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
/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)/
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. Using Eq. (6) and the definitions given following Eqs (1-4), along with the relation for the total current / = 2na50<5z>G, we can eliminate /30 and iS0 from Eq. (5) in favour of 7(0) and /. We find a 1 <£ ^004. ,-0.02 7—0.04TY/-V\1.09 7—101 (H\ pe = l JO a jn £eff 1 (0) i . (/)
It is not surprising that the form of Eq. (7) closely resembles the definition of poloidal
beta for constant UN; ie., /3e<*
Eq. J), which applies to a time-averaged steady state, suggests that Pe is a
ust function of the initial conditions as represented by fa and So. However, fi0 and So are J functions of the normalization parameters (a, 7Q, no, and Bo) and in steady state /?e must not retain a memory of the initial state (it will depend only on the geometry and the boundary conditions, for example Rla and ©). Thus, for an achieved steady state, we are free to replace 70 with T(0). This is merely a change of normalization. The definitions of
Po and So then allow us to use Eq. (6) to relate the on-axis temperature 7(0) (in eV) to the total current /. The result is
7(0) = 0.071a-°V "Z^2/056. (8)
Substituting Eq.(8) into Eq.(7) yields
ft. — 8 fi ~-0.20.,0.10 70.20 r-0.40 ,r\\ Eqs. (8) and (9) are scaling laws for steady state on-axis temperature T(0) and poloidal
14 beta /3e at a constant value of UN (= 2.8-10' A-m). For comparison, the experimental
5 93 0 46 A1 scalings T «= lln™ (or <* f for constant UN) and /3e~ / -°- w - oc / -° were recently found on the Extrap T2 RFP [9]. Earlier confinement experiments on the smaller Extrap
067 016 051 Tl RFP gave the dependence fie « r n oc Z' [10]. The unfavourable /3e scaling with / is due to ohmic heating becoming less effective as compared to parallel heat conduction. The rise in temperature does not compensate for the rise of poloidal flux as the current is increased, thus f3g decreases. The strong heat transport in the radial direction is caused by parallel heat transport along the stochastic field lines. These in turn are caused by overlapping magnetic islands due to the magnetic field fluctuations of the dynamo.
4. ENERGY CONFINEMENT TIME
We now consider the scaling of rE, the energy confinement time. Following [11] we define
2 2 (with Vp = 27I Ra , the plasma volume)
(10) and using the definitions of poloidal beta and parallel Spitzer resistivity, there results
(11)
This is the classical expression for the confinement time (in s). The factor fG stems from the screw-shaped current path in the RFP, and can be approximated by fG « 2.2(5+6© )/(10+9 ) [12]. Eq.(l 1) is derived in the classical transport limit and is clearly an overestimate for the turbulent RFP plasma. It serves, however, as a useful benchmark for comparison with code results. A simple form results after normalizing to the resistive J.5 time TR = Hoa It], with r\ = 5.4-10" ZeJjLnAJT" (Spitzer resistivity):
(12)
In Fig.2 energy confinement times, obtained from the numerical simulations, are compared to the limiting values of Eq. (12). Although the corresponding ratio is at least as large as 0.36, confinement becomes considerably deteriorated for large / because of the degradation of classical confinement limit with decreasing fi0.
JO tu
0.3
Fig. 2. Energy confinement time ratio x^jx£ against fie for different /3Q.
