Exploring the Castaing Distribution Function to Study Intermittence in the Solar Wind at L1 in June 2000 Miriam A. Forman1 and Leonard F. Burlaga2 1 Department of Physics and Astronomy, SUNY/Stony Brook, NY 11794-3800 USA 2 Code 692, NASA/Goddard Space Flight Center, Greenbelt MD 20771, USA Abstract. We considered 31,561 consecutive 64-second values of radial solar wind speed reported by the SWEPAM instrument (D. McComas, Los Alamos, Principal Investigator) on the ACE spacecraft at L1 upstream of the earth's bow shock beginning day 157 of the year 2000. Running values, moments and probability density functions (pdfs) were calculated for the speed differences over a range of lags from 64 seconds to several days. Running values show local intermittency in their amplitudes, and correlate with local solar wind speed. Moments of order greater than 6 are dominated by the largest values, which increase slowly with lag in the inertial range causing the exponent of the structure function at large q to be a straight line. The pdfs are compared to “Castaing distributions” which are superpositions of Gaussians whose standard deviations are log-normally distributed. Although the Castaing distribution does not and in principle cannot precisely fit actual pdfs of velocity increments in the solar wind, it looks good and provides a basis for a handy two-parameter description of the pdfs. The run of those two parameters with scale provides a further handy description of the intermittency in the cascade that is independent of any particular model of the cascade. Proving the disability of the Castaing distribution, the third moment of the longitudinal velocity increments does exist and it scales consistent with Kolmorogov’s 4/5 law that is a requirement for all theories of the inertial cascade. than the pdf at any particular scale. See Fig. 6. INTRODUCTION The solar wind is turbulent on a wide range of scales. Furthermore, its turbulence is intermittent: the local amplitude of the fluctuations fluctuates irregularly from time to time and place to place. Figure 1 illustrates this fluctuation in the amplitude of fluctuations. Some, but not all, is related to the local average solar wind speed. Power spectra do not reveal intermittence. Apparently, intermittence occurs because the cascade of turbulent amplitudes from large to small scales distributes the energy non-uniformly in space at smaller scales (see all of the references). Intermittence is usually studied with statistics of the increments ∆VL(t) ≡ V(t+L)-V(t) such as those shown in Figure 1. The set of moments M(q,L)≡<(∆VL(t))q> is called the structure function. It measures the shape of the probability distribution functions (pdf), such as shown in Figure 2, of ∆VL(t) at different L Its exponent ζ(q) ≡ ∂log(M(q,L))/∂log(L) which is independent of L where scaling occurs [1,2,3, and references therein], measures the evolution of the pdf with scale rather FIGURE 1. Data used in this study. (1) Values of solar wind radial component measured by the SWEPAM instrument on ACE at L1 every 64 seconds. (2) ∆VL ≡ V(t + L)-V(t) for lag, L = 64 seconds, shifted up by 300 km/sec for clarity. (3) as (2), for lag = 1024 seconds, shifted 200 km/sec. (4) as (2) for lag = 16384 seconds, shifted 100 km/sec. (5) as (2) for lag = 262144 seconds (72.8 hours), unshifted. CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference, edited by M. Velli, R. Bruno, and F. Malara © 2003 American Institute of Physics 0-7354-0148-9/03/$20.00 554 Various conceptions of the uneven (intermittent) nature of the cascade of turbulent energy yield theoretical predictions for the detailed shape of ζ(q) [4,5,6,7,8,9,10,11]. However, the finiteness of data sets makes measured moments at large q depend mostly on the single largest data point (e.g. Figure 4), so ζ(q) generally show a straight line for large enough q (e.g., Figure 6). In some sense this scaling relates to the turbulence physics [10], but not in the same way as theories of ζ(q) for a continuous, infinite data set. Figure 4 shows that ζ(q) at q>6 should not be used to compare this data set of 31,561 values with theories for infinite data sets, that is, with any theory except possibly [10]. 1.E+25 1E+25 moment q, (km/sec)^q 1E+20 moment q, (km/sec)^q 1024-sec lag 64-second lag 1E+15 1E+10 100000 1 1.E+20 1.E+15 1.E+10 1.E+05 1.E+00 0 2 4 6 8 10 0 2 moment number, q 4 6 8 moment number, q FIGURE 4. Moments of the distribution of velocity differences. Large filled circles: moments calculated from data in Figure 1. Full line: Castaing model moments, fitted to second and fourth moment of this data. Fits are very good from q=1 to q = 5. Dotted line: moments calculated using only the single largest value (straight line). Dot-dashed line: moments of a Gaussian with the same standard deviation. 