PHYSICS OF PLASMAS
VOLUME 6, NUMBER 9
SEPTEMBER 1999
On the stabilization of neoclassical tearing modes by electron cyclotron waves G. Ramponi,a) E. Lazzaro, and S. Nowak Istituto di Fisica del Plasma, Ass. EURATOM-ENEA-CNR, Via Cozzi 53, 20125 Milano, Italy
共Received 9 February 1999; accepted 14 June 1999兲 The control of neoclassical tearing modes in tokamaks by means of electron cyclotron current drive and heating is investigated. The nonlinear evolution of the amplitude in absence and in presence of the stabilizing terms of an auxiliary current inside the island and of the associate heating is solved self-consistently with the evolution of the rotation frequency for International Thermonuclear Experimental Reactor 共ITER兲 reference magnetic equilibrium 关ITER-JCT and Home Teams, Plasma Phys. Controlled Fusion 37, A19 共1995兲兴. It is shown that, unless the wall braking torque is neutralized by external means, neoclassical tearing modes in ITER will be locked in a very short time. On the other hand, for rotating islands, the beneficial effect of modulating the current source in phase with the island rotation is pointed out, after an analysis of the time scales of the relevant phenomena 共time response of the driven current, island rotation frequency, power pulse duration, and inductive response of the plasma兲. Consideration is given to different effects that may reduce the efficiency of the control of the flux reconnection rate and to the benefits of wall stabilization associated to the island rotation frequency. A quantitative assessment of the EC 共electron cyclotron兲 power required to keep the island width at a reasonable level is given, both in absence and in presence of wall stabilization. © 1999 American Institute of Physics. 关S1070-664X共99兲03809-4兴
I. INTRODUCTION
boundary conditions imposed on the magnetic perturbation by the eddy currents induced in the resistive tokamak vessel.17 Thus the vessel provides a definite reference frame and the stabilization process depends on the island movement, which cannot be eliminated by a Galilei transformation. Consequently, an efficient amplitude control by an external rf source requires a contribution constant in the rotating island frame. It is therefore convenient to modulate the rf power in phase with the island rotation. This fact makes the efficiency of the stabilization process depend also on the island rotation frequency and on the dynamical plasma response to the current source. In this paper the dominant effects of localized current drive and heating on neoclassical tearing modes by oblique injection of EC waves are discussed. To avoid misunderstandings it is necessary to consider carefully the geometry of the problem and the related time scales of the basic physical phenomenon of externally controlled reconnection of helical magnetic perturbations. The geometrical distribution of the rf driven electromotive force 共e.m.f.兲 density depends crucially on the fact that field lines cover ergodically a magnetic island surface but not its volume, with the nearly instantaneous uniformizaton by parallel transport of the rf power density and rf-driven c.m.f. along the field lines, i.e., on the surface of each of the nested flux tubes of which an island consists. In Sec. II the spatial distribution of the rf power absorbed on flux tubes around the O-point of a rotating island and the consequences on relevant time scales are discussed. In Sec. III the relevant equations for flux reconnection in presence of electron cyclotron current drive 共ECCD兲 and a modification of the local resistivity due to heating are pre-
The relevance of neoclassical effects in the description of magnetohydrodynamic 共MHD兲 tearing modes in tokamak discharges has been pointed out by several authors.1–3 As tokamaks begin operating in low collisionality, high  and large boostrap current regimes, neoclassical tearing modes will play an important role in limiting the stored energy and reducing confinement. In particular it has been pointed out that they could set the  limit of long pulse discharges of the proposed International Thermonuclear Experimental Reactor 共ITER兲 well below the ideal limit.4,5 This type of tearing mode occurs when an initial perturbation 共the so-called ‘‘seed island’’ having a width w s 兲 in an otherwise tearingstable plasma leads to the formation of a magnetic island big enough to flatten the local pressure, so that the loss of boostrap current inside the island sustains the perturbation or even leads to further growth. Suppression and/or control of magnetic islands by local interaction of radio-frequency 共rf兲 waves with the plasma has been proposed by a number of authors.6–16 We examine here the case of electron cyclotron 共EC兲 waves as a candidate for driving a current inside the island to locally replace the missing boostrap current, retaining also the associated electron heating. In fact, local heating inside the islands may itself promote the stabilization, both through the current perturbation, owing to the enhancement of the electrical conductivity, and by directly affecting the boostrap current. As is well known, in the laboratory frame these islands rotate with a frequency whose evolution depends on the a兲
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© 1999 American Institute of Physics
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Phys. Plasmas, Vol. 6, No. 9, September 1999
