# Proton and Electron Temperatures in the Solar Wind and Their Correlations with the Solar Wind Speed

Chen Shi, Marco Velli, Roberto Lionello, Nikos Sioulas, Zesen Huang, Jasper

S. Halekas, Anna Tenerani, Victor Réville, Jean-Baptiste Dakeyo, Milan

Maksimović, et al.

► To cite this version:

Chen Shi, Marco Velli, Roberto Lionello, Nikos Sioulas, Zesen Huang, et al. Proton and Electron Temperatures in the Solar Wind and Their Correlations with the Solar Wind Speed. The Astrophysical Journal, 2023, 944, 10.3847/1538-4357/acb341. insu-04473230

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https://insu.hal.science/insu-04473230

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# Proton and Electron Temperatures in the Solar Wind and Their Correlations with the Solar Wind Speed

Chen Shi (时辰)<fnref n="1" /> ID, Marco Velli<fnref n="1" /> ID, Roberto Lionello<fnref n="2" /> ID, Nikos Sioulas<fnref n="1" /> ID, Zesen Huang (黄泽森)<fnref n="1" /> ID, Jasper S. Halekas<fnref n="3" /> ID, Anna Tenerani<fnref n="4" /> ID, Victor Réville<fnref n="5" /> ID, Jean-Baptiste Dakeyo<fnref n="6" /> ID, Milan Maksimović<fnref n="6" /> ID, and Stuart D. Bale<fnref n="7" />,<fnref n="8" /> ID

Received 2022 November 11; revised 2023 January 12; accepted 2023 January 13; published 2023 February 14

## Abstract

The heating and acceleration of the solar wind remains one of the unsolved fundamental problems in heliophysics. It is usually observed that the proton temperature $T_i$ is highly correlated with the solar wind speed $V_{SW}$, while the electron temperature $T_e$ shows anticorrelation or no clear correlation with the solar wind speed. Here, we inspect both Parker Solar Probe (PSP) and WIND data, and compare the observations with simulation results. PSP observations below 30 solar radii clearly show a positive correlation between the proton temperature and the wind speed and a negative correlation between the electron temperature and the wind speed. One year (2019) of WIND data confirm that the proton temperature is positively correlated with the solar wind speed, but the electron temperature increases with the solar wind speed for slow wind, while it decreases with the solar wind speed for fast wind. Using a 1D Alfvén-wave-driven solar wind model with different proton and electron temperatures, we find, for the first time, that if most of the dissipated Alfvén wave energy heats the ions instead of the electrons, a positive $T_i-V_{SW}$ correlation and a negative $T_e-V_{SW}$ correlation arise naturally. If the electrons gain a small but finite portion of the dissipated wave energy, the $T_e-V_{SW}$ correlation evolves with the radial distance to the Sun, such that the negative correlation gradually turns positive. The model results show that Alfvén waves are one of the possible explanations for the observed evolution of the proton and electron temperatures in the solar wind.

Unified Astronomy Thesaurus concepts: Solar wind (1534); Alfvén waves (23); Solar coronal heating (1989)

## 1. Introduction

The solar wind is the plasma flow that is ejected from the solar corona, filling interplanetary space. It carries a large amount of mass and energy out of the Sun and serves as a medium for various physical processes and structures, such as waves and turbulence, coronal mass ejections, and magnetic reconnection. The solar wind continuously interacts with the Earth and injects energy into the Earth's magnetosphere, causing strong disturbances in the Earth's magnetosphere. Understanding the generation of the solar wind as well as the dynamics of the plasma in the solar wind are necessary not only for better space weather predictions, but also for deeper insights into fundamental plasma astrophysics.

A significant amount of energy is needed for the plasma to escape solar gravity and accelerate to a supersonic speed. Early works (Parker 1958, 1964a, 1964b, 1965) show that an isothermal solar corona with strong thermal conduction can generate the observed solar wind speeds. However, in situ measurements imply that in the inner heliosphere, both the ion and electron temperatures of the solar wind decay with radial distance to the Sun, and the radial profiles of the temperatures can be fitted with polytropic relations (Marsch et al. 1982;

Richardson et al. 1995; Štverák et al. 2015; Boldyrev et al. 2020; Shi et al. 2022). The polytropic indices deduced from the in situ measurements are in general smaller than the adiabatic index 5/3, implying that in situ heating mechanisms are important. One possible source of the in situ heating is the turbulence energy cascade. In the solar wind, predominantly outward-propagating Alfvén waves exist (Belcher & Davis 1971), while smaller-amplitude inward-propagating Alfvén waves are generated by processes including wave reflection, induced by the gradient of the Alfvén speed (Heinemann & Olbert 1980), and large-scale stream shears (Roberts et al. 1992; Shi et al. 2020). The nonlinear interaction between the outward- and inward-propagating waves leads to a turbulence energy cascade (Kraichnan 1965), which eventually dissipates the energy of the electromagnetic fields into plasma energy, through wave-particle interactions (Kasper et al. 2013; Kobayashi et al. 2017) and intermittent structures (Osman et al. 2012; Matthaeus et al. 2015; Sioulas et al. 2022b).

It has been long observed that in the solar wind, the proton temperature has a strong positive correlation with the solar wind speed (Burlaga & Ogilvie 1973; Lopez & Freeman 1986; Matthaeus et al. 2006; Démoulin 2009; Elliott et al. 2012; Shi et al. 2021; Hofmeister et al. 2022). This positive correlation indicates the possibility of a particular mechanism contributing to the proton heating and solar wind momentum simultaneously. On the contrary, the electron core temperature usually shows anticorrelation with the solar wind speed (Marsch et al. 1989; Halekas et al. 2020; Maksimovic et al. 2020), and this

<figure><img src="image_2.png" /></figure>

Original content from this work may be used under the terms of the [Creative Commons Attribution 4.0 licence](https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA 90095, USA; cshi1993@ucla.edu</footnote>

<footnote n="2">²Predictive Science Inc, San Diego, CA 92121, USA</footnote>

<footnote n="3">³Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242, USA</footnote>

<footnote n="4">⁴Department of Physics, The University of Texas at Austin, TX 78712, USA</footnote>

<footnote n="5">⁵IRAP, Université Toulouse III—Paul Sabatier, CNRS, CNES, 31400 Toulouse, France</footnote>

<footnote n="6">⁶LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, F-92195 Meudon, France</footnote>

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

<footnote n="8">⁸Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA</footnote>
<figure><img src="image_2.png" /></figure>

# Proton and Electron Temperatures in the Solar Wind and Their Correlations with the Solar Wind Speed

Chen Shi (时辰)<fnref n="1" /> ID, Marco Velli<fnref n="1" /> ID, Roberto Lionello<fnref n="2" /> ID, Nikos Sioulas<fnref n="1" /> ID, Zesen Huang (黄泽森)<fnref n="1" /> ID, Jasper S. Halekas<fnref n="3" /> ID, Anna Tenerani<fnref n="4" /> ID, Victor Réville<fnref n="5" /> ID, Jean-Baptiste Dakeyo<fnref n="6" /> ID, Milan Maksimović<fnref n="6" /> ID, and Stuart D. Bale<fnref n="7" />,<fnref n="8" /> ID

Received 2022 November 11; revised 2023 January 12; accepted 2023 January 13; published 2023 February 14

## Abstract

The heating and acceleration of the solar wind remains one of the unsolved fundamental problems in heliophysics. It is usually observed that the proton temperature $T_i$ is highly correlated with the solar wind speed $V_{SW}$, while the electron temperature $T_e$ shows anticorrelation or no clear correlation with the solar wind speed. Here, we inspect both Parker Solar Probe (PSP) and WIND data, and compare the observations with simulation results. PSP observations below 30 solar radii clearly show a positive correlation between the proton temperature and the wind speed and a negative correlation between the electron temperature and the wind speed. One year (2019) of WIND data confirm that the proton temperature is positively correlated with the solar wind speed, but the electron temperature increases with the solar wind speed for slow wind, while it decreases with the solar wind speed for fast wind. Using a 1D Alfvén-wave-driven solar wind model with different proton and electron temperatures, we find, for the first time, that if most of the dissipated Alfvén wave energy heats the ions instead of the electrons, a positive $T_i-V_{SW}$ correlation and a negative $T_e-V_{SW}$ correlation arise naturally. If the electrons gain a small but finite portion of the dissipated wave energy, the $T_e-V_{SW}$ correlation evolves with the radial distance to the Sun, such that the negative correlation gradually turns positive. The model results show that Alfvén waves are one of the possible explanations for the observed evolution of the proton and electron temperatures in the solar wind.

Unified Astronomy Thesaurus concepts: Solar wind (1534); Alfvén waves (23); Solar coronal heating (1989)

## 1. Introduction

The solar wind is the plasma flow that is ejected from the solar corona, filling interplanetary space. It carries a large amount of mass and energy out of the Sun and serves as a medium for various physical processes and structures, such as waves and turbulence, coronal mass ejections, and magnetic reconnection. The solar wind continuously interacts with the Earth and injects energy into the Earth's magnetosphere, causing strong disturbances in the Earth's magnetosphere. Understanding the generation of the solar wind as well as the dynamics of the plasma in the solar wind are necessary not only for better space weather predictions, but also for deeper insights into fundamental plasma astrophysics.

