Plasma Physics Reports, Vol. 27, No. 9, 2001, pp. 727–732. Translated from Fizika Plazmy, Vol. 27, No. 9, 2001, pp. 771–776. Original Russian Text Copyright © 2001 by Savrukhin. TOKAMAKS Analysis of Quasistatic MHD Perturbations and Stray Magnetic Fields in a Tokamak Plasma P. V. Savrukhin Russian Research Centre Kurchatov Institute, pl. Kurchatova, Moscow, 123182 Russia Received October 23, 2000; in final form, April 4, 2001 Abstract—Mechanisms for the development of quasistatic MHD perturbations in a viscous rotating tokamak plasma are considered. The influence of stray magnetic fields on the stability of MHD modes in the plasma of the TFTR tokamak is analyzed. © 2001 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION involve the three-dimensional modeling of magnetic Recently, the problem of quasistatic MHD perturba- fluxes (including the fluxes induced in the tokamak tions in large tokamaks has attracted considerable inter- conductors) at given positions of the controlling coils. est [1, 2]. An analysis shows that these perturbations Unfortunately, such modeling does not provide the can arise either due to the rotation of magnetic islands required accuracy in determining the stray fields or due to the destabilization of static (locked) MHD because of the complicated spatial structure of the modes. The stopping of rotation is usually accompa- induced fluxes and the mechanical deformation of the nied by a rapid increase in the perturbation amplitude tokamak construction during the experiment. Another and, sometimes, by discharge disruption (see [3]). One method is the direct measurement of helical perturba- of the factors favorable for the development of quasi- tions of the magnetic field with the help of magnetic static MHD perturbations is the breaking of the symme- probes and saddle loops. However, under the experi- try of the tokamak equilibrium magnetic configuration mental conditions, magnetic probes are situated outside (the generation of stray magnetic fields). The threshold the plasma (outside the vacuum chamber of the toka- amplitude of the stray magnetic field at which quasi- mak), which hampers the determination of the local static MHD perturbations develop decreases substan- field structure inside the plasma column. Magnetic tially as the tokamak size increases, the plasma rotation fields inside the vacuum chamber can be measured if slows down, and the plasma pressure increases (see [1, the tokamak is equipped with special movable probes 3]). For these reasons, this kind of instability may be (e.g., in experiments in the DIII-D tokamak [1]). How- dangerous for future tokamak reactors with a slowly ever, such measurements cannot be carried out during rotating plasma under burning conditions [3]. The pri- the discharge. The accuracy of the measurements of mary cause for the appearance of stray magnetic fields, stray magnetic fields can be substantially improved if which is associated with the imperfect fabrication of the tokamak is equipped with additional coils for gen- the tokamak magnetic system and the nonsymmetric erating helical magnetic fields with given amplitudes configuration of conductors, can be minimized by and spatial orientations [2]. In this case, the stray fields improving the accuracy of the device assembly. How- can be deduced by analyzing the thresholds for the ever, there are a number of factors that are fundamen- destabilization of quasistatic MHD perturbations at dif- tally unavoidable. First of all, there is the asymmetry of ferent amplitudes and phases of the external helical the tokamak mechanical elements (such as the divertor, magnetic fields. diagnostic ports, and neutral beam injectors), the inho- Unfortunately, experiments with helical magnetic mogeneity of internal plasma perturbations, and the fields produced by additional coils present serious local character of the interaction of the plasma with the problems to large tokamaks because of the high equip- chamber wall during major disruptions (the excitation ment cost and the rigorous schedule of the device oper- of halo-currents and the injection of impurities [4]). ation. In such a situation, the stray fields can be The diversity of sources exciting inhomogeneous mag- deduced by analyzing the dynamics of internal MHD netic fields makes it difficult to predict the onset of perturbations [5]. In this paper, we describe a procedure instabilities and hampers the development of stabiliza- of determining the dominant harmonics of external tion systems for future experiments. helical perturbations based on the numerical modeling At present, there are several methods for identifying of tearing modes in a viscous rotating plasma. By ana- the structure of stray fields. A direct method is based on lyzing internal perturbations in the TFTR plasma as an the calculations of magnetic fields excited by currents example, we identify the m = 2, n = 1 harmonic (where in the tokamak