Suppression of internal plasma oscillations by trapped high energy nuclei

Volume 132, number 5

PHYSICS LETTERS A

10 October 1988

SUPPRESSION OF INTERNAL PLASMA OSCILLATIONS BY TRAPPED HIGH ENERGY NUCLEI B. COPPP, R.J. HASTIE2 S. MIGLIUOLO1, F. PEGORARO and F. PORCELLI

JETJoint Undertaking, Abingdon, Oxon., 0X14 3EA, UK

Received 13 June 1988; accepted for publication 19 July 1988 Communicated by R.C. Davidson

The suppression of macroscopic oscillations (so-called “sawteeth”) ofthe central region of a magnetically confined plasma column is related to the effect ofmagnetically trapped energetic nuclei produced by the injection ofrfwaves orneutral beams. We evaluate the threshold for, and describe the process involved in, the transition of m°= I modes, which trigger the crash phase of sawtooth oscillations, to a regime that is shown to be stable.

In experiments carried out by the JET machine, the internal electron temperature relaxation oscilations (the so-called “sawtooth” oscillations), which are associated with the excitation of a mode with m°=I, n°=1 poloidal and toroidal mode numbers, have been suppressed for periods of up to 3.2 s during high power injection of ion-cyclotron frequency waves (with coupled power P~?~ 4 MW) and/or neutral beams (with PNBI ~ 7 MW) [11. This observation cannot be explained within the framework of ideal or resistive mhd models, even when the stabilising effects of the ion diamagnetic frequency and of the electron drift wave frequency are considered [2]. In fact, according to these models, stability would occur either when the value q0 of the q parameter on the magnetic axis exceeds unity, or for sufficiently small values of the plasma poloidal beta and low collisionality. Instead, Faraday rotation measurements [31indicate that q0~ 0.7, while high poloidal betas fl~(ro)often in excess ofthe ideal mhd instability threshold [4] P~are reached during sawtooth-free periods in JET. Here, fir, (r0) [8,t/ B~(ro)][ —p(r0)], where B~is the poloidal magnetic field, r0 is the mean radius of the q= 1 surI

2

. Massachusetts Institute ofTechnology, Cainbndge, MA 02139, USA. UKAEA, Culhaxn Laboratory, Abingdon, Oxon., 0X14 3DB, UK.

face, p is the plasma pressure, and is its average value within the q= 1 volume. Specifically we find [5] fl~,(r 0) >0.2 at moderate plasma currents (I~2 MA), while taking into account the effects of plasma shaping [5,6], fl~~0.l—O.2. JET discharges which exhibit sawtooth suppression are characterised by the presence ofanisotropic energetic nuclei, accelerated by radio frequency fields, and/or produced by energetic neutral beam injection (with injection energy ~ ~ 80 keV). In preimmary analyses [7,8] it was pointed out that the energetic trappednuclei can indeed play a stabilising role. Possibly the best experimental clue is provided by “ICRH switch off” experiments [9]. In these cxperiments, the stable period is terminated by a sudden interruption of the applied if fields (no neutral beams are injected in these cases). A time delay of the order of 100 ms is observed between the rfswitch off time and the collapse of the central electron temperature. This time delay is a finite fraction of the fast nuclei slowing down time, therefore suggesting that the loss of these nuclei is the dominant destabilising effect*I. Plasma stabilisation by energetic particle po~ulations was proposed and analysed for the case of Astron [101 and ion ring devices [11], and for the ~ For footnote see next page.

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PHYSICS LETTERS A

ELMO bumpy torus [12]. In axisymmetric toroidal configurations, stability against ballooning modes was theoretically shown to improve in the presence of magnetically trapped fast nuclei (see, e.g. ref. [13]). For the case ofthe m°=I mode, energetic nuclei were

10 October 1988

Hw),

12j(Ymhd

[(O(WWdI)]’

(2)

where H~ It 3s 0

~oflph

—

(3)

~

WDh

initially considered [14,15] in order to show that they can destabilise a branch of the m°=1 dispersion relation. In fact, following the analysis of ref. [2], the relevant dispersion relation in the absence of suprathermal particles is

