J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)156–159
000–000

Physical Properties of Plasma Ion Dynamics in Various Equilibriaof Field-Reversed Configuration
TAKAHASHI Toshiki, UBUKATA Mieko, IWASAWA Naotaka1 and KONDOH Yoshiomi
1Gunma University, Kiryu 376-8515, Japan
Satellite Venture Business Laboratory, Gunma University, Kiryu 376-8515, Japan(Received: 9 December 2003 / Accepted: 28 January 2004)

Abstract
Single particle orbits are traced in the prescribed equilibria inside the separatrix of Field-Reversed Configurations(FRCs). Equilibria inside the separatrix with the peaked, flat and hollow current profile are calculated semi-analytically.
The surface of section plot is used to visualize the regular and stochastic ion motion. In particular, stochasticity in a
kinetic ion (i.e., the large gyro-radius ion) and magnetohydrodynamic (MHD) ion (i.e., the small gyro-radius ion)motion is studied; the majority of the kinetic ions are found to exhibit an adiabatic motion. It appears that the azimuthal
drift velocity for the stochastic and kinetic ions in the elliptic FRC is larger than for the adiabatic ions because of acurvature drift in the neighborhood of the midplane. In the racetrack FRC, on the other hand, no significant difference
is observed in the drift velocity in spite of the stochastic motion.
Keywords:
FRC, single particle motion, drift velocity, stochasticity, adiabaticity, equilibrium, current profile

1. Introduction

and how the equilibrium field affects the particle orbit is stillunclear.
In the present study, the ion dynamics in the various
equilibria of FRC inside the separatrix is studied.

The tilt mode stability is the major physical issue ofField-Reversed Configurations (FRCs) [1-5]. Although theMHD predicts the FRC plasmas are unstable against the tiltmode [6,7], however, the several experimental measurements
show their resilient feature [8]. Since the averaged beta valueof FRC is near unity and thus an ion Larmor radius iscomparable to the scale length, a fluid model approximation
breaks and the velocity distributions become more important
to expect an FRC’s global behavior. In the FRCs, there arethree types of particle trajectories depending on its velocity.
A faster ion with large canonical angular momentum (Pθ ≡mvθ r + qψ (r, z), where m and q are the ion mass and charge,vθ is the azimuthal velocity, ψ is the flux function, and r andz are the radial and axial position) exhibits betatron orbit
encircling the geometric axis. A fast ion but with a smallerPθ than the betatron particle draws a figure-8 orbit. The
direction of its gyrating motion is changed due to the fieldreversal. These two peculiar trajectories may cause a kinetic
stability of FRCs, and therefore the properties of ion
dynamics should be investigated. On this viewpoint,
Hayakawa et al. studied stochasticity or adiabaticity of ion
motion in the deuterium-helium 3 fueled fusion plasma [9].
The existence of adiabatic and trapped particle in the curved
magnetic line region was shown in [9], and it may cause astabilizing flow to the tilt mode activity. However, only Hill’svortex model [10] was employed for the equilibrium of FRC,

2. Computational model
2.1 Equilibrium
The FRC equilibrium state is obtained by solving theGrad-Shafranov (G-S) equation for the flux functionr∂ ⎛ 1 ∂ ψ ⎞ ∂ 2ψ2 dP,⎜
⎟ + 2 = − µ0 r
∂r ⎝ r ∂ r ⎠ ∂ zdψ

dP ∆P ⎡ψ ⎤=⎢γ + 2(1 − γ )⎥,dψ ψ ax ⎣ψ ax ⎦(1)

where ∆P is the pressure difference between the magnetic axis
(i.e., the field-null O-point) and the separatrix and ψax is theflux function at the magnetic axis. The parameter γ controls
the current profile. Peaked current profiles are obtained for0 ≤ γ < 1, the flat one is for γ = 1, and the hollow current
profiles are for 1 < γ ≤ 2. The G-S equation is solved
analytically and numerically using the method developed
earlier [11,12]; the method is slightly modified here in orderto obtain the peaked current profile. The normalized G-Sequation becomes©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: t-tak@el.gunma-u.ac.jp1156

Takahashi T. et al., Physical Properties of Plasma Ion Dynamics in Various Equilibria of Field-Reversed Configuration4∂ 2ψ 1 ∂ 2ψ⎤⎡γ+= −σ ⎢ + (1 − γ )ψ ⎥ ,22∂ρρ ∂z⎦⎣2to the racetrack FRC, a curvature of the field line is foundaround the midplane in the elliptic case. The difference may
affect the azimuthal drift velocity.

