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REVIEW OF RESEARCH: IN SEARCH OF THE "MAGIC MAGNETIC BOTTLE"

NICOLAS DOMINGUEZ CCNF-920728-2 Fusion Division, Oak Ridge National Laboratory, Oak Ridge, 77V 37831-8071, USA DE92 019045

ABSTRACT We summarize the current on being carried out out in fusion research laboratories around the world. The theoretical aspects of the stellarator research are emphasized. *'~

1. Introduction

The steadily increasing need for energy makes it imperative to look for new sources of energy for the future. Fusion energy is one of the most promising posibilities. One of the main approaches to harnessing fusion energy is magnetic confinement. In this approach, the thermonuclear containing the to be fused is confined in a magnetic trap, where it can be heated to the high temperatures necessary for nuclear processes to occur1. Confining a plasma involves the creation of gradients of density, temperature and pressure. The presence of those gradients implies the creation of free . These free energies have the potential to destroy the magnetic confinement through magnetohydrodynamic (MHD) instabilities and microinstabilities. The central issue for magnetic confinement is to create a magnetic bottle that confines the plasma in such a way that the free energies are not too dangerous for confinement. It is fair to say that the history of fusion research for more than 30 years has been the search for a "magic magnetic bottle" -one with excellent confinement properties at low cost. A number of concepts have been developed as part of this search. Among them we can list the , the mirror machines, the pinches, the tandem mirrors, the , the reversed-fisld pinches (RFPs), the , the spherical tokamaks, and the advanced stellarators. Experimental devices based on these concepts have been constructed serving as valuable plasma containers in which different plasma physics theories have been tested. Most of them are no longer in operation, not because the physics of the plasma they contained was fully understood, but mainly because of budget constraints. Different countries have played important roles in the search for fusion devices. In the United States, Oak Ridge National Laboratory (ORNL) has been one of the most innovative leaders, with research on mirror machines, the bumpy torus, tokamaks, spherical tokamaks and stellarators. At present, the is by far the reigning concept, but a number of new stellarators are in operation, under construction or being designed.

OF THfS QOCUMENT IS UNLIMITED Several of the new stellarator concepts look quite complicated but very attractive. In the world of stellarator devices, one notable characteristic is the propensity for designing machines with different number of field periods: 3 (Australia), 4 (Kharkov. Ukraine; Spain), 5 (Germany; Auburn, USA), 6 (ORNL; Spain), 7 (Wisconsin, USA), 8 (;, 9 (Kharkov, Ukraine), 10 (Japan), 12 (ORNL), 14 (Moscow, Russia) and 19 (Japan). The stellarator, a concept first proposed some time ago,2 can be seen as an alternative to the tokamak. The principal characteristics that differentiate stellarators from tokamaks are that stellarators . are three-dimensional (3-D) geometries . are free of disruptions, and . have the capability for steady-state operation. The stellarator should be able to operate, in a steady state and the disruptions that tokamaks can suffer are absent in stellarators because the toroidal net current is zero. The experimental results from tokamaks look a lot better than those from the stellarators because the tokamaks are a lot bigger. The tokamaks have enjoyed a lot of popularity and economic support for the last several decades. However, the experimental results show that the plasma performance in stellarator plasmas is comparable to that in tokamak plasmas of similar size. Because stellarators are inherently 3-D configurations an obvious question arises From the infinite number of possibilities, which 3-D geometries are the best? The question cannot be answered by intuition alone. The first step in trying to answer the question is to carry out optimization studies to determine the best configurations in a space of parameters. The step is followed by realization of a plasma device and finally by carrying out the experiment. The information from the experiments can be used to propose promising new configurations, and so on. The optimization studies are constrained by two factors: . The engineering problems, such as the capability to build the complicated coils and the cost. It has been proven at ORNL, at the Max Planck Institute for Plasma Physics in Garching, Germany, and at other research institutes in many countries that the engineering difficulties can be overcome. Despite the latest engineering feats we have the cost of the devices. The cost of the devices constrains the size of the machines. It is well known nowadays mat the bigger the machines, the better the results, but big machines are very expensive. . Plasma physics. Stellarators must be designed in such a way to obtain the highest possible stable plasma betas and the minimum transport of energy and particles. A great deal of work has been devoted to addressing the second constraint, not only because it is important and interesting to understand the plasma behaviour in magnetic traps but also because reducing the constraints imposed by the plasma physics can help to reduce the engineering constraints, the cost, and the number of possibilities examined in searching for new confinement devices. In addition, an understanding of stellarator plasmas can help to improve the understanding of the plasma behaviour of the most popular toroidal confinement configuration, the tokamak: so the exploration of plasma physics in stellarators c^n also be seen as complementary to tokamak research. Thus, the study of stellarator plasmas is a promising tool for indicating directions towards innovations for tokamak devices. The construction of the Advanced Toroidal Facility (ATF) and the Wendelstein VII-AS (W VII-AS) is proof that the engineering required to build stellarators exists, and there has been some economic support. Construction of TJ-II and the (LHD) is currently under way, and a proposal for the construction of W VII-X is being considered. In this paper we address studies of stellarator plasmas in progress all around the world with the aim of obtaining the best plasmas in 3-D geometry. Stellarator researchers have been very imaginative, exploring ways to attain flexibility in the 3-D configurations and taking different approaches to increasing the limits and to reducing the plasma transport. The physics point of view regarding stability has been to understand the importance of particular stabilizing mechanisms, and to study the mechanisms which can reduce the destabilizing contributions. In the optimization studies there is always a tension between the configuration with the best stability properties and the configuration with the minimum plasma transport; thus, there have been trade-offs regarding some of the equilibrium quantities. Theoretical optimization studies have been done analytically and numerically using reduced two-dimensional (2-D) geometries and lately numerically considering full 3-D geometries. In the last five years there has been a lot of progress in developing tools to consider the 3-D stellarator geometries. This has been possible with the advent of powerful codes and fast computers. New codes to study equilibrium and stability of these 3-D configurations are being used in different parts of the world; much of this effort is concentrated at the Centre de Recherche en Physique des Plasmas, Ecole Polytechnique Federale de Lausanne, Switzerland3*4. Most of the codes are based on the VMEC code.5"7 The main conclusion regarding the MUD stability studies is that stellarator configurations must be studied without several approximations used in the past.8"10 The Helias concept can be realized in W VII-X, which is based on the quasi-helically symmetric configurations proposed by Nuehrenberg.21'12 The bootstrap current has been measured in ATF,13 and its effects for the devices of the next generation have been calculated. An empirical scaling law for energy confinement time has recently been proposed; it is mainly based on Heliotron-E experimental data.14 This LHD scaling law agrees very well with the gyro-Bohm scaling.15-16 The agreement between the ATF experimental data and the LHD scaling gives confidence in the empirical scaling and makes it possible to predict confinement times for devices under design. Density and temperature profiles are also being calculated in computer simulations, and the results are consistent with the LHD scaling.15 In Section 2 we give definitions and rudiments of MHD equilibrium and stability. In Section 3 the latest studies carried out in different parts of the world are summarized. Section 4 presents our final remarks.

2. Basics of Equilibrium and Stability

2.1. Definitions and Equilibrium Relations M, number of toroidal field periods A multipolarity R, major radius of the configuration a, average minor radius v, collision frequency A = R/a, aspect ratio of the configuration In this section we assume that nested flux surfaces exist. 0, magnetic toroidal flux divided by 2 K X, magnetic poloidal flux divided by 2n s, flux label (normalized toroidal flux in this work)

0b, value of 0 at the boundary p ~ I—, average radius of a flux surface

.., tdl . „ V = j—, magnetic well *, rotational transform c di di . , 5 oc oc — t magnetic shear P, plasmd0 a pressurds e /, J plasma poloidal and toroidal net currents, respectively B, magnetic field [in Tesla (T)]

J, plasma current

Equilibrium which has to be satisfied in each point of the flux surface a = —y» parallel current 2u P , peak beta, where B is the magnitude of the magnetic field at the magnetic = ° ° o axis for the particular value of beta considered and Po is the plasma pressure at the magnetic axis. B1 (/?}, average beta, the ratio of the -averaged pressure over ——. 2^ The definition used for (fi) can vary from place to place.

^»(O)> toroidal ^^ shift a We define the equilibrium beta limit as the one satisfying AT(0) = 50%.

2.2. VMEC code One of the main contributions to theoretical research in the last years has been the implementation of the VMEC code5"7 developed by S. P. Hirshman. This code has been used all around the world to carry out studies in equilibrium, stability and transport in stellarator geometries. The code calculates the full 3-D equilibria of 3-D configurations. There are no approximations in the calculations of equilibria regarding . number of field periods, . aspect ratio, . plasma beta values. Thus the VMEC code finds the plasma equilibrium without any of the approximations used in the stellaratcr expansion approach.17 The fine structure of the equilibrium quantities is known. The code finds fixed and free boundary equilibria and has been used very heavily since 1986 in the design and/or plasma physics studies of several devices. VMEC has been mainly used for ATF,8'10-18"21 Low-aspect ratio torsatrons,22 URAGAN,23"24 LHD,25"27 TJ-II,28 H-l,29 Helias.30-31 In the fixed boundary version of the code, the plasma boundary is prescribed in the form J()ng), (2)

Z = f,Zmii(s)sin(m6-ng). (3) The number of harmonics in these sums depends on the number of modes necessary to describe the boundary of the configuration under study, which is for example, 6 for a tokamak; 10 for a ripple tokamak [such as the Texas Experimental Tokamak (TEXT)]; 40 for ATF, low-aspect-ratio torsatrons, and LHD; 70 for Uragan-2M, 110 for TJ-II and more than 110 for the Australian H-l Heliac configurations. VMEC is a very fast code in which the coordinate system is such that the number of harmonics to describe any equilibrium is a minimum. Because of this peculiarity, the VMEC equilibrium must be transformed to another coordinate system with straight magnetic field lines, for example the Boozer coordinate system,32 to carry out stability and transport studies. The only constraint in the calculation of the equilibrium is the assumption of nested closed flux surfaces, although this constraint is being removed. 2.3. Transformation from VMEC to Boozer Coordinates The transformation from VMEC coordinates (&,Q) to the Boozer coordinate system (9. c)32 is carried out flux surface by flux surface. We choose the flux coordinate, s, to be the same in ooui coordinate systems. The transformation of the angles is given by31 where - - I + X'J' -0'1 + X'J' The flux surfaces are expressed in terms of the Boozer coordinates as

Likewise, the geometric toroidal angle is expressed as

Here the array of mode numbers m and n in Boozer coordinates is obviously different from that used in the VMEC coordinate system.

2.4. Ballooning Mode Equation The ballooning mode equation can be written as33'35 A2, (4) dq dq where a gives the stabilizing contribution from the bending of the field line and D is the driving term of the instability. The right hand side gives the inertial contribution, and Tis the growth rate. The solutions must vanish at q —> ±°°. For more details see, e. g., Refs. 8, 10 and 31.

