A Measurement of the Effective Mean-Free-Path of Solar Wind Protons

Weakly collisional plasmas are subject to nonlinear relaxation processes, which can operate at rates much faster than the particle collision frequencies. This causes the plasma to respond like a magnetised fluid despite having long particle mean-free-paths. In this Letter the effective collisional mechanisms are modelled in the plasma kinetic equation to produce density, pressure and magnetic field responses to compare with spacecraft measurements of the solar wind compressive fluctuations at 1 AU. This enables a measurement of the effective mean-free-path of the solar wind protons, found to be 4.35 $\times 10^5$ km, which is $\sim 10^3$ times shorter than the collisional mean-free-path. These measurements are shown to support the effective fluid behavior of the solar wind at scales above the proton gyroradius and demonstrate that effective collision processes alter the thermodynamics and transport of weakly collisional plasmas.

The escaping solar corona, known as the solar wind, expands into interplanetary space as a super-Alfvénic and turbulent plasma [6][7][8]. In situ measurements of particle distribution functions and electromagnetic fields enable fundamental plasma physics observations [9]. The Spitzer-Härm proton-proton collision frequency ν SH p,p decreases with radial distance from the Sun, and by a few solar radii, is much smaller than other characteristic frequencies.
While the Spitzer-Härm collision frequency appears incompatible with the fluidlike behavior of the solar wind, weakly collisional plasmas are also subject to nonlinear processes that prevent extreme departure from equilibrium [38][39][40][41][42]. Solar wind observations present substantial evidence of temperature anisotropy instabilities constraining the particle distribution functions [2,40,[43][44][45][46] suggesting they experience pitch-angle scattering by plasma waves [47]. These processes can play a similar role to collisions i.e., they are effective collision processes.
This Letter presents a measurement of the effective mean-free-path of the solar wind by comparing observations of compressive wave-mode polarizations to numerical solutions of varying effective collisionality. It is shown that the transition from fluid to collisionless dynamics in the solar wind occurs at scales several orders of magnitude below the classical Spitzer-Härm mean-free-path, explaining the fluidlike behavior of the weakly collisional solar wind.
Theory and numerical solutions.-The kinetic MHD equations with the Bhatnagar-Gross-Krook (BGK) collision operator [48,49] produce dispersion relations and plasma fluctuations (e.g., magnetic field and pressure) that span between the collisionless and collisional limits [50][51][52]. They describe a nonrelativistic, magnetized plasma of arbitrary collision frequency [51,52]. The specific equations, which we refer to as KMHD-BGK, model both the proton and electron responses with the kinetic equation. The BGK operator is used here to model relaxation processes, not particle collisions, so we use the language of an effective proton collision frequency ν eff or mean-free-path λ eff mfp = v p th /ν eff , where the proton thermal speed is v p th (see Supplemental Material).
Assuming plasma motions are slow compared to the gyrofrequency Ω p , the second moment of the kinetic equation and the ideal induction equation leads to, where d/dt is the convective derivative and the quantities are the proton density n p , magnetic-field strength B, (perpendicular) parallel (p p ⊥ ) p p proton pressure, field parallel flux of (perpendicular) parallel (q p ⊥ ) q p proton heat, and the unit magnetic field vectorb = B/B [53,54]. The Alfvén speed is v A = B/ 4πn p m p , the proton gyroradius is ρ p = v p th /Ω p , and the ion-acoustic speed is c s = (3k B T p + k B T e )/m p , where the parallel proton (electron) temperature is T p (T e ).
Equations (1) are often discussed when the right hand sides are zero and are then referred to as the double adiabatic equations or Chew-Goldberger-Low (CGL) invariants [53]. The focus here is on how the CGL invariants are broken, for example, by the heat flux terms in the collisionless limit, and by the collisional terms (∝ ν eff ). Therefore, the relative conservation of the CGL invariants provides a sensitive test of the equation of state [55].
