# Extended Cyclotron Resonant Heating of the Turbulent Solar Wind

Trevor A. Bowen, Ivan Y. Vasko, Stuart D. Bale, Benjamin D. G. Chandran, Alexandros Chasapis, Thierry Dudok de Wit, Alfred Mallet, Michael Mcmanus, Romain Meyrand, Marc Pulupa, et al.

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## To cite this version:

Trevor A. Bowen, Ivan Y. Vasko, Stuart D. Bale, Benjamin D. G. Chandran, Alexandros Chasapis, et al. Extended Cyclotron Resonant Heating of the Turbulent Solar Wind. The Astrophysical Journal Letters, 2024, 972, 10.3847/2041-8213/ad6b2e. insu-04689010

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# Extended Cyclotron Resonant Heating of the Turbulent Solar Wind

Trevor A. Bowen<fnref n="1" />, Ivan Y. Vasko<fnref n="2" />, Stuart D. Bale<fnref n="1" />,<fnref n="3" />, Benjamin D. G. Chandran<fnref n="4" />, Alexandros Chasapis<fnref n="5" />, Thierry Dudok de Wit<fnref n="6" />,<fnref n="7" />, Alfred Mallet<fnref n="1" />, Michael McManus<fnref n="1" />, Romain Meyrand<fnref n="8" />, Marc Pulupa<fnref n="1" />, and Jonathan Squire<fnref n="8" />

Received 2023 November 29; revised 2024 July 14; accepted 2024 August 5; published 2024 August 23

## Abstract

Circularly polarized, nearly parallel propagating waves are prevalent in the solar wind at ion-kinetic scales. At these scales, the spectrum of turbulent fluctuations in the solar wind steepens, often called the transition range, before flattening at sub-ion scales. Circularly polarized waves have been proposed as a mechanism to couple electromagnetic fluctuations to ion gyromotion, enabling ion-scale dissipation that results in observed ion-scale steepening. Here we study Parker Solar Probe observations of an extended stream of fast solar wind ranging from ~15 to 55 $R_⊙$. We demonstrate that, throughout the stream, transition range steepening at ion scales is associated with the presence of significant left-handed ion-kinetic-scale waves, which are thought to be ion cyclotron waves. We implement quasilinear theory to compute the rate at which ions are heated via cyclotron resonance with the observed circularly polarized waves given the empirically measured proton velocity distribution functions. We apply the Von Kármán decay law to estimate the turbulent decay of the large-scale fluctuations, which is equal to the turbulent energy cascade rate. We find that the ion cyclotron heating rates are correlated with, and amount to a significant fraction of, the turbulent energy cascade rate, implying that cyclotron heating is an important dissipation mechanism in the solar wind.

*Unified Astronomy Thesaurus concepts: Plasma astrophysics (1261); Space plasmas (1544); Plasma physics (2089); Interplanetary turbulence (830); Solar wind (1534); Fast solar wind (1872); Solar physics (1476); Solar coronal heating (1989)*

## 1. Introduction

Weakly collisional plasmas, which are common in astrophysical environments, are fundamentally governed by kinetic processes (Marsch 2006). Our understanding of kinetic processes responsible for turbulent dissipation, heating, and energy transfer in collisionless environments is relatively incomplete and is necessary to explain phenomena such as solar wind acceleration and coronal heating (Parker 1958; Richardson et al. 1995; Hellinger et al. 2013; Fox et al. 2016), as well as analogous astrophysical processes.

Recent work on the near-Sun solar wind has highlighted the significant presence of circularly polarized ion-scale waves (Bale et al. 2019; Bowen et al. 2020d) and their association with nonthermal features in particle distributions (Verniero et al. 2020, 2022; Klein et al. 2021). These waves are characterized by their quasi-parallel propagation along the magnetic field (Jian et al. 2014; Boardsen et al. 2015; Bowen et al. 2020d; Liu et al. 2023). Electric field measurements suggest that these waves predominantly propagate outward from the Sun (Bowen et al. 2020a), similar to the observed propagation direction of larger-scale Alfvénic turbulent

fluctuations (Roberts et al. 1987; Tu & Marsch 1995; Bavassano et al. 1998; McManus et al. 2020).

While ion-scale waves are often associated with processes related to kinetic plasma distributions, e.g., instabilities and resonant damping (Gary 1993; Isenberg & Lee 1996; Hollweg & Markovskii 2002; Marsch 2006; Klein et al. 2018, 2021), they may also serve as a mechanism to transfer turbulent energy via cyclotron resonance to particle thermal motion (Hollweg & Johnson 1988; Tu & Marsch 1997; Cranmer 2000, 2014; Hollweg & Isenberg 2002). Signatures of quasilinear cyclotron resonance in the observed proton distribution functions have been suggested in various spacecraft observations (Marsch & Tu 2001b; He et al. 2015; Verniero et al. 2022; Bowen et al. 2022). Furthermore, steepening of turbulent spectra at ion cyclotron resonant scales has been interpreted as a signature of cyclotron resonant damping (Denskat et al. 1983; Woodham et al. 2018; Lotz et al. 2023) and is correlated with the presence of circularly polarized signatures (Goldstein et al. 1994; Leamon et al. 1998; He et al. 2011; Lion et al. 2016; Zhao et al. 2021). These helical signatures are correlated with both proton temperature anisotropy and the turbulent amplitudes (Telloni et al. 2019), suggesting that they may be associated with turbulent dissipation.

The quasilinear damping of the in situ population of ion cyclotron waves (ICWs) has been measured with a heating rate accounting for 10%–20% of the turbulent energy flux (Bowen et al. 2022). Further studies have shown the direct transfer of

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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.

<footnote n="1">¹Space Sciences Laboratory, University of California, Berkeley, Berkeley, CA 94720-7450, USA; tbowen@berkeley.edu</footnote>

<footnote n="2">²William B. Hanson Center for Space Sciences, University of Texas at Dallas, Richardson, TX 75080, USA</footnote>

<footnote n="3">³Physics Department, University of California, Berkeley, Berkeley, CA 94720-7300, USA</footnote>

<footnote n="4">⁴Department of Physics & Astronomy, University of New Hampshire, Durham, NH 03824, USA</footnote>

<footnote n="5">⁵Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO 80303, USA</footnote>

<footnote n="6">⁶LPC2E, CNRS and University of Orléans, Orléans, France</footnote>

<footnote n="7">⁷ISSI, Bern, Switzerland</footnote>

<footnote n="8">⁸Department of Physics, University of Otago, 730 Cumberland St., Dunedin 9016, New Zealand</footnote>
energy from waves to protons using wave-particle correlation methods (Vech et al. 2020; Luo et al. 2022). The presence of these waves has been observed to correlate with both turbulent features such as the large-scale cross-helicity and sub-ion-scale intermittency (Bowen et al. 2024).

In this Letter, we study an extended stream of fast solar wind observed by Parker Solar Probe (PSP) from ~15 to 55 $R_⊙$, with persistent signatures of left-hand-polarized waves. We show that left-handed polarization directly corresponds to turbulent steepening in the ion-kinetic-scale transition range (Kiyani et al. 2009; Sahraoui et al. 2009; Bowen et al. 2020c; Duan et al. 2020, 2021), which has historically been interpreted as a signature of dissipation (Denskat et al. 1983; Goldstein et al. 1994; Leamon et al. 1998; Smith et al. 2006, 2012; Lion et al. 2016; Bowen et al. 2020c, 2024). Using the cold plasma dispersion to determine the internal energy and Poynting flux of the measured wave spectrum suggests that the energy to generate waves is stored within the turbulent fluctuations, indicating that the waves are a pathway to ion-scale turbulent dissipation. By applying drifting bi-Maxwellian fits to the distribution function, we estimate the empirical quasilinear heating rate of the local ion-scale waves (Kennel & Engelmann 1966; Isenberg & Lee 1996; Bowen et al. 2022). We similarly estimate local turbulent dissipation rates via turbulent amplitudes and the Von Kármán decay law to determine the decay of the largest, outer-scale turbulent fluctuations (Hossain et al. 1995; Wan et al. 2012; Bandyopadhyay et al. 2020; Wu et al. 2022). We demonstrate strong correlations between the quasilinear heating and turbulent dissipation rates. These strong correlations are evident both in global scaling and local fluctuations in the turbulent cascade and in quasilinear heating rates. These results suggest that cyclotron heating plays an important role in extended solar wind heating. These results may have strong implications for the nature of ion-scale heating in the corona and other weakly collisional astrophysical plasmas.

## 2. Methods and Results

We focus on a stream of fast solar wind observed in the inner heliosphere by PSP from 2021 November 16 to 2021 November 20. During this interval, PSP was flying inward toward the Sun in near corotation with the solar surface, sampling a relatively singular source region over the range of 14–55 $R_⊙$ (Badman et al. 2023; Davis et al. 2023). We use measurements from the PSP Solar wind Electron Alpha and Proton experiment's Solar Probe Analyzer (SPAN; Livi et al. 2022). The PSP FIELDS experiment (Bale et al. 2016) provides measurements of the magnetic field from DC to sub-ion kinetic scales using merged search coil (Jannet et al. 2021) and fluxgate magnetometer (SCaM) measurements (Bowen et al. 2020b).

