On the Heating of the Slow Solar Wind by Imbalanced Alfvén-wave Turbulence from 0.06 to 1 au: Parker Solar Probe and Solar Orbiter Observations

In this work we analyze plasma and magnetic field data provided by the Parker Solar Probe and Solar Orbiter missions to investigate the radial evolution of the heating of Alfvénic slow wind by imbalanced Alfvén-wave (AW) turbulent fluctuations from 0.06 to 1 au. in our analysis we focus on slow solar-wind intervals with highly imbalanced and incompressible turbulence (i.e., magnetic compressibility C B = δ B/B ≤ 0.25, plasma compressibility C n = δ n/n ≤ 0.25, and normalized cross helicity σ c ≥ 0.65). First, we estimate the AW turbulent dissipation rate from the wave energy equation and find that the radial profile trend is similar to the proton heating rate. Second, we find that the scaling of the empirical AW turbulent dissipation rate Q W obtained from the wave energy equation matches the scaling from the phenomenological AW turbulent dissipation rate Q CH09 (with Q CH09 ≃ 1.55Q W ) derived by Chandran & Hollweg based on the model of reflection-driven turbulence. Our results suggest that, as in the fast solar wind, AW turbulence plays a major role in the ion heating that occurs in incompressible slow-wind streams.


INTRODUCTION
Previous in-situ measurements at heliocentric distances near and above 0.3 au revealed the existence of ubiquitous turbulent fluctuations throughout the interplanetary medium (see, e.g., the review of Bruno & Carbone (2013)).Most of these turbulent fluctuations have been found to be Alfvénic in nature and to propagate mainly outward from the Sun in the fast solar wind (Belcher & Davis 1971;Tu & Marsch 1995).Remotesensing observations have also revealed the presence of Alfvén waves in the lower corona with sufficient energy to power the solar wind (De Pontieu et al. 2007;McIntosh et al. 2011).In addition, recent studies have investigated the heating of the solar wind by AWs turbulence using PSP measurements (Adhikari et al. 2021;Bandyopadhyay et al. 2023).
Generally, the solar wind is classified either as fast or slow wind with a typical speed of V > 500 km/s or how AW turbulence and wave reflections might heat and accelerate the solar wind.
The fractional variations in the density and magneticfield strength are often quite small in the fast solar wind, and the turbulent fluctuations in the fast solar wind are often observed to be imbalanced, in the sense that most of the turbulent fluctuations consist of Alfvénwave-like fluctuations propagating away from the Sun in the plasma frame.Although some slow solar-wind streams are more balanced and compressible than the fast wind, other slow solar-wind streams have similar levels of compressibility and imbalance as the fast wind.This latter category of slow wind is referred to as the Alfvénic slow wind (ASW).Numerous intervals of ASW have been observed, for example, by the Parker Solar Probe (PSP ; Raouafi et al. 2023a) (see also, Bale et al. 2019;Kasper et al. 2019;Chen et al. 2021;Bourouaine et al. 2020Bourouaine et al. , 2022)).
Although there is broad agreement that AW turbulence plays a key role in the origin of the fast solar wind, the origin of the slow solar wind is still a highly debated topic (Abbo et al. 2016).Recently, Raouafi et al. (2023b) argued that the solar wind is driven by jetting at the source.The magnetic reconnection at the corona naturally generates Alfvén waves that might heat and accelerate the solar wind at higher altitudes.Previous studies suggested that the solar origin of ASW comes from the boundaries of open coronal fields (D'Amicis & Bruno 2015; Bale et al. 2019), and non-Alfvénic slow wind could emanate from coronal streamers at the boundary of the heliospheric current sheet (Szabo et al. 2020;Chen et al. 2021).
Most previous studies of solar-wind heating by AW turbulence focused on the fast solar wind (e.g., Cranmer et al. 2009;Chandran et al. 2011;Adhikari et al. 2021;Bandyopadhyay et al. 2023).A notable exception to this was the recent study by Halekas et al. (2023), who quantified the radial profiles of the various contributions to the solar-wind energy flux in both the fast solar wind and slow solar wind.
In this paper, we analyze a set of measurements from PSP (for heliocentric distance r = [0.06,0.3] au) and Solar Orbiter (SO) (for r = [0.3,1] au) to investigate the heating of incompressible slow solar-wind streams by imbalanced AW turbulence.We use the steady-state electron and the proton energy equations to estimate the electron and proton heating rates.For the estimation of the AW turbulent dissipation rate, we use the fluctuation-energy equation in the steady state given in Perez et al. (2021).Our analysis differs from that of Halekas et al. (2023) in that we isolate the effects of AW dissipation and plasma heating, rather than quan-tifying the total plasma energization resulting from the combination of AW dissipation and the work done by AWs.In section 2, we describe our methodology and data analysis for the estimation of the plasma and turbulent parameters.The main results are presented in section 3. Finally, in section 4, we summarize and discuss our findings.

