Three-dimensional equilibria in axially symmetric tokamaks

Garabedian, P. R.

doi:10.1073/pnas.0609569103

Cited by 16 publications

(18 citation statements)

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“…An axisymmetric equilibrium is computed followed by a second minimization for stability with a linear constraint imposed. A nonlinearly stable solution is finally obtained which differs from the original (axisymmetric in the case of tokamak studies) equilibrium [21,22]. Solutions were found for tokamaks and stellarators that correspond to low order ballooning-type modes that do not alter the original magnetic axis position.…”

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confidence: 96%

Tokamak Magnetohydrodynamic Equilibrium States with Axisymmetric Boundary and a 3D Helical Core

et al. 2010

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Magnetohydrodynamic (MHD) equilibrium states with imposed axisymmetric boundary are computed in which a spontaneous bifurcation develops to produce an internal three-dimensional (3D) configuration with a helical structure in addition to the standard axisymmetric system. Equilibrium states with similar MHD energy levels are shown to develop very different geometric structures. The helical equilibrium states resemble saturated internal kink mode structures. The essential confinement of particles and energy in magnetically enclosed plasmas is described by magnetohydrodynamics (MHD). The tokamak is the leading fusion energy research concept in which the plasma is expected to be contained in an essentially axisymmetric configuration with relatively small ripple effects from the finite toroidal coils. Symmetry of the equilibrium state along the toroidal coordinate grants the tokamak many appealing properties such as toroidal mass flow, which is known to reduce MHD instability and reduce non-MHD transport of particles and energy, as well as important conservation properties of single particles, thus enhancing confinement on various scales.It is shown in this Letter that despite imposing axisymmetry (toroidal symmetry) at the edge of the plasma, enforced by the geometry of the coil system, the preferred lowest energy state of MHD equilibrium can be nonaxisymmetric in the plasma center. The computation of threedimensional (3D) helical cores constitutes a paradigm shift for the description of tokamak equilibria that opens the way for the application of theoretical and simulation tools developed for stellarator analysis of MHD stability, guiding center particle orbits, kinetic stability, wave propagation or heating, neoclassical transport, gyrokinetics, etc., to determine the impact of these novel 3D states on a large range of magnetic confinement physics phenomena.The ignorable toroidal angle coordinate in axisymmetric magnetic confinement systems allows a simplified description of the MHD equilibrium state through the GradShafranov equation [1,2]. A more sophisticated approach invokes the minimization of the MHD energy to achieve an equilibrium state which can be naturally extended to model 3D systems with the imposition of nested magnetic flux surfaces and a single magnetic axis [3][4][5][6][7][8][9]. Tokamak devices, though nominally axisymmetric, display internal plasma reorganization phenomena that can break the symmetry of the system. In the tokamak à configuration variable (TCV) [10], a transition is observed where core sawteeth relaxations are replaced by global oscillations with low poloidal and toroidal mode numbers [11,12]. In these discharges, the inverse rotational transform q profiles are nearly flat or slightly reversed. One possible explanation for this transition is that q min , the minimum value of safety factor q within the plasma, becomes greater than unity. Equilibria with q min near unity are of interest in the present contribution. Similarly saturated ideal modes in the MAST device have also been re...

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confidence: 96%

Tokamak Magnetohydrodynamic Equilibrium States with Axisymmetric Boundary and a 3D Helical Core

et al. 2010

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“…The method has been applied to the Doublet III-D tokamak at General Atomic and to the Large Helical Device (LHD) stellarator in Japan (9)(10)(11)(12). Comparison with observations is good in both cases.…”

Section: Computational Science Of Magnetic Fusionmentioning

confidence: 98%

Three-dimensional analysis of tokamaks and stellarators

Garabedian

2008

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

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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 ...

