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Probability density functions for the variable solar wind near the solar cycle minimum

Unconditional and conditional statistics is used for studying the histograms of magnetic field multi-scale fluctuations in the solar wind near the solar cycle minimum in 2008. The unconditional statistics involves the magnetic data during the whole year 2008. The conditional statistics involves the magnetic field time series splitted into concatenated subsets of data according to a threshold in dynamic pressure. The threshold separates fast stream leading edge compressional and trailing edge uncompressional fluctuations. The histograms obtained from these data sets are associated with both large-scale (B) and small-scale ({\delta}B) magnetic fluctuations, the latter corresponding to time-delayed differences. It is shown here that, by keeping flexibility but avoiding the unnecessary redundancy in modeling, the histograms can be effectively described by a limited set of theoretical probability distribution functions (PDFs), such as the normal, log-normal, kappa and logkappa functions. In a statistical sense the model PDFs correspond to additive and multiplicative processes exhibiting correlations. It is demonstrated here that the skewed small-scale histograms inherent in turbulent cascades are better described by the skewed log-kappa than by the symmetric kappa model. Nevertheless, the observed skewness is rather small, resulting in potential difficulties of estimation of the third-order moments. This paper also investigates the dependence of the statistical convergence of PDF model parameters, goodness of fit and skewness on the data sample size. It is shown that the minimum lengths of data intervals required for the robust estimation of parameters is scale, process and model dependent.

here that the skewed small-scale histograms inherent in turbulent cascades are better described by the skewed log-kappa than by the symmetric kappa model. Nevertheless, the observed skewness is rather small, resulting in potential difficulties of estimation of the third-order moments. This paper also investigates the dependence of the statistical convergence of PDF model parameters, goodness of fit and skewness on the data sample size. It is shown that the minimum lengths of data intervals required for the robust estimation of parameters is scale, process and model dependent.