⬘ . sented, leading to the expression of the stabilizing term ⌬ EC In Sect. IV the nonlinear dynamics of magnetic islands are considered, both for the amplitude and for the rotation frequency. Numerical evaluations show the possibility of considering two regimes: The first one where the islands are rotating and stabilization must be obtained with modulated in phase rf power, the second where the islands are locked because of wall braking action. In the latter case we shall argue that it is worth considering either an external momentum input 共e.g., via neutral beams兲 to keep steady rotation of the plasma and the islands viscously coupled to it, or to design a ‘‘slow feedback’’ scheme based on the creation of an ad hoc adjustable ‘‘helical magnetic trap’’ to lock the mode at the correct spatial position of absorption of the EC wave power.18 II. GEOMETRY OF THE RF INTERACTION WITH ROTATING MAGNETIC ISLANDS AND RELEVANT TIME SCALES
A q⫽m/n surface is threaded by lines of force of the magnetic field which close onto themselves after m toroidal and n poloidal turns; when magnetic radial perturbations with resonant helical pitch change, by resistive diffusion and reconnection, the topology of these surface into finite size nested helical flux tubes 共magnetic islands兲, these flux tubes are closed upon themselves and covered ergodically by the field lines. As a matter of fact, the field lines winding number on the flux tubes inside the island separatrix can be calculated in terms of a local q * which is a continuous function of the distance from the ‘‘O’’ point to the separatrix and takes the value q * ⫽m/n on the ‘‘O’’ point.19 First we consider, in the simplest scenario of magnetic perturbations that are static in the laboratory frame, what happens immediately after the switch-on of an external source injecting rf power to drive a current at a q rational surface. Even if the rf illuminated zone has a finite extension ⌬ rf in the helical coordinate ⬵m ⫺n , in the very short time 储 ⬇r s2 / 储 of parallel thermal conduction the absorbed power and the associated momentum distribute uniformly on the flux tubes where they build up in a kinetic time . This last, being the current driven by EC waves mainly a consequence of an asymmetric collision rate, is essentially determined by the effective collision time of resonant electrons, i.e., ⬇ c ( v )⫽ 关 e ( v / v th) 3 兴 /(5⫹Z), where e is the thermal electron collision time, v th the thermal velocity and Z the effective ion charge.16 As typically 储 Ⰶ , this means that a back-electromotive force 共induced electric field兲 uniform in and depending only on the radial coordinate will rise in the island in a time . Later on, the radial profile of the induced electric field, embracing a number of closed helical flux tubes on which it produces its stabilizing effect, will decay locally by radial diffusion on a time scale D , which is a fraction of the resistive diffusion time ( D ⫽ 0 a 2 / 21 ), where 21 has a value depending on the radial extent of the electric field profile, 共see Appendix B兲. For a pulse length rf of the rf power such that ⬍ rfⰆ D , an efficient prevention of the mode growth can be expected without modification of the local current density profile.
Ramponi, Lazzaro, and Nowak
If the island is rotating at constant frequency , and the rf power is constant in time in the laboratory frame, the rf driven electromotive force density will appear in the island frame as an oscillating source with a consequent cyclic reduction of some of its effects of contrasting the rate of the flux reconnection process. Only a perfect phasing in time of the modulated rf power can be seen in the reference frame of the island as a constant correction of the reconnection rate and minimize the required control power.
III. FLUX RECONNECTION IN PRESENCE OF ECCD AND ECH
In the usual right handed polar coordinate system (r, , ), we consider a tokamak equilibrium with finite size rotating magnetic islands described by the contours of constant helical flux in the Reduced MHD description with the standard ordering r/R 0 Ⰶ1, B /B Ⰶ1 ⌿⫽⫺x 2
冉 冊
B ⫹ s cos , 2L s
共1兲
where L s ⫽((q 2 R)/(q ⬘ r s )) is the magnetic shear length, q(r)⫽rB /R 0 B the cylindrical safety factor, (t)⫽m ⫺n ⫺ 兰 t0 (t ⬘ )dt ⬘ is the instantaneous phase of the island rotating with pulsation , x⫽r⫺r s defines a slab coordinate from the rational surface where q(r s )⫽m/n, and s (t) is the helical flux reconnected on the rational surface which is proportional to the square of the island width 关 s ⫽(w 2 /16) ⫻(B /L s ) 兴 . The field reconnection process characterizing resistive instabilities implies a time variation of the 共helical兲 magnetic flux, hence an electric field according to Faraday–Ohm’s law. The process can be altered by the presence of contrasting electric fields associated with the mechanism of ECCD and a perturbation of resistivity ␦ due to electron cyclotron resonant heating 共ECRH兲. . E 储 ⫽ 0 J 储 ⫹ ␦ J 储 ⫹E 储 boot⫹E CD 储
共2兲
The boostrap e.m.f. density term, due to the electron parallel viscosity can be represented by E 储 boot⬵⫺
冉 冊冓 冔
e R ⌿ ⬘兩 rs e
pe , x
共3兲
e , e being the viscous and collisional coefficiens and p e the electronic pressure. The incompressibility of J leads to the following expression: J 储 ⫽J I ⫹I 共 ⌿ 兲 ,
共4兲
where I(⌿) is determined by invoking Faraday law and averaging over the island with the nonlinear operator in the instantaneous island frame.20,21
冕 冕
arcsin
具¯典⫽
⫺arcsin
共¯兲
冑1⫺ ⫺2 sin2
d
arcsin
d
⫺arcsin
冑1⫺ ⫺2 sin2
,
Phys. Plasmas, Vol. 6, No. 9, September 1999
On the stabilization of neoclassical tearing modes by . . .
