Magnetic Turbulence, Fast Magnetic Field line Diffusion and
Small Magnetic Structures in the Solar Wind
G. Zimbardo , P. Pommoisy and P. Veltri
y Center
Dipartimento di fisica, Università della Calabria (Italy)
for High Performance Computing, Università della Calabria (Italy).
Abstract. The influence of magnetic turbulence on magnetic field line diffusion has been known since the early days of space
and plasma physics. However, the importance of “stochastic diffusion" for energetic particles has been challenged on the basis
of the fact that sharp gradients of either energetic particles or ion composition are often observed in the solar wind. Here we
show that fast transverse field line and particle diffusion can coexist with small magnetic structures, sharp gradients, and with
long lived magnetic flux tubes. We show, by means of a numerical realization of three dimensional magnetic turbulence and by
use of the concepts of deterministic chaos and turbulent transport, that turbulent diffusion is different from Gaussian diffusion,
and that transport can be inhomogeneous even if turbulence homogeneously fills the heliosphere. Several diagnostics of field
line transport and flux tube evolution are shown, and the size of small magnetic structures in the solar wind, like gradient
scales and flux tube thickness, are estimated and compared to the observations.
INTRODUCTION
TURBULENT VERSUS GAUSSIAN
DIFFUSION
The magnetic turbulence found in many plasmas causes
a magnetic field line random walk which “destroys" the
magnetic surfaces [1, 2, 3, 4, 5, 6], and causes a fast
plasma transport across the magnetic field structure, a
phenomenon which is sometimes called stochastic diffusion. Recently it was shown that magnetic field line diffusion gives a natural explanation for the Ulysses observations of energetic particles at high southern heliographic
latitudes [7]. Nevertheless, the relevance of stochastic
diffusion in space plasmas has been challenged on the basis of the fact that sharp gradients of particular ion composition are often observed in space (see e.g. Ref. [8]).
Also, sharp intensity variations in impulsive energetic
particle events seen by the SWICS instrument on ACE
may suggest that turbulent diffusion is not smoothing out
the gradients [9]. Yet, it was shown by Giacalone et al.
[10], by means of a numerical simulation, that magnetic
field line random walk is consistent with both fast diffusion and small-scale gradients in energetic particle intensity.
In this paper we argue that this is a general feature of
the transport of a passive tracer by a turbulent field, that
turbulent diffusion is different from Gaussian diffusion,
and show by a numerical simulation that fast magnetic
field line and plasma transport can be found simultaneously to small magnetic flux tube structure and field line
mixing. Moreover, we argue on the type of anisotropy
present in the solar wind turbulence.
In a normal, Gaussian diffusion process, the diffusing
particles move randomly in all directions, because of either molecular diffusivity, in an ordinary fluid, or collisions, in a plasma. Such random motion tends to smooth
out a density gradient, and a point like initial concentration of, say, blue dye, evolves with the well known
Gaussian smooth profile. Also, a Gaussian random walk
is characterised by a finite mean square step length, and
by random, uncorrelated directions of motion. In such a
case normal diffusion results, h x2i i
Di t, where Di
is the diffusion coefficient and t the time, and, in particular, the diffusing particles moves in all direction with
equal probability. This leads to the smoothing of density
or concentration gradients.
We consider here turbulent diffusion as the result of
the advection of a passive tracer by a turbulent velocity
(or magnetic) field. The transport of the passive tracer
(or of the test particles in a turbulent magnetic field) is
due to the chaotic evolution of the non linear dynamical system. Such evolution can be very complex and
“unpredictable", but still deterministic. This means that
each passive tracer does not move randomly, but follows
a flow (or field) line for long distances regardless of the
density or concentration gradients. Basically, this is what
happens when you are mixing a blue dye with a white
paint: at least for some time (until molecular diffusion
becomes important) blue stripes stand out on a white
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
409
=2
background. In a turbulent medium, long range correlations among the fluctuation occurr, and, indeed, anomalous transport regime can be found, h x 2i i
Di ti
(i x;y ), with i > (see Pommois et al. [5, 6]). The
main features of such chaotic (not random) transport can
be grasped by considering the evolution of a magnetic
flux tube in the presence of magnetic turbulence. In a uniform background field, a flux tube with an initially circular cross section will be elongated in one direction and
squeezed in the other (because of r , this is the
typical behaviour). Then the flux tube cross section will
be distorted and further on it will develop branches (see
figures by Rechester and Rosenbluth [3]; and Isichenko
[11]). A very complex and ramified structure can be obtained, with the outer size of the flux tube growing fast,
but with many “voids" in the cross section. A numerical
simulation of the flux tube evolution is reported here.