We proceed by constructing measures that describe the correlation and amplitude ratios of the left hand sides of Eqs. (1), where δχ = χ − χ is the fluctuation about the average χ . The method compares predictions for Eqs. (2) derived from the slow-mode eigenmodes of the linearized KMHD-BGK system (e.g., δp p ⊥ , δB etc.) to solar wind measurements. See Supplemental Material.
The model's free parameters are the propagation angle θb ,k and proton effective mean-free-path λ eff mfp , i.e., they can be measured. The wavenumber k and the proton beta β = 8πk B p p /B 2 where p p = 2p p ⊥ /3 + p p /3 are set to measured values. The species temperature ratio is set to a typical value for the solar wind T p /T e = 1 and the effective mean-free-path species ratio is set to λ eff mfp /λ eff mfp,electrons = 5; see Supplemental Material at for details of electron physics.
The top panels of Fig. 1 demonstrate the ability of the KMHD-BGK equations to resolve the dynamics of the compressive slow-mode from collisional (lighter blue) to collisionless (black) [51]. Numerical predictions for Eqs. (2a), (2c) (bottom panels of Fig. 1) show distinct differences at β > 1 for k λ eff mfp , which can be compared to observations. The MHD/collisionless limits are illustrated in magenta, for C ⊥ (bottom right panel) these two limits produce similar trends, therefore it is necessary to make comparisons at multiple k , to measure λ eff mfp . Measurements.-The dataset consists of Wind spacecraft measurements of the pristine solar wind during years 2005-2010. The electrostatic analyzer 3DP records onboard moments of the proton density, velocity and pressure tensor, and the magnetometer MFI records the magnetic field, at a nominal ∼ 3s cadence [57,58].
The dataset is restricted to time intervals satisfying three criteria: (i) 95% of the data is available (the remaining is then linearly interpolated); (ii) the median density must be greater than 1 particle per cm −3 ; and (iii) the average norm of the non-gyrotropic tensor (Π p = p p −bb p p − (1 −bb)p p ⊥ ), must be less than 30% of the average norm of the pressure tensor p p . To probe a set of wavenumbers we measure the four quantities in Eqs. (2), the average radial solar wind velocity V SW , and average proton beta for a set of time intervals τ = [30s, 1min., 2mins., ..., 128mins.]. The time scales are converted to wavenumber k SW = 1/τ V SW via Taylor's frozen-in-flow (TFF) assumption [59]. Three bins of equal probability density are obtained where the median of each bin is k SW = [0.288, 1.41, 6.34] × 10 −5 km −1 , which lie within the inertial range of the magnetic field power spectrum at 1 AU [60]. The wavenumber bins contain [2.98, 16.6, 70.0] × 10 5 samples.
For bin k SW = 0.288 × 10 −5 km −1 the β-conditioned probability functions of Eqs. (2), all mapped to a common color bar, are displayed in Fig. 2. The β-trend lines in magenta (see caption) capture statistically significant differences between β ≶ 1. From the correlations C , C ⊥ it is clear that the CGL invariants are rarely conserved (C , C ⊥ = 1), but display similar trends to the theoretical expectations (Fig. 1). The amplitude ratios A , A ⊥ demonstrate a relative decrease in fluctuation amplitude of the pressure components at β > 1.
Comparison of measurements and numerical solutions.-To compare theoretical predictions of Eqs. 2 to observations, a degeneracy in parametrization must be dealt with: the numerical solutions primarily depend on k λ eff mfp = k cos θb ,k λ eff mfp [51]; such that λ eff mfp , θb ,k are degenerate. To address this we introduce a scale dependent anisotropy model (k ∼ k α ⊥ ), which relates k and θb ,k , where k iso is the isotropic wavenumber (k ⊥ = k ) and α is the anisotropy exponent, generalised from turbulence models [61]. Comparing solutions parameterized by λ eff mfp , α, k iso across multiple wavenumbers k = k SW clears the degeneracy.