where 2 ⫽1/2(⫺⌿/ s )⫹1/2, and the range 0⬍⬍1 describes the island interior, while ⬎1 the exterior. This allows the identification of the flux function I(⌿) I 共 ⌿ 兲 ⫽⫺ ⫺1
s 具 cos 典 ⫺ 具 J I 典 t
⫹ ⫺1 具 E 储 boot典 ⫹ ⫺1 具 E CD 典, 储
共5兲
and substitution in the Ohm’s law and using Ampere’s law leads to a Grad–Shafranov equation 共for large R/a兲 for the island helical flux equilibrium with a nonlinear time dependent source term with nonlocal effects due to averaging over the island region
再
⌬⌿⫽⫺ 0 J 储 ⫽⫺ 0 J I ⫺ 具 J I 典 ⫺ ⫺1
冉
⫹ 1⫺
冊
冉
s 具 cos 典 t
冊
冎
␦ ⫺1 ␦ ⫺1 CD 0 具 E 储 boot典 ⫹ 1⫺ 0 具E储 典 . 共6兲
Equation 共6兲, associated with boundary and initial conditions derived from Eq. 共1兲, gives a formulation of the problem of the externally controlled helical flux nonlinear configuration in the tearing region. The main physical effects can be discussed in terms of the customary ‘‘lumped’’ variables description of the magnetic field reconnection process by averaging and integration operations 具¯典 and 具具 ¯ 典典 ⬁ ⬅r s 兰 ⫺⬁ dx 兰 ⫺ (¯)d in the island frame 共see Appendix A兲. The EC power, which injects momentum in the electrons with an equivalent electromotive force E CD 储 CD CD ⫽ 0 J ⫽⫺共⌿ )/( t) and reduces the resistivity through heating, modifies the rate of flux reconnection according to
s 2 具 cos 典 ⫽ 共 0 ⫹ ␦ 兲 ⫺1 0 ⵜ ⌿⫹ 共 0 ⫹ ␦ 兲关 J I ⫺ 具 J I 典 兴 t
冓
冔
⌿ CD共 x,t 兲 ⫹ 具 E 储 boot典 ⫺ . t
共7兲
We draw attention to the fact that the interpretation of Eq. 共7兲 as a competition of electric fields of different origin is a representation of the actual physical phenomenon preferable to the conventional descriptions in terms of current densities. As anticipated, the stabilization task requires that, in the frame of an island rotating with an instantaneous frequency , the externally applied counter-electromotive field ⫺ 具 关 ⌿ CD(x,t) 兴 / t 典 be seen as constant: This requires that in the laboratory frame the source should be modulated in phase with the rotating island. In order to have an effective controlling action, the field ⫺ 具 关 ⌿ CD(x,t) 兴 / t 典 should have a suitably prompt rise and should not decay faster than one rotation period. As shown in Appendix B, the rf term reaches its maximum in a time of the order of and tends to decay diffusely over a time D ⫽ a / 21 . Therefore, to have the maximum efficiency in the compensation of the reconnection process with a rf source modulated 共in the laboratory frame兲 in phase with the island at a frequency m ⫽2 /T m , it
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should be ⰆT m Ⰶ D . For EC waves and ITER relevant parameters at the q⫽2 surface, ⬇1 ms, and D ⬇4 s. In the evaluation of the contribution of a local heating to the rate of flux reconnection, which is presented here for completeness, we note that it depends on the effective time scale of the perpendicular heat transport inside the island separetrix. From Spitzer resistivity and the equilibrium solution of a thermal balance ruled by the transverse heat conductivity ⬜ , we have
␦ 2 3 ␦Te 3 ⬜ P EC 2 ⵜ ⌿⬅⫺ J ⬇⫺ ⵜ ⌿, 0 2 Te 储 2 0n eT e
共8兲
where ⬜ is an effective radial diffusion time for the electron temperature inside the island. For a flat profile of the power density P EC with a radial width ␦ H , it scales as13 w2 8 ⬜
for w⬍ ␦ H ,
w␦H ⬜ ⬵ 8 ⬜
for w⬎ ␦ H .
⬜ ⬵
共9兲
Now the nonlinear stage of the process of field line reconnection, which leads to a state of lower magnetic energy, can be described as an extension of Rutherford equation for the effective island width w( s ⫽(w 2 /16)(B /L s )) taking cos moments of Eq. 共7兲. This leads to the following expressions for the heating and current drive stabilizing terms:
⬘ ⫽6r s ⌬H
冉 冊冋 Lq I p共 r s 兲
册
Pˆ EC ¯ 兲, J 共 r 兲␦ F 共 w n e ⬜ T e 储 s H H
共10兲
¯ ⫽w/ ␦ H and where w ¯ 兲 ⫽g H ⫻ F H共 w
再
¯ ⬍1 for w ¯ ⭓1. for w
¯ w 1
Similarly the ECCD term is obtained
⬘ ⫽ ⌬ CD
冉 冊
32 I CDtot L q ¯ 兲, F 共w I p 共 r s 兲 ␦ 2j j
共11兲
¯ ⫽w/ ␦ j and where w
再
¯ ⬍1 for w ¯ ⬎1. for w
¯ 1/w gJ ⫻ ¯2 g J0 1/w
¯ 兲⫽ F J共 w
I CDtot is the total current driven within the annular strip of area 2 r s ␦ J and the factors g H,J,J 0 are averages of the heating and current profiles f H ( ), f J ( ) given in terms of the normalized flux surface label g H⫽
g J⫽
冕
0
冕
g J0⫽
⬁
⬁
0
冕
d f H 共 兲 具 cos 典
d f J 共 兲 具 cos 典
⬁
0
d f J共 兲
冕
冕
冕
arcsin
d
⫺arcsin
冑1⫺ ⫺2 sin2
arcsin
d
⫺arcsin
冑1⫺ ⫺2 sin2
arcsin
d
⫺arcsin
冑1⫺ ⫺2 sin2
.