A significant amount of energy is needed for the plasma to escape solar gravity and accelerate to a supersonic speed. Early works (Parker 1958, 1964a, 1964b, 1965) show that an isothermal solar corona with strong thermal conduction can generate the observed solar wind speeds. However, in situ measurements imply that in the inner heliosphere, both the ion and electron temperatures of the solar wind decay with radial distance to the Sun, and the radial profiles of the temperatures can be fitted with polytropic relations (Marsch et al. 1982;

Richardson et al. 1995; Štverák et al. 2015; Boldyrev et al. 2020; Shi et al. 2022). The polytropic indices deduced from the in situ measurements are in general smaller than the adiabatic index 5/3, implying that in situ heating mechanisms are important. One possible source of the in situ heating is the turbulence energy cascade. In the solar wind, predominantly outward-propagating Alfvén waves exist (Belcher & Davis 1971), while smaller-amplitude inward-propagating Alfvén waves are generated by processes including wave reflection, induced by the gradient of the Alfvén speed (Heinemann & Olbert 1980), and large-scale stream shears (Roberts et al. 1992; Shi et al. 2020). The nonlinear interaction between the outward- and inward-propagating waves leads to a turbulence energy cascade (Kraichnan 1965), which eventually dissipates the energy of the electromagnetic fields into plasma energy, through wave-particle interactions (Kasper et al. 2013; Kobayashi et al. 2017) and intermittent structures (Osman et al. 2012; Matthaeus et al. 2015; Sioulas et al. 2022b).

It has been long observed that in the solar wind, the proton temperature has a strong positive correlation with the solar wind speed (Burlaga & Ogilvie 1973; Lopez & Freeman 1986; Matthaeus et al. 2006; Démoulin 2009; Elliott et al. 2012; Shi et al. 2021; Hofmeister et al. 2022). This positive correlation indicates the possibility of a particular mechanism contributing to the proton heating and solar wind momentum simultaneously. On the contrary, the electron core temperature usually shows anticorrelation with the solar wind speed (Marsch et al. 1989; Halekas et al. 2020; Maksimovic et al. 2020), and this

<figure><img src="image_3.png" /></figure>

Original content from this work may be used under the terms of the [Creative Commons Attribution 4.0 licence](https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, CA 90095, USA; cshi1993@ucla.edu</footnote>

<footnote n="2">²Predictive Science Inc, San Diego, CA 92121, USA</footnote>

<footnote n="3">³Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242, USA</footnote>

<footnote n="4">⁴Department of Physics, The University of Texas at Austin, TX 78712, USA</footnote>

<footnote n="5">⁵IRAP, Université Toulouse III—Paul Sabatier, CNRS, CNES, 31400 Toulouse, France</footnote>

<footnote n="6">⁶LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, F-92195 Meudon, France</footnote>

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

<footnote n="8">⁸Space Sciences Laboratory, University of California, Berkeley, CA 94720-7450, USA</footnote>
anticorrelation may originate at the source of the solar wind, due to interchange reconnection (Fisk 2003; Gloeckler et al. 2003). In situ measurements of the fractions between heavy ions, which are good proxies for the freeze-in temperatures of the solar corona, reveal that the coronal electron temperature has an anticorrelation with the solar wind speed (Geiss et al. 1995; KO et al. 1997; Gloeckler et al. 2003; von Steiger & Zurbuchen 2011).

Here, we propose the understanding that the different temperature-speed correlations for protons and electrons are possibly generated in situ, as the solar wind propagates. The major factor has to do with how the energy source for the acceleration of the solar wind is differently deposited into protons and electrons. As the turbulence energy cascades from MHD inertial scales toward ion kinetic scales, various kinetic processes may arise that determine how the electromagnetic energy eventually heats the ions and electrons. Two major processes are kinetic Alfvén waves (KAWs), which mainly heat the electrons through Landau damping, and ion cyclotron waves (ICWs), which heat the ions through cyclotron resonance. Both KAWs (e.g., Salem et al. 2012; Podesta 2013) and ICWs (e.g., Jian et al. 2010, 2009) have been identified in the solar wind, and the recent observations made by the Parker Solar Probe (PSP) show that the two modes may coexist (Huang et al. 2020). Gyrokinetic theory and simulations have shown that the ratio between the ion heating and the electron heating during the dissipation of Alfvénic turbulence is positively correlated with the ion beta (the ratio between the ion thermal pressure and magnetic pressure; Howes et al. 2008; Schekochihin et al. 2009; Howes 2010; Kawazura et al. 2019). Moreover, the existence of a compressive component in the turbulence will increase the ratio between the ion and electron heating (Kawazura et al. 2020). Hybrid-kinetic simulations, using realistic solar wind parameters at 1 au, have shown that 75%-80% of the cascaded turbulence energy heats the ions (Arzamasskiy et al. 2019). Recent theoretical work has suggested that conservation of the magnetic helicity prevents the turbulence energy from cascading toward sub-ion scales, thus most of the cascaded energy is absorbed by the ions (Squire et al. 2022). Bacchini et al. (2022) have shown that during the transition from Alfvén waves to KAWs at sub-ion scales, ions can gain more energy than electrons, because the kinetic energy of the waves is mostly accessible to ions, instead of electrons. In addition to the wave-particle interaction, other effects, such as intermittency (e.g., Osman et al. 2012) and stochastic heating (e.g., Chandran et al. 2010), may also contribute to the differential heating process. Sioulas et al. (2022a), using PSP measurements, have shown that protons can gain more energy from intermittent structures than electrons. The stochastic heating is positively correlated with the turbulence strength (Vech et al. 2017) and is shown to be significant throughout the inner heliosphere (Martinović et al. 2019, 2020), contributing to the perpendicular temperature of ions.

In this study, we show that for an Alfvén-wave-driven solar wind, if most of the wave energy dissipates into ions, the positive $T_p-V_{SW}$ (where $T_p$ is the proton temperature and $V_{SW}$ is the solar wind speed) correlation and negative $T_e-V_{SW}$ (where $T_e$ is the electron temperature) correlation are naturally generated. The paper is organized as follows. In Section 2, we present PSP and WIND observations of the solar wind and show the correlations between temperatures and the solar wind

speed. In Section 3, we describe the 1D Alfvén-wave-driven solar wind model that is used in this study and present the simulation results. In Section 4, we discuss the underlying mechanism that explains the numerical results. In Section 5, we conclude the study.

## 2. PSP and Wind Observations

We use PSP and WIND data to investigate the correlation between the solar wind speed and various solar wind parameters. For PSP, we use data from the first 9 orbits, and we only select data that have been collected below 30 solar radii. For the proton measurements, we mainly use the data from the electrostatic analyzer (SPAN-Ion), but we use the Faraday cup (SPC) for the first orbit when high-quality SPAN-Ion data are unavailable (Fox et al. 2016; Kasper et al. 2016). The electron temperature is the core temperature that is derived by fitting the electron velocity distribution functions measured by SPAN-Electron (Halekas et al. 2020). The time cadence of the SPAN data is typically 7–14 s, while the time cadence of the SPC data is around 0.44 s. The magnetic field data is collected by the fluxgate magnetometer, with a time cadence of 3.4 ms (Bale et al. 2016; Fox et al. 2016). For WIND, we use 1 yr of data, collected in 2019, and we have verified that the 2020 data give very similar results. The proton data are from the 3D Plasma Analyzer (3DP) electrostatic analyzers, with a cadence of 3 s (Lin et al. 1995), while the electron data are from the Solar Wind Experiment electron instruments, with a cadence of 6–12 s (Ogilvie et al. 1995), and the magnetic field data are from the WIND Magnetic Field Investigation fluxgate magnetometers, with a cadence of 3 s (Lepping et al. 1995).

In Figure 1, we show the distribution of the PSP data in the left column and the distribution of the WIND data in the right column. From top to bottom, the rows are the proton number density $N_p$, the proton temperature $T_p$, the electron temperature $T_e$, and the amplitude of the magnetic field fluctuations $|\delta B|$ as functions of the radial solar wind speed $V_{SW}$. For the PSP data, we use 1 minute time windows to calculate the average values of the quantities and the fluctuation strengths, defined as the rms of the magnetic field. For the WIND data, we use 5 minute time windows. The gray squares show the median values and the error bars show the rms values of the binned data. In both data sets, it is clear that a higher solar wind speed generally corresponds to a lower density, a higher proton temperature, and stronger magnetic field fluctuations. As previously shown in Maksimovic et al. (2020), Halekas et al. (2020), Salem et al. (2021), and Dakeyo et al. (2022), the electron temperature has a strong negative correlation with the solar wind speed, as observed by PSP. However, at 1 au, $T_e$ decreases with $V_{SW}$ for winds faster than about 520 km s⁻¹, but seems to increase with $V_{SW}$ for slower wind streams.