Finally, the predictions of Eqs. (2) from the numerical solutions are normalized to the measured βconditioned mean value (dashed magenta lines in Fig.  2) of C , C ⊥ , A , A ⊥ at β ≃ 10 −1 . This is to account for the fact that linear polarizations are only approximately observed in strong turbulence [9].
To make a quantitative comparison, we compute the "goodness of fit", whereŷ i (ȳ i ) is the local numerical solution (local measured mean), summed over i, denoting the ith β-bin. R(k SW ; α, k iso , λ eff mfp ) is calculated for each wavenumber k SW , where the meanȳ i is respective to the wavenumber bin. The R-values are inverted for unnormalized weights (w = R −1 ), divided by the maximum weight, then summed over wavenumber W = k w(k; α, k iso , λ eff mfp ) to break the aforementioned degeneracy. From this volume, weighted geometric means µ x , covariances σ 2 x,y , and two sigma confidence intervals CI x are calculated (see Supplemental Material) [63,64]. This is the method we employ to measure the quantities λ eff mfp , α, k iso , providing the main results of the Letter.
To visualize the weighted parameter space for C ⊥ , Fig.  3 illustrates the weight volume W(α, k iso , λ eff mfp ), numerically integrated over each parameter axis χ, where χ n , χ 0 are limits of the range. The weighted means in Fig. 3 lie in the maximum regions of W χ , within the confidence intervals, indicating the weighted geometric statistics are a good representation of the observations. To check the scale dependence, Fig. 4 displays the observed β-conditioned means of Eqs. (2) and the numerical solutions corresponding to the maximum W. The numerical solutions and observations trend similarly with wavenumber indicating the scale dependence of the effective collisionality has been well modeled. The parameters of the maxima (recorded in the panels of Fig. 4) do not correspond exactly to the weighted geometric means of W (seen in Fig. 3) reflecting the statistical nature of the measured quantities.
The method of calculating statistics for λ eff mfp , α, k iso displayed in Fig. 3 for C ⊥ produces similar statistics for C , A , A ⊥ (see Supplemental Material). Therefore, in Table I combined statistics are reported. The measured effective mean-free-path and mean proton thermal speed (measured with this data set) gives an effective collision frequency of ν eff = v p th /λ eff mfp = 1.11 ×10 −4 s −1 . The transition frequency, where ν eff ≃ ω can be estimated with ν eff = v p th /λ eff mfp and the ion-acoustic dispersion relation ω IA = k c s , giving the parallel transition wavenumber k trans = v p th /c s λ eff mfp [37,51,65]. Using the wavenumber model (Eq. (3)), the transition wavenumber   is, Inserting the combined statistics from Table I, using a typical value of v p th /c s = 1/2 for the solar wind, and using the TFF assumption, the transition wavenumber in spacecraft-frame frequency at 1 AU is V SW k trans = f trans = 0.19 Hz, and CI f trans = [0.046, 0.33] Hz. The uncertainties are propagated from V SW and the four estimates of k trans from C , A , C ⊥ , A ⊥ .
Discussion.-We have measured the relative nonconservation of the CGL invariants and modeled the behavior with the slow-mode branch of the KMHD-BGK equations to measure the effective mean-free-path of the solar wind protons and the scale dependence of the slowmode wavenumber anisotropy (Table I reports the statistics of these measurements). The primary result of this Letter is the measured effective proton mean-freepath that is ∼ 10 3 times smaller than the Spitzer-Härm mean-free-path (λ SH mfp = 1.14 × 10 8 km, measured with this dataset). Therefore, the fluidlike range in the solar wind extends to much smaller scales than predicted based on particle collisions. In addition, the scale dependent anisotropy of the compressive fluctuations (α ≃ 0.4) is consistent with previous measurements [9,62], being more anisotropic than the Alfvénic fluctuations.
