TRITA SE"00363 Report ISSN 1102-2051 VETENSKAP ISRN KTH/ALF/R--99/6--SE OCH KONST m KTH Numerical Simulation of Reversed-field Pinch Confinement J. Scheffel and D.D. Schnack ~~ Research and Training programme on CONTROLLED THERMONUCLEAR FUSION AND PLASMA PHYSICS (Association EURATOM/NFR) FUSION PLASMA PHYSICS ALFVEN LABORATORY ROYAL INSTITUTE OF TECHNOLOGY 3 1-04 SE-100 44 STOCKHOLM SWEDEN TRITA-ALF-1999-06 ISRN KTH/ALF/R--99/6-SE Numerical Simulation of Reversed-field Pinch Confinement J. Scheffel and D.D. Schnack VETENSKAP OCH KONST Stockholm, August 1999 The Alfven Laboratory Division of Fusion Plasma Physics Royal Institute of Technology SE-100 44 Stockholm, Sweden (Association EURATOM/NFR) Numerical Simulation of Reversed-field Pinch Confinement J. Scheffel Fusion Plasma Physics, Alfven Laboratory (Association EURATOM-NFR), Royal Institute ofTechnology, SE-100 44 Stockholm, Sweden http://www.fusion, kth. se D. D. Schnack Science Applications International Corporation., 10260 Campus Point Dr., San Diego, CA, 92121 USA ABSTRACT A series of high resolution, 3-D, resistive MHD numerical simulations of the reversed- field pinch are performed to obtain scaling laws for poloidal beta, energy confinement and magnetic fluctuations at Lundquist numbers approaching 106. Optimum plasma conditions are attained by taking the transport coefficients to be classical, and ignoring 40 radiation losses and resistive wall effects. We find that poloidal beta scales as fie «= / ~° and that energy confinement time scales as T£ °= / °'34 for fixed UN. The scaling of magnetic fluctuations is shown to critically depend on two parameters, rather than on Lundquist number alone. Keywords: Confinement, RFP, scaling law, beta, energy confinement. 1. BACKGROUND The reversed-field pinch (RFP) plasma is characterized by a nonlinear dynamo mechanism that continuously converts poloidal flux supplied from the external circuit into toroidal flux in the plasma core in such a way as to sustain the mean magnetic field configurations against resistive diffusion [1]. The additional energy required to sustain the magnetic fluctuations associated with this dynamo action must be supplied by the external circuit. This results in input power requirements that are in excess of those associated with Ohmic dissipation. In present experiments the dynamo fluctuations are thought to be responsible for anomalously large radial particle and energy transport. Early experimental results at low values of the Lundquist number S = TR/TA (where TR is the resistive diffusion time and TA is the Alfven transit time) indicated that these fluctuation levels may decrease with increasing Lundquist number [2], implying improved confinement properties in high temperature reactor relevant regimes. Later experiments with Lundquist numbers approaching 106 indicated a weaker scaling of the fluctuation level with Lundquist number [3], In this work we investigate the confinement properties of RFP plasmas by carrying out a series of high resolution, three-dimensional, nonlinear, resistive MHD numerical simulations whose aim is to determine the dependence of the poloidal beta fie and the energy confinement time tE on experimental parameters, such as the plasma current. These calculations include the effects of finite pressure, ohmic heating, convection and anisotropic heat conduction. By using plasma parameters relevant for present day RFP experiments, scaling laws that may be useful for predicting performance at higher temperatures and plasma currents are obtained. Throughout we have attempted to obtain configurations with optimal confinement properties, so that the results of this study should be considered to be optimistic. The scaling results so obtained indicate that the "conventional" RFP (ie., with no flow or current profile control) may not extrapolate well to thermonuclear conditions, primarily because of persistent magnetic fluctuations that are large enough to make the core magnetic field stochastic. Techniques for enhancing confinement, such as current profile and sheared flow control [4,5,6] may thus be required to establish the reactor potential of the RFP. Fluctuation scaling is a long standing and debated issue for the RFP. Because of the large number of resonant surfaces, the resistive instabilities that are the fundamental dynamo mechanism, destroy any nested flux surfaces locally. The resulting saturated magnetic islands are large enough to overlap, and the magnetic field becomes stochastic. A magnetic field line may then wander from the core to the edge of the plasma, instead of being confined to a flux surface. This establishes channels for particles and energy to escape in the radial direction along the magnetic field lines. It is thus desired that the amplitudes of the characteristic magnetic fluctuations decrease sufficiently rapidly with increasing Lundquist number (increasing