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Rotating a tokamak plasma with waves for impurity control
Volume 93A, number 9
PHYSICS LETTERS
14 February 1983
ROTATING A TOKAMAK PLASMA WITH WAVES FOR IMPURITY CONTROL H. SUGAI Department of Electrical Engineering, Nagoya University, Nagoya 464, Japan Received 10 May 1982 Revised manuscript received 2 August 1982
A new scheme for radio-frequency impurity control is proposed, in which a tokamak plasma is rotated with uni-directional waves in the ion-cyclotron frequency range. The rapid toroidal rotation can be induced with acceptable power dissipation by the same waves used in ICRF heating.
The use of momentum transfer due to neutral, beam injection to expel impurity ions out of a tokamak plasma has been proposed [ 1] and the idea has recently been tested in some experiments on the impurity study experiment (ISX-B) and Princeton large torus (PLT) tokamaks [2,3]. The early theories which considered only the direct beam=impurity interaction suggested that counter-injection expels impurities while co-injection drives them in. However, experimental results [2,3] have shown outward flow of impurities with co-injection. This apparent paradox has been explained in the framework of neoclassical theories [4,5] by taking into account the rapid plasma rotation induced by the tangentially injected beam. Inducing a toroidal rotation could be the basis of an impurity-control technique for tokamak fusion reactors. In this paper, we propose to use the momentum and/or energy of a wave, instead of neutral-beam injection, to rotate the plasma. In this scheme of RF impurity control, wave-particle interactions generate the toroidal rotation of the ions which drag impurity ions by friction. This parallel impurity flow can give rise to outward impurity convective flow, if it is sufficiently large. The underlying physical basis of the wave-particle interactions is common to current with mass (m) in the derivation o f J / P d in the current generation theory [6] so that c/e a
: ,i [~. v(oJv)]/0, r E ) .
(1)
Here G is the momentum density parallel to the static 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
magnetic field in the z direction, ~ is the unit vector along the displacement in velocity space, E = m(V2z + 02)/2 is the kinetic energy of resonant panicles, oz and oz are the parallel and perpendicular velocities with respect to the magnetic field, respectively, and v is the effective collision frequency. Two types of wave-particle interactions are distinguished: the "momentum input" type where sllz, and the "energy input" type where }J.~. The first corresponds to a Landau damping process, say, for lower hybrid waves, where both momentum and energy are fed to the particles. The second type corresponds to a cyclotron damping process, as in electron cyclotron heating, where the perpendicular gyration energy predominantly increases while the parallel momentum remains unchanged. An accurate determination of v is momentarily deferred; suppose, however, that v 0 -3 in terms of the particle speed v = (02 + 02) 1/2 . Then, eq. (1) can be rewritten as a / P d = (8 + 3 v J v ) / z , o ,
(2)
where ~ = 1 for the momentum input type and 5 = 0 for the energy input type. The resonant velocity oz is far larger than the thermal velocity and o 2 ~ 02 ~, 02. Thus, it follows from eq. (2) that G/P d is 4/3 times larger in the momentum input type than in the energy input type. In view of G/Pd, it is straightforward to compare the momentum G e induced by the wave-electron interaction with the momentum G i due to the w a v e 483
Volume 93A, number 9
PHYSICS LETTERS
ion interaction, when the same amount of power is dissipated. Suppose a plasma composed of electrons, main ions, and impurity ions. Their masses, charge states, densities, and thermal velocities are denoted by me, Zo, no, and OTo, respectively, for the species o = e, i, I. For resonant electrons with a velocity oz much larger than the electron thermal velocity ore, two coUisional scattering rates are given from the Fokker-Planck equation in the high-velocity limit; an energy loss rate vE = Voe/2u3 and a momentum destruction rate v/v1 = (2 + Z 2 + nlZ2/ne)VE, where V0e = 6o4 In A/2rrneO3Te and u = O/Ore. An exact determination of v has been made along the lines outlined in ref. [6] to give v = v0e(5 + Z? + nlZ2/ne)/2u 3 .
