Three-dimensional equilibria in axially symmetric tokamaks

Abstract

The NSTAB and TRAN computer codes have been developed to study equilibrium, stability, and transport in fusion plasmas with three-dimensional (3D) geometry. The numerical method that is applied calculates islands in tokamaks like the Doublet III-D at General Atomic and the International Thermonuclear Experimental Reactor. When bifurcated 3D solutions are used in Monte Carlo computations of the energy confinement time, a realistic simulation of transport is obtained. The significance of finding many 3D magnetohydrodynamic equilibria in axially symmetric tokamaks needs attention because their cumulative effect may contribute to the prompt loss of α particles or to crashes and disruptions that are observed. The 3D theory predicts good performance for stellarators.

Formulation of the Problem

Computational science has led to significant progress in the theory of equilibrium, stability, and transport for fusion plasmas in three dimensions (1, 2). This has made it possible to design advanced tokamaks with net current producing a poloidal field, or stellarators with twisted coils generating rotational transform, as candidates for a fusion reactor. The research we describe is based on the NSTAB and TRAN computer codes (3–5). They enable us to calculate transport in tokamaks by using bifurcated equilibria that do not have two-dimensional (2D) symmetry. For both tokamaks and stellarators, correct solutions of the relevant differential equations have discontinuities associated with islands and current sheets.

We apply the magnetohydrodynamic (MHD) variational principle to calculate equilibrium and stability of toroidal plasmas in three dimensions. Partial differential equations are solved in a conservation form that describes force balance correctly across islands that are treated as discontinuities. The method has been applied to the Doublet III-D (DIII-D) tokamak and also to the Large Helical Device (LHD) stellarator in Japan. Comparison with observations is favorable in both cases (6, 7). Sometimes the solution of the equations turns out not to be unique, and there may exist bifurcated equilibria that are nonlinearly stable when other theories predict linear instability. The calculations are consistent with high values of the plasma parameter β = 2p/B² observed in the LHD.

The NSTAB code captures islands correctly despite a nested surface hypothesis made in the numerical algorithm that is used. The reliability of the code has been established by using it to study stellarators without pressure where islands are known to exist in equilibria found from potential theory (8). We have made convincing calculations of this phenomenon in which the rotational transform ι changes sign so that a sizeable island appears where it vanishes, but the computations are sensitive to details about the profile of ι.

Computational Analysis

3D computer codes are an accepted tool for the study of equilibrium and stability in plasma physics. Good resolution is achieved by implementing the spectral method in flux coordinates describing toroidal geometry. Because the LHD experiment exceeded theoretical β limits, more accurate predictions from runs of the NSTAB code have been considered in the ARIES-CS power plant study (9). Compact stellarators with either two or three field periods have been found that have desirable properties for a reactor (10). The coils for two periods are less twisted because their distance from the separatrix is smaller in units of the plasma radius. Transport of ions and α-particles is similar to that in a comparable tokamak.

Our 3D methods to analyze equilibrium, stability, and transport in quasisymmetric stellarators have been applied to advanced tokamaks like the International Thermonuclear Experimental Reactor (ITER). We have computed bifurcated solutions in axially symmetric tokamak configurations and performed convergence studies to examine the relationship between numerical approximations with 3D asymmetries and the more conventional theory that assumes uniqueness of the 2D solution (cf. Fig. 1). The physical relevance of the bifurcated equilibria has been examined to test a contention that transport and the prompt loss of α-particles depend in a significant way on 3D terms in the magnetic spectrum. There is agreement between estimates of thermal transport from runs of the TRAN code and experimental observations in both tokamaks and stellarators (11). An algorithm determining the electric potential from quasineutrality in 3D equilibria is applied to calculate the energy confinement time at low collision frequencies.

Fig. 1.

Poincaré section of the flux surfaces of a bifurcated ITER equilibrium at β = 0.02 with p = p₀(1 − s^1.1)^1.1 and with net current bringing the rotational transform into the interval 0.9 > ι > 0.4. There are helical islands in this fully converged 3D solution of an axially symmetric MHD problem.