We break the 4-day stream into a set of intervals of 128 s with 50% overlap to study the evolution of the turbulent spectra alongside ion-scale waves. SCaM data are sensitive well into the sub-ion kinetic scales (Dudok de Wit et al. 2022) but are only available for two axes; as a result, the SCaM data are only used to measure the shape of the spectra. The fluxgate magnetometer data provide three-component magnetic field measurements that enable comprehensive study of the properties of observed waves. The fluxgate magnetometer and SCaM data are resampled to a uniform 146.4845 Sa s⁻¹ rate. Data were discarded when discontinuities or changes in instrumental modes led to artifacts. In total 6737 intervals of 128 s were analyzed.

<figure><img src="image_5.png" /><figcaption>**Figure 1.** (a) Measured spectral densities from 2021 November 16 to 20 computed for 6737 intervals of 128 s. Every 40th spectra is plotted. Colors represent radial distance of PSP from the Sun, ranging from ~14 to 55 $R_⊙$, with light colors representing small distances becoming darker to indicate increasing distance. (b) Local spectral index $\alpha$ of measured spectral densities computed in a moving window. (c) 3PL fit to observed spectra for each interval. (d) Local spectral index $\alpha$ of 3PL spectral densities computed in a moving window.</figcaption></figure>

**Turbulent Spectra**—We compute spectral densities $E(f)$, with units nT² Hz⁻¹, from the two-component SCaM data through ensemble averaging over 4096-point spectral densities computed via fast Fourier transform in each 128 s interval. Each spectral density is interpolated onto 320 logarithmically spaced frequencies. Figure 1(a) shows spectral densities computed from the stream intervals, though for clarity only every 40th spectra is plotted. The colors correspond to radial distance from the Sun, which is between 14.1 and 55.0 $R_⊙$; lighter yellow colors are closer to the Sun, whereas darker colors correspond to further heliocentric distances.

An estimate of the local spectral index for each frequency is obtained by performing a linear least-squares fit of $\log_{10}E(f)$ in a moving window consisting of 60 neighboring logarithmically spaced frequencies. The slope of a linear least-squares fit in log space provides a measure of the local spectral index $\alpha(f)$. Figure 1(b) shows the locally measured $\alpha$ as a function of frequency for spectra in Figure 1(a). The observed spectra are consistent with three-power-law (3PL) spectra often reported in the solar wind (Alexandrova et al. 2008; Sahraoui et al. 2009;
Bowen et al. 2020c): each spectrum has a characteristic low-frequency inertial range scaling $\alpha_I$, a transition range with steepened index $\alpha_T$, followed by flattening to an index $\alpha_K$ at higher frequencies. The shape of the spectra is consistent but shifts to higher frequencies closer to the Sun. The index of the steep transition range spectra varies and can have values of up to -10. The extremely steep transition range indices in this stream are predominantly associated with parallel spectra as shown in Duan et al. (2021). As we demonstrate, this steepening is also associated with significant ICW populations (Bowen et al. 2020d, 2024).

To determine the slope of the transition range $\alpha_T$, we implement a piecewise, 3PL fit, following methods developed in Bowen et al. (2020c). Each of the three ranges is modeled as a power-law spectrum with indices $\alpha_I$, $\alpha_T$, and $\alpha_K$ separated by break-point frequencies $f_{IT}$ and $f_{TK}$, referring to the respective inertial to transition and transition to kinetic breaks. Figure 1(c) shows the 3PL fits, corresponding to the spectra in Figure 1(a). Additionally, we apply the procedure to measure the local spectral index $\alpha(f)$ to the 3PL fits, with the results shown in Figure 1(d). There is good agreement between the observations and the 3PL fits.

**Break Scales and Cyclotron Resonance**—We compare the measured break scales $f_{IT}$ and $f_{TK}$ to various physical ion kinetic scales: the ion gyroradius, $\rho_i = v_{thi}/\Omega_i$, where the ion thermal speed is $v_{thi} = \sqrt{2k_B T_i/m_i}$ and the ion gyrofrequency is $\Omega_i = eB_0/m_i$; and the ion inertial scale $d_i = \rho_i/\sqrt{\beta_i}$, where $\beta_i = 2n_0\mu_0 k_B T_i/B_0^2$. The fundamental charge is given as $e$, $\mu_0$ is the permeability of free space, $T_i$ is the proton temperature, $B_0$ is the average background magnetic field, and $n_0$ is the background number density. The wavenumbers corresponding to the ion inertial scale and ion gyroscale are $k_{\rho_i} = \rho_i^{-1}$ and $k_{d_i} = d_i^{-1}$, respectively. We approximate the wavenumber corresponding to cyclotron resonant interactions as $k_{\Omega} = \Omega_i/(v_A + v_{thi})$, where $v_A$ is the Alfvén speed $v_A = B_0/\sqrt{\mu_0 m_i n_0}$ (Leamon et al. 1998; Wicks et al. 2016; Woodham et al. 2018; Bowen et al. 2020d). This approximation for $k_{\Omega}$ is derived via the cyclotron resonance condition between outward-going ICW and protons flowing inward in the plasma frame, $\omega + k_{\parallel}v_{thi} = \Omega_i$ (Leamon et al. 1998), and $\omega = k_{\parallel}v_A$, which is the low-frequency limit to the ICW dispersion for parallel-propagating waves:

$$ \left( \frac{\omega_{\text{ICW}}}{\Omega_i} \right)^2 = \left[ \frac{k_{\parallel} d_i}{2} \left( \sqrt{k_{\parallel}^2 d_i^2 + 4} - k_{\parallel} d_i \right) \right]^2 . \quad (1) $$

For the ion scales $k_{d_i}$, $k_{\rho_i}$, and $k_{\Omega}$, each wavenumber is converted to an effective spacecraft frequency $f$ first using the Taylor hypothesis, $2\pi f \approx k v_{sw}$, to associate ion kinetic scales with spacecraft frequencies $f_{d_i}$, $f_{\rho_i}$, and $f_{\Omega}$. For the cyclotron resonant scale, $k_{\Omega}$, we also consider a correction to the Taylor hypothesis made by incorporating the Doppler shift equation

$$ 2\pi f = \omega(\mathbf{k}) + \mathbf{k} \cdot \mathbf{v}_{\text{sw}}, \quad (2) $$

to improve on our estimate of the frequency corresponding to cyclotron resonant interactions as

$$ 2\pi f'_{\Omega} = \omega_{\text{ICW}}(k_{\Omega}) + k_{\Omega}v_{\text{sw}}, \quad (3) $$

which includes the contribution from $\omega_{\text{ICW}}$ evaluated at $k_{\Omega}$. For calculating $f'_{\Omega}$, we assume that the angle between the mean magnetic field and flow direction, $\theta_{BV}$, is small such that

<figure><img src="image_6.png" /><figcaption>**Figure 2.** (a) Frequencies corresponding to ion kinetic scales computed with the Taylor hypothesis for 6737 intervals as a function of solar radius, ion inertial scale $f_{d_i}$ (red), ion gyroscale $f_{\rho_i}$ (orange), and cyclotron resonant scale $f_{\Omega}$ (blue). Additionally, we include the ICW dispersion to correct the Taylor hypothesis approximation to the the cyclotron resonant scale $f'_{\Omega}$ (green). Break points from spectral fits are also shown for $f_{IT}$ (black) and $f_{TK}$ (teal). The gyrofrequency $\Omega/2\pi$ is shown in purple. (b) Histograms show the distribution of frequency ratios $f_{d_i}/f_{IT}$ (red), $f_{\Omega}/f_{IT}$ (blue), $f'_{\Omega}/f_{IT}$ (green), and $f_{\rho_i}/f_{TK}$ (orange).</figcaption></figure>

$\cos\theta_{BV} \approx 1$ and thus $\mathbf{k} \cdot \mathbf{v}_{sw} \approx k_{\Omega}v_{sw}$ for parallel-propagating ICWs.

Figure 2(a) shows the computed kinetic-scale frequencies alongside the measured breaks from the 3PL fitting algorithm against $R_{\odot}$. Figure 2(b) shows distributions ratios of $f_{d_i}/f_{IT}$, $f_{\Omega}/f_{IT}$, $f'_{\Omega}/f_{IT}$, and $f_{\rho_i}/f_{TK}$. These distributions provide a measure of agreement between break frequencies and the measured ion kinetic scales. Recent work has suggested that the cyclotron scale $f_{\Omega}$ corresponds well with $f_{IT}$ (Vech et al. 2018; Woodham et al. 2018; Duan et al. 2020; Lotz et al. 2023). Here we find that $f_{\Omega}$ is statistically lower than $f_{IT}$; however, the corrected $f'_{\Omega}$ frequency well approximates the $f_{IT}$ break. The $f_{IT}$ transition range break additionally approximates $f_{d_i}$, such that we cannot distinguish the inertial scale from the cyclotron resonant scale. We find that $f_{\rho_i}$ is within the transition range, which is consistent with previous results (Bowen et al. 2020c).

Importantly, $f_{\Omega}$ and $f'_{\Omega}$ are estimates to a single cyclotron resonant scale corresponding to resonance between particles at
the thermal speed $v_{th}$, with low-frequency ICWs, $\omega = kV_A$. However, observations from the solar wind suggest that ICWs occur over a range of frequencies corresponding to $kd_i \approx 1$ (Bowen et al. 2020d). Empirical determination of circular polarization as a function of spacecraft frequency, along with the cold plasma dispersion for parallel-propagating ICWs given by Equation (1), provides information regarding the wavenumbers at which cyclotron resonant waves occur (Bowen et al. 2020a).

