The Opposite Behaviors of Proton and Electron Temperatures in Relation to Solar Wind Magnetic Energy: Parker Solar Probe Observations

Solar wind heating is an outstanding issue that has been discussed for decades. Research on the connection between solar wind particle temperatures and turbulence may provide insight into this issue. Based on Parker Solar Probe observations, this paper investigates the properties of solar wind proton and electron temperatures in relation to turbulent magnetic energy, via the calculation of correlation coefficients (CCs) between particle temperatures and magnetic energy. The calculations are regulated by the spatial scale, plasma beta (β), and the angle between the solar wind velocity and background magnetic field, where the plasma beta is the ratio of plasma thermal to magnetic pressure. Results show that the correlation between proton temperature and magnetic energy is positive and can be strong with a CC exceeding 0.8. The strong correlation preferentially occurs at ion scales, with the wind velocity and background magnetic field quasi-perpendicular and over a wide beta range (β < 3.0). On the other hand, the correlation between electron temperature and magnetic energy is commonly negative, often with an intermediate or negligible CC, accordingly. The CC with an amplitude up to 0.8 can arise at larger scales with the wind velocity and background magnetic field quasi-(anti)parallel and in the low-beta case (β < 0.6). The implication of these findings on the physics of turbulent heating in the solar wind is discussed.


Introduction
The solar wind originates from the corona of the Sun.It streams outward and expands into the interplanetary space (Cranmer & Winebarger 2019;Rouillard et al. 2021).The expansion has been shown to be nonadiabatic, indicating that a heating process must take place in the solar wind.Many energy sources, such as ion cyclotron waves, magnetosonic waves, and coherent structures with current sheets, are proposed to contribute to the heating, while turbulence has attracted much attention (e.g., He et al. 2018;Shoda et al. 2019;Adhikari et al. 2020aAdhikari et al. , 2020b;;Zank et al. 2020;Zhao et al. 2021;Wu et al. 2023;Yang et al. 2023).In the context of turbulence, one crucial question is how turbulence dissipates and heats the plasma (Parashar et al. 2015;Chen 2016;Viall & Borovsky 2020).Addressing this question is not only helpful in understanding the solar wind heating problem but is also important in understanding the mechanism of the formation of the hot solar corona, which has a temperature several hundred times higher than the visible solar surface (Cranmer et al. 2015;Kiyani et al. 2015).Many authors believe that the turbulence in the corona works effectively to maintain the high temperature, although other mechanisms are possible (e.g., Chen et al. 2020;Howson et al. 2020;Kasper et al. 2021;Zank et al. 2021).
Protons and electrons are the two main populations of the solar wind.Research on proton and electron temperatures in relation to solar wind parameters and magnetic fluctuations can provide indications and constraints on the physics of the solar wind heating process.Early observations have shown that higher proton temperature is usually accompanied by faster solar wind speed (e.g., Marsch et al. 1982;Elliott et al. 2012).Later observations by Wind showed that higher proton temperature is also associated with a steeper proton-scale magnetic spectrum (Leamon et al. 1998).Statistical studies by Wind and Advanced Composition Explorer measurements have revealed that higher proton temperature is related to larger inertial-scale magnetic energy (Smith et al. 2006;Vech et al. 2018).In the most recent studies based on Wind and Parker Solar Probe (PSP) observations, it was revealed that higher proton temperature is significantly linked to larger proton-scale magnetic energy.The link exists at 1 au and near 0.2 au and appears to be a common feature of solar wind turbulence (Zhao et al. 2020a(Zhao et al. , 2022b)).For electron temperature, less of a relation has been reported.To the best of our knowledge, only the correlation between electron temperature and solar wind speed has been found for the nascent, young solar wind (Geiss et al. 1995;Maksimovic et al. 2020;Halekas et al. 2022;Shi et al. 2023).
Following the above studies, there is much room for the study of the relation of proton and electron temperatures to turbulent magnetic energy in the solar wind.Zhao et al. (2020aZhao et al. ( , 2022b) ) investigated the correlation between the proton temperature and magnetic energy by calculating their correlation coefficients (CCs).Their results showed that this correlation is scale dependent, with the largest CC occurring at ion scales.The studies by Zhao et al. (2020aZhao et al. ( , 2022b)), however, merely focused on the situation in which solar wind velocity is quasi-perpendicular to the ambient magnetic field (with an angle greater than 60°).A study on the case where the solar wind velocity is quasi-parallel to the ambient magnetic field has not been presented.The dependence of the correlation on the parameter of the plasma beta has not been well 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.
discussed.A correlation of electron temperature with magnetic energy has also not been demonstrated.
