Multiscale structure of the magnetic field and speed at 1 AU during the declining phase of solar cycle 23 described by a generalized Tsallis probability distribution function

Authors


Abstract

[1] This paper describes the multiscale structure of fluctuations in the solar wind speed V and magnetic field strength B at 1 AU in 2003, during the declining phase of the solar cycle 23. As expected, there was a corotating stream and a corotating interaction region (CIR) that recurred every 27 days, but the stream and CIR changed considerably from one solar rotation to the next. There were also other types of physical structures (intermittent turbulence at the smallest scales, compound streams and ejecta at intermediate scales, and the collection of streams and ejecta of different sizes and shapes at the largest scales). A multiscale description of these fluctuations is more appropriate than a model of a recurrent stream with small-scale fluctuations superimposed. The probability distribution functions (PDFs) of changes of V and B on scales from 1 hour to ≈171 days can be described by a generalization of the PDF derived by Tsallis from a nonadditive entropy function in the context of nonextensive statistical mechanics.

1. Introduction

[2] Probability distribution functions (PDFs) were used by Burlaga and Forman [2002] (hereinafter referred to as BF) to describe the multiscale structure of the fluctuations of solar wind speed at 1 AU on scales from 1 hour to 1 year ([Burlaga, 1995, p. 169] “large-scale fluctuations”). They considered the speed fluctuations both in 1999 (near a maximum of solar activity) and in 1985 (near the declining phase of the previous solar cycle). The data analyzed by BF extended from a scale of an hour (where turbulence and discontinuities are important), through scales of days (where streams and interaction regions are dominant), to scales from 27 days (the solar rotation period) to 1 year (where the PDFs are Gaussian).

[3] Burlaga and F.-Viñas [2004] (hereinafter referred to as BV) found that all of the PDFs of the speed fluctuations observed by ACE during 1999, near solar maximum, on scales of 48 s [Forman and Burlaga, 2003] and from 1 hour to 171 days (BF) could be described by a single distribution function. The function is a simple generalization of the Tsallis q distribution that was derived from nonextensive (nonadditive) statistical mechanics [Tsallis, 1988, 2004; Tsallis and Brigatti, 2004]. The fluctuations in speed are related to (1) intermittent turbulence, waves, shocks, and discontinuities on the smallest scales; (2) ejecta, corotating streams, slow flows, interaction regions, etc. on intermediate scales; and (3) systems of interacting flows involving all of these features, on scales greater than the solar rotation period.

[4] This paper extends the work of BV by considering (1) the PDFs of the large-scale fluctuations in the magnetic field strength as well as speed and (2) a part of the declining phase of the solar cycle, the year 2003. Recurrent corotating streams (characterized by high speed, high temperature, and low density) are more prominent during the declining phase of the solar cycle than around solar maximum [Hundhausen, 1977; Burlaga, 1995]. We show that the PDFs of the fluctuations of both V and B observed by ACE (Advanced Composition Explorer) at 1 AU during 2003, from scales of an hour to nearly a year, are given by a generalized form of the Tsallis distribution.

2. Observations

[5] Figures 1a and 1b show hourly observations of the speed and magnetic field strength, respectively, from the ACE spacecraft at 1 AU for days of year (DOY (January 1 = 1)) 1–275, 2003 during the declining phase of solar cycle 23. The hour averages of the speed V used in this study are from the Solar Wind Electron Proton and α Monitor (SWEPAM) instrument [McComas et al., 1998]. These data were obtained from the level 2 ACE SWEPAM data set at http://www.srl.caltech.edu/ACE/ASC/level2/lvl2DATA_SWEPAM.html. A description of the data can be found at the Web site http://www.srl.caltech.edu/ACE/ASC/level2/swepam_l2desc.html. The hour averages of the magnetic field strength B used in this study are from the magnetometer (MAG) instrument on ACE [Smith et al., 1998]. These data were obtained from the level 2 ACE MAG data set at http://www.srl.caltech.edu/ACE/ASC/level2/lvl2DATA_MAG.html. A description of MAG data is at http://www.srl.caltech.edu/ACE/ASC/level2/mag_l2desc.html.

Figure 1.

(a) Hour averages of the solar wind speed fluctuations measured at 1 AU by ACE from days 1 to 275, 2003. (b) Hour averages of the magnetic field strength fluctuations measured at 1 AU by ACE from days 1 to 275, 2003.

[6] Figure 1a shows considerable variability of the speed over the entire range of scales, which we quantify in sections 3 and 5 below. The vertical dashed lines in Figure 1a, spaced at approximately the solar rotation period (27 days), call attention to a “recurrent stream” [Neugebauer and Snyder, 1966] in the time series. One expects to see recurrent streams during the declining phase of the solar cycle, and recurrent streams are often modeled as periodic structures [e.g., Hundhausen, 1972, 1977; Burlaga, 1995]. However, the amplitude, width, and structure of the recurrent stream varied considerably from one solar rotation to the next, so that even this quasiperiodic component has a nonstationary multiscale structure. We shall show that the multiscale structure of the speed profile at 1 AU (Figure 1a) is considerably more complex and dynamic than the model of a recurrent quasi-sinusoidal stream with superimposed small-scale turbulence and waves. Furthermore, we show that in spite of the inherent complexity, some ordered statistical characteristics of the multiscale speed profile are identifiable and are ubiquitous properties of the solar wind structure.

[7] Figure 1b shows the hour averages of the magnetic field strength profile, B(t), measured by ACE from DOY 1 to 275, 2003. As expected, there are large peaks in B in front of the primary recurrent stream. These are related to interaction regions [Burlaga and Ogilvie, 1970a] caused by compression associated with the steepening of streams [Sarabhai, 1963; Hundhausen, 1972]. However, the fluctuations in B are considerably more complex than a periodic series of corotating interaction regions. There is structure on every scale, which we shall quantify in sections 3 and 6 below.

