Pro eedings of ITC/ISHW2007

Analysis of the Alpha Parti le Orbits
in the High Beta Plasma of the Large Heli al Devi e
Ryousuke SEKI , Yutaka MATSUMOTO , Yasuhiro SUZUKI , Kiyomasa WATANABE , andMasafumi ITAGAKI
1)

1)

2)

2)

1)

1)

Graduate S hool of Engineering, Hokkaido University, Sapporo 060-8628, Japan
2)

National Institute for Fusion S ien e, Toki 509-5292, Japan

We investigate the orbits of the 3.5 MeV alpha parti les in the high beta plasma of the Large Heli al Devi e(LHD), in whi h B = 6 T is assumed. For this purpose, we tra e the 100 keV proton in the ase of B = 1 T, theLarmor radius of whi h is almost the same as the Larmor radius of the alpha parti le, by numeri ally solving theguiding- enter equations. As a result, the parti le orbit hara teristi s in the high beta plasma of the LHD are
shown in detail. We point out the importan e to appropriately tra e the re-entering parti les in the analyses of theorbits of the high-energy parti les, su h as the alpha parti les, in the high beta plasma. It is found that the lifetimeof the alpha parti les depends on the lifetime of the haoti orbit parti les almost all of whi h is lost. The haotiorbit parti les are lost before they heat fuels be ause the lifetime of the haoti orbit parti les is mu h shorter
than the alpha-ele tron relaxation time.
Keywords: alpha parti le, parti le orbit, high beta, re-entering parti le, LHD,

1 Introdu tion
The volume-averaged beta-value h i have rea hed up to 5% in the re ent experiments of the Large Heli al Devi e(LHD)[1℄. In a Deuterium (D)-Tritium (T) fusion rea tor, alpha parti les move in the plasma the beta-value ofwhi h is more than 5 %. It is one of the important issues for the realization of a fusion rea tor to investigate
the onnement of the alpha parti les heating the D-T fuels. The magneti eld strength in a future fusion rea tor
is assumed B ' 6 T. The Larmor radius of the 3.5 MeV
alpha parti le in the ase of B = 6 T is almost the same asthat of the 100 keV proton in the ase of B = 1 T. Therefore, we tra e the 100 keV proton in the high beta plasmaof the LHD with Bax = 1 T to investigate the alpha parti leorbits.
In the high beta plasmas of the LHD, the magneti
ongurations hanges from the va uum magneti eld tothe followings. The ux surfa es shift to the dire tion of
the major radius due to the Shafranov shift aused by thenite beta e e t. Additionally, the volume inside the last
losed ux surfa e (LCFS) in the high beta plasma beomes small sin e the ux surfa es in the periphery of thehigh beta plasma are destroyed[2℄. In these ases, the number of the re-entering parti les[3, 4, 5℄, whi h repeatedly
pass into and out of the LCFS, ould be larger than that in
the va uum magneti eld. Thus, the re-entering parti lesmight play important roles for the plasma heating. But, there-entering parti les have been regarded as the loss partiles in the onventional studies on the orbits of the highenergy parti les in the high beta plasma of the LHD[6℄.
author's e-mail: sekifusion.qe.eng.hokudai.a .jp

We appropriately tra e the re-entering parti les in the highbeta plasma of the LHD.
The numeri al model and the initial onditions aregiven in Se tion 2. The results of the al ulations are summarized in Se tion 3. Se tion 4 is devoted to a summary.

2Numeri al Model

In order to tra e the parti les, we use the equilibrium magneti eld ( Bax = 1 T, h i = 2:7 % and Rax ' 3:9 m) alulated by using the three-dimensional magnetohydrodynami (MHD) equilibrium ode, HINT[7, 8℄, in whi h theexisten e of the nested ux surfa es are not assumed.
We tra e the parti les by numeri ally solving theguiding- enter equations in the ollisionless ase. In order
to tra e the re-entering parti les appropriately, the parti leloss boundary must set on the va uum vessel wall, i.e., theparti les rea hing the va uum vessel wall are regarded as
the loss parti les. Therefore, the rotating heli al oordinate system[9℄ is adopted. We use the 6th-order RungeKutta formulas[10℄ and the three-dimensional higher order
spline fun tion[11℄ to a urately tra e the ompli ated orbits of the parti les in the plasma periphery.
The initial onditions are determined as follows. As
mentioned in the pre eding se tion, the tra ed parti le isa proton and its initial energy is assumed to be 100 keV.
The starting points of protons are set on the horizontallyelongated poloidal plane asR= 2:65 + 0:05n (n = 0; 1;    ; 45) m
Z = 0 mRR

Pro eedings of ITC/ISHW2007

1.5

LCFS

ϕ=0magnetic axis

passing particle
banana orbit particle
chaotic orbit particle
prompt loss particle
loss particle

1.0χ0/π0.5Z (m)

LCFS

0.0
−0.5
−1.0
−1.5

R (m)

2.4

2.9

3.4

3.9

4.4

4.9

5.4

R (m)

Fig. 1 Poin are plots of the parti les on the horizontally elongated poloidal plane. Typi al orbits of the passing (blue),the haoti orbit (green), and the banana orbit parti les(yellow) are shown. These parti les are tra ed fromR = 4:1 m; Z = 0 m in the va uum magneti eld. Themagneti eld lines (gray) are also shown.
 = 0;(1)

where (R; Z; ) are the ylindri al oordinates; R is the major radius,  the toroidal angle. The initial pit h angles (0 )are varied from 0:05 to 0:95 with a step size of 0:05.
The protons with su h initial onditions are tra ed for aperiod of 30 ms.