A 64-scale Morlet wavelet transform is applied to each interval to compute a power spectral density $\tilde{B}^2(f, t)$ with units nT$^2$ Hz$^{-1}$ (Farge 1992; Dudok de Wit et al. 2013; Bowen et al. 2020d). We extract circular polarization of the field via the reduced magnetic helicity (Howes & Quataert 2010),

$$ \sigma_B(f, t) = -2 \operatorname{Im}(\tilde{B}_{\perp 1} \tilde{B}_{\perp 2}^*) / (\tilde{B}_{\perp 1}^2 + \tilde{B}_{\perp 2}^2), \quad (4) $$

of the wavelet coefficients perpendicular to the mean field, with left/right-handed polarization represented by positive/negative $\sigma_B$. We calculate the left-hand-polarized power spectra $\tilde{B}_{LH\sigma}^2$ by filtering out power with $\sigma_B < 0.9$. The normalized fractional left-handed spectra are computed as $\tilde{B}_{LH\sigma}^2/\tilde{B}_{total}^2$. Very little right-handed polarization is present in this stream.

Based on previous measurements of the phase speed of circularly polarized waves (Bowen et al. 2020a), which show a strong statistical preference for outward propagation, we can assume that the waves are outward-propagating ICWs, traveling parallel to the mean field. The wavelet transform gives circular polarization as a function of the spacecraft frame frequency, $f$. The use of the Doppler shift Equation (2) in the parallel-propagating limit gives

$$ 2\pi f = \omega_{\text{ICW}} \pm k_{\parallel} v_{\text{sw}} \cos\theta_{\text{BV}}, \quad (5) $$

where we no longer assume either the Taylor hypothesis or that $\cos\theta_{BV} \approx 1$. The combination of Equation (5) with Equation (1) determines the corresponding parallel wavenumber $k_{\parallel}$ for each spacecraft frequency $f$ (Bowen et al. 2020a). Figure 3(a) shows the normalized $\tilde{B}_{LH\sigma}^2/\tilde{B}_{total}^2$ as a function of $k_{\parallel}d_i$ computed from combining Equation (1) with Equation (5). At all radial distances the ICW population appears at $k_{\parallel}d_i = 1$, where ICWs are inherently dispersive, with a finite bandwidth in wavenumber ranging from $k_{\parallel}d_i \sim 0.3$ to 4. The finite bandwidth indicates that, at each time, a range of particle velocities simultaneously satisfy the ICW resonance condition. It is important to note that the spacecraft reaction wheels, highlighted in teal lines in Figures 3(b)–(d), contribute circularly polarized power at higher $k_{\parallel}d_i$ (Bowen et al. 2020d).

Figure 3(b) shows $\tilde{B}_{LH\sigma}^2/\tilde{B}_{total}^2$ as a function of spacecraft frequency at each interval studied as a function of solar radial distance. We plot both the $f_{IT}$ and $f_{TK}$ breaks. The $f_{IT}$ break agrees very well with the circularly polarized regime, while the $f_{TK}$ break bounds the circular polarization at higher frequencies, indicating that the transition range is entirely circularly polarized. We additionally plot the cyclotron scale $f'_{\Omega}$, which approximates the $f_{IT}$ break.

**Cyclotron Waves and Turbulent Dissipation**—Using the cold plasma dispersion, the Doppler shift Equations (1) and (2), and our observations of $\tilde{B}_{LH\sigma}^2(k_{\parallel}d_i)$, we measure both the internal energy density of the waves $U_B$ and the magnitude of the Poynting flux computed in a heliocentric inertial (HCI) frame, $S_{HCI}$, assuming that the ICW propagation is entirely parallel to the radial solar wind flow (Karpman 1974; Shklyar

& Matsumoto 2009):

$$ U'_B = \frac{\tilde{B}_{LH\sigma}^2 v_{ph}}{\mu_0 v_g} = \frac{\tilde{B}_{LH\sigma}^2 (2\Omega_i - \omega)}{2\mu_0 (\Omega_i - \omega)}, \quad (6) $$

$$ S'_{\text{HCI}} = (v_{\text{sw}} + v_g) U'_B, \quad (7) $$

where $v_{ph}$ and $v_g$ are the plasma frame phase and group velocities, respectively. As the units of $\tilde{B}_{LH\sigma}^2$ are nT$^2$ Hz$^{-1}$, both $U'_B$ and $S'_{HCI}$ are defined as spectral densities. The internal energy $U_B$ and Poynting flux magnitude $S_{HCI}$ corresponding to each wavelet coefficient are obtained by multiplying $U'_B$ and $S'_{HCI}$ by the bandwidth of each wavelet $\Delta f = \frac{df}{dk_{\parallel}} \Delta k_{\parallel}$, where the derivative $df/dk_{\parallel}$ and $\Delta k_{\parallel}$ are obtained from Equation (5). Figures 3(c) and (d) show $U_B$ and $S_{HCI}$ at each frequency over the interval.

The total Poynting flux and internal energy can be integrated at each time to give total values $U_{tot}$ and $S_{tot}$ for each interval. Due to the sporadic noise that occurs at low frequencies, as well as high-frequency contributions to polarization from the spacecraft reaction wheels, we zero out contributions to $S_{tot}$ and $U_{tot}$ where $kd_i < 0.3$ and $kd_i > 5$.

We use $U_{tot}$ and $B_{LH\sigma}^{\text{max}} = \max[\tilde{B}_{LH\sigma}^2/\tilde{B}_{total}^2]$ as proxies for the ICWs in each interval. Figure 4 shows how these quantities relate to parameters associated with turbulence, $\alpha_T$, as well as the total rms turbulent amplitude $\langle\delta B^2\rangle$ computed in each 128 s interval. We consider only intervals with $150^{\circ} < \theta_{BV} < 170^{\circ}$ in order to control for effects associated with anisotropy, which affect observational signatures of both the waves and the turbulence (Chen et al. 2010; Bowen et al. 2020d); of the 6736 intervals, 3172 occur with $150^{\circ} < \theta_{BV} < 170^{\circ}$. In each panel of Figure 4 data are plotted with the color scale in Figure 1, with lighter colors corresponding to intervals closer to the Sun. We compute lines of best fit for data within 10–20 $R_\odot$, 20–30 $R_\odot$, 30–40 $R_\odot$, and 40–50 $R_\odot$, which are shown in each panel of Figure 4. Figure 4(a) shows that the transition range slope $\alpha_T$ is strongly correlated with the level of circular polarization, $B_{LH\sigma}^{\text{max}}$ (Bowen et al. 2024), with similar trends observed at all solar radii. Figure 4(b) shows that the internal energy of the waves $U_B$ is anticorrelated with $\alpha_T$, such that steeper slopes contain greater amounts of ICW energy. This correlation is present at all $R_\odot$, though at larger distances the maximum measured $U_{tot}$ are similar to the lowest $U_{tot}$ close to the Sun, such that constraining the data by $R_\odot$ is necessary to see the correlations. Figure 4(c) shows that the internal energy contained within the circularly polarized ion-scale transition range is globally a function of the amplitude of the turbulent fluctuations $dB_{rms}^2$ computed at the 128 s scale, consistent with previous results in Shankarappa et al. (2023).

**ICW Heating Rates**—Our analysis of the transition range reveals signatures of circularly polarized ion cyclotron resonant waves; we apply well-established principles of energy conservation and quasilinear heating to connect these waves to turbulent dissipation.

The Poynting theorem states that the change in internal energy of the ICW population is

$$ \frac{\partial U_B}{\partial t} = -\nabla \cdot \mathbf{S} - Q_{\text{ICW}} + \epsilon_{\text{ICW}}, \quad (8) $$

where $Q_{\text{ICW}}$ is the dissipation rate of the waves and $\epsilon_{\text{ICW}}$ is a driving term. Assuming a steady state, we compute the three terms on the right-hand side of Equation (8). We approximate
<figure><img src="image_7.png" /><figcaption>Figure 3. (a) Fractional left-handed power $\tilde{B}_{LH\sigma}^2/\tilde{B}_{total}^2$ computed from wavelet transform for each of the 6737 intervals as a function of $k_{\parallel}d_i$. (b) Spectrogram of left-handed circularly polarized power as a function of radial distance; $f_{\text{IT}}$ and $f_{\text{TK}}$ for each measurement are plotted in white, and the cyclotron scale $f_{\text{Q}}$ is plotted in green; teal lines show frequencies corresponding to spacecraft reaction wheels (Bowen et al. 2020d). (c) Internal energy $U_B$ and (d) Poynting flux of waves in a HCI frame $S_{\text{HCI}}$ computed from the cold plasma dispersion relation.</figcaption></figure>

$\nabla \cdot S$ in the HCI frame as the radial derivative

$$ \nabla \cdot S \approx \frac{1}{r^2} \frac{\partial}{\partial r} r^2 S_{\text{tot.}} \quad (9) $$

We estimate $Q_{\text{ICW}}$ using the quasilinear heating rate (Kennel & Engelmann 1966) with the empirically observed spectrum of cyclotron waves

$$ I(k_{\parallel}) = \frac{\tilde{B}_{LH\sigma}^2}{B_0^2} \frac{df}{dk_{\parallel}}. \quad (10) $$