## 3. 1D Two-temperature Alfvén-wave-powered Solar Wind Model

### 3.1. Model Description

We utilize a 1D Alfvén-wave-driven solar wind model, with different ion (proton) and electron temperatures. Wave-driven solar wind models have been developed and widely used to analyze the heating and acceleration of the solar wind (e.g., Cranmer & van Ballegooijen 2005; Cranmer et al. 2007; Verdini & Velli 2007; Chandran & Hollweg 2009; Verdini et al. 2009; Lionello et al. 2014; Shoda et al. 2018; Réville et al. 2020).
anticorrelation may originate at the source of the solar wind, due to interchange reconnection (Fisk 2003; Gloeckler et al. 2003). In situ measurements of the fractions between heavy ions, which are good proxies for the freeze-in temperatures of the solar corona, reveal that the coronal electron temperature has an anticorrelation with the solar wind speed (Geiss et al. 1995; KO et al. 1997; Gloeckler et al. 2003; von Steiger & Zurbuchen 2011).

Here, we propose the understanding that the different temperature-speed correlations for protons and electrons are possibly generated in situ, as the solar wind propagates. The major factor has to do with how the energy source for the acceleration of the solar wind is differently deposited into protons and electrons. As the turbulence energy cascades from MHD inertial scales toward ion kinetic scales, various kinetic processes may arise that determine how the electromagnetic energy eventually heats the ions and electrons. Two major processes are kinetic Alfvén waves (KAWs), which mainly heat the electrons through Landau damping, and ion cyclotron waves (ICWs), which heat the ions through cyclotron resonance. Both KAWs (e.g., Salem et al. 2012; Podesta 2013) and ICWs (e.g., Jian et al. 2010, 2009) have been identified in the solar wind, and the recent observations made by the Parker Solar Probe (PSP) show that the two modes may coexist (Huang et al. 2020). Gyrokinetic theory and simulations have shown that the ratio between the ion heating and the electron heating during the dissipation of Alfvénic turbulence is positively correlated with the ion beta (the ratio between the ion thermal pressure and magnetic pressure; Howes et al. 2008; Schekochihin et al. 2009; Howes 2010; Kawazura et al. 2019). Moreover, the existence of a compressive component in the turbulence will increase the ratio between the ion and electron heating (Kawazura et al. 2020). Hybrid-kinetic simulations, using realistic solar wind parameters at 1 au, have shown that 75%-80% of the cascaded turbulence energy heats the ions (Arzamasskiy et al. 2019). Recent theoretical work has suggested that conservation of the magnetic helicity prevents the turbulence energy from cascading toward sub-ion scales, thus most of the cascaded energy is absorbed by the ions (Squire et al. 2022). Bacchini et al. (2022) have shown that during the transition from Alfvén waves to KAWs at sub-ion scales, ions can gain more energy than electrons, because the kinetic energy of the waves is mostly accessible to ions, instead of electrons. In addition to the wave-particle interaction, other effects, such as intermittency (e.g., Osman et al. 2012) and stochastic heating (e.g., Chandran et al. 2010), may also contribute to the differential heating process. Sioulas et al. (2022a), using PSP measurements, have shown that protons can gain more energy from intermittent structures than electrons. The stochastic heating is positively correlated with the turbulence strength (Vech et al. 2017) and is shown to be significant throughout the inner heliosphere (Martinović et al. 2019, 2020), contributing to the perpendicular temperature of ions.

In this study, we show that for an Alfvén-wave-driven solar wind, if most of the wave energy dissipates into ions, the positive $T_p-V_{SW}$ (where $T_p$ is the proton temperature and $V_{SW}$ is the solar wind speed) correlation and negative $T_e-V_{SW}$ (where $T_e$ is the electron temperature) correlation are naturally generated. The paper is organized as follows. In Section 2, we present PSP and WIND observations of the solar wind and show the correlations between temperatures and the solar wind

speed. In Section 3, we describe the 1D Alfvén-wave-driven solar wind model that is used in this study and present the simulation results. In Section 4, we discuss the underlying mechanism that explains the numerical results. In Section 5, we conclude the study.

## 2. PSP and Wind Observations

We use PSP and WIND data to investigate the correlation between the solar wind speed and various solar wind parameters. For PSP, we use data from the first 9 orbits, and we only select data that have been collected below 30 solar radii. For the proton measurements, we mainly use the data from the electrostatic analyzer (SPAN-Ion), but we use the Faraday cup (SPC) for the first orbit when high-quality SPAN-Ion data are unavailable (Fox et al. 2016; Kasper et al. 2016). The electron temperature is the core temperature that is derived by fitting the electron velocity distribution functions measured by SPAN-Electron (Halekas et al. 2020). The time cadence of the SPAN data is typically 7–14 s, while the time cadence of the SPC data is around 0.44 s. The magnetic field data is collected by the fluxgate magnetometer, with a time cadence of 3.4 ms (Bale et al. 2016; Fox et al. 2016). For WIND, we use 1 yr of data, collected in 2019, and we have verified that the 2020 data give very similar results. The proton data are from the 3D Plasma Analyzer (3DP) electrostatic analyzers, with a cadence of 3 s (Lin et al. 1995), while the electron data are from the Solar Wind Experiment electron instruments, with a cadence of 6–12 s (Ogilvie et al. 1995), and the magnetic field data are from the WIND Magnetic Field Investigation fluxgate magnetometers, with a cadence of 3 s (Lepping et al. 1995).

In Figure 1, we show the distribution of the PSP data in the left column and the distribution of the WIND data in the right column. From top to bottom, the rows are the proton number density $N_p$, the proton temperature $T_p$, the electron temperature $T_e$, and the amplitude of the magnetic field fluctuations $|\delta B|$ as functions of the radial solar wind speed $V_{SW}$. For the PSP data, we use 1 minute time windows to calculate the average values of the quantities and the fluctuation strengths, defined as the rms of the magnetic field. For the WIND data, we use 5 minute time windows. The gray squares show the median values and the error bars show the rms values of the binned data. In both data sets, it is clear that a higher solar wind speed generally corresponds to a lower density, a higher proton temperature, and stronger magnetic field fluctuations. As previously shown in Maksimovic et al. (2020), Halekas et al. (2020), Salem et al. (2021), and Dakeyo et al. (2022), the electron temperature has a strong negative correlation with the solar wind speed, as observed by PSP. However, at 1 au, $T_e$ decreases with $V_{SW}$ for winds faster than about 520 km s⁻¹, but seems to increase with $V_{SW}$ for slower wind streams.

## 3. 1D Two-temperature Alfvén-wave-powered Solar Wind Model

### 3.1. *Model Description*

We utilize a 1D Alfvén-wave-driven solar wind model, with different ion (proton) and electron temperatures. Wave-driven solar wind models have been developed and widely used to analyze the heating and acceleration of the solar wind (e.g., Cranmer & van Ballegooijen 2005; Cranmer et al. 2007; Verdini & Velli 2007; Chandran & Hollweg 2009; Verdini et al. 2009; Lionello et al. 2014; Shoda et al. 2018; Réville et al. 2020).
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Left: distributions of the data collected below 30 solar radii during first 9 orbits of the PSP. Right: distributions of 1 yr (2019) of data from WIND. From top to bottom, the rows show the proton number density $N_p$, the proton temperature $T_p$, the electron temperature $T_e$, and the amplitude of the magnetic field fluctuations $|\delta B|$ as functions of the radial solar wind speed $V_{sw}$. For the PSP data, we use 1 minute time windows to calculate the average values of the quantities and the fluctuation strengths. For the WIND data, we use 5 minute time windows. The gray squares show the median values, while the error bars show the rms values of the binned data.</figcaption></figure>

