Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)485–488
000–000
J. Plasma Fusion

Kinetic Simulation for Infinitely Long Cylindrical High-Beta Plasma
with Field-Null Surface
TAKAHASHI Toshiki, MOROHASHI Keisuke, IWASAWA Naotaka 1and KONDOH Yoshiomi
Gunma University, Kiryu, 376-8515, Japan
1Satellite Venture Business Laboratory, Gunma University, Kiryu, 376-8515, Japan(Received: 9 December 2003 / Accepted: 11 May 2004)

Abstract
The kinetic simulation for an infinitely long cylindrical high-beta plasma with field-null surface is carried out. TheVlasov equation is solved in order to study finite Larmor radius (FLR) effect of ions near the field-null region and thefluid equation is used for electrons. The ion distribution function and the radial profiles of macroscopic physical quantities are shown. It is found that the results from the present kinetic scheme are deviated gradually from the one fluid
magnetohydrodynamic (MHD) equilibrium.
Keywords:
high-beta plasmas, field-reversed configuration, field-null, Vlasov equation, rigid rotor, FLR effectand azimuthal velocity components. According to Spivey’swork, the only shifted-Maxwellian distributions that satisfy
the Vlasov equation for systems with cylindrical symmetry
are rigid rotors with constant temperature [1]. The comparison between Spivey’s theory and numerical results are madehere. This study will be applied in near future to the FieldReversed Configurations (FRCs) with several unknown physical issues.

1. Introduction
High beta plasmas exhibit kinetic behaviors due to large
ion gyro-radii compared to their scale length. Existence of afield-null point makes it difficult to predict the plasma behavior because the particle motion near the point is complicated.
A fast ion with the larger canonical angular momentum
draws the betatron orbit and contributes to generate the diamagnetic ion current. On the other hand, a slow ion near thesurface shows a stochastic behavior in its motion and drifts
along the paramagnetic direction. Therefore, obtaining velocity distributions is a key issue for the high-beta plasmas. The
particle simulation or the particle and fluid electron simulation (i.e., the hybrid simulation) are useful numerical tools to
describe a global characteristic of the high-beta plasma. A
physically valid weighting of super-particle is needed in theparticle simulation modeling. Another kinetic method is tosolve the Vlasov and Maxwell’s equations. The time evolutions of the distribution functions are simulated by solving
the Vlasov equation. The electric field is calculated from thefluid equation of motion for massless electrons, and the evolution of magnetic field is calculated from Faraday’s law.
The objectives of this study are to predict a high-beta plasma
structure of an electromagnetically relaxed state and todemonstrate kinetic effects in the plasma with field-null surface by the plasma kinetic theory.
Preliminary results are obtained for infinitely long andcylindrically symmetric plasma whose averaged beta value isabout 0.9. In this case the real space variable is the radial
position alone, and the velocity space variables are the radial

2. Simulation model
We study a high-beta plasma that is infinitely long axially and has an axisymmetry (see Fig. 1). Mathematically, thisrequires ∂/∂θ = ∂/∂ Z = 0. Since the Larmor radius is comparable to the scale length of the high-beta plasma, it is better touse the kinetic equation for ions. On the other hand, MHD is
still applicable for electrons, because effects of the field-nullare localized. Moreover, computation time increases significantly if we solve the kinetic equation also for electrons; thetime interval must be small in order to satisfy the Courant
condition. Therefore, we choose a hybrid scheme where ions’behavior is simulated by solving the Vlasov equation andelectrons’ one are computed with the fluid equation.
Vlasov equation described in the cylindrical coordinate is©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: t-tak@el.gunma-u.ac.jp1485

Takahashi T. et al., Kinetic Simulation for Infinitely Long Cylindrical High-Beta Plasma with Field-Null Surface(FIX) device, is about 0.5 [2]. The initial electric field isassumed to be zero, and then uiθ and ueθ are determined fromEqs. (2) and (3). Therefore,ui θ = −Validity of the electron fluid equation near the wall is questionable, since the density and pressure are almost zero. In
this case, the electric field and flow velocity can never be
fixed by using − ∇pe − en (E + ue × B) = 0. In the presentstudy, the plasma region is defined inside the zero flux point(i.e., the separatrix), and the edge layer is located from theseparatrix to the point where the density is n(ψ = 0) / e. The
electric field and flow velocity of plasma vanish outside theedge layer. This assumption is effective to reduce the numerical noise because of the boundary condition.
The shifted-Maxwellian distribution with azimuthal
flow velocity is introduced for an initial ion distribution:Fig. 1 Schematic view of infinitely long plasma with axisymmetry.