(3)
On the other hand, for resonant ions with the velocity
oz in the range OTl, Ori ~ °z ~ °Te, a similar treatment provides the energy loss rate vE = ~Oi[1 + (mi/mi) × (nlZ2/niZ2)]/2u 3 and the momentum destruction rate vM = v0i [2 + (1 + mi/mi)(nlZ?/niZ?)]/2u 3 , where
z,0i = ~ 4 InA/2rmio3i,
COpi= (n~2e2/mieo)l/2,
U = O/OTi,
and the small electron contribution has been neglected. Using these collision rates, we find v = v0i[5 + (1 + 4mi/ml)(nlZ2/niZ2)][2u 3 .
(4)
Substituting eqs. (3) or (4) into eq. (2) yields
Ge/a i .~ l~0iOTi/l~0eOTe~. (me/mi) (Te/Ti)Z~i "~ 1. Namely, the efficiency G/P d is much higher in the wave-ion interaction than in the wave--electron interaction. Therefore, we should adopt the method to transfer the wave momentum (energy) to ions, not to electrons, in contrast to the current generation using electrons mostly. Since the average ion velocity Fiz is derived from G i = min(fiz, we can estimate the rf power necessary for driving the plasma velocity up to Fiz/or i. In the case of deuterium as the main-ion species, the result is expressed from eqs. (2) and (4) as
484
Pd = (1.0 × 108)
14 February 1983
5 + (1 + 8mp/ml)nlZ2/ni 2u (6u + 3w)
X Tlll/2n24(Oiz/oTi)W/nt3
,
(5)
where w = Oz/OTi, u = V/VTi , m p is the proton mass, the ion temperature/'to is normalized to 10 keV, and the density n14 is nom~alized to 1014 cm -3 . The resonant velocity is usually set in the range 3 < w < 7. To get a simpler expression, we assume w ~ u ~- 5 and the trace-impurity lianit nlZ2/ni Z2 ~ 1. The latter assumption is valid for argon and iron in ISX-B [2] and is marginally satisfied for tungsten in PLT [3]. Then we have Pd = (3.3 X 106)T{J/2n24(Oiz/VTi ) W/m 3 for the energy input type (6 = 0). For exampie, the dissipated power 330 kW/m 3 for T10 = 1 and n14 = 1 yields the plasma velocity Oiz/VTi = 0.1, which is comparable to or greater than the thermal velocity of heavy impurities. The neoclassical theory [5] has shown outward transport of impurities when the velocity of toroidal rotation of the main ions of the plasma exceeds the impurity thermal velocity. In practice, there are several ways to realize the wave-ion interactions. Among them, the most promising one may be the ICRF heating at the second cyclotron harmonic, since waveguides can be used as launcher. Waveguides are very convenient structures for bringing fast waves into the plasma without introducing impurities from the launcher. In the ICRF heating, the wave-ion interaction can be regarded as the energy input type where no net parallel momentum is injected. The momentum of resonant ions is destroyed dominantly by the bulk ions in the traceimpurity limit. By momentum conservation, the bulk ions must drift in the opposite direction to the resonant ions, with the magnitude °iz given in eq. (5). This means that, strictly speaking, the average ion velocity vanishes if the ion velocity is integrated all over the distribution function. In the impurity transport problem, however, the bulk ions act as the dominant friction source for impurities since they are very slow and dense compared with the resonant ions. Thus, we regard the "average" velocity viz in eq. (5) as the "effective" velocity. In present day tokamak experinaents, the ICRF heating at the secc,nd harmonic has not been established yet. Another ICRF heating scheme called twoion hybrid resonance (minority ion) heating is rather
Volume 93A, number 9
PHYSICS LETTERS
well understood [7]. So, it is interesting to check the possibility of an induced plasma rotation in the minority ion heating. Suppose minority protons are introduced in a plasma containing impurities. The region of interest here is characterized by the resonant proton velocity oz in the range VTi, 0Tp < oz
X POp/2U3,
(6)
where
POp = Uoi(mi/mp)l/2(Ti/Tp)3/2 /Z~i ,
u = o/OTp .
