Download Propagation of Charged Particles through Helical

Propagation of Charged Particles through Helical Magnetic Fields
C. Muscatello, T. Vachaspati, F. Ferrer
Dept. of Physics
CWRU 10900 Euclid Ave., Cleveland, OH 44106
METHOD cont’d
ABSTRACT
The study of mobile charged particles through stochastic magnetic fields has both
cosmological and astrophysical implications. Magnetic fields are known to exist in
intragalactic space and are theorized to exist in intergalactic space. More specifically,
helical magnetic fields exist in a number of systems such as galactic jets and possibly
the primordial magnetic field. The current interest is to study the propagation of charged
particles in helical magnetic fields and to determine if the results can be used as a probe
of magnetic helicity. Through a Monte-Carlo simulation, a random helical magnetic
field is generated on a mesh according to magnetic field power spectra given in Fourier
space. The Fourier transform of the field then converts it to a field in Cartesian space.
Particles will then permeate the magnetic field according to a kinetic algorithm and their
characteristic trajectories are recorded.
FFT
Figure 1. Above Left: Magnetic field in Fourier space with power spectra S(k)=k2 and A(k) =k2 / 3.
Above Right: Magnetic field in Cartesian space. In both representations, similar magnitude
vectors share similar colors.
Figure 1. Left: Depiction of helical
field line twisting around a toroidal
axis[1].
In order to easily generate three components of the magnetic field, we should want each
of the components to be independent of each other. If we choose two mutually orthogonal
vectors to k (say l and m) then the off-diagonal components of the first term in the brackets
on the RHS of the correlation function (eq.1) are zero.
It then turns out that bl and bm are simply generated directly according to S(k) while bk is
zero (eq.3). In order to introduce helicity, two more mutually orthogonal vector components
of the magnetic field should be generated bu and bv. These two components are independent
of k and can be written in terms of bl and bm. bu and bv are generated as Gaussian deviates
with variances S(k)+A(k) and S(k)-A(k), respectively. Once bu and bv are calculated, it is a
trivial matter to solve for bl and bm and equally trivial to transform the b vector components
to the home (k1,k2,k3) Fourier basis.
Using the aforementioned method, we can generate magnetic field vectors for a chosen
range of k values given some numerical spacing. To convert the magnetic field into
Cartesian space, the method of Fourier transform is numerically implemented. According to
Fourier analysis, the magnetic field’s components are transformed independently according
to (where the index indicates the field component),
1
ik x
Ba (x) 
b
(k
)e
dk
a
(4)

2
The preceding procedure is repeated for all three spatial components using a 3-dimensional
Fast Fourier Transform (FFT) algorithm, and the values are masked onto the numerical grid.
ii. Particle Propagation
METHOD
Overall, the basic theme is to find a mechanism that will allow us to detect magnetic
helicity in space. In general, we will study the effect a helical component has on the
trajectory and momentum of charged particles.
Particles propagate the magnetic field, and their trajectories are determined by the usual
Lorentz force equation where the electric field is assumed to be zero. A slight manipulation
is made to more easily calculate the incremental change in momentum of particles at each
step through the mesh. The following is calculated for every particle at every step,
i. Magnetic Field Generation
dp  q(dl  B)
Along with the large-scale magnetic fields that exist in astrophysical systems, there
are often small-scale components for which theoretical models can provide a magnetic
field power spectrum in Fourier space (k-space) which contains all necessary
information about the field. Our first task is to find an expression for the two-point
correlation function describing the relationship between two components of the
magnetic field. The expression is as follows,
ki k j 


*
3
l
bi (k )b j (k ')   (k  k ')  S(| k |)   ij 
 A(| k |)i  ijl k 
(1)
2

|k | 


where the first term in the brackets is the symmetric part with S(k) as the symmetric
magnetic field power spectrum and the second term in the brackets is the asymmetric
part with A(k) as the helical magnetic field power spectrum. The second factor in the
first term in the brackets arises from the requirement that no magnetic monopoles exist,
or
 B 0
Because the magnitude of the magnetic field is very small in systems where this method is
applicable, a perturbative method is used to calculate the total momentum of each particle on
its exit from the mesh. Foreach particle, the following is calculated at each grid space i,
p  po   dpi
Figure 2. Above Left: Diagram depicting how delta momentum vectors for each particle trajectory
are determined. po is the unperturbed momentum of the particle across the mesh, p is the
particle’s average momentum under the influence of the magnetic field, and dp represents
the shift in the particle’s momentum across the field (the difference between po and p) .
Above Right: Delta momentum (dp) vectors of flux of particles traveling parallel to the x2
direction (out of the page).
REFERENCES

kB0

(3)
(6)
i
(2)
Since the magnetic field in Cartesian space is the Fourier transform of the magnetic
field in k-space, an equivalent expression for the zero divergence condition is,
(5)
[1] Berger, Mitchell A 1999 Plasma Phys. 41 B167.
RESULTS AND CONCLUSIONS

Figure 1 shows one possible configuration of the magnetic field in both Fourier and
Cartesian spaces. Figure 2 left shows how eq.6 is graphically represented for each particle,
and Figure 2 right shows the momentum vector field for a flux of particles traveling in the x2
direction. The code written for this project allows a flux of particles to travel in any one of the
three principal directions. By studying the change in momenta for a collection of particles, we
may be able to determine if there is a general trend for particle propagation direction through
helical fields. If this work were to be continued, the next step would be to investigate the
relationship between the configuration of the momentum vectors and the helicity of the field.
Overall, we have taken an active approach to create a methodology to model particle
propagation through a magnetic field with given analytic power spectra.