Error field locked modes thresholds in rotating ...

PHYSICS OF PLASMAS

VOLUME 9, NUMBER 9

SEPTEMBER 2002

Error field locked modes thresholds in rotating plasmas, anomalous braking and spin-up E. Lazzaro Istituto di Fisica del Plasma del CNR, Assoc. EURATOM-ENEA-CNR per la Fusione, Via R. Cozzi 53, Milan, Italy

R. J. Buttery and T. C. Hender EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon OX14 3DB, United Kingdom

P. Zanca CONSORZIO RFX Assoc. EURATOM-ENEA-CNR per la Fusione, C.so Stati Uniti 4, Padova, Italy

R. Fitzpatrick Institute for Fusion Studies, Department of Physics, University of Texas at Austin, Austin, Texas 78712

M. Bigi EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon OX14 3DB, United Kingdom

T. Bolzonella CONSORZIO RFX Assoc. EURATOM-ENEA-CNR per la Fusione, C.so Stati Uniti 4, Padova, Italy

R. Coelho ˜ o EURATOM/IST, Centro de Fusa ˜ o Nuclear, 1096 Lisbon, CODEX, Portugal Associaca

M. DeBenedetti CRE ENEA Assoc. EURATOM-ENEA per la Fusione, Via E. Fermi 7, Frascati, Italy

S. Nowak Istituto di Fisica del Plasma del CNR, Assoc. EURATOM-ENEA-CNR per la Fusione, Via R. Cozzi 53, Milan, Italy

O. Sauter CRPP/EPFL Assoc. EURATOM-Switzerland, Lausanne, Switzerland

M. Stamp and contributors to the EFDA-JET work programme EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon OX14 3DB, United Kingdom

共Received 26 March 2002; accepted 5 June 2002兲 The mechanisms of nonlinear interaction of external helical fields with a rotating plasma are investigated analyzing the results of recent systematic experiments on the Joint European Torus 共JET兲 关A. Gibson et al., Phys. Plasmas 5, 1839 共1998兲兴 that widen the previous data base collected on Compass-D 关T. C. Hender et al., Nucl. Fusion 32, 2091 共1992兲兴, Doublet III-D 关R. J. La Haye et al., Nucl. Fusion 32, 2119 共1992兲兴 and JET. The empirical scaling laws governing the onset of ‘‘error field’’ locked modes are re-assessed and interpreted in terms of existing driven reconnection theories and with new models. In particular the important mechanisms of plasma rotation braking, and spin up, associated with error fields are analyzed in detail and interpreted. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1499495兴

I. INTRODUCTION

of resistive locked magnetic islands caused by the helical component of unavoidable ‘‘error fields.’’ The basic mechanism to be understood is the nonlinear response of magnetic reconnection at rational-q surfaces driven by resonant errorfields and the possible screening effect provided by plasma rotation. There have been a number of experimental studies1– 4 of locked modes driven by error fields showing this screening effect implies a threshold in error field amplitude to form a locked mode. These comments anticipate the motivation of the systematic work undertaken recently on the Joint European Torus 共JET兲 device5 to widen existing data bases on the subject. These studies which are relevant for the International Toka-

The ideal tokamak configuration of nested toroidal isobaric magnetic surfaces desired for plasma confinement, is well known to be susceptible to nonaxisymmetric magnetic perturbations that are resonant with closed field line surfaces. On dissipative time scales even an ideally stable plasma can respond to imperfections of the magnetic geometry that tear the ideal configuration deteriorating particles and energy confinement and leading frequently to disruption of the discharge. In addition error field locked modes can seed neoclassical tearing modes 共NTMs兲 and lead to a limit for high ␤ p performance. Therefore, it is important to understand the appearance 1070-664X/2002/9(9)/3906/13/$19.00

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Phys. Plasmas, Vol. 9, No. 9, September 2002

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FIG. 1. JET saddle coils system used in the experiments described. An external field with m, n helical components can be generated by suitable coil connections.

mak Experimental Reactor 共ITER兲 design,6 aim to establish, in particular, the threshold of the amplitude of external helical magnetic fields as a function of the main plasma equilibrium parameters. The appropriate scaling laws are identified including as a novel feature the dependence on the plasma toroidal rotation frequency ( ␻ 0 ). Further motivation comes from the comparison of the effects of ‘‘externally’’ driven modes with those resulting from ‘‘spontaneous’’ reconnection. Namely it has long been known,7 but not yet conclusively explained, that the onset of locked modes often has a strong ‘‘nonlocal’’ effect on the plasma toroidal rotation profile. This aspect is related to the ‘‘anomalous’’ nature of the perpendicular plasma viscosity as well as to the possible appearance of a toroidal viscous force due to broken axisymmetry.8 The paper is organized into a section presenting the set-up of the JET experiment, a description of the data analysis employed to obtain the threshold power laws in comparison with the best known theoretical models and previous results from other experiments. The following section discusses the limitations of conventional models and points out the discrepancies with the experimental observations, isolating the key questions to be addressed. Sections III and IV contain the outline and a discussion of possible models for the anomalous plasma braking and spin-up. Section V gives conclusions and perspectives for understanding and controlling realistic forced reconnection phenomena in tokamak experiments and fusion devices.

II. ACTION AND RESPONSE EXPERIMENTS WITH HELICAL ERROR FIELDS ON JET

The flexibility of the JET device9 has been used to improve the understanding of the response of tokamaks to very small accidental 共‘‘error’’兲 external field perturbations that are generally observed to lock rotating modes causing rapid amplitude growth to disruptive conditions on a variety of tokamaks.1– 4,7,10 The JET tokamak is fitted, within the vacuum vessel, with a system of saddle coils 共see Fig. 1兲 that can be fed with variable polarity by an appropriate power supply and controller, and were originally meant to test concepts of electro-dynamic feedback control of tearing modes.11 The objective of the work presented here is however to single out as much as possible the essential physics of the linear and nonlinear reconnection response on the basis of systematic experiments in which a dc external helical field having a large component with poloidal and toroidal pitch numbers (m⫽2,n⫽1) is applied by the saddle coils 共Fig. 2兲. The coils in the present arrangement are hardwired to produce only fields with an n⫽1 toroidal number, therefore, modes with fractional mode numbers ratio 共e.g., m/n⫽3/2兲 have not been driven directly. Previous experimental results and a number of theoretical arguments indicate that when the amplitude of the static external perturbation resonant at a certain q⫽m/n surface exceeds a critical value, the driven linear reconnection process bifurcates, leading to a nonlinearly amplified state form-

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FIG. 2. 共m, n兲 external field radial component b rm,n , normalized over saddle coils current I EFC vs poloidal mode number m; open squares are for n⫽1, diamonds for n⫽⫺1.

