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(a) Distribution densities of all the normalized values of the parameters shown in Figure 1 (following the same order and color-coding). (b) Standard deviations of each distribution.
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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 ene...
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... 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. ...
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Density irregularities are responsible for the scattering of radio waves in the solar wind and astrophysical plasmas. These irregularities significantly affect the inferred physical properties of radio sources, such as size, direction, and intensity. We present here a theory of angular broadening due to the scattering of radio waves by density irre...
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... 2013b;Livadiotis & Desai 2016;Livadiotis 2019c). That is, c = ÿ, when no correlations exist among particles, and c = * , when significant correlations exist among particles beyond their nearest neighbors. ...
In this paper, we develop the transport equation of kappa, the fundamental thermodynamic parameter that labels kappa distributions of particle velocities. Using the recently developed concept of entropy defect, we are able to formulate the transport equation of kappa as a function of a general, positive or negative, rate of entropy change. Then, we derive the particular case of exchanging plasma ions with low-dimensionality, newly born pickup protons, which interact and decrease the entropy of the flow of otherwise kappa-distributed plasma protons. Finally, we apply the transport equation of kappa to the solar wind plasma protons, which leads to the radial profile of kappa values, as well as the evolution of the kappa distributions through the heliosphere. The results show that the solar wind kappa decreases with increasing heliocentric distance, corresponding to plasmas residing in stationary states far from classical thermal equilibrium. Moreover, in the outer heliosphere and the heliosheath, kappa reaches its lowest values and is spread across the far-equilibrium region of 1.5 < κ < 2.5, which coincides with independent observations provided by NASA’s Interstellar Boundary Explorer mission.
... We identify three potential mechanisms that could possibly provide external thermal energy sources or sinks to the electrons in the form of nonzero Ξ: turbulent heating, instabilities, and collisions. The turbulent cascade transfers energy from large scales to kinetic scales, where kinetic processes dissipate the energy in the form of heat (Tu & Marsch 1995;Breech et al. 2009;Schekochihin et al. 2009;Bruno & Carbone 2013;Goldstein et al. 2015;Livadiotis 2019;Franci et al. 2022). This form of turbulent dissipation leads to an irreversible deposition of thermal energy. ...
We present an observational analysis of the electron thermal energy budget using data from Parker Solar Probe. We use the macroscopic moments, obtained from our fits to the measured electron distribution function, to evaluate the thermal energy budget based on the second moment of the Boltzmann equation. We separate contributions to the overall budget from reversible and irreversible processes. We find that an irreversible thermal energy source must be present in the inner heliosphere over the heliocentric distance range from 0.15 to 0.47 au. The divergence of the heat flux is positive at heliocentric distances below 0.33 au, while beyond 0.33 au, there is a measurable degradation of the heat flux. Expansion effects dominate the thermal energy budget below 0.3 au. Under our steady-state assumption, the free streaming of the electrons is not sufficient to explain the observed thermal energy density budget. We conjecture that the most likely driver for the required heating process is turbulence. Our results are consistent with the known nonadiabatic polytropic index of the electrons, which we measure as 1.18 in the explored range of heliocentric distances.
... We identify three potential mechanisms that could possibly provide external thermal energy sources and sinks to the electrons in the form of non-zero Ξ: turbulent heating, instabilities, and collisions. The turbulent cascade transfers energy from large scales to kinetic scales where kinetic processes dissipate the energy in the form of heat Goldstein et al. 2015;Livadiotis 2019;Bruno & Carbone 2013;Tu & Marsch 1995;Schekochihin et al. 2009;Franci et al. 2022). This form of turbulent dissipation leads to an irreversible deposition of thermal energy. ...
We present an observational analysis of the electron thermal energy budget using data from Parker Solar Probe. We use the macroscopic moments, obtained from our fits to the measured electron distribution function, to evaluate the thermal energy budget based on the second moment of the Boltzmann equation. We separate contributions to the overall budget from reversible and irreversible processes. We find that a thermal-energy source must be present in the inner heliosphere over the heliocentric distance range from 0.15 to 0.47 au. The divergence of the heat flux is positive at heliocentric distances below 0.33 au, while beyond 0.33 au, there is a measurable degradation of the heat flux. Expansion effects dominate the thermal energy budget below 0.3 au. Under our steady-state assumption, the free streaming of the electrons is not sufficient to explain the thermal energy density budget. We conjecture that the most likely driver for the required heating process is turbulence. Our results are consistent with the known non-adiabatic polytropic index of the electrons, which we measure as 1.176 in the explored range of heliocentric distances.
