Needed computations and computational capabilities for stellarators

There is no requirement that the plasma-encircling coils, such as modular coils, be chosen to ensure $\vec{B}\cdot\hat{n}=0$ on the plasma surface. But, they should be chosen to ensure good plasma access, a large separation from the plasma, Section II.2, and may be chosen using other criteria such as simplicity. Construction difficulties associated with maintaining high spatial precision in the coils can be greatly reduced and will be discussed in Section II.4.

Access affects both the cost of fusion power and the speed with which fusion systems can be developed.

The 14 MeV neutrons from deuterium-tritium (DT) fusion reactions degrade materials unacceptably after a certain fluence, which is approximately 10 MW yr/m². Fusion energy is cheaper the greater the power density, MW/m² through the walls until material limits are reached. A major limitation on the fraction of time a fusion power plant can actually produce power, its duty factor, is the speed with which neutron-degraded plasma-chamber components can be replaced. Even in ITER ITER hot Cell: 2017 , the replacement of degraded components is complicated by both their high level of radioactivity and by the requirement of maintaining confinement of tritium. The required replacement time is minimized by open plasma-chamber access.

Open access is not only important for rapid component replacement in power plants but also for testing different designs for in-vessel components during fusion development. These components are a major challenge to fusion.

Two concepts for open access to the plasma chamber are: (1) large spaces between coils and (2) non-plasma encircling coils that can be removed along with a large section of the chamber wall Boozer:2009 , Figure (1).

Large spaces between the coils are possible with helical coils Yamaguchi and are advantageous for heating and diagnostic ports as well as for in-vessel changes. Ripple in the magnetic field strength places difficult constraints on the space between modular coils unless mitigated by coils mounted on removable wall sections.

The use of non-plasma encircling coils mounted on removable wall sections have few fundamental constraints. Coils that can be removed along with the part of the vessel with which they are associated can be represented by current potential patches, which are patches of magnetic dipoles Elder-Boozer:2024 . When a section of the vessel is removed for plasma-chamber access, it must be placed back into position with a high accuracy to obtain a vacuum seal—probably with more accuracy than is required for the placement of the coils.

Computational studies of what is and what is not possible in coil designs with large removable wall sections are needed. Such studies are themselves clearly feasible, and would be a basis for reliable designs for stellarators with open maintenance-access to the plasma chamber.

The thickness of tritium-breeding blankets and neutron shields sets a fundamental limit on the separation between the plasma and the coils. The traditional estimate is approximately 1.5 m. Whatever this scale may be, it sets a minimum spatial scale for a deuterium-tritium (DT) fusion power plant. The larger the coil-plasma separation that can be consistent with feasible coils, the more freedom is available for designing DT power plants having a smaller power and cost per plant.

Maximization of coil-plasma separation requires a simultaneous optimization of three quantities in the choice of coils.

The first is a measure of the desirable physics properties of the magnetic configuration, such as acceptable neoclassical transport. This can be defined by the shape of a reference surface, an outer magnetic surface.

The second is a measure of how efficiently distant coils can support a magnetic surface of that shape. For curl-free magnetic configurations, external coils must produce a field normal to the reference surface $b(\theta,\varphi)=-(\mu_{0}G_{0}/2\pi R)\hat{\varphi}\cdot\hat{n}$, where $\hat{n}$ is the unit normal to the reference surface. Coil efficiency drops exponentially with separation Boozer:RMP . The coefficient in the exponential is approximately given by the wavenumber of the variation of $b$ in the reference surface Landreman:field-gradient . This result follows from a two-dimensional model; $\nabla^{2}\phi=0$ implies $\phi$ has the typical dependence $\sin(k_{y}y)\exp(-k_{x}x)$ with $k_{x}=k_{y}$. The exponential drop in efficiency implies that crude measures, such as the strength of the dipoles required to produce the field, $\vec{d}^{\dagger}\cdot\vec{d}$ of Appendix A, provide a useful measure.

The third is a measure of how the efficiency of the coils represented by dipoles is reduced by enforcing open access requirements. Chamber access and coil-plasma separation interact in determining where the patches of dipoles should be located.