The present study aims to fill this gap by investigating the correlations of both proton and electron temperatures with solar wind magnetic energy in various situations.This paper is organized as follows.Section 2 introduces the data and analysis methods used in this study.Section 3 provides the statistical results of the correlations in various situations.Section 4 presents a summary and discussion.

Data and Analysis Methods
The data used in the present study are from the PSP observations during the first seven close encounters where the data are with high cadences.The magnetic field is obtained from the FIELDS flux-gate magnetometer with a cadence varying between 73.24 and 292.97 Hz (Bale et al. 2016).The proton bulk velocity, density, and temperature are derived by the Solar Probe Cup of the SWEAP investigation with a cadence varying between 1.14 and 4.58 Hz (Kasper et al. 2016).The electron temperature is produced by the Solar Probe ANalyzer-Electron of the SWEAP investigation and typically has a cadence of 0.07-0.14Hz (Halekas et al. 2020;Whittlesey et al. 2020).The choice of the first seven encounters is a compromise between the usage of more data and the applicability of the Taylor frozen-in-flow hypothesis employed in the present study (Taylor 1938).As the number of encounters increases, the perihelia of PSP orbits become closer to the Sun, where the spacecraft speed and Alfvén speed near the perihelia can be considerably high compared to the solar wind speed.(After the seventh encounter, the orbits of PSP have perihelia with heliocentric distances less than 0.075 au and the spacecraft speed can be high up to 150 km s −1 (Szalay et al. 2020.)Consequently, the validity of the Taylor hypothesis will be considerably reduced and less accurate results arise under the Taylor hypothesis (Zhao et al. 2020b;Perez et al. 2021).We hence do not use the data with more encounters in the present study.For a long time period, we divide it into consecutive and overlapping time segments, with each time segment 600 s and the overlap time of 200 s (Zhao et al. 2022c).A total of ∼1.0 × 10 4 time segments with data available are obtained.Proton and electron temperatures refer to average values over a time segment.The magnetic energy (P k ) is scale dependent, and three steps are performed to calculate P k for each time segment (Zhao et al. 2020a(Zhao et al. , 2022b)).First, the energy spectrum of magnetic fluctuations is calculated by the standard Fourier transform technique in the frequency ( f ) domain.Second, the local average operation over a varying window fe −0.5 f fe 0.5 is conducted to produce smoothed magnetic energy.Third, to show the results in spatial scales, the frequency domain is converted to the wavenumber (k) domain in terms of the Taylor hypothesis, 2πf = kU p , where U p is the solar wind speed in the spacecraft reference frame (Zhao et al. 2020b(Zhao et al. , 2022b)).
Figure 1 displays the distributions of parameters used in this study.Panels (a)-(d) show the proton temperature (T p ), electron temperature (T e ), plasma beta (β), and the angle between the solar wind velocity and background magnetic field (θ vb ), respectively, where β is the ratio of plasma thermal to magnetic pressure.The average values of T p and T e in the figure are comparable (∼3.4 × 10 5 K); the distribution of T p is significantly wider than the distribution of T e , with the standard deviations of 1. 8 × 10 5 and 0.4 × 10 5 K for T p and T e , respectively.The β has an average value of 1.3 ± 1.1, and 94.2% of the cases have values less than 3.0.In the present study, the angle θ vb is restricted to be between 0°and 90°for convenience.It has an average value of 37°.4 ± 18°.3; 69.0% of the cases have values less than 45°.This θ vb distribution means that the proton bulk velocity is quasi-(anti)parallel to the background magnetic field in most cases.

Statistical Results
As described in the introduction, previous studies focused on the angle range of θ vb > 60°.They showed that the correlation between T p and P k is positive and enhanced at ion scales with wavenumber kρ p ; 0.5, where ρ p is the proton thermal gyroradius.Figure 2 presents scatter plots to display the correlation of (P k , T p ) at a fixed scale with wavenumber kρ p = 0.5, but for another angle range of 30°< θ vb < 60°.One can see a clear and positive correlation in this angle range at the given scale.Here, we emphasize that T p (and T e ) in the present paper is a background quantity (local average value) and does not depend on kρ p .The correlation of (P k , T p ) depends on kρ p because P k is a function of kρ p .For the sake of convenience, we describe the correlation using kρ p in the present paper.Panels (a)-(c) are for β < 0.6, 0.6 < β < 1.2, and 1.2 < β < 3.0, and the corresponding CCs are 0.82, 0.90, and 0.89, respectively.The comparable CCs suggest that the CC weakly depends on β for β < 3.0, though the CC with β < 0.6 is slightly lower than the other two cases.