ing a magnetic island. The onset of this nonlinear stage is commonly referred to as ‘‘penetration’’ of the external field perturbation, in partial analogy with the phenomenon of penetration of a time varying magnetic field into a conductor2,12 although in the case considered here the characteristic time scales are hybrid expressions involving electrical resistive times, inertial times and fluid viscous times.12 For a fixed aspect ratio R/a and safety factor q a ⫽aB ␸ /RB ␪ , the dependence of the normalized threshold value for penetration b pen /B on plasma parameters can be expected to be reprea aP where sented by a power law scaling b pen /B⬀n a n B a B T e T P aux n is the plasma density, B the toroidal field, T e the electron temperature and P aux the auxiliary, non-Ohmic power density. According to similarity principles applied to tokamaks13 the operational independent variables are B and P aux . From

the point of view of both the theoretical models of forced reconnection and tokamak operation it is preferable to investigate a scaling of the form b pen /B⬀n a n B a B ␻ ␩0 that involves explicitly the density and the plasma rotation frequency, which is related to a condition of local competition of the electro-dynamic and viscous torques that oppose reconnection.12,14,15 In the initial stage of the field penetration the plasma velocity deviates strongly from the equilibrium profile only in the vicinity of the rational surface. The observed plasma rotation is really in the toroidal direction, as in the poloidal direction it is strongly damped by the neoclassical viscous force. Modes are usually observed to rotate in the electron drift direction which for a helical perturbation corresponds to an apparent toroidal rotation counter to the plasma equilibrium current hereafter to be labeled as the ‘‘positive’’ direction. The application of external torque, via neutral beam injection 共NBI兲 or external fields may alter in either way the rotation direction. The experiments on JET considered here were performed in plasmas with low power neutral beams momentum injection, operating at constant q 95⬃3.5 with the toroidal field B in the range from 1 to 3 T, and with densities in the range 1.4 –3.3⫻1019 m⫺3 in a single null separatrix configuration 共see Table I兲. A slow ramp of error field is applied 关Fig. 3共a兲兴 with the helical component m⫽2, n⫽1 reaching an amplitude of B 2,1⬇6.5⫻10⫺4 T at the q⫽2 surface. As apparent from Figs. 3共b兲 and 3共c兲 the amplitude ramp of the magnetic perturbation produces an electro-dynamic torque that brakes the toroidal plasma rotation at the q⫽2 surface located at the major radius position R q⫽2 ⫽3.7 m and when the local angular rotation frequency ␻ 0 ⬅ ␻ (R⫽R q⫽2 ,t 0 ) has been reduced to a critical value ␻ ⬇ ␻ 0 /2 关Fig. 3共c兲兴 the initially linear driven response, measured in the laboratory frame by a set of peripheral magnetic pick-up coils, is nonlinearly amplified 关Fig. 3共b兲兴. The start time of the coils current ramp is t 0 . This indicates field penetration and formation of a magnetic island, that is subsequently ‘‘locked’’ as shown on the temperature profile measured by ECE 共electron cyclotron emission兲, in Fig. 4.

TABLE I. Relevant discharge numbers. Values of toroidal field B in tesla, line averaged density ¯n e (1019), external saddle coils current I E 关A兴 at reconnection, initial toroidal angular frequency ␻ 0 关rad/s兴, spin-up toroidal angular frequency ␻ spin 关rad/s兴, plasma temperature T e 兩 q⫽2 关eV兴 at q⫽2 surface, neutral beam power 共in MW兲. Discharge No.

B 关T兴

¯n e (1019)

I E 关A兴

␻ 0 关rad/s兴

52 066 52 065 52 068 52 069 53 610 52 058 52 059 52 060 52 061 52 062 52 063 52 067

1 1.5 2.0 2.0 2.0 2.5 2.5 2.5 2.5 2.5 2.5 3.0

1.39 1.51 1.93 1.89 1.65 2.27 1.99 2.00 1.85 1.84 1.86 1.90

2200 2275 2672 2559

2500 2264 3092 3028 2242 3000

1744 1631 1735 2243 2844 2253 1849

3231 4848 3199 2681

␻ spin 关rad/s兴 ⫺8.29 38 ⫺149 167 191 ⫺825 ⫺1071 ⫺990 ⫺1172 ⫺1426 ⫺1506

T e 兩 q⫽2 关eV兴 348 651 461 476 862 673 795 705 910 1082 637 627

P NBI 关MW兴 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 1.8 0.9 0.9

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Phys. Plasmas, Vol. 9, No. 9, September 2002

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FIG. 4. Temperature profile flattening in the region of the 共2, 1兲 island formed by driven reconnection 共JET shot 52063兲; the vertical line indicates the location of the q⫽2 surface identified by the magnetic equilibrium reconstruction.

with previous observations in JET discharges with NBI power input.7 Multiple regression analysis of the data gives for the rotation frequency the power law scaling displayed in Fig. 5 with exponents ␣ ⬵0.56, ␤ ⬵0.63. Furthermore it is observed that the density depends on B as n⬀B 0.3. Theory predicts different scaling of the threshold with frequency depending on the prevalent dissipative regimes.12,14,15 In the so-called visco-resistive regime defined by the condition ␻ 0 ⫺2/3 ⫺2/3 Ⰶ ␻ I ⬵ ␶ 1/3 ␶ H , where the time constants appearing in v ␶R the definitions are ␶ v ⫽a 2 / ␮⬜ the anomalous perpendicular viscous time of the order of the energy confinement time, ␶ R ⫽ ␮ 0 a 2 / ␩ the resistive diffusion time and ␶ H ⫽a/ v A , the Alfve´n time.

FIG. 3. 共a兲 Typical external magnetic field wave form. The coils current ramp here starts at t 0 ⫽16.4 s after the discharge start time t⫽40 s. At end of ramp compensation of natural error field allows mode spin-up in e- or i-drift directions. 共b兲 Magnetic signals showing linear and nonlinear response (t ⫽18.04 s). 共c兲 Charge exchange spectroscopy signals showing plasma rotation frequency at different radii normalized on frequency at the start time of the coils ramp. At field ‘‘penetration’’ sudden braking occurs.

When the external field is switched off and the natural error field is suitably compensated 关Fig. 3共a兲兴, the locked mode is observed to spin up either in the e- or the i-drift direction 关Fig. 3共b兲兴 because of viscous restitution of momentum from the underlying plasma flow. The scaling of the threshold on density and field is related to the underlying dependence of the rotation on these variables. For the shots at constant P NBI multiple regression on the data leads to the scaling b pen /B⬀n 0.97⫾0.05B ⫺1.2⫾0.06 in line with that found previously on JET3 in the absence of rotation. With NBI momentum injection of power P NBI the rotation frequency scales as ␻ 0 ⬀B ␣ ( P NBI /n) ␤ in agreement

FIG. 5. Power law scaling of the local angular rotation frequency ␻ 0 ⬅ ␻ (R⫽R q⫽2 ,t 0 ) in rad/s at the q⫽2 surface with n, B, and P NBI .