METHODOLOGY AND DATA ANALYSIS
In our analysis, we use plasma and field measurements from PSP and SO.The data from SO (from August 8 2022 to January 31 2023) are used to study the radial profile of the plasma heating rate and the AW turbulent heating rate between 0.3 au and 1 au.We use data from Encounters 2,4,5,6,7,9 and 10 of PSP for the analysis between 0.06 au and 0.3 au.These Encounters are chosen due to the availability of plasma density that is estimated through the quasi-thermal-noise (QTN) (Moncuquet et al. 2020).

Selection of intervals
In our analysis, we investigate the radial evolution of the solar-wind plasma as well as the AW turbulent dissipation rate.We divide SO and PSP data into 9 hours intervals, only intervals corresponding to 9-hours averaged solar-wind speed V < 500 km/s are initially considered.Both SO and PSP measure solar-wind plasma near the ecliptic; therefore, there will be occasions for some intervals with a mixed field polarity.To eliminate those intervals, we only consider intervals in which at least 70% of the instantaneous B r values have the same sign, where B r is the radial component of the magnetic field.The time duration of each interval is chosen to be T = 9 hours so that the large-scale part of the turbulence can be recovered.For those slow-wind selected intervals, the background (averaged) magnetic-field vector B 0 is properly estimated.Then we estimate the turbulence parameters, such as the plasma compressibility, C n = δn rms /⟨n⟩; the magnetic compressibility, C B = δB rms /⟨|B|⟩; and the normalized cross helicity, Here δn rms and δB rms are the root mean square of the fluctuating number density n = n e (here we assume n e ≃ n p , where n e and n p are the electron and proton density), and the magnetic field strength, respectively.The symbol ⟨• • • ⟩ denotes an average over the time period T for each selected interval.The quantities δV and δb are the fluctuating bulk velocity and the fluctuating Alfvén velocity (i.e., δb = δB/ √ 4πρ 0 , where ρ 0 is the proton mass density averaged over the time interval T .)The quantity B0r(r/ au)2 in nT unit that is proportional to the magnetic field flux.The orange (blue) scatter points are the slow-wind PSP (SO) intervals with more than 70% of either sun outward or inward magnetic field polarity.The black dots correspond to the selected ASW intervals used for our analysis, i.e., intervals that satisfy; CB < 0.25, Cn < 0.25, |σc| > 0.65 and that B0r(r/1 au) 2 ranges between 2 nT and 4 nT.
These parameters allow us to select intervals of Alfvénic slow solar wind (ASW).We consider ASW intervals to be those intervals that are less compressible and where AW turbulence is imbalanced (i.e., by choosing |σ c | > 0.65, C B ≤ 0.25 and C n ≤ 0.25).In Figure 1, we show a scatter plot for σ c (upper left panel), C B (upper right panel), and C n (lower left panel) corresponding to all 9-hour slow-wind intervals satisfying the condition that more than 70% of the instantaneous B r values within the interval have the same sign.Overall, about 60% of |σ c | values are higher than 0.65.Since we are interested in slow-wind intervals with AW turbulence, we apply the low-compressibility condition too such that C B < 0.25 and C n < 0.25.More than 70% of slow-wind intervals have C B < 0.25 and C n < 0.25.To study the evolution of the radial profile of heating and turbulent dissipation rates in ASW we apply another condition to the selected intervals, i.e., we only consider those intervals where the quantity B 0 (r/1 au) 2 cos ψ = |B 0r |(r/1 au) 2 (proportional to the magnetic flux) is between 2 and 4 nT, as indicated by the two horizontal lines in the lower right panel of Figure 1.Here ψ is the Parker angle that depends on the solar-wind speed and heliocentric distance r as tan ψ = Ωr/V , where Ω = 2.7 × 10 −6 rad s −1 is the solar rotation frequency, and B 0r is the radial component of the averaged field B 0 .Overall, only ≃ 28% of the slow-wind intervals seen by PSP and SO (during the data period considered) satisfy the criteria of ASW.

