Speed f luctuations near 60 AU on scales from 1 day to 1 year: Observations and model

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. A10, 1328, doi: /2002ja009379, 2002 Speed f luctuations near 60 AU on scales from 1 day to 1 year: Observations and model L. F. Burlaga Laboratory for Extraterrestrial Physics, NASA-Goddard Space Flight Center, Greenbelt, Maryland, USA J. D. Richardson and C. Wang 1 Center for Space Research, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Received 12 March 2002; revised 16 May 2002; accepted 23 May 2002; published 25 October [1] This paper describes the multiscale, statistical state of the speed observed near 60 AU from mid-1999 to mid-2000 by Voyager 2 (V2), and it shows that a multifluid MHD model can explain the basic features of these observations. The probability distribution functions (PDFs) of the running speed differences (dvn) on scales from 1 day to 256 days provide a relatively complete description of some important properties of the large-scale speed fluctuations. On a scale of 1 or 2 days the PDFs of the positive and negative speed differences observed by V2 are approximately exponential, which is related to jump-ramp structures but might include a contribution from intermittent turbulence. On scales greater than 26 days (the solar rotation period) the PDFs of the speed differences are approximately Gaussian, i.e., quadratic on a semilog scale. On a scale of the order of several days, on which one sees jump-ramp structures in the speed profile, the PDF of the speed differences is cubic on a semilog scale. The standard deviation of dvn increases with increasing scale. The skewness and kurtosis of dvn are relatively large at small scales and decrease to Gaussian values at scales 16 days. The PDFs of speed differences and their lower moments versus scale near 60 AU were also derived from a speed profile predicted by the deterministic, spherically symmetric, multifluid, MHD model of Chi Wang, using ACE observations at 1 AU as the inner boundary conditions. Although the projected speed profile is not the same as the observed speed profile because ACE and Voyager are not radially aligned throughout the 1-year interval, the statistical properties of the observed profiles are essentially the same as the projected speed profiles. Significant evolution of the multiscale statistical properties of the solar wind speed fluctuations occurs between 1 and 60 AU; this evolution can be explained by a deterministic model. INDEX TERMS: 2164 Interplanetary Physics: Solar wind plasma; 2152 Interplanetary Physics: Pickup ions; 2151 Interplanetary Physics: Neutral particles; 2149 Interplanetary Physics: MHD waves and turbulence; KEYWORDS: solar wind speed, speed fluctuations, distant heliosphere Citation: Burlaga, L. F., J. D. Richardson, and C. Wang, Speed fluctuations near 60 AU on scales from 1 day to 1 year: Observations and model, J. Geophys. Res., 107(A10), 1328, doi: /2002ja009379, Introduction [2] A statistical description of the state of the solar wind over large distances and times is needed [Burlaga, 1975, 1995] for several reasons. First, in order to model the solar wind by means of a MHD model, one needs to know the flow conditions (velocity V, density N, temperature T, and magnetic field strength B) on a boundary near the Sun as a function of time. The information about these boundary conditions is incomplete. It will never be possible to determine the boundary conditions with in situ spacecraft data, 1 Also at Laboratory for Space Weather, Chinese Academy of Sciences, Beijing, China. Copyright 2002 by the American Geophysical Union /02/2002JA009379$09.00 and we are far from being able to extrapolate solar measurements in order to obtain the necessary boundary conditions. Second, since it takes 1 year for the solar wind to fill the heliosphere out to 100 AU, one must observe the solar wind at a given distance from the Sun (e.g., 1 AU) for 1 year in order to see the material that fills the heliosphere. Third, since the solar cycle has a period of the order of 11 years, the state of the solar wind at a representative part of the solar cycle is given by data measured during 1 year (1/ 10 of a solar cycle). Fourth, on the scale of a year the structure of the solar wind is very complex, and one needs special methods to describe this complexity. Fifth, the statistical properties of the solar wind on a scale of 1 year evolve significantly with increasing distance from the Sun. [3] The traditional method for modeling observed solar wind flows is to input data from one spacecraft and compute SSH 20-1