Introduction
The solar wind is a complex plasma system in which structures, wavy fluctuations and turbulence co-exist over multiple scales. Since the Sun variably emits a mixture of fast and slow streams and transient ejecta, a single mechanism cannot fully reproduce the observed complexity in the solar wind. Some aspects of this complexity can be described statistically only. Despite the structural and dynamical complexity, field and plasma fluctuations exhibit well defined scalings over specific frequency ranges in power spectral density (PSD) log-log plots. The low-frequency part of the spectrum is believed to reflect the statistics of large-scale structures and their interactions [Burlaga and Goldstein, 1984]. Below the timescale of a day the self-similar inertial range of magnetohydrodynamic (MHD) turbulence is set up, where dissipation is supposed to be negligible and energy cascades towards the small scales (large wave numbers or high frequencies) [Bruno and Carbone, 2013]. At characteristic proton and electron scales energy dissipation in collisionless plasmas takes place leading to non-adiabatic heating of the solar wind [Alexandrova et al., 2013].
Turbulence represents a candidate process which explains the inertial range energy transfer supplying free energy for the kinetic scale dissipation processes. The energy transfer rate ǫ in an MHD turbulent cascade can be calculated from the PSDs, however, these estimations are based on certain theoretical assumptions [Smith et al., 2006;Stawarz et al., 2009]. On the other hand, in hydrodynamics the rigorous Kolmogorovs 4/5 law (Kolmogorov, 1941) represents a way to estimate ǫ from third-order structure functions.
Similarly to the 4/5 law in hydrodynamics, a rigorous third-order moment relation to X -4 VÖRÖS ET AL.: PROBABILITY DENSITY FUNCTIONS calculate ǫ, the "MHD Yaglom's law", was obtained for incompressible MHD case by Politano and Pouquet [1998].
In terms of probability distribution functions (PDFs) non-zero third order moments correspond to skewed PDFs. In hydrodynamic turbulence zero skewness or a non-skewed PDF indicates that the turbulent cascade is absent [Davidson, 2004]. Therefore, the determination of ǫ should preferably be based on the estimation of third-order moments associated with skewed PDFs. However, direct evidence about the (non)stationarity of third-order moments, or equivalently, about the (non)stationarity of the asymmetric shape of PDFs is still missing. In the solar wind the PDFs associated with turbulence are rather symmetric and it is difficult to estimate the skewness [Podesta et al., 2009].
We will study the statistical properties of the large-scale fluctuations on the basis of total magnetic field measurements near the solar cycle minimum in 2008, when alternating fast and slow streams were present. The small-scale statistical properties of two-point magnetic fluctuations will be evaluated on the basis of time-delayed data at the time scale of τ =10 minutes. This is a time scale within the inertial range of turbulence in the solar wind. Parametric model PDFs will be used to fit the histograms obtained from the magnetic field data. The skewness (S) and the estimated parameters of model PDFs will be estimated separately for the large-scale and the small-scale magnetic fluctuations.
We note that the skewness S estimated from B is not straightforwardly associated with the turbulent energy transfer rate ǫ. The third-order moment relation for calculating ǫ in the solar wind contains the components of the Elsssier field (combined magnetic and velocity fluctuations [Politano and Pouquet, 1998] represents a rough approximation only, which is expected to describe certain aspects of MHD turbulence in the highly variable solar wind [Bruno and Carbone, 2013]. Therefore, we believe, it makes sense to further investigate the statistical features of magnetic fluctuations alone, studying also the nonstationarity aspects of their dynamics.
According to [Burlaga and Viñas, 2004] [Vörös et al., 2006]. We argue that conditional statistics applied for both unfiltered and filtered time series can reveal important aspects of multi-scale interactions.
We are going to show that the large-scale and the small-scale PDFs corresponding to the magnetic fluctuations during the whole year in 2008 are different from the conditional PDFs associated with magnetic fluctuations in the leading (compressional) and in the trailing edges of high speed streams. Nevertheless, a few basic types of model PDFs can explain the magnetic fluctuation statistics in the solar wind. These models correspond to additive and multiplicative statistical processes involving also nonzero correlations. We use four types of parametric PDFs, two symmetric distributions (normal and kappa) and two skewed distributions, log-normal and log-kappa. The basic mathematical features of normal and log-normal distributions were described long ago and their mathematical treatment can be find in textbooks. The kappa distribution arises from non-extensive X -6 VÖRÖS ET AL.: PROBABILITY DENSITY FUNCTIONS statistical mechanics describing out of equilibrium systems with long-ranged correlations [Tsallis, 1988;Leubner , 2002]. The log-kappa distribution was introduced recently by Leitner et al. [2009]. As it is shown here, the meaning of the model PDF parameters and of the nonzero skewness depends on the time scale and on the nature of physical processes under consideration. As a consequence, the required sample size for the robust estimation of statistical parameters also varies according to the statistical conditioning.
The organization of the paper is as follows. Section 2 introduces the model PDFs and the methodology of the estimation of statistical parameters. In Section 3 unconditional and conditional statistics is introduced for both large-scale and small-scale magnetic fluctuations. The model PDF parameters and the skewness are estimated for progressively larger data sample sizes. Section 4 summarizes our results.

Probability distribution functions (PDFs) for space plasmas
We study magnetic field fluctuations over the time scales associated with large-scale structures and over the inertial range of scales associated with turbulent interactions.
Our goal is to introduce a minimum number of model PDFs which can explain substantial parts of magnetic field variability in the solar wind. To this end we will use the normal and kappa PDFs and their logarithmic counterparts. The kappa PDFs which arise from non-extensive physics allow us to model magnetic field data exhibiting large-deviations from the normal distribution. Although kappa functions are commonly used to model non-Maxwellian particle velocity distribution functions (VDFs), the underlying physics associated with non-extensive statistics over the MHD and kinetic scales is rather different.
To avoid confusion, we will shortly describe the differences between particle VDFs and magnetic field PDFs.