RESULTS
3.1.Radial profiles of plasma and turbulent parameters for ASW.

Estimation of the proton and electron heating rates in ASW
To estimate the empirical proton heating rate Q p we use the fitting functions for the proton temperature T p and the density n.This is to evaluate the derivatives with respect to r in the steady-state proton energy equation (see e.g., Cranmer et al. (2009)) where k B is Boltzmann constant and ν ep is the frequency of electron-proton Coulomb collisions.T e is the electron temperature that is analyzed using only PSP data.We found that with α e = −0.5.This radial profile of T e will be used to estimate the electron heating rate from 0.06 au to 1 au.Here, V 0 corresponds to the mean solar-wind speed averaged over all ASW intervals, it is V 0 = 370 km/s for ASW SO intervals and V 0 = 315 km/s for ASW PSP intervals.The collision term in the r.h.s of Eq. ( 2) is insignificant but still calculated assuming two isotropic Maxwellian distributions for electrons and protons interacting with one another (Spitzer & Härm 1953).The frequency ν ep is thus estimated from Cranmer et al. ( 2009) Eq. ( 2) can be practically simplified by implementing the fitting functions A 0 (r/r 0 ) −α of n and T p to evalu-ate the derivatives with respect to r.Then, the proton heating rate Q p is given by (5) where α p and α n are the exponents of the power-law fits of T p and n, respectively.Similarly, the radial profile of the electron heating rate Q e will be estimated by implementing power-law fits of T e , and n into the following steady-state electron energy equation ∥,e r 2 cos ψ .(6) Here q ∥,e is the parallel electron heat flux.We estimate q ∥,e based on the collisionless model (Hollweg 1974(Hollweg , 1976) ) as where α H is a dimensionless parameter that is only known approximately.For fast wind analysis Chandran et al. (2011) set α H = 0.75.In our analysis of ASW we evaluate Q e for α H = 0.5, 0.75 and 0.90.In order to evaluate the derivative of the electron heat flux term in Eq. ( 6) we use the conservation of the magnetic field flux B 0 r 2 cos ψ = constant and the fitting functions of T e , n and B 0 that we measured.The electron heating rate is then Figure 3 shows the empirical values of Q p and Q e estimated from PSP (at r < 0.3 au) and SO (at r > 0.3 au).We found that the fitting functions of the radial profiles proton rates are Q p = 0.83 • 10 −17 (r/1 au) −3.92 J m −3 s −1 (Q p = 1.25 • 10 −17 (r/1 au) −3.66 J m −3 s −1 ) at r < 0.3 au (r > 0.3 au) using PSP (SO).In Table (2) we show the electron heating rates for different values of α H .
The proton heating rate needed for ASW is slightly less than the one for the fast solar wind, Q f ast p = 2.4 • 10 −17 (r/1 au) −3.8 J m −3 s −1 found by Hellinger et al. (2011) using Helios measurements.
It is worth-mentioning that formula given in Eq (7) with α H = 0.5, 0.75 and 0.9 guarantees that the conductive heating term is comparable in magnitude to the advective terms in Eq. ( 6) in our analysis.Table 2.The heating rates Qe and Qp for the Alfvénic slow wind (ASW).Here r = (r/1 au)