Conservation Form of Force Balance

Predictions of ballooning instability seem to be less reliable than estimates for modes of relatively low order. Therefore, we look for mathematical difficulties with the numerical methods that have been applied. Comparable problems in fluid dynamics are treated by finite difference schemes that capture shockwaves and contact discontinuities by putting the differential equations of motion in conservation form (12). We illustrate such a theory with a 1D example that models a reversed field pinch (RFP) in slab geometry and is helpful in understanding our numerical method (10).

In NSTAB we implement a continuum model of plasma physics that is defined by differential equations for the magnetic field B, the current density J, and the fluid pressure p. The Maxwell stress tensor

enables one to put the MHD equations

describing force balance in the conservation form

Analogous finite difference equations have the advantage that when they are summed over a test volume, they telescope into an approximate statement of force balance

over the boundary that is globally correct. Numerical calculations of weak solutions that capture discontinuities this way at resonant surfaces in plasma equilibria serve to simulate significant aspects of the physics that are hard to analyze by more refined methods.

The theory can be justified by considering the simplified example of the Burgers equation

subject to three boundary conditions Ψ(−1) = Ψ(1) = 0, Ψ_x(−1) = 1. This models an RFP depending on only one coordinate x, where Ψ is the flux, Ψ_x is the principal component of the magnetic field, Ψ_xx is the current, and η is an artificial resistivity. The conservative difference scheme

for the Burgers equation computes jumps across discontinuities correctly and therefore imposes force balance across a sharp boundary, which occurs at x = 0 in the limiting case where η = 0 so that the solution reduces to Ψ = 1 − |x|. However, a finite difference scheme for the RFP problem that is not in conservation form defines unequal slopes at opposite ends of the solution characterizing a jump in the magnetic field such that force balance fails. Entropy inequalities play an important role in applications of this algorithm to fluid dynamics, but for the plasma physics problem the computations are dominated by a variational principle.

Bifurcated Tokamak Equilibria

The most successful computer codes for the study of equilibrium and stability are based on the MHD variational principle

The variational method leads in a natural way to a representation

of the magnetic field in terms of the toroidal flux s, another flux function θ, and two Clebsch potentials, φ and ζ. If there is no net current and ι stands for the rotational transform, the variables θ + ιφ and φ can be renormalized to become invariant poloidal and toroidal angles on each flux surface s = const., where s is the radial coordinate. We refer to the Fourier coefficients B_mn in the resulting expansion

as the magnetic field spectrum. In this notation it turns out that the value of B_mn can largely be controlled by varying the corresponding coefficient Δ_mn in a suitable formula for the shape of the plasma. Let r, v/Q, and z be cylindrical coordinates about the major axis of our toroidal configuration, where Q stands for the number of field periods. We fix the outer boundary of the plasma at s = 1 and represent it in complex notation by a Fourier series,

where the real coefficients Δ_mn are uniquely determined by the geometry if the poloidal index m is restricted to nonnegative values.

By manipulating the magnetostatic equations, it can be shown that for stellarators there is an expansion

for the parallel current. In three dimensions the small denominators n − ιm are seen to vanish at a dense set of rational surfaces where ι = n/m. This is why smooth solutions of the equilibrium problem that have 3D asymmetry do not in general exist, so that one should only try to construct weak solutions that may not be differentiable. An application of the divergence theorem establishes that any smooth surface of discontinuity of a weak solution must be a current sheet in the sense that it is a flux surface of the magnetic field across which B may have jumps, but B²/2 + p remains continuous. It is our contention that small magnetic islands can be modeled computationally by such contact discontinuities when an adequate finite difference scheme is applied. These current sheets have features in common with transport barriers.

The NSTAB code has been developed to simulate 3D equilibrium and stability in tokamaks and stellarators. Bifurcated solutions of the tokamak problem suggest that 3D effects may be more important than has generally been thought. The most remarkable examples of bifurcated equilibria are 3D asymmetric solutions of the tokamak problem subject to axially symmetric boundary conditions such that a 2D solution exists, too. Weak solutions for axially symmetric tokamaks model neoclassical tearing modes (NTMs) and edge localized modes (ELMs) that have been observed in the DIII-D experiment at General Atomic with the net current limited so that ι < 1.