In order to explore which scales of magnetic energy are the most relevant to the temperature, Figure 3 plots the CC of (P k , T p ) against the wavenumber, showing the dependence of the CC on scales.The plot is also regulated by three θ vb ranges to explore the possible effect of θ vb on the correlation.The three ranges are θ vb < 30°, 30°< θ vb < 60°, and θ vb > 60°, marked by the red, green, and blue lines.In the present study, we use these three ranges to describe the quasi-(anti)parallel, moderately oblique, and highly oblique situations, respectively.Similar to Figure 2, panels (a)-(b) are for β < 0.6, 0.6 < β < 1.2, and 1.2 < β < 3.0.Several results can be found in Figure 3 as follows.First, the CC with θ vb > 60°is much higher than the CC with θ vb < 30°.(We do not discuss the situation in which β < 0.6 and θ vb > 60°because the percentage of the data for this situation is merely 0.4%, significantly less than in other situations.)Second, for a given θ vb range, the peak value of the CC curve in the low-beta case (panel (a)) is slightly lower than those with moderately high beta (panels (b) and (c)).Third, the peak sites of CC curves are around kρ p ; 0.5 in principle, though they slightly shift to a larger kρ p as the β range changes from β < 0.6 through 0.6 < β < 1.2 to 1.2 < β < 3.0.To show the details, Table 1 presents the peak values and sites of CC curves for the various situations shown in Figure 3.
Figure 4 displays the correlation of (P k , T e ) via the scatter plot.It is at the scale of kρ p = 0.1 and in the angle range of θ vb < 30°.Similarly, panels (a)-(c) are for β < 0.6, 0.6 < β < 1.2, and 1.2 < β < 3.0.From panel (a), it can be seen that there is a trend in which T e decreases as P k increases statistically, which indicates a negative correlation.The amplitude of (negative) CC shown in panel (a) is considerable, up to 0.6.The amplitude of CC shown in panels (b) and (c), however, is small/low to 0.42 and 0.22, respectively.This means that the CC depends on β.The CC with β < 0.6 has a larger amplitude of CC than the other two cases.
Following the format in Figure 3, Figure 5 plots the CC of (P k , T e ) against the wavenumber, which is also regulated by the three θ vb ranges, θ vb < 30°, 30°< θ vb < 60°, and θ vb > 60°.The results shown in Figure 5 are somewhat complicated.Nevertheless, several points can be made.First, the CC in Figure 5 is generally negative.Second, the largest amplitude of the CC that occurs is shown by the red curve in panel (a).This means that the correlation between T e and P k is negative and the condition of low β and small θ vb produces the strongest correlation of (P k , T e ), which is significantly different from the results for protons  (Figure 3).Third, also unlike the case of protons, the peak behavior of the CC curves shown in Figure 5 is commonly not obvious.Moreover, for a given CC curve, the scales corresponding to large amplitudes of the CC can be at larger scales with respect to ion scales.They are with kρ p  0.3, as shown by the red curve in panel (a).