3. Spectra of the Fluctuations in V and B

[8] A simple approximate way to describe the multiscale structure of the solar wind is by means of the Fourier power spectrum. The basic idea is to quantitatively describe the structure on different scales by means of a set of similar parts (sine waves) with different sizes (wavelengths) and amplitudes.

[9] Figures 2a and 2b show the Fourier power spectrum of speed fluctuations and magnetic field strength fluctuations, respectively, for the ACE observations from DOY 1 to 275, 2003. There is a peak in the Fourier power spectrum of the speed fluctuations at the solar rotation period (27 days) indicated by the arrow in the top left corner of Figure 2a, which corresponds to the recurrent stream identified in Figure 1a. Two smaller amplitude harmonics of this peak are also seen in Figure 2a. At the highest frequencies considered here, corresponding to periods from 1 to 16 hours, the spectrum of the fluctuations is a power law. The exponent of the spectrum is −1.7 ± 0.1, corresponding to Kolmogorov turbulence [Kolmogorov, 1941; Frisch, 1995]. This component of the speed fluctuations in the solar wind has been studied for decades [Coleman, 1968]. The primary peak at low frequencies and the turbulent spectrum at high frequencies correspond to the simple picture of recurrent streams with superimposed waves and fluctuations.

Figure 2.

(a) Power spectral density (psd) of the speed fluctuations observed by ACE from days 1 to 256, 2003. (b) Power spectral density of the magnetic field strength fluctuations observed by ACE from days 1 to 256. In this figure, f is frequency, t is the time corresponding to a frequency, and d refers to the unit of days.

[10] Note that the power in the turbulence is more than 3 orders of magnitude less than the power in the primary recurrent stream, so there is a clear separation between the turbulence and the recurrent streams. However, there is also power at intermediate scales, from periods of 16 hours to 5.3 days, with a power law spectrum having exponent −2.3 ± 0.1. This regime of the spectrum (where interactions among streams, components of streams, and turbulence occur together) is still largely unexplored and poorly understood. The exponent −2.3 is that which one expects for jump-ramp structures associated with the asymmetrical steepened streams with scales in the range from 1 to 10 days [Burlaga et al., 1989].

[11] Figure 2b shows the spectrum of the magnetic field strength fluctuations for the ACE data from day 1 to 256, 2003. There is no single prominent peak at low frequencies in the spectrum of the magnetic field strength fluctuations. Instead, there are peaks at the solar rotation period and its first three harmonics, all having the same order of magnitude, giving a plateau at low frequencies in the spectrum of B. At high frequencies, corresponding to periods shorter than 16 hours, the spectrum of B is a power law with the slope −2.1 ± 0.1. Such a slope is characteristic of the medium with many discontinuities, which tend to obscure the power associated with waves and turbulence. At intermediate frequencies, corresponding to periods between 16 hours and 0.3 days, the spectrum of B is a power law with slope −2.0 ± 0.1, again suggesting filamentary structures and many nearly discontinuous changes in B at these scales. Such structures can be seen in the time series for hour averages of B shown in Figure 1b.

4. Multiscale Probability Distributions of the Fluctuations

[12] The complexity of the solar wind cannot be fully described by power law Fourier spectra, which are based on the analysis of variance of a stationary time series. The solar wind speed exhibits nonlinear, multifractal behavior at small and intermediate scales [Burlaga, 1995] and Gaussian behavior at large scales (BF, BV). For example, (1) intermittency and a nonlinear cascade of energy is observed in the turbulence at small scales [Burlaga, 1991a, 1993; Marsch and Liu, 1993; Marsch and Tu, 1994, 1997; Carbone and Bruno, 1997; Bruno et al., 1999; Sorriso-Valvo et al., 1999]; (2) nonlinear stream steepening, shocks, and stream-stream interactions are observed at intermediate scales; and (3) Gaussian structure related to interactions of multiple streams of different amplitudes, sizes, and internal structures is observed at larger scales.

[13] There are two things to consider when analyzing a complex multiscale system such as the solar wind on scales from 1 hour to 1 year: (1) the physical (mechanical) properties, which are certainly different at small, intermediate, and large scales and (2) the probability (statistical) structure of the components of the fluctuations seen in time series at the various scales. Thus a natural way to analyze the structure of the large-scale fluctuations of the solar wind is to use the theory of statistical mechanics.

[14] The classical statistical mechanics of Boltzmann and Gibbs can describe an equilibrium system. However, an extension of Boltzmann-Gibbs statistical mechanics is needed to describe a nonlinear, nonequilibrium system such as the solar wind at small and intermediate scales. Tsallis [1988], motivated by the desire to extend Boltzmann-Gibbs statistical mechanics to include systems with scaling properties described by fractal and multifractal structure, introduced a generalization of statistical mechanics. We shall show that a generalization of the PDF of the statistical mechanics of Tsallis can describe the multiscale speed and magnetic field strength fluctuations in the solar wind at 1 AU during 2003.

[15] A key component of this “nonextensive statistical mechanics” is a nonextensive, nonadditive entropy Sq = ∑(piq − 1)/(1 − q) [Tsallis, 1988]. Here pi is the probability of the ith microstate, and q is a constant that measures the degree of nonextensivity. By extremizing this entropy subject to two constraints, Tsallis derived the “Tsallis q-distribution function”:

equation image

where κ ≡ 1/(q − 1) and x is some dimensional physical parameter. In the limit q → 1 (κ → ∞) the statistical mechanics of Tsallis reduce to that of Boltzmann and Gibbs. The function on the right of equation (1) is the traditional kappa function, used for many years in space physics, but without any foundation on first principles, to model speed distribution functions of plasma particles [Olbert, 1968; Vasyliunas, 1968; Maksimovic et al., 1997; Scudder and Olbert, 1979; Leubner, 2002, 2004].