Optimized tokamak plasma confinement also obeys a relation like (12). In this casejfc should be replaced with a neoclassical correction ~ (l-(a/R)05)'2. Since the tokamak /3e is typically at least 10 times larger that of an RFP, the tokamak confinement limit is correspondingly higher, given the same a and T. Using regression analysis, the fitted energy confinement time can be expressed in experimental variables, assuming lnA = 15;
/ . (13)
After inserting T(0) from Eq. (8), there results
j/ (14)
Eqs. (8), (9) and (14) are our principal results. In an extended paper, where we will also include the dependence on UN and 0, specific comparisons with individual experiments will be made. Here we briefly remark on the implications of these results for the fusion potential of the conventional RFP. Using parameter values relevant for a compact RFP reactor (a = 0.6 m, / = 18 MA), Eq. (14) predicts TE ~ 0.004 s. This value for the confinement time is well below the design value of 0.2 s assumed in the TITAN reactor study [13] for the same parameters. Further, Eq.(8) implies an on-axis temperature of only about 1.1 keV as compared to the TITAN design value of 10 keV, contradicting the assumption of ohmic heating to ignition. However, the TITAN study used Q = 1.5 for TITAN, as compared to our value of 1.8. This may yield somewhat better confinement due to a less stochastic plasma core. Care must also be taken when extrapolating to TITAN Lundquist numbers of 10 , by far exceeding the range employed in our study.
For comparison with analytically and experimentally obtained scaling laws, we use Eq. (7) to eliminate 7(0) from Eq. (13). We find (with IIN= 2.8-10"14 A-m)
2A T _1C 1A-7 _1.9 ..0.05 7-0.90 o 1-1.3 /,n T£-O-10 an Leff pe 1 . (15)
10 Connor-Taylor scaling [14], which is based on eory of resistive g-mode fluctuations,
L5 L5 2 predicts the dependence rE «= I (I/N) a . A fit to an international RFP database [15],
L5 L5 2 20 1 using the form TE = c[I (I/N) a f, in units of s, MA, 10 m" and m, results in the parameter values c ~ 6-10"3 and/? = 0.87. Adjusting the temperature dependence to this form, temporarily assuming constant /?& we obtain, from (12), very similar results: c =
15 005 090 2.8£/V /i Ze/ and/? = 0.87. Since fie certainly is not constant, the forms for tE, suggested in [15], are of questionable utility (see also [16]).
The product mE is more informative regarding reactor potential than xE itself. We obtain the following scaling (in s-m"3);
nxE — 1.7• 10 \i a Zeff fig' I' . (16)
05 I7 5 Connor-Taylor scaling results in nxE °= //TV . The fit xE = yt-10 [/W ]^ with the parameters x= 0.91 and k = 2.4 is however more true to experimental data [15]. Using
3 2A 0A0 005 90 (13), our simulations give x = 0.92 and k= \.2-l0 pe a' fi Ze/- .
5. FLUCTUATIONS We now turn briefly to the scaling of magnetic fluctuations in the RFP. Until now, numerical scaling studies have only been performed for pressureless plasmas, displaying a weak decrease of fluctuations with Lundquist number. It is clear from Fig. 3 that finite pressure simulations result in a complex dependence of the magnetic fluctuation level on fio and So, not amenable for expression in simple power law form. It is also seen that the fluctuation levels do not appear to drop below 1 %. This level of fluctuations unfailingly leads to overlapping of magnetic islands and stochastic field lines. The reason for the insensitivity of this result to plasma current and temperature appears to stem from the
11 nature of the basic m = 1, n ~ 2BJa (dynamo) tearing modes of the conventional RFP,
resonant near the reversal region, being robust and essentially unimpaired by finite pressure effects [17].
0.025
0.020 - 0.050 0-151 0.101 ^ 0.015 ~z
E : ^ 0.010 -
0.005 -
0.000 I I 0 2 105 4 105 6 105 8 105 1 106 o
Fig. 3. Globally averaged radial magnetic field component br = -^< br > vs
for different /%.
Our results for magnetic field fluctuations deviate from earlier numerical results [6,7], probably because of the physics introduced by finite beta and energy transport. For example, field reversal is deeper and more consistent with experimental results in the finite pressure simulations.
6. CLOSING STATEMENTS In conclusion, we have determined for the first time theoretical limits for RFP confinement behaviour. The high Lundquist numbers, comparable with those experimentally obtained, are handled numerically by using a fine radial grid and by careful
12 choice of viscosity. In an extended paper we will show that the exact choice of viscosity has little impact on the physical results such as magnetic field fluctuation levels or confinement. The scalings are accomplished by recognizing that there are no more than two governing parameters in the limit of fixed, low UN. In our study an optimum conducting wall case was assumed; further 0 = 1.8, Rla = 1.25, and density is constant. We believe that neither of these restrictions have any strong influence on our conclusions; in particular we have found similar results in test runs at a more realistic aspect ratio of 4.
