Spontaneous
and driven perpendicular
rotation
in tokamaks*
A. B. Hassam,t T. M. Antonsen, Jr., J. F. Drake, P. N. Guzdar, C. S. Liu, D. R. McCarthy, and F. L. Waelbroeck Laboratory for Plasma Research, University of Maryland, CoUegePark, Maryland 20742 (Received 3 December 1992; accepted 4 March 1993) Recent theoretical work pertaining to spontaneously generated or forced plasma rotation in tokamaks is discussed. A description of the spontaneous poloidal spin-up of tokamaks from the Stringer effect is given, highlighting the necessary condition of poloidally asymmetric particle accumulation. The possibility of inducing poloidal rotation using the Stringer effect by poloidally asymmetric particle fueling is suggested. The linear theory of EXB velocity shear stabilization of tokamak microinstabilities is discussed with an emphasis on the general features of the theory and some nonlinear concerns. It is argued that the critical velocity shear for stabilization of microinstabilities in tokamaks required by linear theory may be, in order of magnitude, a universal frequency. The feasibility of driving perpendicular rotation in tokamaks by neutral beam injection, to suppress microturbulence, is assessed for both toroidal and poloidal injection schemes.
I. INTRODUCTION
There has been considerable interest recently in the generation of and effect of large scale EXB rotation of tokamaks. id The spontaneous transition from L to H (low to high) modes of confinement in tokamaks has been found to be accompanied by a large increase in the radial electricfield shear, E:, at the tokamak edge.lF2 An externally imposed torque at the tokamak edge also results in improved conlinement.3 In both cases above, a concomitant suppression in microturbulence is observed, consistent with the improvement in confinement.4 Some important theoretical questions posed by these experiments are: ( 1) How is the E, field generated? (2) How does Ei affect microturbulence? (3) Can these effects of E: be extended to the core of tokamak plasmas? Ion-orbit loss at tokamak edges,5 nonambipolar radial diffusive electron losses,6 flow shear generation by Reynolds stress,7 and spontaneous poloidal spin-up by the Stringer mechanism8’9 are among the mechanisms proposed to address ( 1). It is generally accepted that shear in E, is what affects microturbulence. The effect of E: on existing microturbulence, it has been argued, is to cause reduction in the level of the microturbulence. Also, Ei has been shown to linearly stabilize microinstabilities (see, for example, Refs. 12- 15 ) . Finally, various methods have been proposed to induce E, in tokamaks by external means. These include generation by neutral beams,12’16 magnetic coils,17 and radio-frequency power.‘* In this paper, we elaborate on some of the theoretical ideas mentioned above. In particular, we elaborate on some characteristics of the Stringer spin-up,8’19p20especially as to how the spin-up tendency could be exploited to induce poloidal plasma rotation.2’ We next discuss linear stabilization of microinstabilities by imposed Er and point out some universal features of the theoretical findings.r2,13 Fi*Paper 317, Bull. Am. Phys. Sot. 37, 1409 (1992). ‘Invited speaker. 2519
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0899-8221/93/5(7)/251
nally, we assess the feasibility of using neutral beam injection (NBI) to induce perpendicular rotation in tokamak cores.12,13,16In our discussion of these topics, we have emphasized theoretical ideas and underlying physical mechanisms; we refer, where appropriate, to work reported elsewhere where the corresponding mathematical detail is given. It is important to recall that plasma ion flow, V, in general, consists of flow parallel to B and perpendicular flows made up of EXB and diamagnetic drifts, viz., B EXB BXV*Pi v= VII jj+ B2 + neB2
’
(1)