∂ fi∂t+ vr∂ fi⎡ v2⎤ ∂fq+ ⎢ θ + i (vθ Bz + Er) ⎥ imi∂r ⎣ r⎦ ∂ vr⎡v v⎤ ∂fq+ ⎢ r θ + r (− vr Bz + Eθ ) ⎥ i = 0 .
rm⎣⎦ ∂ vθ
i(1)There is no axial component of force acting on ions in a caseof ∂/∂θ = ∂/∂ z = 0, thus dvz / dt = 0. Consequently, the ion
distribution never deviates in vz from the initial form. The iondistribution function fi is normalized asni =∞∞−∞−∞⎛ m v2 ⎞⎛ m (v θ − u θ )2 ⎞mi ⎞i rii⎟ exp ⎜⎟ exp ⎜⎟ .
⎝ 2 πT ⎠⎝ 2T ⎠⎝⎠
2Tiii⎛

fi (r, vr, vθ ) = n(r) ⎜The Vlasov equation and fluid equations are solved
numerically using the finite difference method. The advection terms in the Vlasov equation are discretized by thefourth order scheme, and a 4-step Runge-Kutta method isemployed for the numerical integration in time. A filteringwith the method of least squares are done for Er and Bz.

∫ ∫ f (r, v , v )dv dv .
ir

θr

θ

Quasineutrarity ni = ne = n is assumed in the present study.
The electromagnetic fields B and E in Eq. (1) are calculated from the fluid equation for massless electrons,E=11( j × B),∇pe − ui × B +enen

3. Results and discussions(2)

The parameters of geometry and plasma we studied arepresented in Table 1. The physical quantities are all normalized as follows:and Maxwell’s equationsj=∂B
1= −∇ × E.
∇ × B,µ0∂t−t ≡(3)−E≡

A solution to one-dimensional (1D) Grad-Shafranov equation
is used as an initial condition. The 1D Grad-Shafranov equation for the flux function ψ isdp
1 ∂ ⎛ 1 ∂ψ ⎞.
⎜⎟ = −µ 0r ∂r ⎝ r ∂r ⎠dψtB
jr −−,−r ≡,B≡,j ≡,21/ω cirw
en0vthi⏐ψ w⏐/rw
E⏐ψ w⏐ω ci /rw, u−i ≡uivthi

, u−e ≡uevthe

where ψw is the flux function on the wall surface, ωci ≡ e|ψw|/(mirw2), and vthα ≡ 2Tα / mα (α = i, e). The upper bar of normalized quantities is omitted from now on for simplicity.
The contours of ion distribution fi (r, vr vθ) are presentedin Figs. 2 and 3. In Fig. 2, contours of (r, vr = 2.5vthi, vθ) are
drawn in r − vθ plane. It appears that the initial distribution ischanged abruptly and two peaks of distribution function areobserved at t = 3/ωci. The radial path length of ions with vr =2.5vthi for the duration of 3/ωci is about 0.45 m, which is comparable to the wall radius. Therefore, the calculation time is(4)The pressure p as a function of ψ is given as follows:⎧ a + a ψ + a2ψ 2 + a3ψ 3 ≡ pin (ψ), ψ ≥ 0p(ψ) = ⎨ 0 1⎩ b0 exp(b1ψ) ≡ pout(ψ), ψ < 0
where a s and b s are coefficients which are set so as the current density and pressure at the separatrix to connect smoothly (i.e., pin (ψ = 0) = pout (ψ = 0), dpin / dψ |ψ=0 = dpout dψ |ψ=0),
and d2pin / dψ 2|ψ=0 = d2pout / dψ 2|ψ=0). Also, we give the valuesof pressure at the separatrix and the wall. The initial density
profile is estimated from the plasma pressurep = n(Ti + Te) = nTtot , Ti =jj
1 ∂ pe+ θ , ue θ = ui θ − θenBz ∂ renen