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FIG. 6. Scaling of the penetration threshold normalized over the toroidal field B vs power law ⬃n 1/2B ⫺5/4␻ 0 where ␻ 0 is the local angular rotation frequency at the q⫽2 surface.

In this regime the expected dependence of b pen /B on frequency is linear4,5 and the best fit of the data with this assumption is b pen /B⬀n 0.55⫾0.03B ⫺1.25⫾0.03␻ 0 as displayed in Fig. 6. A much better fit is given by b pen /B ⬀n 0.58⫾0.02B ⫺1.27⫾0.02␻ 0.5 0 共Fig. 7兲. This nonlinear scaling with ␻ 0 is in agreement with the scaling in the ideal viscous regime12 defined by ␻ 0 ⬎ ␻ I . In Table II a summary of results of different previous experiments and some theoretical predictions are presented for comparison. For the discharges of Table II, the initial rotation is ␻ 0 ⬇ 21 ␻ I within the visco-resistive regime but not far from the boundary ␻ 0 ⬇ ␻ I with the viscous regime.12 In fact the threshold scalings examined appear both plausible but as the rotation is strongly driven by the neutral beams momentum input, the characteristics of the fluid regime may dominate the field penetration mechanism. We can actually show that the scaling shown in Fig. 7 is the most consistent for these experiments. It should be recalled that in absence of rotation the threshold scaling is b pen /B⬀n a n0 B a B0 with a n0 ⫽0.97, a B0 ⫽⫺1.207. Since ␻ 0 ⬀B ␣ /n ␤ , then b pen /B⬀n a a⬘ B a B⬘ ␻ 0.5 with a n⬘ ⫽a n0 ⫹0.5␤ 0 ⫽1.285 and a B⬘ ⫽a B0 ⫺0.5␣ ⫽⫺1.487. Considering that for the present data n⬀B 0.3 and that the data regression of Fig. 7 gives a n ⫽0.58, one can write a B ⫽a B⬘ ⫹0.3(a n⬘ ⫺a n ) that shows ⫽⫺1.275 and obtain b pen /B⬀n 0.58B ⫺1.275␻ 0.5 0 consistency of the present results with those obtained in absence of rotation.4 The analysis of the data shows substantial agreement with the empirical scaling of the threshold of the field penetration found in previous experiments on JET 共see Table II兲 and supports the view that externally driven magnetic reconnection at rational-q surfaces is contrasted by plasma rotation which is, therefore, beneficial. However, behind the global result of this experiment much more complex physics is hidden that needs to be discussed in detail. The process appear-

FIG. 7. Scaling of the penetration threshold vs power law ⬃n 0.58B ⫺1.274␻ 1/2 0 where ␻ 0 the local angular rotation frequency at the q⫽2 surface.

ing as a nonlinear penetration of the external field, with consequent island formation, has been studied in detail in Ref. 12 within a Rutherford regime16 and is associated with the torque balance conditions that may occur in a viscous plasma subject to electrodynamic forces. The experiments confirm strikingly the fact that when the threshold for nonlinear reconnection is reached the rotation frequency is reduced approximately to one half of its initial value, in agreement with the formula12 of the viscoresistive regime

冋 冉 冊册

␻ 1 1 b ext ⫽ ⫹ 1⫺ ␻0 2 2 b thres

2 1/2

共1兲

,

as shown in Fig. 8 for a single discharge 共52 067兲 and in Fig. 9 for a group of discharges. This interpretation is consistent with a picture of the islands’ motion in the plasma fluid similar to that of a rigid body, with no net plasma flow across the separatrix.2,12 In a single fluid plasma model this ‘‘no slip’’ constraint leads to the assumption that the flux surface averaged deviations of the fluid 共poloidal and toroidal兲 angular velocities ⌬ ␻ ␪ ,z (r) from the ‘‘natural’’ values associated with a single magnetic island are hindered by perpendicular TABLE II. Values of exponents of scaling law b pen /B⬀n a n B a B ␻ ␩0 for the normalized reconnection threshold and rotation frequency ␻ 0 ⬀B ␣ ( P NBI /n) ␤ in different experiments 共Refs. 3, 4, and 14兲. For JET-2000 the data of Fig. 7 are considered.

COMPASS 共Ref. 2兲 JET-98 共Ref. 4兲 JET-2000 Theory Ideal-viscous regime 共Ref. 12兲 Visco-resistive regime 共Ref. 12兲

an

aB

␣

␤

␩

0.55 0.97 0.58

⫺2.2 ⫺1.2 ⫺1.274

n.a n.a. 0.562

n.a. n.a. 0.652

1. 0 0.5

2/3

⫺7/3

1/2

7/12

⫺13/6

1

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Phys. Plasmas, Vol. 9, No. 9, September 2002

FIG. 8. Braking of plasma toroidal rotation 共charge exchange diagnostic兲 at the q⫽2 surface. Penetration occurs when the local angular rotation frequency at the q⫽2 surface is ␻ ⫽ ␻ 0 /2. Full dots are the calculated values of theoretical formula 共1兲; circles are the experimentally observed values of ␻ / ␻ 0 . The coils current ramp starts at t 0 ⫽16.4 s from the discharge start time t⫽40 s.

viscosity, and therefore, governed by a diffusion equation with suitable conditions on axis and at the edge. In steady state the resulting toroidal velocity shift is then expected to be rigid body like between the axis and the inboard radius of the island separatrix 共roughly 0⭐r⭐r s 兲 and sheared for larger radii.12

FIG. 9. Occurrence of braking of plasma toroidal rotation at the q⫽2 surface for all the discharges. Full dots are the calculated values of theoretical formula 共1兲; circles are the experimentally observed values of ␻ / ␻ 0 .

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FIG. 10. Experimental plasma toroidal angular velocity profile evolution. There is the clear evidence of the e.m. torque localization at q⫽2 surface and evidence of the global braking after island formation at t⬃18.04 s.

The evidence provided by the JET experiments, as well as that of other machines2,3 is however in contrast with this idealized picture and requires a new point of view that will be described in the next paragraphs. The main observation, presented in Fig. 3共c兲 and in Figs. 10–12 is that as the nonlinear mode amplification 共penetration兲 takes place, the toroidal velocity profile V ␸ (r,t)⫽ ␻ (r,t)R collapses everywhere almost self-similarly, which is incompatible with a viscous decay. In detail Fig. 3共c兲 shows a superposition of the magnetic pick-up coils signal monitoring the growth of the m⫽2, n⫽1 signal and the traces of the charge exchange measurement 共CXSM兲 of plasma rotation. The CXSM traces giving the time history of rotation at all radii are visibly collapsing together at the onset of the nonlinear reconnection 共at t⫽18.4 s from the start of the discharge兲; thus the braking appears sudden and uniformly across the plasma cross section. The rotation frequency is normalized to the value at the starting time point of the ‘‘error field’’ ramp. The interpretation of this fact is not trivial, as it appears also from the CXSM plots of the radial rotation profiles with superposition of the 共2,2兲 共3,2兲, and 共1,1兲 rational surfaces identified by the EFIT magnetic reconstruction code with an expected uncertainty of no more than 5% for the R location; Fig. 10 shows for shot 52 067 that during the subcritical part of the external field ramp, ␶ ⬍ ␶ ric 关Fig. 3共a兲兴 a seemingly diffusive slow reduction of velocity occurs, with clear localization of the braking torque at the resonant radius of the 共2,1兲 surface. At ‘‘field penetration’’ time the braking is abrupt and distributed across the whole plasma body. The nonmonotonic appearance of the rotation profile, with dips around R⫽3.15, 3.45, and 3.7 m could in principle suggest an effect of local

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FIG. 13. Braking of rotation in JET discharge 52 061.