While most previous works assume a one-fluid solar wind, some works have adopted a two-fluid solar wind model, with different proton and electron temperatures (Chandran et al. 2011; Adhikari et al. 2022). The model used in the current study is very similar to the one-fluid model used by Réville et al. (2020), but with independent ion and electron temperatures. The model equations are:

$$B(r) = \frac{A_0}{A(r)} B_0, \qquad (1a)$$

$$\frac{\partial \rho}{\partial t} = -\frac{1}{A} \frac{\partial}{\partial r} (\rho VA), \qquad (1b)$$

$$\frac{\partial V}{\partial t} = -V \frac{\partial V}{\partial r} - \frac{1}{\rho} \frac{\partial}{\partial r} \left( P + \frac{1}{2} \varepsilon \right) - \frac{GM}{r^2}, \qquad (1c)$$

$$\frac{\partial P_i}{\partial t} = -V \frac{\partial P_i}{\partial r} - \gamma_i \frac{1}{A} \frac{\partial (AV)}{\partial r} P_i + (\gamma_i - 1) Q_i, \qquad (1d)$$

$$\frac{\partial P_e}{\partial t} = -V \frac{\partial P_e}{\partial r} - \gamma_e \frac{1}{A} \frac{\partial (AV)}{\partial r} P_e + (\gamma_e - 1) Q_e, \qquad (1e)$$

$$\frac{\partial \varepsilon^+}{\partial t} = -(V + V_A) \frac{\partial \varepsilon^+}{\partial r} - \frac{1}{A} \frac{\partial}{\partial r} (A(V + V_A)) \varepsilon^+ \\ - \frac{1}{2A} \frac{1}{\partial r} \frac{\partial (AV)}{\partial r} \varepsilon^+ + R_+ + D_+, \qquad (1f)$$

$$\frac{\partial \varepsilon^-}{\partial t} = -(V - V_A) \frac{\partial \varepsilon^-}{\partial r} - \frac{1}{A} \frac{\partial}{\partial r} (A(V - V_A)) \varepsilon^- \\ - \frac{1}{2A} \frac{1}{\partial r} \frac{\partial (AV)}{\partial r} \varepsilon^- + R_- + D_-, \qquad (1g)$$

with $P = P_i + P_e$ and $\varepsilon = \varepsilon^+ + \varepsilon^-$. Here, $B(r)$, $\rho(r)$, $V(r)$, $P_i(r)$, and $P_e(r)$ are the radial magnetic field, plasma density, radial solar wind speed, ion thermal pressure, and electron thermal pressure, respectively. $\gamma_{i,e}$ are the polytropic indices for ions and electrons, respectively. We consider a spherically symmetric radial flux tube, such that $A(r)$ is the cross-sectional area of the tube and $A_0$ is the cross-sectional area at the inner boundary. The two Elsässer variables are $\varepsilon^\pm = \frac{1}{4} \rho |z^\pm|^2$ with $z^\pm$. Thus, $\varepsilon^\pm$ represent the energy densities (per volume) of the outward- and inward-propagating Alfvén waves. $V_A = B/\sqrt{\mu_0 \rho}$ is the radial Alfvén speed. $D_\pm$ are the dissipation rates of the two wave populations, due to the nonlinear energy cascade, while $R_\pm$ represent the reflections of the waves, due to the inhomogeneity of the background plasma. $Q_i$ and
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Left: distributions of the data collected below 30 solar radii during first 9 orbits of the PSP. Right: distributions of 1 yr (2019) of data from WIND. From top to bottom, the rows show the proton number density $N_p$, the proton temperature $T_p$, the electron temperature $T_e$, and the amplitude of the magnetic field fluctuations $|\delta B|$ as functions of the radial solar wind speed $V_{sw}$. For the PSP data, we use 1 minute time windows to calculate the average values of the quantities and the fluctuation strengths. For the WIND data, we use 5 minute time windows. The gray squares show the median values, while the error bars show the rms values of the binned data.</figcaption></figure>

While most previous works assume a one-fluid solar wind, some works have adopted a two-fluid solar wind model, with different proton and electron temperatures (Chandran et al. 2011; Adhikari et al. 2022). The model used in the current study is very similar to the one-fluid model used by Réville et al. (2020), but with independent ion and electron temperatures. The model equations are:

$$B(r) = \frac{A_0}{A(r)} B_0, \qquad (1a)$$

$$\frac{\partial \rho}{\partial t} = - \frac{1}{A} \frac{\partial}{\partial r} (\rho VA), \qquad (1b)$$

$$\frac{\partial V}{\partial t} = -V \frac{\partial V}{\partial r} - \frac{1}{\rho} \frac{\partial}{\partial r} \left( P + \frac{1}{2} \varepsilon \right) - \frac{GM}{r^2}, \qquad (1c)$$

$$\frac{\partial P_i}{\partial t} = -V \frac{\partial P_i}{\partial r} - \gamma_i \frac{1}{A} \frac{\partial (AV)}{\partial r} P_i + (\gamma_i - 1) Q_i, \qquad (1d)$$

$$\frac{\partial P_e}{\partial t} = -V \frac{\partial P_e}{\partial r} - \gamma_e \frac{1}{A} \frac{\partial (AV)}{\partial r} P_e + (\gamma_e - 1) Q_e, \qquad (1e)$$

$$\frac{\partial \varepsilon^+}{\partial t} = -(V + V_A) \frac{\partial \varepsilon^+}{\partial r} - \frac{1}{A} \frac{\partial}{\partial r} (A(V + V_A)) \varepsilon^+ \\ - \frac{1}{2A} \frac{1}{\partial r} \frac{\partial (AV)}{\partial r} \varepsilon^+ + R_+ + D_+, \qquad (1f)$$

$$\frac{\partial \varepsilon^-}{\partial t} = -(V - V_A) \frac{\partial \varepsilon^-}{\partial r} - \frac{1}{A} \frac{\partial}{\partial r} (A(V - V_A)) \varepsilon^- \\ - \frac{1}{2A} \frac{1}{\partial r} \frac{\partial (AV)}{\partial r} \varepsilon^- + R_- + D_-, \qquad (1g)$$

with $P = P_i + P_e$ and $\varepsilon = \varepsilon^+ + \varepsilon^-$. Here, $B(r)$, $\rho(r)$, $V(r)$, $P_i(r)$, and $P_e(r)$ are the radial magnetic field, plasma density, radial solar wind speed, ion thermal pressure, and electron thermal pressure, respectively. $\gamma_{i,e}$ are the polytropic indices for ions and electrons, respectively. We consider a spherically symmetric radial flux tube, such that $A(r)$ is the cross-sectional area of the tube and $A_0$ is the cross-sectional area at the inner boundary. The two Elsässer variables are $\varepsilon^\pm = \frac{1}{4}\rho |z^\pm|^2$ with $z^\pm$. Thus, $\varepsilon^\pm$ represent the energy densities (per volume) of the outward- and inward-propagating Alfvén waves. $V_A = B/\sqrt{\mu_0 \rho}$ is the radial Alfvén speed. $D_\pm$ are the dissipation rates of the two wave populations, due to the nonlinear energy cascade, while $R_\pm$ represent the reflections of the waves, due to the inhomogeneity of the background plasma. $Q_i$ and
$Q_e$ are the heating terms for ions and electrons, with both of them consisting of three components, such that $Q_i = Q_{h,i} + Q_{w,i} + Q_{c,i}$ and $Q_e = Q_{h,e} + Q_{w,e} + Q_{c,e}$. Here, $Q_{h,i}$ and $Q_{h,e}$ are the ad hoc heating terms that are significant only at very low altitudes. $Q_{w,i}$ and $Q_{w,e}$ are the heating of ions and electrons by wave dissipation. $Q_{c,i}$ and $Q_{c,e}$ are the heating terms caused by collisionless electron heat conduction (Hollweg 1976).

A comment on the treatment of the Alfvén wave equations and their coupling to the solar wind is in order. Although this model has been used before, it represents a drastic simplification of the true problem, as it writes the evolution equations directly in terms of the separate energy densities of the outward and inward modes, rather than the general second-order moment of the fluctuating fields. The latter would imply at least four equations for the fluctuating energies, rather than two, and a generalized Reynolds stress in the solar wind momentum equation, rather than the simple fluctuating magnetic pressure of Equation (1c). The still unresolved difficulties of this model have been discussed in, e.g., Velli (1993), while the approximation used above is effective in the limit of small reflection—i.e., necessary to trigger nonlinear interactions, but not large enough to require the full second-order moments—that is satisfied by all but the lowest-frequency fluctuations (corresponding to periods of several hours to days).

We use a superradiantly expanding flux tube (Verdini et al. 2009; Lionello et al. 2014): $A(r) = f(r)r^2$, with

$$f(r) = \frac{f_m + f_i \exp(-(r - r_{\text{exp}})/\sigma_{\text{exp}})}{1 + \exp(-(r - r_{\text{exp}})/\sigma_{\text{exp}})} \quad (2)$$

and $f_i = 1 - (f_m - 1)\exp((r_s - r_{\text{exp}})/\sigma_{\text{exp}})$. Here, $r_s$ is the solar radius, $r_{\text{exp}} = 1.31r_s$, and $\sigma_{\text{exp}} = 0.51r_s$. The above expression leads to $f(r_s) = 1$ and $f(+\infty) = f_m$. If $f_m = 1$, we get $f(r) \equiv 1$, which is a radially expanding flux tube. In this study, we set $f_m = 4$, which is a typical value for fast solar wind (Wang & Sheeley 1990). The polytropic indices are $\gamma_i = \gamma_e = 5/3$ (the adiabatic index), so that both species cool as $T_{i,e} \propto r^{-4/3}$, without other heating terms.

