Particle Transport in the Solar Wind Magnetic Turbulence: a
Numerical Investigation
P. Pommoisy, P. Veltriy and G. Zimbardoy
Center for High Performance Computing, Università della Calabria (Italy).
y Dipartimento di fisica, Università della Calabria (Italy)
Abstract. In order to interpret the fluxes of particles accelerated at corotating interaction regions and to forecast the solar
energetic particle events, the propagation of energetic particles is investigated numerically. Particle transport is influenced
in a strong way by the magnetohydrodynamic turbulence in the solar wind. We carried out a simulation in which ions are
injected in a numerical realization of magnetic turbulence superimposed on a background uniform magnetic field. Particle
transport is studied as a function of particle energy, pitch angle distribution, turbulence level, and turbulence anisotropy with
parameter values appropriate to the solar wind. The ratio of perpendicular to parallel diffusion coefficients is estimated. For
strong enough turbulence B=B0 1, as typical of the solar wind, we find that the particle pitch-angle distribution is quickly
isotropized, and that perpendicular transport grows with the particle energy. Varying the turbulence anisotropy, that is the
turbulence correlation lengths lk and l? , we find that in the cases with lk l? (i.e., in the limit of 2D turbulence) the particle
perpendicular transport is faster than in the cases with lk l? (i.e., for the so called slab turbulence).
INTRODUCTION
of solar energetic particle events, we consider in this
study proton energies varying from 100 keV to 1000
MeV. In the solar wind rest frame we can neglect the
electric field, and the particle relativistic equations of
motion can be written as:
The study of particle transport in the presence of magnetic turbulence is of great interest for several physical
problems, like the transport of cosmic rays, the energetic
particles propagation in the solar wind, and the plasma
confinement in laboratory experiments. In strong turbulence analytic treatments are not easily feasible, and it is
necessary to perform numerical simulations. In the solar wind, these require large computational resources, in
order to describe a long turbulence spectrum, which, ideally, should include wave lengths comparable with the
Larmor radius of the considered particles.
In this paper we show the results obtained by a numerical simulation where the particles trajectory are integrated in a three-dimensional (3D) magnetic turbulence. This work is an extension of the study of magnetic field line transport which we performed previously
[1, 2, 3, 4, 5, 6]. Here we present the preliminary results,
for which we kept the ratio between Larmor radius r L
and the shortest turbulence wavelength min small, in order to check whether the findings obtained for field line
diffusion actually hold for particle having small r L .
r
dr = v ; dv = q v ^ B
dt
dt mc
where is the position of the particle having mass m and
charge q , is the particle velocity, and ; v 2 =c2 ;1
is the Lorentz factor which is constant because the only
force is the magnetic force. A 3D magnetic field in a
periodic simulation box is set up (see Pommois et al.
where is the sum of
[2, 3]),
0 static magnetic perturbations
=(1
v
B(r) = B + B(r)
B(r) =
X
k;
)
B(r)
B (k)e() (k)exp i[k r + (k) ]
Considering that the magnetic perturbations propagate
with the Alfvén velocity, the assumption of static perturbations is well satisfied for energetic particles in the
solar wind. The wave vectors are chosen as
n n n 2
x y z
k= N
; ;
l
min x ly lz
NUMERICAL MODEL
in which lx , ly and lz are the correlation lengths (note that
in the axi-symmetric cases, lk lz and l? lx ly ).
The spectrum has a cut-off for both the short and the
=
The kinetic energies of accelerated particles in the solar
wind vary over wide ranges. Having in mind the forecast
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
477
= =
%
30
25
20
15
10
5
0
30
25
20
15
10
5
0
30
25
20
15
10
5
0
10 deg
δB/B0=0.1
10 deg
10 deg
0.5
1
30
25
20
15
10
5
0
50 deg
30
25
20
15
10
5
0
δB/B0=1.0
30
25
20
15
10
5
0
50 deg
30
25
20
15
10
5
0
δB/B0=0.5
-1 -0.5 0
50 deg
30
25
20
15
10
5
0
-1 -0.5 0 0.5
cos(θp)
30
25
20
15
10
5
0
1
90 deg
1.0
1.4
1.0
0.8
90 deg
d
B
d
0.4
0.6
0.4
E=1 MeV
0.2
/rl=91
min
0.2
0.0
90 deg
0.0
0
50
100
150
200
250
t/
300
350
400
450
500
bi
6
3.0
B
-1 -0.5 0
0.5
1
d
5
2.5
4
2.0
3
1.5 B
2
1.0
E=100 MeV
1
/rl=10
min
0.5
d
0
0.0
0
50
100
150
200
250
t*
+
e (k) k = 0
e (k)
=1 2
(k)
2 n2
long wavelengths (band spectrum) with: N min
x
2
2
2
(
)
ny nz Nmax . We have
, where ()
are the polarization unit vectors with ; , and (k)
are random phases. The perturbation amplitude B
represents a self-similar spectrum:
B (k) =
1.2
0.6
FIGURE 1. Distribution of the cosine of the pitch-angle p
for 1000 particle. Here we show the influence of the intensity
of the fluctuation level B=B0 and of the initial pitch angle
on the change of the velocity distribution after an integration
time of 104 =!bi , of particle with 1MeV energy in an isotropic
turbulence spectrum. Left panels: p = 10 ; middle panels:
p = 50 ; right panels: p = 90 . Upper panels: B=B0 = 0:1;
center panels: B=B0 = 0:5; lower panels: B=B0 = 1:0.