... where C = (9π/2) 1/3 /e ≈ 0.89; λ c constitutes the smallest correlation length, interpreted by the interparticle distance b ∼ n −1/ d for collisional particle systems (absence of correlations), or by the Debye length for collisionless particle systems (where long-range interactions induce correlations among particles). The factor g κ,N depends on kappa and the number of correlated particles N , with g κ,N ≈ 1 for large N; the phase-space parcel ÿ c is typically given by Planck's constant, but it was shown to represent a different and larger constant in space plasmas, where Debye shielding limits the distance of correlations, ÿ * = (1.19 ± 0.05) × 10 −22 J s (Livadiotis & McComas 2013c, 2014Livadiotis & Desai 2016;Livadiotis 2016Livadiotis , 2019cLivadiotis et al. 2020); i.e., ÿ c = ÿ in the absence of correlations, while ÿ c = ÿ * when correlations exist among particles beyond the nearest neighbors, as for the majority of space plasmas. ...
We derive annual sky maps of the proton temperature in the inner heliosheath (IHS), and track their temporal evolution over the years 2009–2016 of Interstellar Boundary Explorer observations. Other associated thermodynamic parameters also determined are the density, kappa (the parameter that characterizes kappa distributions), temperature rate, polytropic index, and entropy. We exploit the theory of kappa distributions and their connection with polytropes, to (i) express a new polytropic quantity Π that remains invariant along streamlines where temperature and density may vary, (ii) parameterize the proton flux in terms of the Π invariant and kappa, and (iii) derive the temperature and density, respectively, from the slope and intercept of the linear relationship between kappa and logarithm of Π. We find the following thermodynamic characteristics: (1) temperature sky maps and histograms shifted to their lowest values in 2012 and their highest in 2015; (2) temperature negatively correlated with density, reflecting the subisothermal polytropic behavior; (3) temperature positively correlated with kappa, revealing characteristics of the mechanism responsible for generating kappa distributions; (4) processes in IHS are subisothermal tending toward isobaric, consistent with previously published results; (5) linear relationship between kappa and polytropic indices, revealing characteristics of the particle potential energy; and (6) entropy positively correlated with polytropic index, aligned with the underlying theory that entropy increases toward the isothermal state where the kappa distribution reduces to the Maxwell–Boltzmann description.
... The factor g κ ,N depends on the kappa index κ and the number of correlated particles; for a large number of particles, it becomes g κ ,N ≈ 1. Finally, the phase-space parcel c is typically given by the Planck's constant, but it was shown to represent a different and larger constant in space plasmas, where Debye shielding limits the distance of correlations, * = (1.19 ± 0.05) × 10 −22 J · s [55,[59][60][61][62][63] (namely, c = , when no correlations exist among particles, and c = * , when significant correlations exist among particles beyond their nearest neighbors, as in the case of the majority of space plasmas). ...
This paper develops explicit and consistent definitions of the independent thermodynamic properties of temperature and the kappa index within the framework of nonextensive statistical mechanics and shows their connection with the formalism of kappa distributions. By defining the “entropy defect” in the composition of a system, we show how the nonextensive entropy of systems with correlations differs from the sum of the entropies of their constituents of these systems. A system is composed extensively when its elementary subsystems are independent, interacting with no correlations; this leads to an extensive system entropy, which is simply the sum of the subsystem entropies. In contrast, a system is composed nonextensively when its elementary subsystems are connected through long-range interactions that produce correlations. This leads to an entropy defect that quantifies the missing entropy, analogous to the mass defect that quantifies the mass (energy) associated with assembling subatomic particles. We develop thermodynamic definitions of kappa and temperature that connect with the corresponding kinetic definitions originated from kappa distributions. Finally, we show that the entropy of a system, composed by a number of subsystems with correlations, is determined using both discrete and continuous descriptions, and find: (i) the resulted entropic form expressed in terms of thermodynamic parameters; (ii) an optimal relationship between kappa and temperature; and (iii) the correlation coefficient to be inversely proportional to the temperature logarithm.
... The second term includes the thermal pressure normalized to ρ, while the third term is the magnetic field pressure over ρ. Note that the fraction 1/2 in the third term, comes from averaging over all the possible angles between B and v vectors (e.g., [10,11]). The Bernoulli integral, in terms of the bulk flow speed v, thermal speed v th , and Alfvén speed v A , is: ...
... Sci. 2021,11, 4643 ...
The Bernoulli integral describes the energy conservation of a fluid along specific streamlines. The integral is the sum of individual terms that contain the plasma density, speed, temperature, and magnetic field. Typical solar wind analyses use the fluctuations of the Bernoulli integral as a criterion to identify different plasma streamlines from single spacecraft observations. However, the accurate calculation of the Bernoulli integral requires accurately determining the plasma polytropic index from the analysis of density and temperature observations. To avoid this complexity, we can simplify the calculations by keeping only the dominant terms of the integral. Here, we analyze proton plasma and magnetic field observations obtained by the Wind spacecraft at 1 au, during 1995. We calculate the Bernoulli integral terms and quantify their significance by comparing them with each other. We discuss potential simplifications of the calculations in the context of determining solar wind proton thermodynamics using single spacecraft observations.