The three terms at the right-hand side ofeq. (6) express the momentum destructions by electrons, main ions, and impurity ions. The bulk protons contribute little to p as long as they are a minority (np/n i ~ 1). Minority heating is regarded as an energy input type so that eq. (1) may be rewritten as
G/P d = ~ mpOz(Vi + pI)p2E ,
(7)
where vi and uI indicate the second and third terms respectively in the right-hand side of eq. (6). From eqs. (6) and (7), the power to drive the main ion (deuterium) to the velocity ~iz(= G/mini) in the trace-impurity limit is given by Pd = (5 X 107) (1 + 4X 10-3u3) 2 wu
X Tll/2n24(~iz/vTi)W/m3 ,
(8)
where w = Vz/OTp and the temperatures are assumed to be Te = Ti = T,. For example, w ~ u ~ 5 yields Pd = (4.5 X 106)T~ol/2n24 (~iz/OTi)W/m 3 , which is comparable to the value for the ICRF heating at the second harmonic. A toroidal rotation of the plasma cannot be expected to occur in the present minority-heating experiments [7], since the waves are excited in both directions from the launcher. It is necessary for inducing plasma rotation that the uni-directional wave be launched with a phased waveguide array. If the waves travel to the right, then the minority ions move, due to the
14 February 1983
asymmetric colllsional friction, to the left; the cyclotron resonance velocity is in the.opposite direction to the phase velocity. The collisional friction predominantly comes from the main ions in the trace impurity limit (vI ,~ vi). Since no net parallel momentum is injected into the plasma, the main-ion species must drift to the right. Thus, the plasma rotation (main-ion rotation) takes place in the direction of the wave phase velocity. This relation between the rotating direction and the phase velocity stands also in the second harmonic heating scheme. It must be appreciated that the value Viz does not coincide with the velocity of the plasma rotation in the actual experiment. The extent of the plasma rotation must be determined from momentum-balance equations including the ambipolar potential, momentum sinks, and collisionalfrictions,although such calculations are beyond the scope of the present paper. On the other hand, within the framework of neoclas:sicaltheories [1,4,5], the external momentum input S altersthe radial flux of impurities via the toroidal collisionalfrictioncoupled with the poloidal magnetic fieldBp. Therefore, the direction and the magnitude of S X B_p are the crucial points in the impurity control. If abundant toroidal current was generated together with the plasma rotation, then it would modify the poloidal field,which in turn would change not only the impurity transport but also the main plasma confinement. However, littlecurrent is driven in the ICRF heatings discussed above; again by momentum conservation, the bulk ion current cancels the resonant ion current in the second harmonic heating, while the main-ion (deuterium) current cancels the minority-ion (proton) current because ofZ i = Zp = l in the minority.species heating. In view of the radio.frequency control of the impurity transport* 1,2 there is a possibility to rotate the plasma in the poloidal direction by waves. Another possibility is to inject the wave momentum and energy into the impurity ions directly [10,11]. However, these types of control are beyond the scope of this paper. In conclusion, a new way of controlling impurities , t Solid rotation of a toroidal plasma by magnetic pumping has been investigated in ref. [8]. ,2 A referee noticed a similar independent work [9] on the impurity flow reversal in tokamaks without momentum input.
485
Volume 93A, number 9
PHYSICS LETTERS
with radio-frequency in a tokamak is proposed. It is a three-step rate process. The wave energy is first transferred to the resonant main ions via ICRF heating. A toroidal plasma bulk main ions rotation is then induced by asymmetric friction as a result o f the heating. Finally, the impurities are expelled out o f the plasma via the induced rapid plasma rotation. An estimation o f the dissipated power supports the practicality o f this scheme in a tokamak reactor. Finally, this m e t h o d o f RF-driven tokamak rotation could also be useful to stop neutral beam induced rotation if it should be deleterious for conf'mement. The author would like to thank Professor T. Okuda for his encouragement and Dr. S. Takamura for his interest in this work.
