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Toroidal plasma resonances and electron transit time damping for heating large thermonuclear tori
Volume 49A, number 6
PHYSICS LE~FERS
21 October 1974
TOROIDAL PLASMA RESONANCES AND ELECTRON TRANSIT TIME DAMPING FOR HEATING LARGE THERMONUCLEAR TORI A.M. MESSIAEN* and P.E. VANDENPLAS laboratoire de Physique des ?iasmas, Association “Euratom-Etat beige” Ecole RoyaieMilitaire, 1040 Brussels, Belgium Received 3 September 1974 We advocate a new method consisting of combining the large amplification ofthe r.f. field strength inside the plasma due to a magnetosonic resonance or to a torsional wave resonance of the toroidal cavity with its coilision-less damping resulting from electron transit time damping.
A new generation of large tokamaks intended to test the possibility of attaining physical parameters closer to thermonuclear ignition are being designed; see, e.g., Tb, the Princeton Large Torus (PLT) and the Joint European Torus (JET). Since 1969, we have been stressing [1—3)the importance of bounded plasma resonances (some of which are new) in connection with a realistic evaluation of r.f. heating properties of thermonuclear machines. We are advocating a new method [3, 4] consisting of combining the amplification of the r.f. field strength inside the plasma due to a bounded magnetosonic resonance (or to a torsional wave resonance) with collisionless damping of this high field resulting from electron transit time damping at co k11 Ve. Both ions and electrons are efficiently heated because the energy equipartition time is suffiently small with respect to the energy confIne~ ment time of large machines. As in [3], we consider the cylindrical approximation of a toroidal magnetized plasma column described by hot ion-electron fluid theory, excited by a set of coils placed around the torus and surrounded by an outer metal wall. For brevity’s sake, in this letter we only consider the axisymmetric excitation (n = 0) resulting from 2m circular loops evenly distributed along the torus with 1800 phase difference in the currents flowing in adjacent ioops. Then the toroidicity appears in the exp i [(nO+ k11z) ~~tJcylindrical harmonic expansion of the exciting field by the fact that k11 = hm/R (Ii = ±1, ±3,...) where the z-axis is directed —
*Maltre recherche de t’Institut Interuniversitaire des SciencesdeNuclEaireg.
along the axis of the plasma cylinder, 0 is the azimuthal coordinate around it, and R is the major radius of the torus. The losses in the plasma are described by effective collision frequencies for momenturn transfer v~(a = e for electrons and a = i for ions) whose analytical expressions take into account
(Q ~z
1)
cm A
______________
I
0 L_0015cm
10
=
I i,
-
io~
,~
1
~-----
io~
/
/
(a)_
/ ~
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I W~1t
T
//
“-‘
I
k,,V11 I
0.1 MHZ
I
k~IIVA
/
/ /
I
//
I
1MHz
I
I
/
I
10MHz
I
W121t
Fig 1(a). Plasma loading ~.R11,/A.zversus w with collisionless damping when n = 0 and 7= = 108 K. (b) Same 3, B as (a) except T— 10 K. (C) Wall loadmg ~.R1 [AZ corresponding to (a). Other parameters are: N = 4.5 x i~’~ cm 0= 60kG, deuterium,a=9Ocm,b 110 cm,d 130cm,1. R = 333 cm, h as = m 1, m[3]. 5, a of wall = 3 X i0~(Ii cmi Same symbols 475
Volume 49A, number 6
PHYSICS LE11~ERS
B2.o~,
1
/
0003
001
________________
003
jk,,I=00074cm 1
01
Fig. 2. Frequency of the first n = 0 Alfv~nbounded resonances versus 1k 11 I. The dashed line = 1k11 V~Iindicates the gion of maximum electron 3cm3B transit time absorption. Para- 8K, meters are: N 4 X 10’ 0 = 45 kG, ~ = r. = 10 deuterium, a = 110 cm, d = 150 cm. e 1
Cherenkov absorption by electrons (w k11 Ve) and ions (b., k11 V~)together with cyclotron absorption near w = ~ ~ = eB0/m; V~ = kTalma). Note that electron Cherenkov absorption and electron transit time damping (or Landau) are used here synonymously, i.e., they mean collisionless absorption at 0.) k11 Ve [5, 6]. Fig. 1 displays typical absorption curves for the lowest 1k11 I of the axial Fourier spectrum of a system of 10 coils placed around the torus. ~.R1/~z is the resistance per unit length due to the loading of the coil by the plasma [(a) and (b)J or by the wall (c). Curves, qualitatively not too different, are also obtained when dipolar excitation (n = ±1) is considered [81. The following absorption domains or resonances can be distinguished in these frequency spectra: (i) Ion TTMP: c~r”1k V11. An ion absorption continuum due to ion TTMP is seen around 1k11 J’~Iand values of absorbed power agree with standard values [5, 6). Power absorption by the wall is high. (ii) Electron transit time damping: ~ Ik11V~I.A broad domain of electronic absorption due to electron transit time damping around co = 1k11 ~l is seen in fig. 1. Furthermore bounded plasma resonances due to the excitation of compressional (“B” resonances) and torsional Alfven waves (“A” resonances) exist in this domain and they strongly enhance the absorption. This is an essential phenomenon to be further dis—
476
21 October 1974
‘~
6
10
20
~
w/2Th(M~Z) 40 60
Figplasma to 3. Electron for lowest transit 1k time region (n = 0). Coil loading due 111 mode and also total coil loading resulting from the sum over all k11’s. Same parameters as for fig. 2 plus R = 270 cm, m ~2.
