Proceedings of ITC/ISHW2007

Effects of Coulomb Collisions on the Toroidal Spin­up of a
Field­Reversed Configuration
Toshiki Takahashi, Hidefumi Yamaura, Yoshiomi Kondoh, Tomohiko Asaia, and Tsutomu Takahashia
Department of Electronic Engineering, Gunma University, Gunma 376­8515, JapanaCollege of Science and Technology, Nihon University, Tokyo 101­8308, Japan

A direct conversion process from the magnetic flux to the kinetic angular momentum of plasma ions is apossible mechanism of a toroidal spin­up of a field­reversed configuration (FRC) [T. Takahashi et al., Plasma
Fusion Res. 2, 008 (2007)]. For quantitative discussion, the toroidal rotation velocity of a FRC plasma which is
resistively decaying is calculated numerically. The ion­ion pitch­angle scattering is reproduced by a Monte Carlo
method. The obtained toroidal rotation velocity at the field­null is found to increase linearly up to 0.3v A0 , wherev A0 is the Alfvén velocity defined by the external magnetic field and the ion density at the field­null just after theformation. The result is reasonable in a view of a time evolution of the toroidal flow velocity by the Doppler shiftmeasurement at the Nihon University compact torus experiment­III device. It is found that a classical particle loss
has an insignificant effect on the spin­up of a FRC.
Keywords: field­reversed configuration, toroidal spin­up, n=2 rotational instability, pitch­angle scattering,end loss, classical diffusion

1. Inroduction
The rotational instability with the toroidal mode
number n=2 of a field­reversed configuration (FRC)
plasma is a global instability that is most often observed
experimentally [1­5]. The FRC current just after formation
is primarily carried by electrons, while ions areapproximately at rest [6]. The ions, however, gradually
gain angular momentum in the ion diamagnetic direction
before the onset of the n=2 rotational instability. Rotationof the FRC plasma has been often explained by selective
loss of ions [7­9], end­shorting [10­13], or both [14].
Recent computer simulation shows that a resistive decay of
the internal flux cause selective loss of ions and resultant
spin­up of the FRC plasma [15, 16]. The work is valuable
in that contribution of the flux decay to the spin­up of a
FRC is firstly proposed. However, is a selective loss ofions necessary to explain toroidal spin­up of a FRC
plasma?
Recently, we have proposed an another possible
spin­up mechanism, which is direct conversion from themagnetic flux to the kinetic angular momentum [17].
Comparison between experiment and calculation with asimple model will show validity of our explanation.
Suppose a FRC plasma is axisymmetric. This
assumption is valid until the rotational instability is
triggered. In a collisionless plasma, the canonical angularmomentum Pq = mvq r + qy ( r , z ) of every particle isconserved, where m, q are the mass and charge,
author’s e­mail:t­tak@el.gunma­u.ac.jp

respectively, vq is the toroidal velocity component, andy (r , z ) is the poloidal flux function. If the poloidal flux
decays due to resistivity and toroidal axisymmetry stillholds, thenmD (vq r ) = - qDy .
(1)Equation (1) shows that when the poloidal flux decays,
every ion gains angular momentum in the ion diamagnetic
direction. Generally, the separatrix radius decreases during
the decay phase. If the guiding center r is also decreased,
the toroidal velocity vq is further increased.
Another explanation for FRC plasma rotation can begiven from a viewpoint of particle trajectories; it is notessentially different from discussion above. In a FRC
plasma, small­gyroradius drift orbits, figure­8 orbits, andbetatron orbits are three possible types of trajectories [18,19]. In contrast to the betatron particles, thesmall­gyroradius drift particles and the figure­8 particles
have smaller angular momentum. If the poloidal flux
decays, the figure­8 particles can change abruptly to thebetatron particles due to the increase of the Larmor radius.
The transition of trajectory type results in the increment of
the toroidal angular momentum.
The inductive electric field is also a possible cause of
toroidal spin­up. The betatron particles move around thefield­null. The toroidal electric field always accelerates thebetatron ions in the ion diamagnetic direction. Here, theradial E ´ B drift motion contributes to the betatronoscillation, and therefore the guiding center is fixed at theProceedings of ITC/ISHW2007
equilibrium position at which the centrifugal force of thebetatron motion and the Lorentz force are balanced.
Coulomb collisions of ions, however, breakconservation of Pq . The conversion process given above
may not work in this case. In the present paper, effects of
the ion­ion pitch­angle scattering on the spin­up of a FRC
plasma are investigated. We assume that the plasma ionsare fully thermalized, and therefore the slowing­down
collisions are neglected here.