,
,
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Phys. Plasmas, Vol. 6, No. 9, September 1999
Ramponi, Lazzaro, and Nowak
average electron density in units of 1020 m⫺3 and I CD / P the ratio between the total EC current and absorbed power. This figure of merit is machine dependent and, although from Fig. 2 it seems that the stabilization efficiency related to heating should dominate that related to the current term, for parameters typical of the standard configuration envisaged for ITER it turns out to be
⬘ ⌬ CD ⬘ ⌬H
0.6
⬵
␦ 2J 共 m 兲
冉 冊 冉 冊
FJ FJ ⬵15 ⬎1, FH FH
for not vanishing island width.
IV. A. Nonlinear dynamics of neoclassical tearing modes
FIG. 1. Plots of the weights factors g H , 共a兲, g J 0 , 共b兲, and of the ratio ¯ ⫽w/ ␦ J,H , for Gaussian heating and current g J /g J 0 , 共c兲, as functions of w density profiles f J,H ( ) centered at ⫽0 共i.e., r J ⫽r s 兲.
For realistic Gaussians profiles f J,H ( ) centered at ⫽0 共i.e., ¯ in Fig. r J ⫽r s 兲 the factors g H,J,J 0 are shown as functions of w ¯ ), F J (w ¯ ) and of F J /F H is shown 1 and the behavior of F H (w in Fig. 2. Note that, for full power absorption
⬘ ⌬ CD ⬘ ⌬H
⫽
冉
64 ␥ 20 n e ⬜ T e 3 具 n e典 J储
冊冉 冊冉 冊 1
␦ 2J
FJ , FH
where ␥ 20 is the overall current drive efficiency defined as ␥ 20⫽R 0¯n (I CD / P)(1020 A W⫺1 m⫺2 ), ¯n being the volume
When neoclassical effects such as the boostrap current are included in the Ohm’s law, the nonlinear island evolution is affected by a destabilizing term resulting from a negative perturbed helical boostrap current inside the island. The equation for the non linear evolution of the amplitude w, when obliquely injected EC wave power generates a current at r s and in presence of a conducting wall, turns out to be given by21 4g 1 dW r s2 ⬘ ⫺⌬ GGJ ⬘ ⫺⌬ pol ⬘ ⫺⌬EC ⬘ ⫽ 关 ⌬ 0⬘ ⫹⌬ bs dt R
⬘ 兲兴 , ⫺Re 共 ⌬ wall where the weight resulting from the island averaging is g 1⫽
冕
⬁
0
d 具 cos 典 2
冕
d
arcsin
⫺arcsin
冑1⫺
⫺2
sin 2
⬇0.82
, 4
in agreement with the traditional estimates.20 R ⫽( 0 r s2 )/ nc is the resistive time at the rational surface r s , nc the neoclassical resistivity, ⌬ 0⬘ the usual tearing ⬘ parameter driven by the equilibrium current gradient; ⌬ bs ⫽1.7 p (r s /R) 1/2兩 L q /L p 兩 关 w/(w 2 ⫹w s2 ) 兴 is the destabilizing term associated to the perturbed boostrap current, w s being a threshold island width, related to ( ⬜ / 储 ), 24 below which the mode is stable if ⌬ 0⬘ ⬍0 and L q ⫽q/q ⬘ , L p ⫽p/ p ⬘ the local gradient scale length of the safety factor and of the ⬘ ⫽6  p (r s /R) 2 (L q /r s ) 兩 L q /L p 兩 (1⫺1/q 2 )1/w pressure; ⌬ GGJ is the stabilizing term related to the equilibrium pressure gradients and favorable curvature in the outer part of the ⬘ ⫽7  p (r s /R) 3/2(L q /L p ) 2 r L2 (( / T )⫺1) 2 1/w 3 island,25 ⌬ pol is related to the finite polarization current due to ion inertia effects, r L being the poloidal ion Larmor radius and T ⬘ ⫽⌬ CD ⬘ ⫽T/(eBr s L Te ) the electron drift frequency; ⌬ EC ⬘ is the stabilizing term associated to the injected EC ⫹⌬ H power 关see Eqs. 共10兲 and 共11兲 in Sec. II兴 and ⌬ w⬘ ⫽
FIG. 2. Dependence of the functions F H , 共a兲, F J , 共b兲, and of the ratio ¯ ⫽w/ ␦ J,H , for Gaussian heatF J /F H , 共c兲, on the normalized island width w ing and current density profiles f J,H ( ) centered at ⫽0 共i.e., r J ⫽r s 兲.
共12兲
冉冊
2m r s rs d
2m
共 w 兲 2 ⫹i 共 w 兲 , 1⫹ 共 w 兲 2
is the action of a resistive wall with time constant w . 21 Such effect appears when the island rotates with frequency whose evolution is given by
Phys. Plasmas, Vol. 6, No. 9, September 1999
冋
On the stabilization of neoclassical tearing modes by . . .
