CT/P- Analytical Study of RWM Feedback Stabilisation with Application to ITER Y Gribov ), VD Pustovitov ) ) ITER International Team, ITER Naka Joint Work Site, Japan ) Nuclear Fusion Institute, Russian Research Centre ÒKurchatov InstituteÓ, Russia e-mail contact of main author: gribovy@itergpsnakajaerigojp Abstract An analytical model for studying the feedback control of Resistive Wall Modes (RWMs) in a tokamak with single or double conducting wall is presented The model is based on a cylindrical approximation It is shown that the outer conducting shell, in the case of ITER-like double wall vacuum vessel, does not significantly reduces the RWM instability growth rate but deteriorates the feedback stabilisation It is also shown that six side saddle coils with the nominal voltage 4ÊV per turn is capable of stabilizing the RWM for the expected range of normalized beta Introduction An analytical model for studying the feedback stabilisation of RWM in a tokamak with single or double conducting wall is presented The model is based on a cylindrical approximation - a single mode with poloidal number m and ÒtoroidalÓ number n is considered The model comprises a cylindrical plasma with radius a p, two thin cylindrical conducting shells with radii a, a, thickness d, d, electrical conductivity s, s, and the ideal feedback coils producing the same harmonic (m,ên) Thus, all currents and magnetic fields are proportional to exp[i(mjê-ênv)], where J and V are the poloidal and ÒtoroidalÓ angles, so that r, J and zê=êrv are the cylindrical coordinates (R is euivalent to tokamak major radius) Z, m 6 4 - -4-6 3 4 5 6 7 8 9 R, m FIG ITER plasma, vacuum vessel and their simplified cylindrical models Numerical estimates have been made for ITER, which has elongated plasma and double wall vacuum vessel The upper part of FigÊ shows an ITER plasma of 9ÊMA steady-state scenario and two shells of the vacuum vessel The cylindrical circular model of ITER, used in the analytical study of RWM stabilisation, is shown in the lower part of FigÊ The model has a plasma radius a p Ê=Ê35Êm and shell radii a Ê=Ê35Êa p, a Ê=Ê7Êa p Each shell of the vacuum vessel has a thickness of 6Êmm and resistivity 85ÊmW m Euation for RWM The euation modeling the feedback control of the (m,ên) mode in the double-wall tokamak, derived in [], can be written as: d Br dbr gl -( g -l) - glbr = Bf () dt dt g
CT/P- Here B r is the radial component of the total magnetic field of the mode on the st shell, t = t/ T m is the dimensionless time normalized by resistive time constant of the st shell Tm = ms a d ( m ), B f is the radial component of the magnetic field of the (m,ên) harmonic produced by the feedback coils on the st shell, and g Ê=ÊG T m with G being the growth rate of RWM in the presence of only the st shell without the feedback stabilisation Parameters g and l are, correspondingly, the normalized growth rate and decay rate of the two branches of RWM They depend on the wall parameters and on g : g -l = g -ax, gl = g x( a -), [ ] m m- where x = ( a / a ) - and = + ( s / s )( d / d )( a / a ) - a The model for ITER is characterized by xê=ê66, aê=ê3 for the mê=ê mode and by xê=ê33, aê=ê4 for the mê=ê3 mode 3 RWM without Feedback Control 5 GAMMA(m=3) GAMMA(m=) LAMBDA(m=3) LAMBDA(m=) Without the feedback control, the nd shell with high resistivity (s Ê Ê) would not affect the RWM growth rate, g g (the case of a single wall), whereas it would reduce the growth rate in the opposite case: g g -x when s Ê Ê [] The estimated effect of the nd shell of the ITER vacuum vessel on the RWM growth rate for mê=ê and mê=ê3 is shown in FigÊ The outer shell of the ITER vacuum vessel does not significantly reduce the RWM growth rate (gê»êg ), but, as shown below, it may deteriorate RWM active stabilisation, screening the feedback-produced magnetic field g, l 45 4 35 3 5 5 5 5 5 5 3 35 4 45 5 FIG RWM unstable (g) and stable (l) roots of euation () for RWM modes mê=ê and mê=ê3 in ITER cylindrical model as function of g go 4 Ideal Feedback Coils For active stabilization of the mode (m,ên), the feedback coils must produce the field with the same (m,ên) Static efficiency of this ideal feedback coils can be characterized by a parameter b m,n defined as Bf = bmn, If, where I f is the current in feedback coil producing mode (m,ên) We study the feedback control of RWM assuming T f Ê>>ÊT m, where T f Ê=ÊL f Ê/ÊR f (L f and R f are the effective inductance and resistance of the feedback circuit), since in ITER the time constant of the st shell for mê=ê mode is T Ê»Ê7Ês, while T f Ê» 5Ês The feedback circuit euation can be written as dbf Tm d T B bmntm f Vf t + =,, () f Lf where V f is the voltage applied to the feedback circuit The second term is small under ITER conditions