There are two types of external perturbations: (1) ones that are consistent with the periodicity of the optimized configuration and (2) those that are not. The first type are considered changes in the configuration while the second type are considered to be error fields, the effect of construction inaccuracies. The orders of magnitude differences in configuration sensitivity to changes in different external magnetic field distributions Ku-Boozer:2010 ; Boozer-Ku:control2011 should be recognized in stellarator coil design. Designing coils by minimizing $\vec{B}\cdot\hat{n}$ on a given surface ignores these differences.

Any external magnetic-field distribution to which the plasma is sensitive should be controllable by adjustments to the coil currents Boozer:design . These field distributions could be determined from the Hessian matrix of a plasma optimization.

Unfortunately, there is a mathematical limit on the number of currents that can be adjusted with sufficient accuracy to control physics properties. This limit is approximately given by the condition number of a matrix, Section III.

The orders of magnitude difference in sensitivity to different error-field distributions implies coil systems can be easily designed to greatly reduce the required accuracy of construction. This was explained at a 2010 ITER workshop Boozer:tok-error2011 but ignored in the design of the ITER error-field correction coils.

The calculation of control or error field sensitivities has been largely ignored in stellarator-design studies as well as in tokamak studies. Nevertheless, it is difficult to grasp how a coil system can be considered to be optimized without such a study.

Once a desirable stellarator plasma has been found, a smooth surface can be defined that is on the plasma side of the coils but is otherwise as far from the plasma as possible. Consider the effect of an arbitrary Fourier decomposed current potential on this surface, $\kappa(\theta,\varphi)$. Using a stellarator optimization code, the collection of Fourier amplitudes, with $\oint\kappa^{2}d\theta d\varphi$ fixed, can be determined that maximally degrades the optimization. This defines current potential, $\kappa_{1}$, that gives the worst possible external field error and frequently has an $n=1$ toroidal Fourier dependence. The second worst external field error can be defined by determining the $\kappa_{2}$ with $\oint\kappa_{2}\kappa_{1}d\theta d\varphi=0$ by the same method as $\kappa_{1}$. That is, all the dangerous field errors can be determined in a series ordered by the plasma sensitivity to them.

There are two strategies for reducing the precision requirements of coil construction. First, the plasma optimization can be used to reduce the plasma sensitivity to the worst magnetic-field error, which is defined by $\kappa_{1}$. This efficiently accomplishes what is accomplished by a stochastic optimization Stochastic optimization .

Secondly, error-field correction coils can be installed that can independently drive the various magnetic field distributions to which the plasma is sensitive. These coils can make the total external field in distributions to which the plasma is sensitive zero. External field errors are not cancelled. The external drive just makes the externally-produced magnetic field distributions to which the plasma is sensitive zero. Field errors to which the plasma is not sensitive are driven in this process, so there is maximum number of error field distributions that can be controlled in this way. Fortunately, the plasma sensitivity appears to drop exponentially, so only a few error field distributions need to be controlled to greatly reduce the precision requirements on the coils.

As discussed in Section II.3, plasma sensitivity to small changes in the external magnetic field defines external field distributions that should be controllable in order to have experimental flexibility. However, far more radical changes of the plasma configuration are possible—for example changing the plasma shape from a quasi-isodynamic to a quasi-helically symmetric equilibrium. A fundamental limit on radical changes is set by the plasma not intercepting non-replaceable parts of the chamber walls. More subtle limits are set by the locations of divertors and diagnostics.

Todd Elder’s result that approximately 25% coverage of an enclosing surface by current potential patches allows the accurate definition of a quasi-helically symmetric equilibrium suggests two strategies: (1) Define current-potential patches that are consistent with major changes in the configuration. (2) When the current-potential patches are on removable sections of the chamber walls, one set of wall inserts carrying a current-potential patch could be replaced by another. The first option sounds preferable, but changes in the surrounding walls may be sufficiently great that the second option is required.

One such goal in the design and control of plasma divertors. An important computational study would be what possibilities these remaining degrees of freedom allow for divertors.

There are two basic types of stellarator divertors: (1) the island divertor as on W7-X W7-X:divertor and (2) the non-resonant divertor. The presence of a strong bootstrap current is thought to make island divertors, the type of divertor used on W-7X, impractical. This may be less true in a power plant than in an experimental facility. A quasi-steady-state power plant could operate in a single plasma state with a known edge transform. Both the edge transform and surface shape can be controlled by external coils during startup. The rotational transform can be controlled by poloidal field coils forming a transformer that allows a controlled change in poloidal flux in the central hole of the torus.