FIG. 11. Profile of the toroidal angular velocity shift along the major radius R for JET discharge 52 067 at penetration time t⬃18.04 s. It is apparent that the no slip condition is violated. R⫽2.8 m is the machine axis.

torques at other rational surfaces, but the error in determination of location of, say the q⫽1/1 and 3/2 surfaces 共see Figs. 10–12兲 would exceed largely the error bar and would grossly contradict other independent information from ECE and soft

x-ray emission signals about the location of the magnetic axis and the sawtooth inversion radius. As discussed below the observed profiles are qualitatively consistent with the radial structure of the perturbed poloidal 兩 b ␪ 兩 2 for m⫽2, n⫽1. More precisely as apparent from Fig. 11, for shot 52 067, and in Fig. 12 for several discharges, the no-slip condition on the variation of toroidal angular frequency ⌬ ␻ (r)⬅( ␻ (r,t) ⫺ ␻ 0 (r s ))⬵const. for 0⬍r⬍r q⫽2 2,12 is not met, the variation being larger in the central part than near the q⫽2 surface where the torque has a resonant peak. No other resonant torque appears to influence the rotation profile at other rational surfaces so the local effect of a mode coupling mechanism is not proven in this case, but the role that coupling may have is briefly discussed at the end of the paper, on the basis of the evidence. Data for shot 52 061 共Fig. 13兲 show that after the instant in which the coils have been switched on (t start), there is a quite long phase in which only the q⫽2 (R⫽3.7) surface undergoes a significant braking. This is interpreted as the effect of the localized electromagnetic torque which develops at this surface. The global braking of the plasma velocity happens rapidly, associated with the reconnection of the q ⫽2 surface. Thus the characteristics of the braking observed are not diffusive and indeed an analysis of the scaling of the observed braking rate suggests that the anomalous rotation damping can be described by d 关 V ␸ 共 r,t 兲 /V 0 共 r 兲兴 /dt⫽⫺ 兩 b ␪ /B 0 兩 ␣ ␯ 0 ␸ V ␸ 共 r,t 兲 /V 0 共 r 兲 , 共2兲

FIG. 12. Profile of the relative toroidal angular velocity shift 关 ␻ min(R) ⫺␻0(R)兴/␻0(R) 共normalized over initial profile兲 for several discharges at the time of island formation. R⫽2.8 m is the machine axis. It is apparent that the ‘‘no slip condition’’ is not met, as it would require constant toroidal angular velocity shift.

associated with some mechanism depending on the field perturbation as 兩 b ␪ 兩 ␣ , with ␣ close to 2. In Eq. 共2兲 ␯ 0 ␸ is an appropriate damping coefficient. A nonlinear magnetohydrodynamic 共MHD兲 problem for the m⫽2, n⫽1 mode has been solved numerically for a plasma of constant density, rotating toroidally with an initial

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velocity profile V 0 (r) and with boundary conditions that include an external helical current field perturbation I E . The resistive MHD equations

⳵B ␩ 2 ␩ ⵜ B⫺ⵜ ∧ⵜ∧B, ⫹ⵜ∧ 关 v∧B兴 ⫽ ⳵t ␮0 ␮0 ␳

共3兲

dv ⫽⫺ⵜ p⫹ ␮ 0 ⵜ∧B∧B⫹ ␮⬜ ⵜ 2 v dt

共4兲

关where ␳ is the mass density, ␩ (r) the resistivity and ␮⬜ the perpendicular viscosity兴 have been solved in a customary reduced magnetohydrodynamic 共RMHD兲 model in the tokamak ordering, where the equilibrium and perturbed magnetic and velocity fields B⫽B0 (r)⫹b(r, ␪ , ␸ ,t) and v ⫽V 0 ␸ (r,t)e␸ ⫹v⬜ (r, ␪ , ␸ ,t) are represented through flux and stream functions15 in the form B⫽B 0z ez ⫹B 0 ␪ e␪

ⵜ∧ 共 C m,n ␺ m,n e i 共 m ␪ ⫺nz/R⫹ ␸ 兺 m,n

m,n 兲

em,n 兲

and v⬵V ␸ 0 共 r,t 兲 e␸ ⫹

ⵜ∧ 共 C m,n ␾ m,n e i 共 m ␪ ⫺n/R⫹ ␸ 兺 m,n

m,n 兲

em,n 兲 ,

with suitable direction vectors and metric coefficients tied to the m, n helical path on a cylinder of radius r. Up to four 共complex兲 harmonics of the flux and stream functions are considered and each harmonic is governed by Reduced MHD equations obtained from Eqs. 共2兲 and 共3兲 and also the background toroidal plasma velocity and equilibrium poloidal magnetic field are evolved in time by the quasilinear equations

⳵ V ␸0 n 1 ␳0 Im共 ␺ * m,n ⵜ 2 ␺ m,n 兲 ⫽ ⳵t 2R 0 m,n ␮ 0

兺

⫺n ␳ 0 Im共 ␾ * m,n ⵜ 2 ␾ m,n 兲 ⫹ ␮⬜ ⵜ 2 V ␸ 0 ,

⳵␺0 m ⳵ ␩ 2 1 ⫽⫺ Im共 ␾ m,n ␺ * m,n 兲 ⫹ ⵜ ␺ ⫹E 0 . ⳵t 2 m,n r ⳵ r ␮0 0 0

兺

The numerical simulation of the driven reconnection problem with a simple viscous damping of the toroidal rotation is performed using a finite difference scheme over a suitable nonuniform radial grid and a two step implicit-explicit method in time.17 Even though this model goes beyond the constant ␺ approximation of the Rutherford regime,16 it leads to a result in substantial agreement with the ‘‘no-slip’’ expectation. As shown in Fig. 14 the nonlinear amplification 共‘‘penetration’’兲 of the helical flux Re关␺(r)e i(m ␪ ⫺nz/R) 兴 occurs when V ␸ ⫽V ␸ 0 /2 and the variation of the velocity profile ⌬V ␸ (r,t)⫽⌬ ␻ (r,t)R shown in Fig. 15 tends to be pretty much uniform. The inadequacy of the diffusive model and a qualitative agreement of model 共2兲 with the observations is shown in Figs. 15 and 16 with 兩 b ␪ 兩 2 , for (m⫽0,n ⫽0), or 兩 b ␸ 兩 2 for (m⫽0,n⫽0) for the reasons explained in next section. The discrepancy with the experimental observations and the scaling of the braking rate suggest a conjecture on additional braking mechanisms linked to the radial and helical structure of the perturbation. As discussed below in detail the