25. Mercier Criterion The Mercier criterion36 can be obtained very easily from Eq. (4). This is done by proposing the solution v v 1 v V 3 § = Z +fa)Z - + f2(q)Z -> + US)Z - + -. in the asymptotic limit (q —»«>). The criterion for stability can be written for closed flux sufaces as,37

DM = Ds + Dw + Dj+Da>0. (5) Bauer, Betancourt and Garabedian were the first researchers to study the equilibrium, stability and transport numerically for 3-D geometries.37-38 In Eq. (5), ..U" "*"al liab»"y or re pons - for the accuracy, completeness, or usefuiness of any information, apparatus, producer process disclosed, or represents that its use wou)d not infringe privately owne «««« or reHect those of'he United States Government or any agency thereof. r / The brackets mean integration over angular quantities, explicitly, {(.)) = jjdOdfr). The Mercier criterion for large-aspect-ratio and low-beta tokamaks reduces to39 q > 1, for stability. The Mercier criterion is only a necessary condition for MHD stability. Early studies showing that ballooning modes would be absent in stellarators,40'41 were based on analytical work for the MHD stability of tokamaks.42 However, ideal ballooning modes have been found in stellarators8' 10.31,35,43 by analyzing the full 3-D equilibria.

3. Stellarator Research

A number of stellarator devices are in operation, under construction, or being designed throughout the world. Among these are attractive devices with 3, 4, 5, 6, 7, 8, 9, 10, 12, 14 and 19 field periods. The different devices make use of the stabilizing mechanisms, the magnetic shear and the magnetic well, in different ways. Some of them have very large shear and no magnetic well at vacuum; others have very small shear but a very broad vacuum magnetic well. Because of the diversity of the stellarator devices, most of the research in the different programs is aimed at understanding the peculiarities of specific machines. The compilation of the theoretical and experimental results will give a very good data base for designing the next and more advanced stellarators of the future and for advancing towards a steady-state fusion reactor device. In summary, there are four kinds of 3-D plasma devices in the world . torsatrons and heliotrons, . modular stellarators, and . heliacs. We use the name "stellarator" to refer to all the different kinds. The torsatrons, heliotrons and heliacs have continuous helical coils. From the physics point of view, they are very different in their selection of . magnetic shear, . magnetic wells, and . resonance flux surfaces inside the configuration. We briefly review the major activities of the different fusion research institutions.

3.1. Princeton Plasma Physics Laboratory, New Jersey, USA Princeton is the birthplace of the stellarator, the Model C stellarator was a concept developed in the 1950s by Spitzer.2 In the early 1960s, Greene and Johnson17 developed the first tools for systematically studying the equilibrium and stability of stellarators. They introduced the techniques of the stellarator expansion, which became very popular in the world during the 1980s because of their usefulness. These techniques have been very widely used to design stellarators and understand MHD processes in their plasmas. Johnson has been a very strong leader in the MHD arena of stellarator physics for more than 30 years. His latest work includes the benchmarking of results from different codes in use throughout the world, with the aim of understanding the diffences in results obtained usinj, 2-D and 3-D equilibria44 Other theoretical work at Princeton has been directed toward implementation of a code to obtain equilibrium with islands.

3.2. NYU, New York, USA Bauer, Betancourt and Garabedian37-38 of New York University (NYU) were the first researchers to implement a code to study 3-D MHD plasmas: the BETA code. In their pioneering work, they analyzed the stability and transport of several stellarators that have since been built.

3.3. Madison, Wisconsin, USA Although stellarator research was reduced in the United States following the first experimental results from tokamaks, Shohet and co-workers at the University of Wisconsin continued the stellarator research in small devices. They implemented one of the first codes to study equilibrium for the 3-D configurations. They also studied, in collaboration with the Institute for Fusion Studies in Austin, Texas, Mercier modes and ballooning instabilities in 3-D geometries.35 Their work on this topic35 was the first to show the existence of ideal ballooning modes in stellarators. Lately,45 the group has been very active in proposing the contraction of a quasi-helically symmetric device based on the ideas of Niirenberg. The Wisconsin proposal is under study by the U. S. Department of Energy. The main idea is to find out if the transport properties of the quasi-helical plasma are as predicted by the theory. The experiment will also provide very important information about anomalous transport in quasi-helically symmetric devices. There will be flexibility to study effects of toroidicity in the transport properties by controlling the breaking of the symmetry.

3.4. Institute of General Physics and Kurchatov Institute, Moscow, Russia After the work of Greene and Johnson,17 a great deal of research on stellarators was carried out at the Kurchatov Institute and the Institute of General Physics in Moscow.47 Pustovitov*0 and Sbafranov,41 as well as Kovrizhnykh and Shchepetov,50 were mvoiven in the calculation of equilibrium and stability properties of stellarators using the stellarator expansion. Much of their work led to guidelines for the design of stellarators that have since been built in different countries. These works are characterized by insight into the physics of stellarator plasmas. Results for equilibrium quantities and stability criteria were obtained. The main results of the analytical calculations are documented in Ref. 46. Among the results obtained analytically is the derivation of the equilibrium equation,48

Expressions found analytically are very helpful in obtaining estimates for equilibrium quantities, such as the expressions for the rotational transform considering the effect of quadrupolar fields,48 and the critical elongation49 for the flux surfaces,

Among the useful results of this work are the Shafranov shift in terms of the rotational transform,50

a 2*2 • and the estimate of the equilibrium beta limit,

35. National Institute for Fusion , Nagoya, Japan The Compact Helical System (CHS) device, now in operation, uses a combination of magnetic well and magnetic shear. The CHS experience has been useful to the rest of the stellarator community, in pan because the very small aspect ratio of the device makes it useful for studying the breaking of the flux surfaces and field errors.51 Because of the field errors, an island exists in the ; however, the width of the island can be calculated, and it scales as Aul

MHD stability. Changing the external field ratio BT01BH0 from -0.1 to +0.15 varies *(0) from 0.64 to 0.39, i(a) from 2.6 to 3.1, and a from 0.182 m to 0.252 m. Changing

the vertical field (VF) ratio By0 IBH0 from -0.171 to -0.199 changes the magnetic axis

shift Av from +0.04 m to -0.04 m. Here BTQ, BH0, and BVQ are the central magnetic fields produced by the auxiliary toroidal field coils, the helical coil, and the auxiliary VF coils,

respectively. The magnetic configuration with BT0 / BH0 =0.05 — 0.1 and

BVQIBHQ = -0.192 is more favorable than the standard configuration in terms of both transport and MHD stability.

3.7. ORNL, Oak Ridge, Tennessee, USA The ATF device at ORNL is the largest stellarator in existence. It is characterized by great flexibility and stabilized by a combination of magnetic well and shear. The design of ATF54 was based on studies using the stellarator expansion. The scientific research done in Oak Ridge using the stellarator expansion numerically, combined with the implementation of codes such as VMEC and DKES55 to study equilibrium, stability and transport, and the close coordination with the experimental work have served as the model for research in other stellarators. The remarkable feature of ATF is the flexibility in its equilibrium quantities, which can be varied simply by changing the currents in the external coils. The control of the rotational transform, magnetic well, magnetic shear and fraction of trapped particles has been demonstrated experimentally. A large number of ATF configurations can be obtained by changing the currents in the VF coils. For example, the magnetic axis of the configuration can be displaced horizontally with respect to the geometric center of the coils; as a consequence, the magnetic field line curvature is changed. This is done by changing the magnetic dipole moment of the configuration. The other available control parameter is the quadrupole magnetic moment, which is changed by changing the currents in the mid-VF coils. In this way the ellipticity of the magnetic flux surfaces and, consequently, the rotational transform and the shear can be varied. All of these changes are performed with the constraint of zero current as described by Carreras et al. in Ref. 56. Studies regarding neoclasical transport properties are reported in Ref. 20, and analyses of the MHD stability of these configurations (using the stellarator expansion) are reported in Refs. 54, 57. For the standard configuration the position of the magnetic axis is at 2.1 m and the average minor radius is 27 cm. The standard configuration has a rotational transform, *, of 0.334 at the axis and 0.98 at the edge. The standard configuration was chosen in such a way that *= 1/2 was located inside the well, making the resonance innocuous to low-n modes studied using 2-D equilibria.19 -54«57 The VMEC code has been used to analyze 3-D equilibria. For the studies on equilibrium and MHD stability, the fixed boundary version of VMEC was used; the boundary used as an input for VMEC corresponds to the last closed magnetic flux surface obtained by using a field line following code.58 The boundary is given in terms of the components of the representation given earlier in Eqs. (2) and (3), after being fitted from the output of the field line following code. The studies for the design of ATF were carried out using the RSTEQ code.55 The first step in that procedure was to obtain a 2-D vacuum by first averaging the 3-D vacuum fields and then using the 2-D average vacuum to obtain the finite-beta 2-D equilibria. With the implementation of the VMEC code, the MHD stability of ATF is being revisited. Some interesting results are emerging as a result of analyzing 3-D equilibria. It appears that the stellarator expansion, while giving good agreement for equilibria, is inadequate for ballooning modes. A current issue is how sensitive ideal MHD stability is to quadrupole fields and pressure profile details. To build a stellarator, the coils must be constructed with high precision, about one millimitre in some cases. If that precision is not achieved, field errors can destroy the flux surfaces and consequently the equilibrium and stability of the plasmas. In the best case they will reduce the effective volume. Error fields can be produced from different sources; for example, from ferromagnetic materials in the diagnostic devices or in the structure where the device is located. Sometimes the source of large field errors is unpredictable, as in the case of field errors found in ATF. These field errors were due to dipolar magnetic moments caused by currents that fed the helical coils, and once identified, they were corrected very succesfully.60 Other studies carried out in Oak Ridge have explored dissipative trapped- modes (DTEM). The principal aim has been understanding transport processes, in parallel to the studies being carried out for tokamaks. Theoretical studies of trapped-electron modes in stellarators are reported in Ref. 21. These modes could play an important role in enhancing losses in a toroidal confinement device. However, no direct evidence of these instabilities has been found in such devices. The DTEM in / = 2 torsatrons were studied using the high-n ballooning formalism and considering the full 3-D geometry. From a perturbative analysis, it was found that both toroidally localized and helically localized eigenfunctions exist in the ballooning space for these nonaxisymmetric devices. The helically localized eigenfunctions give the largest growth rates. The helical symmetric limit gives a vciy good description of the helically induced modes for stellaretors such as ATF. The radial width of the unstable modes can be very broad, and therefore it may be possible to detect in future experimental studies with density control. Helically trapped particles interacting with helicity-induced modes dominate the trapped electron instability in the stellarator. These instabilities have relatively broad radial width, because their extent along the field lines is Ar\ = lit IM. They should be easier to detect than the corresponding tokamak instabilities. Toroidally trapped particles interact with helically extended modes in a manner that closely resembles the situation in tokamaks. The density gradient must exceed a critical value for helically localized eigenfunctions. In the helical limit, a condition for nonexistence of helically localized eigenfunctions is \dn\ 3 *M(l 3«) When this condition is not satisfied, the electron temperature gradient should be such that l^i^<0, (7) T, dp n dp which will have the effect of suppressing the helically induced DTEM.