3Results

Based on the results of the parti le orbits tra ing, we lassify the parti les into the four groups: passing parti les, banana orbit parti les, haoti orbit parti les[12℄, and prompt
loss parti les. Figure 1 is the Poin are plot of the parti leson the horizontally elongated poloidal plane. This shows
the typi al orbits of these four groups in the va uum magneti eld, in whi h Bax = 1 T. Both the passing (blue) andthe banana orbit parti les (green) make the losed drift surfa es. The haoti orbit parti les (yellow) repeatedly transit between the lo alized orbit and the blo ked orbit[13℄,and make no losed drift surfa e. Sin e the prompt loss
parti les are lost without any poloidal rotations, we annot
make the Poin are plots of the prompt loss parti les. Basedon this lassi ation, the orbits of the alpha parti les in thehigh beta plasma of the LHD are studied.

3.1 Orbit hara teristi s
Figure 2 shows the parti le lassi ations in the ase of
Bax = 1 T, h i = 2:7 % on the spa e of the starting points
versus the initial pit h angles. Horizontal axis is the major

Fig. 2 Parti le lassi ations on the spa e of the starting points
versus the initial pit h angles. Horizontal axis is R of
the starting points. Verti al axis denotes the initial pit hangles (0 ) divided by . The red squares show the loss
parti les.

radius of the starting points set on the line of Z = 0 m on
the horizontally elongated poloidal plane. Verti al axis denotes the initial pit h angles divided by . The positions of
the magneti axis and the LCFS on the line of Z = 0 m on
the horizontally elongated poloidal plane are also shown.
The red squares denote the loss parti les, i.e., the parti les
rea hing the va uum vessel wall within 30 ms.
Almost all the parti les with 0 ' 0:5 are the banana
and the haoti orbit parti les. Espe ially in the parti lestra ed from near magneti axis, most of the parti les with
0 ' 0:5 are the haoti orbit parti le. Almost all thehaoti orbit parti les are lost within 30 ms. The rangeof the initial pit h angles, with whi h the parti les are thepassing parti les, are maximum near R = 3:6 m. As thestarting points are lose to the LCFS, the su h range of theinitial pit h angles be omes narrow. There are no prompt
loss parti les tra ed from 3:05 m  R  4:0 m. Almost all
the starting points of the prompt loss parti les exist outside
the LCFS. Some parti les with 0 ' 0:35 or 0 ' 0:7are also the prompt loss parti les.

3.2 Re-entering parti le
In order to investigate the e e ts of the re-entering parti es on the onnement of the alpha parti le, we evaluate
the loss parti le ratio. Figure 3 shows the loss parti le ratio
averaged over pit h angles[12℄ at ea h starting points after
tra ing parti les for 30 ms. The bla k line is the loss parti le ratio in the ase of re-entering parti les appropriatelytra ed. The red line represents the loss parti le ratio in thease of the re-entering parti les regarded as the loss parti les. This ase is the same as the onventional study, inwhi h the parti le loss boundary is set on the LCFS. The
positions of the magneti axis and the LCFS on the line ofZ = 0 m on the horizontally elongated poloidal plane arePro eedings of ITC/ISHW2007

magnetic axis

LCFS

LOSS PARTICLE RATIO

1.2

LCFS
re‑entering
no re‑entering

1.0
0.8
0.6
0.4
0.2
0.0

3.0

3.5

4.0

4.5

R (m)

Fig. 3 Loss parti le ratio after 30 ms parti le tra ing. The bla kline is the loss parti le ratio in the ase of re-entering parti les appropriately tra ed and the red line the loss partile ratio in the ase of the re-entering parti les regarded
as the loss parti les. Horizontal axis is R of the starting
points. Verti al axis denotes the loss parti le ratio averaged over the pit h angles at ea h starting points. The positions of the magneti axis and the LCFS are also shown.

also shown.
The signi ant di eren e between the bla k and thered lines an be seen from Fig. 3. This result implies thatmany re-entering parti les exist in the high beta plasma.
Espe ially, the di eren e between the bla k and the redlines is remarkable in the parti les tra ed from R ' 3:3 m.
This means that there exist many re-entering parti les in
the parti les tra ed from R ' 3:3 m.
The loss parti le ratio are overestimated in the aseof the re-entering parti les regarded as the loss parti les.
Thus, it is important to appropriately tra e the re-entering
parti les in the analyses of the orbits of the high-energy
parti les, su h as the alpha parti les, in the high beta
plasma.

