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

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 $\bm{B}_{0}$ is properly estimated. Then we estimate the turbulence parameters, such as the plasma compressibility, $C_{n}=\delta n_{{\rm rms}}/\langle n\rangle$; the magnetic compressibility, $C_{B}=\delta B_{{\rm rms}}/\langle|B|\rangle$; and the normalized cross helicity,

Here $\delta n_{{\rm rms}}$ and $\delta B_{{\rm rms}}$ are the root mean square of the fluctuating number density $n=n_{e}$ (here we assume $n_{e}\simeq n_{p}$, where $n_{e}$ and $n_{p}$ are the electron and proton density), and the magnetic field strength, respectively. The symbol $\langle\cdots\rangle$ denotes an average over the time period $T$ for each selected interval. The quantities $\delta{\bm{V}}$ and $\delta{\bm{b}}$ are the fluctuating bulk velocity and the fluctuating Alfvén velocity (i.e., $\delta{\bm{b}}=\delta{\bm{B}}/\sqrt{4\pi\rho_{0}}$, where $\rho_{0}$ is the proton mass density averaged over the time interval $T$.)

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 $|\sigma_{c}|>0.65$, $C_{B}\leq 0.25$ and $C_{n}\leq 0.25$).

In Figure 1, we show a scatter plot for $\sigma_{c}$ (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 $|\sigma_{c}|$ 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 $B_{0}(r/1\mbox{~{}au})^{2}\cos\psi=|B_{0r}|(r/1\mbox{~{}au})^{2}$ (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 $\psi$ is the Parker angle that depends on the solar-wind speed and heliocentric distance $r$ as $\tan{\psi}=\Omega r/V$, where $\Omega=2.7\times 10^{-6}$ rad s^-1 is the solar rotation frequency, and $B_{0r}$ is the radial component of the averaged field ${\bm{B}_{0}}$. Overall, only $\simeq 28\%$ of the slow-wind intervals seen by PSP and SO (during the data period considered) satisfy the criteria of ASW.

Using the selected ASW intervals that fulfill all conditions stated above, we can now estimate the radial profiles of the electron density $n$, average proton temperature $T_{p}$, the strength of the average magnetic field $|\bm{B}_{0}|$, the energy densities of the anti-sunward (sunward) AWs $E_{\rm out}=\rho_{0}\langle\delta Z^{2}_{\rm out}\rangle/4$ ($E_{\rm in}=\rho_{0}\langle\delta Z^{2}_{\rm in}\rangle/4$), and the energy densities of the fluctuating Alfvén velocity $E_{b}=\rho_{0}\langle\delta b^{2}\rangle/2$ and bulk velocity $E_{V}=\rho_{0}\langle\delta V^{2}\rangle/2$. Here $\delta{\bm{Z}}_{\rm out}=\delta{\bm{V}}-\delta{\bm{b}}$ ($\delta{\bm{Z}}_{\rm in}=\delta{\bm{V}}+\delta{\bm{b}}$) if the background vector magnetic field is pointing outward from the sun. However, if the background field is pointing toward the sun then $\delta{\bm{Z}}_{\rm out}=\delta{\bm{V}}+\delta{\bm{b}}$ ($\delta{\bm{Z}}_{\rm in}=\delta{\bm{V}}-\delta{\bm{b}}$). All these quantities are plotted in Figure 2. Each quantity is fit to a power-law function $A_{0}(r/1\mbox{~{}au})^{-\alpha}$ for PSP and SO separately. The fitting parameters are summarized in Table (1) below:

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 $\nu_{ep}$ 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 $\alpha_{e}=-0.5$. 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 $\nu_{ep}$ is thus estimated from Cranmer et al. (2009)

Eq. (2) can be practically simplified by implementing the fitting functions $A_{0}(r/r_{0})^{-\alpha}$ of $n$ and $T_{p}$ to evaluate the derivatives with respect to $r$. Then, the proton heating rate $Q_{p}$ is given by

where $\alpha_{p}$ and $\alpha_{n}$ 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

Here $q_{\parallel,e}$ is the parallel electron heat flux. We estimate $q_{\parallel,e}$ based on the collisionless model (Hollweg, 1974, 1976) as

where $\alpha_{H}$ is a dimensionless parameter that is only known approximately. For fast wind analysis Chandran et al. (2011) set $\alpha_{H}=0.75$. In our analysis of ASW we evaluate $Q_{e}$ for $\alpha_{H}=0.5,\mbox{~{}}0.75\mbox{~{}and~{}0.90}$. In order to evaluate the derivative of the electron heat flux term in Eq. (6) we use the conservation of the magnetic field flux $B_{0}r^{2}\cos\psi={\rm constant}$ 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 $Q_{p}=0.83\cdot 10^{-17}(r/1\mbox{~{}au})^{-3.92}$ J m^-3 s^-1 ($Q_{p}=1.25\cdot 10^{-17}(r/1\mbox{~{}au})^{-3.66}$ 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 $\alpha_{H}$.

The proton heating rate needed for ASW is slightly less than the one for the fast solar wind, $Q^{fast}_{p}=2.4\cdot 10^{-17}(r/1\mbox{~{}au})^{-3.8}$ 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 $\alpha_{H}=0.5,\ \mbox{~{}0.75}$ and $0.9$ guarantees that the conductive heating term is comparable in magnitude to the advective terms in Eq. (6) in our analysis.

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 $F^{\rm out}=(V+V_{A})E_{\rm out}$ ($F^{\rm in}=(V-V_{A})E_{\rm in}$) is the flux density of the anti-sunward (sunward) AWs, $\eta=-\cos\psi dB_{0}/(B_{0}dr)$, $\xi_{r}=E_{V}-E_{b}$ is the average residual energy density. Now, we introduce the measured power-law fitting functions of $E_{\rm out}\propto r^{\alpha_{\rm out}}$, $E_{\rm in}\propto r^{\alpha_{\rm in}}$, $|B_{0}|\propto r^{\alpha_{B}}$ and $n\propto r^{-\alpha_{n}}$ 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 following expression of $Q_{W}$ as

where $C_{\rm out}=1+V_{0}/V_{A}$, $C_{\rm in}=1-V_{0}/V_{A}$; and $\alpha_{B}$, $\alpha_{\rm out}$ and $\alpha_{\rm in}$ are the the power-law fitting exponents of $B_{0}$, $E_{\rm out}$ and $E_{\rm in}$, 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 $\alpha_{H}=0.9$, $0.75$ and $\alpha_{H}=0.5$ as a function of $r$. Overall, It seems that the protons are more heated than electrons in ASW when $\alpha_{H}\gtrsim 0.9$. 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 PSP 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.

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 $Q_{\rm CH09}$ using the ASW intervals selected for our analysis above. In Figure 5, we overplot the measured $Q_{\rm CH09}$ 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 $\pm 0.45$ 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).