where Pi is the ion pressure tensor. The diamagnetic flow is of order (pi/L,) C,, where pi is the ion gyroradius, L, is the scale length of the pressure gradients, and C,= (Ti/M) 1’2 we denote as the sound speed. For tokamaks, the poloidal sound speed, (r/qR ) C,, plays an important role. In the core of tokamaks, the diamagnetic speed is much smaller than the poloidal sound speed since pi/L,E~~. (7) [In the discussion leading up to (6)) an “effective mass,” proportional to a factor of ( 1 + 28), has been omitted for simplicity. To recover this toroidal effect, the V .V term in (3) and the a V,, /at term in (4) must be reinserted.] Thus, for instability of V,, the poloidally asymmetric particle accumulation rate must exceed E times the damping rate from parallel viscosity. We note from (6) that the Stringer spin-up is intimately related to particle density profiles, as evidenced by the n-II’; term. In fact, it is accentuated at tokamak edges because of the l/n dependence. This feature of the spin-up makes it possible to obtain a bifurcation in density equilibria. An essential added ingredient is the hypothesis that velocity shear can stabilize microturbulence. When all these features are present, a bifurcation exists. The physical details are given in Ref. 8. The poloidal spin-up nonlinearly saturates at the poloidal sound speed. 19t20This happens because if the rotation rate, Ve/r, exceeds the sound frequency, CJqR, the particle accumulation in 8 resulting from IYr and S is smoothed out by the poloidal convection, rather than by parallel flow, according to the approximate continuity equation Ve aii - r -=: ae -v,r,+s.
(8)
( 1 ah
-;?T~-(w~)-*df(-v~r~~~e+s~O~e).
where isothermality is assumed. As is well known, tokamak plasmas experience an effective gravity, g,,, which points toroidally outward. If there is a poloidally varying density, g,, will act to force a poloidal acceleration of the plasma in much the same way in which gravity will force an unbalanced bicycle wheel to start rotating. Thus a poloidally varying density causes a poloidal acceleration, according to Z 2520
ah at
Solving for n from (8) and inserting into (5), we find
a gj I(
where geff=Cz/R is the “effective gravity” of tokamaks, directed toroidally outward; the sin 8 comes from taking the 0 component of g,, (f?= 0 is at the outboard midplane, increasing counterclockwise). By solving for n from (4) and inserting into ( 5 ) , and using ( 3 ) for VII we obtain the equation
(5) Phys. Fluids B, Vol. 5, No. 7, July 1993
(9)
The crossover from (6) to (9) occurs for Vdr> CJqR. A more detailed analysis’9’20T24indicates a saturated rotation speed for Vez (r/qR ) C, . The fact that particle sources with S cos 8 > 0 can precipitate the spin-up could be used to induce poloidal spinups in tokamak plasmas.21 This could be accomplished, for example, by pellet injection to the outboard midplane or by NBI with deposition profiles peaked toward the outboard midplane. In either case, the local density replacement rate from the pellet or neutral beam would have to exceed E Hassam et
a/.
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times the poloidal damping rate. The latter rate is typically of order Yii, the ion-ion collision rate. For n= lot9 me3 and T= 10 keV, Yii=: 100 msec. Thus source induced spinups may be possible, at least transiently. A more detailed analysis20’2’ of source-driven spin-up shows that sources with S sin 6#0 can also drive poloidal spinups. An S sin 8 term enters Eq. (6) as an inhomogeneous forcing term for V, . Thus, while S cos 6 precipitates an instability, S sin 8 directly drives V,. Experimental evidence for a source driven Stringer spin up may already exist. For example, there is evidence that H modes are induced by pellet injection:25 the relevant time scales are roughly as required by the theory. In addition, pellets penetrating into the interior of tokamaks seem to exhibit a poloidal smearing upon deposition.26 III. LINEAR STABILIZATION OF MICROINSTABILITY FROM EXTERNALLY IMPOSED PERPENDICULAR FLOW SHEAR