References [ 1 ] T. Ohkawa, Kaku Yugo Kenkyu 32 (1974) 1 ; also available as General Atomic Company Report No. GA-A12926, unpublished.
486
14 February 1983
[2] R.C. Isler et al., Phys. Rev. Lett. 47 (1981) 649. [31 D.R. Eames, Ph.D. thesis, Princeton University (1980), unpublished. [4] W.M. Stacy Jr. and D.J. Sigmar, Nucl. Fusion 19 (1979) 1665. [5] K.H. Burrell, T. Ohkawa and S.K. Wong, Phys. Rev. Lett. 47 (1981) 511. [6] N.J. Fish and A.H. Boozer, Phys. Rev. Lett. 45 (1980) 720. [7] J. Jacquinot, B.D. McVey and J.E. Scharer, Phys. Rev. Lett. 39 (1977) 88; J. Hosea et al., Phys. Rev. Lett. 43 (1979) 1802; K. Odajima et al., Nucl. Fusion 20 (1980) 1330. [8] R. Deicas and A. Samain, in: Proe. 5th Int. Conf. Plasma physics and controlled nuclear fusion research, Tokyo, 1974, Vol. 1 (IAEA, Vienna, 1975) p. 563; G. Requin and A. Samain, in: Proc. 8th Int. Conf. Plasma physics and controlled nuclear fusion research, Brussels, Belgium, 1980, Vol. 1 (IAEA, Vienna, 1981) p. 755. [9] F.L. Hinton, University of Texas report, IFSR No. 43 (Oct. 1981). [10] H. Sugai, Kaku Yugo Kenkyu 47 (1982) 285;Phys. Lett. 91A (1982) 73. [11 ] TFR Group, Association Euratom-CEA sur la Fusion, Fontenay-aux-Roses, Internal Report EUR-CEA-FC 1131 (1981).
PHYSICS LETTERS
14 February 1983
ROTATING A TOKAMAK PLASMA WITH WAVES FOR IMPURITY CONTROL H. SUGAI Department of Electrical Engineering, Nagoya University, Nagoya 464, Japan Received 10 May 1982 Revised manuscript received 2 August 1982
A new scheme for radio-frequency impurity control is proposed, in which a tokamak plasma is rotated with uni-directional waves in the ion-cyclotron frequency range. The rapid toroidal rotation can be induced with acceptable power dissipation by the same waves used in ICRF heating.
The use of momentum transfer due to neutral, beam injection to expel impurity ions out of a tokamak plasma has been proposed [ 1] and the idea has recently been tested in some experiments on the impurity study experiment (ISX-B) and Princeton large torus (PLT) tokamaks [2,3]. The early theories which considered only the direct beam=impurity interaction suggested that counter-injection expels impurities while co-injection drives them in. However, experimental results [2,3] have shown outward flow of impurities with co-injection. This apparent paradox has been explained in the framework of neoclassical theories [4,5] by taking into account the rapid plasma rotation induced by the tangentially injected beam. Inducing a toroidal rotation could be the basis of an impurity-control technique for tokamak fusion reactors. In this paper, we propose to use the momentum and/or energy of a wave, instead of neutral-beam injection, to rotate the plasma. In this scheme of RF impurity control, wave-particle interactions generate the toroidal rotation of the ions which drag impurity ions by friction. This parallel impurity flow can give rise to outward impurity convective flow, if it is sufficiently large. The underlying physical basis of the wave-particle interactions is common to current with mass (m) in the derivation o f J / P d in the current generation theory [6] so that c/e a
: ,i [~. v(oJv)]/0, r E ) .