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“…A computational theory of equilibrium and nonlinear stability, combined with experimental observations, provides convincing evidence that QAS stellarators are safely stable and not subject to the crashes seen in tokamaks (6). Monte Carlo calculations of thermal transport that take into account quasineutrality suggest that confinement in QAS stellarators may be comparable to that in existing tokamaks, but a test is needed to show that these configurations can achieve the high ion temperatures required for ignition in a fusion plasma (9).…”

Section: Design Of Qas Stellaratorsmentioning

confidence: 99%

“…The numerical method we have described has been applied to design a compact stellarator with two field periods as a candidate for a fusion reactor (6,9). We have found 12 smooth coils that generate the external magnetic field (compare Fig.…”

Section: Design Of Qas Stellaratorsmentioning

confidence: 99%

“…That process is facilitated by a correlation between the shape factors ⌬ mn b and corresponding Fourier coefficients B mn , called the spectrum, in an expansion of the magnetic field strength in terms of invariant poloidal and toroidal angles. Good confinement is achieved when the magnetic spectrum has quasiaxial symmetry (QAS) so the column B m0 dominates, or has quasihelical symmetry so the diagonal B mm dominates (6). After the shape of the plasma has been found, one must determine positions of coils on an outer control surface S defined by an expression: r ϩ iz ϭ e iu ⌬ mn c e Ϫimuϩinv like the one for the free boundary.…”

Section: Locating the Coilsmentioning

confidence: 99%

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Choice of coils for a fusion reactor

Alexander¹,

Garabedian²

2007

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

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In a fusion reactor a hot plasma of deuterium and tritium is confined by a strong magnetic field to produce helium ions and release energetic neutrons. The 3D geometry of a stellarator provides configurations for such a device that reduce net toroidal current that might lead to disruptions. We construct smooth coils generating an external magnetic field designed to prevent the plasma from deteriorating.Biot-Savart law ͉ magnetic fusion ͉ magnetohydrodynamics ͉ plasma physics ͉ stellarator A pressing issue of our time is to develop clean and efficient energy sources. One proposed solution of the problem implements the concept of nuclear fusion of hydrogen to form helium, which does not leave radioactive wastes that are as permanent as those involved in fission (1). Therefore, it is important that the possibility of building nuclear fusion power plants be explored by both theory and experiment. The suggested solution is to contain a very hot deuterium and tritium plasma magnetically in a vacuum so that it does not come near solid walls. In the most favored design, the tokamak, the plasma is confined in the shape of an axially symmetric torus. However, without a poloidal magnetic field induced by net current in the plasma itself, charged particles tend to drift out toroidally and leave the machine.The stellarator concept avoids generating a net current by changing the shape of the plasma to an asymmetrical torus with fully 3D geometry (2). Several projects are underway to build experiments based on proposed configurations. The most ambitious is the International Thermonuclear Experimental Reactor, a large tokamak planned for construction in Cadarache, France. Important stellarator experiments include the Wendelstein 7X under construction in Germany, the Large Helical Device in Japan, and the Helically Symmetric Experiment (HSX) at the University of Wisconsin (3-5). We are concerned with mathematical problems involved in the stellarator model. After the magnetic field inside the boundary of a physically desirable plasma has been optimized, we determine the number, shape, and position of coils that are required to generate the external field. Fig. 1 displays a plot of the plasma and coils as they might look in such a reactor. Mathematical ModelWe are primarily concerned with the magnetohydrodynamic model of plasma physics. In particular we have ƒ ⅐ B ϭ 0, where B is the magnetic field. The requirement of force balance put on the plasma follows from the simplified form:of the magnetohydrodynamic variational principle, where p is the fluid pressure and the gas constant is zero (2). In a stable system the energy throughout the volume of plasma must be a minimum. A mathematical consequence of this principle asserts that the Lorentz force must balance the pressure, so that:where T is the Maxwell stress tensorThe divergence theorem can be applied to the equations over any test volume of the plasma because we have put them in a conservation form with each term expressed as an exact partial derivative. It follows that the no...

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Three-dimensional equilibria in axially symmetric tokamaks

Cited by 16 publications

References 19 publications

Tokamak Magnetohydrodynamic Equilibrium States with Axisymmetric Boundary and a 3D Helical Core

Tokamak Magnetohydrodynamic Equilibrium States with Axisymmetric Boundary and a 3D Helical Core

Three-dimensional analysis of tokamaks and stellarators

Choice of coils for a fusion reactor

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