The magnetic field model is
=
1
=2
B=0
B(r) = B e^z + B(r)
0
that is an uniform background field plus zero-average
fluctuations. Here the magnetic fluctuations are constructed in a parallelepipedal box with sides Li ' li ,
where li is the correlation length in the i-th direction
(i x;y;z ). The magnetic fluctuations are represented
as
( )
B
i (k)
k;
4
=
X
B(r) =
(k)e (k)exp [k r +
r=lx
r=lx/5
]
FIGURE 1. Evolution of a magnetic flux tube cross section
in the case of quasi 2-D spectrum with lx =lz = ly =lz = 0:1
(from bottom to top). The magnetic flux tube is starting in the
circle of radius r at bottom of the figure and is traced with
5000 points. Each adjacent line is integrated, and we plotted
here the evolution for interval Z = 0:25Lz lz (where Lz is
the box simulation dimension, and lz is the correlation length
in z direction). The turbulence level is B=B0 = 0:4. Flux
tube on the left, r = 0:05Lz 0:2lz ; flux tube on the right,
r = 0:25Lz lz .
Magnetic turbulence appears to be anisotropic both in the
solar wind [12, 13] and in the interstellar medium [14].
Therefore, the Fourier intensity is given by
B (k) =
(kx
2
lx
2
+ ky
2
ly
C
2
+ kz
2
lz 2 )=4+1=2
where C is a normalization constant. Note that with this
choice of B , the Fourier amplitude depends on all
of lx , ly , lz , and that the isolevels of B
are ellipsoids
in space (see, Pommois et al. [5, 6], for more details).
Once the magnetic field model is set up, we can trace
the magnetic field structure by integrating the magnetic
field line equations d =ds
=j
j. It is usually
found that the level of stochasticity increases with the
turbulence level B=B0 . Here, B is the rms value of
fluctuations. In a similar way the level of stochasticity
increase with the ratio of correlation lengths lz =lx [6,
15, 16]. A suitable parameter to caracterize the level
of stochasticity and the transport regime was shown to
be the Kubo number R Bl z =B0 lx [15, 16]. This
expression of R was found to be valid even when l x 6 ly
[16]. Besides, the study of magnetic field line transport
carried out in [15, 16] shows that anomalous diffusion is
found for R , quasilinear diffusion, i.e. D i / B 2 , is
found for : <
R<
, and the percolation scaling of the
.
diffusion coefficient, Di / B 0:7 , is found for R >
(k)
(k)
k
r
= B(r) B(r)
=
02
1
1
NUMERICAL RESULTS
Figures 1, 2, and 3 show the evolution of a magnetic
flux tube for different realizations of the turbulent magnetic field. All of them have the same fluctuation level
B=B0
: , and different degrees of anisotropy. In
each case, we choose the starting positions of the field
lines on a path representing the boundary of a circular
flux tube cross section. Then, integrating up to 5000 adjacent field lines, we can devise the evolution of the shape
of a flux tube cross section [3, 11, 17]. We argue that the
spatial distribution of not too energetic particles injected
in a magnetic flux tube will follow the shape of the flux
tube, except for finite Larmor radius effects.
In Figure 1 a quasi-2D anisotropy is considered, with
lz
lx
ly . In such a case we have relatively large
=04
=
= 10 = 10
10
410
r=lx/5
r=lx
r=lz/5
FIGURE 2. Same Figure as Figure 1. We use here a Slab-like
spectrum with lx =lz = ly =lz = 10 for the magnetic turbulence,
and the flux tube cross section from bottom to top are plotted
at intervals Z = 2Lz (which correspond to 8Lz ). Flux tube
on the left, r = 0:05Lx lx =5; flux tube on the right, r =
0:25Lx lx .
=(
)(
FIGURE 3. Same Figure as Figure 1. We use here an
anisotropic spectrum with lx =lz = 8 and ly =lz = 1 for the magnetic turbulence, and the flux tube cross section from bottom
to top are plotted at intervals Z = 2Lz (which correspond to
8lz ). Flux tube on the left, r = 0:05Lz lz =5; flux tube on
the right, r = 0:25Lz lz .
)=4
ly = 1, and lz = 1 was
Kubo number, R
B=B0 lz =lx
, and, indeed, a
very fast evolution of the magnetic flux tube is obtained.
In the left “column”, from bottom top, the initial size of
the flux tube, that is the radius r, is much smaller than the
turbulence correlation length, r
: Lz : lz lz .