3 CT/P- 5 Feedback Control with Radial Field Sensors We describe the feedback control of RWM in terms of the radial component B r of the total magnetic field on the st shell, consisting of several parts: B p from the plasma, B from the st shell, B from the nd shell, and B f from the feedback coil, Br = Bp + B + B + Bf The following feedback algorithm is studied: db kb k db k d f r B r =- r - - 3 dt dt dt (3) Here the term proportional to k is a conventional term used for RWM stabilization (see, for example, [] and []) It is shown below that the term proportional to k 3 is needed in the case of double wall for stabilizing a highly unstable RWM The term with k determines the desired level of B r (zero, in this case) This feedback algorithm is euivalent to the voltage control through euation () Using (3) with constant gains k i, one can get from () the following characteristic euation for RWM in the double wall tokamak when radial field sensors are used in the feedback system: 3 s + cs + cs+ c =, (4) c = xa ( -) k 3 - g + ax, c = xa ( -)( k - g ), c = k xa ( - ), s = ST m, where S is the variable of Laplace transformation When the RWM is stabilized, all the coefficients in (4) are positive This would make negative the real parts of the roots of euation (4) Therefore the necessary conditions for RWM stabilization can be fulfilled if xa ( - ) k3 > g - g cr, k > g, k >, g cr = ax (5) In principle, in this case (double wall) stabilization of RWM could be possible without the term d B r /dt in (3) (k 3 Ê=Ê), if the instability would not be too strong (g < g cr ) [] According to (5), with proper k 3 this restriction on g is eliminated In ITER cylindrical model the critical value of RWM instability growth rate, G cr, corresponds to G cr T = mg cr» 4 for mê=ê, 3 Therefore, moderately unstable RWMs having GT < 4 are expected to be stabilised in ITER without a signal proportional to d B r /dt In order to achieve QʳÊ5 at steady state operation, the value of b N should be somewhere between b N (noêwall) and 5[b N (noêwall)ê+êb N (idealêwall)] (Here b N (noêwall)ê»ê6 and b N (idealêwall)ê»ê36 are corresponding limits imposed by kink modes on the value of b N in the cases without conducting wall and with ideally conducting st shell [3]) This range of b N corresponds to moderate unstable RWMs with GT Ê<Ê4 In the case of a single wall (s Ê=Ê, a ), assuming k 3 Ê=Ê, the characteristic euation (4) is reduced to s + ps + p =, p = k - g, p = k (6) The RWM is stabilized when k Ê>Êg and k Ê>Ê These conditions for k and k are the same as those in (5) However, in this case the term with d B r /dt in (3) is not needed even for a very unstable RWM (high value of g ) 6 Feedback Control with Poloidal Field Sensors In this model, a feedback system with sensors measuring the poloidal component B J of the total perturbed magnetic field on the inner side of the st shell cannot be much different from that using the radial sensors because B = ( + g ) [4] Using B J instead of B r in (3), we J B r
4 CT/P- obtain the same B f as in the previous case, if we reduce all k i by a factor of + g Therefore, in this case the stabilization of RWMs in the double wall tokamak will be achieved with gains k i smaller than those in (5) for the feedback with radial sensors: g - g cr g x( a - )k3 >, k >, k > (7) + g + g Similar to the feedback with radial field sensors, a signal proportional to d B J /dt is reuired when the instability is strong (g > g cr ) In the case of a single wall we obtain the same conditions for k and k, but the term with d B J Ê/dt is not reuired even for a very unstable RWM Note that, in contrast to the case of the feedback with radial field sensors, the reuired gain k does not increase proportional to g Therefore, with poloidal sensors, any k > 5 is sufficient for stabilizing the RWM with arbitrary growth rate Smaller gains are better, but that cannot be the only reason of the dramatic difference between the feedback with radial and poloidal sensors [5] The argument above is valid for ideal feedback coils producing a single mode magnetic field However, the conventional array of saddle feedback coils generates, in addition to the necessary (m,ên) harmonic, some side-band harmonics that are not needed for RWM stabilization, but affects the probe measurements The signals measured by the ÒradialÓ sensors in the euatorial plane are affected much stronger than the ÒpoloidalÓ sensors [6] In some cases the ÒradialÓ signal can become zero while the mode is not yet suppressed That is why the conventional feedback system with radial probes can fail At the same time, a similar system with poloidal sensors can be uite efficient in suppressing RWM [6] 7 Feedback Control with Voltage Saturation