Particle VDFs
Particle VDFs in the solar wind exhibit low-energy near-Maxwellian core and highenergy suprathermal tails described by kappa functions [Vasyliūnas, 1968;Scudder , 1992;Leubner , 2004]. Collisionless space plasmas are often out of thermal equilibrium and suprathermal particles can be accelerated by a number of mechanisms [Marsch, 2006;Pierrard and Lazar , 2010;Livadiotis, 2015]. There exists a strong connection between the long-tail kappa distributions or κ indices (see below) and the non-extensive statistical mechanics representing a generalization of Boltzmann-Gibbs statistics for out of equilibrium particle systems exhibiting long-range correlations [Tsallis, 1988;Leubner , 2002].
It has been shown the thermodynamic temperature T can be uniquely defined for the non-extensive out-of-equilibrium systems as well, through the variance of VDFs (mean kinetic energy), or equivalently, through the connection of entropy with internal energy.
[ Livadiotis and McComas, 2010]. The κ index appears as an independent thermodynamic variable with a physical meaning of enhanced correlations (long-range interactions) or as a parameter corresponding to different stationary states in plasma systems with the same T and n (density) residing in a thermal non-equilibrium state [Livadiotis and McComas, 2013]. As a consequence, for the characterization of non-extensive plasmas described by kappa distributions three independent parameters are needed, (n, T, κ). This can be regarded as a generalization of the polytropic law for out of equilibrium plasmas [Livadiotis, 2015].

PDFs for field and plasma statistics
Examination of the field or particle statistics over the scales of MHD turbulence or even over larger scales implies that the real in-situ data contain contributions from different X -8 VÖRÖS ET AL.: PROBABILITY DENSITY FUNCTIONS physical processes, structures or (non-)mixing plasmas with non-unique combinations of parameters (n, T, κ). While the large-scale statistics can reflect important aspects of solar variability or locally generated processes in the solar wind, the variance of PDFs might not correspond to any well defined plasma temperature. For example, turbulence with the same types of plasmas might be confined to flux tubes having characteristic sizes within the inertial range of turbulence [Borovsky, 2008]. Analyzing longer data intervals involving multiple flux tubes can change the turbulence statistics significantly [Borovsky, 2008]. The frequently observed twisted flux tubes can be more unstable against Kelvin-Helmholtz instability or reconnection than unstable flux tubes [Zaqarashvili et al., 2014a, b]. This can lead to abundant local generation of turbulence containing mixing plasmas and boundaries .
The length of the data sets is critical for reaching statistical convergence in parameter estimations. In fact, careful examination of the radial velocity and radial magnetic field statistics in the solar wind has shown that 10 6 -10 7 data points are required for robust estimation of the third-order moments associated with skewed PDFs [Podesta et al., 2009].
In other studies of third-order moment estimations, based on the Elsässier variables, the number of data points was a few thousands only  turbulence. Nevertheless, the authors found that the multi-scale fluctuations can be well fitted by the non-extensive model PDFs [Burlaga and Viñas, 2004].
As one can see from the above studies, long data intervals are needed for the robust estimation of statistical parameters associated with multi-scale structures and turbulence.
On the other hand, the inclusion of structures, boundaries, solar ejecta, compressional regions and turbulence, occurring during long time intervals, would most certainly results in non-unique combinations of (n, T, κ). This would preclude the connection of model kappa PDFs with non-extensive statistics or particle VDFs. There is a clear need for a better understanding of the meaning of kappa PDFs over the larger scales which contain mixtures of different plasmas.