We remark again that the Lundquist number in an RFP reactor plasma will be of order 109, thus much larger than the Lundquist numbers used in the present simulations. Scaling law extrapolation to the reactor regime is uncertain, but may be supported by our result that magnetic fluctuations remain at high levels in conventional RFP's, maintaining a core plasma region with stochastic field lines and associated transport.
Several effects were neglected in these simulations. A resistive wall and field errors would introduce new unstable modes, impurities would cause line radiation, fluctuations in the presence of a density gradient would cause particle transport [18,19], and hot electrons may enhance core energy losses. These would all lead to a more pessimistic picture than the one presented here.
7. ACKNOWLEDGEMENTS We wish to thank Dr. P. Brunsell for informative discussions on RFP experiments. Computer resources from the National Swedish Supercomputer Centre at Linkoping University are gratefully acknowledged. This work has been supported by the European Community under an association contract between EURATOM and the Swedish Natural Science Research Council. One of us (DS) was supported by the U. S. Department of Energy under Grant DE-FG03-91ER54124.
13 8. REFERENCES [I] S. Ortolani and D. D. Schnack, 1993, Magnetohydrodynamics of Plasma Relaxation (Singapore: World Scientific) [2] R. J. Lahaye et. ah, Phys. Fluids 27, 2576 (1984).
[3] G. Fiksel et. al., IAEA-F1-CN-69/EX4/5. [4] J. S. Sarff et.al, Phys. Rev. Lett. 72, 3670 (1994). [5] M. R. Stoneking, Phys. Plasmas 4, 1632 (1997). [6] C. R. Sovinec, Ph. D. Thesis, Univ. Wisconsin, Madison, USA (1995). [7] S. Cappello and D. Biskamp, Nuc. Fus. 36, 571 (1996). [8] S. I. Braginskii, Reviews of Plasma Physics, Vol. 1, Consultants Bureau,
New York, 1965, p. 205. [9] A. Welander, Plasma Phys. Contr. Fusion 40, 1597 (1998). [10] S. Mazur et. al., Nuc. Fus. 34, 427 (1994). II1] S. Hokin et. al., Phys. Fluids B3, 2241 (1991). [12] J. C. Sprott, Phys. Fluids 31, 2266 (1988). [13] F. Najmabadi et. al., "The TITAN Reversed-Field Pinch Reactor Study", Final Report, UCLA-PPG-1200, University of California, 1990, a synopsis of which is in Fusion Eng. Des. 23, 69 (1993). [14] J. W. Connor and J. B. Taylor, Phys. Fluids 27, 2676 (1984). [15] K. A. Werley, J. N. Di Marco, R. A. Krakowski and C. G. Bathke, Nuc. Fus. 36, 629 (1996). [16] Hattori et. al., Fusion Technol. 28, 1619 (1995). [17] R. A. Nebel, E. J. Caramana and D. D. Schnack, Phys. Fluids B 1, 1671, 1989. [18] M. R. Stoneking et. al., Phys. Rev. Lett. 73, 549 (1994). [19] J. Scheffel and D. Liu, Phys. Plasmas 4, 3620 (1997).
14 TRITA-ALF-1999-06
ISRN KTH/ALF/R-99/6--SE
Numerical Simulation of Reversed-field Pinch Confinement
J. Scheffel and D. D. Schnack
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
0 40 radiation losses and resistive wall effects. We find that poloidal beta scales as pg °= /" and that energy confinement time scales as TE «= / ' for fixed UN. The scaling of magnetic fluctuations is shown to critically depend on two parameters, rather than on Lundquist number alone.
14 pages 3 figures
Keywords: Confinement, RFP, scaling law, beta, energy confinement.
15