Spherical stellarator
 By Paul Moroz

In memory of my teacher,
M. S. Rabinovich, the founder of
the Russian Stellarator Program

Double-helix spherical stellarator

Contact information:

Dr. Paul Moroz
Boston, MA, USA

The Spherical Stellarator (SS) is an advanced concept for controlled nuclear fusion with promises of improved comfinement of high temperature and high pressure plasmas in a compact and efficient machine. It represents a low-aspect-ratio compact stellarator approach that uses plasma current (bootstrap current, if plasma pressure is high enough, or other means otherwise) to achieve a significant part of the rotational transform as compared with that produced by the stellarator's coils. This concept is most advantageous at low plasma aspect ratios, A < 3.5. Here, aspect ratio A = R/a, where R and a are, respectively, the average major and minor radii for the last closed flux surface within which the plasma is confined. The closed flux surfaces in a stellarator are more complex than simple toroidal surfaces, but the use of average R and a for their characterization is common. The SS concept emphasizes effectiveness of the plasma (bootstrap) current in reaching high plasma pressure operation and improved confinement. On this way, compact stellarator configurations with extremely low aspect ratios approaching to A=1 were found.

The SS concept was discovered in an attempt to overcome significant difficulties of traditional magnetic fusion approaches that might prevent them from being reasonable prototypes for an efficient fusion reactor.

At present stage of fusion research, tokamaks and stellarators are the two leading traditional fusion concepts. They demonstrated the best plasma parameters so far, although there are a few other well-known fusion concepts as well. Tokamaks have the best plasma confinement properties, and the low-aspect-ratio tokamaks (also called spherical tokamaks) could confine high pressure plasmas (or high beta plasmas, where beta is the ratio of the plasma pressure to the magnetic field pressure). The average plasma beta in spherical tokamaks was already experimentally demonstrated in the excess of 40%. However, tokamaks are intrinsically pulsed devices, which prevents that approach from leading to an efficient fusion reactor (unless an efficient way to drive a steady-state plasma current at the required high plasma density will be found, which is not the case so far). For tokamaks, the lower aspect ratio is -- the more difficult it is to make them steady-state or even to start-up the initial configuration with the plasma current. This is because of a few reasons. Let me mention a couple of them. Firstly, the magnetic field magnitude variation along the field lines and, correspondingly, the so-called trapped particle effect, become much stronger for tokamaks with small aspect ratios, which leads to significant decrease in the efficiency of the RF-driven and/or beam-driven current. Secondly, installation of a current generating transformer, as it is typically done in tokamaks, becomes a problem, as there is not enough space left for it in the center of the machine. For a spherical tokamak-reactor, it would be practically impossible to put inside a current generating transformer.

Stellarators, on the other hand, are intrinsically steady-state devices, but their confinement properties are worse and the highest plasma pressure in the best stellarators is significantly lower (the corresponding beta is about or less than 5%). This is too low for an acceptable fusion reactor. Also, a reactor based on a traditional stellarator concept would be of a monstrous size, because it is based on the high-aspect-ratio approach.

Combining the best properties of a spherical tokamak and a stellarator was behind the intention leading to a spherical stellarator concept.

In a toroidal plasma, there is a current, called bootstrap current, which is self-generated due to plasma pressure gradients. It is strong in tokamaks with the high-beta plasma, but it cannot be generated there unless there is a significant driven current component, called the seed current. In a stellarator, the bootstrap current is smaller, but does not need any seed current and could flow without any externally driven current. Previously considered stellarator concepts have treated the bootstrap current as a negative phenomenon leading to unwanted change in the magnetic configuration depending on plasma pressure. Thus, the stellarator configurations were traditionally designed such that the bootstrap current was small, almost zero.

The SS concept was the first one in a stellarator family that considered substantial positive effects of the bootstrap current and, moreover, its significant advantages and benefits for compact stellarators.

The SS program was aimed at the formulation of the main principles for this approach, the search for and investigation of particular configurations, and configuration optimization.

In developing this new approach, the stellarator concept was chosen because it leads to a steady-state operating reactor. The compact (spherical) aproach was chosen because it leads to a simple and inexpensive design with the large ratio of plasma volume to surface area, as well as to plasma confinement with very high pressure (much higher than that possible in traditional stellarators) thus making an efficient reactor, and the requirement on plasma current came from the discovery (in contradiction with traditional results for large-A stellarators) that it significantly improves parameters of a compact machine. Another discovery--that the self-induced bootstrap current can be the only plasma-current source required for reaching advanced SS regimes--significantly improves application prospects of this approach.

During configuration search and optimization, main attention was paid to the factors directly contributing to reactor efficiency, such as improved plasma confinement and high pressure regimes of operation, as well as simplicity and efficiency of the design. No extra requirements, such as helical, toroidal, or poloidal symmetry, were artificially imposed, because they were found to be unnecessary limiting for compact stellarators including the SS.

Since the discovery of the SS concept by the author, significant activity in that area appeared at a number of research institutions. Some of the researchers were using the so-called "quasi" symmetry, a vague term not really appropriate in most situations for compact stellarators with advanced characteristics. Very often, looking at the configuration that the authors presented as, say, toroidally symmetric, or as poloidally symmetric, or as helically symmetric, you would find that it does not actually have those properties, and moreover, not even close to that.

For compact stellarator configurations, the toroidal term is typically significantly stronger than the other terms in the magnetic field representation, thus making it easier to reach quasi-toroidal symmetry than any other type of symmetry. Compact stellarator configurations with the plasma current and with the type of symmetry close to toroidal symmetry in magnetic coordinates were found in SS research.