FIG. 14. Theoretical RMHD model of driven reconnection of the helical flux ␺ s in presence of toroidal rotation. Nonlinear mode amplification occurs when rotation velocity is halved. Here the velocity is indicated in the units of the corresponding angular frequency and time is a multiple of Alfve´n time.

breaking of axisymmetry due to resonant and nonresonant helical field perturbations can in fact give origin, in general, to a toroidal viscous force, normally identically zero in axisymmetry.18 III. NEOCLASSICAL VISCOUS FORCE

The neoclassical toroidal viscous force8,18 is a good candidate to explain the global braking of the plasma rotation. This effect is present in nonaxisymmetric plasmas only, and in our case it could appear when the magnetic perturbation of amplitude 兩b兩, induced by the saddle coils becomes important. In principle the symmetry breaking effects can be

FIG. 15. Theoretical model of evolution of the plasma toroidal rotation in presence of viscosity. The variation of the velocity profile ⌬V ␸ (r,t) ⫽⌬ ␻ (r,t)R is pretty much uniform, in contrast with the experimental data of Figs. 10 and 11.

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Defining the pressure gradient and the parallel current density profile as g(r)⫽( ␮ 0 /B 20 )d P 0 /dr and ␴ (r)⫽ ␮ 0 (J0 •B0 /B 20 ) Eq. 共7兲 gives

␮ 0 J0 ⫽ 共 0,␴ B 0 ␪ ⫹gB 0 ␾ , ␴ B 0 ␾ ⫺gB 0 ␪ 兲 .

共8兲

The linearized form of the curl of the force balance equation determines the perturbed field b 共 b•ⵜ 兲 J0 ⫺ 共 J0 •ⵜ 兲 b⫹ 共 B0 •ⵜ 兲 j⫺ 共 j•ⵜ 兲 B0 ⫽0.

共9兲

Introducing direction vectors em,n ⫽C m,n (0,nr/R,m) tied to the m,n helical path on a cylinder of radius r the divergenceless perturbed field consistent with torque balance 共9兲 and Ampe`re’s law can be expressed as an expansion of the type FIG. 16. Theoretical model of evolution of the plasma toroidal rotation in presence of a braking effect ⬀ 兩 b ␪ 兩 2 V ␸ , in qualitative agreement with the typical experimental observations 共e.g., Fig. 10兲.

thought to be of order 兩 b/B ␸ 兩 2 Ⰶ1. However, if at a rational surface q⫽m/n field reconnection into islands of finite width W⬀ 冑b r /B takes place, the order of magnitude of the symmetry breaking effects in the spatially nonuniform field B, can be expected to rise to B ⬘ 冑b r /B At this level in general one can expect, for instance, an enhancement of heat and particles fluxes over the banana regime with a ratio (mW/a ␯ ) 2 , when the ion collisionality parameter ␯ ⬍1. * * The effect we consider here is the toroidal drag appearing because of broken symmetry, whose flux-surface averaged expression is18 e ␾ •ⵜB ⳵ b m,n 2 ␲ 1/2p i V␾ 具 e ␾ •ⵜ•⌸ 典 ⫽ B ⳵␾ v Ti m,n⫽0

兺

冓

冔

q , 共5兲 兩 m⫺nq 兩 where p i is the ion pressure, v Ti is the ion thermal speed and B is the magnetic field magnitude, whose perturbations are denoted b m,n ⫻

冉

B 共 r, ␪ , ␾ 兲 ⫽B 0 共 r 兲 1⫹

兺

m,n⫽0

冊

b m,n 共 r, ␪ , ␾ 兲 .

J0 •B0 B20

B0 ⫺ ␮ 0

ⵜ P 0 ∧B0 B 20

.

b m,n e i 共 m ␪ ⫺nz/R⫹ ␸ 兺 m,n 储

⫹

共7兲

m,n 兲

em,n

C m,n ⵜ ␺ m,n e i 共 m ␪ ⫺nz/R⫹ ␸ 兺 m,n

m,n 兲

∧em,n ,

共10兲

where the Fourier coefficients are expressed in terms of a scalar flux function

␺ ⫽ 兺 ␺ m,n e i 共 m ␪ ⫺nz/R⫹ ␸ m,n 兲 , m,n

共11兲

where ␺ m,n is a real amplitude and ␸ m,n its phase. Moreover ␺ ⫺m,⫺n ⫽ ␺ m,n and ␸ ⫺m,⫺n ⫽⫺ ␸ m,n ⫹ ␲ ensure that b is real. The metric coefficients C m,n (r)⫽(m 2 ⫹n 2 ␧ 2 ) ⫺1/2 and ⫽C m,n ␴ ␺ m,n are determined the ‘‘parallel’’ amplitude b m,n 储 by the condition ⵜ•b⫽0. Consequently the explicit perturbed field components for m,n⫽0 are to all orders in ␧ i b rm,n ⫽bm,n •er ⫽ ␺ m,n , r b m,n ␪ ⫽

共6兲

The notation m,n⫽0 means that the toroidal and poloidal mode numbers cannot be simultaneously zero. In 共5兲 we have discarded a term proportional to the plasma poloidal velocity, under the assumption of a strong poloidal flow damping. All the resonant and nonresonant magnetic perturbations produced by the saddle coils must be taken into consideration in 共5兲, including the m⫽0 component. For this reason we have to model the magnetic perturbation in the plasma to higher order than usual in the inverse aspect ratio ␧⫽r/R. To this purpose it is convenient to adopt the notation developed in Ref. 19 for a cylindrical plasma with periodicity 2 ␲ R in the z direction. In this case ␾ ⫽z/R is a simulated toroidal angle and the equilibrium magnetic field B0 ⫽(0,B 0 ␪ (r),B 0 ␾ (r)) and current J0 are related by the force balance equation J0 ∧B0 ⫽ⵜ P 0 and Ampe`re’s law ⵜ∧B ⫽ ␮ 0 J that give ⵜ∧B0 ⫽ ␮ 0

b⫽

冋

1 d ␺ m,n ⫹n␧ ␴ ␺ m,n 2 2 ⫺m m ⫹n ␧ dr 2

⫹n␧g b ␸m,n ⫽

共12兲

册

G m,n m,n ␺ , F m,n

冋

共13兲

1 d ␺ m,n ⫹m ␴ ␺ m,n n␧ m 2 ⫹n 2 ␧ 2 dr ⫹mg

册

G m,n m,n ␺ . F m,n

共14兲

Here F m,n ⫽mB 0 ␪ ⫺n␧B 0 ␾ and G m,n ⫽mB 0 ␾ ⫹n␧B 0 ␪ . Note that m,n are integers. For the present purposes we specialize the above expressions to the tokamak ordering in ␧ 0 ⫽a/R m,n⫽O(1), B 0 ␪ /B 0 ␾ ⫽O(␧ 0 ), ␴ ⫽O(␧ 0 ) and may neglect the finite pressure gradient term g⬇ ␤ /a⫽O(␧ 20 ). However, contributions for which 兩 m 兩 Ⰶ 兩 n␧ 兩 must be considered with care. For 兩 m 兩 ⬎ 兩 n␧ 兩 we have G m,n ⬇mB 0 ␾ ; b m,n ␪ ⬵⫺

F m,n ⫽mB 0 ␪ ⫺n␧B 0 ␾ ⬇o 共 ␧ 0 兲 ,

1 d ␺ m,n , m dr

共15兲 共16兲

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Phys. Plasmas, Vol. 9, No. 9, September 2002

b ␸m,n ⫽

Error field locked modes thresholds in rotating plasmas . . .

