC methods. Top panel: the magnitude and the three components of the magnetic field in the spacecraft frame. Second to fourth panels: distributions of $\sigma_{m_{\perp 1}^{\prime 1} \perp 2}$, $\sigma_{m_{\perp 2}^{\prime 2}}$, and $\sigma_{m_{\parallel 1}^{\prime 1}}$, respectively, from Method II. Fifth to seventh panels: distributions of $\sigma_{m_{\perp 1}^{\prime 1} \perp 2}$, $\sigma_{m_{\perp 2}^{\prime 2}}$, and $\sigma_{m_{\parallel 1}^{\prime 1}}$, respectively, from Method III.</figcaption></figure>

$V_0$ are determined following J. J. Podesta & S. P. Gary (2011)

$$ \mathbf{B}_0(t_i) = \frac{1}{A} \sum_{t_j=t_i-3s}^{t_i+3s} \mathbf{B}(t_j) \exp \left[ -\frac{(t_j-t_i)^2}{2s^2} \right], \quad (A4) $$

$$ V_0(t_i) = \frac{1}{A} \sum_{t_j=t_i-3s}^{t_i+3s} V(t_j) \exp \left[ -\frac{(t_j-t_i)^2}{2s^2} \right], \quad (A5) $$

where $A = \sum_{t_j=t_i-3s}^{t_i+3s} \exp \left[ -\frac{(t_j-t_i)^2}{2s^2} \right]$ is a normalization factor, and the wavelet scale $s$ ranges from $s \approx 0.007$ s to $s = 20$ s, with a total of 47 points in logarithmic space. Given the distinct temporal resolutions of the magnetic field and plasma measurements, $V_0$ is resampled to match the temporal resolution of the magnetic field data. In addition to defining the $\mathbf{B}_0-V_0$ FAC system, the scale-dependent backgrounds $\mathbf{B}_0$ and $V_0$ are used to determine the angle between the solar wind velocity and the magnetic field, $\theta_{VB} = V_0 \cdot \mathbf{B}_0/(|V_0||\mathbf{B}_0|)$, and the angle between the instantaneous magnetic field and the background field, $\theta_{BB_0} = \mathbf{B} \cdot \mathbf{B}_0/(|\mathbf{B}||\mathbf{B}_0|)$. Using the scale-dependent backgrounds $\mathbf{B}_0$ and $V_0$ significantly improves the identification of the wave direction in the presence of rapid variations of the magnetic field and solar wind velocity, as expected during intervals with switchbacks.

The wavelet-transformed magnetic field perturbations are first reconstructed in the $\mathbf{B}_0-V_0$ FAC system, yielding the complex wavelet amplitude $W(f, t)$ at frequency $f = (w_0 + \sqrt{2 + w_0^2})/4\pi s$

(C. Torrence & G. P. Compo 1998). We then apply the $\mathbf{B}_0-\mathbf{k}$ FAC method to obtain the magnetic field perturbations in the FAC system ($\hat{\mathbf{e}}_{\perp 1}$, $\hat{\mathbf{e}}_{\perp 2}$, $\hat{\mathbf{e}}_{\parallel}$).

Using these FAC magnetic field perturbations, the distributions of the three reduced magnetic helicity components are shown in Figure 5. We find that the distributions of $\sigma_{m_{\perp 1}^{\prime 1} \perp 2}$ and $\sigma_{m_{\perp 1}^{\prime 1} \perp 2}$ are nearly identical, whereas the other four helicity components, which are relevant for diagnosing KAWs, differ significantly.

To further compare the results from the two FAC methods, PDFs of the reduced magnetic helicity components are presented in Figure 6.

The PDFs of $\sigma_{m_{\perp 1}^{\prime 1} \perp 2}$ and $\sigma_{m_{\perp 1}^{\prime 1} \perp 2}$ at $f = 5.3$ Hz for all data are shown in Figure 6(a). The two PDFs are consistent, indicating that both FAC methods play an equivalent role in quantifying quasi-parallel coherent waves.

To highlight the differences in quantifying KAWs, we select turbulent wave data using the method described in the next section. The four PDFs at $f = 5.3$ Hz and 50.1 Hz differ substantially, as shown in Figures 6(b) and (c). In particular, the PDFs of $\sigma_{m_{\perp 2}^{\prime 2}}$ are significantly higher than those of $\sigma_{m_{\perp 2}^{\prime 2}}$ when their absolute values exceed 0.5. Moreover, the PDFs of $\sigma_{m_{\perp 1}^{\prime 1}}$ peak around $\sigma_{m_{\perp 1}^{\prime 1}} = 0$, in agreement with theoretical predictions.