[16] The Tsallis entropy generalization extends the traditional Boltzmann-Gibbs statistical mechanics to physical systems where nonlinearity, long-range forces, long-memory effects, and scaling (fractal and multifractal) are important. The Tsallis distribution is kurtotic for small κ and it tends to a Gaussian in the limit κ → ∞ (q → 1). A transition from kurtotic to Gaussian PDFs at relatively small and large scales, respectively, was observed by BF. The advantage of considering a Tsallis distribution, rather than other PDFs such as that of Castaing et al. [1990], is that the Tsallis distribution is based on an entropy principle and can be related to a generalization of statistical mechanics, whereas the latter does not have such a foundation. Since the entropy principle introduced by Tsallis [1988] was motivated by systems with fractal or multifractal structure, and since multifractal structure is found in the solar wind [Burlaga, 1995], it is reasonable to consider the application of the Tsallis distribution in the studies of the solar wind.

[17] The Tsallis q distribution is symmetric and has no skewness, so it cannot model the PDFs of the solar wind speed observed by BF, which were skewed, except in the Gaussian limit. Therefore, following BV, we consider a “generalized Tsallis distribution,” obtained by adding a cubic term to the Tsallis distribution in equation (1), namely,

equation image

Both Bq and Cq are dimensional quantities in this equation. It is useful to introduce the parameter 1/wqequation image that is related to the width of the PDF; wq has the dimensions of x. Cq determines the skewness of the PDF. The sign of Cq in equation (2) dictates the sense of skewness of the distribution. For large negative (positive) Cq values the skewness of the distribution is toward positive (negative) x. Adding the cubic term in equation (2) means that one considers a more complicated effective Hamiltonian in the formalism of nonextensive statistical mechanics. Considering only the quadratic term implies considering only the kinetic energy [Beck et al., 2001]. Our choice of the form for the skew distribution function is not unique. Beck [2001] used skew distribution similar to equation (2) to describe laboratory turbulence, but he included a linear term as well as a cubic term. The cubic term is meaningful only for x not too large. The coefficients Aq, Bq, Cq, and the nonextensivity parameter q itself are functions of scale τ, as is κ ≡ 1/(q − 1).

[18] When fitting equation (2) to data, it is convenient to consider the dimensionless quantity y ≡ (x − 〈x〉)/σq(x), where 〈x〉 is the mean of x and σq is the standard deviation of x having the dimensions of x. Finally, we introduce the dimensionless parameter σq2/wq2 = 1/wq2 = Bq. The PDF that we use to fit the PDFs of the large-scale fluctuations in the speed V and magnetic field strength B is then

equation image

In equation (3), all the quantities are dimensionless. We shall use the dimensionless parameter Cq = σq3Cq as a dimensionless measure of the skewness of Rq(y), but we shall use the dimensional parameter wq = σqw′q as a measure of the width of the physical distribution Rq(x).

5. Speed Fluctuations

5.1. Time Series of Speed Fluctuations for Various Scales

[19] This paper considers two signals, each consisting of a set of observations s(ti), where i = 1 hour to 6600 hours (275 days × 24 hours/day). A standard method for analyzing the multiscale fluctuations of a nonlinear signal s(ti) is to analyze the differences:

equation image

at different lags τn, where τn ≡ 2n (hours) and n = 0, 1, 2… and so on. The lag τn determines the scale of the fluctuations represented by dsn. The scales that we consider in this paper range from τ0 = 1 hour to τ12 = 4096 hours ≈ 171 days = 1.5 × 107 s ≈ 6.8 × 10−8 Hz. This section considers the speed observations, s = V. Section 6 considers the magnetic field strength observations s = B made by ACE during the same period.

[20] The variations dVn(ti) describe the structure of the speed fluctuations on various scales. Figures 3b–3e show four time series, dVn versus ti for n = 2, 4, 6, and 9, corresponding to lags of τ2 = 22 = 4 hours, τ4 = 24 = 16 hours, τ6 = 2.7 days, and τ9 = 21.3 days, respectively. The scales were chosen to illustrate the basic types of solar wind speed fluctuations. The hour averages of V measured by ACE during 2003 are plotted versus time in Figure 3a, so that one can see how the fluctuations in V are related to the speed profile.

Figure 3.

(a) Solar wind speed shown in Figure 1a repeated for reference. (b) Running differences in the speed, dVn, at a lag of 4 hours, showing the intermittent turbulence on this scale. (c) The dVn at a lag of 16 hours at the transition from turbulence to stream-related structures. (d) The dVn at a lag of 2.7 days, showing changes related to streams. (e) The dVn at a lag of 21.3 days, showing Gaussian fluctuations that are characteristic of scales greater than or equal to the solar rotation period.

[21] On a scale of 4 hours (Figure 3b) the fluctuations dV2 are irregular, bursty, and asymmetric (with larger spikes for positive dV1 than for negative dV1). Fluctuations of this form are typical of intermittent turbulence [Kolmogorov, 1962] on a small scale in the solar wind. In this case, discontinuities (such as stream interfaces and shocks) also contribute to the fluctuations dV2.