With present devices, the v.t < 1 regime is accesible at relatively low electron temperatures. The broad radial width associated with the helically induced modes makes them detectable with the present core fluctuation diagnostics. Present stellarators offer a broad range of magnetic parameters; for example, Wendelstein VII-AS has practically zero shear, and shear values in ATF are below the tokamak range. The effect of these modes can also be changed by varying the dipole and quadrupole fields in ATF; the change in the magnetic field changes the fraction of confined trapped particles. Present experimental measurements of density aiid temperature profiles in stellarators indicate that these devices are stable or close to marginal stability for helically induced modes. At the plasma core, the electron density profile is flat or hollow, hence fulfilling the necessary condition given by Eq. (6). At the plasma edge, the density gradient is large, and the electron temperature and its gradient are low enough to marginally satisfy Eq. (7). Studies of DTEM are being conducted on three devices in parallel: ATF, Wendelstein VII- AS and the TEXT tokamak. In one interesting experiment on ATF, the neoclassical theory on bootstrap current developed by Shaing et al.20>61>62 was verified through direct measurements (in the absence of other current) and parametric scans. Neoclassical theory predicts that, in the l low-collisionality limit, the bootstrap current is given by Jb = -3(/, / fc)GbB~ Vp, where /, and fc are the fractions of trapped and circulating particles and Gb is a magnetic geometry factor that depends on the B spectrum on a flux surface and changes with the quadrupole field or dipole component of the vertical field. The toroidal current observed during electron cyclotron heating (ECH) in ATF was predominantly bootstrap current. Figure 4 of Ref. 13 shows the currents observed as a function of the mid-VF coil current ratio, the parameter that changes the quarirupoiai component of the vertical field. Very good agreement was found between the values measured in the experiment and the values calculated for the bootstrap current using neoclassical ilieory. The bootstrap current decreased as the plasma was vertically elongated. There v/as also very good agreement between theory and experiment as the dipole field was changed, as shown in Fig. 5 of Ref. 13. The data base for energy confinement time in stellarators has been extended by the operation of ATF; which has an energy confinement time tE as high as 20-30 ms. This range is about the same as that for the Impurity Study Experiment (ISX-B) tokamak, which had similar magnetic field intensity and poloidal cross section. Dory et al. have studied 16 stellarator confinement times; one of the main conclusions is that the scaling of xE closely follows the empirical LHD scaling formula, as seen in Fig. 4 of Ref. 16. The effect of fluctuations on transport is also being studied at ORNL. Turbulence models that include the effect of the have been implemented to understand the role of the electric field in transport.

3.8. CIEMAT, Madrid, Spain At the Centro de Investigaciones Energebcas, Medioambientales, y Tecnologicas (CIEMAT), the TJ-II device, a flexible heliac is under construction. It has four field periods and a flat rotational transform (0.22 <*/4 < 0.65), and the magnetic well and magnetic shear can be changed by changing the currents in the external coils. The configuration flexibility provided in the TJ-II heliac by its helical hard core permits significant changes in the equilibrium properties of the plasma. One of the principal theoretical efforts to date is the calculations of Mercier modes carried out by Varias et al.28 These numerical calculations can be used as a guide for equilibrium and stability studies in other heliacs. Some of the results are displayed in Figs. 1-3, 6, 8, 14 and 15 of Ref. 28.

3.9. Australian National University, Canberra, Australia The H-l heliac under construction in Australia has a flat rotational transform profile and a deep magnetic well. An earlier heliac device, Sheila, was built and operated in Australia under the direccion of Hamberger. The H-l heliac device has three field periods, the lowest number of any stellarator in the world. The flattened rotational transform leads to Mercier stability being dominated by the magnetic well and the geodesic curvature contribution. Gardner and Blackwell29 have found in their very systematic study that, because of the destabilizing contributions from the geodesic curvature, the stability beta limit given by interchange modes will be around (fi) ~ 1%. Their study of the effects of Pfirsch-Schliiter currents on equilibrium and Mercier stability is one of the most complete in the literature. 3.10. Kharkov Physical-Technical Institute, Kharkov, Ukraine. The Uragan-2M device, now under construction at Kharkov, uses a combination of magnetic well and magnetic shear. The four device will have the flexibility to change the magnetic equilibrium properties as a result of the consideration of ihe toroidal coils in its design. The latest studies regarding the equilibrium and stability of the Uragan-2M configuration are reported in Refs. 23 and 24.

3.11 Max Planck Institute fur Plasma Physik, Garching, Germany The Helias family of configurations developed in Garching is characterized by flat rotational transform, a magnetic well across the whole cross section, and reduction of Pfirsh-Schliiter currents. W VII-AS is operating, and the quasi-helically symmetric W VII- X is in the approval process. The construction of W VII-AS has proven the feasibility of modular coils for stellarators. This engineering achievement is being incorporated into the design of WVn-X. The German researchers have chosen a very promising concept.11"12 Stability is provided by a shallow magnetic well of 2%, which extends across the whole cross section, I j 2\ and the reduced Pfirsch-Schluter currents (-™-}

B{s, d, 0) = BH (s, 6 - M0) + B, (s, 6,0), where the part of B designated B, introduces 1 / v neoclassical losses. It is very important to notice that the variation of B(s,6,) does not change very much with finite beta, therefore, the neoclassical transport will not change very much with beta. It is convenient to use the magnetic field of a conventional stellarator,

B = B0[l - e, cos 6 + eh cos(£6 - M0)], to illustrate the differences in the diffusion coefficient DVv between W VII-X and conventional stellarators. The diffusion coefficient is equal to:

. £>1/v=0 in a quasi-helically symmetric stellarator,

2 67 . Dv v = const • d]' ^- in W-VII-X,

where 5t =0.01,

2 . DVv= const • el' —- in a classical stellarator,

where ek =0.1. One of the claims of the researchers in Garching is that the anomalous transport in W VII-X will be very small, based on the fact that experimental results obtained in the W VII- A experiment show that the anomalous thermal conduction of is given by x 2n X,M ~ ri~ T~ .Thus , for the parameters of W VII-X they expect that anomalous transport will not be important. The Helias concept is indeed very promising, and the attractiveness of the Helias concept can increase if properties similar to those found for W VII-X can be found for smaller aspect ratio configurations. The actual aspect ratio for W VII-X is A - — = 10. a 3.12. Centre de Recherches en Physique des Plasmas, Ecole Polytechnique Federate de Lausanne, Switzerland The theory group in Lausanne has a long tradition of fine numerical work, which is continuing with the implementation of new and powerful tools to examine the equilibrium and stability of stellarator configurations. The effort has been aimed at studying equilibria with anisotropic pressure and the stability of high-n and low-n ideal MHD perturbations. A number of equilibrium and stability calculations for stellarators were based on approximations using analytical expansions or reduced geometries. These approximations are noi used iii ihe calculations using the VMEC code. Thus, more realistic stability limits for the equilibrium and stability have been found using new codes. One of these new codes, TERPSICHORE,34 is a 3-D linear ideal MHD stability code extended to solve the full MHD equations. It has been used mainly to calculate the growth rates of non-local low- n modes for / = 2 torsatron configurations like ATF and LHD. A correlation between the stability boundaries obtained from the Mercier criterion and the stability boundaries obtained from low-n modes in equilibria using the stellarator expansion was found, and the Mercier stability beta limits were demonstrated to be below the stability beta limits given by low-n modes19. Fu et al.9-68-69 have found, however, that without the approximations of the stellarator expansion, the results are different. They showed that the stability beta limits given by Mercier modes can be lower or higher than those given by low-n modes and that finite growth rates of the low-n modes correspond to finite values of the Mercier parameter. Near the Mercier marginal stability boundary, the low-n modes tend to be weakly unstable with very small growth rates. However, the stability of global-type low-n modes is decorrelated from that of Mercier modes. The low-n modes with global radial structures can be more stable or more unstable than Mercier modes. One of the important results obtained in Lausanne is that the threshold for low-/i modes in ATF can be lower than that obtained by using equilibria obtained with the stellarator expansion9-69. Work by Cooper et al. showed that there are ballooning modes in self-consistent / = 2 stellarator equilibria. The main results regarding this point can be seen in Figs. 8-16 of Ref. 9. The conclusions from the studies using ATF-like equilibria indicate that not only high-n modes but also ideal low-n modes can restrict the stability beta limit in ATF. The stability limits obtained from these theoretical considerations could be tested in the ATF device if it is restarted and equipped with adequate . This would provide an interesting test of MHD theory in 3-D configurations. The calculation of equilibria with anisotropic pressure profiles in 3-D systems4 is pan of the latest numerical work from Lausanne. Equilibria are obtained by using an energy principle considering the functional W = jjj d3x\ + ~fr— • The minimization of W ^2/i0 *-lj using an inverse spectral method reproduces the force balance relations that govern the

MHD equilibrium properties of 3-D plasmas with Py * PL that have nested magnetic flux surfaces. Numerical procedures already developed for the scalar pressure model can be easily extended to the anisotropic pressure model. Specifically, a steepest descent procedure coupled with the application of a preconditioning algorithm to improve the convergence behavior was employed to minimize the energy with highly localized anisotropic pressure distributions to attest to the robustness of the method of solution considered. The new code can be used to obtain more realistic equilibria that can exist easily in experimental conditions, for example, in plasmas under going neutral beam injection. Stabilization or destabiHzation by a sunr^thermal species in 3-D plasmas in magnetic configurations with shear could be explored with the new code.