Chaos is unimportant for the magnetic field lines in the W7-X island-divertor but is essential for understanding non-resonant divertors. Magnetic field lines are chaotic when within a region of non-zero volume each field line has lines that are infinitesimally separated from it that have a separation that increases exponentially with distance along the lines.

The dictionary definition of chaos as complete disorder is misleading in the application of the concept to non-resonant divertors. Chaotic magnetic field lines can lie in well collimated flux tubes that extend from the plasma edge to the locations on the surrounding walls. These tubes define a non-resonant divertor, and the locations where they strike the walls are remarkably robust to changes in the plasma configuration Bader:2023 .

Non-resonant divertors appear naturally in optimized stellarators due to locations of sharp curvature of the magnetic surfaces near the plasma edge, Bader et al Bader:2018 . Due to $\vec{\nabla}\cdot\vec{B}=0$, these are also the locations at which the separation between neighboring magnetic surfaces becomes large. Boozer and Punjabi Boozer-Punjabi:2018 used these features to derive an analytic magnetic-field-line Hamiltonian, the poloidal magnetic flux as a function to the toroidal magnetic flux and two angles, $\psi_{p}(\psi,\theta,\varphi)$. They have used this Hamiltonian in studies of topological features of non-resonant divertors, such as Cantori, turnstiles, and magnetic field line chaos.

An important physics issue for non-resonant stellarator divertors is the electric field that is required for quasi-neutrality and the associated transport in chaotic magnetic field regions Boozer:divertor . Unlike the coil issues in curl-free stellarator models, the way to proceed in addressing this issue is far from certain.

The magnetic surfaces of the well-confined plasma are separated from the sea of chaotic field lines that surrounds it by Cantori, which are like irrational magnetic surfaces that have localized holes through which magnetic field lines can pass, with equal fluxes in both directions, and reach the chamber walls. These holes are caused by nearby unstable fixed points—field lines that exactly close on themselves after making a number of transits of the torus but from which neighboring lines exponentiate. In helical symmetry, these would be the X-points of islands, but in the absence of symmetry, the field lines in the neighborhood of an X-point become chaotic.

Using a map that models a non-resonant divertor, Boozer and Punjabi Boozer-Punjabi:2018 found that the probability that a field line will be lost from a Cantorus in one toroidal transit has the form

where $\psi$ is the toroidal flux enclosed by the Cantorus, $\psi_{o}$ is the flux enclosed by the outermost magnetic surface, $w_{\ell}$ is a constant, and $d\approx 2$. But, the outward and inward flux tubes through a Cantorus need not be adjacent. Adjacent versus non adjacent turnstiles can have different $d$’s Punjabi:2022 .

The number of toroidal transits required for a field line to be lost can be written as

The number of toroidal transits that would be required for plasma to diffuse to the Cantorus at which a field line is lost is $N_{\ell}(\psi_{\ell})=(\psi_{\ell}-\psi_{o})^{2}/D_{p}$. Equating the $N_{\ell}$ from this equation with that from Equation (5) gives

$P_{\ell}$ gives the size of each of the two escaping flux tubes that carries plasma to the walls. Each has a magnetic flux is $\delta\psi=\psi_{\ell}/N_{\ell}$.

In a non-resonant divertor the plasma flows to the wall along one or more pairs of flux tubes, which appear to be very narrow. The compactness of the flux tubes that carry the plasma to the divertor may seem surprising. The same narrowness is important in the runaway electron problem in post-disruption tokamaks. Relativistic electrons from a central region of chaotic field lines exit the plasma along narrow flux tubes when the field evolves slowly compared to the time required for a relativistic electron to cover the chaotic region but exits the plasma over a broad region when the evolution is fast runaway:2016 . Both effects are seen in tokamak experiments.

A way to study the collimation of the narrow flux tubes carrying plasma to the divertor is to cover the surrounding chamber wall with a grid. In almost all grid cells, the normal magnetic field to the wall will be non-zero, and the field lines can be followed from the location where they enter the chamber to the place where they exit after $N_{t}$ toroidal transits. Equation (8) says the probability of a field line staying in as $N_{t}\rightarrow\infty$ is zero. Nonetheless there can be pairs of points on the chamber wall where field lines make $N_{t}\rightarrow\infty$ transits. These lines with $N_{t}\rightarrow\infty$ come extremely close to a magnetic surface, which can be the outermost magnetic surface of the confining plasma and are the lines to be used in divertors. Another possibility is that lines with $N_{t}\rightarrow\infty$ come extremely close to the outermost surface of a magnetic island that lies outside the plasma.