2 SSH 20-2 BURLAGA ET AL.: LARGE-SCALE SPEED FLUCTUATIONS BETWEEN 1 AND 60 AU the flow that would be observed by another spacecraft at some other location. For a meaningful comparison this method requires that the spacecraft be radially aligned with the Sun. In general, any two spacecraft are at different latitudes and longitudes and at different distances from the Sun. Radially aligned spacecraft will quickly cease to be radially aligned with the Sun because the inner spacecraft has a higher angular velocity. Hence one can consider only certain brief events, rather than data over a period of a year, for comparison of observed and predicted profiles of the flow fields. [4] Burlaga and Forman [2002] showed that the speed fluctuations over a wide range of scales at 1 AU could be described by a set of probability distribution functions (PDFs) of the speed differences at different scales. Specifically, they studied the PDFs of dvn dvn(t i ; t n ) V(t i + t n ) V(t i ), where t n 2 n (hours), n =0,1,...,13for the hour averages of t i measured by the SWEPAM instrument on ACE during They found that the PDFs of dvn varied with scale from a nearly exponential form at small scales to a Gaussian form at scales greater than the solar rotation period. The skewness S and kurtosis K were relatively large at small scales, and reached their Gaussian values (S =0,K = 3) at the scale of the solar rotation period. The standard deviation of dvn increased sigmoidally from small to large scales. [5] The aim of this paper is to present a quantitative description and model of the multiscale structure of the large-scale solar wind speed fluctuations [Burlaga, 1984] at 60 AU on the basis of limited observations. We shall present two major results. First, it is shown that one can use the PDFs of dvn and their moments as a function of scale to describe the large-scale speed fluctuations measured by Voyager 2 (V2) near 60 AU over the course of 1 year. Second, it is shown that one can use a deterministic, spherically symmetric, multifluid MHD model, with ACE observations at 1 AU as input, to compute the speed profile V(t) at the distance of V2 and from this derive the statistical properties of the large-scale speed fluctuations measured by V2 after evolving between 1 and 60 AU. 2. Observations and Model [6] We shall analyze the speed fluctuations measured by the plasma (PLS) instrument on Voyager 2 [Bridge et al., 1977] in the interval corresponding to the observations made by ACE at 1 AU throughout Considering the propagation speed of the solar wind at 1 AU and the deceleration en route to 60 AU owing to the production of pickup protons, the appropriate analysis interval for the V2 data is approximately day 200, 1999 to day 260, This interval corresponds to solar wind leaving the Sun during the rising phase of solar cycle, approaching the time of maximum sunspot number in There were many transient flows during 1999 [Richardson et al., 2000]. During the interval under consideration V2 moved from 58.4 to 62.0 AU, and its heliographic latitude changed from 20.5 to [7] The Voyager 2 spacecraft is tracked in real time, typically for 8 to 12 hours per day. We use daily averages for this study because one needs a nearly continuous time series in order to calculate the speed differences accurately. We analyze the variations of the radial component of daily averages of the solar wind velocity, V(t i ), measured by V2; t i t is the time measured in days from the beginning of the interval from day 200, 1999 to day 260, [8] We shall compare the multiscale statistical properties of the speed fluctuations V(t) observed by V2 at 60 AU with the projected speed fluctuations, V m (t), at that distance, which are computed from a MHD model using the hour averages of the ACE plasma and magnetic field observations at 1 AU as input. During the analysis interval, ACE moved 360 around the Sun in the ecliptic, while the heliographic inertial longitude of V2 changed by only 0.5. Since ACE and V2 were radially aligned during just a small fraction of the analysis interval, it is not possible to project the flow profiles observed by ACE to the position of V2 throughout the interval. Therefore we compare the multiscale statistical properties of V(t) and V m (t) rather than the speed profiles themselves. We assume that the solar wind speeds sampled by ACE in its orbit around the Sun are representative of the speeds of the solar wind that moves along a line from the Sun to V2. This assumption allows us to use an azimuthally symmetric model. [9] Since the heliographic latitude of V2 ( 21.1 ) is not greatly different from the latitude of ACE (±7.5 ), we can use a spherically symmetric model as a first approximation near solar maximum, when the latitude variations are relatively small compared with those at solar minimum [Phillips et al., 1995]. The pickup protons make a significant contribution to the internal pressure of the solar wind beyond 20 AU [Axford, 1972; Holzer, 1972; Vasyliunas and Siscoe, 1976; Whang et al., 1996; Burlaga et al., 1994, 1996] and their production decelerates the solar wind [Wang et al., 2000a, 2000b; Richardson et al., 1995], so the model must include pickup protons. One must use an MHD model, since the magnetic field plays an important role in the dynamics of the heliosphere [e.g., see Burlaga, 1995]. [10] Wang and Richardson [2001, 2002] developed a deterministic, spherically symmetric, multifluid MHD model that has all of the essential properties that are required. The model accounts for the interactions of solar wind protons and pickup protons. The neutral hydrogen is included self-consistently by a hydrodynamic approach. Since the equations of the model are hyperbolic, one can specify conditions on a spherical inner boundary at 1 AU as a function of time. The quantities that must be specified are the bulk speed V, density N, proton temperature T, and magnetic field B. The model considers the proton and electron temperatures to be the same, and it uses a polytropic equation of state. [11] We use hour averages of these quantities measured by ACE as input to the model, since a nearly continuous set of observations is available and since one needs hour average resolution to describe the steep gradients that are dynamically important near 1 AU. The hour averages of V for 1999 are from the SWEPAM instrument on ACE [McComas et al., 1998]. These data were obtained from the level 2 ACE SWEPAM data set at caltech.edu/ace/asc/level2/lvl2data_swepam.html. A description of the data is at ACE/ASC/level2/swepam_l2desc.html. The principal investigator for the SWEPAM instrument is D. McComas.

