Stellarator / Helical Approach

The stellarator is a magnetic confinement approach that confines plasma using external coils alone. Whereas a tokamak relies on a current driven through the plasma itself, a stellarator needs no such current. The coil shapes become correspondingly complex, but this brings a major advantage: current disruptions cannot occur in principle, making the approach well suited to steady-state operation. This page explains, step by step, from the core idea to helical devices, W7-X, and the latest research on optimizing the magnetic field.

Start with Intuition (High School)

When you try to confine plasma in the shape of a doughnut (torus), one troublesome thing happens. The magnetic field is stronger on the inside of the doughnut and weaker on the outside, and because of this difference, positive particles slip gradually upward and negative particles slip downward. This sideways slipping is called drift. Left alone, the particles hit the wall and escape.

It is known that to prevent this, you should not run the field lines straight around the doughnut, but instead twist them into a helix. As a field line goes once around the doughnut, it should also circle around the cross section from top to bottom and bottom to top. Then a particle that tries to slip upward eventually rides onto the path on the lower side, and a particle that tries to slip downward rides onto the upper side, so the slipping cancels out and the plasma stays together. This “degree of twist in the field lines” is called rotational transform.

So how do you twist the field lines? A tokamak drives a large current through the plasma and uses the field created by that current to produce the twist. A stellarator, on the other hand, shapes the external coils themselves so as to create a twisted field from the outset. To use an analogy, a tokamak is the approach of “having someone inside spin a top,” while a stellarator is the approach of “twisting the road on the outside beforehand.”

This idea of doing everything from the outside has a decisive advantage that may look modest. Because there is no need to keep driving a current through the plasma, the device can run continuously without stopping (steady-state operation). Moreover, the tokamak’s weak point, the accident in which the current suddenly vanishes and the plasma collapses all at once (disruption), does not occur, because there is no current to drive in the first place. Named with the wish to create a star (stellar) on the ground, this is the stellarator. It was invented by Lyman Spitzer in 1951.

Understand the Physics (Undergraduate)

The rotational transform $\iota$ (iota) is defined as the angle by which a field line rotates in the poloidal direction (the direction of the cross-sectional loop) as it goes once around in the toroidal direction (the direction of the doughnut’s large loop). It is often expressed as the number of rotations per turn, written as $\iota / 2\pi$. This quantity determines the very force that confines the particles.

The essential difference between a tokamak and a stellarator lies in where this $\iota$ comes from. In a tokamak, the poloidal field created by the plasma current $I_p$ produces the rotational transform. In other words, the rotational transform depends on the state of the plasma, and if the current vanishes, the confinement vanishes too. In a stellarator, $\iota$ is created solely by the geometry of the external coils, so closed magnetic surfaces exist in vacuum even before the plasma has been established. This fact that “the vacuum magnetic surfaces are complete from the start” supports both ease of start-up and steady-state behavior.

The safety factor $q$ of a stellarator is the inverse of the rotational transform, $q = 2\pi / \iota$, and, as in a tokamak, magnetic islands tend to form near rational values. However, unlike a tokamak, how $\iota$ varies in the radial direction (the magnetic shear) can also be built in through coil design, giving the stellarator vastly greater design freedom.

Let us organize the representative configurations.

• Heliotron/torsatron: rotational transform is created by continuous coils wound in a helix. Japan’s Large Helical Device (LHD) is a representative example.
• Modular stellarator: the field is created by arranging individually shaped three-dimensional coils. Germany’s Wendelstein 7-X (W7-X) is a representative example.
• Heliac: a configuration in which the magnetic axis itself is twisted into a helix, with Spain’s TJ-II as a representative example.

Because the three-dimensionally twisted field is created by external coils alone, the coil shapes are inevitably complex. This complexity is both the technical difficulty of the stellarator and, at the same time, the source of its strength, the ability to “design the magnetic field freely.” The outlook for confinement performance becomes more vivid when understood together with the scaling-law ideas covered in the transport chapter.

Deepen the Theory (Graduate)

The freedom to create the field with external coils alone raises, conversely, the question of “what kind of field should one aim to design.” That is because a naively constructed three-dimensional field has a serious weakness. When regions of strong and weak toroidal field arise, trapped particles are born there. These are particles caught in the weak-field valleys, bouncing back and forth. In an axisymmetric system like a tokamak, the orbits of these trapped particles close near the magnetic surface and stay there. In a general three-dimensional stellarator, however, the orbits do not close, and the particles drift systematically away from the magnetic surface. This worsens neoclassical transport, becoming especially serious in the hot, low-collisionality regime.

The key to solving this problem is the concept of quasi-symmetry, organized from the 1980s onward. From the argument of the adiabatic invariant in the Boltzmann equation, it is known that what determines particle confinement is not the direction of the magnetic field vector itself, but the distribution of the magnetic field strength $B = |\mathbf{B}|$ that the charged particles feel. Therefore, even if the shape in real space is three-dimensional, if the field can be designed so that the field strength $B$ appears symmetric in appropriate coordinates (Boozer coordinates), then good particle confinement equivalent to that of an axisymmetric system can be recovered. This “symmetry of strength” is quasi-symmetry.