1 5 scaled 64-second pdf (dimensionless number) scaled 1024 sec pdf 0.8 scaled 16384 sec pdf scaled 262144 sec pdf 0.6 Gaussian 0.4 0.2 ln(sigma zero) 4 ln(sigma) 3 lambda^2 2 1 2 1 3 0 0 -3 -2 -1 0 1 2 0 3 FIGURE 2. Probability density functions (pdfs) of the velocity increments in figure 1. 10 15 FIGURE 5. Castaing parameters calculated from the second and fourth moments of velocity differences in the entire data set in fig. 1, using equations 2 and 3. The increase of λ2 towards smaller scales describes the systematic deviation from a Gaussian shown in figure 2. When portions of these curves are straight lines, the Castaing model implies that the scaling of the structure function is lognormal, with quadratic coefficient given by 0.5dλ2/dln(scale) ≈ 0.011 in the inertial range. Strictly speaking, if the fit to Castaing pdf is very good, and plots in this figure are straight lines, the scaling is log-normal in that range of q and of scales. 1 P(x), normalized to unity 5 ln(lag time) del V/standard deviation at that lag 0.1 0.01 0.001 histogram of del V for 64-second lag Castaing with same variance and kurtosis 0.0001 -12 -6 0 6 DATA SET AND PRELIMINARY ANALYSIS 12 x = delta V/sigma zero Level 2 data from the SWEPAM instrument on the ACE spacecraft are provided on the ACE website at <http://www.srl.caltech.edu/ACE/ASC/level2/lvl2DA TA_SWEPAM.html>. We took as our data set, 64second values of the radial component of the ion speed beginning at 00:00:09 on day 157 of the year 2000, through day 180 at 08:59:49. We ended the data series then because of a data gap of over 10 minutes. We filled the few gaps of mostly one or two data points by FIGURE 3. Fit of a “Castaing pdf” to the 64-second increments, curve 2 in figure 1 and the most peaked pdf in figure 2. Although the fit looks good to the eye, note that the data falls short of the Castaing pdf at very small and very large velocity increments, as if the smallest and largest values of σ are missing from Equation (2). A closer look, using the cumulative distribution, shows the pdf is slightly skewed to positive increments (as it must be in the inertial range to satisfy Kolmogorov’s 4/5 law [3]) and is a powerlaw at large increments. 555 10 linear interpolation. We had then a set of 31,561 consecutive values of radial solar wind speed. (Fig. 1) RESULTS Figure 3 shows a pdf and Cpdf for our data. Figure 4 shows the extent of the fit of equation (3) to data in the inertial range, and figure 5 shows how the Castaing parameters vary with scale. In figure 5, λ2 describes the deviation of the pdf at each scale from Gaussian; it is very large in the inertial range below an hour or so, but still increasing slowly with decreasing scale. The large value of λ2 denotes how intermittent the fluctuation amplitudes in Figure 1 are, and how big the wings on the pdfs in figures 2 and 3 are. The steady change in λ2 with scale indicates that the turbulence cascades intermittently in that range. That slope, = d(λ2)/d(lnL) = ζ(2)/2 - ζ(4)/4, is a simple robust measure of local intermittency in the cascade process independent of any detailed model, or the size of a (reasonably large) data set and even independent of the assumption that the Cpdf is a reasonable representation of observations. In this analysis, we make no distinction between episodes of fast and slow solar wind, or any other macro or meso-scale attributes of the data, but treat the whole set and look for its statistical properties at different scales. We formed sets of running differences at time t, and different lags, L: ∆VL(t,L) = V(t+L)-V(t). Defined this way, -∆VL is the longitudinal velocity increment which is the subject of Kolmogorov’s 4/5 law [3], in the inertial range. We took lags from one data point (31,560 64-second increments) to 214 data points. Figure 1 also shows the running values of four of the 15 data sets so formed. CASTAING PDF BASICS Sorriso-Valvo, et al [12] first used Castaing distributions [13,14] for the solar wind. Pdfs of ∆VL in the inertial range are highly kurtotic, and look like the Castaing distribution (Figs. 2 and 3). The Castaing pdf (Cpdf) is a convolution of a parent Gaussian of width σ, with a lognormal distribution of σ, whose width is λ. The parent Gaussian may be thought of as pertaining to very small sub-sets of the data, or as the distribution of ∆V at large scale. The Cpdf is given by ∫ (∆V )2 e 2σ 2 2 πσ (ln σ − ln σ )2 0 − 2 λ 2 e 2 πλ d ln σ . (1) The odd moments of the Cpdf are identically zero because the Cpdf is symmetric. The moments of the absolute values are 1E+36 2.5 14 1E+32 q 2 λ2 = a (q )e 2 12 1E+28 (σ 0 ) q M(q,Lag), (Km/sec)^q M(q , L ) ≡ ∆V q (2) where a(q) is the qth moment of a Gaussian of unit width. We evaluated the Castaing parameters σ0 and λ2 from Eq. 2, using the second and the fourth moment of the velocity increments at each scale, without going through a tedious curve-fitting process. 