3 BURLAGA ET AL.: LARGE-SCALE SPEED FLUCTUATIONS BETWEEN 1 AND 60 AU SSH 20-3 Figure 1. (a) Daily averages of the speed measured by ACE during (b) Daily averages of Voyager 2 observations. (c) Daily averages of speed projected from ACE to Voyager 2 using a model. [12] The speed profiles that we shall consider in this paper are shown in Figure 1. The daily averages of the speed observed by ACE at 1 AU during 1999 are shown in Figure 1a. Large-amplitude streams with a scale of the order of a few days are present; a jump-ramp structure is not prominent. The range of the speeds at 1 AU is from 300 to 900 km/sec. The speed fluctuations at 1 AU are not homogeneous throughout the year; in particular, the ampli-

4 SSH 20-4 BURLAGA ET AL.: LARGE-SCALE SPEED FLUCTUATIONS BETWEEN 1 AND 60 AU tudes of the speed fluctuations appear to be larger during the latter half of 1999 than during the first half of the year. Smaller-scale fluctuations are superimposed on both the streams and the low speed regions. [13] The daily averages of the speed measured by V2 from day 200, 1999 to day 260, 2000 are shown in Figure 1b. A key feature is the jump-ramp structure, with a scale of the order of several days, which is repeated in a variety of forms with different amplitudes for the jump and different durations for the ramp. Small fluctuations are superimposed on the jump-ramp structure. The jump-ramp structure is superimposed on large-scale variations. The speeds are distributed within a relatively small range, from km/sec. A decrease in the amplitudes of the speed fluctuations with increasing distance from the Sun was reported by Wolfe [1972]. [14] The projected speed profile, obtained from the model with ACE observations as input, is shown in Figure 1c. As expected, there is not a one-to-one correspondence between the projected and observed speed profiles, since ACE is not radially aligned with V2 throughout the analysis interval. However, there is a strong similarity in the form of the two speed profiles. The projected speed profile also shows a jump-ramp structure with a scale of the order of 10 days. Small fluctuations are superimposed on the jump-ramp structure, and the speeds are distributed within a relatively small range, from 350 km/sec to 430 km/sec. 3. Speed Fluctuations Near 60 AU [15] We will use the same methods to analyze both the speed fluctuations observed by V2 and those of the projected speed profile produced by the model. It is well known that one can describe the fluctuations of V on various scales (lags) t n by studying the running differences dvn dvnðt i ; t n ÞVðt i þ t n Þ Vðt i Þ [Burlaga, 1995]. There is one PDF of the numbers dvn(t i ) for each lag t n. The lag t n (where t n 2 n (days), n =0,1,..., 8) determines the scale of the fluctuations represented by dvn(t i ). The scales range from t 0 =2 0 = 1 day to t 8 =2 8 = 256 days = sec. [16] The speed fluctuations observed by V2, dvn, are plotted as a function of time for the scales 2, 4, 16, and 64 days (n = 1, 2, 4, and 6, respectively) in Figure 2. Figure 2a shows the speed profile observed by V2 for reference. At scales of 2 days and 4 days the fluctuations are intermittent and asymmetric about dv1 = 0, like the results reported at smaller scales by Burlaga [1991a, 1991b, 1992, 1993], Marsch [1991], Marsch and Liu [1993], and Matthaeus et al. [1983] associated with intermittent turbulence. Burlaga and Goldstein [1984] showed that the turbulence tends to extend to larger scales with increasing distance from the Sun beyond 1 AU, and Burlaga et al., [1987] showed that inertial range turbulence can be seen at scales of 5 days or more if it is not obscured by shocks. Thus one might expect turbulence extending to a scale of 1 or 2 days to be present at 60 AU. At a scale of 2 days the fluctuations dv1 (Figure 2b) are large and positive at the jumps (as a result of the jumps themselves) and they are small in the ramps. At a scale of 16 days the fluctuations dv4 are more symmetric about dv4 = 0, and they are related to the scale of the ramps. At a scale of 64 days the fluctuations dv6 are very irregular. However, there is a trend for dv6 to increase with time from negative to positive values over the 1 year interval, which reflects the tendency for V to decrease during the first half of the year and increase during the second half of the year at Voyager 2. [17] The projected daily average speed fluctuations computed with a model using hour averages of ACE data as input are shown in Figure 3, in the same format as Figure 2. The projected speed profile at the position of V2 (60 AU) is shown in Figure 3a for reference. The properties of the projected speed fluctuations are qualitatively the same as those of the observed speed fluctuations shown in Figure 2a. At scales of 2 days and 4 days the fluctuations are intermittent and asymmetric about dv1 = 0, largely the result of the jump-ramp structure but possibly including the effects of intermittent turbulence. At a scale of 2 days the fluctuations dv1 are large and positive at the jumps (as a result of the jumps themselves), and they are small in the ramps. At a scale of 16 days the fluctuations dv4 are more symmetric about dv4 = 0, and they are related to the scale of the ramps. At the scale of 64 days the fluctuations dv6 are very irregular. Note that the fluctuations of dv6 have larger amplitudes during the second half of the year than during the first half of the year. 4. PDFs of Speed Fluctuations Near 60 AU [18] Given the nine data sets dvn(t i ; t n ), n =0,1,2,...8, we can determine the PDF of dv for each set. These PDFs provide a quantitative and detailed description of the speed fluctuations at the respective scales. PDFs of speed fluctuations are used extensively to describe turbulence [Sreenivasan, 1991; Sorriso-Valvo et al., 1999; Castaing et al., 1990; Marsch and Tu, 1994]. However, we use PDFs here primarily to describe flow-related structures rather than just turbulence. Figures 4 and 5 show PDFs for six of the nine scales, namely, 1, 2, 4, 8, 16, and 64 days. Bins of 5 km/s are used for lags of 1, 2, and 4 days; bins of 10 km/sec are used for lags of 8, 16, and 64 days. The bins are plotted on a linear scale, and the fraction of the total number of counts in each bin is plotted on a logarithmic scale. The same ranges for the scales are used in all of the panels in Figures 4 and 5 below. [19] The PDFs for the speed fluctuations observed by V2 on various scales are shown in Figure 4. The PDF for speed fluctuations dv0 at a scale of 1 day is shown in Figure 4a. This distribution is narrow, reflecting that the smallest-scale fluctuations considered, and has relatively small amplitudes. The PDF is skewed, with more large positive values of dv than negative values. The large positive values of dv are largely the result of the jumps in the speed profile, although intermittent turbulence produces a similar structure [Burlaga, 1991a, 1991b; Carbone and Bruno, 1997] and might be a contributing factor. To first approximation, the PDF for a lag of 1 day can be described by two exponential functions (two straight lines with different slopes on the semilog scale in Figure 4a, one for positive dv and another for negative dv ). Such a structure was observed by V2 at 45 AU