+
B
0.8
300
350
400
450
500
bi
FIGURE 2. Evolution of the dimensionless magnetic moment d (full line) and of the local magnetic field modulus (dashed line), for particle having 1 MeV energy (upper
panel) and 100 MeV energy (lower panel), in a turbulent magnetic field with B=B0 = 1:0. Here d = =0 where 0 =
v02 =2mB0 .
C
(kx2 lx2 + ky 2 ly 2 + kz 2lz 2)=4+1=2
order to better discriminate the effects of different energies. Two types of initial particle distribution were used
in the simulation: conic distribution (with fixed injection
pitch angle p between the initial velocity 0 and magnetic field 0 ) and isotropic distribution (where the direction of 0 is random).
During the integration the conic distribution is evolving and is progressively isotropised, see Fig. 1. The factors that speed up pitch angle “diffusion" are found to be:
The intensity of the magnetic fluctuation B=B0 ; and the
energy of the particles.
From Fig. 1, it can be seen that there is no particular
(e.g.
difficulty for pitch angle diffusion across p
[7]), and it is shown that stronger magnetic fluctuation
give a flatter distribution. Also shown is the way in
which a different initial pitch angle can influence the final
;1.
distribution, after the same integration time of 10 4 !bi
=3 2
where = is the Kraichnan spectral index. More
details on the numerical simulation can be found in Refs.
[1, 2, 3, 4, 5, 6]. We report in the following on the
main features of particle transport in such a magnetic
configuration. Indeed, the above equations of motion are
integrated in the presence of various levels of magnetic
turbulence and various kinds of turbulence anisotropy.
nT is assumed, and time
A background field B0
is measured in units of the inverse of the nonrelativistic
proton gyrofrequency, ! bi qB0 =mc.
B
v
= 10
=
v
= 90
EVOLUTION OF THE PARTICLE
VELOCITY DISTRIBUTION
In the runs presented here the injection energy was constant (in other words, the distribution function corresponds to an energy shell rather than to a Maxwellian), in
478
Quasi-2d
Slab-like
0.0001
0.01
Kx,Ky
1
Kx,Ky
0.001
<dµ>/µ
1e-05
0.0001
1e-06
0.1
1e-05
0
5e+05
1e+06
0.01
0.1
1
10
100
0
5e+05
1e+06
0.001
Kz
1000
Kz
E (MeV)
0.0001
FIGURE 3. Relative change of the magnetic moment in
function of the particle energy. Here for each points, the simulation are done for 100 particles integrating up to 500/!bi , and
with fluctuation level B=B0 = 1:0.
0.001
1e-05
0
5e+05
1e+06
2
0
5e+05
αx,αy,αz
CONSERVATION OF THE MAGNETIC
MOMENT =
2 (r)
1.5
1
1
x
y
z
x
y
z
0.5
0
0
0
5e+05
1e+06
6
U
U
U
4
3
2
1
0
0
5e+05
time*ωbi
0
5e+05
16
14
12
10
8
6
4
2
0
K x, K y, K z
5
(r)
1e+06
U
U
1e+06
U
K x, K y, K z
0
5e+05
time*ωbi
1e+06
FIGURE 4. Evolution of Ki , i , and KiU in function of
time. The 1 MeV particles are integrated in a spectrum with
quasi-2D turbulence (lk l? , panel on the left) and Slab-like
turbulence (lk l? , panel on the right) with a fluctuation level
B=B0 = 1:0.
10
02
TRANSPORT PROPERTIES
100
0 03
αx,αy,αz
1.5
0.5
We find two types of changes for the magnetic moment
2 = B during the
(the first adiabatic invariant) mv?
integration: first, a periodic variation associated with cyclotron motion in a non uniform magnetic field; this variation is more or less the same for the different energies
(see Fig. 2). Here B
is the modulus of the local total
magnetic field. Second, a non adiabatic variation associated with large magnetic field changes (strong magnetic
gradients). This variation increases with energy (that is
with the Larmor radius rL ) as it should. Indeed, it can
be seen that the change in can be large even when
rL min = , and that the change in is particularly
strong when B is weak.