(2)−where ρ ≡ (r/rs)2, z/rs → z, ψ /ψax → ψ, and ψax ≡ rs2Bs /√σ .
The quantities rs and Bs are the separatrix radius and themagnetic field at the separatrix and midplane, respectively.
The parameter σ is obtained from the condition that thenormalized flux function at the magnetic axis is unity. Thesolution to Eq. (2) is

2.2 Calculation of ion orbit
Ion orbits in the prescribed equilibrium magnetic field
are traced by solving the equation of motion

mkmax

(3)

Bz =k =1

The coefficients ak are numerically calculated by solving thefollowing recurrence formula:1⎞ π
ν2⎛, ν = ⎜n + ⎟ ,
2⎠ E
8⎝

(4)

where E is the separatrix elongation. The function A(ρ) isdifferent with the current profile control parameter γ, and iswritten as follows:A( ρ ) =1 ∂ψ
1 ∂ψ, Br = −.
r ∂rr ∂z(6)If the semi-analytical solution eq. (3) is directly used, thecomputation to trace the ion orbit needs a lot of time becauseof the summation with respect to k and n in eq. (3). Therefore,
the flux function is calculated with the aid of eq. (3) only at
the grid mesh points in the r – z plane. The magnetic field at
an ion’s position is calculated by the interpolation method.
To observe the ion motion, the surface of section plot isemployed [9]. The surface intersecting the minimum B point
along the line of forces is chosen as a plotting surface for thesmall-gyroradius and figure-8 particles, because due to theconservation of the magnetic moment, the force toward theσ (1 − γ )ν2ak = ak +1 ,( k + 1) ( k + 2 ) ak + 2 +44
a1 = 1, a2 =(5)The cylindrical coordinate system (r, θ, z) is used in thepresent study. The magnetic field in eq. (5) is calculated fromthe flux functionN1⎞ π ⎤⎡⎛ψ ( ρ , z ) = ∑ Cn R( n ) ( ρ ) cos ⎢ ⎜ n + ⎟ z ⎥ + A( ρ ) ,2⎠ E ⎦n =0⎣⎝R ( n ) ( ρ ) = ∑ ak ρ k .

dv= q ( v × B) .
dt

⎡ ⎛ σ (1 − γ ) ⎞ ⎤γρ ⎟ − 1⎥⎢cos ⎜⎟ ⎥2 (1 − γ ) ⎢ ⎜⎝4⎠ ⎦⎣0 ≤ γ < 1 (for the peaked current profile),σA( ρ ) = − ρ 2
16A( ρ ) =γ = 1 (for the flat current profile),⎡⎛ σ (γ − 1) ⎞ ⎤γ
⎢cosh ⎜ρ ⎟ − 1⎥⎜⎟ ⎥2 (1 − γ ) ⎢4⎝⎠ ⎦⎣1 < γ ≤ 2 (for the hollow current profile).

Unknown expansion coefficients Cn are determined
numerically by the boundary condition to set the flux function
to be zero on the separatrix surface. The separatrix shape isgiven in the form [12],Fig. 1

Various equilibria inside the elliptic and racetrack
separatrix (a) (c) (e). The elliptic separatrix shape (µ = 1)(b) (d) (f). The racetrack separatrix shape (µ = 4). Figures
are shown for the (a) (b) peaked (γ = 0.5), (c) (d) flat (γ =10), and (e) (f) hollow (γ = 0.5) current profile.

Fig. 2The current profile on the midplane. The peaked (solidline), flat (dashed line), and hollow (dotted line) currentprofiles are shown.

2µ⎛z⎞ρs ( z ) = 1 − ⎜ ⎟ .
⎝E⎠When µ = 1, the separatrix is in the elliptic shape. The
racetrack separatrix is obtained for µ > 1. The obtained
equilibrium states are shown in Fig. 1, where E = 5.0 for all
the cases and µ = 4 for the racetrack separatrix and µ = 1 forthe elliptic separatrix. The parameter γ is chosen as 0.5, 1.0
and 1.5 for peaked, flat and hollow current profile. The
corresponding current profiles on the midplane (z = 0) are
shown in Fig. 2. Since the contour lines for the hollow current
profile are concentrated near the separatrix, therefore a steep
density gradient is found to be located near the separatrix.
On the other hand, uniform density is observed near the fieldnull O-point for the hollow current profile. On the contrary2157

Takahashi T. et al., Physical Properties of Plasma Ion Dynamics in Various Equilibria of Field-Reversed Configuration

minimum B point acts on all these ions. On the other hand,
we can find the betatron particles that never cross theminimum B surface. Therefore, for the betatron particles, thefollowing plotting surface is used, which satisfies,2( P − qψ (r , z ) )∂U= 0, U (r , z; Pθ ) ≡ θ∂r2mr 2where U is the effective potential known as the Störmer
potential [13].
In a plasma without the electric field, the kinetic energy
K ≡ mv 2/2 and the canonical angular momentum Pθ areconstants of motion, and then these are good measures tocharacterize the ion motion. The new constants s̄ and α
instead of them are defined:s ≡∫α≡rsRr dr2mK, ρi =rs ρ iqBPθ,qψ (r = R, z = 0)

Fig. 3where R is the field-null circle radius. The constant s̄ is ameasure to discuss the finite Larmor radius (FLR) effect andα is a measure to give location of a guiding center and anorbit type (i.e., betatron etc.). Note that s̄ is slight different
from its definition customary used [1]; the Larmor radius in
the customary definition is determined by the plasma
temperature.