We note that the power spectra are normalized such that total left-handed variance

$$ dB_{LH}^2 = \int_0^\infty \tilde{B}_{LH\sigma}^2 df. $$

For each SPANi measurement, we perform a drifting bi-Maxwellian fit in the proton core frame with the form

$$ g_p(v_{\perp}, v_{\parallel}) = \frac{n_c}{\pi^{3/2} w_{c,\perp}^2 w_{c,\parallel}} \exp \left[ -\frac{v_{\perp}^2}{w_{c,\perp}^2} - \frac{v_{\parallel}^2}{w_{c,\parallel}^2} \right] + \frac{n_b}{\pi^{3/2} w_{b,\perp}^2 w_{b,\parallel}} \exp \left[ -\frac{v_{\perp}^2}{w_{b,\perp}^2} - \frac{(v_{\parallel} - v_D)^2}{w_{b,\parallel}^2} \right], \quad (11) $$

which has anisotropic thermal speeds perpendicular and parallel to the mean field ($\perp$ and $\parallel$) for the beam and core (subscripts b and c), $w_{c,\perp}$, $w_{c,\parallel}$, $w_{b,\perp}$, $w_{b,\parallel}$, and a relative drift, $v_D$, parallel to the mean magnetic field (Marsch 2006; Klein et al. 2021). We compute the average fit parameters over the 128 s interval such that an average gyrotropic $\tilde{g}(v_{\perp}, v_{\parallel})$ is
<figure><img src="image_8.png" /><figcaption>**Figure 4.** (a) Transition range spectral index $\alpha_T$ plotted against maximum circular polarization max[$\tilde{B}_{L\alpha/\tilde{B}_{total}}$]. (b) Total internal energy of cyclotron waves $U_B$ plotted against $\alpha_T$. (c) Total internal energy of cyclotron waves $U_B$, which is bandwidth limited and filtered by left-hand polarization at ion kinetic scales, plotted against total rms amplitude of magnetic field $dB_{rms}^2$ computed at 128 s. Data are colored by $R_\odot$, with larger heliocentric distances in darker shades. Solid lines show the linear regression for 10–20 $R_\odot$ (yellow), 20–30 $R_\odot$ (orange), 30–40 $R_\odot$ (red), and 40–50 $R_\odot$ (purple); darker colors correspond to larger distances.</figcaption></figure>

computed. Using $\tilde{g}(v_\perp, v_\parallel)$ and the observed spectrum of cyclotron waves $I(k_\parallel)$, we use the cold plasma dispersion, Equation (1), with wavenumbers defined from Equation (5), to determine the volumetric heating rate in the local interval:

$$ 
\begin{aligned}
Q_{\text{ICW}} &= \int \frac{m_p v^2}{2} \frac{\partial \bar{g}}{\partial t} d^3\mathbf{v} \\
&= \frac{\pi e^2 B_0^2}{4m_p} \int d^3\mathbf{v} \left\{ v^2 \int_0^\infty dk_\parallel \frac{1}{v_\perp} \hat{G}_k[v_\perp \delta(\omega_k - k_\parallel v_\parallel) - \Omega_{p,\parallel}} \right. \\
&\qquad \left. \frac{\omega_k^2}{k_\parallel^2 c^2} I(k_\parallel) \hat{G}_k \bar{g}(v_\perp, v_\parallel) \right\},
\end{aligned}
\quad (12) $$

with

$$ \hat{G}_k = \left( 1 - \frac{k_\parallel v_\parallel}{\omega_k} \right) \frac{\partial}{\partial v_\perp} + \frac{k_\parallel v_\perp}{\omega_k} \frac{\partial}{\partial v_\parallel} \quad (13) $$

(Kennel & Engelmann 1966). A gyrotropic differential volumetric heating rate $Q(v_\perp, v_\parallel)$ can be defined from $Q_{\text{ICW}} = \int Q(v_\perp, v_\parallel) dv_\parallel dv_\perp$. We outline how to calculate $Q_{\text{ICW}}$ and $Q(v_\perp, v_\parallel)$ in Appendix A by using the $\delta$-function to evaluate the integral over wavenumber and the subsequent numerical integration of $Q(v_\perp, v_\parallel)$ over velocity coordinates.

Figures 5(a) and (b) show $Q(v_\parallel)$, the volumetric heating rate as a function of resonant parallel velocity, which is computed from integrating $Q(v_\perp, v_\parallel)$ over perpendicular velocities, as a function of solar radius. Figure 5(a) shows positive $Q(v_\parallel)\Delta v_\parallel$, at resonant velocities less than the thermal speed (shown in yellow), corresponding to wave absorption and consequent plasma heating. The $\Delta v_\parallel$ term, which is the resolution of the numerical integration $1 \text{ km s}^{-1}$, is included to normalize the differential $Q(v_\parallel)$ in terms of a volumetric heating rate $\text{W m}^{-3}$. Figure 5(b) shows wave emission, which, consistent with Bowen et al. (2022), is significantly less energetically relevant than the regions with positive $Q(v_\parallel)\Delta v_\parallel$ wave absorption, which indicates net resonant heating of the plasma.

Importantly, the use of a bi-Maxwellian model to approximate the distribution may not capture nonthermal features that resonate with the observed populations of waves (Dum et al. 1980; Viñas & Gurgiolo 2009). In Appendix B, we implement two well-understood nonparametric techniques to model the observed distributions, Hermite polynomials, and radial basis functions (RBFs), to verify the independence of our results from the bi-Maxwellian model. While there are slight differences in the level of heating predicted by each model, the qualitative interpretation of ICW resonance heating is found in each case, and the average values closely follow that of the drifting bi-Maxwellian.

*Connecting ICW Heating to the Turbulent Cascade*—The importance of resonant heating can be understood by comparing the net heating rates $Q_{\text{ICW}}$ to the turbulent cascade rate $\epsilon$. Following Bandyopadhyay et al. (2020) we compute the energy cascade rate of the turbulence assuming a von Kármán decay law:

$$ \epsilon^{\prime \pm} = \alpha \frac{(\delta z^{\pm})^2 \delta z^{\mp}}{L^{\pm}}, \quad (14) $$

where $\delta z^{\pm}$ are rms fluctuation amplitudes of the Elsässer variables $\delta z^{\pm} = \delta v \pm \delta \mathbf{B}/\sqrt{\mu_0 \rho}$. The correlation length $L^\pm$ is determined from the solar wind advected spatial scale corresponding to the time lag for which the correlation reduces below a factor $e^{-1}$. The coefficient $\alpha = 0.03$ is determined from numerical means (Hossain et al. 1995; Wan et al. 2012; Usmanov et al. 2014; Bandyopadhyay et al. 2018) and chosen to agree with Bandyopadhyay et al. (2020) and Wu et al. (2022). However, this value of $\alpha$ is derived assuming low cross-helicity, while the near-Sun solar wind is characterized by imbalance with $\sigma_c \approx 1$ (McManus et al. 2020). Note that the units for $\epsilon'$ in Equation (14) are J s$^{-1}$ kg$^{-1}$; we define $\epsilon$ with units of W m$^{-3}$, which are comparable to the volumetric heating rates in Equation (12), by multiplying $\epsilon'$ by the background mass density $\epsilon = \epsilon' m_i n_0$.

Measurement of $\epsilon'^+$ via Equation (14), which we assume approximates the total cascade rate $\epsilon'$, requires the Elsässer amplitude $\delta z^{\pm}$ at the outer, energy-containing scales of the turbulence. Davis et al. (2023) have studied the evolution of the outer scales in this same stream and found that the outer-scale spectral break evolves from $10^{-2}$ Hz close to $14 R_\odot$ to $10^{-3}$ Hz
<figure><img src="image_9.png" /><figcaption>**Figure 5.** (a) Volumetric heating rates, $Q(v_\|)\Delta v_\|$, with $\Delta v_\| = 1\text{km s}^{-1}$, which have been integrated over perpendicular velocity, as a function of resonant parallel velocity and solar radial distance. (b) Same as panel (a), but for negative values of $Q(v_\|)\Delta v_\|$. Yellow circles in panels (a) and (b) show the measured parallel thermal speed. (c) Measured turbulent cascade rate $\epsilon$ from von Kármán decay law (black), average cyclotron heating rate from quasilinear theory ($Q_{\text{ICW}}$) (red), and $(\nabla \cdot \mathbf{S})$ for total cyclotron waves (blue). Power-law fits, $\epsilon'$ and $Q_{\text{ICW}}'$, are shown with dashed lines. (d) $(Q_{\text{ICW}}' - Q_{\text{ICW}}')$ plotted against $\epsilon - \epsilon'$, where + and $\Delta$ correspond to data with $(Q_{\text{ICW}}' - Q_{\text{ICW}}')$ and $\epsilon - \epsilon'$ greater and less than zero, respectively. (e) 2D histogram of $Q_{\text{ICW}}'$ against $\epsilon$ interpolated onto 128 s cadence. (f) 2D histogram of $Q_{\text{ICW}}'/\epsilon$ against $R_\odot$; dashed black lines show $Q_{\text{ICW}}'/\epsilon$ at levels of 0.1, 0.5, and 1; the average value of $Q_{\text{ICW}}'/\epsilon$ is shown in black; $Q_{\text{ICW}}'/\epsilon > 1$, indicating that quasilinear heating at levels greater than the turbulent cascade may arise owing to uncertainty in the estimates of both $Q_{\text{ICW}}'$ and $\epsilon$. The distributions in panels (e) and (f) are column normalized to the maximum value in each data column.</figcaption></figure>

at $\sim 50 R_\odot$. Accordingly, we compute the cascade rate $\epsilon^+$ in successive 30-minute intervals (corresponding to $\approx 5 \times 10^{-4}$ Hz) with 50% overlap. In each 30-minute interval we then compute the average heating rate $\langle Q_{\text{ICW}} \rangle$ and $(\nabla \cdot \mathbf{S})$.