2. Calculation Model
To investigate the time evolution of toroidal flow
velocity of a FRC plasma, plasma ions are traced
numerically. The fields are resistively decaying, and they
are modeled as the poloidal flux decay given by¶y= - rh J q .
(2)¶tHere, the electric resistivity h equals f Ah cl ,where f A is the anomaly factor and h cl is the classicalresistivity. The flux lifetime is controlled by theparameter f A . By integrating Eq. (2) by means of theRunge­Kutta method, the flux function y at a calculation
point is found. The electromagnetic fields are then writtenby the obtained y asæyö
B = Ñ ´ A = Ñ ´ ç eq ÷ ,
r
è
ø
1
J=
Ñ ´ B , E = hJ .
(3)
m0
For a detailed calculation, we often use
E = hJ - u e ´ B - Ñpe / (ene ) . When ions are at rest andJ ´ B = Ñpe + Ñpi is satisfied, then E » hJ + Ñpi / (ene ) .
The azimuthal electric drift due to the ion pressure gradientis in paramagnetic direction. Therefore, it is impossible toexplain ion spin­up by the electric drift due to the ionpressure gradient. The electric field due to both the Lorentz
force acting on electrons and the electron pressure gradientis neglected in calculating orbits of ions. Initially, ions areloaded in the r­z plane uniformly. The ion as asuper­particle is weighted by the non­shifted Maxwellian
distribution, and therefore the ions are initially at rest. The
ion temperature is set to be uniform inside the separatrix,
and it decreases with the flux function exponentially inopen field region. The initial electron density is calculated
to satisfy the Grad­Shafranov equation. The equation of
motion is solved for ions in B and E fields given by Eq.
(3), and the ion density and toroidal flow velocity are
obtained by a particle in cell method at each calculation
time step.
The ion­ion pitch­angle scattering is reproduced by aMonte Carlo method [20]. The pitch­angle is calculated as

[(

) ] ,

l n = lo (1 - n dt ) ± 1 - l2o n dt

1/ 2

v //
= cos q p , -1 £ cos q p £ 1 .
(4)v
Here, n d is the deflection collision frequency and t is thetime interval of random number generation. The
subscripts n and o mean new and old variables. We
choose either sign equally by the uniform random number
at the sign ± .
Calculation results are compared with the experiment
data measured on the NUCTE (Nihon University Compact
Torus Experiment)­III device. Typical experimentparameters of the NUCTE­III device are the external
magnetic field Bex =0.4 T, the coil radius rc = 0.17 m,
and the typical plasma parameters are the separatrix
radius rs = 0.05 m , the separatrix length l s = 0.40 m , theion temperature Ti » 100 eV, the electron temperature
Te » 100 eV and the density at the field­null
n0 = 2.0 ´ 10 21 m­3. A measured flux F at the field­null isshown in Fig. 1, where the flux F is equal to 2py . Gradualdecrease of the flux can be seen after completion of FRC
formation at about 8 ms. Decrement of the flux reaches
0.32 mWb by 50 ms, which corresponds to 35t A0 after
formation, where t A0 º rc / vA0 and vA0 º Bex / m 0 mi n0 .
When f A is 10, decrement of the calculated fluxfor
35t A0is
estimatedby
DF = 2p (Dy )» 2p (0.01y w ) » 0.31 mWb ; it is comparable to the
NUCTE­III experiment. Therefore, orbits of ions arecalculated at f A of 10, and then the resultant toroidal flow
velocity is compared with the NUCTE­III experiment.

l=Fig. 1 The time evolution of the maximum poloidal flux
measured at the NUCTE­III device.

3. Results and Discussion
To study an effect of flux decay on toroidal spin­up ofa FRC plasma, orbits of ions are calculated for thepresence of and absence of flux decay. The calculated time
evolution of the toroidal flow velocity normalized by
vA0 is shown in Fig. 2, where flux decay is present for Fig.
2(a) and is absent for Fig. 2(b). Here, the flow velocities atthe separatrix (the solid line) and at the field­null o­point(the dotted line) are presented separately. Gradual increases
are found at both the separatrix and o­point in Fig. 2(a),Proceedings of ITC/ISHW2007
whereas no increase can be seen in Fig. 2(b). By the timeof 35t A0 , the FRC plasma is found to rotate with thetoroidal velocity up to 0.35v A0 at the separatrix and up to
0.25v A0 at the o­point.

through the axial end. It is found that there is little
difference between two cases, because near the separatrix
few thermal ions are lost by 35t A0 even in the decaying
plasma. Therefore, it appears the effect of the ion loss on
spin­up of FRC plasma is negligible.