magnetic system. These calculations m and n are the transverse and longitudinal wavenum- 1063-780X/01/2709-0727 $21.00 © 2001 MAIK “Nauka/Interperiodica” 728 SAVRUKHIN bers, respectively), which plays a key role in the initi- one-fluid MHD theory [6], which describes the electro- ation of the discharge disruption. In order to improve magnetic effects in a plasma with finite viscosity and the accuracy of the analysis, the parameters of the inertia. The geometry of the model is shown schemati- numerical model are determined by comparing the cally in Fig. 1. results of calculations with the results of previous External magnetic field perturbations and MHD experiments in tokamaks equipped with external coils perturbations (modes) are represented in the form of (JET, DIII-D) at prescribed helical perturbations of the helical harmonics B = B exp( jmχ ) and B = magnetic field [5]. em e e rm χ χ ϑ ϕ ω χ Brexp( jm m). Here, m m = m Ð n + ∫ mdt , m e = ϑ ϕ ω χ χ 2. NUMERICAL MODEL OF MHD m Ð n + ∫ edt (where m and e are the phases and PERTURBATIONS CONTROLLED ω ω m and e are the angular frequencies of the modes and BY EXTERNAL MAGNETIC FIELDS external fields, respectively), and θ and φ are the trans- The evolution of MHD perturbations under the verse (poloidal) and longitudinal (toroidal) coordinates. ω action of external helical magnetic fields is analyzed The amplitude Br and the instantaneous frequency m = χ using a phenomenological model of tearing modes in a d m/dt are described by the equations viscous rotating plasma [5]. The model is based on the B dB /dt= c ∆' B r r 1 frb r (1) ()ω τ 2 ()ω 2 τ 2 ()χ χ 2 1 – c2 Br m w /1+ m w + c3 Be cos e – m , ω ()ω ω Brd m/dt= c4 p – m (2) 2ω τ ()ω 2 τ 2 ()χ χ – c5 Br m w/1+ m w + c6 Be sin e – m ; ∆ where 'frb is the stability parameter of the tearing τ mode in a plasma with a free boundary, w is the time ω constant of the conducting tokamak chamber, p is the instantaneous rotation frequency of the bulk plasma surrounding the magnetic island, and ci (i = 1Ð6) are numerical factors calculated using the measured plasma parameters. MHD perturbations were modeled and stray mag- netic fields were then identified using the PLASCON 3 program [5] in the MATLAB programming environ- ment [7]. The block diagram of the program is shown ω (‡) in Fig. 2. The TEARING MODE block models the growth and rotation of MHD perturbations by numeri- cally solving Eqs. (1) and (2) for given values of B and ω e p0 χ ∆ e. The stability parameter of the tearing mode 'frb is ω p specified using the current density profile calculated with the TRANSP code [4]. The transmission charac- (b) teristics of the set of magnetic probes (see below) are ω modeled by the MAGNETIC block. The calculated val- m ω Ω ues of Br, m (denoted by BR and m), and the signals r from magnetic detectors (BR-LMD and BP-MM) are rs rp rw compared with the measured values. The amplitude and phase of external fields are determined by fitting the Fig. 1. Schematic illustration of the tearing mode model and calculated values to the experimental results. the profiles of the angular plasma rotation velocity (a) in a ω quasi-stable configuration and (b) upon the onset of MHD The instantaneous angular frequency m and the perturbations. Magnetic islands ( ) are located in the vis- 1 perturbation amplitude Br depend on the mode phase cous plasma (2) confined in a chamber with conducting ω ω with respect to the external magnetic field [see Eqs. (1), walls (3). Here, m and p are the instantaneous angular χ χ frequencies of the mode and the plasma, respectively; r , r , (2)]. Depending on the phase shift ( e Ð m), the mode s p periodically accelerates and decelerates during the rota- and rw are the minor radii of the resonant magnetic surface, the bulk plasma, and the conducting wall, respectively; and tion period. At large amplitudes of the external field, ω p0 is the plasma rotation frequency in the absence of tear- such nonuniform rotation produces characteristic saw- ing modes. tooth signals from magnetic detectors, which allows us PLASMA PHYSICS REPORTS Vol. 27 No. 9 2001 ANALYSIS OF QUASISTATIC MHD PERTURBATIONS 729 to identify the phase of the stray field under given Experimental data experimental conditions. Magnetic detectors 3. EXPERIMENTAL RESULTS Plasma parameters BLMD, Bp AND COMPARISON WITH CALCULATIONS ω R0, rp, rw, p, The diagnostic complex of the TFTR tokamak [8] ∆' , B , T , frb τt e makes it possible to identify external and internal MHD ne, w perturbations and, concurrently, to measure the plasma MHD modes m/n, , , ω parameters used in numerical calculations (see table rs wi Br m and Fig. 3). The spatial structure of internal MHD modes (wavenumbers m and n), the angular frequency, Model and the magnetic island width are measured with the χ help of two microwave polychromators situated in the TEARING MODE Be, e cross sections separated by an angle of 126¡ in the tor- oidal direction.