Turbulent Heating in Solar Wind Thermodynamics

This paper considers the concept of wave-particle thermodynamic equilibrium in order to improve our understanding of the role of turbulent heating in the solar wind proton plasma. The thermodynamic equilibrium in plasmas requires the energy of a plasmon—the quantum of plasma fundamental oscillation—to be balanced by the proton-magnetized plasma energy, that is, the magnetic field and proton kinetic/thermal energy. This equilibrium has already been confirmed in several prior analyses, but also in this paper, by analyzing (i) multi-spacecraft data sets along the radial profile of the inner heliosphere, and (ii) representative data sets of a variety of 27 different space and astrophysical plasmas. Recently, it was shown that the slow mode of the near-Earth solar wind plasma is characterized by a missing energy source that is necessary for keeping the energy balance in the plasmon–proton-magnetized plasma. Here we show strong evidence that this missing energy is the turbulent energy heating the solar wind. In particular, we derive and compare the radial and velocity profiles of this missing energy and the turbulent energy in the inner heliosphere, also considering other minor contributions, such as the temperature of pickup protons. The connection of the missing plasmon–proton energy with the turbulent energy provides a new method for estimating and cross-examining the turbulent energy in space and astrophysical plasmas, while it confirms the universality of the involved new Planck-type constant that implies a large-scale quantization.


Introduction
Space and astrophysical plasmas are a ubiquitous form of matter in the universe, nearly always found to be turbulent. The turbulence is a chaotic, stochastic process that alters the characteristics of the plasma fluctuations. The entire heliosphere is closely linked to the properties of plasma turbulence. Solar wind protons flow throughout the supersonic heliosphere under the influence of expansive cooling and two primary groups of turbulent heating sources: (i) the solar-origin large-scale energy fluctuations; and (ii) the excitation of plasma waves by newborn interstellar pickup ions (e.g., Smith et al. 2001Smith et al. , 2006Adhikari et al. 2015). This paper investigates the interplay and partition of these turbulent heating sources in solar wind thermodynamics.
In thermodynamically stable space plasmas, the waveparticle thermodynamic equilibrium requires the energy of a plasmon-the quantum quasi-particle of plasma oscillationsto be balanced by the energy of the proton-magnetized plasma, that is, the field and proton average energy. This has been verified in a number of analyses in space and astrophysical plasmas (Livadiotis & McComas 2013a, 2014aWitze 2013;Livadiotis 2015, 2016, 20172018a;Livadiotis & Desai 2016;Livadiotis et al. 2018). However, in the case of the expanding solar wind in the inner heliosphere, a difference between the plasmon and proton plasma energies has been observed, which decreases with the wind speed and the heliocentric distance R (Livadiotis & Desai 2016). We speculate that this energy difference comprises the turbulent energy, responsible for heating the solar wind proton plasma, which was not considered in the plasmon-proton plasma energy balance. In particular, we may ask the following question: Is there a solid connection between the solar wind turbulent energy and the plasmon-proton-magnetized plasma energy balance?If yes, then, the solar wind thermodynamic equilibrium can be used for developing a new method of estimating the turbulent heating of solar wind.
The purpose of this paper is to improve our understanding of the following subjects: (i) the nature of the missing energy, that is, the difference in the balance between plasmon and protonmagnetized plasma energies; (ii) the connection of the missing energy to the mechanisms of heliospheric turbulent heating; (iii) the partition of the energies involved in the solar wind thermodynamic equilibrium; and (iv) the thermodynamic equilibrium and the energy balance between a plasmon and a proton-magnetized plasma, interwoven with the concept of large-scale quantization constant. Thermodynamic equilibrium and large-scale quantization will be examined and further developed in Section 2. In Section 3, we investigate the velocity and radial profiles of the missing energy, while in Section 4, we examine all the components contributing to the missing energy in the plasmon-proton-magnetized plasma energy balance, focusing on the (i) turbulent energy, (ii) temperature of pick-up ions, and (iii) gravitational potential energy. In Section 5, we formulate the missing energy by assembling all the previously discussed components, and then compare the constructed missing energy with the observed turbulent energy. The results lead to rewriting the protonmagnetized plasma thermodynamic equilibrium, and provide a new method for estimating and cross-examining the turbulent energy in space and astrophysical plasmas. Finally, Section 6 summarizes the results.