Large scale perpendicular flow shears in tokamaks are of interest because such flow shears could suppress or stabilize microturbulence. The experimental evidence was mentioned earlier.4 Theoretical studies have shown that given a level of convective cell turbulence, the addition of an externally imposed flow shear, in tending to shear the cells, reduces the level of turbulence.“>” The theoretical question could also be asked another way-in linear theory, are unstable modes rendered stable beyond a critical applied flow shear? If so, is this critical flow shear also sufficient to effect a nonlinear stabilization (i.e., are the modes also nonlinearly stable?). In this section, we describe some work done to address these questions.12P13 The interesting result that emerges from linear stability theory is that not only does perpendicular flow shear stabilize several tokamak microinstabilities, but also that roughly the same critical flow shear is sufficient to stabilize all wave numbers and all these microinstabilities. This “universal” feature makes it possible to assess the flow shears that would be required to suppress microturbulence significantly.‘2 The “universality” arises as follows. In their work on the finite Larmor radius stabilization of the interchange mode, Rosenbluth, Krall, and Rostoker2’ found that the unstable interchange mode with growth rate rg was stabilized if the propagation introduced by the diamagnetic drift speed, Vo, was strong enough to cause the mode to propagate cross field faster than it had time to grow, i.e., stabilization was achieved for k, * V, >.Y~ . [The correspondrelation schematically, ing dispersion V,) = - r’,.] We postulze, by analogy, that w(w+k, EXB flow shear will stabilize a mode with frequency w if it causes shearing of the mode faster than w-l, i.e., the stability criterion is l
kl .V$x>
[WI,
(10)
where Yk=dV,/dx, and hx is the linear mode width. As is stands, (10) suggests that the critical V, is proportional to 1o 1/k, A.x and thus dependent on the mode in 2521
Phys. Fluids B, Vol. 5, No. 7, July 1993
FIG. 1. Growth rates plotted vs velocity shear. The upper curve is the resistive g mode, the lower is the drift wave. The velocity shear is normalized differently for each mode: (g/L,)“2 for the former, C/L, for the latter.
question, as well as the wave number. However, the majority of tokamak electrostatic microinstabilities are “soundlike” in the sense that linear mode widths are determined by the “sound point” given by kll (x)C,= I o I. Since k,, (x)=ki Ax=kh Ax/L,, the ratio Iol/kl Ax is independent of both kl and the details of the mode, being equal to CJL,. The stability criterion becomes’2.13
(11)
v;> C,/L,
( L, is the scale length of the magnetic shear). Electrostatic microinstabilities that are “soundlike” include the drift wave,‘2T’3 the vi mode,‘2-‘5 the Vi mode,13 and the resistive-g mode.12 For all except the last mode, condition ( 1 1 ), within numerical factors, holds, as we will show. For the resistive-g mode, CJL, is replaced by -CJR, although only the slab version of this mode has been considered. In Fig. 1, we show the results of a numerical study” of linear resistive-g modes and the drift wave (the so-called “i-4” model). The growth rate as a function of normalized Vk is plotted. Respective normalizations for the resistive g and the drift wave are (g/L,) 1’2 and CJL, . The modes are seen to be stabilized beyond the critical flow shear given by Eq. ( 10). A small amount of perpendicular viscosity was necessary in the case of the drift wave to give bounded modes and complete stabilization. The drift wave has also been considered analytically by Waelbroeck et al. and similar results were found.13 The vi mode has been considered by Hassam, l2 Waelbroeck et al., l3 Hamaguchi and Horton,15 and Wang et al.,14 and the CJL, scaling obtained. Finally, the Vi mode, an qrtype instability driven by parallel flow shear, was considered by Waelbrock et al. l3 and also found to be stabilized, according to ( 11). It is important to note that while the linear theory shows that beyond a critical velocity shear, -C/L,, modes are linearly stable, the nonlinear theory of Biglari et al., on the other hand, indicates that a given level of microturbulence is reduced beyond a critical flow shear that depends on the characteristics of the turbulence itself. Nonlinear destabilization aside (see below), the foregoing would suggest that for Vk> CJL,, complete suppression would be obtained but that even before this critical level is reached, nonlinear suppression would result in partial reHassam et al.