3.13. National Institute for Fusion , Toki, Japan. The Large Helical Device (LHD)70 is a big I - 2 stellarator, stabilized by a combination of magnetic well and shear, with superconducting coils. It is being constructed at the newly established National Institute for Fusion Science under the directorship of A. Iiyoshi. The goal of the LHD is to demonstrate high energy confinement and high beta in a helical device, which are necessary steps toward a helical reactor system. The overall objective of this project is to clarify the physics and engineering issues important in designing a future helical reactor by studying the behavior of currentless plasmas in large-scale experimental helical device. Among the goals are . To carry out a wide range of studies on transport under high-ntT plasma conditions that can be extrapolated to reactor plasmas. . To obtain the basic data necessary to realize steady-state operation through experiments that demonstrate quasi-steady plasma control using a . This is a unique feature because existing stellarators rely on the natural divertor generated by the magnetic field created by the external coils. . To study the behavior of high-energy particles in the helical magnetic field and to conduct alpha-particle simulation experiments. . To increase the comprehensive understanding of toroidal plasmas by carrying out studies complementary to those in tokamaks. The theoreticians working for LHD are very versatile and imaginative. Their research program is one of the most comprehensive, covering MHD equilibrium and stability, neoclassical transport and anomalous transport. We summarize part of their work. The work of Nakamura, Wakatani et al.;25 is an excellent example of how optimization studies are done in search of the best configurations in a space of parameters. The MHD and trapped energetic particle confinement studies were carried out for configurations with 10 to 14 toroidal field periods. Optimization of both stability to ideal Mercier and low-n modes and transport properties was pursued under the condition of plasma aspect ratio A > 7. It was found that it was possible to obtain stable MHD plasmas with a beta limit of 4% to 5% for M < 12. One very interesting result of the studies was that the addition of a quadrupole field improves the confinement of trapped particles at zero pressure, but particle losses increase with increasing beta. However, it was found that the loss is less severe if the vacuum magnetic axis is shifted slightly inward. The conclusion of the studies of Ref. 25 was that a configuration with M = 10 was optimum in terms of both MHD properties and energetic particle confinement. The configuration is characterized by a coil pitch parameter Pc in the range from 1.25 to 1.30 and an added quadrupole field. The studies in Ref. 25 were the basis for the design of the next generation of large torsatron/heliotron devices. An example of such a device is shown in Figs. 21 nnd 22 of Ref. 25. Tb'iz device has a major radius of 4 m and a helical coil current density of 40.4 / mm2. The rotational transform is 0.32 at the origin and 1 at the edge. With an appropriate quadrupole field such that the current in the VF coils is 1.4 times the helical coil current, the energetic particle confinement is good enough to permit neutral beam injection and cyclotron range of frequencies heating, while maintaining favorable MHD properties. The flexibility of this optimum configuration can be further increased by the addition of a vertical field. Nakamura, Ichiguchi, Wakatani et al.26 systematically studied the stability of ideal and resistive low-n and localized pressure-gradient-driven modes for I = 2 heliotron/torsatron configurations. Furthemore, they investigated numerically the relation between the appearance of the low-/i ideal modes and the beta limit for the Mercier criterion. The effects on MHD stability of vertical and toroidal fields for Heliotron-E were studied for realistic experimental conditions. They found that an additional toroidal field can suppress the instabilities resonant at the * = 1 / 2 and * = 2 / 3 surfaces in the central region. They also developed techniques to consider higher beta effects in two dimensions in a form that can be used as input for the 2-D STEP code. They compared the new VMEC code results with the STEP code results for Heliotron-E. The main results of these studies can be seen in Figs. 1-4 of Ref. 26. Three-dimensional MHD equilibria that self-consistently include the bootstrap current were studied for LHD in Ref. 27. The magnitude of the bootstrap current is sensitive to the magnetic axis position or Shafranov shift in LHD, which is proportional to the plasma pressure and inversely proportional to the rotational transform. Since the bootstrap current enhances the rotational transform (in LHD) and reduces the Shafranov shift, the self- consistent MHD equilibrium is crucial to estimating the bootstrap current. The remarkable feature of the neoclassical bootstrap current in stellarators/torsatrons is its strong dependence on the magnetic field configuration. The magnitudes and profiles of bootstrap currents in LHD with R = 5 m have been examined by Nakajima et al.,71 who reveal that shifting the plasma by means of dipole field control is effective and that the ellipticity of magnetic surfaces, which can be controlled by the quadrupole field, has a remarkable influence on the bootstrap current; a small outward shift of the plasma and vertically elongated magnetic surfaces are favorable for a reduction of the bootstrap current. Total bootstrap current easily exceeds 100 kA for LHD plasmas with (/3)>1% under the assumption that birth electrons and belong to the 1 / v regime in the whole plasma region. Effects of the pressure profile on the bootstrap current have been also investigated. Recently, a model transport coefficient based on the gyro-Bohm scaling law has been incorporated into a one-dimensional (1-D) transport code to predict the plasma parameters and uieii profiles for the proposed LHD machine parameters.14*71 The obtained plasma profiles are used to calculate the bootstrap current and the beam driven current. A 1-D transport code is used to obtain the plasma profiles reproducing the LHD empirical scaling law. Goldston et al.72 derive a scaling similar to the LHD scaling by using a mixing-length argument for the collisionless drift wave turbulence. Assuming that the spatial variation of

the density fluctuation kLh is not larger than that of background density n ILA in the quasilinear theory of collisionless drift waves, the turbulent diffusivity X,~Y^1 *s

calculated using y =

eB a The turbulent difusivity is like the Bohm diffusivity but greatly reduced by the factor pj a. Thus, with

2 2 and Vp =2n Ra K, with K the elongation and P the absorbed power, rf = 0.25P^6n°-6B0iR06a2AKA^2. As explained in Section 3.7, the experimental data of ATF are consistent with this scaling. Dory et al.16 compared the LHD empirical scaling with the gyro-Bohm scaling and concluded that the gyro-Bohm scaling predicts practically the same plasma parameters as those of the LHD empirical scaling. Okamoto et al.15 have found that the global energy confinement, which is defined as the divided by the input power

r£ = (Wt + Wt) IP, is 97 ms from their simulations and 102 ms using the LHD scaling. It is known that in the 1 / v regime the ripple diffusion is remarkably decreased by the inward shift of the plasma in terms of the vertical magnetic field. One of the important results of Ref. 15 is that their transport code indicates that not the ripple diffusion but instead the anomalous transport xf in the plasma edge has the dominant effect on the global plasma confinement for the plasma with the magnetic axis shifted 15 cm inward.

3.14 Next-Generation Siellarators: Looking beyond the stellarators now in operation, under construction, or being designed, we find the next-generation devices. One of these is the ATF-II device studied at ORNL.73 This device is projected to have high plasma heating power (P = 10 to 15 MW) to test beta limits, low-collisionality transport, impurity control, and edge power handling at higher plasma parameters. The main constraints placed on the ATF-II design studies are low aspect ratio (A = 3 to 5), reasonable device cost [==$50 million to $100 million on the device-only cost basis that yields a cost of $20 million for ATF (power supplies, heating, controls, diagnostics, utilities, etc., extra)], and steady-state capability at 4 T. The studies are still in the design seeping stage and have not yet resulted in a final concept for ATF-II.

4. Final Remarks

With the construction of CHS, W VII-AS and ATF, several questions regarding the engineering feasibility of coils for the realization of advanced stellarators have been answered. The devices that are now in operation, or will soon begin or resume operation or in near future operation will provide experimental data to test MHD and neoclassical transport as well as anomalous transport theories. The first two theories were the basis of the optimization principles used in the design of the existing devices. So far, the performance of the stellarators has proven to be as good as that of tokamaks of similar sizes. In the present generation of stellarators it has been necesary to develop techniques to reduce field errors, for example in ATF and CHS. The energy confinement seems to follow the gyro-Bohm scaling, which is close to the empirical LHD scaling law. The next generation of stellarators, Wendelstein VII-X, LHD and perhaps ATF-II, will have plasma parameters similar to those of present tokamaks and will provide a data base for the design of stellarator reactors. In the theoretical area, new codes (using the VMEC code) implemented on fast new have demonstrated that stability calculations using die classical stellarator expansion must be revisited. Steilarator configurations must be considered in ideal MHD stability calculations as the 3-D configurations they are. Most of the work on stellarators has been carried out in support of optimizations based on MHD theory and neoclasssical theories. However, new work has been done in other areas such as drift waves and anomalous transport, mainly at ORNL. Studies of stellarator plasmas like those currently under way for tokamak plasmas using particle simulation are still well into the future. The stellarator program is very versatile; if the present designs are realized in machines there will be a splendid data base to find the scaling laws of bigger stellarator plasmas. This will have two immediate benefits: . There will be a better basis for selecting one or more of the present concepts for the next optimization iteration. . Tokamak plasmas could be better understood because the studies being carried out in stellarators are complementary to those carried out for tokamaks. The history of the world fusion program is characterized by the search for the "magic magnetic bottle." A number of configurations have been tested, and sometimes one has looked better than the others, so it is very difficult to guess which kind of device will become an effective reactor for the future. From what we know of magnetic confinement today, we can only speculate that it could be a tokamak, a very low-aspect-ratio tokamak, a stellarator, an innovative tokamak or perhaps a hybrid, although we know that the first attempts will be with the magnetic bottle we know the best, the tokamak. Acknowledgements

The author acknowledges several useful discussions with all the members ot the theory and experimental sections of the Fusion Energy Division at ORNL, especially with K. C. Shaing. The author gratefully acknowledges the input of W. A. Cooper, H. Gardner, V. E. Lynch, J. Nuremberg, V. D. Pustovitov, A. Varias and M. Wakatani. Specially thanks go to H. Gardner, who provided the author with copies of his unpublishea works. This research was sponsored by the Office of Fusion Energy, U.S. Department of Energy, under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc.

References

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'The submitted manuscnpl has been authored by a contractor of the U.S. Government under contract No. DE-AC05-S4OR21*00 Accordingly. ihe U.S. Government retains a nonexclusive royatty-lree license to publish or reproduce the published torm ot this contribution, or allow others to do so. lor U.S. Government purposes.*

REVIEW OF STELLARATOR RESEARCH: IN SEARCH OF THE "MAGIC MAGNETIC BOTTLE"

Nicolas Dominguez Vergara FUSION ENERGY DIVISION, ORNL

The author gratefully acknowledges enlightening discussions with all of the theorists and experimentalists at the Fusion Energy Division in ORNL, especially with K. C. Shaing. The author would also like to thank W. A. Cooper, H. Gardner, J. Nuehrenberg, V. E. Lynch, V. D. Pustovitov, A. Varias, and M. Wakatani for their contributions.

V LATIN AMERICAN WORKSHOP ON PLASMA PHYSICS MEXICO CITY, JULY 1992

'Research sponsored by the Office of Fusion Energy, U. S. Department of Energy, under contract DE-AC05-84OR21400 with the Martin Marietta Energy Systems, Inc.