These comparisons between the two FAC methods demonstrate that our improved approach provides a more accurate quantification of KAW signatures in solar wind turbulence.
<figure><img src="image_9.png" /><figcaption>**Figure 6.** Comparison of the two FAC methods based on the PDFs of the reduced magnetic helicity. (a) PDFs of $\sigma_{m_{\perp 1 \perp 2}}$ (red) and $\sigma_{m_{\perp 1 \perp 2}'}$ (black) at $f = 5.3$ Hz for all data. (b) PDFs of $\sigma_{m_{\perp 2 \parallel}}$ (red), $\sigma_{m_{\perp 2 \parallel}}$ (blue), $\sigma_{m_{\parallel 1}}$ (purple), and $\sigma_{m_{\parallel 1}'}$ (yellow) at $f = 5.3$ Hz for the identified turbulent waves. (c) The same four PDFs at $f = 50.1$ Hz for the identified turbulent waves.</figcaption></figure>

## Appendix B
### Identification of Turbulent and Coherent Waves

During the time interval of interest, both turbulent waves and quasi-parallel coherent waves are present, as shown in Figures 1 and 5. To isolate the intervals dominated by turbulent waves, a statistical identification method based on the signature of $\sigma_{m_{\perp 1 \perp 2}}$ is developed for this purpose.

The primary motivation for our identification method is derived from the distributional properties of the PDFs of $\sigma_{m_{\perp 1 \perp 2}}$, as shown in Figure 7. To better capture the statistical characteristics, the values of $\sigma_{m_{\perp 1 \perp 2}}$ are averaged over a smoothed frequency band between $f_L$ and $f_U$, where $f_U/f_L = 2$.

Furthermore, to quantitatively characterize these PDFs, we calculate their statistical moments, including the mean value $\bar{\sigma} = \sum_{i=1}^N \sigma_i / N$ in Figure 7(b), the standard deviation $\sigma_{\text{std}} = \sqrt{\sum_{i=1}^N (\sigma_i - \bar{\sigma})^2 / N}$ in Figure 7(c), the skewness $S = \frac{1}{N} \sigma_{\text{std}}^3 \sum_{i=1}^N (\sigma_i - \bar{\sigma})^3$ in Figure 7(d), and the kurtosis $K = \frac{1}{N} \sigma_{\text{std}}^4 \sum_{i=1}^N (\sigma_i - \bar{\sigma})^4$ in Figure 7(e), where $N$ denotes the number of data points. Among these moments, the skewness quantifies the asymmetry of the distribution, with positive (negative) values indicating a longer tail to the right (left) of the mean, while the kurtosis describes the peakedness and tail heaviness of the distribution.

The PDFs of $\sigma_{m_{\perp 1 \perp 2}}$ in Figure 7 exhibit approximately Gaussian shapes in two distinct frequency ranges: $f = 0.08-0.8$ Hz and $10-100$ Hz. In both ranges, the skewness $S$ is close to zero, as expected for an ideal Gaussian distribution. The kurtosis is approximately 2.8-3.0 in the low-frequency range ($f = 0.08-0.8$ Hz), slightly below the kurtosis value of three expected for a Gaussian distribution, while in the high-frequency range ($f = 10-100$ Hz) the kurtosis gradually increases toward three as the frequency increases.

However, a distinct departure from Gaussian behavior is observed in the intermediate-frequency range $f = 0.8-10$ Hz. Specifically, the distributions exhibit pronounced heavy tails at large values of $|\sigma_{m_{\perp 1 \perp 2}}|$, predominantly attributable to the presence of quasi-parallel coherent waves such as ICWs and FMWs. If these coherent wave events are excluded from the dataset within this frequency range, one would expect the resulting PDFs to resemble those observed in the low- and

high-frequency ranges, displaying Gaussian-like features consistent with stochastic fluctuations.

Based on this observation, we develop a systematic procedure to process the $\sigma_{m_{\perp 1 \perp 2}}$ data and identify coherent wave events. First, we exclude data points with $|\sigma_{m_{\perp 1 \perp 2}}|$ smaller than the threshold $\sigma_t$, defined as $\bar{\sigma}_{LR} \approx 0.26$, which corresponds to the mean value of $\sigma_{\text{std}}$ within the low-frequency range (0.08-0.8 Hz). This step effectively removes weak, incoherent fluctuations. Next, at each time step, we determine the widest continuous frequency band containing the remaining data and identify key wave parameters, including the lower and upper frequency boundaries ($f_{LB}$ and $f_{UB}$, respectively) and the frequency $f_{\text{max}}$ at which the maximum $|\sigma_{m_{\perp 1 \perp 2}}|$ occurs (denoted as $\sigma_{\text{max}}$). Finally, to classify strongly coherent waves, we apply a helicity criterion of $|\sigma_{\text{max}}| > 3\bar{\sigma}_{LR}$ and the frequency band criterion of $f_{UB}/f_{LB} > 2$.