Plasmon-Proton Plasma Energy Balance
The wave-particle thermodynamic equilibrium in plasmas requires the energy E pl of a plasmon (energy quantum) to be balanced by the proton-magnetized plasma energy E p , that is, the magnetic field and proton kinetic/thermal energy, namely: [ · ( )] (e.g., Thejappa et al. 1993Thejappa et al. , 2012. Plasmons with only ω∼ω pl occur in the approximation of spatial scales quite larger than the Debye length. The proton plasma energy density (in the reference frame of the flow) is mainly given by the sum of its thermal energy density for a compressible flow, [γ/(γ−1)]nk B T, and the magnetic energy density, B 2 /(2μ 0 ); note that the fraction 1/2 comes from averaging sin 2 (α), where α is the angle between particle velocity to the magnetic field (e.g., Park et al. 2019). This proton plasma energy density, divided by the proton number density, gives the proton plasma energy per (proton) particle, which can be simply referred to as proton energy E p (Livadiotis & McComas 2014a).
In our approximation we will consider solar wind and pickup protons; other particles, such as alphas and pickup helium, are low in density, thus they do not significantly contribute to the total plasma energy density. (Parameter symbols: m e , m p : electron and proton masses; e: elementary electric charge; ε 0 : permittivity; μ 0 : permeability; k B : Boltzmann constant.) Therefore, we have the energies: where "other" means smaller energy contributions that will be examined in Section 5. Therefore, the plasma thermodynamic equilibrium in Equation 1(a), E pl =E p , is materialized by the balance of plasmon energy and proton-magnetized plasma energy, given in Equations 2(a) and (b), that is, Before describing other smaller contributions to the plasma energy E p , let us examine the concept of large-scale quantization constant that is unfolded by the star subscript of Planckʼs constant in Equation (3).

Large-scale Quantization Constant
A number of analyses have already confirmed that for space plasmas, the ratio between ionʼs average energy E p and plasma frequency ω pl is constant-as expected for the equality of with E p shown in Equation (3). Surprisingly, however, the constant value of the ratio E p /ω pl is not equal to the Planck constant =´- 1.05... 10 J s 34 · ; instead, it is shown that space plasmas lead to indeed a constant value, but ∼12 orders of magnitude larger, (1.19±0.05)×10 −22 J·s (Livadiotis & McComas 2013a, 2014aWitze 2013;Livadiotis 2015, 2016, 20172018a;Livadiotis & Desai 2016;Livadiotis et al. 2018); this large-scale analog of Planckʼs constant is noted by  * . Figure 1 demonstrates the large variation of the representative average values and uncertainties of the plasma parameters , ionosphere (io), aurora (au), plasma sheet (ps), plasmasphere (pl), sunspot plume (sp), shock example by Burlaga & King (1979) (sb), shock example by Gopalswamy & Yashiro (2011) (sg), magnetosheath (ms), inner heliosheath (ih), magnetosphere-average (ma), magnetosphere -Cluster (mc), outer corona (oc), inner corona (ic), coronal holes (ch), Van Allen belts (va), Jovian magnetosphere-average (jm), termination shock (ts); (for details on the data sets used, see Appendix A). The plotted color-coded parameters are: (a) density (gray), (b) temperature (light blue), (c) magnetic field strength (magenta), (d) plasma beta (brown), (e) Alfvén speed (yellow), (f) fast magnetosonic speed (deep blue), (g) Debye number (green), and (h) the ratio E p /ω pl (red). All parameter values are normalized to 1 (by dividing each of the 27 values with the maximum between them). The variation of all the plotted parameters in contrast to the constancy of E p /ω pl is clear.
of 27 space and astrophysical plasmas, while the respective values of the ratio E p /ω pl remain almost constant (see also Livadiotis & McComas 2013a, 2014a. (Note that all 27 values for each parameter are normalized to their maximum value.) Quantitative comparison of the distribution and variance of these values is shown in Figure 2. We observe that the standard deviation of the normalized values of log E p /ω pl is 10-30 orders of magnitude smaller than the standard deviations of the other normalized parameters. (For details on the data sets used, see Table 1 and details in Appendix A.) Having verified the small variability of the values of log E p /ω pl for the examined 27 types of space and astrophysical plasmas, it is straightforward to apply Equation (3) to derive the value of  * . All the values of log E p /ω pl and their uncertainties are plotted in Figure 3 We have seen the constancy of the ratio E p /ω pl by examining the representative parameter values from a variety of 27 space and astrophysical plasmas. The constancy of the ratio E p /ω pl can be also shown by examining a single space plasma. As an example we use actual measurements of solar wind proton plasma to again derive the ratio E p /ω pl . Voyager 1 and 2 measurements of the solar wind-a largely variant plasma-reveal a quasi-fixed value of the ratio E p /ω pl . Figure 4 plots the derived values of the ratio E p /ω pl against the heliocentric radial profile from 2 to 10 au. These plots, as well as the histograms on the right side confirm the constancy of the ratio E p /ω pl .