Table 1 Geometry and plasma parameters
Parameters

1αTtot , Te =Ttot
1+α1+αwhere total temperature Ttot is uniform and the parameter isconstant. For the plasma in the FRC Injection Experiment
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Symbols

ValuesExternal magnetic fieldBex

0.05 TPlasma temperature

Ttot = Ti + Te

150 (100+50) eV

Reference number density

no5.0 × 1019 m3

Wall radius

rci0.4 m

Takahashi T. et al., Kinetic Simulation for Infinitely Long Cylindrical High-Beta Plasma with Field-Null Surface

Fig. 2 Contours of the ion distribution function at (a)t = 0,(b)1ωci, (c)2/ωci, and (d)3/ωci,. The figures are drawn invθ − r plane, and the radial velocity vr is fixed at 2.5vthi.

long enough for the plasma to transform the velocity distribution. The distortion in fi (r, vr vθ) around the separatrix r ≈0.6rw is found significant. It is found from Fig. 3, the distribution deviates from the isotropic Maxwellian. Hence ion
dynamics in high-beta plasmas, in particular the plasmas withthe field-null point, can cause a behavior that is never predicted by an MHD model.
Radial profiles of various physical quantities Bz, jθ , n,Eθ , Er, uir, uiθ , ueθ , are shown in Fig. 4. Results from the 1D
Grad-Shafranov equation and the fluid equation of motion

Fig. 4 Radial profiles of (a)BZ, (b)jθ , (c)n, (d)Er, (e)Eθ , (f)uir, (g)uiθ , and (h)ueθ at t = 0 (solid line), 1/ωci (dashed line),
2/ωci (dotted line), and 3/ωci (thin solid line). Here, B0 ≡|ψw |/rw2, j0 ≡ en0vthi, E0 ≡ |ψw |ωci /rw.

are for t = 0. Figure 4(f) shows that ions start to move inward
inside the field-null surface and outward outside the surface,which flattens the density profile (cf. Fig. 4(c)).
Consequently, the density gradient is increased at the separatrix compared with the initial profile. The azimuthal electricfield Eθ is generated due to this radial ion flow, which is
identical to the E × B drift velocity (see Figs 4(e) and (f)). It
appears that the generated radial electric field Er by the effectof the Hall term is localized near the separatrix; the fieldrotates the plasma ions. According to Spivey’s work, the only
shifted-Maxwellian distributions that satisfy the Vlasov equation for systems with cylindrical symmetry are rigid rotors uiθ
= rωiθ with constant temperature [1]. The present calculationshows that the velocity distribution is gradually deviatedfrom the initial shifted-Maxwellian, and thus the ion’s rotation velocity profile is different from the rigid rotors.
Moreover, inclusion of the plasma wall interaction in a calculation model can slow down the rotation velocity of plasmaions.

4. Conclusions
The kinetic simulation for an infinitely long cylindricalhigh-beta plasma with field-null surface has been carried out.
The Vlasov equation is solved in order to study finite Larmor
radius effect of ions near the field-null region and the fluid
equation is used for electrons. The ion distribution function
and the radial profiles of macroscopic physical quantities

Fig. 3 Contours of the ion distribution function at (a)t = 0,(b)1ωci, (c)2/ωci, and (d)3/ωci. The figures are drawn invθ − vr plane, and the radial position r is fixed at 0.4rw.

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Takahashi T. et al., Kinetic Simulation for Infinitely Long Cylindrical High-Beta Plasma with Field-Null Surface

have been shown. It has been found that the velocity distribution is deviated from the initial shifted-Maxwellian distribution. Therefore, the kinetic simulation is necessary to predictthe high-beta plasma behavior. The electric field has beenfound to generated, which results from the ion drift motion
and the effect of the Hall term.
The gross stability of FRC will be studied with a use of
the present hybrid code. It is not easy, however, to obtain thevelocity distribution functions that are described by the variables in the six-dimensional phase space. If the finite differ-ence method is applied, the number of grid meshes becomesenormous. The effective technique to reduce the number ofgrid meshes in the velocity space significantly will be needed.

References[1] B. Spivey, Ph.D. thesis, University of California, Irvine,
1992.
[2] T. Asai, Ph.D. thesis, Graduate School of Engineering,Osaka University, Japan, 2001.

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