5 BURLAGA ET AL.: LARGE-SCALE SPEED FLUCTUATIONS BETWEEN 1 AND 60 AU SSH 20-5 Figure 2. (a) Daily averages of Voyager 2 observations of speed. The remaining panels show how the running differences of the speed with the indicated time lags vary with time.

6 SSH 20-6 BURLAGA ET AL.: LARGE-SCALE SPEED FLUCTUATIONS BETWEEN 1 AND 60 AU Figure 3. (a) Daily averages of the projected speed observations. The remaining panels show how the running differences of the speed with the indicated time lags vary with time.

7 BURLAGA ET AL.: LARGE-SCALE SPEED FLUCTUATIONS BETWEEN 1 AND 60 AU SSH 20-7 cubic fit km/s km/s cubic fit km/s km/s quadratic fit km/s km/s Figure 4. PDFs of the speed differences observed by Voyager 2 at the indicated lags.

8 SSH 20-8 BURLAGA ET AL.: LARGE-SCALE SPEED FLUCTUATIONS BETWEEN 1 AND 60 AU Figure 5. PDFs of the speed differences obtained from the speed profile projected to the distance of Voyager 2 using ACE observations and the spherically symmetric MHD model (see text).

9 BURLAGA ET AL.: LARGE-SCALE SPEED FLUCTUATIONS BETWEEN 1 AND 60 AU SSH 20-9 during 1994 and at 50 AU during 1996 [Burlaga and Ness, 1998] and on a smaller scale at 1 AU [Burlaga and Ogilvie, 1970]. The PDF for speed fluctuations with a 2-day lag (Figure 4) is similar to that for a 1-day lag, consistent with the intermittent structure of the two profiles dv1(t) and dv2(t) shown in Figure 2. A secondary peak is observed in the dv1 distribution (Figure 4b) at dv 20 km/sec and a shoulder is observed in the dv2 distribution (Figure 4c) near dv 20 km/sec. These features might correspond to the jumps in speed profile in Figure 2a, which have a characteristic size of the order of 20 km/sec. [20] The PDFs for speed fluctuations at scales of 8 and 16 days have a different structure than those at smaller scales. Recall that dv4(t) in Figure 2 is more symmetric than both dv0 and dv1 described above, and the fluctuations have scales comparable to those of the ramps. In this case the PDFs no longer have a quasi-exponential structure; rather, they can be described by cubic polynomials as shown by the fits in Figures 4d and 4e. Figure 4e might also be described by a quadratic polynomial. The widths of the PDFs for the scales of 8 and 16 days are larger than those for smaller scales, since the jump-ramp structure seen at several days is more prominent than those of the smaller scale fluctuations. Finally, at a scale of 64 days, the PDF is approximately Gaussian, as indicated by the quadratic fit to the data plotted on a semilog scale in Figure 4f. [21] Now consider the PDFs for the speed fluctuations of the projected (model) speed profile, which are shown in Figure 5 in the same format as Figure 4. We emphasize that these PDFs are derived from a projection of the ACE data profiles at 1 AU to 60 AU using a deterministic, multifluid MHD model that does not explicitly include turbulence. The qualitative features of the PDFs of the projected speed differences in Figure 5 are the same as those of the PDFs of the speed differences observed by V2. At small scales (1 and 2 days), the PDFs are narrow, skewed and quasiexponential. The model predicts the observed peak and shoulder in the distributions of dv1 and dv2 near dv =20 km/sec discussed above in reference to Figure 4 (see Figures 5b and 5c). These features probably correspond to the jumps in the computed speed profile in Figure 3a, which have scales and magnitudes of the order of 20 km/sec. At intermediate scales (8 and 16 days) the PDFs are broader and less skewed than at the small scales, and at a scale of 64 days the PDF is Gaussian, as indicated by the quadratic fit to the data on a semilog scale in Figure 5f. 5. Moments of the PDFs Near 60 AU as a Function of Scale [22] Some of the basic properties of the PDFs of the speed differences as a function of scale discussed in section 4 can be summarized by plotting the lowest order moments of dvn (standard deviation SD, skewness S, and kurtosis K ) as a function of scale n =log 2 (days). These quantities are defined as follows: SD {[(1/(N 1)] P (x i hx i i) 2 } 1/2, S {[1/(N 1)] P P (x i hx i i) 3 }/SD 3, and K {[1/(N 1)] (xi hx i i) 4 }/SD 4 where hx i i is the mean of x i dvn(t i ), N is the number of points in the sample, and the sum is over x i from 1 to N. For a Gaussian distribution, S = 0 and K =3. The skewness measures the asymmetry between positive and negative jumps in speed. The kurtosis measures the non-gaussian tails in the distribution of dvn; anomalously large jumps in dvn (intermittency) are present when S > 0 and K >3. [23] Let us first compare the statistics measured by Voyager 2 near 60 AU with those measured by ACE at 1 AU (the latter are discussed in more detail by Burlaga and Forman [2002]). SD(n), S(n) and K(n) for dvn were calculated from daily averages of the speed measured by Voyager 2 near 60 AU and by ACE at 1 AU. They are plotted as open circles and closed triangles, respectively, as a function of scale n = log 2 (days) in Figure 6. Two important results are evident in Figure 6. First, there was a significant decrease of SD(n) between 1 and 60 AU, which is primarily the result of stream interactions which decelerate fast flows and accelerate slow flows. Second, for lags between 1 and 8 days the skewness and kurtosis increase with increasing distance from the Sun; this represents the jump-ramp structure (or saw-tooth structure) that develops as a consequence of stream steepening, shock formation, and shock interactions between 1 and 60 AU, and it might include a contribution from the intermittent turbulence. At scales 16 days the skewness and kurtosis of both the Voyager 2 and ACE data are zero, consistent with Gaussian distributions. At scales between 8 and 128 days the standard deviation of the ACE data remains relatively constant while that of the Voyager 2 data continues to increase. Note, however, that ACE measurements show a small peak in SD(t) atn = 6 (64 days). [24] Now, let us compare the multiscale statistics measured by Voyager 2 near 60 AU with those predicted by the model using the measurements obtained by ACE at 1 AU as input. Figure 7 shows SD(n), S(n), and K(n) for dvn measured by Voyager 2 (open circles) and the predictions of the model (closed squares) as a function of scale n. Significantly, the model predicts the basic qualitative features of the statistics as a function of scale observed by Voyager 2: (1) the monotonic increase in the standard deviation at scales from 1 to 64 days, (2) the positive skewness at scales from 1 to 16 days and the nearly zero skewness at larger scales, and (3) the positive kurtosis at scales from 1 to 8 days and a kurtosis of 3 at larger scales. Since Voyager 2 and ACE are not radially aligned with the Sun, one would not expect to find perfect agreement between the observations and the model s predictions even if the measurements and model were perfect. In general, the difference between the observed and predicted points is less than 25%. We conclude that the radial evolution of the statistics between 1 and 60 AU can be described by the model to this degree of accuracy. [25] The SD of the V2 speed fluctuations increases with increasing scale from 6 km/sec at 1 day to a plateau at 30 km/s at scales 64 days. The calculated SD versus scale, computed from the projected speed profile has the same functional form as the SD for the V2 observations. The SD for the projected data shown in Figure 7 are described by the sigmoidal function y A2 +(A1 A2)/ (1 + e (x xo)/dx ) with the parameters A1 = 5.4 ± 3.9, A2 = 26.8 ± 2.0, xo = 2.87 ± 0.55, and dx = 0.89 ± The V2 observations of SD(n) are described by a sigmoidal function with A1 = 4.1 ± 3.7, A2 = 36.1 ± 1.9, xo = 3.0 ± 0.4 and dx = 1.1 ± 0.4. The parameter xo is the value of x at which y(xo) =(A1 +A2)/2, and dx measures the width over which the x