DISTRIBUTION Or Ti ,:j DOCUMENT !S UNLIMITED OUTLINE

INTRODUCTION

*MHD MODES -High-n Modes -Low-n Modes

*BOOSTRAP CURRENT

'SCALING OF ENERGY CONFINEMENT IN STELLARATORS

^LINEAR DISSIPATIVE TRAPPED ELECTRON MODES

W-VII-X

*FL\AL REMARKS

DIFFERENT FORMAT FROM THE PAPER FOR PROCEEDINGS) INTRODUCTION

AZTECS NEVER HAD PROBLEMS OBTAINING ENERGY FROM FUSION

EVERY YEAR WITH ENOUGH HUMAN SACRIFICES ASKED FOR ANOTHER YEAR OF UFE FOR THEIR

! i

Florenrino Codex

ENOUGH FREE ENERGY TIMES HAVE CHANGED THE GODS ALSO CHANGED Nothing Last forever - Netzahualcoyotl

AND WE NEED MORE ENERGY

WE COULD ASK FOR ONE MIRACLE:

A PLASMA WHICH LASTS FOREVER

-EQUILIBRIUM -STABILITY -TRANSPORT

MIRACLES DO NOT HAPPEN OFTEN .BETTER NOT TO WAIT FOR ONE MEANING OF CONFINEMENT

CONFINEMENT MEANS : TO TRAP THE PLASMA

i. e.; TO CREATE GRADIENTS OF DENSITY Vn TEMPERATURE VT PRESSURE VP i. e. TO CREATE FREE ENERGIES I DESTROY THE CONFINEMENT

THEN, WHY TO CONFINE THE PLASMA? ' NO ALTERNATIVE T = 10-20 KeV

n =2-3 x 1020 m-}

P-H Rebut, "perspective on Nuclear Fusion'. JET-P(1992), March 1992.

THE EFFORT IS AIMED TO CONFINE THE PLASMA IN SUCH A WAY THAT Vn, VT, VP ARE XOT TOO DANGEROUS FOR CONFINEMENT CONFINEMENT DEVICES

WHAT'S THE BEST BOTTLE TO CONTAIN THE PLASMA?

The history of the fusion research in more than 30 years has been the search for the MAGIC MAGNETIC BOTTLE" Concept -C Stellarator 1951

Mirrors (ORNL, Russia, Japan...) Elmo Bumpy Torus

Tandem Mirrors (Livermore)

Tokamaks, STX, Peng-omak (Russia, USA, Germany, etc., etc;)

3 * 2. Drakon (Shafranov's idea) r< THE STELLARATOR ALTERNATIVE

ALTHOUGH THE TOKAMAK CONCEPT IS NOW THE REIGNING CONCEIT, THE STELLARATOR IS AN ALTERNATIVE PROMISE

* 3- D GEOMETRIES * FREE OF DISRUPTIONS *THEY HAVE CAPABILITY OF STEADY STATE OPERATION

*THEY HAVE BEEN MADE WITH DIFFERENT NUMBER OF FIELD PERIODS

3 H-I.AUSTRALIA 4 URAGAN 2-M, TJ-II, SPAIN 5 HELIAS CONCEPT. W-VII-AS. W-VII-X. GERMANY 6 STORM, SPAIN. ATF-II, ORNL 7 IMS. WISCONSIN, USA $ CHS. SPAIN 9 KHARKOV.UKRAINE 10 LHD. JAPAN 12 ATF. ORNL 14 MOSCOW. RUSSIA 19 HELIOTRON-E. JAPAN OPTIMIZATION STUDIES

AS THE STELLARATORS ARE INHERENTLY 3-D CONFIGURATIONS, THE OBVIOUS QUESTION IS AGAIN:

FROM THE INFINITE NUMBER OF POSSIBILITIES WHICH 3-D GEOMETRIES ARE THE BEST?

OPTIMIZATION STUDIES IN A SPACE OF PARAMETERS

REALIZATION OF PLASMA DEVICES

EXPERIMENT I PARTIAL ANSWERS

OPTIMIZATION STUDIES ARE CONSTRAINED BY

1) ENGINEERING PROBLEMS AND COST CAPABILITY OF LABORATORIES TO BUILD THE 3-D GEOMETRIES BIG MACHINES GIVE NICE RESULTS BUT THEY ARE EXPENSIVE

2) PLASMA PHYSICS HIGHEST POSSIBLE BETA VALUE MINIMUM TRANSPORT OF ENERGY AND PARTICLES (COMPLEMENTARY TO TOKAMAK RESEARCH) OUTLINE

*MHD MODEL *EQUILIBRIUM -Vacuum. Flux surfaces, islands, etc. -Finite beta -VMEC code, Full 3-D Equilibrium -Transformation from VMEC coordinates to Boozer coordinates

"LINEAR MHD EQUATIONS AND ENERGY PRINCIPLE

*HIGH-n MODES -Derivation of High-n Resistive Ballooning Modes Equation -Ideal Ballooning Mode Equation For 3-D Geometries -Mercier Criterion -Stability Boundaries in Tokamaks -Shafranov's Conjeture and first 3-D calculations -Comparison between 2-D and 3-D Calculations -High-n Ballooning Modes in stellarators -Conclusions

LOW-n STABILITY ANALYSIS FOR 3-D GEOMETRIES -Linear Low-n Mode Calculations -Comparison of Stability Boundaries from Low-n (using Stellarator Expansion) and Mercier Modes -Full 3-D Calculations. Comparison between Low-n and Mercier mode -Conclusions

*BOOSTRAP CURRENT -ATF, Control of Boostrap. Current. Theory and Experiment. -Conclusions

•SCALING OF ENERGY CONFINEMENT IN STELLARATORS -LHD SCALING *LINEAR DISSIPATIVE TRAPPED ELECTRON MODES

•W-VII-X

* FINAL REMARKS MHD MODEL

dt

; dt

— dt

— = -Vx E dt

J =

EOUILIBRIUM fi 0

Lou)- n LINEAR STABILITY

NONLINEAR STABILITY EQUILIBRIUM, VACUUM SOLUTIONS

/. field Lines

MIRROR MACHINE

BUMPY TORUS

STELLARATOR 2. FLUX SURFACES

* IN GENERAL: GOOD FLUX SURFACES, ISLANDS AND STOCHASTIC REGIONS

Fig. " Crass section 01 magnetic surface* 01 HS-5-" Islands ai me Dounatry conopoad 10 « * 1.0.

;l-SION TECHNOLOGY 1L I" UN I9W '55

ISLANDS CAN BE REDUCED £"t- O.\ 3

'.••>• y. :~i

n ^t T. • ' 'r-

-•i.U m=2 ISLAND IN CHS Nishimura et al.: Fusion Technology 17(1990)87

Fig. 16. Magnetic surface on the fluorescent mesh at B, = 0.0875 T: (a) average minor radius = 17 cm and (b) m — 2 island.

MAIN SOURCE OF THE ISLAND IS THE AMBIENT FIELD DUE TO MAGNETIZATION OF THE BUILDING STRUCTURE

NOT IMPORTAN IN MOST IMPORTANT EXPERIMENTS THAT WILL BE DONE AT B. > IT. SINCE THE WIDTH OF THE ISLAND IS ROUGHLY PROPORTIONAL TO B;V2 ORNL-OWG 89-2469 FED

(b) AFTER (a) INITIAL SYMMETRIZATION

(c) AFTER (d) ISLAND SIZE vs "TROMBONE" POSITION OPTIMIZATION 3, AR=15cm

0 10 20 30 40 50 (trombone)

FIG. 2. (a) Initial coil configuration \before modificationi. showing uncom- pensated feed coils, ib) Symmetrized coil configuration, (c) Opcimized confisura- rion with "trombone" sections shifted 15 •:m radially ouuvard at o = 150°. 1SO°. 270°. and 300°. fd) Reduction in island size as a functic^n of "trombone" positions at o = 150°. 1SO°. 270°. and 300°. ORNL-OWG 88-3344 F£0 0.4 f M00EL EXPERIMENT t - 1/3 0.2 \-

// ; — 0

-0.2 i— V>Ix«tf. V. X-.v' -0.4 1 0.4 \{e) EXPERIMENT EXPERIMENT = 1/2 0.2

N

-0.2

-0.4 1 1.8 2.0 2.2 2.4 (.8 2.0 2.2 2.4

FIG. 1. iai Computed islands ac surfaces of < = 1/3. 1/2. and 2 Z. assum- r.eid errors due to uncompensated currents in HF and VF coii :'ee<;s shown in 2. a i. b>-'d) Corresponding; measured iiux surfaces. 3. MAGNETIC WELLS AT VACUUM

*MAGNETIC WELL 1.5 N/1

R/a ^MAGNETIC HILL

*MAGNETIC WELL AND MAGNETIC HILL

1.8 2.0 2.2 2.4 4. VACUUM MAGNETIC SHEAR

*Shearless

*High-Shear

*Low and High Shear 5. VACUUM ROTATIONAL TRANSFORM

"* Low Shear 1.0

0-7 2/3

o.s O.f - Vf 0.1 0.2.

0.1

High Shear

3

0.8

O.H

0 6. MAGNETIC FIELD

'TOKAMAK

•CLASSICAL STELLARATOR MAGNETIC FIELDS

'QUASI-HELICALLY SYMMETRIC EQUILIBRIUM FINITE P EQUILIBRIUM

EQUILIBRIUM EQUATIONS

v.7 =

WE ASSUME THAT NESTED FLUX SURFACES EXIST

M, number of toroidal field periods £, multipolarity R, Major radius of the configuration a, average minor radius A=R/a, aspect ratio of the configuration 0, magnetic toroidal flux divided by 2K X, magnetic poioidai flux divided by 2 it s, flux label (normalized toroidal flux in this work) <£b, value of 0 at the boundary /? = ,,•—, average radius of a flux surface

\"=&—, magnetic weL 1 B i, rotational transform

5 oc —i. oe —, magnetic shear d

7, plasma current

(I'0'-J'X') = P'V EQUILIBRIUM EQUATION SATISFIED IN EACH POINT OF THE FLUX SURFACE /• B a = —j-, Parallel current

p0 = zh£s.t PEAK BETA. Bo is the magnitude of the magnetic field at the

magnetic axis for the particular value of beta considered and Po is the plasma pressure at the maenetic axis. B2 {/?}, AVERAGE BETA. Ratio of the volume averaged pressure over —2-. The definition used for (p) can vary from place to place.

= R,0(P) /^(O) toroidai axis shift a

EQUILIBRIUM BETA LIMIT. Jr(/?) = 50%.

cos(md-nc)

J'B

—r = —r Y £_. cosimd - nc) °0 mn = JB B dt~t~ B4 B dsd 4g{4{ 9 dq ; dd

ds Bl -n VMEC CODE. FULL 3-D EQUILIBRIUM

*Any number of field periods *Any aspect ratio *No ft restrictions

* Fixed and free boundary equilibria (in general no need to use any other supporting code to calculate finite beta [coils can be set upj). *Used for ATF, URAGAN, LHD, TJII, etc., etc., very heavily used in the world.

*Very fast code

IN FEW WORDS. NOT USING APPROXIMATIONS DONE BEFORE. The fine structure of the equilibrium quantities are known.

"Great radial resolution, 31. 61, 121. 241, 581, etc (whatever you want, if you are patient). The radial resolution of this code has no comparison with resolution of 3-D codes before VMEC (like the BETA code).