For a given g, the value of feedback gains providing a desired uality of RWM control can be easily found from characteristic euations (4) or (6) For example, in the case of a single wall, feedback control with the radial field sensors will have a critically damped regime with a settling time T set when k = g + tset, k = tset, tset = Tset Tm The poloidal field sensors ensures the critically damped regime when these gains are reduced by a factor of + g Even with the appropriate choice of feedback gains, the RWM control can fail, when the voltage reuested by the controller (3) is higher than the limit, V max, established by the power supply For a given value of V max, we can estimate the level of the perturbation B r when the control of the mode is impossible because of the voltage limitation Consider a mode growing till tê=ê without control At tê=ê the feedback control is switched on If Br ( ) = B is high enough, the voltage reuested by the controller via (3) and () is higher than the capability of power supply V max, and only the constant voltage will be applied to the feedback circuit In this case, the RWM evolution is described by () and () with V f = ÐV max Neglecting the term proportional to T m /T f in (), the solution is expressed by the following formula: bmntm Br = ( B - Bcr) e + æ gt g ö -lt ( g + l) ( glt - g + l) l, Bcre + Bcr, Bcr = è ø ( + ) L V max l l g g l g f The evolution of RWM will follow one of the two scenarios If B < B cr, B r will sooner or later reduce to the level when the reuired feedback voltage becomes lower then V max This makes possible feedback stabilization of the RWM In the opposite case, when B ³ B cr, the
5 CT/P- RWM will grow unlimited It should be noted that, in practice, the value of the critical field B cr is reduced by a technical limit imposed on the current in the feedback circuit In ITER, the error field correction coils (CCs) will be used for the feedforward and feedback stabilisation of RWM The system comprises 6 top CCs, 6 side CCs, and 6 bottom CCs Toroidally opposite coils are connected to produce the magnetic field with nê=ê and have a common power supply The feedforward stabilisation is achieved by error field correction using all CCs The feedback stabilisation of RWM will be provided by the voltage applied to the side CCs according to a signal from the magnetic probes located between the plasma and vacuum vessel inner shell A rough estimate for ITER with the design parameters V Ê=Ê4ÊV/turn, bê=êêt/ma turn, L f Ê=Ê5ÊmH/turn gives B cr about ÊmT Assuming that an RWM with the amplitude B r Ê» ÊmT can be detected, the instability can be stabilized with the voltages available for the ITER side CCs 8 Conclusions The study based on the cylindrical model [] has shown that ITER-like double wall vacuum vessel does not significantly reduces RWM instability growth rate (gê»êg ), but deteriorates feedback control There is a critical value of instability growth rate, G cr, above which the feedback voltage should have a term proportional to d B/ dt in addition to the conventional term proportional to db/dt In ITER steady state operation with QʳÊ5, moderately unstable RWMs having GT Ê<Ê4 is expected These instabilities have growth rates less than the estimated value of G cr and therefore their stabilization can be achieved without knowledge of d B/ dt The voltage of 4ÊV/turn in side CCs seems reasonable for control of the RWM having the amplitude less than about ÊmT These analytical results are also supported by numerical calculations [] The poloidal field magnetic sensors located inside the vacuum vessel inner shell are preferable for feedback control For example, in the case of a single wall and ideal feedback coils, RWM stabilization can be achieved with the ÒradialÓ sensors when the gain k is proportional to the instability growth rate g, whereas with the ÒpoloidalÓ sensors, it can be achieved with k independent on g, if kê>ê/ References [] PUSTOVITOV VD, ÒFeedback Stabilisation of Resistive Wall Modes in a Tokamak with a Double Resistive WallÓ, Plasma Physics Reports, 7 () 95-4 [] LIU YQ, et al, ÒActive Control of Resistive Wall Modes in TokamaksÓ, Proc 9 th EPS Conf on Plasma Phys and Contr Fusion (Montreux, 7- June ) ECA 6B () P-6 [3] MEDVEDEV SY, Private communication () [4] PUSTOVITOV VD, ÒComparison of RWM Feedback Systems with Different Input SignalsÓ, Plasma Phys Contr Fusion 44 () 95-99 [5] JOHNSON LC, et al, ÒStructure and Feedback Stabilization of Resistive Wall Modes in DIII-DÓ, Proc 8th EPS Conference on Contr Fusion and Plasma Phys (Funchal, 8- June ) ECA 5A () 36-364 [6] PUSTOVITOV VD, ÒIdeal and Conventional Feedback Systems for RWM SuppressionÓ Report NIFS-73 (National Institute for Fusion Science, Toki, Japan)