Propagation of Charged Particles through Helical Magnetic FieldsC. Muscatello, T. Vachaspati, F. Ferrer

Dept. of PhysicsCWRU 10900 Euclid Ave., Cleveland, OH 44106

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.

ABSTRACT

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.

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.

i. Magnetic Field Generation

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,

(1)

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

(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,

(3)

METHOD

bi (

vk )b j

*(vk ') =δ 3(

vk−

vk') S(|

vk |) δ ij −

kikj

|vk |2

⎛⎝⎜

⎞⎠⎟+ A(|

vk |) iε ijlk

l⎡

⎣⎢

⎤

⎦⎥

€

∇⋅ v

B = 0

€

vk ⋅

v B = 0

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),

(4)

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

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,

(5)

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,

(6)

METHOD cont’d

Ba (

vx) =

12π

ba(vk)e−i

vk⋅vx∫ d

vk

€

dv p = q(d

v l ×

v B )

€

vp =

v p o + d

v p i

i

∑

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.

FFT

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).

Figure 1. Left: Depiction of helical field line twisting around a toroidal axis[1].

REFERENCES

[1] Berger, Mitchell A 1999 Plasma Phys. 41 B167.