[22] On a scale of 16 hours (Figure 3c) the fluctuations dV4 have larger amplitudes than those on a scale of 4 hours. They too are irregular, bursty, and asymmetric with larger spikes for positive dV4 than for negative dV4. The largest spikes have dV4 > 0 and are generally associated with the leading edges of the streams. There are also spikes for dV4 < 0 associated with the trailing edges of the streams. The greater amplitude of the positive spikes than the negative spikes is a consequence of the “steepening” of the streams as the fast flow overtakes the slower flow ahead and moves away from the slower flow behind [Sarabhai, 1963]. This steepening contributes to the skewness of the fluctuations in dVn on a scale of 16 hours.

[23] On a scale of 2.7 days (Figure 3d) the fluctuations dV6 have large amplitudes and are more symmetric than the fluctuations on smaller scales. These fluctuations tend to resemble the streams themselves, because they have scales of the same order of magnitude as those of the streams. The “streams” include corotating streams, ejecta, and slow flows.

[24] On a scale of 21.3 days (Figure 3e) the fluctuations dV9 have large amplitudes, are symmetric, appear to be random, and have no strong relation to V(ti). The magnitudes of these fluctuations are related to the variations of the widths, amplitudes, and structures of the streams on a variety of intermediate scales.

5.2. Probability Distribution Functions of dVn: Observations and Fits

[25] Consider the PDFs of dVn(ti) for the scales n = 0, 2, 4, 6, 7, and 9. The PDFs for dVn(ti) are plotted as histograms of the fraction of counts in bins versus dV on a semilog scale as the points in Figure 4. The corresponding curves in Figure 4 are best fits of the data to the generalized Tsallis distribution function (2) using the Levenberg-Marquardt algorithm [Levenberg, 1944; Marquardt, 1963; Bard, 1974]. The basic qualitative results can be seen readily by inspection of Figure 4. Remarkably, a single function, the generalized Tsallis PDF (equation (2)), provides excellent fits to all of these PDFs.

Figure 4.

Observed PDFs of speed differences dVn at scales 2n hours, with n = 0, 2, 4, 6, 7, and 9 shown by the points. Each PDF is shifted by a factor of 100 from the one below it, for clarity. The lowest points in each PDF correspond to 1 count/bin. Curves are fits of the data to the generalized Tsallis distribution. PDFs are narrow, kurtotic, and skewed at small scales, and they approach a Gaussian distribution (a quadratic on a semilog scale) at the largest scales.

[26] The width of the distributions increases from the smallest scale (1 hour) to a scale of the order of the solar rotation period (29 = 21.3 days). At small scales (1 and 4 hours) the PDFs of dV0 and dV2 have two inflection points and are thus kurtotic; the PDFs are also skewed. At an intermediate scale (16 hours) the PDF dV4 is less kurtotic than dV0 and dV4, and again, one can see skewness. At scales ≥5.3 days the PDFs dV7 and dV9 have no inflection points, appear to be symmetric, and can be fitted with a quadratic polynomial, corresponding to a Gaussian function.

5.3. Fitting Parameters for the Probability Distribution Functions of dVn

[27] The generalized Tsallis PDF (equation (3)) contains the parameters q, wqequation image, and Cq, as well as a normalization constant Aq. These parameters are functions of the scale τ. The values of the q, wq, and Cq derived from the PDFs in Figure 4 are plotted versus scale in Figure 5. These parameters are related to the structure and dynamical processes of the solar wind at different scales.

Figure 5.

Parameters of the generalized Tsallis distribution derived from fits to the observations. (a) Nonextensivity parameter q decreasing from 1.44 at a scale of 1 hour to 1 at scales greater than ≈5 days. A transition at scales of the order of 16 hours occurs where interaction regions are dominant. (b) Width parameter, wq, as a function of lag τ. This parameter tends to be anticorrelated with q, approaching asymptotic values at large and small scales, and with a transition at scales of the order of 1 day. (c) Skewness parameter Cq as a function scale. Skewness parameter is zero at scales ≥2.3 days, and it is larger at small scales.

[28] The value of q decreases from q = 1.44 ± 0.04 at a scale of 1 hour to q = 1.01 ± 0.02 at a scale of 171 days, as shown in Figure 5a. The parameter q is the fundamental “nonextensivity” parameter in the theory of Tsallis. This is related to kurtotic tails seen in the PDFs of dVn in Figure 4, and it describes the exponent of a power law in the limit that the quadratic term in equation (2) is much larger than 1. Turbulence dominates the dynamics of the solar wind speed at a scale of 1 hour at 1 AU. Beck [2001] argued that for aerodynamic turbulence, q approaches 1.5 at very small scales. Thus the ACE observations of q for the PDF of dVn at the smallest scale are probably determined principally by the intermittency of MHD turbulence at small scales in the solar wind. Figure 5a shows a transition from a state with q ≈ 1.44 to q = 1 at scales larger than ≈10 days. Beck et al. [2001] determined q as a function of scale for Couette-Taylor flow in a laboratory experiment, and they found q ≈ 1.185 near the Kolmogorov length scale and ≈1.03 at the largest scales available in their apparatus. The value q = 1 corresponds to a Gaussian distribution, as indicated in Figure 4 for n = 9. Thus the values q = 1 that we find at relatively large scales in the solar wind represent an equilibrium state, the Boltzmann-Gibbs state. This state is determined by the structure of the various types of flows (principally corotating streams, slow flows, and ejecta at 1 AU) at a given phase of the solar cycle.

[29] The widths of the PDFs of dVn increase as wq increases. Figure 5b shows wq as a function of scale for the PDFs in Figure 4. At small scales in the solar wind, wq = 13.7 ± 14.0 is relatively small; at scales greater than ≈10 days, wq = 235 ± 8 is relatively large. The small-scale fluctuations in V are related to turbulence and waves superimposed on the streams, and wq is a measure of their amplitudes. The large-scale fluctuations in dVn at 1 AU are related to the collection of streams in the solar wind at 1 AU during an interval of the order of a year, and wq is a measure of the characteristic amplitude of these streams.

