Magnetic field data from HELIOS 1 and 2 are used to test a stochastic model for Alfvenic fluctuations recently proposed. A reasonable matching between observations and predictions is found. A rough estimate of the correlation length of the observed fluctuations is inferred

Bavassano, B.

Mariani, F.

English

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INTERPLANETARY ALFVENIC FLUCTUATIONS: A STATISTICAL STUDY OF THE
DIRECTIONAL VARIATIONS OF THE MAGNETIC FIELD
1 1,2
B.Bavassano and F.Mariani
i) Istituto di Fisica dello Spazio Interplanetario, CNR, Frascati, Italy
2) Istituto di Fisica "G. Marconi", Universit_ di Roma, Roma, Italy
ABSTRACT
Magnetic field data from HELIOS 1 and 2 are used to test a stochastic
model for Alfv&nic fluctuations recently proposed by A. Barnes. A reasonable
matching between observations and predictions is found. A rough estimate of the
correlation length of the observed fluctuations is inferred.
Introduction
In a recent paper Barnes (1981) proposes a simple stochastic model for
interplanetary Alfv_nic turbulence based on the random wandering of a vector of
constant length. Let us locate this vector with the usual spherical coordinates
e, ¢ and assume it initially along the polar axis (e=o). As time advances
the tip of the vector walks over a sphere, moving at each step of an angle
e along a great circle in a random direction. In other words if the position of
the vector after N steps is (e, ¢), at the (N+l)th step the vector will lie
at an angle g away along a great circle, whose tangent at (e, ¢) makes a
random angle _ with the unit vector in the e direction. A field direction
confinement can be simulated with reflecting boundaries. This model, with
certain qualifications, reproduces the well known observational feature of the
Alfv&nic fluctuations of a well defined direction of minimum variance aligned
with the mean magnetic field. The model predicts also the statistical distribu-
tions of the field directions in terms of a density of probability W(p,t)
and an integrated probability P(p,t), being p=cose. W(p,t)dp gives the probabi-
lity that after a time t the angle e that the vector makes with its initial
position has its cosine between p and p+dp. P(p,t), integral of W between -i
and p, gives the probability that, at the time t, cose lies between -i
and p. Figures 4 and 5 of Barnes (1981) show the evolution with time of these
functions.
An observational test of the temporal evolution of the fluctuating field
direction as predicted by the model has been performed by using interplanetary
magnetic field data (6 s averages) of HELIOS 1 and 2 (Rome/GSFC experiment). We
have selected four periods of 2-4 days in the trailing edge of high velocity
streams characterized by the presence of fluctuations whose Alfv&nic character
had been already established by previous analyses (Bavassano et al., 1982a and
b). Table 1 of the latter reference gives a list of the periods considered.
Three of them refer to HELIOS 2 observations of the same stream during three
successive solar rotations. The fourth period refers to an observation of the
same stream by HELIOS 1 almost contemporary to the second of the HELIOS 2
stream encounters. To obtain a statistical description of the temporal evolu-
99
https://ntrs.nasa.gov/search.jsp?R=19840005008 2020-03-22T07:41:04+00:00Z
tion of the direction of the measured fluctuating field we have divided the
selected periods in intervals of 20 minutes, for each interval we have deter-
mined how the magnetic field vector evolves (starting from its initial position
at the beginning of the interval) as time advances, and finally for each of the
four stream encounters we have grouped the results from the various intervals
to have statlstical distributions for the parameters characterizing, following
the Barnes model, the field evolution with time. The parameters considered are
c, the angle between consecutive vectors (6 s averages), @, the direction
of the field variation as defined in the model, and 0, the angle that the
magnetic vector has with its initial position at the beginning of the interval.
Wewould note that our results do not change significantly if intervals longer
than 20 minutes are taken for the statistics. Finally, all intervals including
discontinuities as identified through automatic selection criteria (Mariani et