8it

$

B~ ( r0) X$dr(—) r

where w is the mode frequency (in the frame of ref-

so= roq’ (r

erence the equilibrium Ex B drift vanishes at the q = where 1 surface), WdIE [—c(dp 1/dr) /enBrJ0 is the bulk ion diamagnetic frequency at r= r0 and Ymhd is the growth rate found by the ideal mhd approximation [4]. When Ymhd
WA

~,

“effective viscosity” arising from the resOnance between this mode and the trappedenergetic nuclei with bounce averaged magnetic drift frequency w~&iual to w. This resonant interaction is most effective when (001 w11~, with thDh the of characteristic wf~ over the distribution the energetic value nuclei,of The “sawtooth-free” regimes in JET are generally characterised [61 by I ~~ < WDh, while the

PJh]~

0) is the dimensionless shear 2is parameter, the Alfvén vA/Ro~J~, and VA = B/ (47tm1n~)” velocity. Acceptable roots ofeq. (2) must satisfy the condition Yrnin HRe (w) >0 for the corresponding eigenfunction to be spatially regular [14]. Purely Oscillatory modes are obtained from (2) for 3~,/4Y2mhd). (4) H?~ He,. WdL (1 — W At H= Her, W = Ymhd/ (1 + H~r)”2~ ~0di and ~ HW = Yrnhd[1—(1+ H;2) 1/2] As H is increased alove Y,,,~,d/w 0~,only the lower frequency root satisfies the condition ymho—Hco> 0. Stabilisation arises because energy must be spent to perturb the energetic trapped nuclei when their magnetic drift velocity is largerthan mode phase velocity. criteWith 2 the (ftfllhd)2] the stability [4] y(4) ~forW,~ E~ [ft non ftp> /J~’° can be rewritten as —

—

—

—

>

aS 0 r0

ratio between the transverse pressure of the trapped energetic nuclei PIh and the bulk pressure p can be as high as the inverse aspect ratio e~ r0/ T0. In this case, as shown by the derivation in the second part of this Letter, the dispersion relation (1) for modes with frequency 0)< (ih-~h is modified into

The short duration of the delay rules out the possibility that

the instability is induced by a significant diffusion ofthe qprofile between the switch-off time and the collapse time. Moreover the delayis too long to suggest that ponderomotive effects are important, as these should cease almost instantaneously after the rf power is switched off. An even stronger argument against ponderomotive effects is that sawteeth at JET have recently been suppressed with neutralbeams only.

268

1/2

(1)

W(W—Wd~)= —Y~hd,

Ill

312d ~_[(L)

~

2

ftp

Dh

(5)

~—‘

where a is a numerical factor of order unity which depends primarily on the q profile. Condition (5) depends on the density ofthe energetic nuclei and on their spatial profiles, and, since thDh ~ A, is independent oftheir mean energy ‘h’ It is consistent with the plasma parameter values (fi~h 0.05, A ~ 100 3, fl keY, O),-~h/Wdj~ 0n~0.2—0.3,and R/r0~6—7)of the regimes where sawtooth oscillations are suppressed in JET. In addition, when off-axis if heating is applied, the profile of p~h may not be sufficiently peaked to fulfill condition (5) and this can explain why sawteeth are difficult to suppress in this case. As the value of H is increased and the marginal

Vàlume 132, number 5

PHYSICS LETTERS A

10 October 1988

stability condition is fulfilled, a term representing the mode particle resonance co= ci$~Qhas to be added on the r.h.s. of eq. (2). Specifically H is to be replaced by H[l+ig(coR/c~h)], where w~~Re(w)and g(w~/a)~,)depends on the velocity distribution of the energetic nuclei. For the model given by eq. (13),

i.e

312. With the addition of the resonant However, (2) isceases contribution, damped. the the to lower modes frequency with frequency root of geq. reduces to (w~/á~h) CU
as

the equilibrium distribution of the energetic nuclei, but generally it involves values of H considerably larger than Hcr in eq. (4). be excited [15]. The second threshold depends on The conclusions given above do not take into account the effect of the small but finite electron resistivity. When this is included, the relevant dispersion relation is [2,141.