One could state that other symmetries, such as poloidal or helical symmetry, are not really appropriate for compact stellarators. That is generally true for any low-aspect-ratio compact configuration, unless it is strongly elongated vertically and is narrow in the major radius direction. However, the later configuration could hardly be called a compact one, although formally its aspect ratio defined through the average minor radius might be small. Such configurations lose one of the important advantages emphasized by the SS that the ratio of the plasma volume to the plasma surface area has to be large. That requirement represents significant advantages for a fusion reactor where power is generated in the plasma volume while all losses happen through the surface.

The best results based on combined benefits of improved plasma confinement and high plasma pressure regimes of operation were obtained for an approach which is very different from any symmetry. Namely, they were obtained for such SS configurations that feature a very low outboard ripple of the magnetic field in comparison with that for the inboard part of the torus. The helical post stellarators, found in this research, were the very first stellarator configurations to feature those advanced characteristics.

The main unique advantages of the SS concept described in the publications below are as follows:

The SS concept was officially announced in January 1996 - the time when my proposal was submitted to the U.S. Department of Energy (USDOE) and a couple of articles were submitted for publication. At that time, there was no research on compact stellarators not only in the USA but in the world. Moreover, the US stellarator program was very weak (it included just a couple of small university experiments and a few people) and probably nobody thought that a low aspect ratio compact stellarator could be efficient.

During the two-year research on the SS, many unusual configurations of this type were discovered. My work on development of this concept was supported by the Office of Fusion Energy Sciences (FES) of USDOE until mid-1998. However, during a late spring of 1996, a team was formed at ORNL that included D. B. Batchelor, B. A. Carreras, S. P. Hirshman, V. E. Lynch, D. A. Spong and J. Whitson. They were interested in compact stellarator research and invited me to work with them as a consultant. I was glad to contribute to that effort because of my enthusiasm for the SS concept and longing for someone to build it. I continued to work both on the ORNL project called at that time SMARTH (Small Aspect Ratio Tokamak Hybrid) and on further development of the SS concept. All SS results were presented to the ORNL team and I was assigned a task of modification of 3D MHD equilibrium code VMEC to include self-consistent bootstrap current calculations in it. That task was successfully accomplished in 1998.

By the end of 1996, when a number of presentations were made, a few articles on this topic were published (see list below), and the benefits of this program became clear, somewhat similar compact stellarator research programs were also initiated at PPPL and at some universities. The first PPPL team on compact stellarator research was small and included R. Goldston, A. Reiman, D. Monticello, H. Mynick, M. Zarnstorff, and maybe a couple of other participants, but later that team grew very quickly. I was invited to visit PPPL for a few times to make presentations. At that time, Paul Garabedian of NYU was developing a quasi-toroidally-symmetric stellarator approach, although then without a plasma current. The PPPL team became interested in developing that approach further, taking the idea of importance of the plasma current from the SS concept.

I was active in making various presentations (including those given in response to FES call for innovative ideas and concepts in fusion) and publishing numerous journal articles. However, growing interest from the National Labs and some universities, and willingness to develop their own compact stellarator programs (that is how the term quasi-symmetry became popular) produced a negative effect on my research: the USDOE funding for the SS research was decreased and then terminated.

At the same time, research on compact stellarators with the plasma current, initiated by the SS program, became one of the main elements in the U.S. fusion program of that period. It was well funded by USDOE in those Labs (NCSX project at PPPL, focusing on configurations with quasi-toroidal symmetry, and a QPS project at ORNL, focusing on configurations with quasi-poloidal symmetry), as well as in a number of other US institutions (see, for example, the CTH (Compact Toroidal Hybrid) project at Auburn University, or ARIES program at UCSD, or Columbia University Non-neutral Torus CNT).

The compact stellarator initiative, started by the SS research, provided a fresh set of ideas for designing advanced stellarators more suitable for a fusion reactor than previously investigated concepts. It is encouraging that there is an interest to some of these ideas in the fusion research community and a few machines of this type were constructed or under construction in the USA. However the designs of those machines so far do not allow realization of spherical stellarator promises mentioned above. For example, the beta limit for the NCSX machine is about 4% (by the way, that level is already reached in experiments on high-aspect-ratio stellarators), and for the QPS concept such a limit is below 2%. Mentioned beta limits are the principal limits for those magnetic configurations and not for the specific machines, so, for example, proportional increase of machine dimensions or heating power will not increase their betas further. Unfortunately for the NCSX program at PPPL, after about 10 years of design and construction, the USDOE funding for NSTX stopped in 2008.

High plasma pressure operation is a significant advantage of the Spherical Stellarator concept. Fusion research has a long history of exploring various plasma configurations. It is time now to concentrate more on configurations that can confine high pressure plasmas, because only those configurations can be relevant to controlled fusion via magnetic plasma confinement. Good confinement properties are of primary importance as well, but they should be good for the high pressure plasmas.

This page was designed as an attempt to collect together some of the initial published results on spherical stellarators. Those results were prepared in a relatively short period of time (the corresponding USDOE funding lasted for about two years) and a lot still awaits to be done. However, that work initiated a field with significant potential for controlled fusion, as well as for some other applications. I believe this potential will be developed further and unleashed one day in an efficient fusion reactor.

Selected publications and presentations on SS
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© 1999-Present, Paul Moroz