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.
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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
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1
0
-1
5.5
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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
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SUMMARY AND CONCLUSIONS
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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
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