册
d 1 dI ⫽ ⫺n 共 T EM⫹T visc兲 ⫺ 共 ⫺ T 兲 , dt I dt
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共13兲
where: I ⫽c g R 3 r s i (r s )w is the effective moment of inertia of the plasma mass associated through viscous drag to the rotation of an island with separatrix width w 共 i being the ion mass density and c g a geometry correction coefficient兲; T EM⫽ 关 (2 2 R)/ 0 兴 r s Im(⌬w⬘ )K2w4 is the electromagnetic torque due to the eddy currents in the wall, (K ⫽(B r s )/(16Rq(r s )L q )); T visc⫽ ␣ ( ⫺ T )/ M is the viscous torque with anomalous viscosity ( ␣ ⫽2.3 i r s2 (R/w)). Before showing the numerical solutions of Eqs. 共12兲 and 共13兲 for ITER relevant parameters, we point out that the problem of stabilization and/or control of a rotating magnetic island is primarily that of the equilibrium solution of the first order ordinary differential equation 关Eq. 共12兲兴, and the role of the external controlling source is to suppress the island growth through the achievement of a ‘‘new’’ stable solution for dw/dt⫽0. As it will be shown in the next section, the new saturated island width decreases as the peak power increases. If the task assumed is to suppress the island, shrinking it below the threshold size, the controlling power may be discontinued, to be turned on again when another trigger event is monitored, adopting a so-called bang–bang control scheme.18 The fully suppressed island, however, becomes undetectable, while the cause of the instability is generally not permanently removed by the compensating action of the external drive. Alternatively, it may be convenient to keep the island size under control at a nonvanishing, detectable equilibrium level, deemed not dangerous. In this case an accurate phase tracking is necessary to reduce the required power.
B. Numerical evaluations of Eqs. „12… and „13…
We base our evaluations on a set of official ITER parameters and corresponding profiles, considered typical for a thermonuclear power generation regime26
FIG. 3. Time evolution of the width 共normalized to the average plasma 2 radius兲 共a兲 and frequency 共normalized to f 0 ⫽1 Khz兲 共b兲 of the 1 mode in ITER 共without stabilizing EC term兲, neglecting the interaction with the tokamak vessel.
value (w sat /a⬵0.19) in t⫽130 s 共⬵0.3* tear , where tear ⫽0.82 R / 兩 r s ⌬ 0⬘ 兩 ) while the frequency after few seconds reaches the value ⫽ T . In Fig. 4 the electromagnetic interaction with the wall is taken into account 共with w ⫽0.3 s兲. It may be observed that the wall effect on the rotation frequency is crucial: After only 3 s the mode is completely locked. In such a case the island amplitude is unaffected by the wall and a feedback control would be possible only by keeping the island in rotation by neutral beam injection 共NBI兲 or external rotating magnetic fields, neutralizing the wall braking torque.27 We now take into account the stabilizing effect of EC waves by considering obliquely injected beams at a frequency f ⫽140 Ghz as an O-mode. The corresponding power and current density profiles, shown in Fig. 5 for P EC⫽50 MW, are computed by taking into account the selfdiffraction of Gaussian wave beams injected by seven horizontal rows of a multibeam launcher.22,23 We notice that a
n e 共 兲 ⫽n e0 共 0.9共 1⫺ 25兲 ⫹0.1; T e ( )⫽T e0 (0.964(1⫺ ) 1.25⫹0.036 with n e0 ⫽1.27⫻1020 m⫺3 ; T e0 ⫽28 KeV; major radius of the magnetic axis R ax ⫽8.36 m, Bt(R ax )⫽5.77 T, plasma minor radius a⫽2.8 m, plasma elongation K⫽1.6, plasma current I p ⫽21 MA. The parameters at the ⫽0.7 surface, where q⫽2, are: r s ⫽2.34 m; s ⫽r s /a⬵0.8; n e ⫽1.27⫻1020 m⫺3 ; T e ⫽7 KeV;  p ⫽0.6. Equations 共12兲 and 共13兲 are solved in a number of different cases for the 2/1 mode with r s ⌬ 0⬘ ⫽⫺2, w s ⫽2.5 cm, L q ⫽L p ⫽1 m, R ⫽1018 s. In Fig. 3 the time evolution of the island width and frequency is shown 共without the stabilizing EC term兲 neglecting the interaction with the tokamak vessel: The width grows reaching w/a⫽0.17 共corresponding to 90% of its saturated
2
FIG. 4. Time evolution of the width 共a兲 and of the frequency 共b兲 of the 1 mode of Fig. 3 when the wall interaction is taken into account 共with wall⫽0.3 s兲.
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Phys. Plasmas, Vol. 6, No. 9, September 1999
FIG. 5. Absorbed power and current density profiles for EC waves launched as O-mode gaussian beams at f ⫽140 Ghz by seven horizontal rows of mirrors on ITER plasma. The total injected power P EC⫽50 MW is evenly distributed on the seven beams who have slightly different toroidal injection angles 共ranging from ⫽19° for the first row 共at the top兲 to ⫽24.5° for the lowest兲 to compensate for the N 储 differences due to different poloidal components.