3D equilibria are calculated by at first imposing and later releasing a suitable constraint in runs of the NSTAB code chosen to find bifurcated solutions that cannot be obtained without introducing discontinuous alterations in the topology of the magnetic surfaces. On relatively crude radial grids, the computations are capable of capturing small islands whose widths are of the same order of magnitude as the mesh size. Fig. 2 shows the decay of the residuals and the growth of the most dangerous mode in an NSTAB run of this kind. It is seen that the residuals increase while the mode is being computed and only later decrease to a level of round-off error establishing that the discrete problem has been solved numerically.

Fig. 2.

Iterations of the NSTAB code to a bifurcated equilibrium for the ITER tokamak problem with axially symmetric boundary conditions. A forcing term was used early in the run to trigger a mode that appears as a discontinuity in the 3D solution (cf. Fig. 1). This double-precision calculation converges to the level of round-off error, showing that the discrete problem has been solved.

Detailed convergence studies have been made of runs of the NSTAB code that calculate 3D solutions of the tokamak problem specified by 2D boundary conditions. Magnetic islands are easiest to find at resonances of the rotational transform ι that occur halfway between two values at adjacent nodes of the radial mesh. The results are sensitive to the profile of ι as well as to the distribution of mesh points. They depend on the pressure profile, too, but only in an expected fashion that becomes more pronounced as β increases. 3D modes seem to be largely driven by net toroidal current. Sometimes they are linearly stable and do not lead to a bifurcated equilibrium unless they are triggered by a large enough forcing term with the right trigonometric structure. The NSTAB iterations diverge when islands tend to overlap and touch each other, violating the nested surface hypothesis of the code.

In Fig. 3, we display a cross-section of the flux surfaces of a bifurcated equilibrium calculated for the DIII-D at General Atomic by running the NSTAB code to simulate a transport barrier by current sheets and islands. We employ a pressure profile p = (1 − s³)⁶ with β = 0.035 that has a steep gradient near the barrier, where there is an NTM with ι = 4/5. Inside that ι is almost constant, while further out another island representing an ELM appears. The computational method attempts to model observations by 3D perturbations in the numerical solution of an MHD problem defined by 2D specifications, but it fails to explain precisely how to produce transport barriers in an experiment. However, placing the maximum of the pressure gradient at the right resonance is helpful. Only differences in the details of the profiles of ι and p and the formula for the separatrix distinguish between the examples presented in Figs. 1 and 3.

Fig. 3.

Poincaré section of the flux surfaces of a bifurcated DIII-D equilibrium at β = 0.035 with p = p₀(1 − s³)⁶ and with net current bringing the rotational transform into the interval 0.83 > ι > 0.50. There are islands and current sheets representing a transport barrier at ι = 4/5, and more islands appear further out describing an ELM, in this solution of a tokamak problem with four fictitious stellarator field periods.

We have computed a variety of bifurcated equilibria in tokamaks. The Kolmogoroff–Arnold–Moser (KAM) theory displays small denominators at rational surfaces of 3D solutions, and analysis of the continuous spectrum shows that the stability of tokamaks is singular. This is consistent with observations of sawtooth oscillations, NTMs and ELMs, and disruptions. Desirable 3D solutions of the MHD equilibrium equations may not exist, may not be unique, and may not depend continuously on the data. Yet success of the DIII-D and LHD experiments fosters a belief that it is possible to design a magnetic fusion reactor. Perhaps a quasisymmetric stellarator is the answer, because some rotational transform should come from the external magnetic field (13). There is substantial numerical evidence for our contention that a productive approach to these issues lies in the numerical calculation of weak solutions of the MHD partial differential equations.