0.._~ -

oH~O

~ H.H,,

• H-3H~

, /

7/

~

1.2 H,,

—--.-

—.

°~

I

05

/ / .

// .

/

0~

[W(W_0)dj)]”2i[ymhd”H0)(l+ig)]

X Q3121’((Q—l)/4)

(6)

where Q2 ico(a— di )w— ~*e) /W~~ with ~ ~c2sO/4ItrOwA the inverse magnetic Reynolds number. In the experiments of interest, ~ is typically 10—v—i 0~.The parameter Ymhd can become [5] larger than the chacteristic growth rate A ~ of resistive m°=I modes, while 0)A I I I I~ with (7).~,e=[(TeC/eBrfl)(dfl/dr)(1+1.7lr7e)]o and 1e=d hi T~/dlnn.The dispersion relation (2) is recovered from (6) to lowest order by taking I Q I <1. In this limit, treating the effect of resistivity as a perturbation, we find that the mode-article resonance overcomes the destabiising effect of resistivitywhen 2(ymh0_HWR)HCoRg>~jWi(CoR (7) At HHcr, eq. (7) reduces to CUdj/E~/3CUA~ (áDh/codl)”2 for g(co~/th~~,) ~ For larger values of Wdj/~cOAwe find a range in H where both roots of eq. (2) are stable. This is illustrated in fig. 1 where the two roots, as obtained from the numerical integration of the eigenvalue problem leadjug to eq. (6), are followed in the complex frequency ~

(w~/á~h)312.

—ic’ 0.0

I

I

I

0.5

I

I 1.0

Fig. I. Path in the complex plane of the two relevant roots of the dispersion relation (6), for increasing Hand forthe following set of parameters: Ymhd 1i2th1,e/3(ThDh/3 (all curves); (Od~/ oA=4.S~(solid curve); Wdi/WA=1.5~’3 (dashedeurves). The solid curves show complete stabilisation for values of 1 .2H~H~314,. The roots with higher values of~¼violate the condition Vmhd~HWR>0 (needed for their spatial regularity) at the indicated x-points.

plane as a function of H for two different values of Wdi/E~13WA. The oscillation frequency is reduced when H is increased above H~ 1.As a consequence, the mode damping due to the resonance becomes weaker. A3,similar effect for values beand thus the occurs dispersion relationOf(6) does low WAV not yield complete stabiisation when However, in this limit and for 0)dj/0)A> 2a~/3,the viscous dissipation resulting from bulk ion collisions effectively replaces the resonant damping and ensures stability [161, independently of the presence of the trapped energetic nuclei. A problem remains for more realistic values ofJET Ymhd

Ymhd~~°.

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Volume 132, number 5

PHYSICS LETTERS A

experimental parameters, corresponding to the dashed curves of fig. 1. In this case the instability is not completelysuppressed even though the ideal mhd growth rate is strongly depressed. However one should bear in mind that in this case the occurrence of magnetic reconnection is necessary and that the linearised mode description breaks down for rather small amplitudes, corresponding to a width of the magnetic island of the order of the mode transition layer. Since this layer becomes of the order of the ion

10 October 1988

[14,19] in terms of their perturbed perpendicular pressure fl± h ( w) ~K (co) ~

J —,t

=

4~2i

—

B~

2

0s0~ Jdr r

dO [e~X1c~Vfi±h(w)]exp(iwt—iP), 2x

—

(9)

where e 11 = B/B and x= (e1~ V )e11. The solution for ~ given above, which is valid outside a small layer around x=0, in a region where the inertial and finite ion Larmor radius being containedin eq. (6) are un.