In a stellarator the helical field $B_{h}$ is much stronger than the net poloidal field $B_{\iota}\propto(B_{h}/B)^{2}B$ that determines the rotational transform Spitzer:1958 . In an unpublished study, Alkesh Punjabi finds that in most grid cells field lines quickly exit, $N_{t}\sim 1$. Although the magnetic field lines are in principle chaotic in the region outside those bounded by magnetic surfaces, once a magnetic field line escapes from the Cantori that surround these surfaces, the line strikes the wall within a few toroidal transits, so the exponential separation of neighboring lines does not occur over a sufficient length to destroy the compactness of the exiting flux tubes.

Understanding how the location of the strike points and the width of the diverted flux tubes are controlled by the freedom in the external coils remains to be established. Studies of curl-free stellarator configurations would clarify how the location of the $N_{t}\rightarrow\infty$ points and the widths the holes in the Cantori, the $w_{\ell}$, are determined. The control of both is very important for the design of attractive non-resonant divertors. Although the location of the $N_{t}\rightarrow\infty$ points is remakably robust, baffles to keep neutrals close to divertor pumps can be better designed the more accurately the location of these points can be defined and the behavior of field lines near these points can be determined, which is related to $w_{\ell}$.

A stellarator configuration Boozer:zoned-stell , which is illustrated in Figure (2), has been suggested as way to achieve three goals: (1) Enhance the confinement time of tritium. (2) Have a sufficient density of high-Z impurities to radiate the thermal power escaping from the core as is needed for an acceptable divertor solution while having an extremely low impurity density in the core. (3) Maintain a large fraction of the plasma in a burning plasma state with an optimal tritium fraction.

When the tritium density equals that of deuterium, the fraction of the tritium that is burnt in an energy confinement time is a fraction of a percent. This follows from the enormous ratio of the energy of the fusion-produced alpha particles to the temperature of the plasma. The tritium must make many passes through the plasma before it is burnt. This worsens a major issue for fusion power, the achievement of tritium self-sufficiency of deuterium-tritium power plants tritium ; tritium2 . Concepts for increasing the number of tritiums produced per neutron would have a major impact on the feasibility of fusion.

There are two ways to increase the tritium burn fraction: (1) lower the tritium fraction in a deuterium-tritium plasma, at the cost of requiring better energy confinement to achieve a self-sustained burn or (2) increase the confinement time of the tritium. Both are discussed in Boozer:zoned-stell , but the zoned concept illustrated in Figure (2) is designed to increase the tritium confinement time.

In standard stellarator reactor designs, the helium and high-Z impurities are flushed out of the plasma by a 50-50 deuterium-tritium mixture. Tritium is many orders of magnitude more expensive than deuterium. The zoned concept tries to avoid the bizarre use of tritium as a flushing fluid.

Before the attractiveness of the zoned concept can be assessed two physics issues must be better understood: (1) the physics of the interdiffusion of different species and (2) the localization and the depth of plasma penetration by pellets.

The consistency of various features of plasma configurations is important to study. An example, is it possible to have a large shear in the rotational transform in a curl-free configuration with small neoclassical transport? Another is the tendency of optimized stellarators to have outer surfaces with high surface curvature. These regions are important for the formation of non-resonant divertors.

Reference Boozer:omnigeneity discussed optimal ion and electron temperatures and their control together with the generality of gyroBohm scaling. These issues have significant impacts on stellarator design and are active areas of research, but the level of certainty that can be achieved is far below that in coil design for curl-free stellarator configurations.

A related issue that should be studied is whether stellarators can be designed so Alfvénic modes can be driven by the energetic alpha particles but only in the central region. Central Alfvénic modes do not directly cause an enhancement of the energetic alpha losses to the walls. To the extent these alpha-driven Alfvén waves are damped by the viscosity associated with background deuterium and tritium ions, direct ion heating is achieved. Normally, the alpha energy is transferred to the electrons, which indirectly transfer it to the ions. This indirect method fails to heat ions when the electron-ion equilibration time is long compared to the ion energy confinement time, which can be the case when the electrons become too hot Boozer:omnigeneity .