M simmd - nc) Whatever number of harmonics is needed 6: tokamaks 10:ripple tokamaks (TEXT) 40: ATF Low aspect ratio torsatrons LHD 60: URAGAN llO.TJII >110: H-l

CURRENTLY BEING IMPLEMENTED TO CALCULATE EQUILIBRIA WITH ISLANDS (S. P. Hirshman, Private Communication.) STELLARATOR EXPANSION (2-D) (Greene and Johnson, Phys. Fluids, 1961) 3-D Vacuum equilibria-* Average vacuum 2-D equilibria -^finite beta 2-D equilibria. ( 'finite beta equilibrium should not be large enough to distort vacuum equilibrium1) LIKE A TOKAMAK

*Good for large aspect ratio devices

*Large number of field periods U R

'Fixed boundary equilibria (in ORNL, Carreras et al.; )also free boundary using STEP (ORNL; Jeff Holmes, PPPL, Johnson).

^ = +0 TiL(+i -io)j Rotational Transform b" E* = i + *° , Critical Elongation

Shafranov's Shift _ 2 (*a l"ii"J "a"1 Equ^ibrium ^eta Limit

*Critera for stability was found by solving a "reduced" ballooning equation which allowed only Mercier modes to be unstable. But not even the Mercier criterion shows the destabilizing effect of the geodesic curvature due to the helical variation of the mag field. TRANSFORMATION FROM VMEC TO BOOZER COORDINATES

From VMEC coordinates (0,

The transformation of the angles is given by [J. Nuehrenberg and R. Zille. Proceedings of the International School of Plasma Physics held at Varenna, . 1987. Page3]: &=d+6,

Where,

and The quantities

X is the poloidal magnetic flux divided by 2K. The flux surfaces are expressed in terms of the Boozer coordinates as:

0 = s •*" X

The transformation of coordinates is carried out flux surface by flux surface. FLUX SURFACE RECONSTRUCTION (Using Boozer Coordinates)

FLUX SURFRCE RECONSTRUCTION 1

w r 91

i.i i.s ;.-•• 1.4 1.3 :.: :.; i.i 2.3 2.1 is 1.1 i.r

FLUX SURPRCE RECONSTRUCTION

'A •A

=1

1.5 ..= :~ ..3 1.9 :.O LINEAR MHD EQUATIONS

dt

dP_ dt dB

J =

LINEARIZED EQUATIONS

v = v0 + v; = v, B = Bo + 5,

7, = v x B\

— n't P=P

ENERGY PRINCIPLE SOLVE EQUATIONS OR MINIMIZE 5W

- 1\\ VP) i, B1 q = V x \VX x B) RESISTIVE BALLOONING EQUATIONS WITH COMPRESSIBILITY EFFECTS

We start from the resistive MHD equation?

dV

= V x J,

Consider perturbations of the form

K = Ve*

= nV(q - q{$ - 9k Note that,

Bo • VS = n§0 • VI = n[fl0 • Vc-BQ({0-dk}B0 • V^ + (750

Thus, Bo • VS = 0 V«S = 0 V • fl, = 0 V • (Be'5 J = 0 ni& o(n)

. =0

Let us do the n-ordering; first V2S. = = -n% V/ • -B,n- o(n2) o(n) o(\) An-) Carrying out the n-ordenng Using that Vj =V,

Assumption

Highest order in (n) plus low-frequency perturbation assumptionj -^-«CO 1 CO, •£?,=()

5, • VS = 0

Therefore, we propose three perpendicular vectors to express the components of the perturbations

xVk + V.b i

For ij = 0 andF=0

5o IDEAL BALLOONING MODE EQUATION

D. Correa-Restrepo (1978), Berk, Rosenbluth and Shohet (1983), R. L. Dewar and A. H, Glasser (1983)

i2l

In Boozer coordinates, the perpendicular wave vector is,

The magnetic filed can be written as, where the covariant and contravariant components are,

B. = -/

B9=J

B=B\

We have defined the metric element.

The Jacobian of the transformation is given by,

The contravariant components of the plasma current are, J'=Q. The curvature of the magnetic field lines is given by,

The covariant components of the curvature are,

The contravariant component JC\ is obtained from the equation,

The local shear is given by,

8"{ 8" ) The rotational transform is written in our notation as,

*. 0' P is the pressure, p the density and / the growth rate. MERCIER CRITERION The mercier Criterion is obtained by proposing the solution

in terms of g and 2 = *g.

The criterion for stability can be written for closed flux sufaces as [F. Bauer, 0. Betancourt, P. Garabedian, Magnetohydrodynamic Equilibrium and Stability of Stellarators (1984)],

and

Low beta and large aspect ratio tokamaks, Shafranov-Yurchenko' stability criterion for interchange instabilities, - —-\ +~—h-n2))o, 4{q dr) rB£ dr K ' or q>l, for stability. SECOND STABILITY AND BETA LIMITS

TWO STABLE REGIONS

I IIII M I M II II Los Angeles

TOKAMAK

S SHEAR

Like Deing in Hawaii

a - PLASMA BETA ONE STABLE REGION —> BETA LIMIT

LhCJE TO SO

UJ

1

. LOCALIZED PERTURBATIONS IN A TOKAMAK

BALLOONING MODES s SAME MODE (DIFFNAME)

MERCER MODES PLASMA BETA

THE BOUNDARIES ARE OBTAINED BY SOLVING THE BME

dr\ dr\

Bending term Curvature

-id x £,

"We do not need to solve two equations' Just one Because it is only one mode with two names BALLOONING MODE

MERCER OR INTERCHANGE MODE TO DETERMINE STABILITY TO MHD LOCALIZED MODES OF ANY DEVICE IT IS NECESSARY TO SOLVE THE BALLOONING EQUATION: d d , ~ - -> - — HTT—& + D0= y'0 drj drj

BECAUSE YOU CAN HAVE DIFFERENT WIDTHS OF THE MODE

We must not use just the Mercier criterion to determine stability: because this is just a limit i. e.; the Mercier criterion is just a necessary condition for stability not a sufficient one.

I s CMNMKAL ATOMICS ,„ ACHIEVED INCREASES WITH

10

CIRCULAR DISCHARGES

a-J C BALLOONING LIMIT ,' ~ o • EXPERIMENTAL

t

\ MA V. •

BROAD PEAKED BALLOONING MODES IN STELLARATORS

*Before 1989 the prevailing concept was that ballooning modes did not exist in stellarators (Shafranov's conjecture; Plasma physics 24(1982)233.)

Even though, before 1989, there were already published papers demonstrating that Shafranov's conjecture was wrong.

*Berk, Rosenbluth and Shohet, Phys. Fluids 26(1983)2616. First counter-example to Shafranov's conjecture. Numerical work.

*Llobet, Berk, Rosenbluth, Phys. Fluids 30(1987)2758. Analytically and numerically for model stellarators.

*Nuehrenberg Proceddings of the International School of Plasma Physics held at Varenna, ltaly.(1987). Page. 3. COMMENTS ON BALLOONING INSTABILITIES ON STELLARATORS "Unfortunately, it seems, as yet rather a complicated problem to determine an accurate value for L for stellarators. It is known that for a Tokamak the value of finn is determine by numerical solution of equilibrium equations and of equations describing the plasma MHD stability. In contrast to Tokamaks, a stellarator with optimal parameters is not principally a two-dimensional system, but rather a three-dimensional system (recall that different helical harmonics are required to produce a magnetic well). Numerical calculation methods for such systems are at present under development (Chodura et at., 1979; Anderson et al, 1979; Herrnegger et al.f 1979). Analytical methods are based on the expansion with respect to a small parameter which is essentially the relative distorsion of magnetic flux surfaces in comparison with circular concentric ones (A I a. er etc. ). It is clear that when extreme values are achieved magnetic surfaces are considerable distorted. In this case it is difficult to obtain an accurate, reaiistic value for f3iai because of the strong Bin dependence on the expansion parameters". Mikhaiiov and Shafranov, Plasma Physics 24(1982) 23.

"Ballooning instabilities also appear in stellarators, even though a simplified analysis predicts stability." Llobet, Berk and Rosenbluth, Phys. Fluids 30(1987)2750.

2 effects *the modulation of the normal curvature caused by helical effects.

* Effects from Pfirsch-Schliiter current. NNERVFCOL8

MNO VF COILS

STRUCTURAL SMELL

VACUUM VESSEL OUTER VF COLS

VACUUM VESta PORTS

SUPPORT COLUMNS

HF COILS ' BASE DIAGNOSTIC AND UTILITY PENETRATIONS"'"

Fig. 1. Line drawing of ATF showing the principal components. = o p- ORNL-DWG 84-18294 FED R AXIS = 2.05 2.10 1

-0.27 L h

z i I I 1 I i O J (.'" ' J U J h - I-

J L

Ot- u. ( J r • >-•.-/; r -I o

j p 0.13 f- r ™w i-

,i i 0.27- 1

FIELD VARIATION Data from "DISSiP-IOTA" EFFECT OF QUADRUPOLAR FIELD ON ROTATIONAL TRANSFORM 6-0

lb-0

2 c o LL z -0.13

< o <

12 "• 0.8

RADIUS 0RNL-0W* M-nm FED n AXIS* 2.05 2.J0 2.15 i I r i i

-0.27 L

I I

-0.J3 h

»X^r;: _ u A/ S I I I _ L ,: 4 >-/ • ; I1I r -A ste -

3.13- •

i ••i !:• ! j "ar UNSTABLE REGIONS. 3-0 CALCULATION

7

6-

S -

4 - 3-0 MERCER

3- A 2-

o.o 0.2 0.4 0.6 0.8

UNSTABLE REGIONS. 2-0 CALCULATION

3 2-D MERCIER

i.8 3-0 MEHCIES CALCULATION (USING VMEC) 3-3

a SHSAR

• GS9SS5CC

: s

CALCULATION (USING S3T=C^

s SH=AS

;M

- i : 5 WELL SHEAR ANO WEIUSHEAR (3-D) 3-3

3 5H=fn

a DM

• "^ Data from "ShifinSc-MERCIER/Papa" 3-D EQUILIBRIUM

a aa m IIIII1I1IIIL

//

/ / G a m a a m T""P"

3- a m a

2- a lUUUUUIHULJ a an . y...... njtn f / 1 - a aan • n nni ii i i i i 7"

0.0 0.2 0.4 0.6 0.8 1.0

NORMALIZED RADIUS

Data from "Shifin5c-MERCJER/Pap«" 2-D EQUILIBRIUM

• 2

4-

2/3

2"

0.0 0.2 0.4 . .o 0.8 1 .0

NORMALIZED RADIUS Dm from RSTEQ and VMEC. Peak betasO.8% 2 unstabia regions from VMEC

• •••• DnvVMEC Dm-ASTEQ

SO 0.90

3A0IUS IDxO STANDARD CONFIGURATION P=Po(1-Nor. Pol Rux)M2 (Probably not optimum press profile)

CSJ.W*

a qziOOS qzi007 a azi010 = qz1011 = QZ1012 0.0 0.1 0.2 0.3 0.4 3.5 0.6 0.7 0.8 0.9 i .0

RADIUS

Data from "STAND-IOTAS"

IT O

0.8 0.9

-••

34 33 32 31 30 29 28- 27- 26- 25- 24- 23- 22- h ^:

IL 20-

•7-j

'2 H •2-j

1 H 3-i

- j 5 i

PEAK BETA (%) ROTATIONAL TRANSFORM AND PRESSURE

1.CS

1.33 • i-flSTEQ

0.95:

:.5c-

0 35-

0.80-

3.75 ••

3.70:

0.85:

0.60^

C.40-J

225 OP TAB - ^ p Z2Z\

' S **

s.VMSC •. C ~ ' " :s J

3.13 0.20 C23 C.iC . = 0 :.so

RADIUS VMEC-Unstablt region driven by geodesic curvature+magnetic hill

Df-VMSC o DwVMEC UJ Di-VMcC Og>VMEC Dm-VMEC

:.:s 3.:: : 23 3 33 :.io : =: :.so :.:: : ss 3.32

RAOtUS CONCLUSIONS

With the implementation of the VMEC code, the MHD stability of ATF is being revisited.