cussed in (iii). The power absorbed by the plasma is higher than that absorbed by the wall. (iii) Alfvén wave bounded resonances. These resonances are very important for this letter and their typical frequency dependence versus k11 is shown in fig. 2. One distinguished: (a) The “A” resonance (labelled A in the figs.) When (k11 VA/wCi) ~ I (VA: Alfve’n velocity), a main resonance, as carefully defined in [2, 3], due to a complex combination of coupling to the torsional A1fv~nwave and to the compressional Abfve’n wave is observed in the vicinity of their confluence at o. 1k11 VA I. (b) Magnetoacoustic resonances (labelled B in the figs.) These magnetoacoustic resonances [7) appear when w> 1k11 VA I and correspond to radial compressiorial Alfve’n waves having a given periodicity along the major circumference of the torus. They occur for the successive zeros of Re {J1~,(k~a)}whereJ~~1 is the Bessel function and k~= f(k11 ,w,JV~coca‘~‘a’Ta) the radial wavenumber of the compressional wave [2, 3]. These resonances are labelled B1,0 (p = 1,2,...) in fig. 2 where n = 0. (iv) Ion cyclotron resonance absorption: co An absorption peak is observed very near ~ and is due to a narrow region of ion cyclotron damping of width 1k11 Vu; the coupling to the r.f. source can be strongly enhanced by a magnetosonic resonance. Summation over all the contributions of the space
Volume 49A, number 6
PHYSICS LETTERS
21 October 1974
harmonics has been carried out for 4 axisymmetric loops (n = 0) placed along the torus. The total loading resistance due to the plasma is presented in fig. 3. That due to the wall (not shown) is of the order of 30—40% of that of the plasma. The intricate magnetoacoustic resonance pattern is due to the k11 -frequency
R.W. Gould for his hospitality and his constructive
dependence (fig. 2) of the various resonances exhibited. Further details and data together with results around lower hybrid (wave-guide excitation) and with
[11 P.E. Vandenplas, A.M. Messiaen, J.-L. Monfort,and
a detailed comparison of the respective merits of the present scheme, of ICRH heating and of ion TTMP, are given in a forthcoming paper [8]. The conclusion is that the excitation of a bounded magnetoacoustic resonance (or of a “torsional” A resonance) at a frequency f~rwhich there is considerable absorption due to electron transit time damping appears as the most promising rf method using coil excitation. Note that this method is possible because for low (3 machine ~ VA when additional heating is required. The manuscript of this paper was written while one of the authors (P.V.) was Visiting Associate at the California Institute of Technology under U.S.A.E.C. Contract No. AT(04-3)-767. He warmly thanks Prof.
comments.
References
J.J. Papier, Phys. Rev. Lett. 22 (1969) 1243; Plasma PhYS. 12 (1970) 391. [2] A.M. Messiaen, P.E. Vandenplas, Nuclear Fusion 11 (1971) 556; Plasma Phys. 15 (1973) 505. [3] A.M. Messiaen, P.E. Vandenplas, R.R. Weynants and R. Koch, VI European Conf. on Controlled fusion and plasma physics, Joint Institute for Nuclear Research, Moscow (1973) 545; Report 61, Lab. Phys. Plasmas, Ecole Royale Militaire (1974), submitted for publication. [4] A.M. Messiaen and P.E. Vandenplas, Discussion of if heating of JET, report 60, Lab. Phys. Plasmas, Ecole Royale Militaire (1973). [5] T.H. Stix, The theory of plasma waves (McGraw-Hill, NewYork 1962), pp. 186—207. [6] E. Canobbio, Nuclear Fusion 12 (1972) 561. [7] D. Frank-Kamenetskii, Soviet Phys. JETP 12 (1961)
469. [8] A.M. Messiaen and P.E. Vandenplas, Paper IAEA-CN33/C4-2, 5th IAEA Conf. on Plasma physics and controlled nuclear fusion, Tokyo (Nov. 1974).