gyroradius, the mode amplitude for which the linear stability analysis breaks down is very modest. Another point to be considered is that the set of equations that we have adopted to describe the possible occurrence of resistive modes may not be adequate, given the low rate of collisionality of the regime of interest. The main analytical steps which lead to conclusions we have given are now outlined. Wethe assume the orderingpj h ~p.In addition, since we limit our considerations to a population of trapped energetic nuclei, we have PIIh ~ h~ The dominant component of the lagrangian displacement is represented by ~=~(r)exp ( iwt+ IF) where P= C—fl, C and 0 are the poloidal and toroidal angles, respectively. The total momentum conservation equation is written in the form —

nm,

d2~

=

—

V’H+

1

(Jx B+ Jx~), C

—

important, can be matched with the relevant asymptotic solution within this layer. Then we arrive at the dispersion relation 2=jwAEAH+~K(O))], (10) [w((0—W*I)1”

of which eq. (2) is a special case, showing that Yhd =

We derive fi~h(w) from the perturbed distribution functions neglecting effects relatedto the presence ofa radial equilibrium electric field. Solving the linearised Vlasov equation in the drift kinetic approximation,3~,can be conveniently split in two parts: )~ =7~+J~. The adiabatic part is 3’ ~,° = VFO~,,with FQh the equilibrium distribution function energetic For w< bh, with cow,ofthe the fast nuclei trappednuclei. bounce frequency, and ne—

(8)

where the pressure tensor H contains the contributions of both the background plasma and the energetic nuclei. Eq. (8) can be reduced to an equation in ~ using a standard procedure [17]. Then we expand ç~in powers of~ order ~we = 0( —x)~!o,where ~0.T isoalowest constant, x=in(r— r find

glecting finite banana orbit effects, the non-adiabatic part to lowest order in ~ is ,7 ~ad j’ ~ad X exp( iwt+ iS), where S= C— qO and —

2~nad— Jh

~r W—W~Th ~ Rwwf~~

[cos(q0)]~°~,

(11)

0) /r0,

and 0(x) is the Heaviside function. To the next relevant order, [14,17] for x.~1, i.e. approaching the we q= 1find surface, d~~r —

dx

— —

H +AK

(w)

ir.s 2 0x

2,which is derivede.g. 0 ( ~o$p) to the mhd energy inThe refs.parameter [4,6,18], ~‘H is ~ proportional functional 6W and is such that A 1~>0when /~1~> $~lhd The parameter )~K 0 (eoflph) represents the

contribution of the energetic nuclei and is expressed 270

with (ôF0/ä ~)W~h = —e11 XVFOh~VP/mhQh, and 2. TheThe superscript “(o)”and indicates particle orbit averaging. phases ofJ~° 7~° differ by f= mhv/ a factor S— P= (1— q) 0, as in the considered frequency range Jgad must be constant to lowest order along the trapped particle orbit, i.e. E The parameter ‘1K is also split in two 1 ‘v7i~° terms,=0. AK(w).~°+2~°(w),

where

Volume 132, number 5

2a0_ K —

J,’ $

0s0R0~ dr

d0j3~\,cos 0,

0

,~nod_ K —

2ic B~0SOR0~ $rdr 0

fd0i5~cos(qO)

—it

(12)

and ~ = ~ fd3v v~jj~. The perturbed perpendicular pressure ~ of the energetic nuclei depends rather sensitively on their equilibrium distribution function Foh. For the. present analysis, we take Foh =

~A) \2

X exp where A

3/2

(

flh (r) K( 1/2) \/~

)s\

[

(R)1”2 \2r

(A

—

1

)]~

(13)

2B 1 /v) 0/B is the pitch angle variable in velocity space, B0 is the value of B on the axis, B0,’ B ~ 1 + (r/R) cos 0, and K(x) is the elliptic integral of the second kind (K( 1/2) ~ 1.85). For simplicity A is taken to be a constant value A 100 keV. This

~K(O)

+Ak~J)/~Dh

+#%K,res

(14) where AK(O)~A~( +)~(O) and AK,res represents the

contribution of the resonant nuclei. We observe [201that I )‘K(O) /t~I ~ 1— q0. In.this Letter, we treat A.K(O) as a correction to )~H. When the simple analytic form of FOb given in eq. (13) is used, the coefficient of the linear term becomes Ak (it! 3s0 )~oPpb which corresponds to the expression for H entering eq. (2). We note that a distribution function which includes passing particles, as would be the case during —

similar to (12), where 2~is contributed, with opposite signs, by both passing and trapped particles, and 2~by trapped particles only. A near cancella2K is smaller. tion of A~occurs for p,~, ~ As a result, at constant energy content of the fast power nuclei, levels are reThis can explain why higher quired to suppress the sawtooth oscillations with quasi-tangential beam injection. In adopting the expansion (14) for 2K~ we have disregarded the possibility that modes with co/