[30] Finally, the skewness of the PDFs of dVn in Figure 4 is represented by Cq as a function of scale, shown in Figure 5c. Figure 5c shows −Cq on the ordinate, which is positive for distributions skewed toward positive dVn. At the scales greater than ≈10 days the value Cq = 0 is consistent with the fact that the PDFs of dVn are Gaussian at these scales, representing an equilibrium state. Significant skewness is observed at the smallest scales (1–4 hours), as also observed by Burlaga and Ogilvie [1970b], related to the convection of turbulence past the spacecraft. There is also measurable skewness at a scale of the order of a day, which is probably related to the skewness of streams produced by the overtaking of slower parts of the streams by faster parts.

5.4. Statistics of dVn(τ)

[31] The multiscale structure of the solar wind speed fluctuations can be described approximately by the statistics (as a function of scale) of the measured time series. Consider the statistics of dVn(ti) for n = 0, 1, 2,… 12, i.e., the statistics of dVn(ti) on scales (lags) τn ranging from 1 hour to 171 days. The standard deviation (SD), skewness (S), and kurtosis (K) are defined by SD ≡ {[1/(N − 1)] ∑(xi − 〈xi〉)2}1/2, S ≡ {[1/(N − 1)] ∑(xi − 〈xi〉)3}/SD3, and K ≡ {[1/(N − 1)] ∑(xi − 〈xi〉)4}/SD4 where 〈xi〉 is the mean of xidVn(ti), N is the number of points in the sample, and the sum is over xi from 1 to N. The kurtosis as defined above is 3 for a Gaussian distribution. These statistics describe the basic features of the PDF of the speed differences at each scale. Together, the three curves SD(τn), Sn), and Kn) provide an overview of the structure of the large-scale speed fluctuations as a function of scale.

[32] Figures 6a, 6b, and 6c show the standard deviation, skewness, and kurtosis of dVn(ti), respectively, as a function of lag τn = 2n for n = 0–12. The results for the ACE data during 2003 are shown by the solid squares. The corresponding results derived from the time series of the speed measured by the Wind spacecraft in 1995 [Ogilvie et al., 1995], during the declining phase of the previous solar cycle (BF), are shown by the open circles. There is good agreement between the ACE and Wind statistics at the smallest and largest scales.

Figure 6.

Statistics of the PDFs of speed differences computed from the time series of dVn as a function of scale. Solid squares show the ACE data for 2003, and the open circles show the Wind observations for 1995. (a) Standard deviation of the speed differences, increasing monotonically from small values at small scales to an asymptotic state at large scales greater than ≈5 days, in the Gaussian range. (b) Skewness, which is relatively large at small scales and declines to zero in the Gaussian range. (c) Kurtosis, which is large at small scales and decreases monotonically to near 3 in a Gaussian range. Also presented are the SD, skewness, and kurtosis obtained from the probability distribution functions of the data (shown by the points marked with a cross) and obtained from the fits to the generalized Tsallis distribution calculated via moments of the PDF (shown by the plus signs).

[33] Figure 6 also presents the SD, skewness, and kurtosis obtained from the probability distribution functions of the data (shown by the points marked with a cross) and obtained from the fits to the generalized Tsallis distribution calculated via moments of the PDF (shown by the plus signs). In addition, we show the statistical SD, skewness, and kurtosis as determined from the velocity difference time series shown by the connected square marks. The agreement among the three estimates is remarkable. The differences that exist among them are due to the fact that the estimated moments from the PDF data contain zero values, whereas those moments estimated by the fitted PDFs are never zero, since they represent predicted values. The moment parameter values differ from those determined by the statistical velocity difference time series because these estimates are not weighted by the PDF values; they only provide approximate values of the moments. Nevertheless, the moment values are in good agreement with the statistics derived from the ACE observations of the velocity differences.

[34] For lags greater than ≈10 days the kurtosis is ∼3 and the skewness fluctuates about 0, for both the Wind and ACE data, consistent with Gaussian distributions of dVn(ti) for n = 9–12. They confirm that the PDFs are Gaussian at scales greater than ≈10 days, as we concluded in section 5.3 from the fact that q = 1 and C = 0 at those scales. Thus we infer that the PDFs of the speed fluctuations at 1 AU at scales greater than or of the order of ≈10 days for the declining phase of the solar cycle describe an “equilibrium” state, in the sense of Boltzmann-Gibbs statistics.

[35] The points for the standard deviation of the ACE data in Figure 6 were fitted to the sigmoidal function FA2 + (A1 − A2)/(1 + (τ/x0)p). This simple growth curve provides a good fit to the standard deviation over the full range of lags, from 1 hour to 171 days. It is significant that over all the scales in Figure 6, the function SD(τ) derived from the ACE observations of V(ti) (solid squares) and from fits of the PDFs of dVn in Figure 4 (crosses) during 2003 agree with one another. They also agree with SD(τ) derived from the Wind data during 1995 (except at scales of the order of 1 day), nearly a solar rotation earlier, during the declining phase of solar cycle 22. The standard deviation at scales >10 days is 168 ± 4 for the ACE data and 161 ± 2 for the Wind data, indicating that the characteristic amplitudes of the fluctuations associated with the various flows were essentially the same during 2003 and 1995.