Nonlinear Stability of Quasiaxially Symmetric (QAS) Stellarators

A good way to assess MHD stability of the toroidal equilibrium of a plasma by the variational method is to introduce a small perturbation in the equilibrium equations and then calculate the corresponding change of the solution. A satisfactory test of nonlinear stability by means of the NSTAB code consists of removing the forcing term after a certain number of cycles, but continuing the iterations to find out whether a bifurcated solution different from the original one can be constructed. Failure of uniqueness sometimes furnishes a plausible proof of instability. The preconditioned accelerated method of steepest descent implemented in NSTAB produces a perturbation of the solution that incorporates much more structure of the unstable mode than had to be input by the displacement. In this sense, the code solves for the most dangerous eigenfunction of linear stability theory without requiring calculation of derivatives that might not exist. The good resolution of the spectral method, combined with excellent accuracy in the radial difference scheme, is what makes the approach a success in practice.

Runs of the NSTAB code show that the LHD stellarator is linearly unstable, but remains nonlinearly stable, at a β of 4.3% achieved experimentally. Predictions of ballooning stability for the LHD are more pessimistic than our estimates from bifurcated solutions calculated over 1, 2, 5, or 10 field periods. In the case of tokamaks, we have made detailed comparisons with observations from the DIII-D experiment, and there is substantial agreement. Examination of 3D bifurcated equilibria that we have calculated confirms that in a standard configuration, the β limit is near 5%, but when shaping coils are used to push the plasma inward and reduce the aspect ratio, then values of β >10% can be attained (14).

Convergence studies have been made to check runs of the NSTAB code used to design QAS stellarators, which are those such that the column B_m0 dominates the spectrum. Fig. 4 displays an example where the residuals of 10⁴ discrete equations have been reduced by 10⁷ iterations of an accelerated method of steepest descent to the level 10⁻¹³ of round-off error in double precision. Eventually, a magnetic island becomes visible at the resonant surface ι = 1/4. The β limit is estimated by performing long, accurate runs to find out whether wall-stabilized ballooning structures appear in the solution. The code has been applied to design a QAS stellarator with two field periods (15). Transport of ions, electrons, and α-particles is not unlike that in a comparable tokamak.

Fig. 4.

Convergence to the level of round-off error in the numerical solution of approximate equations for the MHD equilibrium of a QAS stellarator with two field periods showing that the 3D problem has an exact answer, which has been computed by the NSTAB code. The amplitude of an MHD mode is plotted describing the width of islands that appear in the equilibrium (cf. Fig. 5).

Confinement and Quasineutrality

The Monte Carlo method evaluates confinement time by tracking test particles in the plasma and calculating how long it takes them to leave. Good agreement with observations is achieved by computing the ion and electron confinement times separately and introducing 3D terms to impose quasineutrality by making the results coincide. Our transport computations employ the TRAN Monte Carlo code after applying the NSTAB code to calculate bifurcated equilibria that determine the magnetic field in the plasma. Experimental data have been used to validate the results of numerical computations. At temperatures of 3 keV observed in the LHD experiment, the TRAN code predicts an energy confinement time of 160 ms that agrees with measured values (11).

To introduce quasineutrality as an equation for the electric potential Φ, we drop the highest derivatives in Poisson's equation, defining it in terms of the particle densities n_e and n_i. The theory of singular perturbations suggests that the resulting requirement of quasineutrality must determine the electric field, even though Φ no longer appears explicitly. The electric field is dominated by a radial term, but small 3D terms improve the reliability of our transport calculations.

The algorithm implemented in the TRAN code solves the quasineutrality equation by requiring that Fourier coefficients of the densities n_i and n_e of ions and electrons coincide. The electric potential Φ is expanded as a Fourier series

in the invariant poloidal and toroidal angles θ and φ with coefficients P_mn, measured in units of the temperature, that depend on the flux s. The charge separation is expanded in a similar Fourier series

after dividing out the total number of electrons or ions. Calculation of the Fourier coefficients C_mn is accomplished by using the Monte Carlo method to estimate expected values of suitable trigonometric functions.

Our aim is to determine the coefficients P_mn so that C_mn = 0. It is reasonable to expect wells and hills in the electric potential to cancel similar oscillations in the charge separation n_e − n_i. A natural iteration driving the coefficients C_mn toward zero is

for a suitable choice of the relaxation factor ε. Numerical calculations with this quasineutrality algorithm show that coefficients P_mn of relatively small terms in Φ that depend on the poloidal and toroidal angles θ and φ simulate transport in our model by reducing the electron confinement time τ_e much more than the ion confinement time τ_i.