Parametric PDFs for the magnetic field
We consider four types of parametric distribution functions which correspond to additive, multiplicative, non-extensive statistical processes and their combinations.
The symmetric Gaussian (normal) distribution P n appears as a result of the additive central limit theorem: the sum of identically distributed independent random variables (each having an expected value µ and variance σ 2 ) is normally distributed: It has already been found that PDFs corresponding to the time delayed differences δ(X(t, τ )) = X(t+τ )−X(t), calculated from solar wind time series X(t), do not follow the normal distribution P n over the small time scales τ [Marsch and Tu, 1997;Sorriso-Valvo et al., 1999;Burlaga and Viñas, 2004]. This is the time scale of turbulence, coherent structures, interaction regions, spanning from hours down to seconds. Due to correlations X -10 VÖRÖS ET AL.: PROBABILITY DENSITY FUNCTIONS and nonlinear interactions the small-scale PDFs are peaked, skewed and exhibit fat tails (the largest fluctuations occur more frequently than for normally distributed random fluctuations). However, random fluctuations over longer time scales τ (from hours to days) exhibit Gaussian distribution due to the loss of correlations.
The skewed log-normal distribution P Ln appears as a result of multiplicative central limit theorem [Crow and Shimizu (eds.), 1988] and reads as: Here µ is a scale parameter while σ represents a shape parameter.
When no time delayed differentiation is applied, the magnetic field, density, proton temperature statistics exhibit skewed PDFs which can be approximately modeled through the log-normal distribution [Burlaga and Lazarus, 2000;Bruno et al., 2004;Veselovsky et al., 2010;Dmitriev et al., 2013]. Since log-normal distributions are generated through multiplications of random variables, logarithmic scale invariance in turbulence is often understood in terms of multiplicative cascades [Castaign et al., 1990]. In the solar wind random multiplications beyond turbulence are also foreseen for random amplification/weakening of waves, or for compressional effects increasing the values of proton density or magnetic field [Dmitriev et al., 2013]. In fact, the log-normal model describes rather well the density and temperature fluctuations with the highest probabilities (within ±2σ) or the magnetic field statistics (roughly within ±1.5σ). However, the fat tails present in data distributions show again probabilities higher than the P Ln model [Dmitriev et al., 2013]. This is also valid for the histograms of derived quantities, e.g. for the histogram of the solar wind quasi-invariant (the inverse square of the Alfvén-Mach number), which correlates well with sunspot cycle [Leitner et al., 2011a]. We also mention that instead of a single log- normal PDF two or more different log-normal models are added to form a single skewed one to fit the fluctuation statistics associated with the jumps in magnetic field vector orientation and magnetic field intensity [Bruno et al., 2004]. It is supposed that different P Ln models might be associated with different components of MHD turbulence, such as uncompressive fluctuations of Alfvénic origin or compressive structures. Three component log-normal models can also be associated with the anisotropic magnetic field-aligned (slab) and field-perpendicular (quasi-2D) populations of fluctuations complemented by convected compressive structures [Bruno et al., 2004]. It is not known how many different log-normal PDFs were needed to model the solar wind data.
A superposition of random uncorrelated, normally or lognormally distributed processes can provide a composed skewed PDF with fat tails which can fit the observations reasonably well. However, turbulent space plasmas contain structures, long-range interactions and correlations. Normal (or for particles the Maxwell-Boltzmann) distributions are associated with Boltzmann-Gibbs statistics describing systems with non-interacting or weakly interacting subsystems in thermal equilibrium. For stationary non-equilibrium systems the Boltzmann-Gibbs entropy was generalized and the so-called nonextensive entropy was introduced by Tsallis et al. [1998]. The nonextensive entropy introduces interactions between subsystems (e.g. particles) [Leubner , 2002] and its extremization under certain physically relevant constraints leads to kappa distribution [Tsallis et al., 1998;Livadiotis and McComas, 2013]. A simple physical model based on the generalized Galton board clearly demonstrates the formation of fat tailed kappa-like distributions from Gaussian distributions, when interaction terms and memory effects are switched on [Leitner et al., 2011b]. The peaked, fat tailed data distributions corresponding to turbulence in the solar wind are well modeled by the kappa distribution [Leubner and Vörös, 2005a, b]: The nonextensive κ index represents a measure of correlations or memory in systems out of equilibrium, when κ ≪ ∞.
Since the kappa distribution represents a symmetric distribution, the observed skewed distributions [Burlaga and Lazarus, 2000]  artificial cubic term to their version of P κ which could be used then to fit the observed skewed PDFs. An alternative approach to deal with the skewed PDFs was formally based on the assumption that the κ-distributed correlated fluctuations can also be combined in a multiplicative manner leading to the log-kappa distributions [Leitner et al., 2009]: The normal, log-normal, kappa and log-kappa distributions are compared to each other in Figure 1. The nice feature of these model PDFs is that for κ → ∞, P κ ⇒ P n ( Figure   1a) and P Lκ ⇒ P Ln (Figure 1b). In other words, depending on the values of κ (we show P κ and P Lκ for κ =3, 10, 20), the kappa and log-kappa distributions are capable to reproduce the PDF tails which are fatter than the normal or log-normal PDFs. Although other types of parametric PDFs (e.g. [Consolini et al., 2009]) or their combinations could also work well, we provide evidence that the considered PDFs are sufficient to model substantial part of the observed magnetic fluctuations in the variable solar wind in 2008.
D R A F T October 1, 2015, 1:26am D R A F T