[36] Non-Gaussian structures (K ≠ 0, S ≠ 0) were observed by ACE at small and intermediate scales, as we inferred from the observation that q ≠ 1 at these scales, where turbulence and the steepening of streams are important. The values of the skewness at scales from 1 to 4 hours is the same for the ACE and Wind data. The kurtosis is largest at the smallest lags (1–4 hours) in both the ACE and the Wind data, and it has approximately the same values as the ACE and Wind data at those scales. At intermediate scales the kurtosis decreases monotonically to an asymptotic value K ≈ 3 at scales greater than ≈10 days. The kurtosis decreases somewhat more rapidly with the ACE data than for the Wind data. The skewness has a maximum at a scale 4 hours for the ACE data and 8 hours for the Wind data.

6. Probability Distributions of the Fluctuations in the Changes in B

6.1. Magnetic Field Strength Differences

[37] One can analyze the multiscale structure of B in the same way we analyzed the multiscale structure of V in section 5. We analyze the differences in B at different intervals (lags) τn, dBndBn(ti) ≡ B(ti + τn) − B(ti), where τn ≡ 2n (hours) and n = 0, 1,…, 12. Again, the lag τn determines the scale of the fluctuations represented by dBn(ti).

[38] Figures 7b–7e show dBn versus ti for n = 0, 2, 6, and 9, respectively, corresponding to lags of τ0 = 1 hour, τ2 = 4 hours, τ6 = 2.7 days, and τ9 = 21.3 days, respectively. The hour averages of B(ti) measured by ACE from days 1 to 275, 2003 are shown again for reference in Figure 7a. Although there is a recurrent peak in B at the vertical dashed lines (an interaction region produced by the steepening of the corotating stream), it is not the principal characteristic of the magnetic field strength fluctuations, and the peak is variable from one solar rotation to the next. The important point is that there is a broad spectrum of fluctuations on all scales.

Figure 7.

(a) Magnetic field strength shown in Figure 1b repeated for reference. (b and c) Running differences in B and dBn, at a lag of 1 (Figure 7b) and 4 (Figure 7c) hours, showing the effects of intermittent turbulence, filamentary structure, and boundaries of interaction regions on this scale. (d) The dBn at a lag of 2.7 days, showing changes related to the streams. (e) The dBn at a lag of 21.3 days, showing that the fluctuations of B are spiky (thus non-Gaussian) even at these large scales.

[39] Figures 7b and 7c show spiky, bursty fluctuations of B on scales of 1 and 4 hours, respectively. Unlike the speed fluctuations at these scales, the fluctuations in B appear to be symmetric. Turbulence probably contributes to these fluctuations, but there are important contributions from (a) the gradients in B at the front and rear of interaction regions, (b) tangential discontinuities and filaments, and (c) shocks. On a scale of 2.7 days (Figure 7d), which is of the order of the size of a stream, and on a scale of 21.3 days (Figure 7e), which is close to the solar rotation period (27 days), the fluctuations dB6 and dB9 resemble one another; both are spiky, as one observes at smaller scales. Whereas the speed fluctuations in dV9 (Figure 3e) were Gaussian, the magnetic field strength fluctuations dB9 are not Gaussian.

6.2. Probability Distribution Functions of dBn

[40] Consider the PDFs of dBn(ti) for the scales n = 0, 2, 6, 7, 9, and 11, shown by the points in Figure 8. The PDFs for dBn(ti) are plotted as histograms of the fraction of counts in bins versus dBn on a semilog scale; each PDF is offset vertically from the one below it by a factor of 100. The lowest bin in each histogram corresponds to one count per bin, which accounts for the relatively broad spread of the lowest points in each histogram. The PDFs of dBn are relatively narrow at the smallest scales, they get broader with increasing scale up to ≈5 days, and they remain similar to one another at scales from ≈5–85 days. Each histogram was fitted with the generalized Tsallis distribution, giving the PDFs shown by the solid curves in Figure 8. An important result is evident in Figure 8: The generalized Tsallis distribution fits all the data very well.

Figure 8.

Observed PDFs of magnetic field strength differences dBn at scales 2n hours, with n = 0, 2, 6, 7, 9, and 11 shown by the points. Each PDF is shifted by a factor of 100 from the one below it. Lowest points in each PDF correspond to 1 count/bin. Curves are fits of the data to the generalized Tsallis distribution. PDFs are narrow at small scales and broader at larger scales. PDFs are kurtotic at all scales; they do not approach a Gaussian distribution (a quadratic on a semilog scale) at the largest scales.

[41] There are significant qualitative differences between the PDFs of the speed and magnetic field strength fluctuations. The PDFs of dVn approach a Gaussian (a quadratic function on a semilog scale) at n ≥ 7, whereas the PDFs of dBn do not approach anything like a Gaussian even at n = 11 (τ = 171 days). The PDFs of dVn are skewed at small and intermediate scales, whereas the PDFs of dBn all seem to be relatively symmetric. These differences between the PDFs of the speed and magnetic field strength fluctuations can be expressed quantitatively by the parameters of the generalized Tsallis distribution (sections 5.3 and 6.3, respectively) and by the low-order statistics of the temporal fluctuations as a function of scale (sections 5.4 and 6.4, respectively).

6.3. Fitting Parameters for the Probability Distribution Functions of dBn

[42] The parameters q, wqequation image, and Cq as a function of scale were derived from fits of the generalized Tsallis PDF (equation (3)) to the PDFs of dBn(τ) in Figure 8. The parameters q, wq = σqwq, and Cq are plotted versus scale in Figures 9a, 9b, and 9c, respectively. These parameters and their variation with scale are determined by the structure of the magnetic field fluctuations at different scales and the dynamical processes that produce them.

Figure 9.

Parameters of the generalized Tsallis distribution derived from fits to the observations. (a) Nonextensivity parameter q decreasing from ≈1.83 at a scale of 1 hour to ≈1.7 at scales greater than ≈5 days. Transition between 4 hours and 2.7 days occurs where interaction regions are dominant. (b) Width parameter wq as a function of scale mirroring the behavior of q, being small at scales <4 hours and large at scales >2.7 days. (c) Skewness parameter −Cq versus scale. This parameter is consistent with zero within the uncertainties.

