Computational mathematics and physics of fusion reactors

Proc. Natl. Acad. Sci. U.S.A.

2008

Self Cite

The NSTAB equilibrium and stability code and the TRAN Monte Carlo transport code furnish a simple but effective numerical simulation of essential features of present tokamak and stellarator experiments. When the mesh size is comparable to the island width, an accurate radial difference scheme in conservation form captures magnetic islands successfully despite a nested surface hypothesis imposed by the mathematics. Three-dimensional asymmetries in bifurcated numerical solutions of the axially symmetric tokamak problem are relevant to the observation of unstable neoclassical tearing modes and edge localized modes in experiments. Islands in compact stellarators with quasiaxial symmetry are easier to control, so these configurations will become good candidates for magnetic fusion if difficulties with safety and stability are encountered in the International Thermonuclear Experimental Reactor (ITER) project. magnetic fusion ͉ numerical methods ͉ plasma physics M agnetic fusion reactors confine a plasma of electrons and of deuterium and tritium ions in a strong magnetic field so at high temperatures the ions can fuse to form helium and release neutrons as a practical source of energy. The geometry of such devices is usually toroidal to allow the hot particles to circulate with a mean free path exceeding by far the size of the reactor. The most thoroughly studied configurations are tokamaks, which are axially symmetric and require a destabilizing net current in the plasma to prevent charged particles from drifting out toroidally. There is an alternate concept called the stellarator, which uses fully three-dimensional (3D) coils to create a poloidal field eliminating the toroidal drift. Our interest lies in magnetohydrodynamic (MHD) partial differential equations governing equilibrium and stability of the plasma, and in guiding center orbits of the ions and the electrons that lead to numerical simulations of transport. The method we apply to this problem in computational science is to run the NSTAB equilibrium and stability code and the TRAN Monte Carlo code written by Octavio Betancourt and Mark Taylor (1-4). We calculate physically realistic predictions of limits on the average ratio ␤ ϭ 2p/B 2 of the fluid and magnetic pressures and reliable estimates of the energy confinement time, which is a measure of pressure times volume divided by power in magnetic fusion experiments. Computational Science of Magnetic FusionUsing the Maxwell stress tensor, we write the MHD equations for equilibrium and stability of a plasma in the conservation formwhere B is the magnetic field and p is the fluid pressure. To obtain an accurate numerical approximation of force balance across discontinuities like islands or current sheets, analogous difference equations are represented in the NSTAB code by a similar scheme that telescopes down through addition over grid points to an integral statement of equilibrium related to the divergence theorem. Elementary examples from shock capturing show that ignoring conservation form results in failure ...

Section: An Optimized Qas Stellaratormentioning

confidence: 99%

“…Some remarkable results of the calculation are shown in Figs. 2 and 3, and more detailed information has been documented elsewhere (5,17).…”

Section: An Optimized Qas Stellaratormentioning

confidence: 99%

Three-dimensional analysis of tokamaks and stellarators

Proc. Natl. Acad. Sci. U.S.A.

2008

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“…The ␤ 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 ␣-particles is not unlike that in a comparable tokamak.…”

Section: Nonlinear Stability Of Quasiaxially Symmetric (Qas) Stellaramentioning

confidence: 99%

Three-dimensional equilibria in axially symmetric tokamaks

Proc. Natl. Acad. Sci. U.S.A.

2006

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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 ␣ particles or to crashes and disruptions that are observed. The 3D theory predicts good performance for stellarators.magnetic fusion ͉ magnetohydrodynamics ͉ plasma physics ͉ stellarator ͉ weak solution 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 ␤ ϭ 2p/B 2 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 changes sign so that a sizeable island appears where it vanishes, but the computations are sensitive to details about the profile of . Computational Analysis3D computer codes are an accepted tool for the study of equilibrium and stability in plasma physics. Good resolution is achiev...

“…β tend to be linearly unstable, but nonlinearly stable, so that a better understanding of bifurcated solutions becomes desirable [11].…”

Section: Computation Of Force Balancementioning

confidence: 99%

“…The NSTAB code captures islands successfully despite a nested surface hypothesis made in the coordinate system that is employed [11]. The resolution of the code can be checked by applying it to the vacuum field of stellarators where islands are known to exist in equilibria found by line tracing [14]. Figures 3 and 4 display calculations of an example of this phenomenon in which the rotational transform changes sign so that a sizeable island appears at ι = 0.…”

Section: Magnetic Islandsmentioning

confidence: 99%

Bifurcated equilibria and magnetic islands in tokamaks and stellarators

2006

CAMCoS

The magnetohydrodynamic variational principle is employed to calculate equilibrium and stability of toroidal plasmas without two-dimensional symmetry. Differential equations are solved in a conservation form that describes force balance correctly across islands that are treated as discontinuities. The method is applied to both stellarators and tokamaks, and comparison with observations is favorable in both cases. Sometimes the solution of the equations turns out not to be unique, and there exist bifurcated equilibria that are nonlinearly stable when other theories predict linear instability. The calculations are consistent with recent measurements of high values of the pressure in stellarators. For tokamaks we compute three-dimensionally asymmetric solutions that are subject to axially symmetric boundary conditions. MSC2000: 82D10, 70K50, 82C40.