~ 1 become significant [151.The reason for this is that the actual distribution function Foh is cxpected, unlike expression (13), to have a spread in the pitch angle variable A. Preliminary results from our numerical analysis [71 indicate that the threshold value of I1,.,~,for the onset of modes in this high frequency range is significantly larger than the value CODh

(v

distribution function represents trapped energetic nuclei with a magnetic turning point at 00=x/2. It can be used as a rough model for the anisotropic energetic nuclei tail which is formed during ICRH heating in a toroidal magnetic confinement configuration, since trapped nuclei having the tips oftheir banana orbits on the if resonant layer along a vertical chord across the centre of the plasma column tend to absorb relatively more power from the launched wave compared with the other nuclei. For 0)/thDh < 1, AK(w) becomes simply ‘%K((0)

10 October 1988

neutral beam injection, yields an expression for ‘1K

ro

2n

B~

PHYSICS LETTERS A

required to stabilise modes with (o< ~ It was the discovery of this “macroscopic stability window” which induced us to first report [7,81the possibility that sawtooth oscillations are suppressed by the presence of energetic trapped particles. Useful discussions with D. Campbell, D.F. Dllchs and J. Wesson are gratefully acknowledged. Two of us (B.C. and S.M.) have been sponsored in part by the U.S. Department of Energy.

References [I] D. Campbell et al., Phys. Rev. Lett. 60 (1988) 2148. [2] 0. Ara, B. Basu, B. Coppi, 0. Laval, M.N. Rosenbluth and B.V. Waddell, Ann. Phys. (NY) 112 (1978) 443. [3]J. O’Rourke et at., in: Proc. 15th Eur. Conf. on Controlled

fusion and plasma heating, Vol. 3 (Dubrovnik, Yugoslavia, 1987) p. 155. [4] M.N. Bussac, R. Pellat, D. Edery and J.L. Soulé, Phys. Rev. Lett.35 (1975) 1638. [5] F. Pegoraro, F. Porcelli, W. Han, A. Taroni and F. Tibone, to be published. [6] F. Porcelli, B. Coppi and S. Migliuolo, Bull. Am. Phys.Soc.

32 (1987) 1770. [8] F. Porcelli and F. Pegoraro, II European fusion theory meeting (Varenna, December 1987). [9] D. Campbell et al., in: Proc. 15th Eur. Conf. on Controlled fusion and plasma heating, Vol. 1 (Dubrovnik, Yugoslavia,

1987) p. 377.

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[10] N.C. Christofilos, in: Proc. 2nd UN Int. Coal’. on the Peaceful uses of atomic energy, Vol. 32 (UN, Geneva, 1958) p.

279. [ll)R.N. SudanandE.Ott,Phys~Rev. Lett. 38 (1974) 355. [12] R.A. Dandl et al., in: Plasma physics and controlled nuclear fusion research 1974, Vol. 2 (IAEA, Vienna, 1975) p. 141. [l3]M.N. Rosenbiuth, S.T. Tsai, J.W. Van Dam and M.G. Engquist, Phys. Rev. Lett. 51(1983)1867. [14] B. Coppi and F. Porcelli, Phys. Rev. Lett. 57 (1986) 2272. [15] L. Chen, R.B. White and M.N. Rosenbiuth, Phys. Rev. Lett. 52 (1984) 1122.

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[16] F. Porcelli and S. Migliuolo, Phys. Fluids 29 (1986) 1741. [17] B. Coppi, R. GalvAo, R. Pellat, M.N. Rosenbiuth and PH. Rutherford, Fiz. Plazmy 2 (1976) 961 [Soy.J. Plasma Phys. 2(1976)533].

[18) G. Crew and J.J. Ramos, Phys. Fluids 26 (1983) 2621. [19] B. Coppi, S. Migliuolo and F. Porcetli, Phys. Fluids, to be published. [20] L. Chen and R.J. Hastie, Bull. Am. Phys. Soc. 30 (1985) 1422.