Figure 5(c) shows $\epsilon$, $\langle Q_{\text{ICW}} \rangle$, and $(\nabla \cdot \mathbf{S})$ as a function of radial distance. A surprisingly good agreement is found between $\epsilon$ and $\langle Q_{\text{ICW}} \rangle$, while the divergence of the Poynting flux term is significantly smaller. The relatively small $(\nabla \cdot \mathbf{S})$ indicates that the wave dynamics are dominated by the source, $\epsilon_{\text{ICW}}$, and dissipation, $Q_{\text{ICW}}$, and that the waves do not propagate significantly. This result is consistent with previous studies arguing for in situ driving of these waves (Bowen et al. 2020d; Vech et al. 2020; Liu et al. 2023).

We find a Spearman, nonparametric, ranked correlation between $\epsilon$ and $\langle Q_{\text{ICW}} \rangle$ of 0.79; we further compute the Pearson correlation of the logarithmic quantities as 0.88. Both of these values indicate strong correlations. Furthermore, we fit each of $\epsilon$ and $\langle Q_{\text{ICW}} \rangle$ to a power law in $R_\odot$, noted as $\epsilon'$ and $Q'_{\text{ICW}}$. We compute the local variations from the global power-law trends using $\epsilon - \epsilon'$ and $\langle Q_{\text{ICW}} \rangle - Q'_{\text{ICW}}$ as a measure of how closely the local quasilinear heating rate follows the turbulent energy

cascade rate. Figure 5(d) shows the local fluctuations $\langle Q_{\text{ICW}} \rangle - Q'_{\text{ICW}}$ against $\epsilon - \epsilon'$. We measure a Spearman ranked correlation of 0.56.

Additionally, we interpolate $\epsilon$, measured at a 30-minute cadence, onto the 128 s cadence of the $Q_{\text{ICW}}'$ measurements. Figure 5(e) shows a 2D histogram of $Q_{\text{ICW}}'$ against the interpolated $\epsilon$; the distribution is column normalized to the measured cascade rate. Good correlations are obtained, with a Spearman ranked correlation of 0.58. These results suggest that characteristically the quasilinear heating rate is proportional to the energy cascade rate, with a constant of proportionality near unity. Figure 5(f) shows a 2D histogram of $Q_{\text{ICW}}'/\epsilon$, where the interpolated $\epsilon$ is again used, against solar radius $R_\odot$; the distribution is column normalized to $R_\odot$. Implicitly, this suggests that much of the turbulent cascade flux enters the ICW population such that $Q_{\text{ICW}}' = \chi\epsilon^+$ with $\chi \approx 0.1-1$ over the vast majority of the observed stream. There is some radial trend, with $0.1 < \chi < 0.5$ at $\approx 15-20 R_\odot$ and $\chi \approx 1$ at larger distances. Values with $\chi > 1$ suggest dissipation in excess of the turbulent cascade rate; we believe that this is predominantly due to uncertainty in the measurements of $Q_{\text{ICW}}'$ and $\epsilon$.
### 3. Conclusions

The nature of the steepening of turbulent energy spectrum at ion kinetic scales has long been debated and is an important signature in understanding collisionless turbulent dissipation (Denskat et al. 1983; Smith et al. 1990; Goldstein et al. 1994; Leamon et al. 1998). Studies of the location of the ion-kinetic-scale break frequency suggest that the most likely candidate scale corresponds to resonance between outward-going ICWs and the thermal ion population (Vech et al. 2018; Woodham et al. 2018; Duan et al. 2020; Lotz et al. 2023). Previous observations from 1 au have suggested that ICWs play a role in dissipating solar wind turbulence (Leamon et al. 1998; Lion et al. 2016; Telloni et al. 2019; Woodham et al. 2019; Zhao et al. 2021; Luo et al. 2022). In this Letter, we concretely demonstrate that ion-scale spectral steepening is associated with circular polarization. The break between the inerital and transition ranges follows the regime of ICWs, and the spectral break between the transition range and sub-ion-scale turbulence bounds the circularly polarized waves exactly at higher frequencies. We suggest that the transition range corresponds to cyclotron waves that are associated with turbulent dissipation and ion-scale heating (Bowen et al. 2024). Our results show that internal energy and momentum of the waves are correlated with the amplitude of the turbulent fluctuations (Figure 3), suggesting that the energy contained in, and transported by, the waves originates from the turbulent fluctuations. These results are consistent with earlier observations from 1 au, suggesting that the level of steepening in the turbulent cascade may relate to the level of Alfvénicity and the turbulent cascade rate (Smith et al. 2006; Bruno et al. 2014).

In this Letter, we compute two measures of energy transfer: first, the quasilinear heating rate of kinetic-scale ICWs, $\langle Q_{ICW} \rangle$, and second, the von Kármán turbulent decay rate $\epsilon^+$. The measurements of $\langle Q_{ICW} \rangle$ are obtained through integrating kinetic phase-space densities measured by SPANi, while $\epsilon^+$ is determined from outer-scale turbulent fluctuations. It is striking that these quantitative estimates of energy transfer, $\epsilon^+$ and $Q_{ICW}$, obtained via entirely different methods at vastly different spatial scales show significant correlations. Such strong correlations suggest that parallel cyclotron resonance dissipates significant amounts of turbulent energy into the solar wind ion populations. The correlation between energy flux and quasilinear heating rate is observed both in the global scaling of the quantities and in local variations, i.e., fluctuations in the turbulent cascade rate are typically accompanied by correlated variations in the quasilinear heating rate. These novel measurements, which support a radially extended cyclotron heating mechanism, are important in understanding how turbulent dissipation results in heating the expanding solar wind. These results further indicate significant progress on PSP's objective "to trace the flow of energy that heats and accelerates the solar corona and solar wind" (Fox et al. 2016). Furthermore, the extended ICW heating, which is observed from 15 to 55 $R_\odot$, may have significant implications for the role of ICW heating in the corona (Hollweg & Johnson 1988; Cranmer 2000).

Recent simulations by Squire et al. (2022) suggest that the dissipation of strongly imbalanced Alfvénic turbulence may result in polarization signatures similar to those measured in the solar wind (He et al. 2011; Podesta & Gary 2011; Huang et al. 2020). The Squire et al. (2022) simulations, as well as the underlying idea of a helicity barrier whereby imbalanced

turbulence is prevented from cascading to small scales owing to a conserved generalized helicity (Meyrand et al. 2021), provide a novel framework to understand the connection between turbulence and ICWs. These theoretical ideas have found support in observations showing that ion-scale ICWs preferentially occur when large-scale fluctuations are highly Alfvénic (Bowen et al. 2024) and that sub-ion-scale intermittency depends largely on the level of ICWs present, which is in turn correlated with the cross-helicity, suggesting that ICWs play a role in the dissipation of imbalanced turbulence.

The idea that parallel-propagating ICWs dissipate a predominantly perpendicular cascade challenges our current understandings of solar wind turbulence and dissipation. It has long been suggested that the turbulence should drive fluctuations into smaller perpendicular scales (Shebalin et al. 1983), and observations regularly indicate significant anisotropy perpendicular to the mean field (Horbury et al. 2008; Chen et al. 2010; Duan et al. 2021). The hybrid simulations of the helicity barrier by Squire et al. (2022) suggest that oblique ICWs are the primary heating mechanism and that parallel ICWs are emitted as a secondary process (Chandran et al. 2010) through the Alfvén/ion cyclotron instability (Gary 1993).

While the secondary emission of parallel ICWs from oblique ICW heating provides a mechanism to generate parallel ICWs from turbulent heating, in such a mechanism the plasma is heated by oblique ICWs but cooled by the parallel ICWs (since they are emitted as an instability). In contrast, our measurements indicate that the parallel ICWs are robustly heating the plasma, suggesting that they should be directly driven by the turbulence, as opposed to generated via oblique ICW heating. If our present measurements, consistent with previous observations (Bowen et al. 2022), are correct, this presents a conundrum, given the difficulty of sourcing quasi-parallel waves from perpendicular turbulent structures. Our use of nonparametric representations of $g(v)$ (Appendix B) further complicates this issue, as our measurement of ICW heating is not simply due to the incorrect parameterization of the SPANi observations with a drifting bi-Maxwellian fit.

A possible source of systematic error is in the assumption of a cold plasma. We assume a cold plasma dispersion (Stix 1992), which is likely not entirely accurate for the solar wind. Implementation of warm plasma dispersion solvers with arbitrary distribution functions (Verscharen et al. 2017; Walters et al. 2023) may enable greater understanding of parallel wave generation. Furthermore, inclusion of beam populations (Verniero et al. 2020; Ofman et al. 2022) and $\alpha$-particles may affect instabilities (McManus et al. 2024) and may help further understanding of these dynamics.

Outside of uncertainties in the distribution function and wave dispersion relations, a possible explanation for the in situ production of ICWs that result in solar wind heating could include the adiabatic evolution associated with expansion. Adiabatic evolution associated with expansion (Chew et al. 1956) should drive distribution functions that are unstable and emitting ICWs toward stability. The extent to which expansion may enable ICW heating rather than cooling within the helicity barrier framework remains largely unstudied.