冋

1 d ␺ m,n G m,n m,n m,n ⫹m n␧ ␴ ␺ ⫹mg ␺ m2 dr F m,n

3915

册

⬇o 共 ␧ 0 兲 b ␪m,n .

共17兲

But for m⫽0 a strong scaling of the field perturbation with ␧ ⫺1 appears F 0,n ⫽⫺n␧B 0 ␾ ; G 0,n ⫽n␧B 0 ␪ ⬇o 共 ␧ 20 兲 ,

共18兲

b 0,n ␪ ⫽

1 G 0,n ␴ 0,n ␴ ␺ 0,n ⫹g 0,n ␺ 0,n ⬵ ␺ , n␧ F n␧

共19兲

b ␸0,n ⫽

1 d ␺ 0,n . n␧ dr

共20兲

冋

册

FIG. 17. Radial extension of typical m⫽2, n⫽1 b r , and b ␪ perturbations; the plasma toroidal rotation is affected by a braking effect ⬀ 兩 b ␪ 兩 2 V ␸ , over a radial range wider than the island size 共see Fig. 15兲.

The b field is a real quantity, so we can write b r 共 r,t 兲 ⫽⫺ b ␪ 共 r,t 兲 ⫽

兺 m,n⫽0

共21兲

兺

b ␪m,n cos共 m ␪ ⫺n ␾ ⫹ ␸ m,n 兲 ,

共22兲

兺

b ␾m,n cos共 m ␪ ⫺n ␾ ⫹ ␸ m,n 兲 .

共23兲

m,n⫽0

b ␾ 共 r,t 兲 ⫽

␺ m,n 共 r 兲 sin共 m ␪ ⫺n ␾ ⫹ ␸ m,n 兲 , r

m,n⫽0

It is easy to show that in the formula 共6兲 B 0 ⫽ 冑B 02 ␾ ⫹B 02 ␪ and b m,n ⫽

冉

B 0␪ B 20

共24兲

b ␪m,n ⫹

B 0␾ B 20

冊

b ␾m,n cos共 m ␪ ⫺n ␾ ⫹ ␸ m,n 兲 . 共25兲

Therefore, the general expression for toroidal viscous force arising from helical perturbations is

具 e ␾ •ⵜ•⌸ 典 ⫽

␲ 1/2p i V R 0 v Ti ␾ m,n⫽0

兺

⫻

共 B 0 ␪ b ␪m,n ⫹B 0 ␾ b ␾m,n 兲 2

B 40

•

具 e ␾ •ⵜ•⌸ 典 m⫽0 ⫽

n 2q . 兩 m⫺nq 兩

共26兲

Considering now the tokamak ordering with m, n⫽O共1兲 we have for m⫽0 the following contribution to the toroidal viscous force is:

冉 冊

␲ 1/2p i ␺ m,n V␾ ␧2 具 e ␾ •ⵜ•⌸ 典 m⫽0 ⬵ R 0 v Ti rB ␾ m⫽0,n⫽0

兺

冋

␴ r gr q ⫻ ⫹ 2 m␧ ␧ m⫺nq

cancels the b m,n contribution. The residual term ⬀g presents ␪ an apparent singularity; however, in the nonlinear stage with a magnetic island, due to the pressure profile flattening (p ⬘ ⫽ p ⬙ ⫽0) across the island width it is expected that g⬇(m ⫺nq) 2 . Then to this order the viscous force in the island region would vanish, but nonresonant components 共m⬍0, n⬎0, and m⬎0, n⬍0兲 in the summation would still give a contribution. For m⫽2, n⫽1 the contribution to this viscous force due to the b ␪m,n ⬵⫺(1/m)(d ␺ m,n /dr) term influences a region of plasma substantially wider8 than the island formed at reconnection at the rational surface as can be seen qualitatively from Figs. 16 and 17 contributing to damping of rotation over a wide radial interval. A more significant effect is due to the m⫽0 perturbations for which we have obtained different expressions. In this case 共19兲 and 共20兲 give for the m⫽0 contribution to the toroidal viscous force the expression

冉

␲ 1/2p i 1 d ␺ 0,n /dr V␾ 2 R 0 v Ti B␾ n⫽0 兩 n 兩 ␧

兺

⬅K•V ␾ .

2

共28兲

If we assume constant kinetic terms, also the quantity K is constant across the minor radius. In fact for m⫽0 the marginal stability mode condition 共9兲 reduces to

冉

冊

d 1 d ␺ 0,n ⫽0→ ␺ 0,n ⫽A 0,n ⫹B 0,n r 2 . dr r dr

2

冊

共29兲

A 0,n ⫽0 in order to avoid a singularity of the radial field at r⫽0 关see definition 共12兲兴. Therefore, we have

册

m⫺nq r d ␺ m,n 2 n 2q ⫺ • . 2 m,n m q ␺ dr 兩 m⫺nq 兩 共27兲 Note that the quantity in the squared bracket is O共1兲. In the vicinity of the rational surfaces there are helical current sheets related to the jump in d ␺ m,n /dr and the radial current perturbation is negligible; in this limiting case Ampe`re’s law gives b ␸m,n ⬇⫺(n␧/m)b m,n 共see also Ref. 12兲 that in Eq. 共26兲 ␪

K⫽

冉

4 ␲ 1/2p i 1 ␺ 0,n 共 a 兲 2 R 0 v Ti n⫽0 兩 n 兩 ␧ 0 aB ␾

兺

冊

2

,

共30兲

with ␧ 0 ⫽a/R. The neoclassical viscous force is associated to the damping rate 具 e ␾ •ⵜ•⌸ 典 / ␳ V ␾ . A comparison between Eqs. 共27兲 and 共30兲 shows that the m⫽0 contribution to the neoclassical viscous force is a factor O(1/␧ 4 ) greater than the m⫽0 term. Assuming typical JET values ␧⫽0.37, T i ⫽3 keV and a normalized perturbation ␺ /aB ␾ ⫽10⫺3 , which is actually the maximum reasonable value, for the