The surface of section plots for ions with α = 0.8 in theelliptic FRC. (a) s̄ = 2 (the kinetic case) and γ = 0.5 (thepeaked current profile), (b) s̄ = 2 and γ = 1.5 (the hollow
current profile), (c) s̄ = 10 (the MHD case) and γ = 0.5,and (d) s̄ = 10 and γ = 1.5.

Table 1 The percentages of adiabatic particles

3. Results and discussion
All the calculation is done for the separatrix elongation
E = 5. The current profile control parameter γ is 0.5 for thepeaked profile, 1.0 for the flat and 1.5 for the hollow profile.
The FLR parameter s̄ is 2 for the kinetic case and 10 for theMHD case. The solution of the flux function to the G-Sequation is applicable only inside the separatrix, and thus theparticle orbit outside the separatrix is never traced. Ourcalculation therefore is restricted inside the separatrix.
The results from the surface of section plot are shown inFig. 3. The kinetic ions (i.e., s̄ = 2) tend to exhibit an adiabatic
motion. On the other hand, the MHD ions (i.e., s̄ = 10) arefound to move stochastically. The orbit type of the MHD ion
with large axial velocity is changed, when it passes throughthe curved magnetic line region. For instance, when a gyrating
ion enters the region where the magnetic line is curved, its
orbit is changed to the figure-8 or betatron. This is an originof stochastization due to randomization of gyro-phase. On thecontrary to the MHD ions, a kinetic ion encircles thegeometric axis and exhibits a betatron motion; the type oforbit is never changed. Hence it moves adiabatically. There isno significant effect of the current profile on the behavior ofion motion as shown in Fig. 3. The percentages of adiabaticions in the surface of section plot are summarized in Table 1.
Although they show the number of test ions, however, itrelates with the macroscopic number density. More adiabaticions are found in the kinetic case (s̄ = 2) than the MHD case(s̄ = 10) for all the equilibria. The ions with larger canonicalangular momentum (i.e., the ions with large α) are found toexhibit an adiabatic motion. They are confined only near thefield-null O-point and have betatron orbits (α ≥ 1).
The azimuthal drift velocity of ions contributes to theglobal characteristic of FRC, and then its dependence on α (orPθ) is shown in Fig. 4. There is no effective drifting motionin the MHD case, if they have a small gyro-radius or figure-8orbit. A slight difference among equilibria can be seen; thedrift velocity is the largest in the peaked current profile. Inthe kinetic case, even figure-8 and small-gyroradius ions (i.e.,α < 1) drift in the diamagnetic direction. Though stochasticity
does not affect the drift motion in the MHD case, however,the stochastic fast ions in the elliptic FRC are found to havea larger drift velocity due to the curvature of the magneticfield.

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Takahashi T. et al., Physical Properties of Plasma Ion Dynamics in Various Equilibria of Field-Reversed Configuration

Fig. 4

The azimuthal drift velocity of ions vs. the normalized canonical angular momentum α (a) s̄ = 2 and elliptic FRC, (b) s̄ = 2 andracetrack FRC, (c) s̄ = 10 and elliptic FRC and (d) s̄ = 10 and racetrack FRC. The lines in Figs. (a) and (b) are drawn linearly witha use of least square method. Although the curves in Fig. (c) are also fitted by the least square, however the lines in (d) aredrawn only to guide the reader’s eye.

4. Conclusions

References

Single particle orbits have been traced in the prescribedequilibria inside the separatrix of Field-Reversed
Configurations (FRCs). Equilibria with the peaked, flat andhollow current profile have been calculated semi-analytically.
The surface of section plot was used for the classification ofion motion, and it is found that the kinetic ions (i.e., fast ions)tend to move adiabatically. Their betatron orbits are neverchanged, and therefore there is no rapid change of orbit (e.g.
from the figure-8 to the small-gyroradius orbit, and viceversa) as a source of randomization process. No azimuthal
drift velocity has been found for the slower ions, although
they move in non-uniform magnetic field. This suggests thatthe averaged azimuthal motion due to the diamagnetic
curvature drift near the curved magnetic field region is almost
the same as the azimuthal motion by the paramagnetic gradB drift in the neighborhood of midplane. It is found that thestochastic fast ions in the elliptic FRC have a larger drift
velocity due to the curvature of the magnetic field. A selfconsistent study for the tilt mode stability of FRC with usinga hybrid or particle scheme is the subject in a near future.

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