3.3 Connement of alpha parti le
We investigate the onnement of the alpha parti les generated by the D-T fusion rea tion. As mentioned above,
the Larmor radius of the 3.5 MeV alpha parti le in the aseof B = 6 T is almost the same as that of the 100 keV protonin the ase of B = 1 T. The results of the 100 keV protonin the ase of B = 1 T ould be regarded as that of the 3.5
MeV alpha parti le in the ase of B = 6 T.
The birth points of the alpha parti les produ ed by theD-T fusion rea tion depend on the plasma pressure prole.
The number of alpha parti les is large in the region of thehigh plasma pressure, namely, the plasma ore. Thus, wefo us on the parti les tra ed from the plasma ore (3:3 m R  4:2 m). It is seen from Fig. 2 that the prompt loss

parti les hardly exist in the parti les tra ed from 3:3 m R  4:2 m. Both the passing and the banana orbit parti lesare onned in the va uum vessel wall for 30 ms. On theother hand, almost all of the haoti orbit parti les is lost.
Therefore, the lifetime of the alpha parti les is determinedby the lifetime of the haoti orbit parti les.
The lifetime of the haoti orbit parti les is estimatedas  10 3 s in the present study. In order to ompare
this lifetime with the ollision times, we assume the D-T
plasma with the ion temperature T i = 10 keV, the ion density ni = 1021 m 3 , the ele tron temperature T e = 10 keV,and the ele tron density ne = 1021 m 3 . In su h plasma,
the alpha-ion dee tion time d i ' 0:9 s and the alphaele tron relaxation time r e ' 1:6  10 2 s. The lifetimeof the haoti orbit parti les is mu h shorter than theseollision times. Therefore, the haoti orbit parti les arelost before they heat fuels. In addition, the passing andthe banana orbit parti les onned in our al ulations beome the haoti orbit parti les by the pit h angle s attering during many toroidal and/or poloidal rotations. Thus,
the passing and the banana orbit parti les ould not suÆiently heat the fuels by the Coulomb ollision.
We have analyzed the orbit of the alpha parti le in theurrent LHD (the major radius R0 = 3:9 m, the averaged
plasma minor radius ap ' 0:64 m[14℄). But, the size ofthe heli al rea tor, whi h satises the ignition ondition, isassumed as R0 = 10 m and ap = 2 m[15℄. In su h a heli alrea tor, the region, in whi h the haoti orbit parti les anmove, be omes large. Therefore, the haoti orbit parti lelost in the present study ould be onned for a long timeenough to heat the fuels.
Instead of the plasma heating by the Coulomb ollision, the plasma heating alled the alpha- hanneling isproposed[16℄. In the alpha- hanneling, the fuels are heatedby the wave in the plasma, whi h the alpha parti le amplies during the time s ale shorter than ollision times. If
the alpha- hanneling is used, the alpha parti les ould heat
the fuels independent of the lifetimes of the alpha parti les.

4Summary

We have investigated the orbits of the 3.5 MeV alpha partiles in the high beta plasma of the LHD, in whi h B = 6 T.
For this purpose, we have tra ed the 100 keV proton in thease of B = 1 T, the Larmor radius of whi h is almost thesame as the Larmor radius of the alpha parti le, by numeri ally solving the guiding- enter equations. The following
information has been obtained.
The passing parti les play a large part in the parti les
tra ed from the plasma ore. On the other hand, mostof the parti les with 0 ' 0:5 are the haoti orbit
parti les. They are lost within 30 ms.
 It is important to appropriately tra e the re-entering
parti les in the analyses of the orbits of the highPro eedings of ITC/ISHW2007

energy parti les, su h as the alpha parti les, in thehigh beta plasma.
 The lifetime of the alpha parti les depends on the lifetime of the haoti orbit parti les almost all of whi his lost.
 The haoti orbit alpha parti les are lost before theyheat fuels be ause the lifetime of the haoti orbit alpha parti les is mu h shorter than the alpha-ele tron
relaxation time.
 In the future heli al rea tor, thehaoti orbit alphaparti le lost in the present study ould be onned fora long time enough to heat the fuels be ause the sizeof the future heli al rea tor will be larger than that of
the LHD.
 Through the alpha- hanneling, the alpha parti lesould heat the fuels independent of the lifetimes of
the alpha parti les.
In the near future, the alpha parti les in the higher beta
plasma (h i > 5 %) will be studied. We will investigate thealpha parti le orbit by numeri ally solving the equation ofmotion be ause the Larmor radius of the alpha parti le islarge.
This work is suported by the ollaborative resear hprogram of NIFS (NIFS06KLHH302).
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