al., 1983) and visual inspection of the data have been rejected.
Experimental Results
In Figure 1 we show the histograms of the angles e and _ at the four
heliocentric distances considered. The 20 minutes intervals contributing to the
statistics are (in order of decreasing R) Iii, ii0, 104 and 235 respectively.
The histograms of c show that most of the values fall between 0° and 15° • A
dependence on heliocentric distance is clearly apparent, the average value of
being 6.1 ° at 0.87 AU and becoming 11.0 ° at 0.29 AU. These histograms should
be comparedwith the fixed step length assumption in the model. The histograms
of _ do not show on the contrary any dependenceon the heliocentric distance.
In Figure 1 we give only the curve for 0.87 AU, representative within i% for
all distances. To a good approximation @has a uniform distribution, in
agreement with the assumption of the model that the direction of each step is
random.
I
I I
I
I
I0
,[
L
It" ,r _"
(
I K_)
-- IZ I.|/
_- IZ 0.66
-- III ,&l
-- IOZ L n
Figure i. Frequency
distributions of e and _.
F H[LIOS Z
"1- 11-0.81 AU li-0.65 AU
I
i ,_1""
N-
I-
t u I _
11-029 AU •c (rain)
-- I
__ ¢
--- u
M
_ Cos O
Figure 2. Frequency distributions of cose.
Let us consider now the evolution with time of the magnetic field
i00
direction in terms of the angle e between the current vector and its initial
position at the beginning of the 20 minutes interval. In Figure 2 we show the
frequency distributions of cose for different delay times T (time elap-
sed from the beginning of each interval). These histograms correspond to the
curves of the function W of the model of Barnes (see his Figures 4 and 5),
taking care that W is a probability density. The features of the temporal
evolution of cose are perhaps more clearly seen in Figure 3 in terms of the
integrated frequency distributions, giving the number of cases for which cose
is between - 1 and a given value. These curves should be comparedwith those of
the function P given by Barnes in his Figures 4 and 5. One is easily convinced
that the experimental curves look like those of the model. Whenthe delay time
T is small (e.g., T=I min) most of the values of cose are close to I. For
increasing • the distributions spread out in a quite regular way. The final
state is not very different at the various distances but it is reached more
rapidly near the Sun. This is clearly seen in Figure 3 where at 0.29 AU the
curve for _=i0 min is almost coincident with that for _=20 min (i.e., final
state is already reached after about i0 minutes), whereas at 0.87 AU they
differ noticeably (i.e., after i0 minutes some evolution is still necessary
before reaching final situation). This different time scale can be related to
the radial gradient of the angle c (see Figure i), the evolution being more
rapid near the Sun where the angle between consecutive vectors is larger.
Another experimental evidence is that our curves do not reach a uniform distri-
bution, as those of the model, and this holds also if • increases up to 1 hour.
S
10-
N
|
M-
HELLOS Z
II-O,17 AIJ - . • • ( "
n_.)
/
z
i "
L
I i
.........................................
1 is o i u e _ o5 e cos (}
Figure 3. Integrated fre-
quency distributions of
cose.
The evolution of the frequency distributions of cose has been also seen
in terms of variation of their moments. Figures 4 and 5 show the variation with
the delay time x of the average value of cose, <cose>, and of its standard
deviation, a(cose), respectively. The changes in the histograms of cose
shown in Figure 2 are here seen as a regular decrease of <cos8> and a
corresponding increase of _(cose). Again it is clearly apparent that the varia-
tion is more rapid near the Sun (heavy line). These trends almost completely
i01
disappear after 10-15 minutes, with <cose> remaining around 0.7 and
e(cose) around 0.3 for delay times up to 1 hour. In other words most of the
variation of <cose>and a(cos8) takes place in the first minutes (_ i0 near
the Sun and _15 near the Earth), after that only a very slight dependencefrom
is found. Wewould notice that the observations made at 0.65 AU by HELIOS2
and at 0.41 AUby HELIOS1 are almost contemporary and at the sameheliographic
latitude. This allows us to exclude that the observed trends are due to slow