In the regions adjacent to the identified coherent waves, the measured helicity signatures of nearby waves can be affected by the coherent waves due to the finite width of the wavelet window used in the wavelet transform. Moreover, waves within these adjacent regions may exhibit moderate coherence, either as a result of contamination from nearby coherent waves or because they possess weak intrinsic coherence. To address this issue, we define an exclusion region around each coherent wave event, spanning from $t_{\text{coh}} - \Delta t$ to $t_{\text{coh}} + \Delta t$, where $t_{\text{coh}}$ denotes the time location of the coherent wave and $\Delta t = 1/f_{\text{max}}$ represents the characteristic timescale associated with the effect of the coherent wave. By excluding data within these intervals, we ensure that the remaining dataset for turbulent wave analysis is minimally affected by adjacent coherent wave signatures.

By applying the aforementioned procedures, we classify the dataset into coherent waves, waves in regions adjacent to coherent waves (termed adjacent waves), and turbulent waves. The resulting distribution of wave populations comprises 16% coherent waves, 44% adjacent waves, and 40% turbulent waves.

The validation of the identified results is shown in Figures 8 and 9.

Figure 8 presents the statistical properties of the three identified types of waves. Figures 8(a) and (b) display the PDFs of the identified turbulent, coherent, and adjacent waves at two representative frequencies, 2.2 Hz and 5.3 Hz, within the intermediate-frequency range. The PDFs of the coherent
<figure><img src="image_10.png" /><figcaption>**Figure 7.** The PDFs of $\sigma_{m_{\perp 1}\perp 2}$ and their characteristic statistical parameters for all waves. (a) The PDFs of $\sigma_{m_{\perp 1}\perp 2}$ at frequencies between 0.08 and 84.31 Hz, where each PDF is multiplied by a factor of $0.8^{[4 \times \log_2(84.31/f)]}$ to improve visualization. (b) The mean value $\bar{\sigma}$, (c) the standard deviation $\sigma_{\text{std}}$, (d) the skewness $S$, and (e) the kurtosis $K$ corresponding to each PDF.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 8.** The PDFs of $\sigma_{m_{\perp 1}\perp 2}$ for the identified coherent (purple), adjacent (blue), and turbulent (red) waves, as well as for all waves (yellow) at (a) $f = 2.2$ Hz and (b) $f = 5.3$ Hz. Panels (c)–(f) show the same parameters as those shown in Figure 7, but for the identified coherent (purple), adjacent (blue), and turbulent (red) waves, along with all waves (yellow).</figcaption></figure>

waves exhibit pronounced deviations from a Gaussian distribution, with significant values of $\sigma_{m_{\perp 1}\perp 2}$. In contrast, the PDFs of turbulent waves closely follow a Gaussian-like shape. The PDFs for adjacent waves show intermediate behavior between these two populations. The corresponding statistical

moments, shown in Figures 8(c)–(f), further support these observations. The turbulent waves maintain relatively stable and Gaussian-consistent moment values across the entire frequency range, while the coherent waves display pronounced enhancements in standard deviation, skewness, and kurtosis
<figure><img src="image_12.png" /><figcaption>Figure 9. The distributions of the reduced magnetic helicities $\sigma_{m_{\perp 1}\perp 2}$ and $\sigma_{m_{\perp 2}\parallel}$ for coherent waves (a)–(b), adjacent waves (c)–(d), and turbulent waves (e)–(f).</figcaption></figure>

near their dominant frequencies. The adjacent wave data exhibit moderate departures from Gaussianity.

Figure 9 shows the distributions of the two reduced magnetic helicities for the identified categories of coherent, adjacent, and turbulent waves. Coherent waves in Figures 9(a) and (b) exhibit localized, intermittent high-$\sigma_{m_{\perp 1}\perp 2}$ signatures over broad frequency ranges. Adjacent waves in Figures 9(c) and (d) display moderate $\sigma_{m_{\perp 1}\perp 2}$, while turbulent waves in Figures 9(e) and (f) maintain stochastic $\sigma_{m_{\perp 1}\perp 2}$ distributions. These results demonstrate the effectiveness of the wave identification criteria in distinguishing the three wave categories.