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mode resonant at the q⫽2 surface 共therefore, adding both the harmonics m⫽2, n⫽1 and m⫽⫺2, n⫽⫺1兲 the damping time scale is of several second, which is too long. While the same calculation for the m⫽0, n⫽1 and m⫽0, n⫽⫺1 modes brings the time scale to ⬃ 50 ms. Moreover the m⫽0 term produces a uniform influence across the minor radius, which virtually explains the global braking of the plasma velocity, while the m⫽0 term varies substantially with r. As shown in Fig. 2 the mode spectrum produced by the saddle coils contains m⫽0 harmonics with non negligible amplitudes. Therefore, the neoclassical viscous force associated with the m⫽0 modes is a good candidate to explain the experimental braking data. The fluxsurface averaged equation of motion along the toroidal direction is then written as r␳

冋

册

⳵␻ ␾ ⳵␻ ␾ ⳵ T␾ ⫺ ␮⬜ r ⫹rK ␻ ␾ ⫽rS ␾ /R 20 ⫹ , ⳵t ⳵r ⳵r 4 ␲ 2 R 30 共31兲

where ␻ ␾ ⫽V ␾ /R 0 is the toroidal angular velocity, T ␾ the flux-surface integrated electromagnetic torque and S ␾ a momentum source, which provides the equilibrium velocity profile ␻ 0

b

c

⳵ ⳵␻ 0 ␮ r ⫹rS ␾ /R 20 ⫽0. ⳵r ⬜ ⳵r

共32兲

Assuming ␳, ␮⬜ constant and considering that the electrodynamic torque is resonantly peaked on the q⫽2 rational surface then for the bulk of the plasma outside T ␾ ⬵0. The experimental data during the braking phase suggest a separable solution of Eq. 共31兲

␻ ␾ 共 r,t 兲 ⬵ ␻ 0 共 r 兲 •y 共 t 兲 ;

y 共 0 兲 ⫽1.

共33兲

Thus S␾ dy K ⫽0. ⫹ y⫹ 共 y⫺1 兲 2 dt ␳ R 0 ␳␻ 0

共34兲

Note that the m⫽0 modes gives just a K constant with r. Defining S ␾ /R 20 ⫽ ␣␮⬜ ␻ 0 , from 共32兲 one gets ␣ ⫽( j 0,1 /a) 2 , j 0,1 being the first positive zero of the Bessel function J 0 . So we can write 共34兲 in the form y dy y⫺1 ⫹ ⫹ ⫽0, dt ␶ D ␶␮

共35兲

2 . The consistency of this where ␶ D ⫽ ␳ /K, and ␶ ␮ ⫽ ␳ a 2 / ␮ j 0,1 equation implies that the coefficients are independent of r. Note that the m⫽0 modes gives just a K constant with r and we have also assumed K independent of time. The solution of 共35兲 is

y 共 t 兲⫽

␶␮ ␶D e ⫺t 共 1/␶ ␮ ⫹1/␶ D 兲 ⫹ . ␶ ␮⫹ ␶ D ␶ ␮⫹ ␶ D

共36兲

A necessary condition for substantial slowing down of the plasma is

␶ D⭐ ␶ ␮ .

共37兲

In this case the time asymptotic value of Eq. 共36兲 is y brak ⭐1/2 and the exponential braking time is

␶ brak⫽

冉

1 1 ⫹ ␶␮ ␶D

冊

⫺1

⭐ ␶ ␮ /2,

共38兲

which are both consistent with the experimental observations shown in Figs. 8 and 9. As can be seen from 共30兲 the dominant contributions to K come from the n⫽1 and n⫽⫺1 harmonics. Adding these two terms the criterion 共37兲 becomes

␺ 0,1共 a 兲 ␧ ⭓ aB ␾ 2 ␲ 1/4

冑

R . v Ti␶ ␮

共39兲

An estimate with the values ␧⫽0.37, T i ⫽3 keV, ␶ ␮ ⫽0.5 s, gives ␺ 0,1/(aB ␾ )⭓3.8•10⫺4 , which is indeed in qualitative agreement with experiment. Moreover the inclusion of higher toroidal harmonics in K increases the braking effect and lowers this threshold. The fact that strong rotation damping occurs at penetration of the m⫽2 mode suggests that an enhancement of the nonresonant m⫽0 spectral component can be due to the elliptical deformation of the plasma cross section that couples m⫾2 sidebands to the m⫽2 external field perturbations.20,21 A complete discussion of this point is postponed to a subsequent paper. An apparently similar slowing down of plasma rotation associated with a transit time magnetic pumping 共TTMP兲 effect of nonresonant helical fields has been reported in the DIII-D tokamak.3,22

IV. MODE UNLOCKING AND SPIN-UP

In about 50% of cases, after the external field is turned off, removing the electromagnetic braking torque, the locked mode is observed 共on Mirnov coils signals兲 to spin up in the e- or i- drift direction by viscous restitution of momentum by the rest of the plasma. Simultaneously the island decays away on a resistive time scale since it is no longer driven by the external field and it is damped by the effect of the eddy currents generated in the vessel wall. A simple interpretation is given in terms of the standard Rutherford model16 for the evolution of magnetic islands of width W m,n ⬀ 冑rb rm,n , that after switch off of external fields is governed by the equation 0.822␶ R

冋

冉冊

d 共 W m,n /r s 兲 rs ⫽ ⌬ 0⬘ r s ⫺2m dt d

2m

册

共 ␻␶w兲2 . 1⫹ 共 ␻ ␶ w 兲 2 共40兲

Here island rotation effects and resistive wall boundary conditions2 are included. In the customary notation m,n are ⬘ and ␻ the tearing mode instability the mode numbers, ⌬ 0m index and island rotation frequency, r s and d the resonant rational surface and vessel minor radii and ␶ R , ␶ w the plasma and vessel wall resistive time constants. The last term on the RHS represents the wall eddy current effect on the mode amplitude. Before the application of the external field, for 0⬍t ⬍t on , the plasma can be assumed to be 共marginally兲 stable, with ⌬ 0⬘ (W)⭐0 for all W.

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Phys. Plasmas, Vol. 9, No. 9, September 2002

Error field locked modes thresholds in rotating plasmas . . .

3917

Therefore, the mode spin-up can be positive 共e-drift兲 or negative 共i-drift兲 depending on the rate of viscous restoration of bulk fluid rotation 共ion direction兲 versus the restoration of pressure gradients increasing d ␻ /dt 共electron direction兲. * The partition of e- and i-drift cases according to magnetic field illustrated in Fig. 18 does not appear to have a causal relation with the threshold for reconnection. V. CONCLUSIONS

FIG. 18. Partitioning of e-drift and i-drift modes according to B. The ordinates are the averaged mode spin-up angular frequencies of the Mirnov signals averaged and the abscissae are the values of toroidal field scanned in the experiment.