temporal variations or heliographic latitude dependence.
U
14 u-
IlAtl)
-- II U;
-- ,I I1_
.... II Ul
-- llt i_
IS .... , ............
: (rain)
Figure 4. <cosO> vs. _.
U
®
io
I1
U
/ - <.
! (AU)
-- |1 o|;
-u n! ,05
- i1 L¢t
S l! Is zl
•((rain)
Figure 5. _(cose) vs. _.
Concluding Remarks
As regards the observational test of the Barnes model we can summarize the
results as follows: a) the direction of the variation of the magnetic field, as
given by the angle #, has a uniform distribution, in agreement with the
assumption of the model that the direction of each step is random; b) the angle
between consecutive vectors (6 s averages) is of the order of 5°-10 ° and
increases near the Sun (in the model this angle is taken as a constant); c) the
frequency distribution of cos8 spreads out when • increases but does not
reach a uniform distribution (as in the model). In conclusion the model of
Barnes, although essentially local, can be considered in reasonable agreement
with the experimental results. On the other hand, as already indicated by the
author himself, the model can be considerably improved, for example by using a
variable step length and by simulating the confinement of the field around the
spiral direction. In this way a better matching with the observations could be
obtained.
The time scale of the variation of cose can give some information about
the correlation time of the fluctuations. Barnes (1981) shows that, in terms of
population statistics, the autocorrelation function of the temporal series of
random fluctuating vectors is just given by the average value of cose.
His computations show also that, with a reflecting boundary, the average value
102
of cose initially decreases as time advances but beyond the correlation time
tends towards a constant value. Figure 4 shows that the decrease of <cosO>
lasts I0-15 minutes, the evolution being faster near the Sun, after that this
trend almost completely disappears. We can then infer that the correlation time
of the observed fluctuations is of the order of i0 minutes near the Sun and 15
minutes near the Earth. With the knowledge of the average solar wind speed in
the different periods this gives a correlation length of _65 RE near the
Sun and _90 RE near the Earth. Our estimate for the correlation length at 0.87
AU agrees with previous determinations near the Earth's orbit (Chang and
Nishida, 1973; Fisk and Sari, 1973; Sari and Valley, 1976; Crooker et al.,
1982). The decrease of the correlation length approaching the Sun should be
essentially related to the greater angular variability of the magnetic field
near the Sun.
Acknowledgements. Use of hourly averages of solar wind data from the
HELIOS 1 and 2 plasma experiment (H. Rosenbauer and R. Schwenn) is acknowled-
ged. We thank Mrs. A.M. Cialoni for typing the manuscript.
REFERENCES
Barnes, A., Interplanetary Alfv&nic fluctuations: a stochastic model, J. Geo-
phys. Res., 86, 7498, 1981.
Bavassano, B., M. Dobrowolny, G. Fanfoni, F. Mariani, and N. F. Ness, Statisti-
cal properties of MHD fluctuations associated with high speed streams from
HELIOS 2 observations, Solar Phys., 78, 373, 1982a.
Bavassano, B., M. Dobrowolny, F. Mariani, and N. F. Ness, Radial evolution of
power spectra of interplanetary Alfv&nic turbulence, J. Geophys. Res., 87,
3617, 1982b.
Chang, S. C., and A. Nishida, Spatial structure of transverse oscillations in
the interplanetary magnetic field, Astrophys. Space Sci., 23, 301, 1973.
Crooker, N. U., G. L. Siscoe, C. T. Russell, and E. J. Smith, Factors
controlling degree of correlation between ISEE 1 and ISEE 3 interplanetary
magnetic field measurements, J. Geophys. Res., 87, 2224, 1982.
Fisk, L. A., and J. W. Sari, Correlation length for interplanetary magnetic
field fluctuations, J. Geophys. Res., 78, 6729, 1973.
Mariani, F., B. Bavassano, and U. Villante, A statistical study of MHD
discontinuities in the inner solar system: HELIOS 1 and 2, Solar Phys., in
press, 1983.
Sari, J. W., and G. C. Valley, Interplanetary magnetic field power spectra:
mean field radial or perpendicular to radial, J. Geophys. Res., 81, 5489,
1976.
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Interplanetary Alfvenic fluctuations: A statistical study of the directional variations of the magnetic field

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