[43] Note that the dimensionless skewness parameter Cq is equal to zero, within the uncertainties, at all scales. Thus the skewness of the PDFs of dBn is small at all scales from 1 hour to ≈85 days. This implies that the classical Tsallis distribution itself (equation (1)) describes the magnetic field fluctuations over this broad range of scales, within the uncertainties of the fits to the generalized Tsallis distributions. Thus, to good approximation, the state of large-scale magnetic field strength fluctuations at 1 AU during 2003 is described by two parameters, q and wq, related to the nonextensivity and widths of the measured PDFs, respectively.

[44] The value of q for the magnetic field strength fluctuations is in the range 1.66 < q < 1.83 for scales from 1 hour to ≈85 days. It is always larger than the largest value of q (≈1.44) observed for the speed fluctuations. In particular, q for the PDF of dBn describes something other than turbulence, even at the smallest scales. Significantly, q never approaches 1, even at the largest scale ≈85 days. Thus the PDF of dBn does not approach a Gaussian distribution (an equilibrium state) at large scale, unlike that of dVn, as we noted in the discussion of Figure 8. On the other hand, q does not vary much between scales of ≈5 and 85 days, so that the fluctuations in B might be in a metastable state at these scales.

[45] The parameter wq, a measure of the width of the PDFs of dBn(τ), is relatively small at small scales (narrow PDFs) and relatively large (broader PDFs) at intermediate and large scales (Figure 9b). This is a quantitative expression of the observation made in reference to Figure 8 that the PDFs are narrow at small scales and broader at larger scales. The value of wq tends to be anticorrelated with that of q. The value of q is larger at small scales than it is at larger scales (Figure 9a).

6.4. Statistics of dBn(τ)

[46] Let us now consider the statistics of dBn(ti) for n = 0, 1, 2,… 12, i.e., the statistics of dBn(ti) on scales (lags) τn ranging from 1 hour to ≈171 days. The three curves SD(τn), Sn), and Kn) provide an overview of the structure of the magnetic field strength fluctuations over this range of scales from days 1 to 275 during 2003. The solid squares in Figures 10a–10c show SD, S, and K, respectively, derived from the time series dBn(ti) as a function of lag τn = 2n for n = 0–12. Figure 10 also presents the SD, skewness, and kurtosis obtained from the probability distribution functions of the data (shown by the points marked with crosses) and obtained from the fits to the generalized Tsallis distribution calculated via moments of the PDF (shown by the plus signs). As we found for dVn, the agreement among the three estimates is very good. The differences that exist among them are due to the fact that the estimated moments from the PDF data contain zero values, whereas those moments estimated by the fitted PDFs are never zero, since they represent predicted values. The moment parameter values differ from those determined by the statistical velocity difference time series because these estimates are not weighted by the PDF values, and therefore they only provide an approximate value to the moments. Nevertheless, the moment values are in good agreement with the ACE observations.

Figure 10.

Statistics of the field strength differences dBn as a function of scale. Solid squares show results computed from the time series, and the plus signs show the results computed to the PDFs. (a) Standard deviation of the speed differences increasing monotonically from small values at small scales to constant values at scales >5.3 days. (b) Skewness, which is generally relatively small and does not show any clear trend with increasing scale. (c) Kurtosis, which is large at small scales and decreases monotonically to a nearly constant value at scales greater than ≈2.7 days.

[47] The points for the standard deviation of the ACE data in Figure 10a were fitted to the sigmoidal function FA2 + (A1 − A2)/(1 + (τ/x0)p). This simple growth curve provides a good fit to the standard deviation over the full range of scales, from 1 hour to 171 days. The curve has the same form that we found for dVn versus τ in Figure 6b, but it approaches an asymptotic value (4.14 ± 0.05) nT.

[48] The skewness 0 ≤ S(dBn) < 1 at all scales, but there is no simple scale dependence. The kurtosis as a function of scale, K(τ), in Figure 10c shows the same qualitative behavior that we observed for the kurtosis of dVn (Figure 6c). There is a monotonic decline of kurtosis from relatively high values at small scales to a plateau at larger scales ≥2.7 days. However, the asymptotic value is not K = 3, as one would expect for a Gaussian distribution. Rather, the distribution of dBn remains kurtotic at scales >5.3 days, consistent with the hypothesis that the fluctuations in B at scales >5.3 days are in a metastable state.

7. Scaling Structure

[49] Scaling structure is observed in the speed and magnetic field strength profiles at large distances from the Sun throughout the solar cycle [Burlaga, 1995; Burlaga et al., 2003] and at 1 AU near solar maximum [Burlaga, 1993] and during the declining phase of the solar cycle [Burlaga, 1991b, 1992]. Given that the choice of an entropy function made by Tsallis [1988] was motivated by the desire to describe systems with scaling symmetry (fractal or multifractal), we consider whether the fluctuations on V and B observed by ACE during 2003 had a scaling symmetry.