Fitting and error statistics
The Levenberg-Marquardt nonlinear least-square method [Press et al., 1992] will be used to fit the parametric model PDFs to the histograms of the large-scale and time-delayed magnetic field fluctuations. The algorithm initially uses the gradient descent method to iteratively update the fitting parameters in large steps, then it adaptively switches to Gauss-Newton method minimizing the sum of squares in smaller steps. Together with the fitted curves the 95-percent confidence intervals of the fits will be plotted. The error measure for the parameters is estimated as the squared-root of the diagonal of the parameter covariance matrix. The goodness of fit Q is calculated from the incomplete gamma function gammq as Q = gammq(0.5ν, 0.5χ 2 ), where ν is the number of degrees of freedom and χ 2 is the "chi-square" parameter calculated from minimizing the sum of squares divided by the standard deviation [Press et al., 1992]. The dependence of the model PDF parameters on the length of the time series will be calculated as well. Additionally, the skewness S (describing the asymmetry of PDFs) and its standard error will be estimated using a bootstrap method. The bootstrap is a Monte-Carlo simulation method treating the original time series as a pseudo-population from which new populations are obtained via random resampling [Martinez and Martinez , 2007].

Data analysis
In this paper we use OMNI 1 minute high resolution plasma and magnetic data ([King P n statistics indicate that the underlying magnetic fluctuations might be associated with successive 27-day quasi-periodic recurrent streams. This is also valid for the large-scale compressional statistics (black curves in Figure 5). After more than 10 days of data Q becomes larger than 0.05 (Figure 5d). However, due to the B(P dyn > 3) [nPa] condition, the compressional data are collected from multiple streams during the whole year. Since the log-kappa and log-normal PDFs are rather similar (Figure 4 b1) the κ parameter is large. At Q > 0.05 (Figure 5d) κ = 8 ± 4 (Figure 5a). The relatively large errors (±4) are caused by the small amount of compressional data (∼ 24000 data points).
The skewness (S) associated with the conditional and unconditional statistics is depicted in Figure 5d. The same color code is used as above. The error bars calculated from bootstrap are negligible. In accordance with the skewed PDFs (Figures 4 a1 and 4 b1), both the yearly and compressional data show positive skewness S ∼ 2, while the skewness associated with normal PDF (Figure 4 c1) is zero (Figure 5d).
The results of the model PDF fits together with S are in Table 1. The values of parameters correspond to the full lengths of data sets.  Table 2. The values of parameters correspond to the full lengths of data sets.

Small-scale PDFs
A nonzero ǫ should be associated with skewed distributions, however, our PDFs are rather symmetric (Figure 6). Podesta et al. [2009] have pointed out that the third order moments or the skewness represent signed moments subject to cancelation effects. Indeed, the skewness S calculated for the different δB data change sign in Figure 7e. Also, the error bars are of the same order as the mean values indicating the occurrence of strong non-D R A F T October 1, 2015, 1:26am D R A F T stationarity effects. By increasing the sample size the cancelation of successive positive and negative skewnesses can effectively lead to S ∼ 0 hamstringing the calculation of the actual value of ǫ. In fact, S approaches zero for the largest sample sizes in Figure   7e. In hydrodynamics a skewness of zero would indicate no vortex stretching and no energy cascade in a turbulent flow [Davidson, 2004]. Positive or negative S could be associated with typical coherent structures such as sheets or tubes generated by vortex stretching or compression [Davidson, 2004]. In our case S simply changes sign when A larger S can be obtained by considering subsets of the data. As an example, we selected three subsets of trailing edge data in 2008, each with S ∼ −0.38 at τ = 10 minutes.
Bootstrap calculations for the concatenated data, consisting of 25000 data points, give S = −0.38 ± 0.08. Figure 8 shows the histogram of time delayed magnetic fluctuations shifted by 4 [nT] to positive values. The shift does not affect the skewness, however, it allows a comparison of symmetric and skewed PDF models. Figure 8a shows the P κ (symmetric, black line) and P Lκ (asymmetric, red line) model fits. The absolute residuals in Figure 8b show that the central part of the histogram is fitted by the P κ and P Lκ models equally well. The tails of the histogram are described slightly better by the skewed