Velocity Profile of the Missing Energy
The constancy of the ratio E p /ω pl , and thus, the plasmonproton plasma thermodynamic equilibrium, has been confirmed by various space plasma measurements in previous years. Nevertheless, the thermodynamic equilibrium appears to be violated in the case of the slow and near-Earth measurements of the solar wind (e.g., Livadiotis & Desai 2016). More precisely, it has been observed that near 1 au, the ratio of the proton energy over the plasma frequency, E p /ω pl , deviates from the constant value  * that characterizes space plasmas. This deviation is larger for smaller solar wind speed; indeed, the ratio E p /ω pl undergoes a continuous transition from the slow to the fast solar wind, tending asymptotically toward the known value of ÿ * ( Figure 4). The observed deviation of the ratio E p /ω pl from the constant  * is caused by a difference between plasmon and proton energies: According to Figure 5, the missing energy ΔE is larger for low solar wind speeds and smaller for high solar wind speeds, and is actually negligible for speeds higher than V SW >550 km s −1 . This dependence of the missing energy ΔE on solar wind speed is similar to the behavior of the turbulent energy in the interplanetary space. Indeed, turbulence is more intense in the slow rather than the fast solar wind (Hadid et al. 2017).

Radial Profile of the Missing Energy
We examine in detail the plasmon-proton plasma thermodynamic equilibrium, as well as its violation observed in the slow and near-Earth solar wind. Using Equation 4(b) we calculate the missing energy, log ΔE/m p , and illustrate it as a function of solar wind speed V sw and the heliocentric distance R (Figure 6). In panel (a), the logarithm of the missing energy, ΔE per proton mass, as formulated in Equation 4(b), is depicted as a function of the solar wind speed V sw , for each radial bin of the heliocentric distances R from 0.29 to 5.41 au; each radial bin is color-coded.
In particular, for each radial bin, we perform a second binning among the values of the solar wind speed V sw (with constant width of bins D = V 10 km sw s −1 ). Then, for each V sw -bin we estimate the mean value and standard error of log ΔE/m p . The central value and half-width of each V sw -bin determine the mean value and error of V sw , respectively. Furthermore, we perform a linear fitting of the points {V sw ±δV sw , logΔE/m p ±δlogΔE/m p } within each radial bin. The intercept and slope-and their errors-derived from these fits are plotted in panels (b) and (c), respectively, as a function of R (again, the central value and half-width of each radial bin determine the mean value and error of R in these panels). We observe that on average both the intercept and slope decrease when R increases.

Turbulent Energy
There are three primary sources of turbulence in the heliosphere: (1) turbulence driven by shear due to the interaction between fast and slow solar wind streams (Coleman 1968;Roberts et al. 1992), (2) compressional sources of turbulence due to stream-stream interactions and shock waves (Whang 1991), and (3) turbulence due to pickup ions created by charge exchange between solar wind protons and interstellar neutral hydrogen (Williams & Zank 1994). The sources can be divided into two groups: (1) solar-origin large-scale energy fluctuations (stream shears and shock waves) driven turbulence, and (2) interstellar pickup ion driven turbulence (e.g., Smith et al. 2001Smith et al. , 2006Adhikari et al. 2015). Both of these groups of sources contribute to the solar wind heating, but (1) is dominant in the inner heliosphere and (2) is dominant in the outer heliosphere.
The turbulent energy, developed along the solar wind radial expansion, is given by:   The Elsässer vector variable + Z corresponds to Alfvénic modes with an outward radial direction of propagation (in the solar wind frame). The outward propagating turbulent energy radial profile in the inner heliosphere, + E t (R) for R<5.5 au was derived for Helios 1 and 2 and Ulysses data sets (from 0.29 to ∼5.4 au) by Bavvasano et al. (2000) and later by Adhikari et al. (2015). There is some difference in the results of these two analyses, caused by the different lengths of data intervals (hour versus days), thus we use their weighted average. This was performed by (i) binning both the radius R and energy + E t -on log-log scales, and then, (ii) averaging at each bin the results of the two papers. The results are shown in Figure 7.
Note that the other Elsässer vector -Z corresponds to Alfvénic modes with an inward radial propagation direction. The corresponding turbulent energy

