Non-existence of normal tokamak equilibria with negative central current

Hammett, G. W.; Jardin, S.C.; Stratton, B.

doi:10.1063/1.1608935

Cited by 19 publications

(9 citation statements)

References 16 publications

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“…Of course, it is also possible to keep a nested equilibrium invariant under perturbations, but then the values for these must be taken along the bifurcation set only and must obey relations similar to the one in Eq. (9). This recovers the idea of conditioned parameter variation [10], mentioned in the previous brief review.…”

Section: Nested Equilibria As a Measure-zero Setsupporting

confidence: 75%

“…Moreover, differentiating the previous equation with respect to ρ, the derivative d p/dρ is found to vanish at the PFR layer. Taken together, these two facts suggest that only some special hollow pressure profiles are allowed, as confirmed later using a version of the virial theorem [9]. That the pressure profile needs to be hollow is a mere consequence of p(ψ) being assumed to be a smooth function of the non-monotonic poloidal flux, while the restriction imposed onṗ(ψ) = d p/dψ over ρ L does not have any natural justification.…”

Section: A Brief Reviewmentioning

confidence: 93%

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Tokamak equilibria with toroidal current reversal: properties and computational issues

Rodrigues¹,

Bizarro²

2006

AIP Conference Proceedings

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Several properties of axisymmetric plasma equilibria with toroidal-current reversal (TCR) are discussed using some unifying concepts from catastrophe theory. Namely, those of structural stability of functions near critical points, singularity unfolding by small perturbations, and model parameter-space division by bifurcation sets are found to be of particular usefulness. Magnetic configurations displaying, simultaneously, TCR and nested flux surfaces are thence shown to be necessarily degenerate and structurally unstable, meaning that they are easily transformed into non-nested ones by small perturbations in the model parameter set. This should lead to a new paradigm when discussing TCR equilibria, as most of present knowledge relies mainly on the properties of nested solutions, which is expected to favor the study of the broader class of non-nested configurations that recently attracted a considerable discussion in the fusion community. In addition, it is also shown how TCR imposes some constraints on plasma profiles, and how these may be dealt with computationally while keeping the ability to manipulate the shape of the inner island system.

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Section: Nested Equilibria As a Measure-zero Setsupporting

confidence: 75%

Section: A Brief Reviewmentioning

confidence: 93%

Tokamak equilibria with toroidal current reversal: properties and computational issues

Rodrigues¹,

Bizarro²

2006

AIP Conference Proceedings

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“…The conditions above are clearly necessary and sufficient for the GTorIsEq to be a RCEq, and therefore they automatically imply the necessary condition of existence of a RCEq with nested flux surfaces derived in Ref. 2:…”

Section: Reversal Current Equilibriamentioning

confidence: 94%

Analytical examples of reversal current, zero core current, and surface current, toroidal magnetostatic equilibria with nested flux surfaces

Aly¹

2012

Physics of Plasmas

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We present exact analytical examples of three types of axisymmetric toroidal magnetostatic equilibria with nested flux surfaces: (1) current reversal equilibria, for which the net toroidal current switches from a negative to a positive value when moving away from the magnetic axis; these equilibria have a non-monotonic pressure profile, in accordance with Hammett et al.'s theorem stating that the pressure on the current reversal surface has to exceed the volume-averaged pressure within that surface; (2) zero core current equilibria, in which the toroidal current density vanishes inside some flux surface; and (3) surface current equilibria, constituted of an arbitrary number of nested layers inside which the plasma pressure is constant and the magnetic field forcefree, with two adjacent layers being separated by a current sheet. All these configurations are obtained by shaping in an adequate way the arbitrary function which intervenes in the class of generalized isodynamic equilibria first constructed by Palumbo and recovered later on by Bishop and Taylor. A derivation of these equilibria by a method slightly different from Palumbo's is given in an Appendix.

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“…This is likely to be physically not realistic. Eventhough it may have appeared as possible, the actual stability of these configuration is doubtful, see for instance [29][30][31] .In order to generate a profile that could be more physically relevant, we may have again a look at Eq. (10).…”

Section: Cea Irfm F-13108 St Paul-lez-durance Cedex Francementioning

confidence: 99%

Tailoring steep density profile with unstable points

et al. 2019

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The mesoscopic properties of a plasma in a cylindrical magnetic field are investigated from the view point of test-particle dynamics. When the system has enough time and spatial symmetries, a Hamiltonian of a test particle is completely integrable and can be reduced to a single degree of freedom Hamiltonian for each initial state. The reduced Hamiltonian sometimes has unstable fixed points (saddle points) and associated separatrices. To choose among available dynamically compatible equilibrium states of the one particle density function of these systems we use a maximum entropy principle and discuss how the unstable fixed points affect the density profile or a local pressure gradient, and are able to create a steep profile that improves plasma confinement.Being able to sustain a steep density profile in hot magnetized plasma is one of the major key points to achieve magnetically confined fusion devices. These steep profiles are typically associated with the emergence in the plasma of so-called internal transport barriers (ITB) [1, 2]. Both the creation and study of these barriers have generated numerous investigations mostly numerical using either a fluid, or magnetic field or kinetic perspective or combining some of these. In this paper starting from the direct study of particle motion, we propose a simple mechanism to set up a steep profile which may not have been fully considered yet. Indeed, charged particle motion in a non-uniform magnetic field [3-9] is one of main classical issues of physics of plasmas in space or in fusion reactors. To tackle this problem the guiding center [7] and the gyrokinetic [8] theories are developed to trace the particle's slower motion by averaging the faster cyclotron motion. These reductions suppress computational cost and they are widely used to simulate the magnetically confined plasmas in fusion reactors [6]. These reduction theories assume existence of an invariant or an adiabatic invariant of motion associated with the magnetic moment. Meanwhile, this assumption does not always hold true. Then, recently, studies on full particle orbits without any reductions are done to look into phenomena ignored by these reductions and to interpolate the guiding center orbit. There exists a case that a guiding center trajectory and a full trajectory are completely different [10]. Further, it is found that the assumption of the invariant magnetic moment breaks [11,12] due to the chaotic motion of the test particles. * shun.ogawa@riken.jp † Xavier. Leoncini@cpt.univ-mrs.fr Let us quickly review the single particle motion and adiabatic chaos. We consider a model of charged particle moving in a non-uniform cylindrical magnetic field B(r ) = ∇ ∧ A 0 (r ). The vector potential A 0 (r ) is given bywhere the cylinder is parametrized with the coordinate (r, θ, z), B 0 is strength of the magnetic field, z has 2πR perperiodicity, e θ and e z are basic units for each direction, and q(r ) is a winding number called a safety factor of magnetic field lines. The Hamiltonian of the particle H = v 2 /2...

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Non-existence of normal tokamak equilibria with negative central current

Cited by 19 publications

References 16 publications

Tokamak equilibria with toroidal current reversal: properties and computational issues

Tokamak equilibria with toroidal current reversal: properties and computational issues

Analytical examples of reversal current, zero core current, and surface current, toroidal magnetostatic equilibria with nested flux surfaces

Tailoring steep density profile with unstable points

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