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Calculation of Ion Radial Transport in the GAMMA10 A-divertor
GAMMA10 A-divertor?: = >0I!F< !4CO !

9Shun Masaki, Isao Katanuma, Shuhei Sato, Kazuto Sekiya and Tsuyoshi Imai,
Plasma Research Center, University of Tsukuba
1-1-1 Tennoudai, Tsukuba, Ibaraki 305-8577, Japan305-8577

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1-1-1

Numerical calculation of ion orbits in the GAMMA10 A-divertor magnetic configuration was carried out. By incorporating Monte Carlo code into the orbit calculation, the effects of Coulomb collision were taken into account. Underanisotropic electrostatic potential, the neoclassical diffusion of ions was enhanced.

1 Introduction
Plasma Research Center, University of Tsukuba is
now planning replacement of west anchor cell in theGAMMA10 tandem mirror with an axisymmetric divertor configuration shown in Figure 1. One of the
purpose of this plan is the simulation experiment of adivertor in large torus devices, utilizing some existingheating systems. Core plasma with high temperatureand density is artificially evacuated to dipole region,
where a divertor plate is installed. The electrostatic
fluctuation is considered as a way of artificial exhaustand in our investigation we seek anisotropic electrostatic potential which cause efficient evacuation, calculating ion trajectories under the potential. The particle orbit calculation code was to include the effectsof Coulomb collision and ions diffused radially underanisotropic electrostatic potential.
Calculation meshes coincide with magnetic surface coordinates and the magnetic field is stored on the meshpoints. We calculate the following drift equations using the predictor-corrector method and tracked a guiding center.
dψdtdθdtdζdt∂φ c[2(ǫ − qφ) − µB] ∂B−∂θqB∂θ∂φ c[2(ǫ − qφ) − µB] ∂B= c+∂ψqB∂ψ= −c= vk B

Here ∇ζ = ∇ψ ×∇θ = B and ∇ψ ·∇ζ = ∇θ ·∇ζ =0 are satisfied in the coordinates. ǫ and µ mean total energy and the magnetic moment of a test particle
respectively and the other notations are conventional.

∇ψ · ∇ζ = ∇θ · ∇ζ = 0 under the assumption of axisymmetric magnetic field and the normal componentof magnetic field line curvature is obtained from therelation∂BBr ∂ 2 z
Bz ∂ 2 r−=∂ψrB ∂l2
rB ∂l2where, ∂/∂l is the derivation along a magnetic fieldline.
Before introducing Coulomb collisional effects, theverification of the Monte Carlo code was done. The
linearized Fokker-Planck equation is used on this codeand change in velocity due to collisions satisfies thefollowing three equations.
∂uu µk=(1)∂tu δt2σ⊥∂∆u2⊥ =(2)∂tδtσk2∂2∆uk =(3)∂tδtwhere,rXmT haα 1µk = δt1+2exp{−aα u2 }mπuααi√1− 2 erf{ aα u} ΓαuXh2δt1 1σk2 =exp{−aα u2 }−√u απaα ui√
1+ 2 erf{ aα u} Γα2u aα
Xh 1 12δt2
σ⊥=exp{−aα u2 }√u απaα uin√
1 oerf{ aα u} Γα1− 22u aα

Present GAMMA10(a)4.38

coil(b)theoritical line with slope

107[cm/s]

magnetic field lines
New device with a divertor

1.93

hu i : ensemble average of velocityµkδt1014[cm2/s2]

ensemble average : h∆u2⊥ itheoretical line with slope

1.5800Standard magnetic field of GAMMA10B[msec]

104(c)

0-50z

5102.001014[cm2/s2](m)

104(d)

7.38-10[msec]

2
σ⊥δt

[keV]

ensemble average : h∆u2k itheoretical line with slope

Fig. 1: GAMMA10 A-divertor

00[msec]

Tk~200eV(2 degrees of freedom)T⊥

σk2~100eV

δt0

104

0

[msec]

104

Fig. 2: Time evolution of particle velocity and temperature

Here uk and u⊥ mean respectively parallel and perpendicular components of a velocity vector before collisions. particle velocities after Coulomb collisions is2
assumed to follow a normal distribution and µk , σ⊥
and σk2 are its mean and variances (two directions)
respectively. Change in velocity is probabilisticallycalculated by normal random numbers. When particle trajectories are traced, velocity changes calculatedby the Monte Carlo method should be added to particle velocities determined in the orbit calculation code,
which includes the effects of Coulomb collisions effectively.
2 Calculation Results
As the verification of the Monte Carlo code, a large
number of ions with initial energy 1keV collided withfield electrons with temperature Te = 100eV and density n = 1012 /cm3 , excluding field ions. The time
evolution of the ensemble average of u, ∆u2⊥ and ∆u2k
is respectively plotted in the figure 2(a), (b) and (c).
The straight line in three figures corresponds to thetheoretical value of slowing down and the pitch angle scattering which ions with 1keV receive from thisfield. It was found that calculation results satisfied
equations (1), (2) and (3) because both lines accordedin the linear region of each figures. In addition, figure 2(d) shows that both Tk and T⊥ was relaxed to thefield temperature.
Then, the Monte Carlo code was embedded in the particle trajectory calculation code and ions were traced.
Figure 3(a) represents a two-dimensional distributionof non-axisymmetric electrostatic potential model, thatis, the axisymmetric component of φ = 1000V on thez-axis with its perturbation of the mode number m =3. Here, the model potential is assumed to be constant along a magnetic field line. The orbits of ionswere calculated including Coulomb collisional effects(a)Equi-contourof φ(r,θ)(b)

z = 520 cm
median plane of A-div

1.0
initial position :r = 15 cm
r= 47.7 cm

[kV]
0.0

Fig. 3: Electrostatic potential and Poincare map

under this non-axisymmetric potential. Figure 3(b) is
the Poincare map of the median plane of A-divertor.
Ti = 1keV Te = 10eV and n = 1012 /cm3 wereassumed in the calculation and some of Maxwellian
particles with initial mean energy 450eV were plotted in the map. Because ions are deeply trapped in
the axisymmetric magnetic divertor, there is no radial
diffusion under axisymmetric electrostatic potential,
while they diffused radially under non-axisymmetricone. This result represents that total energy ǫ and themagnetic moment µ of test particles were updated bycollisions and that the particle trajectory at the median plane of A-div changed from transit orbit which
doesn’t excurse the initial magnetic surface to banana
orbit with a large radial deviation, diffusing radiallythere.

!

3Conclusion
We practiced the verification of Monte Carlo codeand it was validated to include Coulomb collision effects in the enough accuracy. The radial diffusion ofions due to Coulomb collision arised in the anisotropic
potential.