[50] Following the procedure described by Burlaga [1991a, 1995] and Paladin and Vulpiani [1987], we search for scaling behavior of moments m and p of dVn and dBn, respectively (〈dVnm〉 and〈dBnp〉). We found that the moments 〈dVnm〉 increase linearly with scale in the range 2 ≤ τ ≤ 64 hours and the moments 〈dBnp〉 increase linearly with scale in the range 2 ≤ τ ≤ 128 hours. The slopes sV and sB of linear fits to these moments in the scaling range are shown as a function of m and p in Figures 11a and 11b, respectively. For m and p > 4 the slopes increase only linearly with m and p, possibly owing to limitations of the number and accuracy of the observations, so that we cannot determine if the fluctuations in dVn and dBn were multifractal. On the other hand, for both the moments of dVn and dBn the slopes sV and sB increase nonlinearly with the moment m and p for m and p ≤ 4, indicating scaling behavior. In fact, the slope functions sV and sB are quadratic for 0 < m ≤ 4 and 0 < p ≤ 4, indicating the presence of intermittency and scaling behavior at these scales [Mandelbrot, 1972, 1989; Kolmogorov, 1962].

Figure 11.

(a) Slope sV of moments of 〈dVnm〉 versus scale τ in the range 2–64 hours as a function of m. (b) Slope sB of moments of 〈dBnp〉 versus scale τ in the range 2–128 hours as a function of p. In this figure, Bh and Vh refer to hour averages of B and V, respectively. The slope of linear fit to the second moment versus scale is s(2). The coefficients of the quadratic fits to the observations are a, b, and c.

8. Summary and Discussion

[51] We analyzed the fluctuations in the speed V and magnetic field strength B in the solar wind at 1 AU during the declining phase of solar cycle 23 from days 1 to 275, 2003. These fluctuations in V and B on scales from 1 hour to at least ≈85 days can be described by a single function, a generalization of the Tsallis distribution of nonextensive statistical mechanics. The Tsallis probability distribution was derived in the framework of a generalization of Boltzmann-Gibbs statistical mechanics, by extremizing a new entropy function subject to two constraints. The physical processes determine the parameters in the generalized Tsallis distribution as a function of scale in the particular system under consideration. Together, this PDF and the multiscale behavior of its parameters are properties of the statistical mechanics of the solar wind.

[52] On small scales, of the order of hours, the PDFs of the speed fluctuations have a value of the nonextensivity parameter q = 1.44, close to the value q = 1.5 expected for intermittent aerodynamic turbulence at very small scales. Thus the generalized Tsallis PDF provides a quantitative description of the metastable state related to intermittent MHD turbulence in the solar wind. On relatively large scales, greater than ≈10 days, the PDF of velocity fluctuations has q = 1, corresponding to a Gaussian distribution, the Boltzmann-Gibbs limit. At these scales the PDF describes an equilibrium state of the solar wind consisting of a collection of various types of flows (corotating streams, ejecta, and slow flows) with various sizes, speeds, and structures, interacting weakly with one another. This equilibrium state might vary with the solar cycle, radial distance from the Sun, and latitude. At intermediate scales of the order of days, 1 < q ≤ 1.44, so one does not have an equilibrium (Gaussian) state, but the generalized Tsallis PDF describes the statistical properties of the individual flows.

[53] We considered an interval during the declining phase of the solar cycle when equatorial extensions of the polar holes produced corotating streams. One corotating stream was observed recurring with a period of 27 days. However, the stream evolved from one solar rotation period to the next in size, amplitude, and structure; there was also considerable variability on all scales related to other sources. Thus it is not appropriate to think of the speed profile as a simple periodic component with small-scale turbulence superimposed. The multiscale PDFs provide a more accurate description of the multiscale structure of the solar wind speed fluctuations and the dynamics producing it.

[54] The fluctuations in the magnetic field strength were also described quantitatively by the generalized Tsallis distribution at all the scales considered. In fact, the skewness of the PDFs was small, so that the Tsallis distribution itself describes the magnetic fluctuations to a good approximation. For the PDFs of the magnetic field strength fluctuations we found 1.66 < q < 1.83, in contrast to the values 1.01 ≤ q < 1.44 for the PDFs of speed fluctuations. The values of q for the small-scale fluctuations were not the values that one expects for intermittent turbulence; the fluctuations were more kurtotic (bursty) than turbulence. Perhaps this is related to the shocks, discontinuities, and filamentary structure of the magnetic field strength observed at many scales, but further study is needed to test this hypothesis. At the relatively large scales, ≈85 days, q did not approach 1; thus the fluctuations in B did not approach a Boltzmann-Gibbs equilibrium state, a Gaussian distribution. The PDFs of the magnetic field strength fluctuations were kurtotic at even the largest scales considered. On the other hand, q did not change much for scales between ≈5 and 85 days, suggesting a possible metastable equilibrium state at these scales.

[55] The generalized statistical mechanics introduced by Tsallis provide a new quantitative and meaningful framework to describe the multiscale structure of the solar wind in terms of PDFs and their parameters as a function of scale. The PDFs of the large-scale fluctuations of V and B observed by Voyager at various distances and epochs of the solar cycle were explained by a deterministic multifluid MHD model with ACE data as input [see, e.g., Burlaga et al., 2003, and references therein]. We are extending that work to determine whether the observed and predicted multiscale PDFs beyond 1 AU are related to the generalized Tsallis distribution.

[56] There is no model that predicts the multiscale statistical structure of the solar wind at 1 AU. The results of this paper provide a set of observations that such a model must predict. These observations include (1) the PDFs of the changes and the speed and magnetic field strength and on scales from 1 hour to ≈171 days; (2) the parameters of these PDFs as a function of scale; (3) the standard deviation, skewness, and kurtosis of changes in the speed and magnetic field strength on scales from 1 hour to 171 days; and (4) the power spectra of the fluctuations of the speed and magnetic field strength.

Acknowledgments

[57] D. McComas and R. Skoug provided the speed data from the SWEPAM instrument on ACE used in this work. N. Ness and C. Smith provided the magnetic field data from the MAG instrument on ACE.

[58] Shadia Rifai Habbal thanks the referee for his/her assistance in evaluating this paper.

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