(1)
Here G is the momentum density parallel to the static 0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
magnetic field in the z direction, ~ is the unit vector along the displacement in velocity space, E = m(V2z + 02)/2 is the kinetic energy of resonant panicles, oz and oz are the parallel and perpendicular velocities with respect to the magnetic field, respectively, and v is the effective collision frequency. Two types of wave-particle interactions are distinguished: the "momentum input" type where sllz, and the "energy input" type where }J.~. The first corresponds to a Landau damping process, say, for lower hybrid waves, where both momentum and energy are fed to the particles. The second type corresponds to a cyclotron damping process, as in electron cyclotron heating, where the perpendicular gyration energy predominantly increases while the parallel momentum remains unchanged. An accurate determination of v is momentarily deferred; suppose, however, that v 0 -3 in terms of the particle speed v = (02 + 02) 1/2 . Then, eq. (1) can be rewritten as a / P d = (8 + 3 v J v ) / z , o ,
(2)
where ~ = 1 for the momentum input type and 5 = 0 for the energy input type. The resonant velocity oz is far larger than the thermal velocity and o 2 ~ 02 ~, 02. Thus, it follows from eq. (2) that G/P d is 4/3 times larger in the momentum input type than in the energy input type. In view of G/Pd, it is straightforward to compare the momentum G e induced by the wave-electron interaction with the momentum G i due to the w a v e 483
Volume 93A, number 9
PHYSICS LETTERS
ion interaction, when the same amount of power is dissipated. Suppose a plasma composed of electrons, main ions, and impurity ions. Their masses, charge states, densities, and thermal velocities are denoted by me, Zo, no, and OTo, respectively, for the species o = e, i, I. For resonant electrons with a velocity oz much larger than the electron thermal velocity ore, two coUisional scattering rates are given from the Fokker-Planck equation in the high-velocity limit; an energy loss rate vE = Voe/2u3 and a momentum destruction rate v/v1 = (2 + Z 2 + nlZ2/ne)VE, where V0e = 6o4 In A/2rrneO3Te and u = O/Ore. An exact determination of v has been made along the lines outlined in ref. [6] to give v = v0e(5 + Z? + nlZ2/ne)/2u 3 .
(3)
On the other hand, for resonant ions with the velocity
oz in the range OTl, Ori ~ °z ~ °Te, a similar treatment provides the energy loss rate vE = ~Oi[1 + (mi/mi) × (nlZ2/niZ2)]/2u 3 and the momentum destruction rate vM = v0i [2 + (1 + mi/mi)(nlZ?/niZ?)]/2u 3 , where
z,0i = ~ 4 InA/2rmio3i,
COpi= (n~2e2/mieo)l/2,
U = O/OTi,
and the small electron contribution has been neglected. Using these collision rates, we find v = v0i[5 + (1 + 4mi/ml)(nlZ2/niZ2)][2u 3 .
(4)
Substituting eqs. (3) or (4) into eq. (2) yields
Ge/a i .~ l~0iOTi/l~0eOTe~. (me/mi) (Te/Ti)Z~i "~ 1. Namely, the efficiency G/P d is much higher in the wave-ion interaction than in the wave--electron interaction. Therefore, we should adopt the method to transfer the wave momentum (energy) to ions, not to electrons, in contrast to the current generation using electrons mostly. Since the average ion velocity Fiz is derived from G i = min(fiz, we can estimate the rf power necessary for driving the plasma velocity up to Fiz/or i. In the case of deuterium as the main-ion species, the result is expressed from eqs. (2) and (4) as
484
Pd = (1.0 × 108)
14 February 1983
5 + (1 + 8mp/ml)nlZ2/ni 2u (6u + 3w)
X Tlll/2n24(Oiz/oTi)W/nt3
,
(5)