With the external field ramp forced reconnection eventually occurs at a critical value of the external current I E and mode grows to a forced saturation amplitude W s1 where 兩 ⌬ 0⬘ (W s1 ) 兩 ⫽0. After turn off of the external field ramp the locked mode with amplitude W s1 finds itself again with ⌬ 0⬘ (W s1 )⬍0 and decreases as it spins, due to the wall eddy currents term. The decay of the velocity shift profile during spin-up can then be explained in elementary terms by considering the solution of Eq. 共34兲 when the external torques due to the external field are off d 共 y⫺1 兲 1 ⫹ 共 y⫺1 兲 ⫽0 dt ␶␮

for t⭓t off ,

␻ ⫺ ␻ 0 ⫽ 共 ␻ 共 t off兲 ⫺ ␻ 0 兲 e ⫺ 共 t⫺t off兲 / ␶ ␮ .

共41兲

共42兲

If there is a mode in the laboratory frame its observed magnetic Mirnov signal frequency is ␻ Mirnov and if the plasma is rotating toroidally with frequency ␻ then ␻ Mirnov⫽ ␻ ⫹ ␻ , * where ␻ is the diamagnetic terms from the dispersion rela* tion for the mode, so for a tearing mode will be related to the electron diamagnetic frequency. From Fig. 3 it appears that at t⫽t 0 , before mode onset, ␻ Mirnov兩 t 0 ⫽0 but ␻ ⫽ ␻ 0 ⫽V ␸ 0 /R. Also at the end of the braking ramp, t⫽t off , ␻ Mirnov兩 t off⫽0, and therefore, the local plasma rotation is ␻ t off⫽⫺ ␻ 兩 t offⰆV ␸ 0 /R 共nearly vanishing * in accord with pressure and temperature flattening in the island region兲. Then from Eq. 共42兲, in the laboratory frame we get, shortly after t⬎t off

冉

冊

ACKNOWLEDGMENTS

This work was performed under the European Fusion Development Agreement and was partly funded by Euratom, the UK Department of Trade and Industry and the Italian Consiglio Nazionale delle Ricerche. R. Fitzpatrick and T. Hender, Phys. Fluids B 31, 1003 共1991兲. T. C. Hender, R. Fitzpatrick, A. W. Morris, P. G. Carolan, R. D. Durst, T. Edlington, J. Ferreira, S. J. Fielding, P. S. Haynes, J. Hugill, I. J. Jenkins, R. J. La Haye, B. J. Parham, D. C. Robinson, T. N. Todd, M. Valovic, and G. Vayakis, Nucl. Fusion 32, 2091 共1992兲; P. Leahy, P. G. Carolan, R. J. Buttery, and K. D. Lawson, Proceedings of the 26th EPS Conference on Controlled Fusion and Plasma Physics, Maastricht, 1999, edited by B. Schweer, G, Van Oost, E. Vietzke 共European Physical Society, Petit-Lancy, 1999兲, Vol. 23J, p. 157. 3 R. J. La Haye, A. W. Hyatt, and T. J. Scoville, Nucl. Fusion 32, 2119 共1992兲. 4 R. J. Buttery, M. De Benedetti, T. C. Hender, and B. J. Tubbing, Nucl. Fusion 40, 807 共2000兲. 5 A. Gibson and the JET Team, Phys. Plasmas 5, 1839 共1998兲. 6 R. Aymar, V. A. Chuyanov, M. Huguet, Y. Shimomura, ITER Joint Central Team, and ITER Home Teams, Nucl. Fusion 41, 1301 共2001兲. 7 D. Stork, A. Boileau, F. Bombarda et al., Proceedings of the 14th EPS Conference on Plasma Physics and Controlled Fusion, Madrid, 1987 共European Physical Society, Petit-Lancy, 1987兲, Vol. 1, p. 306; J. A. Snipes, D. J. Campbell, P. S. Hughes et al., Nucl. Fusion 28, 1085 共1988兲. 8 A. I. Smolyakov, A. Hirose, E. Lazzaro, G. Re, and J. Callen, Phys. Plasmas 2, 1581 共1995兲. 9 A. Tanga, K. Behringer, A. E. Costly et al., Nucl. Fusion 27, 1877 共1987兲. 10 J. A. Wesson, R. Gill, M. Hugon, F. Schu¨ller, J. A. Snipes, D. Ward, D. Bartlett, D. Campbell, P. Duperrex, A. Edwards, R. Granetz, N. Gottardi, T. Hender, E. Lazzaro, P. Lomas, N. Lopez-Cardozo, F. Mast, M. N. Nave, N. Salmon, P. Smeulders, P. Thomas, B. Tubbing, M. Turner, and A. Weller, Nucl. Fusion 29, 641 共1989兲. 11 A. Santagiustina, S. Ali Arshad, D. J. Campbell, G. D’Antona, M. Debenedetti, A. M. Edwards, G. Fishpool, E. Lazzaro, R. J. La Haye, A. W. 1

2

that gives explicitly

␻ Mirnov共 t 兲 ⬇

A series of systematic experiments with external error field in JET has led to the reassessment and understanding of the empirical scaling laws governing the onset of error field locked modes and has shed light on the important physics of plasma rotation braking associated with a toroidal viscous drag originating from broken axisymmetry. In particular a very satisfactory match of basic theoretical concepts with data has been achieved, namely concerning the nonlinear mode amplification occurring after slowing down to half the original frequency. Also the representation of the error field threshold is improved by a power law expression including the plasma rotation frequency. Finally the global plasma braking effect associated with mode locking has been examined in detail. A new theory involving the role of helical modes has been developed, which seems consistent with observations. The relationship between global plasma braking and mode penetration requires an extensive treatment that will be presented fully elsewhere.23

d␻ ␻ * ⫹ 0 共 t⫺t off兲 . dt ␶␮

共43兲

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3918

Lazzaro et al.

Phys. Plasmas, Vol. 9, No. 9, September 2002

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18

K. C. Shaing, S. P. Hirshman, and J. D. Callen, Phys. Fluids 29, 521 共1986兲. 19 R. Fitzpatrick, Phys. Plasmas 6, 1168 共1999兲. 20 D. Edery, J. L. Soule R. Pellat et al., Proceedings of the 8th IAEA Conference on Plasma Physics and Controlled Fusion Research, Bruxelles 1980, Nucl. Fus. Supplement 共1981兲 Vol. 1, p. 269. 21 R. Fitzpatrick, R. J. Hastie, T. J. Martin, and C. M. Roach, Nucl. Fusion 33, 1533 共1993兲. 22 R. J. La Haye, Bullettin of the 43rd Am. Phys. Soc, Div. of Plasma Phys. Ann. Phys. Long Beach, Ca., 2001. 23 E. Lazzaro, T. Hender, and P. Zanca, Proceedings of the 29th EPS Conference on Controlled Fusion and Plasma Physics, Montreux, Switzerland 2002 共European Physical Society, Petit-Lancy, in press兲.

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