Temperature of Pickup Ions
Pickup ions (PUIs) play an essential role in the thermodynamic energy balance of the solar wind. The internal particle energy of the solar wind is dominated by PUIs beyond ∼20 au from the Sun  . Velocity and radial profiles of the missing energy. (a) log ΔE/m p is depicted as a function of the solar wind speed V sw , for each bin of the heliocentric distances R from 0.29 to 5.41 au, as shown on the graph. The means and standard errors of log ΔE/m p are calculated for each V SW -bin (=10 km s −1 ). The linear fit of each radial data subset {V sw ±δV sw , logΔE/m p ±δlogΔE/m p } estimates the intercept and slope-and their errors-corresponding to a certain distance R placed on the middle of the radial bin, with δR characterizing the half-width of this bin; thus, the radial profiles of (b) intercept and (c) slope are plotted, indicating a clear radial decrease. As R increases, the intercept decreases, while the slope is negative and becomes steeper.
The average energy of a proton must take into account the energy of a solar wind proton as well as the energy of the pickup proton. Thus, before we compare the energy missing from the plasmon-proton plasma balance, ΔE, with the turbulent energy, + E t , we must include the PUI energy contribution into the proton energy E p in Equation 2(b). Below we show how we derive the PUI temperature and blend it in The distribution function of speeds in the solar wind frame, f pui (u), was transformed to the S/C frame and its best fit to data was determined by χ 2 minimization. Once f pui (u) was derived, the PUI temperature was determined by the second statistical moment á ñ u 2 of f pui (u), i.e., T pui =[m p /(3k B n pui )]·á ñ u 2 . The PUI average energy E pui is derived as follows: the total proton pressure sums the solar wind and pickup proton partial pressures, = + P P P , i.e ., 6a p,tot p pui ( ) The total temperature T p,tot , derived from mixing solar wind and pickup protons, replaces the temperature in Equation 2 where we consider that n pui =n p for the examined radial range up to 6 au (as shown in Figures 6-8). Thus, the PUI energy is

Gravitational Potential Energy
Solar wind protons are also subject to gravitational potential energy; that is, another minor contribution in Equation 2(b): (Note that gravitational potential energy contributes to the proton plasma energy globally, throughout the heliosphere. In addition, other potential energies may also exist but contribute only locally and/or occasionally to the proton plasma energy, e.g., the electrostatic potential energy; Cuperman & Harten 1971;Lacombe et al. 2002;Livadiotis 2018b;Nicolaou & Livadiotis 2019.) In the next section, we will use the turbulent energy given by Equation (5), the PUI energy given by Equation 8(b), and the gravitational potential energy given by Equation (9), in order to improve the proton energy E p in Equation 2(b).

Formulation of the Missing Energy
Having estimated the minor contributions of the PUI energy and the gravitational potential energy, the total proton plasma energy in Equation 2(b) becomes where the non-turbulent part of the proton plasma energy is

Comparison between the Constructed Missing Energy and the Observed Turbulent Energy
Next, we compare the missing energy ΔE/m p with the turbulent energy. We have already plotted the radial profile of turbulent energy in Figure 7. Therefore, we need to calculate the radial profile of the missing energy. Then, we will compare the two radial profiles. For this, we calculate the missing energy ΔE/m p using daily averages of the solar wind and interplanetary magnetic field data taken from Helios 1 and 2, Wind, and Ulysses S/C, for the heliocentric distance from 0.29 to 5.41 au. Then, we construct the radial profile of the missing energy ΔE, and compare this result with the radial profile of the turbulent energy shown in Figure 7. Finally, the two radial profiles are shown in Figure 8.
The missing energy ΔE, derived from Equation (11), and the turbulent energy, derived by Bavvasano et al. (2000), Adhikari et al. (2015), and averaged as shown in Figure 7, are coplotted in Figure  The p-value of the statistical hypothesis that the two data sets describe the same statistical population is very high (∼0.4), thus the hypothesis is statistically confident.

Rewriting the Proton-magnetized Plasma Thermodynamic Equilibrium
We have shown that the energy balance between the plasmon and the proton plasma magnetized energy is written as

Conclusions
This paper considered the concept of thermodynamic equilibrium between plasmons and proton-magnetized plasma and determined their energy balance in order to quantify the contribution of the turbulent energy. This equilibrium was shown and confirmed in several prior publications, but also in this paper, by analyzing (i) multi-spacecraft data sets along the radial profile of the inner heliosphere (R<10 au), and (ii) representative data sets of a variety of 27 different space and astrophysical plasmas.
The near-Earth solar wind plasma, observed in the slow wind mode, is characterized by a small deviation from the plasmonproton-magnetized plasma energy balance (Livadiotis & Desai 2016). This is expressed as a missing energy that prevents the plasmon-proton-magnetized plasma energy balance. The paper performed theoretical and space plasma data analyses in order to improve our understanding of the origin and nature of the missing energy. In particular, we investigated the velocity and radial profiles of the missing energy along the inner heliosphere. We also examined the interplay and partition of the turbulent heating sources in solar wind thermodynamics, and showed that radial profiles of the missing energy coincide with the radial profile of the turbulent energy.
In addition, the thermodynamic equilibrium and the energy balance between a plasmon and a proton-magnetized plasma are interwoven with the concept of large-scale quantization constant. Recently, strong evidence has shown that space and astrophysical plasmas are linked to a Planck-like constant, but ∼12 orders of magnitude larger. The plasmon-proton energy balance is described confirming the universality of this largescale quantization constant.