477
PHYSICS LE~FERS
21 October 1974
TOROIDAL PLASMA RESONANCES AND ELECTRON TRANSIT TIME DAMPING FOR HEATING LARGE THERMONUCLEAR TORI A.M. MESSIAEN* and P.E. VANDENPLAS laboratoire de Physique des ?iasmas, Association “Euratom-Etat beige” Ecole RoyaieMilitaire, 1040 Brussels, Belgium Received 3 September 1974 We advocate a new method consisting of combining the large amplification ofthe r.f. field strength inside the plasma due to a magnetosonic resonance or to a torsional wave resonance of the toroidal cavity with its coilision-less damping resulting from electron transit time damping.
A new generation of large tokamaks intended to test the possibility of attaining physical parameters closer to thermonuclear ignition are being designed; see, e.g., Tb, the Princeton Large Torus (PLT) and the Joint European Torus (JET). Since 1969, we have been stressing [1—3)the importance of bounded plasma resonances (some of which are new) in connection with a realistic evaluation of r.f. heating properties of thermonuclear machines. We are advocating a new method [3, 4] consisting of combining the amplification of the r.f. field strength inside the plasma due to a bounded magnetosonic resonance (or to a torsional wave resonance) with collisionless damping of this high field resulting from electron transit time damping at co k11 Ve. Both ions and electrons are efficiently heated because the energy equipartition time is suffiently small with respect to the energy confIne~ ment time of large machines. As in [3], we consider the cylindrical approximation of a toroidal magnetized plasma column described by hot ion-electron fluid theory, excited by a set of coils placed around the torus and surrounded by an outer metal wall. For brevity’s sake, in this letter we only consider the axisymmetric excitation (n = 0) resulting from 2m circular loops evenly distributed along the torus with 1800 phase difference in the currents flowing in adjacent ioops. Then the toroidicity appears in the exp i [(nO+ k11z) ~~tJcylindrical harmonic expansion of the exciting field by the fact that k11 = hm/R (Ii = ±1, ±3,...) where the z-axis is directed —
*Maltre recherche de t’Institut Interuniversitaire des SciencesdeNuclEaireg.
along the axis of the plasma cylinder, 0 is the azimuthal coordinate around it, and R is the major radius of the torus. The losses in the plasma are described by effective collision frequencies for momenturn transfer v~(a = e for electrons and a = i for ions) whose analytical expressions take into account
(Q ~z
1)
cm A
______________
I
0 L_0015cm
10
=
I i,
-
io~
,~
1
~-----
io~
/
/
(a)_
/ ~
\
I W~1t
T
//
“-‘
I
k,,V11 I
0.1 MHZ
I
k~IIVA
/
/ /
I
//
I
1MHz
I
I
/
I
10MHz
I
W121t
Fig 1(a). Plasma loading ~.R11,/A.zversus w with collisionless damping when n = 0 and 7= = 108 K. (b) Same 3, B as (a) except T— 10 K. (C) Wall loadmg ~.R1 [AZ corresponding to (a). Other parameters are: N = 4.5 x i~’~ cm 0= 60kG, deuterium,a=9Ocm,b 110 cm,d 130cm,1. R = 333 cm, h as = m 1, m[3]. 5, a of wall = 3 X i0~(Ii cmi Same symbols 475
Volume 49A, number 6
PHYSICS LE11~ERS
B2.o~,
1
/
0003
001
________________
003
jk,,I=00074cm 1
01
Fig. 2. Frequency of the first n = 0 Alfv~nbounded resonances versus 1k 11 I. The dashed line = 1k11 V~Iindicates the gion of maximum electron 3cm3B transit time absorption. Para- 8K, meters are: N 4 X 10’ 0 = 45 kG, ~ = r. = 10 deuterium, a = 110 cm, d = 150 cm. e 1
Cherenkov absorption by electrons (w k11 Ve) and ions (b., k11 V~)together with cyclotron absorption near w = ~ ~ = eB0/m; V~ = kTalma). Note that electron Cherenkov absorption and electron transit time damping (or Landau) are used here synonymously, i.e., they mean collisionless absorption at 0.) k11 Ve [5, 6]. Fig. 1 displays typical absorption curves for the lowest 1k11 I of the axial Fourier spectrum of a system of 10 coils placed around the torus. ~.R1/~z is the resistance per unit length due to the loading of the coil by the plasma [(a) and (b)J or by the wall (c). Curves, qualitatively not too different, are also obtained when dipolar excitation (n = ±1) is considered [81. The following absorption domains or resonances can be distinguished in these frequency spectra: (i) Ion TTMP: c~r”1k V11. An ion absorption continuum due to ion TTMP is seen around 1k11 J’~Iand values of absorbed power agree with standard values [5, 6). Power absorption by the wall is high. (ii) Electron transit time damping: ~ Ik11V~I.A broad domain of electronic absorption due to electron transit time damping around co = 1k11 ~l is seen in fig. 1. Furthermore bounded plasma resonances due to the excitation of compressional (“B” resonances) and torsional Alfven waves (“A” resonances) exist in this domain and they strongly enhance the absorption. This is an essential phenomenon to be further dis—
476
21 October 1974
‘~
6
10
20
~
w/2Th(M~Z) 40 60
Figplasma to 3. Electron for lowest transit 1k time region (n = 0). Coil loading due 111 mode and also total coil loading resulting from the sum over all k11’s. Same parameters as for fig. 2 plus R = 270 cm, m ~2.