## Appendix C

### Theoretical Predictions for the Reduced Magnetic Helicity of Kinetic Alfvén Waves

We use the two-fluid model to examine the magnetic helicity $\sigma_{m_{\perp 2}\parallel}$ of KAWs. The plasma and magnetic field parameters used are as follows: the number density $n_p = n_e = 385 \text{ cm}^{-3}$, the proton temperature $T_p = 15 \text{ eV}$, the electron temperature $T_e = 29 \text{ eV}$, and the magnetic field strength $B_0 = 86 \text{ nT}$. The wave properties in the plasma rest frame and in the two typical spacecraft frames are illustrated in Figures 10, 11, and 12. We note that the wave properties predicted by the plasma kinetic model under these conditions are qualitatively consistent with those from the two-fluid model. A key advantage of using the two-fluid model is that it allows us to easily identify waves with the conventional definition of wave frequency $\omega$ being positive.

In the plasma rest frame, the magnetic helicity $\sigma_{m_{\perp 2}\parallel}$ is positive for KAWs propagating along $B_0$ and negative for those propagating against $B_0$, as depicted in Figure 10. Positive $\sigma_{m_{\perp 2}\parallel}$ has been previously discussed in the literature (S. P. Gary 1986; J. S. Zhao et al. 2014). The differing signs

of $\sigma_{m_{\perp 2}\parallel}$ for KAWs traveling in opposite directions can be understood through the relation

$$ \frac{dB_{\perp 1}}{dB_{\parallel}} = -\frac{k_{\parallel}}{k_{\perp 1}}, $$

derived from the condition $\nabla \cdot dB = 0$, where $dB = dB_{\perp 1}\hat{e}_{\perp 1} + dB_{\perp 2}\hat{e}_{\perp 2} + dB_{\parallel}\hat{e}_{\parallel}$. Here, $dB_{\perp 1}$, $dB_{\perp 2}$, and $dB_{\parallel}$ represent the magnetic perturbations of KAWs in the $\hat{e}_{\perp 1}$, $\hat{e}_{\perp 2}$, and $\hat{e}_{\parallel}$ directions, respectively. For KAWs propagating along ($k_{\parallel} > 0$) and against ($k_{\parallel} < 0$) the background magnetic field, $dB_{\parallel}$ exhibits different correlations with $dB_{\perp 1}$. Since $dB_{\perp 1}$ shares the same correlation with $dB_{\perp 2}$ for counterpropagating KAWs, the correlation between $dB_{\perp 2}$ and $dB_{\parallel}$ varies based on the propagation direction of the KAWs.

Given that the solar wind speed is much greater than the local Alfvén speed, both outward and inward low-frequency waves are convected outward by the solar wind flow.

In the outward sector of the solar wind magnetic field, where the magnetic field points away from the Sun, outward KAWs maintain their propagation direction in the spacecraft frame. This implies that the correlations of $dB_{\perp 2}$ with $dB_{\parallel}$ and $dB_{\perp 1}$ remain the same. Conversely, for inward KAWs in the plasma rest frame, their propagation direction is reversed in the spacecraft frame, altering the correlation between $dB_{\parallel}$ and $dB_{\perp 1}$. Furthermore, this change in wave direction results in a reversed correlation between $dB_{\perp 1}$ and $dB_{\perp 2}$, as illustrated in Figure 11. Consequently, the correlation between $dB_{\perp 2}$ and $dB_{\parallel}$ for inward KAWs remains consistent in both the plasma rest frame and the spacecraft frame.

The changes in the correlations among the three magnetic field components of KAWs in the inward sector (where the solar wind magnetic field points toward the Sun) mirror those in the outward sector, as shown in Figure 12.
<figure><img src="image_13.png" /><figcaption>**Figure 10.** Properties of KAWs with wave normal angles $\theta = 80^{\circ}$ (red) and $100^{\circ}$ (blue) in the plasma rest frame. (a) The wave frequency $\omega$ normalized by the proton cyclotron frequency $\omega_{cp}$, where the dotted curves show eigenmode waves beyond the two KAWs of interest. (b) The ratio of $dV_{p\perp 2}/V_A$ to $dB_{\perp 2}/B_0$. (c)–(d) The argument and absolute value of $dB_{\perp 1}/dB_{\perp 2}$. (e)–(f) The argument and absolute value of $dB_{\parallel}/dB_{\perp 2}$. (g)–(h) $\sigma_{m\perp 1\perp 2}$ and $\sigma_{m\perp 2\perp}$.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 11.** Properties of KAWs in the spacecraft frame, where the solar wind streams at a speed of $|V_0| = 2V_A$ along the magnetic field. The red and blue curves respectively represent waves with $\theta = 80^{\circ}$ and $100^{\circ}$ in the plasma rest frame. The descriptions of the figure are the same as those shown in Figure 10.</figcaption></figure>