10 SSH BURLAGA ET AL.: LARGE-SCALE SPEED FLUCTUATIONS BETWEEN 1 AND 60 AU Figure 6. Moments of the probability distribution functions for Voyager 2 speed fluctuations and the corresponding ACE speed fluctuations as a function of scale. (a) The standard deviation as a function of scale. (b) The skewness as a function of scale. (c) The kurtosis as a function of scale. The amplitude of the speed fluctuations, as indicated by the standard deviation, decreases considerably with increasing distance from the Sun. The skewness and kurtosis increase with increasing distance from the Sun for lags from one to several days.

11 BURLAGA ET AL.: LARGE-SCALE SPEED FLUCTUATIONS BETWEEN 1 AND 60 AU SSH Figure 7. Moments of the probability distribution functions for Voyager 2 speed fluctuations and corresponding predictions of the model with ACE speed fluctuations as input. (a c) The standard deviation, skewness, and kurtosis, respectively.

12 SSH BURLAGA ET AL.: LARGE-SCALE SPEED FLUCTUATIONS BETWEEN 1 AND 60 AU variable changes. Thus the SD(n) for the projected speed variations and for the speed variations observed by V2 agrees quantitatively within the uncertainties, except for A2 which suggests a small systematic difference. [26] The skewness S(n) observed by V2, shown by the open circles in Figure 7b, is positive at scales 16 days at both 60 and 1 AU. However, the peak in the skewness occurs at 2 days at 60 AU in contrast to 4 hours at 1 AU [Burlaga and Forman, 2002]. The skewness at scales >1 day is due to the jump-ramp structure and possibly turbulence at 60 AU. At scales greater than the solar rotation, the skewness observed by both ACE and V2 is zero, consistent with Gaussian distributions of the dvn. The model (solid squares in Figure 7b) predicts a positive skewness at scales <16 days and a skewness close to zero at scales >16 days, in agreement with the observations. [27] The kurtosis K(n) observed by V2, shown by the open circles in Figure 7c, is positive at scales <16 days and it is 0 at scales 16 days at 60 AU. At 1 AU a large kurtosis at scales <1 day at 1 AU is associated with intermittent turbulence [Burlaga and Forman, 2002]. It is not clear what causes the kurtosis observed at 60 AU at scales from 1 4 days. It is significant that the model does predict positive kurtosis close to the observed values in this range (except at 1 day, where the predicted kurtosis is larger than observed). [28] The preceding results discuss the multiscale structure of the speed fluctuations at 60 and 1 AU as well as the relationship between the two. It is natural to ask how the statistical structure of these fluctuations varies as a function of distance from the Sun, R. The complete answer to this question is beyond the scope of this paper, but we give a partial answer which offers some important insights. SD(n), computed using the model discussed in section 2, is shown for several values of R (R = 1, 5, 10, 20, 30, 40, 50, and 60 AU) in Figure 8. Figure 6a shows that SD(n) decreases by a factor of the order of 5 10 (depending on scale) between 1 and 60 AU, and Figure 8 shows that approximately half of this decrease occurs between 1 and 5 AU; in this region the pickup proton pressure is negligible compared with the solar wind thermal and magnetic pressures. Between 5 and 60 AU, SD decreases much more slowly with increasing R. Only a small fraction of the decrease in SD with R occurs between 30 and 60 AU, where pickup protons contribute significantly to the internal pressure of the solar wind. We conclude that pickup protons are not the dominant factor in the evolution of the speed fluctuations between 1 and 60 AU. The evolution of the speed fluctuations at all scales (as described by the SD) is rapid within 5 AU (where the streams are eroded primarily by the expansion of the interaction regions ahead of them) and slower at larger distances (where interactions among streams, interaction regions, and shocks are dominant and change the qualitative structure of the solar wind [Burlaga, 1995]). [29] A peak in the SD at n = 6 (64 days) was observed by ACE at 1 AU (Figure 6a), and a peak in SD at n = 6 was predicted by the model between 