In Fig. 3, we plot the relative variation of the magnetic
moment < d > = (which correspond to the non adiabatic variation) as a function of particle energy. We note
that for the 1 MeV particles, < d > = : which
is quite high considering that the smallest length in the
spectrum min is about
rL. This shows that turbulence has a strong influence on particle motion and that
guiding center approximation is not suitable for particles
having energies equal or larger than 1 MeV with the parameter used in these simulations: B0 nT and a correlation length lc : AU, which is tipical in solar
wind at 1 AU from the Sun. In particular, we find that the
first adiabatic invariant is not conserved. This shows
that the many small variations of that a particle undergoes when moving in the presence of turbulence can give
rise to the non conservation of the first adiabatic invariant
even if rL min . This effect should be even more important in the actual solar wind turbulence because of the
presence of much smaller wavelengths than those found
in our numerical model.
1e+06
2
In order to determine quantitatively the transport properties, we compute the variances h x2i i, where xi
xi ; x(0)
i (i x;y;z ), as a function of time t. Then
we make a fit of h x2i i, with the anomalous transport law h x2i i
Ki ti and determine i and Ki
when t is large enough to attain asymptotic values (see
[1, 2, 3, 5]. The results presented here were obtained with
6 ! ;1 ; in physical units, this corresponds to more
t
bi
than 10 days for B
nT. Here, the exponent i characterizes the transport law: i
in the diffusive regime
in the ballistic regime;
(Gaussian random walk); i
i < in the case of trapping (subdiffusive regime), and
< i < in the case of Lévy random walk (superdiffusive regime). The results for Ki , i and for the kurtosis
KiU h xi 4 i=h xi 2 i2 are shown in Fig. 4. The Gaussian value of KiU is 3.
=
10
= 2
= 10
1
479
= 10
1
2
=
=1
=2
=
2
1.5
1.5
αz
<αx>
2
1
0.5
1
0.5
1
10
100
0
0.1
1000
12
12
10
10
8
8
KUz
U
<K x>
0
0.1
6
4
1
10
100
1000
1
10
100
1000
6
2
1
10
100
Energia (MeV)
1000
0.1
We studied particle transport for 3 different cases of magnetic turbulence (isotropic, 2D and Slab spectrum), and
for different particle energies. The main results are: (1)
fast pitch angle diffusion leads to a rapid isotropization
of the velocity distribution function; (2) the first adiabatic invariant is not well conserved if rL =min exceeds ;2 ; (3) motion along the average field is not
ballistic but diffusive, even in the small Larmor radius
cases considered up to now; (4) anomalous transport are
found for lk l? ; (5) perpendicular transport increase
with particle energy, and with the ratio lk =l? ; (6) the ratio K? =Kk varies in the range reported in the literature,
between ;4 and ;1 [8].
10
4
2
0.1
CONCLUSIONS
10
10
Energia (MeV)
FIGURE 5. Exponent and Kurtosis for different particle
energy and different magnetic turbulence. Full line and pluses:
quasi-2D with B=B0 = 1:0; long-dashed line and crosses:
isotropic spectrum with B=B0 = 0:5; short-dashed line and
stars: isotropic spectrum with B=B0 = 1:0; dotted line and
squares: Slab-like with B=B0 = 1:0. The left panels show
this coefficient averaged in the x and y directions which are
perpendicular to the mean magnetic field direction.
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.
In general we obtain Gaussian diffusion for the spectrum having larger lk =l? (that is quasi 2D turbulence, see
Fig. 4 left panels). In such cases the coefficient Ki are
the diffusion coefficients. We found also non Gaussian
cases for lower energy, an example is shown in Fig. 4,
right panels, with superdiffusion along the direction z ,
and subdiffusion along x and y . The values of the Kurtosis, reported in the bottom panels of Fig. 4, confirm
that we are in Gaussian regime for quasi-2D turbulence,
and in anomalous, non Gaussian regimes for Slab-like
turbulence. The superdiffusion along z indicates that the
particle motion along the magnetic field is intermediate
between diffusive and scatter free.
Figure 5 shows the values of i and KiU as a function
of particle energy. It can be seen that the larger the
energy, the closer we are to a Gaussian regime.
The present simulation allows to directly evaluate the
ratio K? =Kk at least in the cases of Gaussian diffusion.
This ratio is often required in astrophysical studies of
particle transport (e.g., [8]). We note that K ? =Kk increases with the particle energy, with the turbulence fluctuation level, and passing from Slab turbulence to 2D
turbulence spectrum, that is with the ratio lk =l? . More
complete results will be reported elsewhere.
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