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FIG. 2. Time sequence of contours of constant density in x- z space showing Rayleigh-Taylor turbulence as it is affected by an applied velocity shear. The torque driving the velocity shear is initiated after the turbulence has reached a steady level, as in Fig. 2(a).
duction. The amount of partial reduction attained from a Vk which is a given width away from CJL, is an important question that would be relevant to confinement improvement. The question of nonlinear destabilization also needs to be resolved: Modes linearly stabilized by Vk could possibly be nonlinearly destabilized. There is, indeed, evidence to this effect.28 To this end, we have done a preliminary study for two systems that we report here. The sequence of figures shown in Fig. 2 shows a time series of density contours corresponding to Rayleigh-Taylor turbulence. In the simulation box shown in the figures, a gravitational field points from right to left (the x coordinate), a magnetic 2522
Phys. Fluids B, Vol. 5, No. 7, July 1993
field points into the page, and an isothermal plasma is present everywhere in the box. Plasma is fed in at x=0.85 and removed at the same rate at x=0.15-thus, in the absence of turbulence, a density gradient would be set up between these two x coordinates. Hard wall boundary conditions are used at x=0 and 1 and periodic boundary conditions are used at z=O and 1. Compressible magnetohydrodynamics with the four field variables n, B, V is used. Small amounts of diffusion are allowed. The system is Rayleigh-Taylor unstable and, to begin with, it is allowed to reach the strongly turbulent stable shown in Fig. 2(a). At that time, an external torque is applied to the system and held in place. The torque is applied as an external force Hassam et al.
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to the z component of the momentum equation with the force being linear in x and zero at x=0.5. The boundary condition on V, was (JVJ~X)~=O at x=0 and 1. Thus, given a small viscosity, the laminar solution to V, would vary as 6x/4- (SX)~/~ where 6x=x-0.5. Note that in the central regions, V, is largely linear. Figures 2(b) to 2( 1) show that as the V, velocity builds up to its laminar value, the turbulence is correspondingly suppressed until a laminar state is reached asymptotically. The magnitude of the torque was adjusted so that the final flow speed shear, Vi, had a maximum value at x =0.5 of 1.8yg. This is consistent with the stability criterion given in Eqs. (IO) and (19) of Ref. 29. No evidence of nonlinear destabilization was found. [Note that in this numerical experiment, we minimized I$‘, the second derivative. This was done to prevent onset of the Kelvin-Helmholtz instability. The RayleighTaylor is also subject to self-stabilization3’ by the Reynolds Stress mechanism (see, for example, Ref. 7). This effect was not predominant in the parameters chosen for the above numerical experiment.] In Ref. 29, an analytic calculation was presented to show that the Rayleigh-Taylor instability is nonlinearly stabilized by application of a velocity shear. The critical shear required for stabilization was found to be yg My$k: ~u)l”~,where p is the perpendicular viscosity. A numerical experiment similar to the one described above was also conducted for the resistive ballooning mode. A three-dimensional (3D) numerical code set up to study resistive ballooning turbulence3’ was used. Again, preliminary results indicated that existing resistive ballooning turbulence was suppressed for an applied torque roughly consistent with linear theoretical predictions. Further work on these nonlinear studies is clearly needed; in particular, a study of the rate of decrease of turbulent amplitudes as a function of subcritical applied torque would be useful.