... We employ two physical properties of the expanding solar wind plasma for improving our understanding of the turbulent heating of solar wind at 1 au, i.e., (a) wave-particle thermodynamic equilibria (Livadiotis 2019a), and (b) transport of the rate of entropy change (Adhikari et al. 2020). ...
... Recently, it was shown that the energy transfer between plasma particles and waves is governed by a new and unique relationship: the ratio between the energy per particle over the plasma frequency is constant, that is, a large-scale analog of Planckʼs constant denoted by ÿ * (Livadiotis & McComas 2013, 2014a, 2014bWitze 2013;Livadiotis 2015, 2016, 2017Livadiotis & Desai 2016;Livadiotis et al. 2018;Livadiotis 2019a). As an example, Figure 1 demonstrates the large variation of the representative average values and uncertainties of the plasma parameters of 27 space and astrophysical plasmas (Livadiotis 2019a), which are stable and residing at stationary states (Livadiotis 2018a(Livadiotis , 2018b, while the respective values of the ratio E p /ω pl that approaches the value of ÿ * remain almost constant. ...
... Recently, it was shown that the energy transfer between plasma particles and waves is governed by a new and unique relationship: the ratio between the energy per particle over the plasma frequency is constant, that is, a large-scale analog of Planckʼs constant denoted by ÿ * (Livadiotis & McComas 2013, 2014a, 2014bWitze 2013;Livadiotis 2015, 2016, 2017Livadiotis & Desai 2016;Livadiotis et al. 2018;Livadiotis 2019a). As an example, Figure 1 demonstrates the large variation of the representative average values and uncertainties of the plasma parameters of 27 space and astrophysical plasmas (Livadiotis 2019a), which are stable and residing at stationary states (Livadiotis 2018a(Livadiotis , 2018b, while the respective values of the ratio E p /ω pl that approaches the value of ÿ * remain almost constant. ...
The paper presents a new method for deriving turbulent heating of the solar wind using plasma moments and magnetic field data. We develop the method and then apply it to compute the turbulent heating of the solar wind proton plasma at 1 au. The method employs two physical properties of the expanding solar wind plasma, the wave-particle thermodynamic equilibrium, and the transport of entropic rate. We analyze plasma moments and field data taken from Wind S/C, in order to compute (i) the fluctuating magnetic energy, (ii) the corresponding correlation length, and (iii) the turbulent heating rate. We identify their relationships with the solar wind speed, as well as the variation of these relationships relative to solar wind and interplanetary coronal mass ejection plasma.
... In the inner heliosphere, the solar-origin large-scale energy fluctuations constitute the primary source of turbulent heating. This turbulent energy decreases with heliocentric distance as [26][27][28]: ...
This paper improves our understanding of the interplay of the proton plasma turbulent heating sources of the expanding solar wind in the heliosphere. Evidence is shown of the connections between the polytropic index, the rate of the heat absorbed by the solar wind, and the rate of change of the turbulent energy, which heats the solar wind in the inner and outer heliosphere. In particular, we: (i) show the theoretical connection of the rate of a heat source, such as the turbulent energy, with the polytropic index and the thermodynamic process; (ii) calculate the effect of the pick-up protons in the total proton temperature and the relationship connecting the rate of heating with the polytropic index; (iii) derive the radial profiles of the solar wind heating in the outer and inner heliosphere; and (iv) use the radial profile of the turbulent energy in the solar wind proton plasma in the heliosphere, in order to show its connection with the radial profiles of the polytropic index and the heating of the solar wind.
The paper derives the one-to-one connecting relationships between plasma heating and its polytropic index, and addresses the consequences through the transport equation of temperature. Thermodynamic polytropic processes are classified in accordance to their polytropic index, the exponent of the power-law relationship of thermal pressure expressed with respect to density. These processes generalize the adiabatic one, where no heating is exchanged between the system and its environment. We show that, in addition to heating terms, the transport equation of temperature depends on the adiabatic index, instead of a general, nonadiabatic polytropic index, even when the plasma follows nonadiabatic processes. This is because all the information regarding the system's polytropic index is contained in the heating term, even for a nonconstant polytropic index. Moreover, the paper (i) defines the role of the polytropic index in the context of heating; (ii) clarifies the role of the nonadiabatic polytropic index in the transport equation of temperature; (iii) provides an alternative method for deriving the turbulent heating through the comparably simpler polytropic index path; and, finally, (iv) shows a one-component plasma proof-of-concept of this method and discusses the implications of such derived connecting relationships in the solar wind plasma in the heliosphere.
The recently developed concept of “entropic defect” is important for understanding the foundations of thermodynamics in space plasma physics, and more generally for systems with physical correlations among their particles. Using this concept, this paper derives the basic formulation of the distribution function of velocities (or kinetic energies) in space plasma particle populations. Earlier analyses have shown how the formulation of kappa distributions is interwoven with the presence of correlations among the particles’ velocities. This paper shows, for the first time, that the reverse is true: the thermodynamics of particles’ physical correlations are consistent only with the existence of kappa distributions.