5 and 60 AU, possibly tending to move toward n = 7 beyond 20 AU (Figure 8). The peak is most pronounced at 10 AU, where merged interaction regions and relatively strong shocks tend to form and strongly erode streams [Burlaga et al., 1985]. Voyager 2 observed a peak in SD at n = 7. Since the corresponding peak was observed by ACE, its origin was in the input conditions; the model shows its growth and decay with increasing distance from the Sun, not its formation. It is likely that the source of this peak is in the relatively broad and fast streams observed during the second half of 1999, as we discussed in reference to Figure 1. The fast flows observed by ACE between approximately days 225 and 300 evolved into a part of a global merged interaction, a quasi-spherical shell with strong magnetic fields and high densities, that produced a major step-decrease in the cosmic ray intensity [Burlaga et al., 2002]. 6. Power Spectra [30] A standard method of describing fluctuations is by means of power spectra of a time series. Power spectra were introduced in solar wind studies to identify turbulence in the solar wind [Coleman, 1968] and to identify the effect of discontinuities in the interplanetary magnetic fields [Sari and Ness, 1969]. Power spectra have been used to describe speed fluctuations throughout the solar wind, from near the Sun to the distant heliosphere [see, e.g., Burlaga and Mish, 1987; Burlaga et al., 1987]. Power spectra place a constraint on models, and they allow us to compare the Voyager 2 observations with many other observations of speed fluctuations. However, one must bear in mind that power spectra correspond to the analysis of variance, and they do not describe the fluctuations completely. A more complete description of the fluctuations is provided by the PDFs described above and by the multifractal spectra [Mandelbrot, 1972, 1989; Meneveau and Sreenivasan, 1987; Paladin and Vulpiani, 1987; Tel, 1988]. The latter approach was used to describe the solar wind [see Burlaga, 1995; Burlaga and Forman, 2002, and references therein], but it too is an incomplete description of the solar wind fluctuations. [31] Figure 9a shows the power spectra of the speed fluctuations observed by Voyager 2 from mid-1999 to mid-2000, and Figure 9b shows the projected speed fluctuations computed from the corresponding speed profile observed by ACE during The scales of the two panels are the same. The spectra were computed using the first 256 days of each interval, since the algorithm that was used is based on intervals equal to powers of 2. One can see that the two spectra are similar; both panels show linear behavior on a log-log scale, indicating power law behavior. The slope of the power spectrum for the Voyager 2 observations is 2.3 ± 0.2, which is consistent with the slope of the power spectra for the projected speed profile, 2.7 ± 0.2 within the relatively large uncertainties. Such slopes are characteristic of jump-ramp structure in time series [Burlaga et al., 1989]. If intermittent turbulence is present in the V2 observations, its power level is less than that contributed by the jumpramp structure. Since the spectra in Figure 9 were computed from daily averages, it is possible that the spectrum of a shock dominated structure with a spectral slope of 2.0 would resemble that of a jump-ramp structure with a steeper slope, as a result of the lower resolution associated with the averaging. Spectra with exponents of 2 have been observed at 5 AU by Burlaga and Mish [1987] and at 15 AU by Burlaga et al. [1987]. The observed and predicted power law behavior in Figure 9 extends from

13 BURLAGA ET AL.: LARGE-SCALE SPEED FLUCTUATIONS BETWEEN 1 AND 60 AU SSH Figure 8. The standard deviation of the speed fluctuations as a function of scale from 1 to 60 AU. The triangles show the ACE observations at 1 AU. The open circles showed Voyager 2 observations at 60 AU. The remaining curves show the predictions of the model at intermediate distances; the predictions for 60 AU are shown by the asterisks. The standard deviation diminishes most rapidly in the region between 1 AU and 5 AU, where the effect of pickup protons is negligible.