Furthermore, estimates of the cascade rate are notoriously difficult to estimate via single-spacecraft measurements (Bandyopadhyay et al. 2018, 2020). Care must be taken in
understanding these cascade rates, especially in strongly imbalanced states, in which a stationary energy flux may not exist, or a negative cascade rate may occur (Smith et al. 2009; Meyrand et al. 2021). This is especially important in the regime of imbalanced turbulence frequently observed by PSP (Meyrand et al. 2021; Squire et al. 2022). However, recently Wu et al. (2022) found good agreement between the Von Kármán decay rates and measured perpendicular proton heating rates, suggesting that the turbulent dissipation can be studied via these methods. While care has to be taken in further studies of cascade rates, the general correspondence between the measured heating rates and the von Kármán decay rates obtained in this present work and by Wu et al. (2022) is promising.

In any case, these robust observations of the extended cyclotron resonant process in the inner heliosphere suggest that kinetic-scale waves are strongly coupled to the turbulent cascade and provide an important pathway to the dissipation of turbulence in the solar wind and corona.

## Acknowledgments

T.A.B. acknowledges NASA grant No. 80NSSC24K0272. The work of I.V. was supported by NASA grant No. 80NSSC22K1634. B.D.G.C. acknowledges the support of NASA grant 80NSSC24K0171. J.S. and R.M. acknowledge the support of the Royal Society Te Apārangi, through Marsden-Fund grants MFP-UOO2221 (J.S.) and MFP U0020 (R.M.), as well as through the Rutherford Discovery Fellowship RDF-U001804 (J.S.). A.C. acknowledges NASA grant No. 80NSSC23K0737.

# Appendix A

## Numerical Determination of Quasilinear Heating Rate

Determination of $Q_{\text{ICW}}$ and $Q(v_{\perp}, v_{\|})$ is obtained by integrating Equation (12) by parts to obtain

$$Q_{\text{ICW}} = -\pi^2 \Omega_p^2 m_p \int_{-\infty}^{\infty} dv_{\|} \int_{0}^{\infty} dv_{\perp} \times \left[ \int_{0}^{\infty} dk_{\|} v_{\perp}^2 \delta(\omega_k - k_{\|} v_{\|} - \Omega_p) \frac{\omega_k^2}{k_{\|}^2} I(k_{\|}) \hat{G}_k \bar{g}(v_{\perp}, v_{\|}) \right]. \quad (A1)$$

The $\delta$-function in Equations (12) and (A1) corresponds to the ICW resonance condition and defines a single parallel wavenumber that resonates with particles at each parallel velocity. Change of variables yields a $\delta$-function for resonant wavenumber $k_{\text{res}}(v_{\|})$ as

$$\delta(\omega_k - k_{\|}v_{\|} - \Omega_p) = \frac{\delta(k_{\|} - k_{\text{res}})}{|v_g(k_{\text{res}}) - v_{\|}|}, \quad (A2)$$

where $v_g(k_{\text{res}}) = \partial\omega_k/\partial k|_{k_{\text{res}}}$ is the group speed evaluated at each $k_{\text{res}}$, which is the wavenumber that satisfies the delta function. For positive $k_{\|}$, taken as outward-going ICWs, there is no resonance unless $v_{\|} < 0$, limiting our concern only to negative parallel velocities. We use the $\delta$-function to substitute $k_{\text{res}}(v_{\|})$ for $k_{\|}$, thereby evaluating the integral over

wavenumber, and Equation (A1) becomes

$$Q_{\text{ICW}} = -\pi^2 \Omega_p^2 m_p \int_{-\infty}^{0} dv_{\|} \int_{0}^{\infty} dv_{\perp} \times \left[ \frac{v_{\perp}^2}{|v_g(k_{\text{res}}) - v_{\|}|} \frac{\omega_{k_{\text{res}}}^2}{k_{\text{res}}^2} I(k_{\text{res}}) \hat{G}_{k_{\text{res}}} \bar{g}(v_{\perp}, v_{\|}) \right], \quad (A3)$$

where the spectrum $I(k_{\|})$ is interpolated from $k_{\|}$ to $k_{\text{res}}$, constructing $I(k_{\text{res}})$. The integrand in Equation (A3) is entirely a function of $v_{\|}$ and $v_{\perp}$ and can be calculated via a numerical integration over velocity space. We further identify the differential heating rate

$$Q(v_{\perp}, v_{\|}) = -\pi^2 \Omega_p^2 m_p \frac{v_{\perp}^2}{|v_g(k_{\text{res}}) - v_{\|}|} \frac{\omega_{k_{\text{res}}}^2}{k_{\text{res}}^2} I(k_{\text{res}}) \hat{G}_k \bar{g}(v_{\perp}, v_{\|}). \quad (A4)$$

To perform the numerical integral in Equation (A3), we evaluate $Q(v_{\perp}, v_{\|})$ on a $1 \text{ km s}^{-1} \times 1 \text{ km s}^{-1}$ grid, which was found to be sufficient for convergence of Equation (A3). The range of integration is set to $-1000 < v_{\|} < 0 \text{ km s}^{-1}$ and $0 < v_{\perp} < 1000 \text{ km s}^{-1}$. We have removed contributions to the spectra with $k_{\|}d_i < 0.3$ and $k_{\|}d_i > 5$, which for $\beta_i = 0.1$ corresponds to an approximate range of thermal speeds between 0.02 and 7.5 $v_{\text{th}}$. Figures 5(a) and (b) shows that the parallel thermal speeds hover near $50 \text{ km s}^{-1}$ over this entire interval. Thus, the $1000 \text{ km s}^{-1}$ limits are sufficiently large with respect to the thermal and Alfvén speeds for this stream that there are no significant contributions in portion of phase space out of the bounds of integration. The heating rate as a function of parallel resonant velocity $Q(v_{\|})$ is computed by integrating $Q(v_{\perp}, v_{\|})$ over perpendicular velocities.

These methods have previously been used to obtain heating rates $Q_{\text{ICW}}$ of order ~10% of the turbulent cascade rate (Bowen et al. 2022), though this previously studied interval contained significantly fewer left-hand-polarized waves than this stream. Our results shown in Figure 5 suggest that a large amount of the turbulent energy likely enters the plasma via cyclotron resonant heating.

# Appendix B

## Application of Nonparametric Representations

While in situ observations of the proton velocity distribution have long been recognized to have a mostly Gaussian shape (Marsch & Goldstein 1983) and drifting bi-Maxwellian approximation has regularly been used to approximate ion velocity distributions (Marsch et al. 1982; Marsch & Tu 2001a; Tu & Marsch 2001; Heuer & Marsch 2007; Klein et al. 2021; Verniero et al. 2022; McManus et al. 2024), it is clear that deviations from nonthermal structure in the velocity distribution can affect wave-particle resonant processes (Dum et al. 1980; Bowen et al. 2022; Walters et al. 2023). Furthermore, it is well known that the equilibrium kinetic contours for dispersive ICW resonance do not coincide with Maxwellian curvature (Isenberg & Lee 1996), such that relaxation via ICW resonant processing cannot be entirely captured via bi-Maxwellian approximations (Heuer & Marsch 2007). In this sense it is important to test the heating rates computed from the drifting bi-Maxwellian model in Equation (11) against heating
rates computed from nonparametric models of the velocity distribution.

### B.1. Hermite Polynomial Interpolation

Bowen et al. (2022) previously used linear least-squares fits of orthogonal Hermite polynomials to estimate non-Maxwellian features in the velocity distribution; following this previously developed method, we perform a linear least-squares fit to the average proton velocity distribution in each 128 s interval using the Hermite polynomials $H_n$ and Hermite functions $\phi_m$,

$$ \bar{g}^H(v_\perp, v_\parallel) = \sum_{m,n} A_{mn} \phi^m(v_\perp/w_\perp\text{th}) \phi^n(v_\parallel/w_\parallel\text{th}), \quad (B1) $$

$$ H_n(v) = (-1)^n e^{v^2} \frac{d^n}{dx^n} e^{-v^2}, \quad (B2) $$

$$ \phi^m(v) = \frac{H^m(v)}{\sqrt{2^{m\pi^{1/2}}m!}} e^{-v^2}. \quad (B3) $$

We fit an orthogonal set of Hermite functions to each velocity distribution by constructing the matrix $A$, which consists of the coefficients $A_{mn}$ corresponding to each 2D Hermite polynomial combination. We choose maximum sixth-order Hermite polynomials, which performed reasonably better at replicating the observed distributions than the bi-Maxwellian fits in Bowen et al. (2022). Linear least-squares fitting of $\Phi$, with elements $\phi^m \phi^n$, to the observed velocity distribution, written as matrix $g$, such that inversion of $g = A\Phi$ gives

$$ A = g\Phi^T(\Phi W\Phi^T)^{-1}. \quad (B4) $$

A singular value decomposition (SVD) determines the pseudo-inverse of $\Phi\Phi^T$. The matrix $A$ then gives the best-fit coefficients for the Hermite polynomial composition.

We furthermore adopt errors on $g$ associated with Poisson counting, such that the uncertainty at each velocity coordinate $k$ is $\sigma_{gk} = \sqrt{g_k}$ (Wüest et al. 2007). Poisson noise between energy bins is uncorrelated such that we construct the diagonal-weight matrix as $W$ with entries $\sigma^2$ that can be included in weighted linear least-squares fitting as $gW = A\Phi W$ such that

$$ A = gW\Phi^T(\Phi W\Phi^T)^{-1}. \quad (B5) $$

The coefficients $A$ are then used to construct $\bar{g}^H(v_\perp, v_\parallel)$ using Equation (B1).

Using $\bar{g}^H(v_\perp, v_\parallel)$, which is an analytic, differentiable function, it is possible to separately estimate $Q_{ICW}$ without the assumption of a drifting bi-Maxwellian structure to the velocity distribution.