The first three terms on the right-hand sides of Equations (1f) and (1g)) correspond to the Wentzel–Kramers–Brillouin evolution of the wave amplitudes (Alazraki & Couturier 1971; Belcher 1971; Hollweg 1974). The reflection term is written as

$$R_{\pm} = C_R \times \left| (V \mp V_A) \frac{\partial}{\partial r} \ln \sqrt{\rho} \right| \varepsilon^{\mp}, \quad (3)$$

with $C_R = 0.1$ being a constant coefficient. The nonlinear dissipation is

$$D_{\pm} = -\frac{1}{8} \rho \frac{|z^{\mp}||z^{\pm}|^2}{\lambda} = -\frac{\sqrt{\varepsilon^{\mp} \varepsilon^{\pm}}}{\sqrt{\rho} \lambda}, \quad (4)$$

where $\lambda(r)$ is the perpendicular correlation length (Réville et al. 2020), which is modeled as

$$\lambda(r) = \lambda_0 \sqrt{\frac{A(r)}{A_0}}. \quad (5)$$

In this study, we set $\lambda_0 = 6 \times 10^7 m$, similar to the typical size of large supergranules (Verdini & Velli 2007; Verdini et al. 2009; Réville et al. 2020). Since the dissipated wave energy heats the protons and electrons, we have

$$Q_{w,i} + Q_{w,e} = -(D_{+} + D_{-}). \quad (6)$$

In the model, a free parameter $C_w \in [0, 1]$ controls the portion of the dissipated wave energy that heats the ions:

$$Q_{w,i} = -C_w(D_{+} + D_{-}), \quad Q_{w,e} = -(1 - C_w)(D_{+} + D_{-}). \quad (7)$$

The ad hoc heating is

$$Q_{h,(i,e)} = Q_{0,(i,e)} \frac{A_0}{A} \exp\left(-\frac{r - r_s}{r_s}\right), \quad (8)$$

and we set $Q_{0,i} = 5 \times 10^{-7} \text{ J m}^{-3} \text{ s}^{-1}$ and $Q_{0,e} = 2 \times 10^{-7} \text{ J m}^{-3} \text{ s}^{-1}$. These terms represent contributions from processes such as nanoflares (e.g., Cargill & Klimchuk 2004), which are important in the low corona. We choose $Q_{0,e} < Q_{0,i}$ because remote-sensing observations reveal that the electron temperature is smaller than the proton temperature in the coronal holes (Cranmer 2009). The collisionless heat conduction term is written as

$$Q_{c,i} = 0, \quad Q_{c,e} = -\frac{1}{A} \frac{\partial}{\partial r} (A q_c), \quad (9)$$

where $q_c = \frac{3}{2}P_e V$ (Hollweg 1976). Note that the collisionless heat conduction takes effect for electrons only.

The simulation domain is $r \in [1, 215]r_s$, with a nonuniform grid. The spatial resolution is $\Delta x = 0.001 r_s$ at the inner boundary and $\Delta x = 0.1 r_s$ at the outer boundary, and the total number of grid points is $N = 2934$. Dirichlet boundary conditions are imposed for $(B, \rho, P_i, P_e, \varepsilon^+)$ at the inner boundary, and the outer boundary is open, so that the wind and waves can propagate out of the domain freely. We note that no inner-boundary conditions are needed for $V$ and $\varepsilon^-$ because the sonic point and the Alfvén point implicitly impose two constraints for them (Parker 1958; Barkhudarov 1991; Velli 1993). In all simulations, we set $B = 5G$, $n = 1 \times 10^8 \text{ cm}^{-3}$, and $T_i = T_e = 1 \text{ MK}$ at the inner boundary, where $n$ is the number density of the plasma and $T_i$ and $T_e$ are the ion temperature and the electron temperature, respectively. We carry out five sets of simulations, with $C_w = [1, 0.95, 0.9, 0.85, 0.8]$. For each $C_w$, we conduct a series of runs with varying $\varepsilon^+(r_s)$, such that $|z^+| = [0, 5, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90, 100] \text{ km s}^{-1}$ at the inner boundary. We run each simulation until all the fields reach a stationary state ($\partial_t = 0$) and acquire the radial profiles of the fields.

## 3.2. Results

In Figure 2, we plot the radial profiles of the various quantities in two sets of runs. The left column shows the runs with $C_w = 1$ and the right column shows the runs with $C_w = 0.8$. From top to bottom, the rows are the solar wind speed, the plasma number density, the ion temperature, and the electron temperature. In each panel, the dark to light colors correspond to runs with increasing values of the inner-boundary wave amplitude, from 0 to 100 km s$^{-1}$. A larger wave energy input leads to a higher solar wind speed, because of a stronger wave pressure gradient and greater heating of the plasma. The wave amplitude does not change the density profile much. The left column clearly shows that a larger wave amplitude leads to a higher ion temperature and a lower electron temperature. For $C_w = 0.8$, the ion temperature still increases with the wave amplitude, but the behavior of the electron temperature is more complicated. Close to the Sun ($r \lesssim 20r_s$), the electron temperature decreases with the wave amplitude. Farther away from the Sun, the electron temperature increases with the wave amplitude. Figure 3 shows the
$Q_e$ are the heating terms for ions and electrons, with both of them consisting of three components, such that $Q_i = Q_{h,i} + Q_{w,i} + Q_{c,i}$ and $Q_e = Q_{h,e} + Q_{w,e} + Q_{c,e}$. Here, $Q_{h,i}$ and $Q_{h,e}$ are the ad hoc heating terms that are significant only at very low altitudes. $Q_{w,i}$ and $Q_{w,e}$ are the heating of ions and electrons by wave dissipation. $Q_{c,i}$ and $Q_{c,e}$ are the heating terms caused by collisionless electron heat conduction (Hollweg 1976).

A comment on the treatment of the Alfvén wave equations and their coupling to the solar wind is in order. Although this model has been used before, it represents a drastic simplification of the true problem, as it writes the evolution equations directly in terms of the separate energy densities of the outward and inward modes, rather than the general second-order moment of the fluctuating fields. The latter would imply at least four equations for the fluctuating energies, rather than two, and a generalized Reynolds stress in the solar wind momentum equation, rather than the simple fluctuating magnetic pressure of Equation (1c). The still unresolved difficulties of this model have been discussed in, e.g., Velli (1993), while the approximation used above is effective in the limit of small reflection—i.e., necessary to trigger nonlinear interactions, but not large enough to require the full second-order moments—that is satisfied by all but the lowest-frequency fluctuations (corresponding to periods of several hours to days).

We use a superradiantly expanding flux tube (Verdini et al. 2009; Lionello et al. 2014): $A(r) = f(r)r^2$, with

$$f(r) = \frac{f_m + f_i \exp(-(r - r_{\text{exp}})/\sigma_{\text{exp}})}{1 + \exp(-(r - r_{\text{exp}})/\sigma_{\text{exp}})} \quad (2)$$

and $f_i = 1 - (f_m - 1)\exp((r_s - r_{\text{exp}})/\sigma_{\text{exp}})$. Here, $r_s$ is the solar radius, $r_{\text{exp}} = 1.31r_s$, and $\sigma_{\text{exp}} = 0.51r_s$. The above expression leads to $f(r_s) = 1$ and $f(+\infty) = f_m$. If $f_m = 1$, we get $f(r) \equiv 1$, which is a radially expanding flux tube. In this study, we set $f_m = 4$, which is a typical value for fast solar wind (Wang & Sheeley 1990). The polytropic indices are $\gamma_i = \gamma_e = 5/3$ (the adiabatic index), so that both species cool as $T_{i,e} \propto r^{-4/3}$, without other heating terms.

The first three terms on the right-hand sides of Equations (1f) and (1g)) correspond to the Wentzel–Kramers–Brillouin evolution of the wave amplitudes (Alazraki & Couturier 1971; Belcher 1971; Hollweg 1974). The reflection term is written as

$$R_{\pm} = C_R \times \left| (V \mp V_A) \frac{\partial}{\partial r} \ln \sqrt{\rho} \right| \varepsilon^{\mp}, \quad (3)$$

with $C_R = 0.1$ being a constant coefficient. The nonlinear dissipation is

$$D_{\pm} = -\frac{1}{8} \rho \frac{|z^{\mp}||z^{\pm}|^2}{\lambda} = -\frac{\sqrt{\varepsilon^{\mp} \varepsilon^{\pm}}}{\sqrt{\rho} \lambda}, \quad (4)$$

where $\lambda(r)$ is the perpendicular correlation length (Réville et al. 2020), which is modeled as

$$\lambda(r) = \lambda_0 \sqrt{\frac{A(r)}{A_0}}. \quad (5)$$

In this study, we set $\lambda_0 = 6 \times 10^7 m$, similar to the typical size of large supergranules (Verdini & Velli 2007; Verdini et al. 2009; Réville et al. 2020). Since the dissipated wave energy heats the protons and electrons, we have

$$Q_{w,i} + Q_{w,e} = -(D_{+} + D_{-}). \quad (6)$$

In the model, a free parameter $C_w \in [0, 1]$ controls the portion of the dissipated wave energy that heats the ions:

$$Q_{w,i} = -C_w(D_+ + D_-), \quad Q_{w,e} = -(1 - C_w)(D_+ + D_-). \quad (7)$$

The ad hoc heating is

$$Q_{h,(i,e)} = Q_{0,(i,e)} \frac{A_0}{A} \exp\left(-\frac{r - r_s}{r_s}\right), \quad (8)$$

and we set $Q_{0,i} = 5 \times 10^{-7} \text{ J m}^{-3} \text{ s}^{-1}$ and $Q_{0,e} = 2 \times 10^{-7} \text{ J m}^{-3} \text{ s}^{-1}$. These terms represent contributions from processes such as nanoflares (e.g., Cargill & Klimchuk 2004), which are important in the low corona. We choose $Q_{0,e} < Q_{0,i}$ because remote-sensing observations reveal that the electron temperature is smaller than the proton temperature in the coronal holes (Cranmer 2009). The collisionless heat conduction term is written as

$$Q_{c,i} = 0, \quad Q_{c,e} = -\frac{1}{A} \frac{\partial}{\partial r} (A q_c), \quad (9)$$

where $q_c = \frac{3}{2}P_e V$ (Hollweg 1976). Note that the collisionless heat conduction takes effect for electrons only.