Note that the above results are based on the data covering a heliocentric distance range of 0.14-0.24au and potentially suffer from radial effects (due to radial evolution of the solar wind).To test the radial effects, we investigate the data with narrower heliocentric distance ranges.According to our test results, the radial effects appear to be relatively weak for protons in situations in which θ vb > 30°, while they can be strong for electrons because they can considerably affect the CC of (P k , T e ) in all situations of θ vb .To demonstrate this point, Figures 6 and 7 present the data corresponding to distance ranges of 0.14-0.17(top panels), 0.17-0.20 (middle panels), and 0.20-0.24au (bottom panels), where the data size is ∼0.3 × 10 4 for each distance range; only those curves with a sample number larger than 80 are plotted.Figures 6 and 7 follow the format as in Figure 3 (or Figure 5), and are for protons and electrons, respectively.One can see that the CC shown in Figure 6 is comparable to that shown in Figure 3, in principle, except for the cases shown by the red curves in planes (b), (c), and (g) that are in the situation in which θ vb < 30°.The CC in Figure 7, however, can change much with respect to that in Figure 5. From panels (d)-(f) in Figure 7, it can be seen that the amplitude of the CC is overall much larger than that in Figure 5. Consequently, both low and moderately high β (<3.0) can contribute to an intermediately large CC amplitude (∼0.6).Peaks can be found in some of the curves shown in Figure 7.For instance, the green curves in panels (d) and (e), corresponding to the situation of 30°< θ vb < 60°, have Figure 3. CC of (P k , T p ) against wavenumber k for (a) β < 0.6, (b) 0.6 < β < 1.2, and (c) 1.2 < β < 3.0, where red, green, and blue lines mark the ranges of θ vb < 30°, 30°< θ vb < 60°, and θ vb > 60°, respectively.peaks at kρ p ; 0.28 and 0.40, respectively.In particular, a peak at kρ p ; 0.25 is visible in the red curve in panel (d), corresponding to the situation where θ vb < 30°and β < 0.6.This peak presents the largest CC amplitude (0.86), as shown in Figure 7.To show the details responsible for this largest CC amplitude, the scatter distribution of (P k , T e ) is plotted in Figure 8, where a clear correlation can be seen.

Summary and Discussion
The present study first investigates the correlation between T p and P k with different ranges of θ vb for the data during the heliocentric distance range of 0.14-0.24au.The results show that the CC of (P k , T p ) strongly depends on θ vb .According to Figure 3, the condition θ vb < 30°corresponds to the peak values (∼0.6) of the CC curves, while θ vb > 60°contributes to the significantly large peak values (∼0.9) of CC curves for moderately high β.On the other hand, the CC of (P k , T p ) weakly depends on β.The peak values of CC curves, for a given θ vb range, slightly increase when β changes from the case of β < 0.6 to the case of β > 0.6.Nearly irrespective of the θ vb and β ranges, the scales corresponding to the peak values of CC curves are typically at kρ p ; 0.5.Further investigation of the data with the narrower distance ranges suggests that radial effects on the correlation are weak if θ vb > 30°is satisfied (Figure 6).
The present study also explores the correlation between T e and P k , which has been rarely discussed before.The results show that the CC of (P k , T e ) is commonly negative.It suffers from considerable radial effects, and often has an amplitude less than 0.6.For the situation with the distance range 0.14-0.24au, the intermediately large CC amplitude (∼0.6) arises merely when both θ vb < 30°and β < 0.6 are satisfied.In this situation, there are no obvious peaks for CC curves of (P k , T e ).The intermediately large CC amplitude occurs at larger scales with kρ p  0.3.For the situations with the narrower distance ranges, the CC amplitude can become larger and an intermediately large CC amplitude can occur for moderately high β.Some peaks of the CC curves are visible.The peak with CC amplitude up to 0.86 occurs at kρ p ; 0.25, for the situation where θ vb < 30°and β < 0.6 (panel (d) in Figure 7).
It should be notable that the behaviors of T p and T e in relation to P k are opposite.One is a positive correlation, while the other is a negative correlation.The strongest correlation for protons occurs at the ion scales, with the wind velocity and background magnetic field highly oblique and in the case of moderately high plasma beta.The strongest correlation for electrons, however, arises at larger scales, with the wind velocity and background magnetic field quasi-(anti)parallel and in the case of low beta.
The above results provide observational implications on the physics of turbulent heating in the solar wind.Because the strongest correlation for protons occurs at ion scales with the wind velocity highly oblique to the background magnetic field, an implication is perhaps that the proton heating tends to be controlled by ion-scale k ⊥ spectra of magnetic energy, where k ⊥ refers to the wavenumber perpendicular to the background magnetic field.In contrast, the electron heating seems to be related to larger-scale k ∥ spectra in the low-beta solar wind, where k ∥ refers to the wavenumber parallel to the background magnetic field; a lower electron temperature would be accompanied by a stronger k ∥ spectrum.