Runs of the TRAN code have been compared with measurements from tokamak experiments (4). These simulations demonstrate that a 2D model cannot explain observations. The Monte Carlo method takes into account the complex geometry of the drift surfaces, so it works better. If in the calculations, 3D perturbations of Φ and 1/B² are set equal to zero and no iteration of the electric potential is performed, then the confinement times do not come into line with measurements. However, when quasineutrality is successfully imposed to determine the 3D terms, then the numerical answers agree well with observations.

Among many bifurcated equilibria that exist for tokamaks, only those satisfying the condition of quasineutrality are realistic. In the example from Fig. 3 that used bifurcated equilibria to model transport barriers, the quasineutrality algorithm implies that flow of the plasma compatible with a nontrivial electric potential Φ should be encountered in relevant experiments.

Design of Fusion Reactors

We have described applications of the NSTAB and TRAN codes to tokamak and stellarator equilibria with parameters of major experiments. Good correlation of the computations with observations in the DIII-D and the LHD serves to validate predictions about equilibrium, stability, and transport for a QAS stellarator with two field periods. A reactor configuration has been designed that has major radius 7 m and plasma radius 2.8 m. The β limit is 4%, and 12 moderately twisted coils defined by the Biot–Savart law

provide robust magnetic surfaces with islands whose width is small (16). There is access for maintenance through ports between the coils, and the gap between the separatrix and filaments defining the coils is everywhere >140 cm. Theory and practice show that it is only when smooth coils are found that the structure of the field lines outside the plasma becomes well enough organized to allow for the construction of an effective divertor. That can be achieved by employing a judiciously filtered set of harmonics in the implementation of the Biot–Savart law.

Refinement of the numerical method implemented in the NSTAB code has produced a reliable diagnostic for the presence of magnetic islands in both tokamaks and stellarators. When this technique is applied to ITER, it shows that bifurcated equilibria may cause disruptions or degrade confinement. If the damage is significant, one might switch to a similar QAS stellarator as the optimal concept for fusion reactors. In that case, NSTAB computations predict that the effect of islands is negligible provided that the rotational transform avoids the most dangerous resonances (compare Figs. 4 and 5).

Fig. 5.

Cross-sections of flux surfaces over one field period in a QAS stellarator modeling a reactor. Bootstrap current raises ι from 0.44 at the magnetic axis to a peak of 0.51, but lowers it to 0.49 at the edge of the plasma. Small m = 4 islands are visible in the solution that may be less damaging than 3D effects in a bifurcated equilibrium for ITER.

The TRAN code has been modified to find the prompt loss of α-particles in a fusion plasma at reactor conditions, which is the percentage of α-particles that escape from the plasma during one slowing down time. Samples of between 128 and 1,024 particles are adequate to give a meaningful answer. For stellarators, the spectrum plays a decisive role in the computations. Experience shows that only a significant improvement in the quasisymmetry required for satisfactory thermal transport produces a loss of α-particles as low as 10% that might be acceptable in the design of a reactor.

The next step in magnetic fusion research should be an experiment to show that the ion confinement time in a stellarator can be raised to the level necessary for fusion. A further challenge is the heat load on the first wall of the reactor, but that problem might be overcome by introducing a stellarator configuration with larger aspect ratio (17, 18). Because questions remain open at the present stage about the feasibility of a fusion reactor, it is important to keep in mind the advantages available from fully 3D geometry, even if that adds to the cost of electricity. The mathematical theory we have expounded provides computational tools with which to address these fusion energy issues in a timely fashion.

Abbreviations

MHD

magnetohydrodynamic

ELM

edge localized mode

NTM

neoclassical tearing mode

QAS

quasiaxially symmetric

DIII-D

Doublet III-D

LHD

Large Helical Device

ITER

International Thermonuclear Experimental Reactor

RFP

reversed field pinch.