cussed in (iii). The power absorbed by the plasma is higher than that absorbed by the wall. (iii) Alfvén wave bounded resonances. These resonances are very important for this letter and their typical frequency dependence versus k11 is shown in fig. 2. One distinguished: (a) The “A” resonance (labelled A in the figs.) When (k11 VA/wCi) ~ I (VA: Alfve’n velocity), a main resonance, as carefully defined in [2, 3], due to a complex combination of coupling to the torsional A1fv~nwave and to the compressional Abfve’n wave is observed in the vicinity of their confluence at o. 1k11 VA I. (b) Magnetoacoustic resonances (labelled B in the figs.) These magnetoacoustic resonances [7) appear when w> 1k11 VA I and correspond to radial compressiorial Alfve’n waves having a given periodicity along the major circumference of the torus. They occur for the successive zeros of Re {J1~,(k~a)}whereJ~~1 is the Bessel function and k~= f(k11 ,w,JV~coca‘~‘a’Ta) the radial wavenumber of the compressional wave [2, 3]. These resonances are labelled B1,0 (p = 1,2,...) in fig. 2 where n = 0. (iv) Ion cyclotron resonance absorption: co An absorption peak is observed very near ~ and is due to a narrow region of ion cyclotron damping of width 1k11 Vu; the coupling to the r.f. source can be strongly enhanced by a magnetosonic resonance. Summation over all the contributions of the space
Volume 49A, number 6
PHYSICS LETTERS
21 October 1974
harmonics has been carried out for 4 axisymmetric loops (n = 0) placed along the torus. The total loading resistance due to the plasma is presented in fig. 3. That due to the wall (not shown) is of the order of 30—40% of that of the plasma. The intricate magnetoacoustic resonance pattern is due to the k11 -frequency
R.W. Gould for his hospitality and his constructive
dependence (fig. 2) of the various resonances exhibited. Further details and data together with results around lower hybrid (wave-guide excitation) and with
[11 P.E. Vandenplas, A.M. Messiaen, J.-L. Monfort,and
a detailed comparison of the respective merits of the present scheme, of ICRH heating and of ion TTMP, are given in a forthcoming paper [8]. The conclusion is that the excitation of a bounded magnetoacoustic resonance (or of a “torsional” A resonance) at a frequency f~rwhich there is considerable absorption due to electron transit time damping appears as the most promising rf method using coil excitation. Note that this method is possible because for low (3 machine ~ VA when additional heating is required. The manuscript of this paper was written while one of the authors (P.V.) was Visiting Associate at the California Institute of Technology under U.S.A.E.C. Contract No. AT(04-3)-767. He warmly thanks Prof.
comments.
References
J.J. Papier, Phys. Rev. Lett. 22 (1969) 1243; Plasma PhYS. 12 (1970) 391. [2] A.M. Messiaen, P.E. Vandenplas, Nuclear Fusion 11 (1971) 556; Plasma Phys. 15 (1973) 505. [3] A.M. Messiaen, P.E. Vandenplas, R.R. Weynants and R. Koch, VI European Conf. on Controlled fusion and plasma physics, Joint Institute for Nuclear Research, Moscow (1973) 545; Report 61, Lab. Phys. Plasmas, Ecole Royale Militaire (1974), submitted for publication. [4] A.M. Messiaen and P.E. Vandenplas, Discussion of if heating of JET, report 60, Lab. Phys. Plasmas, Ecole Royale Militaire (1973). [5] T.H. Stix, The theory of plasma waves (McGraw-Hill, NewYork 1962), pp. 186—207. [6] E. Canobbio, Nuclear Fusion 12 (1972) 561. [7] D. Frank-Kamenetskii, Soviet Phys. JETP 12 (1961)
469. [8] A.M. Messiaen and P.E. Vandenplas, Paper IAEA-CN33/C4-2, 5th IAEA Conf. on Plasma physics and controlled nuclear fusion, Tokyo (Nov. 1974).
477