Estimation of turbulent dissipation rate Q W in ASW
To investigate the radial dissipation rate Q W of the AW (imbalanced turbulence) in the slow-solar wind, we use the steady-state wave-energy equation given in Perez et al. (2021) but we account for the Parker spiral field geometry for the flux-tube, thus we have where is the flux density of the anti-sunward (sunward) AWs, η = − cos ψdB 0 /(B 0 dr), ξ r = E V −E b is the average residual energy density.Now, we introduce the measured powerlaw fitting functions of E out ∝ r αout , E in ∝ r αin , |B 0 | ∝ r α B and n ∝ r −αn to estimate the turbulent heating rate Q W from Eq. ( 9).The power-law functions and the magnetic-field flux conservation will allow us to simplify the derivatives in Eq. ( 9).Also, by considering that solar wind speed is nearly constant we now can use the  10) for ASW intervals for r < 0.3 au (r > 0.3 au) using PSP (SO).Red dots represent the proton heating rate Qp for ASW.Solid green line represents the estimated electron heating rate Qe when αH = 0.9.Bottom panel: proton-to-total heating ratio versus heliocentric distance when αH = 0.9, αH = 0.75 and αH = 0.5, and the solid lines are the same ratio obtained from the powerlaw fits of Qp and Qe.
following expression of Q W as where and α B , α out and α in are the the power-law fitting exponents of B 0 , E out and E in , respectively.
In Figure 4, we plot the estimated Q W (based on Eq. ( 10) as a function of r using PSP (for r < 0.3 au) and SO (for r > 0.3 au) for ASW intervals.Interestingly enough, the trend of the turbulent dissipation rate Q W for ASW follows the trend of the proton heating rate Q p over the entire range of heliocentric distances r we studied.On average, the turbulent heating rate Q W is about 1.5 times higher than the proton heating rates Q p .The lower panel in Figure 4 shows the ratio Q p /(Q p +Q e ) for α H = 0.9, 0.75 and α H = 0.5 as a function of r.Overall, It seems that the protons are more heated than electrons in ASW when α H ≳ 0.9.Here we did not consider the collisional Spitzer-Härm (SH) electron heat flux as the majority of the observed electron heat flux near-Sun region using P SP data lie below the SH limit (Halekas et al. 2021).Using SH heat flux in our analysis would lead to electron cooling instead of electron heating.

3.4.
Comparison with a phenomenological model of the heating rate in reflection-driven AW turbulence Dmitruk et al. (2002) derived a phenomenological turbulent heating rate for reflection-driven turbulence in coronal holes in the absence of background flow.They assumed that there is much more energy in waves propagating away from the Sun than waves propagating towards the Sun, and that the energy cascade timescale of Sunward-propagating waves is comparable to or shorter than their linear wave period.Chandran & Hollweg (2009) generalized this model to allow for the background flow of the solar wind, thereby taking into account the work done by the waves on the solar wind.The essence of these models is to balance the reflection rate of the produced sunward waves against the rate at which these waves cascade and dissipate via interactions with the anti-sunward waves.Chandran & Hollweg (2009) found that the turbulent heating rate in reflection-driven turbulence is where V is the solar-wind outflow velocity and v A is Alfvén speed.We have estimated the turbulent energy dissipation rate Q CH09 using the ASW intervals selected for our analysis above.In Figure 5, we overplot the measured Q CH09 together with the estimated Q W values obtained earlier from the turbulent energy equation ( 10).This figure shows that the two rates scale in a similar way with r and are comparable to each other with where the uncertainty ±0.45 is the propagation error due to the power-law fits.This result provides good observational evidence supporting the reflection-driven turbulence model for turbulent heating in ASW, and is consistent with a related analysis using PSP data by Chen et al. (2020).