where w = Oz/OTi, u = V/VTi , m p is the proton mass, the ion temperature/'to is normalized to 10 keV, and the density n14 is nom~alized to 1014 cm -3 . The resonant velocity is usually set in the range 3 < w < 7. To get a simpler expression, we assume w ~ u ~- 5 and the trace-impurity lianit nlZ2/ni Z2 ~ 1. The latter assumption is valid for argon and iron in ISX-B [2] and is marginally satisfied for tungsten in PLT [3]. Then we have Pd = (3.3 X 106)T{J/2n24(Oiz/VTi ) W/m 3 for the energy input type (6 = 0). For exampie, the dissipated power 330 kW/m 3 for T10 = 1 and n14 = 1 yields the plasma velocity Oiz/VTi = 0.1, which is comparable to or greater than the thermal velocity of heavy impurities. The neoclassical theory [5] has shown outward transport of impurities when the velocity of toroidal rotation of the main ions of the plasma exceeds the impurity thermal velocity. In practice, there are several ways to realize the wave-ion interactions. Among them, the most promising one may be the ICRF heating at the second cyclotron harmonic, since waveguides can be used as launcher. Waveguides are very convenient structures for bringing fast waves into the plasma without introducing impurities from the launcher. In the ICRF heating, the wave-ion interaction can be regarded as the energy input type where no net parallel momentum is injected. The momentum of resonant ions is destroyed dominantly by the bulk ions in the traceimpurity limit. By momentum conservation, the bulk ions must drift in the opposite direction to the resonant ions, with the magnitude °iz given in eq. (5). This means that, strictly speaking, the average ion velocity vanishes if the ion velocity is integrated all over the distribution function. In the impurity transport problem, however, the bulk ions act as the dominant friction source for impurities since they are very slow and dense compared with the resonant ions. Thus, we regard the "average" velocity viz in eq. (5) as the "effective" velocity. In present day tokamak experinaents, the ICRF heating at the secc,nd harmonic has not been established yet. Another ICRF heating scheme called twoion hybrid resonance (minority ion) heating is rather
Volume 93A, number 9
PHYSICS LETTERS
well understood [7]. So, it is interesting to check the possibility of an induced plasma rotation in the minority ion heating. Suppose minority protons are introduced in a plasma containing impurities. The region of interest here is characterized by the resonant proton velocity oz in the range VTi, 0Tp < oz
X POp/2U3,
(6)
where
POp = Uoi(mi/mp)l/2(Ti/Tp)3/2 /Z~i ,
u = o/OTp .
The three terms at the right-hand side ofeq. (6) express the momentum destructions by electrons, main ions, and impurity ions. The bulk protons contribute little to p as long as they are a minority (np/n i ~ 1). Minority heating is regarded as an energy input type so that eq. (1) may be rewritten as
G/P d = ~ mpOz(Vi + pI)p2E ,
(7)
where vi and uI indicate the second and third terms respectively in the right-hand side of eq. (6). From eqs. (6) and (7), the power to drive the main ion (deuterium) to the velocity ~iz(= G/mini) in the trace-impurity limit is given by Pd = (5 X 107) (1 + 4X 10-3u3) 2 wu
X Tll/2n24(~iz/vTi)W/m3 ,
(8)
where w = Vz/OTp and the temperatures are assumed to be Te = Ti = T,. For example, w ~ u ~ 5 yields Pd = (4.5 X 106)T~ol/2n24 (~iz/OTi)W/m 3 , which is comparable to the value for the ICRF heating at the second harmonic. A toroidal rotation of the plasma cannot be expected to occur in the present minority-heating experiments [7], since the waves are excited in both directions from the launcher. It is necessary for inducing plasma rotation that the uni-directional wave be launched with a phased waveguide array. If the waves travel to the right, then the minority ions move, due to the
14 February 1983