rather good localization at r s is achieved both for the current density and for the power deposition profile 共full width ␦ J ⬇ ␦ H ⬇0.2 m兲. In Fig. 6 the reduction of the amplitude of the growing island of Fig. 3 is shown, for two values of the total injected power and for current density and power deposition profiles centered at r s 共on the O-point兲 and. The width is reduced toward a new saturated level whose value is related to the EC power. It should be noticed that the heating effect 共calculated with ⬜ ⫽0.4 m2/s, as obtained from Refs. 5 and 26兲, is
FIG. 6. Time evolution of the amplitude of the magnetic island of Fig. 3 when oblique EC waves are injected on its O-point, for power and current density profiles centered, as in Fig. 5, at r s with ␦ J ⬇ ␦ H ⬇0.2 m and ⬜ ⬘ )兴. 共a兲: P EC⫽0; 共b兲: ⫽0.4 m2/s, 关neglecting the stabilizing term Re(⌬ wall ⬘ ⫽⌬CD ⬘ ⫹⌬ H⬘ 共continous line兲, ⌬ EC ⬘ ⫽⌬CD ⬘ 共dotted P EC⫽37.9 MW and ⌬ EC line兲; 共c兲: P EC⫽63 MW. It can be noted that the heating effect, as expected, ⬘ ). is much smaller than that of the current (⌬ H⬘ ⬵0.1 ⌬ CD
Ramponi, Lazzaro, and Nowak
FIG. 7. Values of w sat /a vs P EC共MW) for the same conditions of Fig. 6; 共a兲 ⬘ ⫽⌬CD ⬘ ; 共b兲 for ⌬ ⬘EC⫽⌬CD ⬘ ⫹⌬ H⬘ . for ⌬ EC
⬘ ⬵0.1⌬ CD ⬘ ), and much smaller than that of the current (⌬ H that the power values needed to reduce the amplitude are mostly related to the current drive efficiency which, in turn, depends on local values of electron density and temperature. In Fig. 7 the values of w sat /a vs P EC共MW) are shown in the same conditions of Fig. 6, whereas in Fig. 8 the effect of a radial shift of the location of the current and power deposition profiles is pointed out. In Fig. 9 the beneficial effect of the wall on the amplitude of the rotating island is shown: The saturated amplitude value is reduced to w/a⫽0.14 for P EC⫽0, and lower values of power are necessary to keep the island amplitude at reasonable values, as also shown in Fig. 10. All calculations leading to Figs. 5–10 have been made under the hypothesis of a perfect phase matching. In Fig. 11 it is shown that the control of the mode amplitude is crucially dependent on keeping a minimum and constant phase shift between the island rotation and the induced electric field.
FIG. 8. As in Fig. 7, for power and current density profiles not centered at r s 关 (r s ⫺r J )⫽2.4% r s 兴 .
Phys. Plasmas, Vol. 6, No. 9, September 1999
⬘ ) in the amplitude evoluFIG. 9. As in Fig. 6, including the term Re(⌬ wall tion with wall⫽0.3 s.
V. DISCUSSION AND CONCLUSIONS
We have discussed the formulation of the problem and the interpretation of the physical mechanism of controlled reconnection by interaction of localized rf wave beams with magnetic islands in tokamaks including consideration of the island rotation regime, which is an essential part of the physical mechanism. An essential result is that the correct time scales for the build up on the electromotive force contrasting the flux reconnection rate must be determined considering the role of ergodicity of field lines on the island flux tubes and parallel heat transport time on the distribution of rf power. We conclude that, for the required physical process, these time scales are compatible with an rf power modulation in the laboratory frame, phase locked to the instantaneous island
On the stabilization of neoclassical tearing modes by . . .
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FIG. 11. Effects on the island amplitude evolution of 共constant兲 phase shifts between the island rotation and the induced electric field.
rotation, that represents the most efficient, technically feasible, control scheme. We have also recalled and evaluated the beneficial reduction of the required rf power obtained by keeping the islands in rotation, through the stabilizing effect of the vessel eddy currents. As open problems we recall that the rf current has an m spectrum and the m⫽0 component may be large enough to modify the q profile11 or drive the natural ⌬ ⬘o ⬎0, and that compensation of the bootstrap drive is insufficient if the island width is above threshold for nonlinear coupling with other modes.27 For application to an ITER sized device it is shown that the crucial problem is that the time scales of the mode locking 共due for instance to interaction with eddy currents in the vessel兲 is much shorter than that of the mode amplitude growth. The assessment of required power presented here has been considered in the acceptable range of the ITER power availability although the results may appear in fact economically discouraging. More attractive scenarios could be explored varying the temperature and density profiles, but this would have far reaching consequences for the whole reactor operating regime and is obviously beyond the scope of the present work. In any case it is worth noting that, in general, more acceptable power requirements to restrain the island growth are possible by keeping the islands in rotation by NBI or external rotating magnetic fields, neutralizing the wall braking torque.
APPENDIX A: ISLAND GEOMETRY AND NONLINEAR AVERAGES
FIG. 10. Comparison between the values of w sat/a vs P EC共MW) computed neglecting, 关curve 共a兲兴, or including, 关curve 共b兲兴, the wall effects in the island amplitude evolution.
Here we recall the basic definitions and notation related to magnetic island geometry in large aspect ratio case and the fundamental integral operations performed in ‘‘island coordinates’’ in the island reference frame. A magnetic island is a spatial domain within a given 共slowly varying兲 constant magnetic flux contour ⌿ * ⫽⫺ s 共Fig. 12兲 and nested helical flux tubes described by Eq. 共1兲.