The simulation domain is $r \in [1, 215]r_s$, with a nonuniform grid. The spatial resolution is $\Delta x = 0.001 r_s$ at the inner boundary and $\Delta x = 0.1 r_s$ at the outer boundary, and the total number of grid points is $N = 2934$. Dirichlet boundary conditions are imposed for $(B, \rho, P_i, P_e, \varepsilon^+)$ at the inner boundary, and the outer boundary is open, so that the wind and waves can propagate out of the domain freely. We note that no inner-boundary conditions are needed for $V$ and $\varepsilon^-$ because the sonic point and the Alfvén point implicitly impose two constraints for them (Parker 1958; Barkhudarov 1991; Velli 1993). In all simulations, we set $B = 5G$, $n = 1 \times 10^8 \text{ cm}^{-3}$, and $T_i = T_e = 1 \text{ MK}$ at the inner boundary, where $n$ is the number density of the plasma and $T_i$ and $T_e$ are the ion temperature and the electron temperature, respectively. We carry out five sets of simulations, with $C_w = [1, 0.95, 0.9, 0.85, 0.8]$. For each $C_w$, we conduct a series of runs with varying $\varepsilon^+(r_s)$, such that $|z^+| = [0, 5, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90, 100] \text{ km s}^{-1}$ at the inner boundary. We run each simulation until all the fields reach a stationary state ($\partial_t = 0$) and acquire the radial profiles of the fields.

## 3.2. Results

In Figure 2, we plot the radial profiles of the various quantities in two sets of runs. The left column shows the runs with $C_w = 1$ and the right column shows the runs with $C_w = 0.8$. From top to bottom, the rows are the solar wind speed, the plasma number density, the ion temperature, and the electron temperature. In each panel, the dark to light colors correspond to runs with increasing values of the inner-boundary wave amplitude, from 0 to 100 km s$^{-1}$. A larger wave energy input leads to a higher solar wind speed, because of a stronger wave pressure gradient and greater heating of the plasma. The wave amplitude does not change the density profile much. The left column clearly shows that a larger wave amplitude leads to a higher ion temperature and a lower electron temperature. For $C_w = 0.8$, the ion temperature still increases with the wave amplitude, but the behavior of the electron temperature is more complicated. Close to the Sun ($r \lesssim 20r_s$), the electron temperature decreases with the wave amplitude. Farther away from the Sun, the electron temperature increases with the wave amplitude. Figure 3 shows the
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Radial profiles of various quantities in two sets of simulations. The left column has $C_w = 1$, i.e., all the dissipated wave energy heats the ions. The right column has $C_w = 0.8$, i.e., 20% of the dissipated wave energy heats the electrons. From top to bottom, the rows are the solar wind speed, the plasma number density, the ion temperature, and the electron temperature, respectively. In each panel, the curves with different colors correspond to the different wave amplitudes at the inner boundary. From dark to light, the colors correspond to increasing wave amplitudes of [0, 5, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90, 100] km s⁻¹.</figcaption></figure>

contributions to ion heating (solid curves) and electron heating (dashed curves) per unit mass, i.e., $Q/ρ$, by different mechanisms, in the run with $C_w = 0.8$ and $|z^+| = 100$ km s⁻¹ at the inner boundary. The blue curves show the ad hoc heating, the orange curves show the wave heating, and the green curve shows the collisionless electron thermal conduction. The wave heating is weaker than the ad hoc heating close to the Sun, but the radial extent of the significant wave heating is larger than that of the ad hoc heating. In addition, the collisionless thermal conduction is comparable to or even larger than that of the wave heating far away from the Sun.

In Figure 4, we show how the plasma number density (top panels), the ion temperature (bottom panels, blue lines), and the electron temperature (bottom panels, orange lines) vary with the solar wind speed at 1 au, in runs with different $C_w$. From left to right, the columns are $C_w = 1, 0.9, $ and $0.8$, respectively. The behavior of the plasma density does not depend on $C_w$, as how the dissipated wave energy is distributed among the ions and the electrons does not affect the radial profile of the solar wind speed or density much. As we increase the wave amplitude, the density drops at first, then starts to increase, though only with small variation (n varies between 12.5 and 14.5 cm⁻³). Since the mass conservation law gives $n(r) = n_0 \times (A_0V_0/A(r)V(r))$, as

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Radial profiles of different heating terms (per unit mass) in the run with $C_w = 0.8$ and $|z^+| = 100$ km s⁻¹ at the inner boundary. The blue curves show the ad hoc heating, the orange curves show the wave heating, and the green curve shows the collisionless heat conduction. The solid curves show the heating of ions and the dashed curves show the heating of electrons. The minimum of the collisionless heating is around $-6.1 \times 10^5$ m² s⁻³, at $r = 1.85 R_s$.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Radial profiles of various quantities in two sets of simulations. The left column has $C_w = 1$, i.e., all the dissipated wave energy heats the ions. The right column has $C_w = 0.8$, i.e., 20% of the dissipated wave energy heats the electrons. From top to bottom, the rows are the solar wind speed, the plasma number density, the ion temperature, and the electron temperature, respectively. In each panel, the curves with different colors correspond to the different wave amplitudes at the inner boundary. From dark to light, the colors correspond to increasing wave amplitudes of [0, 5, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90, 100] km s⁻¹.</figcaption></figure>

contributions to ion heating (solid curves) and electron heating (dashed curves) per unit mass, i.e., $Q/ρ$, by different mechanisms, in the run with $C_w = 0.8$ and $|z^+| = 100$ km s⁻¹ at the inner boundary. The blue curves show the ad hoc heating, the orange curves show the wave heating, and the green curve shows the collisionless electron thermal conduction. The wave heating is weaker than the ad hoc heating close to the Sun, but the radial extent of the significant wave heating is larger than that of the ad hoc heating. In addition, the collisionless thermal conduction is comparable to or even larger than that of the wave heating far away from the Sun.

In Figure 4, we show how the plasma number density (top panels), the ion temperature (bottom panels, blue lines), and the electron temperature (bottom panels, orange lines) vary with the solar wind speed at 1 au, in runs with different $C_w$. From left to right, the columns are $C_w = 1, 0.9, $ and $0.8$, respectively. The behavior of the plasma density does not depend on $C_w$, as how the dissipated wave energy is distributed among the ions and the electrons does not affect the radial profile of the solar wind speed or density much. As we increase the wave amplitude, the density drops at first, then starts to increase, though only with small variation (n varies between 12.5 and 14.5 cm⁻³). Since the mass conservation law gives $n(r) = n_0 \times (A_0V_0/A(r)V(r))$, as

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Radial profiles of different heating terms (per unit mass) in the run with $C_w = 0.8$ and $|z^+| = 100$ km s⁻¹ at the inner boundary. The blue curves show the ad hoc heating, the orange curves show the wave heating, and the green curve shows the collisionless heat conduction. The solid curves show the heating of ions and the dashed curves show the heating of electrons. The minimum of the collisionless heating is around $-6.1 \times 10^5$ m² s⁻³, at $r = 1.85 R_s$.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Plasma number density (top), ion temperature (bottom, blue lines), and electron temperature (bottom, orange lines) as functions of the solar wind speed at 1 au. From left to right, the panels show runs corresponding to $C_w = 1$, 0.9, and 0.8, respectively.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 5.** $T_i-V$ (top) and $T_e-V$ (bottom) correlations at different radial distances to the Sun, in runs with $C_w = 1$ (left), 0.9 (middle), and 0.8 (right). In each panel, from dark to light, the colors correspond to $r = 10, 25, 50, 100$, and 215 solar radii.</figcaption></figure>

the wave amplitude increases, if $V_0$ increases more slowly than $V(r)$, $n(r)$ has an anticorrelation with $V(r)$, and vice versa. Hence, the small variation of the density with the wave amplitude indicates that the waves modify $V_0$ and $V(r)$ in similar proportions. Similar to the density, the ion temperature is not significantly modified by $C_w$, either. On the contrary, the electron temperature is quite sensitive to $C_w$. As $C_w$ decreases, the negative $T_e-V$ correlation gradually turns positive, consistent with what is shown in Figure 2.