On the right side of Figure 1, the initial size of the flux
tube is comparable to the turbulence correlation length,
r : Lz lz . It can be seen that very elongated and
ramified structures quickly form. This is in agreement
with the magnetic field line evolution reported by Zank
et al. [18]. In Figure 2 a quasi-slab anisotropy is considered, with lz lx =
ly = . In such a case we have a
relatively small Kubo number, R
: , and the evolution of the flux tube is slower. Indeed, it can be seen that
the flux tube is distorted, but this process is much slower
than in the previous case. It appears that the Kubo number also quantifies the speed of ramification of the flux
tubes.
In Figure 3, an anisotropic spectrum with l x
,
= 0 05
used, as appropriate to the solar wind case (see [13, 7]). In this case the Kubo number
: . In Figure 3, the flux tube has been followed
is R
for larger distances, up to 32l z , in order to make a direct comparison with the observations in the solar wind.
Indeed, considering that at 1 AU from the Sun, the correlation length is about 0.03 AU [7], we can argue that
to travel for about 1 AU in the heliosphere (the typical
distance where most of the spacescraft take measures)
we need to cover about 30 correlation lengths l z Each
plot, from bottom to top, is spaced by a distance in the
z direction which corresponds to 8lz . At the top of the
flux tube cross section of left panel of Figure 3, we can
observe filamentation of the curves, which corresponds
to small structures in the magnetic field. If we consider
that a particular ion composition or energetic particles
are injected at the base of the magnetic flux tube, we can
understand how sharp gradients can be found in the solar
wind. Here two filaments are separed by 2–3lz , which
= 0 05
02
= 0 25
= 10 = 10
r=lz
= 0 04
=8
411
are the largest structures observable in the plot, even
though smaller structures are present. Converted in the
solar wind dimension this corresponds to structures of
about 0.06–0.09 AU and is consistent with the findings
of Giacalone et al. [10]. Clearly the smaller structures
will give rise to shorter time variations.
On the other hand, the flux tube evolution represented
in Figures 1, 2, and 3 shows that this is very sensitive
to the anisotropy of the turbulence spectrum. Different
kinds of anisotropy give rise to rather different evolution (see also [4, 18]). In particular, the details of the
flux tube fine structure can be different for different kind
of anisotropies. Hence, we argue that an accurate study
of the morphology of impulsive energetic particle events
when compared to the simulation results, can give information on the anisotropy of solar wind magnetic turbulence.
Indeed, when the Kubo number is large (like for quasi
2-D anisotropy) the evolution of the flux tube is very fast,
with the formation of fine scale structures. Consequently,
the energetic particle intensity time profile will exhibit
many short time peaks, possibly with a hierarchy of
durations. (Indeed, a fractal structure is often the result
of the evolution of a strongly nonlinear system). On the
other hand, when the Kubo number is small (like for slab
turbulence, or when l x lz ), the evolution of the flux
tube is slow, and only a few, well defined periods of high
energetic particle fluxes will be found in the intensity
profile.
dients. At the same time, fast spreading out, that is fast
perpendicular transport, is obtained with respect to the
average magnetic field. Further study may help to use
the observation of the fine structure in the ion composition or energetic particle fluxes to infer information on
the anisotropy of turbulence.
ACKNOWLEDGMENTS
This work is part of a research programme which is financially supported by the Ministero dell’Università e
della Ricerca Scientifica e Tecnologica (MURST), the
Agenzia Spaziale Italiana (ASI), contract no. I/R122/01,
and the High Performance computing center at the University of Calabria.
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CONCLUSIONS
In this paper we have pointed out the different features of
normal diffusion due to collisions and of turbulent diffusion due to the convection of a passive tracer by turbulence. We have shown by means of a numerical simulation of the evolution of a magnetic flux tube that fast
magnetic field line transport and the formation of small
structures in the flux tube cross section go together. We
also have found that the rate of small structure formation
depends on the anisotropy of turbulence, and in particular on the Kubo number.
If we assume that either energetic particles accelerated
by a flare, or particular ion compositions are injected
from the corona in the solar wind in a given flux tube,
we can imagine that the charged particles in their propagation will follow to a good extend the magnetic field
lines. Hence, the spatial distribution of particles will be
organized as the cross sections of the magnetic flux tubes
shown in Figures 1 to 3. As a consequence, when such
structure is convected over the spacecraft from the solar
wind flow, sudden variations of energetic particles or ion
concentration will be seen, corresponding to sharp gra-
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