Furthermore, one may note that the negative correlation of (P k , T e ) appears to be inconsistent with direct anticipation.Intuitively, larger magnetic energy (turbulence amplitude) would contribute to higher proton and electron temperatures if one speculates that turbulence with a larger amplitude has a greater ability to heat protons and electrons.Existing simulations support this speculation because simulation results show that larger initial turbulence amplitudes lead to higher proton and electron temperatures (or heating rates; Wu et al. 2013;Hughes et al. 2017).Consequently, positive correlations could be anticipated for not only protons but also electrons.An explanation for this inconsistency may be based on the recently discovered helicity barrier effect for imbalanced magnetized turbulence in a low-beta plasma (Meyrand et al. 2021;Squire et al. 2022).The barrier effect inhibits turbulent energy from cascading past the scales of the barrier (around ρ p ) and causes energy growth above those scales.This process favors the heating of protons but weakens the heating of electrons since less energy can cascade to electron scales where electron heating efficiently occurs (Sahraoui et al. 2009;Chen et al. 2019).If the barrier effect is stronger, the turbulence amplitude at large scales (above ρ p ) would be larger due to more energy accumulation above the scales of the barrier, and meanwhile, the degree of weakening of the electron heating is stronger.In this regard, a negative correlation between T e and P k would be produced.
One may also note that our results are in line with the recent finding by Bandyopadhyay et al. (2023).The authors investigated the empirical proton and electron heating rates and showed that proton heating becomes increasingly dominant over electron heating as heliocentric distances decrease.In particular, they found that the protons receive about 80% of the total plasma heating near the Sun, significantly larger than that for electrons, about 20% (Bandyopadhyay et al. 2023).
Finally, several remarks are in order.First, it is unclear at present why the peak sites of the CC curves of (P k , T p ) are typically at kρ p ; 0.5, i.e., near the scale of double of the proton gyroradius.(In our calculations, kρ p = 1.0 means a scale equal to ρ p .)This result tends to imply that some heating processes become efficient at kρ p ; 0.5, and stochastic heating might be relevant (Voitenko & Goossens 2004;Martinović et al. 2020;Zhao et al. 2022a).Existing studies have shown that stochastic heating has the highest heating rate near the ion (proton) gyroradius scale (Chandran et al. 2010;Cerri et al. 2021).We speculate that a large CC might be attributed to not only a high heating rate but also a large magnetic energy, and it is possible that the largest CC of (P k , T p ) would occur at kρ p ; 0.5, although the heating rate would not be not the highest, the magnetic energy is larger than that at smaller scales.Second, the CC curves of (P k , T p ) with θ vb < 30°can vary, as shown in Figure 6, and the reason is also unclear.It is known that ion-scale electromagnetic waves are frequently observed in the solar wind when the wind velocity is quasi-(anti)parallel to the background magnetic field (Jian et al. 2009;Zhao et al. 2019;Bowen et al. 2020).Hence, the possible role of the presence of the ion-scale waves on the correlation between T p and P k needs to be investigated in further studies.Third, the CC curves of (P k , T e ) in Figure 7 show that the correlation between T e and P k is more complex than that for protons.The electron temperature would depend on many Figure 6.CC of (P k , T p ) against wavenumber k for the data during heliocentric distance ranges of 0.14-0.17(top panels), 0.17-0.20 (middle panels), and 0.20-0.24au (bottom panels), respectively.The format in Figure 3 is followed.
processes in the solar wind, for instance, the electron heat flux, kinetic Alfvén waves, ion acoustic waves, and intermittent structures (Gary et al. 1998;Sahraoui et al. 2009;Mozer et al. 2022;Hubert et al. 2023;Phillips et al. 2023).The present study tends to imply that some ion-scale process of turbulence may affect the electron temperature since the considerable correlation between T e and P k exists near ion scales.The underlying physical mechanism, however, remains to be studied.
In summary, this paper investigates the properties of proton and electron temperatures in relation to solar wind magnetic energy in various situations and reveals their opposite behaviors.A negative correlation between electron temperature and magnetic energy is particularly found.We propose that the helicity barrier effect may be causing the negative correlation.Further investigation on this issue is desirable.
Figure 7. CC of (P k , T e ) against wavenumber k for the data during heliocentric distance ranges of 0.14-0.17(top panels), 0.17-0.20 (middle panels), and 0.20-0.24au (bottom panels), respectively.The format in Figure 5 is followed.
Figure 8. Scatter plot of (P k , T e ) for the data during the heliocentric distance range of 0.17-0.20 au, where θ vb < 30°, β < 0.6, and kρ p = 0.25 are used.

Figure 1 .
Figure 1.Panels (a)-(d): distributions of proton temperature (T p ), electron temperature (T e ), plasma beta (β), and the angle between the solar wind velocity and background magnetic field (θ vb ).

Table 1
Peak Values and Sites of CC Curves in Figure3