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具具 ¯ 典典 ⫽r s
冕
⫽r s w
冖
⬁
⫺⬁
dx
共 ¯ 兲d
冕 冕 ⬁
d
⫺arcsin
0
共¯兲
arcsin
冑
2
⫺sin2 /2
d 共 /2兲 . 共A4兲
Applying 具具¯典典 to Faraday–Ampere’s law 共7兲 multiplied by cos produces Eq. 共8兲 for the nonlinear evolution of the magnetic island width.
s 具具具 cos 典 cos 典典 t FIG. 12. Magnetic island flux contour in (r, ) coordinate; r s indicates the 2 m⫽ 1 resonant surface, r 1 and r 2 the radial strip boundary of width w.
⫺1 2 2 ⫽ 具具 0 ⫺1 0 ⵜ ⌿ cos 典典 ⫹ 具具 ␦ 0 ⵜ ⌿ cos 典典
⫺ A widely used notation sets ⍀⬅(⫺⌿ * / s ) and the range ⫺1⭐⍀⭐1 describes the island interior, ⫺1 corresponding to the island axis 共‘‘O’’ point兲 and 1 to the island separatrix 共boundary兲. The squared distance from the island axis is x 2⫽0 ⫽(4L s /B 0 ) s and, therefore, instantaneous value of the reconnected flux is r elated to the island width by
s⫽
冉 冊
w2 B0 , 8 2L s
共A1兲
The nested contours in an island can be described by
x ⫾ ⫽⫾
w 2
冑
2
⫺sin2 , 2
共A2兲
where ⍀⫽2 2 ⫺1, 0⬍⭐1, ⫽0 on the axis and ⫽1 on the separatrix. Consequently for the closed contours within the island separatrix boundary ⫺arcsin ⭐/2⭐arcsin . Two important integral operators must be defined to obtain the nonlinear evolution equations of Rutherford type. The first is the angular averaging operator of any quantity on any contour of label within the island, defined as 具 ¯ 典 ⫽养(¯)d /⌿ x* /养 d /⌿ x* and calculable 共often in terms of elliptic integrals兲 as
冕 冕
冔冔
⫹ 具具 共 0 ⫹ ␦ 兲关 J I ⫺ 具 J I 典 兴 cos 典典 .
共A5兲
The averages over the island region produce constant coefficients in the island frame, not necessarily in the lab frame. If the driving rf source in the lab frame is modulated with an arbitrary phase CD(t)⫽ t⫹ ␦ with respect to the island target ‘‘O’’ point 共although with at the same frequency of the rotating island兲, the rf terms appear oscillating in the island frame and their change of sign will produce alternating stabilizing and destabilizing effects. If the control action starts with the correct phase, but there is no phase tracking, the subsequent destabilizing effect might however be less dramatic due to the relatively long diffusive decay time of the stabilizing e.m.f. As illustration lets consider the CD driving term ( ⌿ CD/ t)⫽㜷 CD( ) f (t) as a sequence of periodic pulses in the lab frame. For the sake of argument we consider only one harmonic term: f (t)⬇cos CD(t), with CD(t)⬵ t⫹ ␦ . Then the EC driving e.m.f. term is
冓冓
⌿ CD cos t
冔冔
⫽ 具具 㜷 CD共 兲 f 共 t 兲 cos 典典 ⫽ 具具 㜷 CD共 兲 cos CD共 t 兲 cos 典典 ,
and since ⫽m ⫺ isl , one has cos cos CD共 t 兲 ⫽cos共 m ⫺ isl兲 cos CD ⫽ 12 关 cos m cos共 isl⫹CD兲
arcsin
具¯典⫽
冓冓
⌿ CD cos t
⫺arcsin
共¯兲
冑 2 ⫺sin2 d
arcsin
⫺arcsin
冑
2
⫺sin m sin共 isl⫹CD兲兴
d .
共A3兲
⫺sin2
The second integral operator defines the moments of any function of 共,兲 over the full island region. We define it as ⬁ dx 兰 ⫺ 具具 ¯ 典典 ⬅r s 兰 ⫺⬁ (¯)d and transforming from the (x, ) coordinates to the 共,兲 island coordinates, the operator 具具¯典典 takes the form
⫹ 12 关 cos m cos共 isl⫺CD兲 ⫺sin m sin共 isl⫺CD兲兴 . The high frequency terms 共⫹兲 can be ignored, and the 共⫺兲 terms are oscillating functions of the 共time dependent兲 phase difference between the source and the island. Only a control of the phase difference which keeps ␦ (t)⬅const produces a constant driving term in the island frame, and only perfect phase matching ␦ (t)⬅0 guarantees the maximum contribution to the stabilization process.
Phys. Plasmas, Vol. 6, No. 9, September 1999
On the stabilization of neoclassical tearing modes by . . .
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( 21 ⬇246) for values of x 1 , x 2 , w relevant for ITER, leading to characteristic times D ⫽ a / 21 shorter than the basic resistive time. From the solution of the boundary value initial value problem in cylindrical geometry the expression for the flux rate of change turns out to be, for x 1 ⭐x⭐x 2
共 x, 兲 共 x,0兲 ⫺ 2 ⬵ e 1 ⬁
⫹ ⫻ FIG. 13. Plot of R( ) vs , showing the value of the first 共smallest兲 root 1 which determines the reduction of the local diffusive time scale D with respect to the global diffusion time a .