In conclusion, the values of $\sigma_{m\perp 2\parallel}$ for inward and outward KAWs in the plasma rest frame are equivalent to those of the corresponding waves in the spacecraft frame. An observational confirmation of this theoretical prediction is that the sign of $\sigma_{m\perp 2\parallel}$ remains consistent between switchback structures and the surrounding plasma environment, as shown in Figure 13. A schematic illustration of this observation is provided in Figure 14.

In addition, based on Figures 10–12, Table 1 summarizes the typical properties of KAWs in both the plasma rest frame and the spacecraft frame. To identify the propagation direction of KAWs in the plasma frame, both $\text{Cor}(\delta V_{\perp 2}, \delta B_{\perp 2})$ (the correlation between the perpendicular velocity and magnetic field perturbations) and $\sigma_{m\perp 2\parallel}$ serve as reliable diagnostic indicators.
<figure><img src="image_15.png" /><figcaption>**Figure 12.** Properties of KAWs in the spacecraft frame, where the solar wind streams at a speed of $|V_0| = 2V_A$ against the magnetic field. The red and blue curves respectively represent waves with $\theta = 80^\circ$ and $100^\circ$ in the plasma rest frame. The descriptions of the figure are the same as those shown in Figure 10.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>**Figure 13.** (a)–(b) Distributions of (a) $\bar{N}(\sigma_{m\perp2}, f)$ and (b) $R(f)$ for turbulent waves observed during magnetic field reversal intervals of switchbacks, where only reversal intervals lasting longer than 20 s are included. Data bins with durations shorter than 10 s are excluded in panel (a). (c)–(e) An example of an interval containing switchbacks (05:30–08:30 UTC on 2018 November 5), showing (c) the radial magnetic field $B_r$ and the magnetic helicity components (d) $\sigma_{m\perp1\perp2}$ and (e) $\sigma_{m\perp2}$.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Figure 14.** A schematic illustration of the sign of $\sigma_{m\perp2}$ for KAWs in the solar wind in the presence of switchback structures.</figcaption></figure>
<table><caption>**Table 1**

Typical Properties of Kinetic Alfvén Waves in the Plasma Rest Frame and in the Outward Sector and Inward Sector in the Spacecraft Frame</caption><thead><tr><th>Parameters</th><th>OutSec:<sup>1</sup> k<sub>||</sub> > 0</th><th>OutSec: k<sub>||</sub> < 0</th><th>InSec:<sup>2</sup> k<sub>||</sub> > 0</th><th>InSec: k<sub>||</sub> < 0</th><th>PRF: k<sub>||</sub> > 0</th><th>PRF: k<sub>||</sub> < 0</th></tr></thead><tbody><tr><td>Direction Relative to Sun</td><td>outward</td><td>inward</td><td>inward</td><td>outward</td><td>...</td><td>...</td></tr><tr><td>Cor(dV<sub>⊥2</sub>, dB<sub>⊥2</sub>)</td><td>negative</td><td>positive</td><td>negative</td><td>positive</td><td>negative</td><td>positive</td></tr><tr><td>σ<sub>m⊥12</sub></td><td>positive</td><td>negative</td><td>negative</td><td>positive</td><td>positive</td><td>positive</td></tr><tr><td>σ<sub>m⊥2||</sub></td><td>positive</td><td>negative</td><td>positive</td><td>negative</td><td>positive</td><td>negative</td></tr></tbody></table>

**Note.** Here, Cor(dV<sub>⊥2</sub>, dB<sub>⊥2</sub>) denotes the correlation between dV<sub>⊥2</sub> and dB<sub>⊥2</sub>, and k<sub>||</sub> > 0 and k<sub>||</sub> < 0 represent the waves propagating along and against the background magnetic field in the plasma rest frame ("PRF"), respectively.

¹ Outward sector ("OutSec").

² Inward sector ("InSec").

## ORCID iDs

Jinsong Zhao https://orcid.org/0000-0002-3859-6394

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

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

Chen Shi https://orcid.org/0000-0002-9166-1036

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

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

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