14 SSH BURLAGA ET AL.: LARGE-SCALE SPEED FLUCTUATIONS BETWEEN 1 AND 60 AU Figure 9. (a) The power spectral density of the speed profile measured by Voyager 2 as a function of frequency. (b) The power spectral density of the projected speed profile computed using a model (see text) with ACE observations as input.

15 BURLAGA ET AL.: LARGE-SCALE SPEED FLUCTUATIONS BETWEEN 1 AND 60 AU SSH one day to 26 days, and the level of the power at 26 days is comparable in both cases. 7. Summary [32] We have examined the multiscale statistical state of speed fluctuations observed by Voyager 2 near 60 AU on scales from 1 day to 1 year. Fluctuations occur on scales smaller than one day, but we neglect their role in the dynamics of the solar wind to first approximation. The statistical structure of the large-scale fluctuations is described by the PDFs of speed differences on scales of 1 day, 2 days, 4 days, and so on, up to a scale of 256 days. At the smallest scales considered the speed distribution resembles that of turbulence in the inertial range with an exponential structure for positive and negative dvs, but it is largely due to the jump-ramp structure. At scales of the order of several days the speed distribution is cubic on a semilog scale. And at scales greater than 26 days, the solar rotation period, the speed distribution is Gaussian, i.e., quadratic on a semilog scale. [33] Three functions, namely the standard deviation, skewness, and kurtosis as a function of scale, provide an approximate but simple description of the multiscale statistical state of the speed fluctuations. The standard deviation of the speed differences increases sigmoidally with increasing scale. The kurtosis decreases sigmoidally with increasing scale, reaching a value of 3 at a scale of 16 days, consistent with a Gaussian distribution of speed differences at large scales. The skewness also decreases with increasing scale, reaching the value of zero at 16 days, also consistent with a Gaussian distribution. Thus at scales <16 days, the PDFs of the speed differences have a positive skewness and a kurtosis >3. [34] In general, deterministic models can be used to project conditions observed near 1 AU to the position of another spacecraft only when the spacecraft are radially aligned. However, it is impossible for two spacecraft at significantly different radial distances to be radially aligned for a time of the order of 1 year. Moreover, it is impossible at present to determine the input conditions on the scale of 1 year on a surface near the Sun either from solar observations or from spacecraft observations. However, we have shown that the multiscale statistical state of the speed fluctuations on various scales observed by Voyager 2 near 60 AU can be predicted from the ACE observations of the plasma and magnetic field, using a deterministic model. We used the model of Chi Wang, which is a one-dimensional, multifluid, spherically symmetric, MHD model that includes pickup protons and the neutral interstellar gas. [35] The speed profile predicted near 60 AU by the model, from the ACE observations at 1 AU, is qualitatively similar to the velocity profile observed by Voyager 2. The statistical structure of the projected speed profile is quantitatively the same as that of the speed profile observed by Voyager 2 to good approximation. At small scales the distribution of the speed differences is exponential to first approximation for positive and negative dvs. At a scale of the order of 8 days the distribution of speed differences is cubic on a semilog scale. And at scales 26 days, the distribution of speed differences is Gaussian, or quadratic on a semilog scale. The predicted kurtosis and skewness are relatively large at scales 16 days and decrease sigmoidally toward larger scales. The skewness reaches zero and the kurtosis goes to 3 at the scales 16 days, suggesting that the jump-ramp structure is not dominant on these scales. In addition, we found that the power spectra of the velocity fluctuations observed by Voyager 2 are similar to the power spectra of the projected velocity fluctuations computed from ACE using the spherically symmetric model. Thus we demonstrated that a deterministic model using the observations from the spacecraft at 1 AU can explain the basic features of the multiscale statistical structure of the speed fluctuations in the distant heliosphere near 60 AU. [36] These results and approach of this study are significant for understanding the structure and dynamics of the heliosphere, and they should be extended to other epochs and variables. The results provide a new approach for interpreting the modulation of cosmic rays, the acceleration and propagation of energetic particles, the motions of the termination shock associated with fluctuations in the solar wind, and the interaction of the solar wind with interstellar neutrals. The multiscale statistical state approach discussed in this paper should be applicable to astrophysical problems such as the structure of stellar winds, planetary nebulae, supernovae, and astrophysical jets. [37] Ultimately, one would like to have a statistical physics model of the heliospheric structure that is based on statistical state variables themselves. One can imagine one class of multiscale models that describe the evolution of the PDFs. Alternatively, one might consider a class of multiscale models that describe only the lowest order moments: the average, standard deviation, skewness, and kurtosis of the physical fields as a function of scale, position, and time. Statistical models and descriptions are complementary to deterministic models and descriptions; they are not mutually exclusive. Both are needed in order to fully understand the solar wind. [38] Acknowledgments. 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