temperature), so that these anomalous losses may diminish to acceptable levels for reactor plasmas. Numerical studies of the fluctuation levels of nonlinearly evolving RFP plasma have shown that the saturated fluctuation amplitudes decrease with increasing S. In one study [7] the scaling law SB °= S *0'22 was obtained for Lundquist numbers in the range 3-103<5< 105. Here SB is the r.m.s. value of the variation of the total volume averaged magnetic field in time, including all poloidal and toroidal modes. Another study [6] obtained a fluctuation scaling of S '°18 in the range 2.5-103 < S < 4-104. These studies were carried out for zero pressure plasmas without thermal transport. In contrast, we find that these results do not persist when finite pressure effects are included; instead the high magnetic fluctuation levels remain. In order to address the fusion potential of the RFP, we have striven to attain upper limits of confinement states; no impurity radiation is assumed, and we use a stabilizing conducting wall and classical values for transport coefficients. These simulations were carried out for plasma currents in the range 18-252 kA and densities in the range (0.5-7.0)-10" m"3. This resulted in on-axis temperatures in the range 26-105 eV and on-axis Lundquist numbers in the range 2-104 < S(0) < 7-105. Numerical studies of transport scaling have not been previously performed. This may be because of the apparent difficulties arising from the number of independent parameters appearing in the problem. In contrast, in this study we have found that no more than two dimensionless parameters may be needed to determine optimised confinement. 2. MODEL The resistive MHD equations, when normalized to the Alfven time and pinch radius, are ^ = 0 (1) VxA = B, ^ = vxB--^VxB (2) dt So i!X _( .v)v + -(VxB)xB--^Vo + -V2v (3) dt = v p 2p p (4) po\ 2 We have defined #> = 4^n0To/B0 and So =TRITAO = The dependent variables have their usual meanings. Classical, radially dependent L5 2 2 5 transport parameters [8] are used; r\- T" (Spitzer resistivity), K± = K10 n f(B 7°' ), 2 5 2 5 and /c/ = K/ZQT ' . Classical ion viscosity is given by v = VQT " ! n, with v0 = 4 2$ O5 8.5-10 /uTo /(aBono lnA). The local Lundquist number and poloidal beta are given by S = SQBT In and (3g = fio<p>/B0 (1), respectively. Normalized pressure p = nT obtains 2 through using p0 = 2n0T0 = B Our studies have shown that confinement properties are largely independent of the exact choice of viscosity. Thus, for reasons of numerical stability, we choose local viscosity to be the largest of classical viscosity or a value sufficient to produce an effective Reynolds' number (usually =30) at the scale length of a grid that is small enough to avoid transition to a turbulent state. In all cases, we have chosen the plasma current to line density ratio to be IIN= 2.8-10"14 A-m. This is close to the experimental lower limit, similar to the Greenwald limit for tokamaks. It may also be shown that K,/Q cannot be chosen smaller than the value given below without further decreasing UN. Thus we may use the classical value for KUQ to obtain the best reasonable confinement, i.e. small values of parallel heat conduction. The five parameters (30, So, KJ_Q, K//0, and v0 are not independent, since K10 - 05 05 10.0/Jo/i , and K//o = 32.5voS0/i" (expressed in SI units with fi being the ion to proton mass ratio). Thus only the two parameters /3Q and So are required for computing optimal confinement within the resistive MHD model, and we have performed a series of runs using the DEBSP code [1,6] with varying /Jo and SQ. We have used the aspect ratio Rla = 1.25, a perfectly conducting wall close to the plasma, 0 = B^a)/<BZ> =1.8 and p(a)/p(0) = 0.1. The mass density is maintained constant throughout, rather than being evolved using Eq.(l). The grid resolution is 300 radial mesh points, 42 axial and 6 poloidal modes. All calculations are run until a quasi-steady state with a self-sustaining dynamo is reached, and time-averaged values of poloidal beta and energy confinement time are determined. 3. POLOIDAL BETA AND ON-AXIS TEMPERATURE Values of poloidal beta for various values of /Jo and So are displayed in Fig.l. These data are obtained by averaging over a sawtooth period, which usually lasts a few percent of the resistive time. Using linear regression analysis, the data points in Fig.l can be well represented as a power law (indicated as solid curves in Fig. 1) 0.407 ±0.011 -0.165±0.007 = 3.65 A'0 (5) 0.30 0.25 - 0.00 2 105 4 105 6 105 8 105 1 106 o Fig.l. Numerical results for poloidal beta (/3g) as function of the basic parameters ft and SQ. Solid lines represent power law fit, given by Eq. (5), to data points. In a similar manner, the on-axis temperature T(0) can also be found as a function of (3Q and So- The result is = 0.221 a efffJ0 (6) The numerical parameters fa and SQ can now be related to physical quantities.