Some interesting results are emerging as a result of analyzing 3-D equilibria.

It appears that the steilarator expansion, while giving good agreement for equilibria, is inadequate for ballooning modes.

A current issue is how sensitive ideal MHD stability is to quadrupoie fields and pressure profile details. LOW-nMODE CALCULATIONS 5Wp , = 0

SWp : Perturbed of the Plasma : Perturbed in the Vacuum Region 2 -Qi SWk : Perturbed of the Plasma CO : Frequency of the Perturbation

'•c

5WV = IjJJ

1-1

Where

and D =

|^|

uJ{s, $,

$ :the potential energy matrix $ is the kinetic energy matrix x is the eigenfunction, co2 is the eigenvalue Negative eigenvalues correspond to unstable modes. The matrix is solved with an iteration scheme t.00

FIG. 1. Examples of dominant components of the perturbed electrostatic eigenmode ($„)„, versus radius, in the i = 2 wrsatron configurations studied. ia) m = I component of the n = I mode for the M * 19 configuration and currentless equilibrium with 80 = 6.2%. ib) m = 8 component of the n = 6 mode for the M = 10 configuration and flux-conserving equilibrium with do = 12.0%. This figure illustrates the global and localized of the low-n modes studied. —>_ M- 10 RSTEQ —-— M- -,0 VMEC _ -c- - M - 14 RSTEQ ----- M - 14 VMEC

•2 14 '6

FIG. 2. Toroidal shifts versus oeaic oeia on from the 2-D (RSTEQ) ana 3-D I VMEC) equilibrium calculations for configurations with \f = 10 and M — 14 in me zero-current mode of operation for pressure profile P oe nn-4 M.10 P-d-¥)2 0.05 ZERO CURRENT - 30-6.7%

0.04 - ~n . 2 • \

Y 0.03 -

0.02 - - \\

0.01

0 - v M-10 P «(1 • v)2 FLUX CONSERVING

0.15

0.10

0.05 |-

-4.5 -3.5 -2.5 -1.5 -0.5 0.5 INWARD SHIFT OF MAGNETIC AXIS (AT VACUUM) (%)

FIG. 5. Growth rates versus inward shift of the magnetic axis from the low-n stability calculations for the M = 10 configurations with pressure profile P ce i] - V)2 and toroidal mode numbers n=2 and n = 4. 'at Zero-current sequences. J,, = 6. 7%. (b) Flux-conservine a sequences. dn - 6.7 c. 10.6% and 14.8%. 15 i r (a)

M-12

10 5/4 4/3 7/5 6/4'

J L 15 i I i r (b)

P « (i - oj2 5/4 M-12 4/3 10 \J

I I ! I 0 0.1 02 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 P FIG. 7. Merrier unstable regions for the flux-conserving M = 12 configuration, (a) P « (1 - *J2 and (b) P « (1 - $)2. The locations of the unstable low-n resonances are indicated bv thick lines. 1 ' I ' I M- 14 P« (1 - yj2

6 - /FLUX / CONSERVING

2 - / ^N \ZERO n-3n3 J CURRENT // I 10 15 20

FIG. 9. Growth rates versus 3o. determined from the low-n stability calculations, for the M * 14 configuration with P oe (1 - V)2, for n * 3 and n = 6 in the zero-current mode of operation and for n * / in the flux-conserving sequences. DOMINGUEZ et aJ.

14 i i (a) 12 ZERO CURRENT 10 h M-14

6f" j 4 h-

2 r-

0 "—

2 I-

0 0.1 0.2 0.3 0.4 0.5 0.6 C.7 0.8 0 9 1.0 P

FIG. 10. Mercier unstaoie reeions ror the M = 14 configuration with P at (/-+r. iai Zero-current mode or operation.

7 M-19

6

5 I-

4

3

2

1 U

3 t- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 03 C 3 - 3 i

FIG. 12. Mercier unstable region for the zero-current M = 19 configuration. The thick curves within the Mercier boundan indicate the locations of the mm = 514 and mm = 1/1 resonant es. BOOSTRAP CURRENT

fc

Gb: GEOMETRIC FACTOR, FUNCTION OF B

a o

(b) 80 - /\ - -0.13

60 /

40 /

/ 0.13 20 *^' 0.23^--^

1 i 0.2 0.4 0.6 0.8 1.0 pit K. C. Shaing, et al.; Phys. Fluids B1 (1989)1663. BOOSTRAP CURRENT EFFECTS

CONVENTIONAL t-=i STELLARATOR

Nr£ IT dV r iTtr1 d

* BOOSTRAP CURRENT INCREASES THE ROTATIONAL TRANSFORM IN i = 2 STELLARATORS i EQUILIBRIUM PROPERTIES CHANGE i is STABILITY PROPERTIES CHANGE

THERE IS NEED TO CONTROL THE BOOSTRAP CURRENT (1 = stellarators)

*IN QUASI-HELICALLY SYMMETRIC DEVICES TOROIDAL FIELDS HAVE TO BE INTRODUCED TO INCREASED THE ROTATIONAL TRANSFORM BOOSTRAP CURRENT CONTROL

1.0

0.8

0.6

0.4

0.1 —— WITH BOOTSTRAP CURRENT - NO BOOTSTRAP CURRENT — — WITH BOOTSTRAP CURRENT AND OUAORUPOLE FIELD

0.2 0.4 0.8 0.8 1.0 RADIUS

WITH •OOTSTRAP CURRENT NO MOrSriMP CUMICNT WITH BOOTSTRAP CURRENT ANO QUAOftUWLE FIELD ' I 0.4 r - 0 I

-0 4 H

0.4 - J

j •0.4 (- h '66 2.06 2.46 '66 2 06 2 46 '66 2.06 Z «6 "(ml »(ml B(m)

7HE BOOSTRAP CURRENT CAN BE CONTROLLED BY USING THE OUADRUPOLAR FIELDS EXPERIMENT: DIRECT MEASUREMENTS OF BOOSTRAP CURRENT

'NEGLIGIBLE OHMIC HEATING 'NEGLIGIBLE ECH DRIVE 1) Microvave power is launched perpendicular to the magnetic field 2) the current responds to reversal of the helical coil current ant to changes in n, (i. e., not an ECH-driven current). THE RESULTS CONFIRM THE PARAMETRIC DEPENDENCES ON VERTICAL FIELD THAT ARE PREDICTED BY NEOCLASSICAL THEORY

OHNL-OWG MM-30MA2 FED

QUADRUPOLE SCAN

CRNL-OWQI9M-300IA FED

O EXPERIMENT • THEORY ECH (0.95 T) DIPOLE SCAN

•5-4-2 0 2 4

-itf tern) ROTATIONAL TRANSFORM PROFILE

CONVENTIONAL 1 = 2 STELLARATOR R =

Z= r{s)sind + rh{s)sin{Q-M0)

FROM THE CONDITION 2n(Be) = IT(s)

lT(s): net toroidal current enclosed in surface "s"

Near the magnetic axis Mr2 iH = —r*- : helical transform / dV iT = —^— : tokamak transform lizr- dO

Here, *,, is the helical transform, and *r is the tokamak transform. Therefore,

. Vr? | IT dV r livr d0 SCALINGS OF ENERGY CONFINEMENT IN STELLARATORS

EMPIRICAL SCALING: LHD SCALING

*Gross energy confinement time,

= 0.17P^5SnO69BOMa-°ROJ5

P is absorbed power (MW), n is the line average electron density (lO2Om~3), B is the magnetic field strength on the plasma axis (T).

*Empirical scaling of the density limit obtainable under optimum condition:

*From ri and «: a beta limit can be derive:

Sudo, Takeiri, Zushi, Sano, Itoh et al; Nucl. Fusion, 30(1990)11. EMPIRICAL ENERGY CONFINEMENT SCALING OF CURRENTLESS S/H DEVICES

T(»0.17P-°"n°*f80-1 Oqor ;MWHKJwnftin (m) / / 1 V(ECH)

/ 10" L o

V7A F . ^ L2

^^HOR

J 1Q- •0"* T.-(S)

'Or

3.11-

O.I 10

Sudo, Takeiri, Zushi, Sano, Itoh et al; Nucl. Fusion, 30(1990)11 GLOBAL CONFINEMENT IN ATF

' IS wp, diamagnetically measured plasma stored energy

* IT IS CONSISTENT WITH THE LHD SCALING

OHNL-OWG •0M4ON FED 0.025

0.005 0.010 0.015 0.020 0.025

Global energy confinement vs LHD scaling.

* DATA ALSO FIT THE Gyro-reduce Bohm scaling

-(lx • -0.2 OflNL-OWQ MM4100 FED rf U.UO 1.90 T 0.9ST 0 2NBI 0.020 • a 1 NBI A ECH • -. 0.015 S m Hi A >

" 0.010 a * .**

0.005 & -j

0 i 0 0.005 0.010 0.015 0.020 0.025 0.030 :^° (s) M. Murakami, et al., 1990. Global energy confinement vs drift wave turbulence scaling. DISSIPATIVE TRAPPED ELECTRON MODES IN STELLARATORS

From the work: Dissipative Trapped Electron Modes in Torsatrons by N. Dominguez, B. A. Carreras, V. E. Lynch and P. H. Diamond. Phys. of Fluids. In print.