In Figure 5, we show $T_i-V$ (top panels) and $T_e-V$ (bottom panels) at different radial distances to the Sun, for runs with $C_w = 1$ (left), $C_w = 0.9$ (middle), and $C_w = 0.8$ (right), respectively. In each panel, from dark to light, the colors correspond to $r = 10, 25, 50, 100$, and 215 solar radii. For

$C_w = 1$, a positive $T_i-V$ correlation and a negative $T_e-V$ correlation are well established at very close distances to the Sun, and maintained as the wind propagates. For $C_w = 0.9$ and $C_w = 0.8$, the ion temperature is not modified much, while the $T_e-V$ correlation evolves as the solar wind propagates. Close to the Sun, a negative $T_e-V$ correlation is produced, while as $r$ increases, the $T_e-V$ correlation gradually turns positive. This trend is similar to the in situ measurements from multiple satellites (Maksimovic et al. 2020).

## 4. Discussion

The positive $T_i-V$ correlation is easy to understand: with more wave energy being injected from the inner boundary, the solar
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Plasma number density (top), ion temperature (bottom, blue lines), and electron temperature (bottom, orange lines) as functions of the solar wind speed at 1 au. From left to right, the panels show runs corresponding to $C_w = 1$, 0.9, and 0.8, respectively.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 5.** $T_i-V$ (top) and $T_e-V$ (bottom) correlations at different radial distances to the Sun, in runs with $C_w = 1$ (left), 0.9 (middle), and 0.8 (right). In each panel, from dark to light, the colors correspond to $r = 10, 25, 50, 100$, and 215 solar radii.</figcaption></figure>

the wave amplitude increases, if $V_0$ increases more slowly than $V(r)$, $n(r)$ has an anticorrelation with $V(r)$, and vice versa. Hence, the small variation of the density with the wave amplitude indicates that the waves modify $V_0$ and $V(r)$ in similar proportions. Similar to the density, the ion temperature is not significantly modified by $C_w$, either. On the contrary, the electron temperature is quite sensitive to $C_w$. As $C_w$ decreases, the negative $T_e-V$ correlation gradually turns positive, consistent with what is shown in Figure 2.

In Figure 5, we show $T_i-V$ (top panels) and $T_e-V$ (bottom panels) at different radial distances to the Sun, for runs with $C_w = 1$ (left), $C_w = 0.9$ (middle), and $C_w = 0.8$ (right), respectively. In each panel, from dark to light, the colors correspond to $r = 10, 25, 50, 100$, and 215 solar radii. For

$C_w = 1$, a positive $T_i-V$ correlation and a negative $T_e-V$ correlation are well established at very close distances to the Sun, and maintained as the wind propagates. For $C_w = 0.9$ and $C_w = 0.8$, the ion temperature is not modified much, while the $T_e-V$ correlation evolves as the solar wind propagates. Close to the Sun, a negative $T_e-V$ correlation is produced, while as $r$ increases, the $T_e-V$ correlation gradually turns positive. This trend is similar to the in situ measurements from multiple satellites (Maksimovic et al. 2020).

## 4. Discussion

The positive $T_i-V$ correlation is easy to understand: with more wave energy being injected from the inner boundary, the solar
<figure><img src="image_8.png" /><figcaption>**Figure 6.** $T_i-V$ (top) and $T_e-V$ (bottom) correlations at 1 au, for three sets of runs with different amplitudes of the electron ad hoc heating. The blue curves show $Q_{0,e} = 2 \times 10^{-7} \text{ J m}^{-3} \text{ s}^{-1}$, the orange curves show $Q_{0,e} = 1 \times 10^{-7} \text{ J m}^{-3} \text{ s}^{-1}$, and the green curves show $Q_{0,e} = 0$. All runs have $C_w = 1$.</figcaption></figure>

wind speed increases, due to the larger wave pressure and the larger thermal pressure gradient. Meanwhile, because most of the dissipated wave energy heats the ions, the ion temperature also increases, resulting in a positive $T_i-V$ correlation. The cause of a negative $T_e-V$ correlation is more complicated. If we consider the most simple case, where the electron fluid is polytropic, such that $T_e(r) = T_{e0} \times (\rho_0/\rho(r))^{\gamma-1}$, the $T_e-V$ relation should be similar to the $n-V$ relation. However, the left column of Figure 4 shows that even in the large-$V$ regime, where the density increases with $V$, $T_e$ still decreases with $V$. This indicates that the ad hoc heating may play an important role in forming the $T_e-V$ anticorrelation. By observing the equation for electron pressure (Equation (1e)), we see that, in a stationary state ($\partial_t = 0$), and without Alfvén wave heating of the electrons, the radial gradient of the electron pressure can be written as

$$ \frac{\partial P_e}{\partial r} = C_1 P_e \frac{1}{A} \frac{\partial A}{\partial r} + C_2 P_e \frac{1}{V} \frac{\partial V}{\partial r} + C_3 \frac{Q_{h,e}}{V}, \quad (10) $$

where $C_1$, $C_2$, and $C_3$ are constants, depending on $\gamma_e$ and the collisionless heat conduction strength. Close to the inner boundary, the dominating term is the ad hoc heating term (Figure 3). Because $Q_{h,e}(r)$ is a given function of $r$, the contribution of this term to the increment of pressure is inversely proportional to the solar wind speed. That is to say, the faster the plasma is ejected, the less internal energy it gains during its propagation. This is why a stronger Alfvén wave injection leads to a lower electron temperature at close distances to the Sun. In Figure 6, we show the $T_i-V$ (top) and

$T_e-V$ (bottom) relations at 1 au, for three sets of runs with $C_w = 1$ and varying amplitudes of the electron ad hoc heating. The blue curves correspond to $Q_{0,e} = 2 \times 10^{-7} \text{ J m}^{-3} \text{ s}^{-1}$, the orange curves correspond to $Q_{0,e} = 1 \times 10^{-7} \text{ J m}^{-3} \text{ s}^{-1}$, and the green curves correspond to $Q_{0,e} = 0$. One can see that as $Q_{0,e}$ is decreased, the negative $T_e-V$ correlation gradually vanishes, implying that the ad hoc heating is necessary for the negative $T_e-V$ correlation. However, if the electrons gain a portion of the wave energy during the solar wind expansion, the anticorrelation between $T_e$ and $V$ is gradually destroyed, because farther away from the Sun the contribution of the ad hoc heating gradually becomes less important, compared to the contribution from the wave dissipation (Figure 3). This explains why there is a radial evolution of the $T_e-V$ relation. Based on this scenario, the in situ observations (Figure 1) indicate that in the slow solar wind, the electrons get more heating during the solar wind propagation compared to the electrons in the fast solar wind, leading to different $T_e-V$ correlations for the slow and fast streams (the right column of Figure 1). The underlying mechanisms, however, require further study.

## 5. Conclusion

Through a 1D Alfvén-wave-driven solar wind model with different ion and electron temperatures, we have successfully reproduced two important features of the solar wind: a positive correlation between the ion temperature ($T_i$) and the solar wind speed ($V$), and a negative correlation between the electron temperature ($T_e$) and the solar wind speed. In our simulations, the different $T_i-V$ and $T_e-V$ relations are a result of the fact that most of the dissipated Alfvén wave energy heats ions instead of electrons (e.g., Arzamasskiy et al. 2019; Bacchini et al. 2022; Squire et al. 2022), meaning that the electron heating close to the Sun by the ad hoc heating term, which represents mechanisms such as magnetic reconnection, is less efficient, due to the faster wind speed. With a small but finite portion of the dissipated wave energy heating the electrons, the simulations also reproduce the observed radial evolution of the $T_e-V$ relation, i.e., the initially negative correlation gradually turns into a positive one (Maksimovic et al. 2020), because the contribution from the ad hoc heating term gradually becomes negligible, compared to that from the Alfvén wave heating, as the wind propagates.