CONCLUSION AND DISCUSSION
In this work we analyzed plasma and magnetic-field data collected by PSP and SO to study the radial evolution of the plasma heating of the slow solar wind by incompressible Alfvénic turbulence.We selected intervals lasting approximately 9 hours in which at least 70% of the measured magnetic field has the same polarity -i.e., the same sign of B r .We further restricted our analysis to comparatively incompressible intervals (C B < 0.25  11) compared to the turbulent dissipation rate QW obtained from Eq. ( 10) (black dots).All measured using PSP for r < 0.3 au and SO at r > 0.3 au. and C n < 0.25) with a high degree of imbalance (i.e., σ c > 0.65) -i.e., to ASW.We then analyzed those selected intervals to estimate the turbulent heating (dissipation) rate and the plasma (electron and proton) heating rate.We found that the radial profile of the proton heating rate in ASW correlates well with the turbulent heating rate Q W which suggests that Alfvénic turbulence plays a major role in the heating of the protons in ASW, as it does in the fast solar wind.We also found that the measured AW turbulent heating rate agrees well with the phenomenological heating rate proposed by Chandran & Hollweg (2009) for reflection-driven AW turbulence.
Although our results show that AW turbulence can account for much of the heating of ASW between 0.06 au and 1 au, this does not mean that AW turbulence is the dominant mechanism for accelerating the ASW over this range of radii.Indeed, Halekas et al. (2023) used PSP data to examine the radial profiles of the various components of the solar-wind energy flux between 13R ⊙ and 100R ⊙ .Because the solar-wind accelerates over this range of radii, the bulk-flow kinetic energy of the solar wind accounts for an increasing fraction of the total energy flux as r increases.For the slow solar wind, Halekas et al. (2023) showed that this radial increase is offset primarily by a decrease in the fraction of the energy flux carried by the electron enthalpy flux and heat flux.In other words, it is the electron enthalpy flux and heat flux that account for most of the acceleration of the slow solar wind between 13R ⊙ and 100R ⊙ , not AW turbulence.Our results indicate that AW turbulence is likely the dom-inant heating mechanism, at least in ASW.AW turbulence may heat mainly the protons in ASW through distinct possible heating mechanisms such as resonantcyclotron heating by high-frequency ion-cyclotron waves (see e.g., Hollweg & Isenberg 2002) or by low-frequency AW turbulence (see, e.g., Bourouaine et al. 2008;Chandran et al. 2013;Bourouaine & Chandran 2013).The generation of the high-frequency waves in the solar wind could be triggered by low-frequency turbulence as shown in the numerical simulation work done by Squire et al. (2022).
SB and JCP acknowledge support from NASA grant 80NSSC21K1768.BC acknowledges the support of NASA grants 80NSSC21K1768, NNN06AA01C, and 80NSSC24K0171.VKJ acknowledges support from the Parker Solar Probe mission as part of NASA's Living with a Star (LWS) program under contract NNN06AA01C.Parker Solar Probe was designed, built, and is now operated by the Johns Hopkins Applied Physics Laboratory as part of NASA's Living with a Star (LWS) program (contract NNN06AA01C).Support from the LWS management and technical team has played a critical role in the success of the Parker Solar Probe mission.

Figure 1 .
Figure 1.Top left panel: The normalized cross-helicity |σc|.Top right panel: The magnetic compressibility CB.Bottom left panel: The plasma compressibility Cn.Bottom right panel:The quantity B0r(r/ au) 2 in nT unit that is proportional to the magnetic field flux.The orange (blue) scatter points are the slow-wind PSP (SO) intervals with more than 70% of either sun outward or inward magnetic field polarity.The black dots correspond to the selected ASW intervals used for our analysis, i.e., intervals that satisfy; CB < 0.25, Cn < 0.25, |σc| > 0.65 and that B0r(r/1 au) 2 ranges between 2 nT and 4 nT.

Figure 2 .
Figure 2. Top panels from left to right: The plasma density n and the proton temperature Tp.Middle panels from left to right: The strength of the average magnetic field B0 and the energy densities of the anti-sunward Eout sunward Ein AWs.Bottom panels from left to right: The energy densities of the kinetic energy density EV and the magnetic energy density E b in velocity unit.All plotted as a function of the radial distance r.

Figure 4 .
Figure 4. Top panel: Black dots are the estimated turbulent dissipation rate QW from Eq. (10) for ASW intervals for r < 0.3 au (r > 0.3 au) using PSP (SO).Red dots represent the proton heating rate Qp for ASW.Solid green line represents the estimated electron heating rate Qe when αH = 0.9.Bottom panel: proton-to-total heating ratio versus heliocentric distance when αH = 0.9, αH = 0.75 and αH = 0.5, and the solid lines are the same ratio obtained from the powerlaw fits of Qp and Qe.

Figure 5 .
Figure 5.The estimated turbulent dissipation rate, QCH09 from the turbulence reflection model of Chandran & Hollweg (2009) (blue dots) from Eq. (11) compared to the turbulent dissipation rate QW obtained from Eq. (10) (black dots).All measured using PSP for r < 0.3 au and SO at r > 0.3 au.