*FOR Tt»Ti AND Vn*0 TRAPPED ELECTRON MODES ARE RELEVANT WHENEVER: i) Fraction of Trapped Particles is significant ii) v.,

vei : Electron Collision Frequency

cob, = viThll^ : Trapped Electron Bounce Frequency

vtn\ Thermal Velocity Lj: Effective Connection Length iii) o).«coit

* STELLARATORS SUPPORT AT LEAST 2 CLASSES OF TRAPPED PARTICLES i) Helically Trapped ii) Toroidally Trapped THE VARIOUS CLASSES OF TRAPPED PARTICLES INTERACT QUITE DIFFERENTLY WITH STELLARATOR DRIFT MODES

^HELICALLY TRAPPED PARTICLES ACCESS THE v.. < 1. AT LOWER

VALUES OF Tt, THAN TOROIDALLY TRAPPED PARTICLES BECAUSE

\)L « R: connection length fortokamaks ll)L = R/M: connection length for stellarators DRIFT WAVE EQUATION FOR STELLARATORS

MODEL BASED ON FLUID IONS AND ADIABATIC ELECTRONS

electron response diamagnetic drift

Sound Wave polarization drift curvature drift

cs :sound velocity, c; ~TJ m,,

Te : electron temperature m\ :ion , f>,=cj Q;: sound Larmor radius Q the ion cyclotron frequency Va is the perpendicular wave vector 77 is the length along the magnetic field line

The diamagnetic and curvature drift velocities are

ds

IT - ib*K

nn : is the electron density DRIFT WAVE POTENTIALS

Tlin DR11T WAVIi EQUATION CAN BE WR1TEN AS A SCHROD1NGER EQUATION

TOKAMAK STRAIGHT STELLARATOR STELLARATOR SOLUTIONS OF DRIFT WAVES IN STRAIGHT STELLARATORS

THE DRIFT WAVE SOLUTIONS CAN BE LOCALIZED OR EXTENDED DEPENDING ON THE SHEAR LENGTH, DENSITY LENGTH, WAVENUMBER, ETC

ORNL-OWQ01M4S24 FED 1.002

1.001

a? 0.6 • 1.000

*CQ

" 0.999

0.998 LINEAR THEORY OF THE DISSIPATIVE TRAPPED ELECTRON MODE IN STELLARATORS

The DTEM regime is characterized by cob > viff xo»co.t > a>d Nonadiabatic trapped electron response from:

/ *\ • CO -(co - cod

r\e =dlnTJdlnne: ratio of electron density and temperature scale lengths

The perturbed electron density is obtain using the solution of the equation.

: : l + « |VajV- 1(2) = — co' dri~ Q\ : bounce average along he field line Assuming that the nonadiabatic electron response is small,

5co . 4 co.. to, vn vs,f where

11. B i SOLUTION OF THE DRIFT WAVE EQUATION IN HELICALLY SYMMETRIC SYSTEMS

)p| V For localized eigenfunctions, MA, < l.

l/2

' MK3SJ [X(O ) S a the solution of the dispersion relation.

The condition for existence of these helically localized solutions for electron density profiles with negative gradient is

For Localized Soultions, the linear growth rate is given by

MA. fl is the fraction of helically trapped particles Thus, for Helically induced modes the condition for stability is 17. <0 The radial width of the eigenfunction is given by ,1/2

Because of the constrain on L,, we found that Ar » ps NUMERICAL RESULTS FOR 3-D CONFIGURATIONS

A way to visualize the helically trapped particle fraction is to plot the contours of constant B^ = minjB(p,efl,C8)| in the plane (pcose,,psin6s). This plot indicates the departure of the deeply trapped particle orbits from the flux surfaces. The fractional area enclosed by the last closed B^ contour correlates with the fraction of trapped panicles confined.18

ORNI. DWG 91-3538 FED AREA = 0.6894 AREA = 0.2485 AREA = 0.1736 = -0.13 Ib=0.13 Ib=0.23

\ , . \ / \ 1 / \ i • , ! ; • j 1 \ \ i ! / !U 1 ] / / I .• \ i \ \\ ORNL-0WQ91M-3628 FED

12 I-

0.2 3.4 0.6 0.8 ORNL-0WQ91M46Z7 FED

•O *- 2.0 2.3 4.0 5.0 C'J 0RNl-0WGS1M4SaS FED

-2 p L r,, a = i pLn/a =2

=0 LINEAR GROWTH RATE AS A FUNCTION OF RADIUS FOR ATF CONFIGURATIONS WITH DIFFERENT QUADRUPOLAR MAGNETIC FIELD COMPONENTS

2 2 n{p)=n,[l-(p/af], pL. /a =[l-(p/a) ]/2 = 0.3%

ORNC-0W3 92M-23SS FED

p/£T EIGENFUNCTIONS FOR /, =-0.13

ORNL-OWQ91M-3634 FED ANGULAR WIDTH OF EIGENFUNCTIONS

0RNL-0WQS1M463S FED

0.2 .3 0.9

Only for /^ = -0.13 does the eigenfunction remain localized for all radial positions. RADIAL WIDTH OF EIGENFUNCTIONS

OrWL>0WQt1l

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 CONCLUSIONS REGARDING TO DTEM

* Helically trapped particles interacting with be'icity-induced modes are the dominant trapped electron instability in the stellarator. These instabilities have relatively broad radial width, because their extent along field lines is &r\ = 2JC / M. They should be easier to detect than the corresponding tokamak instabilities.

* Toroidally trapped particles interact with helically extended modes in a manner that closely resembles the situation in tokamaks.

* The helical symmetric limit gives a good description of the helically induced modes for stellarators such as ATF. The density gradient must exceed a critical value for helically localized eigenfunctions. In the helical symmetric limit, a condition for nonexistence of helically localized eigenfunctions is

1 dn M p n dp RQ 3

When this condition is not satisfied, the electron temperature gradient should be such that

Te dp n dp and in this way suppress the helically induced DTEM.

With present devices, the v.e < l regime is accessible at relatively low electron temperatures. The broad radial width associated with the helically induced modes makes them detectable with present core fluctuation diagnostics. W-VII-AS AND W-VII-X

* FLAT ROTATIONAL TRANSFORM *SHALLOWWELL,2% *REDUCnON OF PFIRSCH-SCHLUTER CURRENTS *FIVE PERIODS * —, — = 0.1, HARMLESS FIELD ERROS

HELIAS EQUILIBRIUM, (/3)=15%, ^

SHAFRANOV SHIFT OF HELIAS CONFIGURATIONS IS VERY SMALL. QUASI-HELICALLY SYMMETRIC EQUILIBRIUM

PHYSICAL INSIGHT: IT IS POSSIBLE TO HAVE A TOROIDAL EQUILIBRIUM WITH TRAx\5PORT PROPERTIES OF A HELICALLY SYMMETRIC EQUILIBRIUM.

W7-X W7-AS

- 1 3.

(-10)

( vi a 1 0.5 0.0 CONTOURS OF MAGNETIC FIELDS FOR STANDARD STELLARATOR AND QUASI-KELICALLY SYMMETRIC EQUILIBRIUM

HELIAS 4-12 HELIAS 5-6 (5081)

e

toroidal inql* §oql»

1 / v LOSSES DISSAPEAR —* LIKE A TOKAMAK + ADVANTAGE OF OPERATING IN STEADY STATE.

VERY PROMISING CONCEPT REDUCTION OF BOOSTRAP CURRENT

ALTHOUGH THE HELICALLY SYMMETRIC EQUILIBRIUM IS VERY PROMISING, TOROIDAL EFFECTS HAVE TO EE INCLUDED TO REDUCE BOOSTRAP CURRENT. OTHERWISE THE ROTATIONAL TRANSFORM DECREASES HITTING LOW-ORDER RESONANCES

MAGNETIC FIELD COMPONENTS FOR HEUCALLY SYMMETRIC EQUILIBRIUM AND MAGNETIC FIELD FOR EQUILIBRIUM WITH REDUCED BOOSTRAP

CURRENT: B(s, 0,) = B{s, 6-M) + Bx(s, 6, )

(0.0)

0.0 1.0

WENDELSTEIN-VII-X IS STABLE TO IDEAL BALLOONING MODES UP TO DIFFUSION COEFFICIENTS FOR STELLARATORS IN THE l / v REGIME

CONSIDERING

B = BQ[l - e, cos 6 + eh cos{£9 - M

DVv = 0 In a quasiheiically symmetric stellarator Dhv = 0

W VII X Dvv= const. Sr^T ^ - - where Se =0.01.

3 lZ Dv v = const • £ h — In a classical stellarator

where eh = 0.1.

ANOMALOUS THERMAL CONDUCTIVITY OF ELECTRONS IN W- VII-A y - n~xT Ae.an lk x THEREFORE, RESEARCHERS IN GARCHING HOPE IS TO HAVE SMALL ANOMALOUS TRANSPORT IN W-VII-X. (G. Grieger et ah; Phys. Fluids B4( 1992)2081.

THIS CONCEPT WOULD LOOK EXTREMELY ATRACTIVE IF THE ASPECT RATIO IS REDUCED (The actual aspect ratio for W-VII-X is FINAL REMARKS

* CONSTRUCTION OF CHS, W-VU-AS AND ATF i FEASIBILITY OF COILS

* EXPERIMENTS WILL PROVIDE DATA BASE TO TEST -MHD THEORY -Neoclassical Transport Theory -Anomalous Transport Theories

* FIELD ERRORS CAN BE SUCCESSFULLY CORRECTED (CHS, ATF)

* LHD, W-Vll-X AND ATF-II WILL HAVE PLASMA PARAMETERS AS PRESENT TOKAMAKS

* LHD, W-VII-X, AND ATF-il WILL PROVIDE DATA FOR DESIGN OF STELLARATOR REACTORS

* NEW CODES IMPLEMENTED (using the VMEC code) ARE DEMONSTRATING THAT STABILITY CALCULATIONS USING THE CLASSICAL STELLARATOR EXPANSION HAVE TO BE REVISITED

* OPTIMIZATION STUDIES HAVE BEEN BASED ON - MHD Theory - Neoclassical Transport Theory - New Areas of Theoretical Research: Drif Waves —* Anomalous Transport Parallel Studies with Tokamaks *IF MACHINES ARE RE-STARTED AND DESIGNS REALIZED IN DEVICES, THERE WILL BE DATA BASE TO FIND SCALING LAWS OF BIGGER STELU\RATOR PLASMAS

1) Better data base to choose one or more of the present concepts for the next optimization iteration

2) Tokamaks could be better understood because the studies being carried out in stellarators are complementary to those carried out for tokamaks

*IN THE SEARCH FOR THE "MAGIC MAGNETIC BOTTLE", SOME CONCEPTS LOOKED BETTER THAN OTHERS, BUT WHICH ONE WILL BE AN EFFECTIVE REACTOR?

TOKAMAK

INNOVATED TOKAMAK STELLARATOR

HYBRID

ALTHOUGH WE KNOW THATTHE FIRST ATTEMPTS WILL BE WITH THE MAGNETIC POTTLE WE KNOW THE BEST, THE TOKAMAK