Conclusions
The main goal of this paper was to show that a limited number of parametric model PDFs can describe major parts of the magnetic fluctuations over multiple scales in the solar wind. The range of scales from minutes to several months predetermines the nature of physical processes occurring near a solar cycle minimum in the solar wind.
The large scales are dominated by a mixture of corotating streams, corotating interaction regions, solar ejecta, discontinuities and turbulence [Burlaga and Viñas, 2004]. The large-scale statistics is based on the histograms of the total magnetic field fluctuations B obtained in 2008. This paper shows that the histogram associated with the large-scale mixture of physical processes is best described by the log-kappa P Lκ model ( Figure 4).
Naturally, the κ parameter has a completely different meaning for the large-scale PDFs than for the almost instantaneous particle VDFs. Although the non-extensive interpretation of the κ index for the large-scale magnetic histograms is not straightforward, the log-kappa PDF model seems to be superior to the log-normal model even for longer time intervals encompassing a solar cycle [Leitner et al., 2011a]. This indicates that the κ index could have a reasonable interpretation over the large-scales as well. In order to further decipher the physics behind the large-scale log-kappa model we used conditional PDFs,  Table 1) and the log-kappa and log-normal PDF models are very close to each other. The "logarithmic" models associated with compressive fluctuations indicate the occurrence of multiplicative processes which has already been found by Veselovsky et al. [2010] and Dmitriev et al. [2013]. However, when the yearly PDF is considered, the mixture of normally distributed trailing edge and the lognormally distributed leading edge processes clearly forms a logkappa distribution with κ = 3.5 ± 0.5 (Figure 4), providing a better fit than the P Ln model. The other two parameters associated with P Ln and P Lκ models (µ and σ in Table   1) are similar. At this stage, however, it is impossible to interpret the κ index in terms of the non-extensive physics. The mixture of large-scale processes cannot be associated with a combination of (n,T,κ) parameters. Also, it is neither possible nor relevant to interpret the skewness associated with yearly or compressional large scale fluctuations (S = 2.4 ± 0.1 and S = 2 ± 0.1, Table 1) (Table 2). According to the non-extensive approach the kappa values are restricted to the interval κ ∈ (1.5, ∞], simply because for κ < 1.5 the temperature is not determined for kappa VDFs [Pierrard and Lazar , 2010;Livadiotis and McComas, 2013]. The limiting value κ = 1.5 corresponds to the furthest state from equilibrium (or "anti-equilibrium") while κ → ∞ corresponds to the equilibrium state associated with a Maxwellian VDF [Livadiotis and McComas, 2013]. It would be tempting to interpret the estimated values scale τ are associated with the structures repeatedly generated by turbulence introducing correlations to observed fluctuations. We mention that correlations in a generalized Galton board lead to peaked and fat tailed kappa-like distributions, while in the absence of correlations Gaussian distributions are obtained [Leitner et al., 2011b]. Figure 6 and the fitting results in Table 2 show that the κ and σ parameters are significantly different for the yearly and for each conditional PDFs. Although each histogram is peaked and fat tailed, the leading edge compressional data seem to be associated with different type of turbulence than the trailing edge data. This might indicate that the large-scale structures which are selected by the conditional statistics are affecting the small-scale turbulence.
The goodness of fit parameter Q depends on the data sample size. It is not surprising that for the robust estimation of large-scale model PDF parameters longer data sets are needed than for the small-scale model PDF parameters (Figures 5 and 7). The proper sample size ensuring a good fit, however, does not guarantee that the skewness S can be reliably calculated. S is a signed moment therefore subject to cancelation effects. The consideration of longer data sets of δB containing alternating positive and negative S leads to S → 0 which can be seen in Figure 7e. When only those subsets of data are considered for which the skewness is, for example, equal and negative, the skewed P Lκ model describes the tails of the histogram slightly better than the symmetric P κ model.
Finally, we mention that for the description of normal or peaked, fat tailed or skewed histograms the log-kappa and kappa PDF models might suffice. From these models the normal and log-normal distributions can be obtained by taking the limit κ → ∞.      d. e. [nT] [nT]  σ, µ are in [nT] for the P n and the P κ models.