### B.2. Radial Basis Function Interpolation

We further include another approximation to the velocity distributions through interpolation via RBFs (Broomhead & Lowe 1988; Bowen et al. 2023) to model the distribution function via a summed set of interpolating functions given by

$$ g_p^{\text{RBF}}(v_\perp, v_\parallel) = \sum_{i}^{N_{\text{RBF}}-1} R_i \psi_i(\zeta_i), \quad (B6) $$

with $\zeta_i = v - v_i^c$, where $v_i^c$ is the center of each interpolating RBF, denoted by $\psi_i$. The RBF method requires choice of a basis function, $\psi$, which we choose to be an isotropic 2D bi-

Maxwellian

$$ \psi_i(v_\perp, v_\parallel) = \frac{1}{\sqrt{\pi^3} w_{\text{RBFi}}^3} \exp \left[ - \frac{(v_\perp - v_{\parallel i}^c)^2}{w_{\text{RBFi}}^2} - \frac{(v_\parallel - v_{\parallel i}^c)^2}{w_{\text{RBFi}}^2} \right]. \quad (B7) $$

The number of interpolating functions, $N_{RBF}$, must be specified along with the thermal speed, $w_{RBFi}$, and central location, $v_i^c$, of each of the $\psi_i$ implemented in the interpolation. In principle, $w_{RBFi}$ does not need to be isotropic and can vary for each $\psi_i$. We determine that setting a Maxwellian RBF at each measured SPAN energy bin with nonzero phase-space density gives suitable results; on average $N_{RBF} \approx 50$ basis functions, with a maximum of $N_{RBF} = 238$; all in all, 90% of the 6737 intervals have $N_{RBF} > 30$. We uniformly set $w_{RBF} = 40 \text{ km s}^{-1}$ for each basis function, which we find is suitable for interpolating the proton distribution.

The interpolated velocity distribution via RBFs is then given by

$$ \bar{g}^{\text{RBF}}(v_\perp, v_\parallel) = \sum_{i}^{N_{\text{RBF}}-1} R_i \psi_i(\zeta_i), \quad (B8) $$

with $\zeta = \vec{v} - \vec{v}_c$. Determination of the weights $R_i$ is again performed through SVD estimation for the pseudo-inverse giving a least-squares fit of $\bar{g}^{\text{RBF}}$ to the observed velocity distribution. $N_{RBF}$ is by construction less than the total number of SPANi energy bins; in this case, the system of equations is overdetermined. We again compute the RBF on each 128 s average velocity distribution and include weighted errors in the least-squares fit corresponding to Poisson noise analogous to Equation (B5).

### B.3. Comparing Model Distributions

To verify the heating of the plasma via ICW resonant interactions, we recompute Equation (12) using the Hermite polynomial and RBF approximations to the proton distribution, $\bar{g}^H(v_\perp, v_\parallel)$ and $\bar{g}^{\text{RBF}}(v_\perp, v_\parallel)$; we refer to the heating rates computed from Equation (12) via the drifting bi-Maxwellian, Hermite polynomials, and RBF interpolation as $Q_{BM}$, $Q_{HMTE}$, and $Q_{RBF}$ respectively; and we also compute the average heating rate of the three terms $Q_{AVG}$. Note that all quantities used to determine $I(k_\parallel)$ are independent of the kinetic phase-space distribution used; thus, variations in the integral in Equation (12) depend only on the action of the quasilinear operator on the modeled distribution $\hat{G}\bar{g}$.

Figures 6(a)–(c) respectively show $Q_{HMTE}$, $Q_{RBF}$, and $Q_{AVG}$ against $Q_{BM}$. While all computed heating rates have the same qualitative behavior, i.e., positive valued indicating the absorption of ICWs and plasma heating via cyclotron resonance, $Q_{HMTE}$ tends to systematically underestimate the heating rates predicted by $Q_{BM}$, whereas $Q_{RBF}$ tends to overestimate $Q_{BM}$. The average of all heating rates $Q_{AVG}$ closely follows $Q_{BM}$; thus, $Q_{BM}$ was chosen as the value reported in the body of the manuscript.

The bottom panels show statistics on the quality of the various models. To compare the quality of the relative models, we compute two parameters. Figure 6(d) shows the distribution of measured Pearson correlation coefficients between the logarithm of the observed distribution $\text{Log}_{10}g_p(v_\perp, v_\parallel)$ and the three models $\text{Log}_{10}\bar{g}^{\text{BM}}(v_\perp, v_\parallel)$, $\text{Log}_{10}\bar{g}^H(v_\perp, v_\parallel)$, and
<figure><img src="image_10.png" /><figcaption>**Figure 6.** (a) 2D histogram of $Q_{\text{HMTE}}$ against $Q_{\text{BM}}$ for each of the 6737 studied intervals; data are column normalized, and the black lines show $Q_{\text{HMTE}}/Q_{\text{BM}}$ at values of 0.1, 1, and 10. (b) Same as panel (a), but with $Q_{\text{RBF}}$ against $Q_{\text{BM}}$. (c) Same as panel (a), but with the average of all three heating rates $Q_{\text{AVG}}$ against $Q_{\text{BM}}$. Contours in each plot show counts at 10, 20, and 50 levels. (d) Histogram showing distributions of Pearson $R^2$ correlation between the 6737 modeled distributions and observed SPANi distributions; drifting bi-Maxwellian (BM) in black; Hermite (HMTE) in teal; RBF in orange. (e) Histogram showing distributions of $\chi^2_{\text{RBF}}/\chi^2_{\text{BM}}$ (orange) and $\chi^2_{\text{HMTE}}/\chi^2_{\text{BM}}$ (teal) on a logarithmic scale.</figcaption></figure>

$\text{Log}_{10}g^{\text{RBF}}(v_{\text{L}}, v_{\|})$. High levels of correlation indicate that the model correctly matches the observed distribution. The use of the logarithm of the distribution in computing the Pearson $R^2$ correlation coefficient allows for weighting of the entirety of phase space and includes the regions where ICW resonant interactions are most important; without the logarithmic weighting, the correlation coefficient is mostly dominated by the model's performance at the peak of the distribution. We find that $R^2$ is largest for the RBF interpolation and smallest for the parametric drifting bi-Maxwellian fit. Interestingly, the level of heating computed between these models remains relatively

constant even though the $R^2$ values vary; this suggests that variations between the three models are mostly occurring in regions that are not resonant with the outward-going ICWs. We speculate that these differences are likely due to the model's ability to approximate the ion beam population.

We also compute the $\chi^2$ of each of the models. Given the nature of the nonlinear fitting, there is a lack of a clear number of degrees of freedom. Furthermore, errors in the model are not likely to obey Gaussian statistics. For these reasons, hypothesis testing via the $\chi^2$ statistic is not possible. To determine the quality of the fits, we compare the ratio of $\chi^2$ with the value
compared for the bi-Maxwellian, $\chi_{BM}^2$. Figure 6(e) shows the distribution of the logarithmic ratio of $\chi_{RBF}^2/\chi_{BM}^2$ and $\chi_{HMTE}^2/\chi_{BM}^2$. The RBF model tends to perform much better than the drifting bi-Maxwellian with a uniform $\chi_{RBF}^2/\chi_{BM}^2 < 1$. In terms of $\chi^2$, the Hermite polynomial approximation tends to have a similar performance to the drifting bi-Maxwellian fit in this stream, though the Pearson correlation is significantly better than the bi-Maxwellian fit. Again, the ability of each of the models to reproduce similar heating rates, but with significantly different $\chi^2$, suggests that the main difference in the models may be in reproducing the ion beam. Our future work will investigate further the role of the ion beam in these processes.

In any case, these various parametric and nonparametric models for the observed proton distribution function all recover heating rates that suggest that ICWs are absorbed into the core of the proton distribution, resulting in plasma heating via ICW resonance.

## ORCID iDs

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

Ivan Y. Vasko https://orcid.org/0000-0002-4974-4786

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

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

Alexandros Chasapis https://orcid.org/0000-0001-8478-5797

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

Alfred Mallet https://orcid.org/0000-0001-9202-1340

Michael McManus https://orcid.org/0000-0001-6077-4145

Romain Meyrand https://orcid.org/0000-0002-8327-5848

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

Jonathan Squire https://orcid.org/0000-0001-8479-962X

## References

Alexandrova, O., Carbone, V., Veltri, P., & Sorriso-Valvo, L. 2008, ApJ, 674, 1153

Badman, S. T., Riley, P., Jones, S. I., et al. 2023, JGRA, 128, e2023JA031359

Bale, S., Badman, S., Bonnell, J., et al. 2019, Natur, 576, 237

Bale, S. D., Goetz, K., Harvey, P. R., et al. 2016, SSRv, 204, 49

Bandyopadhyay, R., Goldstein, M. L., Maruca, B. A., et al. 2020, ApJs, 246, 48

Bandyopadhyay, R., Oughton, S., Wan, M., et al. 2018, PhRvX, 8, 041052

Bavassano, B., Pietropaolo, E., & Bruno, R. 1998, JGR, 103, 6521

Boardsen, S. A., Jian, L. K., Raines, J. L., et al. 2015, JGRA, 120, 10207

Bowen, T. A., Bale, S. D., Bonnell, J. W., et al. 2020a, ApJ, 899, 74

Bowen, T. A., Bale, S. D., Bonnell, J. W., et al. 2020b, JGRA, 125, e27813

Bowen, T. A., Bale, S. D., Chandran, B. D. G., et al. 2024, NatAs, 8, 482

Bowen, T. A., Chandran, B. D. G., Klein, K. G., et al. 2023, XXXVth General Assembly and Scientific Symp. of the Int. Union of Radio Science (URSI GASS, 335) (Piscataway, NJ: IEEE), 335