asymmetric colllsional friction, to the left; the cyclotron resonance velocity is in the.opposite direction to the phase velocity. The collisional friction predominantly comes from the main ions in the trace impurity limit (vI ,~ vi). Since no net parallel momentum is injected into the plasma, the main-ion species must drift to the right. Thus, the plasma rotation (main-ion rotation) takes place in the direction of the wave phase velocity. This relation between the rotating direction and the phase velocity stands also in the second harmonic heating scheme. It must be appreciated that the value Viz does not coincide with the velocity of the plasma rotation in the actual experiment. The extent of the plasma rotation must be determined from momentum-balance equations including the ambipolar potential, momentum sinks, and collisionalfrictions,although such calculations are beyond the scope of the present paper. On the other hand, within the framework of neoclas:sicaltheories [1,4,5], the external momentum input S altersthe radial flux of impurities via the toroidal collisionalfrictioncoupled with the poloidal magnetic fieldBp. Therefore, the direction and the magnitude of S X B_p are the crucial points in the impurity control. If abundant toroidal current was generated together with the plasma rotation, then it would modify the poloidal field,which in turn would change not only the impurity transport but also the main plasma confinement. However, littlecurrent is driven in the ICRF heatings discussed above; again by momentum conservation, the bulk ion current cancels the resonant ion current in the second harmonic heating, while the main-ion (deuterium) current cancels the minority-ion (proton) current because ofZ i = Zp = l in the minority.species heating. In view of the radio.frequency control of the impurity transport* 1,2 there is a possibility to rotate the plasma in the poloidal direction by waves. Another possibility is to inject the wave momentum and energy into the impurity ions directly [10,11]. However, these types of control are beyond the scope of this paper. In conclusion, a new way of controlling impurities , t Solid rotation of a toroidal plasma by magnetic pumping has been investigated in ref. [8]. ,2 A referee noticed a similar independent work [9] on the impurity flow reversal in tokamaks without momentum input.
485
Volume 93A, number 9
PHYSICS LETTERS
with radio-frequency in a tokamak is proposed. It is a three-step rate process. The wave energy is first transferred to the resonant main ions via ICRF heating. A toroidal plasma bulk main ions rotation is then induced by asymmetric friction as a result o f the heating. Finally, the impurities are expelled out o f the plasma via the induced rapid plasma rotation. An estimation o f the dissipated power supports the practicality o f this scheme in a tokamak reactor. Finally, this m e t h o d o f RF-driven tokamak rotation could also be useful to stop neutral beam induced rotation if it should be deleterious for conf'mement. The author would like to thank Professor T. Okuda for his encouragement and Dr. S. Takamura for his interest in this work.
References [ 1 ] T. Ohkawa, Kaku Yugo Kenkyu 32 (1974) 1 ; also available as General Atomic Company Report No. GA-A12926, unpublished.
486
14 February 1983
[2] R.C. Isler et al., Phys. Rev. Lett. 47 (1981) 649. [31 D.R. Eames, Ph.D. thesis, Princeton University (1980), unpublished. [4] W.M. Stacy Jr. and D.J. Sigmar, Nucl. Fusion 19 (1979) 1665. [5] K.H. Burrell, T. Ohkawa and S.K. Wong, Phys. Rev. Lett. 47 (1981) 511. [6] N.J. Fish and A.H. Boozer, Phys. Rev. Lett. 45 (1980) 720. [7] J. Jacquinot, B.D. McVey and J.E. Scharer, Phys. Rev. Lett. 39 (1977) 88; J. Hosea et al., Phys. Rev. Lett. 43 (1979) 1802; K. Odajima et al., Nucl. Fusion 20 (1980) 1330. [8] R. Deicas and A. Samain, in: Proe. 5th Int. Conf. Plasma physics and controlled nuclear fusion research, Tokyo, 1974, Vol. 1 (IAEA, Vienna, 1975) p. 563; G. Requin and A. Samain, in: Proc. 8th Int. Conf. Plasma physics and controlled nuclear fusion research, Brussels, Belgium, 1980, Vol. 1 (IAEA, Vienna, 1981) p. 755. [9] F.L. Hinton, University of Texas report, IFSR No. 43 (Oct. 1981). [10] H. Sugai, Kaku Yugo Kenkyu 47 (1982) 285;Phys. Lett. 91A (1982) 73. [11 ] TFR Group, Association Euratom-CEA sur la Fusion, Fontenay-aux-Roses, Internal Report EUR-CEA-FC 1131 (1981).