兺 n⫽1
关 J m 共 n x 1 兲 J m 共 n x 兲 ⫺Y m 共 n x 1 兲 Y m 共 n x 兲兴
再冋 冕
储 X 2n 储 x2
x1
册
ˆ E RF 储 共 x ⬘ 兲 X n 共 x ⬘ 兲 x ⬘ dx ⬘ 关 f 共 兲 ⫺D 关 f 共 兲兴兴
冎
ˆ 关 共 兲兴兴 , ⫺ 关 x 2 X ⬘n 共 x 2 兲 ⫺x 1 X n⬘ 共 x 1 兲兴关 s 共 兲 ⫺D s 共B4兲
APPENDIX B: DRIVEN FIELD DIFFUSION IN THE ISLAND REGION
The linear analog of electrodynamic problem of Eq. 共6兲, of evolution of the 共helical兲 magnetic flux driven by an external time dependent electromotive force E EC 储 (x) f (t) consists in the solution in the island ⫻(dF CD/dt)⫽E EC 储 region of the equation
m 1 ⫽⫺E EC 关 x ⬘兴 ⬘⫺ 2 ⫺ 储 共 x 兲 f 共 兲, x x
储 X 2n 储 ⫽
共 r 1 ,t 兲 ⫽⫺ s 共 t 兲 , 共B2a兲
冉 冊
B0 ⌿ * 共 x, , ,0兲 ⫽⫺ 共 x⫺x s 兲 2 ⫹ s 共 0 兲 cos共 m ⫺n 兲 . 2L s 共B2b兲 The solution of Eq. 共B1兲 in the region x 1 ⭐x⭐x 2 (x 1,2⫽x s ⫾w/2) can be sought in the form of a suitable Fourier– Bessel expansion in eigenfunctions of the corresponding homogeneous problem: X n 共 x 兲 ⫽C N 关 J m 共 n x 1 兲 J m 共 n x 兲 ⫺Y m 共 n x 1 兲 Y m 共 n x 兲兴 , with C N a normalization constant and the eigenvalues n being the roots of the relation R 共 n 兲 ⫽ 关 J m 共 n x 1 兲 Y m 共 n x 2 兲 ⫺Y m 共 n x 1 兲 J m 共 n x 2 兲兴 ⫽0. 共B3兲 The plot of R( ) in Fig. 13 shows that, for m⫽2, the first 共smallest兲 root 1 , which determines the reduction of the time scale of the long time diffusive evolution, is rather large
冏冕
x2
x1
dxx 关 J m 共 n x 1 兲 J m 共 n x 兲
冏
⫺Y m 共 n x 1 兲 Y m 共 n x 兲兴 2
1/2
,
and ˆ 关 f 共 t 兲兴 ⫽ D
共B1兲
written in the dimensionless variables x⫽r/a, ⫽t/ a , a ⫽ 0 a 2 / , ¯ ⫽ a . A magnetic island is a spatial domain within a given 共slowly varying兲 constant magnetic flux contour ⌿ * ⫽⫺ s 共Fig. 12兲. Therefore, an adequate choice for 共time dependent兲 boundary and initial conditions for the equation for flux evolution under rf power deposition is
共 r 2 ,t 兲 ⫽⫺ s 共 t 兲 ,
where
冕
0
2
e ⫺ n共 ⫺ 兲 f 共 兲 d
is a delay operator with a dominant time constant D ⫽ a / 21 共⬃4 s for ITER兲. The second term on the right-hand side 共RHS兲 of Eq. 共B4兲 leads to the assessment of the time scales of the effect of the external source in reducing the rate of spontaneous flux reconnection. For a simple switch-on function of the rf power source f ( )⫽1⫺e ⫺t/ , one can see that in the interval x 1 ⭐x⭐x 2 the rf term evolves as
冋冕
x2
x1
册
⫺t/ E EC ⫺e ⫺t/ D 其 , 储 共 x ⬘ 兲 X n 共 x ⬘ 兲 x ⬘ dx ⬘ 兵 e
reaching its maximum at t * ⫽ ln(D /). Therefore, the compensation of the reconnection process with an rf source modulated in phase with the island at a frequency m ⫽2 /T m occurs in a time much smaller than the diffusive time, but, in order to have the maximum efficiency 共i.e., the rise of the counter electromotive field to its maximum兲, it should be T m Ⰷ . 1
R. Carrera, R. D. Hazeltine, and M. Kotschenreuther, Phys. Fluids 29, 899 共1986兲. 2 J. D. Callen, W. X. Qu, K. D. Siebert, B. A. Carreras, K. C. Shaing, and D. A. Spong, in Plasma Physics and Controlled Nuclear Fusion Research, Vol. II 共International Atomic Energy Agency, Vienna, 1987兲, p. 157. 3 T. S. Hahm, Phys. Fluids 31, 3709 共1988兲. 4 H. R. Wilson, M. Alexander, J. W. Connor et al., Plasma Phys. Controlled Fusion 38, A149 共1996兲. 5 O. Sauter, R. J. La Haye, Z. Chang et al., Phys. Plasmas 4, 1654 共1997兲. 6 J. A. Holmes, B. Carreras, H. R. Hicks, S. J. Lynch, and B. V. Waddell, Nucl. Fusion 19, 1333 共1979兲.
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