Bowen, T. A., Chandran, B. D. G., Squire, J., et al. 2022, PhRvL, 129, 165101

Bowen, T. A., Mallet, A., Bale, S. D., et al. 2020c, PhRvL, 125, 025102

Bowen, T. A., Mallet, A., Huang, J., et al. 2020d, ApJS, 246, 66

Broomhead, D., & Lowe, D. 1988, Complex Systems, 2, 321

Bruno, R., Trenchi, L., & Telloni, D. 2014, ApJL, 793, L15

Chandran, B. D. G., Pongkitiwanichakul, P., Isenberg, P. A., et al. 2010, ApJ, 722, 710

Chen, C. H. K., Horbury, T. S., Schekochihin, A. A., et al. 2010, PhRvL, 104, 255002

Chew, G. F., Goldberger, M. L., & Low, F. E. 1956, RSPSA, 236, 112

Cranmer, S. R. 2000, ApJ, 532, 1197

Cranmer, S. R. 2014, ApJS, 213, 16

Davis, N., Chandran, B. D. G., Bowen, T. A., et al. 2023, ApJ, 950, 154

Denskat, K. U., Beinroth, H. J., & Neubauer, F. M. 1983, JGZG, 54, 60

Duan, D., Bowen, T. A., Chen, C. H. K., et al. 2020, ApJS, 246, 55

Duan, D., He, J., Bowen, T. A., et al. 2021, ApJL, 915, L8

Dudok de Wit, T., Alexandrova, O., Furno, I., Sorriso-Valvo, L., & Zimbardo, G. 2013, SSRv, 178, 665

Dudok de Wit, T., Krasnoselskikh, V. V., Agapitov, O., et al. 2022, JGRA, 127, e30018

Dum, C. T., Marsch, E., & Pilipp, W. 1980, JPIPh, 23, 91

Farge, M. 1992, AnRFM, 24, 395

Fox, N. J., Velli, M. C., Bale, S. D., et al. 2016, SSRv, 204, 7

Gary, S. P. 1993, Theory of Space Plasma Microinstabilities (Cambridge: Cambridge Univ. Press)

Goldstein, M. L., Roberts, D. A., & Fitch, C. A. 1994, JGR, 99, 11519

He, J., Marsch, E., Tu, C., Yao, S., & Tian, H. 2011, ApJ, 731, 85

He, J., Wang, L., Tu, C., Marsch, E., & Zong, Q. 2015, ApJ, 800, L31

Hellinger, P., Trávníček, P. M., Štverák, Š., Matteini, L., & Velli, M. 2013, JGRA, 118, 1351

Heuer, M., & Marsch, E. 2007, JGRA, 112, A03102

Hollweg, J. V., & Isenberg, P. A. 2002, JGRA, 107, 1147

Hollweg, J. V., & Johnson, W. 1988, JGR, 93, 9547

Hollweg, J. V., & Markovskii, S. A. 2002, JGRA, 107, 1080

Horbury, T. S., Forman, M., & Oughton, S. 2008, PhRvL, 101, 175005

Hossain, M., Gray, P. C., Pontius, D. H. J., Matthaeus, W. H., & Oughton, S. 1995, PhFl, 7, 2886

Howes, G. G., & Quataert, E. 2010, ApJL, 709, L49

Huang, S. Y., Zhang, J., Sahraoui, F., et al. 2020, ApJL, 897, L3

Isenberg, P. A., & Lee, M. A. 1996, JGR, 101, 11055

Jannet, G., Dudok de Wit, T., Krasnoselskikh, V., et al. 2021, JGRA, 126, e28543

Jian, L. K., Wei, H. Y., Russell, C. T., et al. 2014, ApJ, 786, 123

Karpman, V. I. 1974, SSRv, 16, 361

Kennel, C. F., & Engelmann, F. 1966, PhFl, 9, 2377

Kiyani, K. H., Chapman, S. C., Khotyaintsev, Y., Dunlop, M. W., & Sahraoui, F. 2009, PhRvL, 103, 075006

Klein, K. G., Alterman, B. L., Stevens, M. L., Vech, D., & Kasper, J. C. 2018, PhRvL, 120, 205102

Klein, K. G., Verniero, J. L., Alterman, B., et al. 2021, ApJ, 909, 7

Leamon, R. J., Smith, C. W., Ness, N. F., Matthaeus, W. H., & Wong, H. K. 1998, JGR, 103, 4775

Lion, S., Alexandrova, O., & Zaslavsky, A. 2016, ApJ, 824, 47

Liu, W., Zhao, J., Wang, T., et al. 2023, ApJ, 951, 69

Livi, R., Larson, D. E., Kasper, J. C., et al. 2022, ApJ, 938, 138

Lotz, S., Nel, A. E., Wicks, R. T., et al. 2023, ApJ, 942, 93

Luo, Q., Zhu, X., He, J., et al. 2022, ApJ, 928, 36

Marsch, E. 2006, LRSP, 3, 1

Marsch, E., & Goldstein, H. 1983, JGR, 88, 9933

Marsch, E., Schwenn, R., Rosenbauer, H., et al. 1982, JGR, 87, 52

Marsch, E., & Tu, C. Y. 2001a, JGR, 106, 227

Marsch, E., & Tu, C. Y. 2001b, JGR, 106, 8357

McManus, M. D., Bowen, T. A., Mallet, A., et al. 2020, ApJS, 246, 67

McManus, M. D., Klein, K. G., Larson, D., et al. 2024, ApJ, 961, 142

Meyrand, R., Squire, J., Schekochihin, A. A., & Dorland, W. 2021, JPIPh, 87, 535870301

Ofman, L., Boardsen, S. A., Jian, L. K., Verniero, J. L., & Larson, D. 2022, ApJ, 926, 185

Parker, E. N. 1958, ApJ, 128, 664

Podesta, J. J., & Gary, S. P. 2011, ApJ, 734, 15

Richardson, J. D., Paularena, K. I., Lazarus, A. J., & Belcher, J. W. 1995, GeoRL, 22, 325

Roberts, D. A., Klein, L. W., Goldstein, M. L., & Matthaeus, W. H. 1987, JGR, 92, 11021

Sahraoui, F., Goldstein, M. L., Robert, P., & Khotyaintsev, Y. 2009, PhRvL, 102, 231102

Shankarappa, N., Klein, K. G., & Martinovic, M. M. 2023, ApJ, 946, 85

Shebalin, J. V., Matthaeus, W. H., & Montgomery, D. 1983, JPIPh, 29, 525

Shklyar, D., & Matsumoto, H. 2009, SGEO, 30, 55

Smith, C. W., Hamilton, K., Vasquez, B. J., & Leamon, R. J. 2006, ApJL, 645, L85

Smith, C. W., Stawarz, J. E., Vasquez, B. J., Forman, M. A., & MacBride, B. T. 2009, PhRvL, 103, 201101

Smith, C. W., Vasquez, B. J., & Hollweg, J. V. 2012, ApJ, 745, 8

Smith, W. C., Matthaeus, H. W., & Ness, F. N. 1990, ICRC (Adelaide), 5, 280

Squire, J., Meyrand, R., Kunz, M. W., et al. 2022, NatAs, 6, 715

Stix, T. H. 1992, Waves in Plasmas (New York: AIP)

Telloni, D., Carbone, F., Bruno, R., et al. 2019, ApJL, 885, L5

Tu, C.-Y., & Marsch, E. 1995, SSRv, 73, 1

Tu, C.-Y., & Marsch, E. 1997, SoPh, 171, 363
Tu, C. Y., & Marsch, E. 2001, JGR, 106, 8233
Usmanov, A. V., Goldstein, M. L., & Matthaeus, W. H. 2014, ApJ, 788, 43
Vech, D., Kasper, J. C., Klein, K. G., et al. 2020, ApJS, 246, 52
Vech, D., Mallet, A., Klein, K. G., & Kasper, J. C. 2018, ApJL, 855, L27
Verniero, J. L., Chandran, B. D. G., Larson, D. E., et al. 2022, ApJ, 924, 112
Verniero, J. L., Larson, D. E., Livi, R., et al. 2020, ApJS, 248, 5
Verscharen, D., Chen, C. H. K., & Wicks, R. T. 2017, ApJ, 840, 106
Viñas, A. F., & Gurgiolo, C. 2009, JGRA, 114, A01105
Walters, J., Klein, K. G., Lichko, E., et al. 2023, ApJ, 955, 97

Wan, M., Matthaeus, W. H., Karimabadi, H., et al. 2012, PhRvL, 109, 195001
Wicks, R. T., Alexander, R. L., Stevens, M., et al. 2016, ApJ, 819, 6
Woodham, L. D., Wicks, R. T., Verscharen, D., & Owen, C. J. 2018, ApJ, 856, 49
Woodham, L. D., Wicks, R. T., Verscharen, D., et al. 2019, ApJL, 884, L53
Wu, H., Tu, C., He, J., Wang, X., & Yang, L. 2022, ApJ, 926, 116
Wüest, M., Evans, D. S., & von Steiger, R. 2007, Calibration of Particle Instruments in Space Physics (Noordwijk: ESA Communications)
Zhao, G. Q., Lin, Y., Wang, X. Y., et al. 2021, ApJ, 906, 123