Quasi-symmetry comes in several kinds depending on direction.

• Quasi-axisymmetry (QA): $B$ is nearly uniform in the toroidal direction. It aims for good confinement close to that of a tokamak.
• Quasi-helical symmetry (QH): $B$ is symmetric in the helical direction. The United States’ HSX achieved this for the first time in the world and demonstrated improved transport.
• Quasi-isodynamic (QI): although not strictly quasi-symmetric, $B$ is designed so that the average drift of trapped particles closes within the magnetic surface. W7-X was designed to aim for this QI configuration.

Modern stellarator design incorporates such conditions on $B$ into an objective function and, through numerical optimization, works backward to find coil shapes that simultaneously satisfy the stability of the MHD equilibrium (the magnetohydrodynamic balance), the maintenance of magnetic surfaces at finite beta (the ratio of plasma pressure to magnetic pressure), and coil feasibility. W7-X is the first large-scale demonstration of an optimized stellarator, a superconducting device with a major radius of about 5.5 m and a field strength of about 3 T, which began operation in 2015. LHD, on the other hand, is one of the world’s largest helical devices, with a major radius of about 3.9 m and a field strength of about 3 T. Since beginning operation in 1998, it has accumulated abundant data from long continuous discharges and deuterium experiments.

Research Frontier (PhD)

Stellarator research is now an active field, driven by the maturing of optimization theory and the growth of computational resources. Here we introduce the main topics within the bounds of established fact.

One is the search for new configurations that approach quasi-symmetry more precisely. It is theoretically thought to be difficult to make perfect quasi-symmetry hold throughout the entire volume, so how closely one can approach it, and in which regions one compromises, are under discussion (precise quasi-symmetry). Using open-source stellarator optimization codes (such as STELLOPT and SIMSOPT), research on optimizing coil and plasma shapes simultaneously is advancing.

Coil engineering is also at the frontier. Three-dimensional coils require high manufacturing precision, and W7-X demanded precision management on the order of a few mm relative to the design values. In recent years, advances in high-temperature superconductors (HTS) have prompted study of the possibility of building more compact devices with stronger fields. There are also studies that include the manufacturability and accessibility of coils, that is, how to assemble and maintain complex coils, in the objective function of the optimization itself.

From the standpoint of reactor engineering, the challenge is designing blankets (the mechanism that catches neutrons and breeds tritium fuel) and divertors (the exhaust-heat and exhaust-gas mechanism) that conform to the three-dimensionally twisted wall shape. Divertor concepts specific to stellarators, such as the island divertor and divertors that use regions where field lines diverge, are being studied.

Furthermore, in recent years, private companies have begun moving toward commercial reactors based on optimized stellarators. Combining numerical optimization with HTS, design studies aiming at more compact devices and earlier practical use are underway. Whether the stellarator can translate its fundamental strength as a “steady-state reactor that does not rely on current” into the reality of engineering and cost is the focus going forward.

Check Your Understanding

Q1. How do a stellarator and a tokamak differ in where they get their rotational transform?

A Both obtain rotational transform from the field created by a current driven through the plasma B A tokamak gets it from the shape of external coils, and a stellarator from the plasma current C A tokamak gets it from the field created by the plasma current, and a stellarator gets it solely from the geometric shape of external coils D Both obtain rotational transform from the shape of external coils alone

Q2. Why can disruptions not occur in principle in a stellarator?

A Because it does not require a large plasma current for confinement, so there is no current that can suddenly vanish and collapse B Because it has a control system that always keeps the plasma current constant C Because the field is weak, so even if the plasma collapses the impact is small D Because the external coils instantly compensate for the loss of current

Q3. Why does neoclassical transport worsen in a naively constructed three-dimensional stellarator field?

A Because the coils generate heat and disturb the field B Because the orbits of trapped particles formed by the field-strength variation do not close on the magnetic surface and drift systematically away C Because the plasma current is too large and increases turbulence D Because the vacuum magnetic surfaces do not exist from the start

Q4. What does quasi-symmetry make symmetric?

A It makes the shape of the coils in real space itself axisymmetric B It makes the distribution of the plasma current symmetric C It makes the distribution of the magnetic field strength B = |B| that charged particles feel symmetric within an appropriate coordinate system D It makes the temperature distribution of the plasma cross section symmetric

Q5. What is each of QA, QH, and QI aiming for?

A QA is symmetric in the helical direction, QH is uniform in the toroidal direction, and QI realizes perfect quasi-symmetry throughout the entire volume B QA is quasi-axisymmetry, nearly uniform in the toroidal direction; QH is quasi-helical symmetry, symmetric in the helical direction; and QI is a quasi-isodynamic configuration designed so that the average drift of trapped particles closes within the magnetic surface C All of them improve confinement by changing the distribution of the plasma current D QA, QH, and QI are all the same field-strength distribution called by different names

Related Topics

• Concepts of Magnetic Confinement - The shared foundation such as rotational transform and magnetic surfaces
• Tokamak Approach - The contrasting approach that creates rotational transform with plasma current
• Large Helical Device (LHD) - One of the world’s largest helical devices
• Transport - The theoretical background of neoclassical transport and confinement performance