We note that the model used here is not fully self-consistent, and that some important characteristics of the observed solar wind are missing from the model results. First, the temperature evolution given by the model agrees with the observations only qualitatively, but not completely quantitatively. The radial decay rate ($\alpha$ in $T_e \propto r^{-\alpha}$) of the electron temperature is large, and it does not vary much with the solar wind speed. In the bottom right panel of Figure 2, $\alpha$ changes from ~0.67 to ~0.65 as the wind speed at 1 au increases from 470 to 660 km s⁻¹. In contrast, $\alpha$ varies between ~0.4 and ~0.2 as the wind speed changes from 400 to 600 km s⁻¹, as estimated using HELIOS data (Maksimovic et al. 2020). This difference implies that other mechanisms that are omitted from our model, e.g., the ambipolar electric field (Boldyrev et al. 2020), play an important role in local electron heating. Second, the density changes only slightly among runs with different wave amplitudes (Figure 2), and thus the density does not vary with
<figure><img src="image_9.png" /><figcaption>**Figure 6.** $T_i-V$ (top) and $T_e-V$ (bottom) correlations at 1 au, for three sets of runs with different amplitudes of the electron ad hoc heating. The blue curves show $Q_{0,e} = 2 \times 10^{-7}$ J m$^{-3}$ s$^{-1}$, the orange curves show $Q_{0,e} = 1 \times 10^{-7}$ J m$^{-3}$ s$^{-1}$, and the green curves show $Q_{0,e} = 0$. All runs have $C_w = 1$.</figcaption></figure>

wind speed increases, due to the larger wave pressure and the larger thermal pressure gradient. Meanwhile, because most of the dissipated wave energy heats the ions, the ion temperature also increases, resulting in a positive $T_i-V$ correlation. The cause of a negative $T_e-V$ correlation is more complicated. If we consider the most simple case, where the electron fluid is polytropic, such that $T_e(r) = T_{e0} \times (\rho_0/\rho(r))^{\gamma-1}$, the $T_e-V$ relation should be similar to the $n-V$ relation. However, the left column of Figure 4 shows that even in the large-$V$ regime, where the density increases with $V$, $T_e$ still decreases with $V$. This indicates that the ad hoc heating may play an important role in forming the $T_e-V$ anticorrelation. By observing the equation for electron pressure (Equation (1e)), we see that, in a stationary state ($\partial_t = 0$), and without Alfvén wave heating of the electrons, the radial gradient of the electron pressure can be written as

$$ \frac{\partial P_e}{\partial r} = C_1 P_e \frac{1}{A} \frac{\partial A}{\partial r} + C_2 P_e \frac{1}{V} \frac{\partial V}{\partial r} + C_3 \frac{Q_{h,e}}{V}, \quad (10) $$

where $C_1$, $C_2$, and $C_3$ are constants, depending on $\gamma_e$ and the collisionless heat conduction strength. Close to the inner boundary, the dominating term is the ad hoc heating term (Figure 3). Because $Q_{h,e}(r)$ is a given function of $r$, the contribution of this term to the increment of pressure is inversely proportional to the solar wind speed. That is to say, the faster the plasma is ejected, the less internal energy it gains during its propagation. This is why a stronger Alfvén wave injection leads to a lower electron temperature at close distances to the Sun. In Figure 6, we show the $T_i-V$ (top) and

$T_e-V$ (bottom) relations at 1 au, for three sets of runs with $C_w = 1$ and varying amplitudes of the electron ad hoc heating. The blue curves correspond to $Q_{0,e} = 2 \times 10^{-7}$ J m$^{-3}$ s$^{-1}$, the orange curves correspond to $Q_{0,e} = 1 \times 10^{-7}$ J m$^{-3}$ s$^{-1}$, and the green curves correspond to $Q_{0,e} = 0$. One can see that as $Q_{0,e}$ is decreased, the negative $T_e-V$ correlation gradually vanishes, implying that the ad hoc heating is necessary for the negative $T_e-V$ correlation. However, if the electrons gain a portion of the wave energy during the solar wind expansion, the anticorrelation between $T_e$ and $V$ is gradually destroyed, because farther away from the Sun the contribution of the ad hoc heating gradually becomes less important, compared to the contribution from the wave dissipation (Figure 3). This explains why there is a radial evolution of the $T_e-V$ relation. Based on this scenario, the in situ observations (Figure 1) indicate that in the slow solar wind, the electrons get more heating during the solar wind propagation compared to the electrons in the fast solar wind, leading to different $T_e-V$ correlations for the slow and fast streams (the right column of Figure 1). The underlying mechanisms, however, require further study.

## 5. Conclusion

Through a 1D Alfvén-wave-driven solar wind model with different ion and electron temperatures, we have successfully reproduced two important features of the solar wind: a positive correlation between the ion temperature ($T_i$) and the solar wind speed ($V$), and a negative correlation between the electron temperature ($T_e$) and the solar wind speed. In our simulations, the different $T_i-V$ and $T_e-V$ relations are a result of the fact that most of the dissipated Alfvén wave energy heats ions instead of electrons (e.g., Arzamasskiy et al. 2019; Bacchini et al. 2022; Squire et al. 2022), meaning that the electron heating close to the Sun by the ad hoc heating term, which represents mechanisms such as magnetic reconnection, is less efficient, due to the faster wind speed. With a small but finite portion of the dissipated wave energy heating the electrons, the simulations also reproduce the observed radial evolution of the $T_e-V$ relation, i.e., the initially negative correlation gradually turns into a positive one (Maksimovic et al. 2020), because the contribution from the ad hoc heating term gradually becomes negligible, compared to that from the Alfvén wave heating, as the wind propagates.

We note that the model used here is not fully self-consistent, and that some important characteristics of the observed solar wind are missing from the model results. First, the temperature evolution given by the model agrees with the observations only qualitatively, but not completely quantitatively. The radial decay rate ($\alpha$ in $T_e \propto r^{-\alpha}$) of the electron temperature is large, and it does not vary much with the solar wind speed. In the bottom right panel of Figure 2, $\alpha$ changes from ~0.67 to ~0.65 as the wind speed at 1 au increases from 470 to 660 km s$^{-1}$. In contrast, $\alpha$ varies between ~0.4 and ~0.2 as the wind speed changes from 400 to 600 km s$^{-1}$, as estimated using HELIOS data (Maksimovic et al. 2020). This difference implies that other mechanisms that are omitted from our model, e.g., the ambipolar electric field (Boldyrev et al. 2020), play an important role in local electron heating. Second, the density changes only slightly among runs with different wave amplitudes (Figure 2), and thus the density does not vary with
the wind speed significantly (the top panels of Figure 4). However, it is well known that the faster solar wind is generally less dense than the slower solar wind, and the mass flux only moderately depends on the solar wind speed (Wang 2010). Both the inner-boundary plasma density and the expansion factor can heavily affect the solar wind density, but they are both constant in the current study. A thorough parametric study —in which all these parameters are treated as variables—is necessary, and will be conducted in the future. Hence, the current study serves as a demonstration of how the Alfvén waves can contribute to the observed *T*–*V* correlations in the solar wind, while other mechanisms (e.g., Fisk 2003) may still be important, and should be incorporated within a complete description of the system.

This work is supported by NASA HTMS 80NSSC20K1275 and NASA Parker Solar Probe Observatory Scientist grant NNX15AF34G. The instruments of PSP were designed and developed under NASA contract NNN06AA01C. We thank Dr Kun Zhang for many useful suggestions.

*Software:* Matplotlib (Hunter 2007).

## ORCID iDs

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

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

Roberto Lionello https://orcid.org/0000-0001-9231-045X

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

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

Jasper S. Halekas https://orcid.org/0000-0001-5258-6128

Anna Tenerani https://orcid.org/0000-0003-2880-6084

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

Jean-Baptiste Dakeyo https://orcid.org/0000-0002-1628-0276

Milan Maksimović https://orcid.org/0000-0001-6172-5062

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

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This work is supported by NASA HTMS 80NSSC20K1275 and NASA Parker Solar Probe Observatory Scientist grant NNX15AF34G. The instruments of PSP were designed and developed under NASA contract NNN06AA01C. We thank Dr Kun Zhang for many useful suggestions.

*Software:* Matplotlib (Hunter 2007).

## ORCID iDs

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

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

Roberto Lionello https://orcid.org/0000-0001-9231-045X

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

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

Jasper S. Halekas https://orcid.org/0000-0001-5258-6128

Anna Tenerani https://orcid.org/0000-0003-2880-6084

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

Jean-Baptiste Dakeyo https://orcid.org/0000-0002-1628-0276

Milan Maksimović https://orcid.org/0000-0001-6172-5062

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

## References

Adhikari, L., Zank, G. P., Zhao, L.-L., & Telloni, D. 2022, ApJ, 933, 56

Alazraki, G., & Couturier, P. 1971, A&A, 13, 380

Arzamasskiy, L., Kunz, M. W., Chandran, B. D. G., & Quataert, E. 2019, ApJ, 879, 53

Bacchini, F., Pucci, F., Malara, F., & Lapenta, G. 2022, PhRvL, 128, 025101

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

Barkhudarov, M. R. 1991, SoPh, 135, 131

Belcher, J. W. 1971, ApJ, 168, 509

Belcher, J. W., & Davis, L. 1971, JGR, 76, 3534

Boldyrev, S., Forest, C., & Egedal, J. 2020, PNAS, 117, 9232

Burlaga, L. F., & Ogilvie, K. W. 1973, JGR, 78, 2028

Cargill, P. J., & Klimchuk, J. A. 2004, ApJ, 605, 911

Chandran, B. D. G., Dennis, T. J., Quataert, E., & Bale, S. D. 2011, ApJ, 743, 197

Chandran, B. D. G., & Hollweg, J. V. 2009, ApJ, 707, 1659

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Elliott, H., Henney, C., McComas, D., Smith, C., & Vasquez, B. 2012, JGRA, 117, A09102

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