Fig. 3 The time evolution of toroidal velocity measured by
the Doppler shift of a spectral line of impurity
carbon at NUCTE­III device in Nihon University.

Fig. 2 The toroidal flow velocity evolution at the separatrix(the solid line) and at the field­null o­point (thedotted line). (a) The poloidal flux decays due toresistivity. (b) The flux decay is absent.

For comparison, the toroidal velocity measured by the
Doppler shift of the impurity line C4+is shown in Fig. 3.
The measured velocity can be read as the average along theline of sight, whose distance of closest approach from thegeometric axis is 4.5 cm; it locates initially between theseparatrix and o­point. As with Fig. 2(a), an almost linear
increase is found by the experimental result. Therefore, aqualitative agreement can be observed between
experimental and computational results of the toroidal
rotation velocity. Because vA0 = 138 km/s , the velocity of
20 km/s corresponds to about 0.15v A0 .A toroidal flow canbe observed even for the absence of flux decay as shown in
Fig. 2(b), although ions are assumed to obey thenon­shifted Maxwellian distribution. The ion diamagnetic
drift and absence of the radial and axial electric field may
play a role of the rotation velocity in Fig. 2(b).
Comparison of the time evolution of toroidal flow
velocity at the separatrix between the presence of andabsence of end­loss ions are made as shown in Fig. 4. The
axial direction of motion is reversed at the axial end of thecalculation region for the case w/o end loss. Therefore, inthis case no ions suffer from the end loss. Calculation oforbit stops for the case with end loss, when ions pass

Fig. 4 Comparison of the time evolution of toroidal flow
velocity at the separatrix between the presence ofand absence of end­loss ions. The axial directionof motion is reversed at the axial end of thecalculation region for the case w/o end loss (thedotted line). Calculation of orbit stops for the casewith end loss (the solid line), when ions passthrough the axial end.

When the ion­ion pitch­angle scattering is considered,
the particle loss fraction gradually increases with time dueto the classical transport process. The particle loss may
enhance the spin­up of a FRC plasma, we compare therotation velocity between the collisional case and thecollisionless case. The time evolution of the particle loss
fraction that is the ratio of the number of end­loss ions to
the number of all ions inside the confinement region isshown in Fig. 5. Although the flux decays, it is found thations are lost only before 20t A0 for the collisionless case.
About 10 % difference between the collisionless andcollisional case is observed until 35t A0 , and it may causea significant difference in the toroidal rotation velocity.
Comparison of the toroidal velocity between thecollisionless and collisional case is shown in Fig. 6.

Proceedings of ITC/ISHW2007
Higher fluctuation level is found for the collisionless case,and reduction of its level for the collisional case is due tothe viscosity. Except time fluctuation, however,insignificant difference is observed. Therefore, theparticle loss by the classical transport process is not be
responsible for the spin­up of a FRC plasma.

Therefore, the conversion process still plays an importantrole of the spin­up of a FRC plasma. The particle loss due
to classical transport has been found to have a small effecton the spin­up process. Though the toroidal flow affects theelectromagnetic fields and the ion motion, this effect isneglected in the present letter and is left as a subject for
future study.

Acknowledgement
This work is supported under the auspices of the
NIFS Collaborative Research Program (NIFS05KNXN033,
NIFS06KDBD003 and NIFS06KKMP003).

References[1]Fig. 5 The time evolution of the particle loss fraction. The
ion­ion pitch angle scattering is taken into accountfor the solid line and is not taken into account for the
dotted line.

Fig. 6 The time evolution of the toroidal rotation velocity at
the radial position of 4.5 cm. The solid and dottedlines are drawn as Fig. 5.

4. Summary
It has been shown that a FRC plasma can spin up
without any loss of ions because of the direct conversion
process from the poloidal flux to the kinetic angularmomentum of ions. Abrupt change of the type of ion
trajectory, such as the betatron, figure­8 andsmall­gyroradius drift orbits, is possible cause of spin­up ofa FRC plasma. The rotation velocity is calculated, and theresults are found to be in qualitative agreement with theexperiment at the NUCTE­III device.
In this paper, Coulomb collisions that breakconservation of the canonical angular momentum have also
been considered. If collisionality increases, the loss of flux
is not always converted to the angular momentum.
Insignificant difference of the toroidal velocity, however, isfound between the collisional and collisionless case.

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