Analytical solutions to the Grad–Shafranov equation for tokamak equilibrium with dissimilar source functions

Carthy, P. J. Mc

doi:10.1063/1.873630

Cited by 148 publications

(170 citation statements)

References 7 publications

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“…A function parameterization algorithm rather than the off-line Grad-Shafranov algorithm is used to calculate the tokamak equilibrium flux surfaces on a 39x69 grid for real time control [12,13]. The 95 plasma position and shape parameters are also calculated in real time.…”

Section: Magnetic Equilibriummentioning

confidence: 99%

Data acquisition and real-time signal processing of plasma diagnostics on ASDEX Upgrade using LabVIEW RT

Giannone¹,

Cerna²,

Cole³

et al. 2010

Fusion Engineering and Design

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The existing VxWorks real time system for the position and shape control in ASDEX Upgrade has been extended to calculate magnetic flux surfaces in real time using a multi-core PCI Express system running LabVIEW RT 8.6. Real time signal processing of bolometers and manometers is performed with the on-board FPGA to calculate the measured radiated power flux and particle flux respectively from the raw data. Radiation feedback experiments use halo current measurements from the outer divertor with real time median filter pre-processing to remove the excursions produced by ELM's. Integration of these plasma diagnostics into the control system by the exchange of XML sheets for communicating the real time variables to be produced and consumed is in operation. Reflective memory and UDP are employed by the LabVIEW RT plasma diagnostics to communicate with the control system and other plasma diagnostics in a multi-platform real time network.

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Section: Magnetic Equilibriummentioning

confidence: 99%

Data acquisition and real-time signal processing of plasma diagnostics on ASDEX Upgrade using LabVIEW RT

Giannone¹,

Cerna²,

Cole³

et al. 2010

Fusion Engineering and Design

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“…4 It should be pointed out, however, that exact analytical solutions can be obtained by the method of separation of variables when F͑u͒ and G͑u͒ are general linear functions of u, say F = F 0 + F 1 u and G = G 0 + G 1 u. 13,14 The direct method, 10 unfortunately, does not result in new solutions when either F 1 0 or G 1 0, at least for the types of similarity reductions considered in this brief communication. Specifically, using x = x͑z͒ in Eq.…”

Section: ͑27͒mentioning

confidence: 99%

“…11 The literature on exact solutions to the Grad-Shafranov equation is extensive. Exact solutions in terms of special functions have been derived for the case of a linear inhomogeneous equation, [12][13][14][15] and series expansion solutions to the homogeneous equation have been developed. 16 Yet interest in new techniques for solving the equation remains strong.…”

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confidence: 99%

A similarity reduction of the Grad–Shafranov equation

Litvinenko

2010

Physics of Plasmas

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A direct method for finding similarity reductions of partial differential equations is applied to a specific case of the Grad-Shafranov equation. As an illustration of the method, the frequently used Solov'ev equilibrium is derived. The method is employed to obtain a new family of exact analytical solutions, which contain both the classical and nonclassical group-invariant solutions of the GradShafranov equation and thus greatly extends the range of the available analytical solutions. All the group-invariant solutions based on the classical Lie symmetries are shown to be particular cases in the new family of solutions. © 2010 American Institute of Physics. ͓doi:10.1063/1.3456519͔The analysis of similarity reductions of partial differential equations plays an important role in many physical applications. Self-similar reductions not only solve problems with specific initial or boundary conditions 1 but also serve as intermediate asymptotic solutions to a much wider class of problems. 2 The classical method for finding a similarity reduction of a given partial differential equation is to use the Lie group method to determine a one-parameter group, admitted by the equation, and the corresponding group-invariant solution. 3 Recently White and Hazeltine 4 have used this approach to calculate the complete symmetry group admitted by the Grad-Shafranov equation 5,6 for the poloidal magnetic flux u͑r , z͒,in the particular case of constants F and G. The equation describes ideal magnetohydrodynamic tokamak equilibria in which both the pressure and the square of the poloidal electric current are linear functions of u͑r , z͒.Here and in what follows, the same notation as in Ref. 4 is adopted, although ͑r , z͒ is more often used in literature to denote the poloidal magnetic flux.White and Hazeltine 4 used the symmetry group to generate some new solutions from available solutions and derived three new group-invariant solutions to the GradShafranov equation. Not all interesting similarity reductions can be obtained using the standard Lie group method for finding group-invariant solutions. The well-known Solov'ev 7 equilibrium, for instance, is not among the Lie groupinvariant solutions to Eq. ͑1͒. A nonclassical method of group-invariant solutions 8,9 generalizes the classical Lie method by analyzing group transformations that do not necessarily map a given partial differential equation into itself. The nonclassical method can lead to additional similarity reductions.A direct method for finding similarity reductions was proposed by Clarkson and Kruskal. 10 The method is relatively simple to implement because it does not use group theory. The resulting similarity reductions, however, have been shown to be of both classical and nonclassical symmetry types. 11 The literature on exact solutions to the Grad-Shafranov equation is extensive. Exact solutions in terms of special functions have been derived for the case of a linear inhomogeneous equation, 12-15 and series expansion solutions to the homogeneous equation have been develope...

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“…∆B/BM SE = 1.59% In order to validate the forward model, a reference discharge has been conducted on ASDEX Upgrade. The discharge parameter were chosen to reflect conditions which have been analysed with the CLISTE equilibrium code 25,44 . Fig.…”

Section: Effect Of Atomic Extension Onto Experimental Quantitiesmentioning

confidence: 99%

Influence of non-local thermodynamic equilibrium and Zeeman effects on magnetic equilibrium reconstruction using spectral motional Stark effect diagnostic

Reimer

Marchuk

Geiger

et al. 2017

Review of Scientific Instruments

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Analytical solutions to the Grad–Shafranov equation for tokamak equilibrium with dissimilar source functions

Cited by 148 publications

References 7 publications

Data acquisition and real-time signal processing of plasma diagnostics on ASDEX Upgrade using LabVIEW RT

Data acquisition and real-time signal processing of plasma diagnostics on ASDEX Upgrade using LabVIEW RT

A similarity reduction of the Grad–Shafranov equation

Influence of non-local thermodynamic equilibrium and Zeeman effects on magnetic equilibrium reconstruction using spectral motional Stark effect diagnostic

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