The measured transition frequency, the scale between fluid behavior (f ≪ f trans ) and collisionless behavior (f ≫ f trans ), of f trans = 0.189 Hz is at the well-known break in power law (k ⊥ ρ p ∼ 1) of the magnetic field power spectrum at 1 AU [8,60,66]. These measurements therefore justify the use of fluid MHD theory at larger scales (k ⊥ ρ p < 1) [7, 9, 13-21, 32, 67]. If the result k ⊥ ρ p ≃ k λ eff mfp turns out to be a general property of weakly collisional plasma, this provides a simple parameterization for the effective collisionality of astrophysical plasmas.
Effective collisional processes have long been studied theoretically and numerically [3,[68][69][70][71][72][73], but it is an open question as to the relevant role of the various mechanisms [2,[40][41][42][74][75][76]) and how they are activated [65,77,78]. Therefore, further studies are necessary to assess exactly what key physics of weakly collisional plasma leads to the measured effective collisionality, since most astrophysical plasmas, being multi-scale and turbulent, will support effective collision mechanisms [79]. The measurements presented here provide constraints to be satisfied by theories of effective collision processes.

OUTLINE OF SUPPLEMENTAL MATERIALS
This document provides supplemental materials for the Letter titled, "Measurement of the Effective Mean-Free-Path of the Solar Wind Protons". Section details the wavenumber model (Eq. 3 of the Letter) and the transition wavenumber (Eq. 6 of the Letter), Section introduces the weighted geometric statistics (reported in Table 1 of the Letter) and shows a figure of the weighted geometric statistics, and Section introduces the Spitzer-Härm collision quantities (referenced in the text of the Letter). Section introduces the collisional-kinetic magnetohydrodynamic equations, shows the linear Fourier analysis, normalisation and the linear system of equations to be solved numerically.

WAVENUMBER MODEL
The model k ∼ k α ⊥ introduced here is generalized from the critical balance model of Alfvénic turbulence (see Ref. [61]). In the Letter it is used to model the compressive wave propagation angle θb ,k . To ensure the isotropic scale is defined correctly we begin with, where α is the anisotropy exponent. Since, k = k cos(θb ,k ), k ⊥ = k sin(θb ,k ) we have, so that at θ * b,k = 45 • , sin(θ * b,k ) = cos(θ * b,k ) = 1/ √ 2, we have, so we have recovered the isotropic scale k = k iso where k ⊥ = k . Writing the wavenumber model This Eq. appears as number 3 of the Letter. Here k depends on θb ,k parametrised by α ∈ [0, 1), k iso . Eq. S4 can be inverted on θb ,k ∈ [0, 90 • ) for k. The wavenumber model is also used in the derivation of Eq. 6 of the Letter. Just above Eq. 6, in the text of the Letter, the relation, is argued to define k trans , which can be compared to measurements with the full wavenumber k trans . Using the model (Eq. S4) to write, v p th c s λ eff Solving for θ tranŝ b,k , Now, k trans can be written, Using the trigonometric identities, yields, (S10) Where, which simplifies to, Now inserting χ, This appears as Eq. 6 of the letter.

WEIGHTED GEOMETRIC STATISTICS
Following Refs. [63,64]. For the ith observation x i , the geometric mean is, where n is the number of observations. It is noted that, where the operator, is the arithmetic expectation value, so that the natural logarithm of the geometric mean is the arithmetic expectation of the natural logarithm of the observations. If the observations have unormalized weights w i , the weighted geometric mean, which reduces to the geometric mean if w i = 1 ∀ i. The same applies for the expectation values, where, is the weighted arithmetic expectation operator. The arithmetic covariance matrix is, It follows then that the weighted geometric covariance matrix, ln σ x,y w.g.
The confidence interval for the arithmetic statistics, where CI x a is the arithmetic confidence interval. For the two sigma geometric confidence, where CI x w.g. is the weighted geometric confidence interval. The statistics detailed here are used to calculate the main results of the Letter, which are reported in Table 1. In the Letter, the subscripts w.g. has been dropped.