2 M ( 4 , L ) 2 . Once λ and σ0 are calculated, 1E+24 10 1E+20 8 1E+16 6 1E+12 4 3 1E+08 2 10000 1 1 10 10,000 lag, seconds λ = 0.25 ln 3[M(2, L)]2 10,000,000 exponent zeta(q) of the structure function P(∆V ; σ 0 , λ ) = − Kolmogorov’s “4/5” law [3] states that the third moment of the signed values of the longitudinal velocity increment is not zero, but = -0.8*E*L, where E in the energy dissipation rate per unit mass, and L is the scale, in the inertial range. This is why all theories of inertial turbulence predict ζ(3)=1. Figure 7 shows our data set fits this relation, with E about 25,000 Joules per second per kilogram. Such results are impossible if the pdf were an intrinsically symmetric Cpdf. In addition, the actual pdf lack both small and large values of σ compared to the lognormal distribution in the Cpdf that fits most of the pdf (Figs. 3 and 4). While this appears mostly due to the finite resolution and length of the data set, self-organized criticality for extreme increments may play a role [10]. 2 1.5 1 min to 1 hour 1 hour to 1 day 1 0.5 0 0 5 10 15 moment number, q FIGURE 6. Left panel: Structure function of the radial component of the solar wind (this data set). Right panel: Exponent of this structure function, for the inertial range (1 min to 1 hour) and for the interaction range. we use Equation (1) to evaluate the whole Cpdf. 556 We looked at the fastest and lowest-speed epochs, each about 36 hours long, in our data. σ and σ0 were about twice as large in the fast epoch, consistent with Fig. 7. The λ2 in the inertial range is about 0.3 in each, also consistent with Figure 7. The high-speed episodes apparent in Figures 1 and 7 are much less intermittent than the recurrent “fast wind” analyzed by Liu and Marsch [15]. This is probably because June 2000 is practically solar maximum, when classic fast wind is rare. longitudinal velocity increments does exist and it scales consistent with Kolmorogov’s 4/5 law that is a requirement for all theories of the inertial cascade [3]. ACKNOWLEDGMENTS We are very grateful to NASA, to the ACE project and SWEPAM experimenters for the beautiful data of the SWEPAM instrument available on the ACE website. We are also grateful to a careful referee. MAF is supported by NASA grant NAG58106. 4 ln (local std dev) 3 REFERENCES 2 1. Burlaga, L.F., J. Geophys. Res., 96, 5847 (1991) 2. Burlaga, L.F., Interplanetary Magnetohydrodynamics, New York: Oxford University Press, 1995 3. Frisch, U., Turbulence, The Legacy of A. N. Kolmogorov, Cambridge: Cambridge University Press, 1995 4. Kolmogorov, A. N., Dokl. Akad. Nauk, SSSR, 30, 299 (1941) 5. Kolmogorov, A.N., J. Fluid Mech., 13, 82 (1962) 6. Obukhov, A.M., J. Fluid Mech., 13, 77 (1962) 7. C. Meneveau and K.R. Sreenivasan, Phys. Rev. Lett., 59, 1424 (1987) 8. Z–S. She and E. Leveque, Phys. Rev. Lett, 72, 336 (1994) 9. B. Dubrulle, Phys. Rev. Lett, 73, 959 (1994) 1 0 -1 5.5 6 6.5 7 ln(solar wind speed, km/sec) Figure 7. The running local standard deviation, σ, of speed increments (in this case 16 64-second increments, 16 points on curve 2 in figure 1) versus the solar wind speed averaged over the same interval of 1024 seconds. Although σ tends to increase with solar wind speed, the (fairly constant) intrinsic spread in lnσ at each speed due to local intermittency is about as large as the variation between the slowest and fastest solar wind in this data set. Both effects contribute to the apparent intermittency in the 24-day data set. Overall, the pdf of the lnσ in this figure is very like a truncated Gaussian, consistent with the deviations from a true Castaing pdf apparent in Figures 3 and 4. Horizontal trails of points at large σ in this figure are due to shocks. 10. Shertzer, D., S. Lovejoy, F. Schmitt, Y. Chigirinskya, and D. Marsan, Fractals, 5, 427 (1997) SUMMARY AND CONCLUSIONS 11. Forman, M.A., Castaing Scaling and Kernels for Theories of the Turbulent Cascade, in Solar Wind Ten, edited by M. Velli et al., AIP Conference Proceedings, Woodbury, NY: American Institute of Physics (this volume) On the basis of a statistical study of 24 days of solar wind radial velocities at a cadence of 64 seconds at L1 in June 2000, we find: 12. Sorriso-Valvo, L., V.Carbone, P.Veltri, G.Consolini, and R. Bruno, Geophys. Res. Lett. 26, 1801 (1999) Although the Castaing distribution does not and in principle cannot precisely fit actual pdfs of velocity increments in the solar wind, it looks good and provides a basis for a handy two-parameter description of the pdfs. The run of those two parameters with scale provides a further handy description of the intermittency in the cascade that is independent of any particular model of the cascade. Proving the disability of the Castaing distribution, the third moment of the 13. Castaing, B., Y. Gagne, and E.J. Hopfinger, Physica D 46, 177-200 (1990) 14. Chabaud, B., A. Naert, J. Pienke, F. Chilla, B. Castaing, and B. Hebral, Phys. Rev. Lett, 73, 3227-3230 (1994) 15. Marsch, E., and S. Liu, Annales Geophysicae, 11, 227238 (1993) 557
© Copyright 2024 Paperzz