IV. PERPENDICULAR ROTATION NEUTRAL BEAM INJECTION
DRIVEN
BY
In this section we address the following question: Based on CJL, being the critical perpendicular flow shear needed for significant turbulence suppression, can neutral beam injection be used to impart sufficient flow shear to improve confinement in the core plasma? We begin by noting that in the core plasma, since the diamagnetic flow is small, perpendicular rotation and EXB drifts are synonymous. Perpendicular rotation (or E,) can therefore be attained by a toroidally rotating plasma or a (largely) poloidally rotating plasma or a combination of the two. There are relative inefficiencies and advantages pertaining to both schemes which we briefly discuss here.‘2.‘3,‘6 Consider first toroidal NBI. If density replacement is small and the final rotation speed is subsonic, the steadystate momentum and energy balance equations are roughly expressible as NMVpJrpp2PdVb, 2523
Phys. Fluids B, Vol. 5, No. 7, July 1993
(12)
(3/2)NT/r,zPb,
(13)
where N is the total number of plasma ions, r,r and rE are the toroidal momentum and ion energy confinement times, Pb and V, are the beam power and beam speeds, and energy transfer to electrons is ignored. Since rp and 7E are approximately equal, we find from ( 12) and ( 13) V&Z3CJVb,
(14)
where C,= ( T/M) 1’2, Since we require V; > CJ L, , and, for toroidal rotation, V,z (B+./B*) V, , we need (15)
V&‘Cs(q’/q).
Toroidal rotation spreads out diffusively over the minor radius, a, regardless of beam deposition. Thus V&- V,+/cz. Using this in ( 15) and using ( 14) for V,+,, we see that achieving condition ( 15) may or may not be possible; a small q’dq is advantageous but CJV, is generally small and represents an inefficiency given the requirement of high-speed beams for penetration. We also note that the assumption of low-density replacement and subsonic rotation have to be revisited-when these are reconsidered, achieving (15) seems like a good possibility.i6 We now consider poloidally rotating plasmas such as would be driven from off-axis NBI with a poloidal component. First, the CJV, inefficiency applies also to poloidal injection. Compared to toroidal NBI, however, poloidal NBI has two advantages and one disadvantage. Since Vl - V. (given Be/B, (Be/B,JC,, an effect not included above. Thus vp would be affected by the size of the final poloidal speed, and we have fP=fP(
vem-
If V, can be made to exceed ( BdB,) C,, magnetic pumping could be reduced considerably, allowing the poloidal component to contribute significantly in Eq. ( 17). As a final remark, we add that the localization of V, to A
(u/A)~. The minimum value that vTp can take is, in fact, for vpr+,- (a/A)2- 1, and so qTp> Be/B,. ACKNOWLEDGMENTS
The work reported in this paper has benefited from discussions with K. H. Burrell, J. M. Finn, A. A. Galeev, R. J. Groebner, R. G. Kleva, R. M. Kulsrud, M. N. Rosenbluth, and R. Z. Sagdeev. This work was supported by the U.S. Department of Energy. ‘R. J. Groebner, P. Gohil, K. H. Burrell, T. H. Osborne, R. P. Seraydarian, and H. St. John, in Proceedings of the Sixteenth European Conference on Controlled Fusion and Plasma Physics, Venice, 1989 (European Physical Society, Petit-Lancy, Switzerland, 1989), Vol. 13B, p. 245. *K. Ida, S. Hidekuma, Y. Miura, T. Fujita, M. Mori, K. Hoshino, N. Suzuki, T. Yamauchi, and the JFT-2M group, Phys. Rev. Lett. 65, 1364 (1990). ‘R. J. Taylor, M. L. Brown, B. D. Fried, H. Grote, J. R. Liberati, G. J. Morales, and P. Pribyl, Phys. Rev. Lett. 63, 2369 (1989). 4E. J. Doyle, C. L. Rettig, K. H. Burrell, P. Gohil, R. J. Groebner, T. K. Kurki-Suonio, R. J. LaHaye, R. A. Moyer, T. H. Osborne, W. A. Peebles, R. Philipona, T. L. Rhodes, T. S. Taylor, and J. G. Watkins, to appear in Proceedings of the 14th International Conference on Plasma Physics and Controlled Nuclear Fusion Research, Wurzburg, 1992 (International Atomic Energy Agency, Vienna, 1993). 5K. C. Shaing and E. C. Crume, Phys. Rev. Lett. 63, 2369 (1989). ‘%.-I. Itoh and K. Itoh, Phys. Rev. Lett. 60, 2276 (1988).
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Hassam et a/.
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