Topic/Type: 2.2 Kinetic methods, Particle-In-Cell and Vlasov, Poster

### Coulomb collisional relaxation process of ion beams in linear magnetic confinement devices

Y.Nishimura

Plasma and Space Science Center, National Cheng Kung University, Tainan 70101, Taiwan

A new algorithm is developed to calculate ion beam trajectories in linear magnetic confinement devices by applying perturbation method [1]. The equation of motion (the newton equation) is solved including the $\small V \times B$ Lorentz force term and Coulomb collisional relaxation term [2]. The frictional force is regarded as a small term, $\small \epsilon \ll 1$. The basic concept is as follows. Normalizing the equation of motion by ion cyclotron frequency,

$\small \dot{V} = V \times B + \epsilon F$ (1)

$\small \dot{X} = V$ (2).

Expanding $\small B = B_0 + \epsilon B_1$,

$\small \dot{V_0} = V_0 \times B_0$ (3)

$\small \dot{X_0} = V_0$ (4)

for the lowest order equation and

$\small \dot{V_1} = V_1 \times B_0 + V_0 \times B_1 + F$ (5)

$\small \dot{X_1} = V_1$ (6)

for the first order equation in $\small \epsilon$. The solution then is given by the summation $\small X = X_0 + \epsilon X_1$, $\small V = V_0 + \epsilon V_1$. The lowest order solution is periodic when $\small B$ is axis-symmetric (Noether\'s theorem) and accordingly the first order solution. The crux in Eq.(5) and (6) are the changes in particle velocity ($\small V_1 \times B_0$) and particle\'s displacement ($\small V_0 \times B_1$) both induced by the small friction force (the $\small F$ term). By storing the first periodic motion of both the lowest and the higher order solution, the algorithm can predict further periodic motion and thus can reduce computation time. The perturbation method is also useful since we only need to change the constant $\small \epsilon$ when the the plasma parameters change e.g. background densities and temperatures (and do not need to recalculate the whole trajectories). We focus on the high energy particles whose temperature is much larger than that of the background thermal plasma. The scheme is useful in studying ion beam heating and macroscopic stability of plasma columns such as magnetic mirrors and theta pinch plasmas.[3] In general, the algorithm can be applied to periodic motion under perturbative frictional forces, such as guiding center trajectories[4] or satellite motion. This work is partially supported by NCKU Top University Project.

[1] Y.Nishimura, Diploma thesis, Osaka University (1991).

[2] K.Miyamoto, Plasma Physics for Nuclear Fusion, (Iwanami, Tokyo, 1986), p.76.

[3] T.Ishimura, Phys. Fluids 27, 2139 (1984).

[4] Y.Nishimura, Contributions to Plasma Physics 48, 224 (2008).