In the Letter it is stated that the methods used to measure the three unobserved parameters α, k iso , λ eff mfp has been repeated for the measurables C ⊥ , C , A ⊥ , A . The left column of Fig S1 displays the statistics. The means and confidence intervals are consistent among all the measurements, and so the combined statistics are reported in the Letter.
The right column of Fig. S1 shows the normalized weighted geometric covariance between the model parameters, They are consistent with each other, so that only the combined statistics are reported in the Letter.

COLLISION LENGTH AND TIME-SCALES
Following Refs. [10,11] the Spitzer-Härm proton-proton collision frequency for a proton-electron plasma with T p ≤ T e , where T p (T e ) is the proton (electron) temperature, where the inequality sign is the typical case for the solar wind plasma, is written, FIG. S1. The weighted geometric mean (Eq. S18) are plotted on the left column as circles. The confidence interval (Eq. S24) is plotted as a vertical line. The right column are the non-diagonal terms of the normalised weighted covariance matrix (Eq. S25).
where n p (cm −3 ) is the proton number density, k B T p is in eV and the Coulomb logarithm is ln|Λ|. The Coulomb logarithm for proton-proton collisions, The dataset described in section Measurements of the Letter provides the following averages, where the proton thermal speed is v p th . With these measurements we calculate, ν SH p,p = 4.23 × 10 −7 (s −1 ), (S31) where λ SH mfp is the Spitzer-Härm proton-proton mean-free-path.

LINEAR COLLISIONAL-KINETIC MAGNETOHYDRODYNAMIC
In this section the eigenvalue problem that is solved to report all numerical quantities is detailed. The kinetic magnetohydrodynamic equations are found in Ref. [52]. The collision operator is the Bhatnagar-Gross-Krook [BGK] introduced in the research article Ref. [49]. This form of equations, now called KMHD-BGK, which appears in Ref. [50], has been studied previously because they offer a linear closure scheme that incorporates Landau/Barnes damping and collisions or a relaxation processes. In Ref. [51] they show the dispersion relations produced from the KMHD-BGK equations transition from collisional (fluid) to collisionless wave mode solutions, which is the primary purpose of our study.
The equations are introduced in Section , linearized and Fourier analyzed in Section , and written as a linear homogenous system of equations in Section . This can be solved with a numerical treatment of the plasma dispersion function and a numerical algorithm for eigenvalue problems. Solutions to these equations form the numerical treatment in the Letter.

Overview of equations
Beginning with the Vlasov equations and transforming the coordinates into a new velocity frame v i → w i = v i − u s i (t, r) shifted by the species "s" bulk velocity u s i (t, r) it can be shown that a gyrotropic distribution function f s = f s (x, w, t) will evolve according to, p s /n s k B . The proton plasma beta β p = v 2 p /v 2 A = p p /p B = 8πnk B T p /B 2 where, p s = (2p s ⊥ + p s )/3 is the background total pressure. Rewriting the equations we obtain, ωñ − k ũ + k ⊥ũ⊥ = 0 (S74) and the complex poles, This system of equations can be solved with a numerical recipe for the plasma dispersion function and a numerical root finder. This system of equations provide the numerical solutions used in the analysis that provide the main results of the Letter. The electron response is modelled with the kinetic equation. The ratio of the species mean-free-paths arises when parametrising this model. In the Letter it is quoted to be λ eff mfp /λ eff mfp,electrons = 5. This reflects the fact the electron gyroscale is much smaller that the protons. Regarding the numerical code, when increasing the ratio λ eff mfp /λ eff mfp,electrons from a value of 1, the dispersion relations do not change drastically, but by about 7, the root finding algorithm begins to fail or the null space of the matrix is not small. Therefore, we kept the parameter to be 5.