On the accuracy of the symmetric ergodic magnetic limiter map in tokamaks

2010 ◽  
Vol 76 (5) ◽  
pp. 777-794 ◽  
Author(s):  
A. R. SOHRABI ◽  
S. M. JAZAYERI ◽  
M. MOLLABASHI
Keyword(s):  
Delta Function ◽  
Mapping Technique ◽  
Error Criterion ◽  
Flux Function ◽  
Energy Error ◽  
Chaotic Field ◽  
Shafranov Equation ◽  
Field Lines ◽  
Poloidal Flux ◽  
Shafranov Shift

AbstractA new symmetric symplectic map for an ergodic magnetic limiter (EML) is proposed. A rigorous mapping technique based on the Hamilton–Jacobi equation is used for its derivation. The system is composed of the equilibrium field, which is fully integrable, and a Hamiltonian perturbation. The equilibrium poloidal flux function is a solution of the Grad–Schlüter–Shafranov equation. This equation is written in polar toroidal coordinate in order to take into account the outward Shafranov shift. The static perturbation field breaks the exact axisymmetry of the equilibrium field and creates a region of chaotic field lines near the plasma edge. The new symmetric EML map is compared with the conventional (asymmetric) EML map which is derived by applying delta-function method. The accuracy of the maps is considered through mean energy error criterion and maximal Lyapunov exponents. For asymmetric and symmetric maps the approximate location of the main cantorus near the edge of plasma is determined with high accuracy by using mean energy error. The forward–backward error criterion is applied to show the relation between the accuracy of the symmetric EML map and the number of EML rings. We also report on the effect of the number of EML rings on the maximal Lyapunov exponent of the symmetric EML map.

scholarly journals Magnetohydrodynamic Equilibrium Models for Rotamak Plasmas

1987 ◽  
Vol 40 (2) ◽  
pp. 175 ◽  
Author(s):  
IJ Donnelly ◽  
EK Rose ◽  
JL Cook
Keyword(s):  
Magnetic Field ◽  
Analytic Solutions ◽  
Toroidal Magnetic Field ◽  
Analytic Method ◽  
Equilibrium Models ◽  
Flux Function ◽  
Compact Torus ◽  
Shafranov Equation ◽  
Poloidal Flux

A semi-analytic method is used to solve the Grad-Shafranov equation for a range of compact torus plasma configurations which have ellipsoidal separatrices, zero toroidal magnetic field and pressure P proportional to the square of the poloidal flux function 1[1. The equilibria are compared with the analytic solutions of the Solov'ev model, for which P ex 1[1.

Effect of plasma flow on the equilibrium of an axisymmetric toroidal magnetic trap

1997 ◽  
Vol 58 (3) ◽  
pp. 421-432 ◽  
Author(s):  
ZH. N. ANDRUSHCHENKO ◽  
O. K. CHEREMNYKH ◽  
J. W. EDENSTRASSER
Keyword(s):  
Plasma Flow ◽  
Magnetic Trap ◽  
Plasma Equilibrium ◽  
Plasma Rotation ◽  
Stationary Plasma ◽  
Shafranov Equation ◽  
Nonlinear Vector ◽  
Analytical Expressions ◽  
Poloidal Rotation ◽  
Shafranov Shift

The effect of finite plasma rotation on the equilibrium of an axisymmetric toroidal magnetic trap is investigated. The nonlinear vector equations describing the equilibrium of a highly conducting, current-carrying plasma are reduced to a set of scalar partial differential equations. Based on Shafranov's well-known tokamak model, this set of equations is employed for the description of a kinetic (stationary) plasma equilibrium. Analytical expressions for the Shafranov shift Δ are found for the case of finite plasma rotation, where two regions of possible plasma equilibria are found corresponding to sub- and super-Alfvénic poloidal rotation. The shift Δ itself, however, turns out to depend essentially on the toroidal rotation only. It is shown that in the case of a stationary plasma flow, the solution of the Grad–Shafranov equation is at the same time also the solution of the stationary Strauss equation.

Demonstration of Shafranov Shift by the Simplest Grad–Shafranov Equation Solution in IR-T1 Tokamak

2009 ◽  
Vol 29 (1) ◽  
pp. 73-75 ◽  
Author(s):  
A. Rahimirad ◽  
M. Emami ◽  
M. Ghoranneviss ◽  
A. Salar Elahi
Keyword(s):  
Equation Solution ◽  
Shafranov Equation ◽  
Shafranov Shift

scholarly journals Magnetized Neutron Stars

2017 ◽  
Vol 45 ◽  
pp. 1760032
Author(s):  
Gibran H. de Souza ◽  
Ernesto Kemp ◽  
Cecilia Chirenti
Keyword(s):  
Magnetic Field ◽  
Neutron Star ◽  
Neutron Stars ◽  
Numerical Solutions ◽  
Internal Magnetic Field ◽  
Shafranov Equation ◽  
Field Lines

In this work we show the results for numerical solutions of the relativistic Grad-Shafranov equation for a typical neutron star with 1.4 solar masses. We have studied the internal magnetic field considering both the poloidal and toroidal components, as well as the behavior of the field lines parametrized by the ratio between these components of the field.

scholarly journals Nonradial and nonpolytropic astrophysical outflows

2018 ◽  
Vol 612 ◽  
pp. A63 ◽  
Author(s):  
L. Chantry ◽  
V. Cayatte ◽  
C. Sauty ◽  
N. Vlahakis ◽  
K. Tsinganos
Keyword(s):  
Black Hole ◽  
Kerr Black Hole ◽  
High Sensitivity ◽  
Polar Axis ◽  
Kerr Metric ◽  
Efficiency Criterion ◽  
Flux Function ◽  
Available Energy ◽  
Efficiency Parameter ◽  
Field Lines

Context. High-resolution radio imaging of active galactic nuclei (AGN) has revealed that the jets of some sources present superluminal knots and transverse stratification. Recent observational projects, such as ALMA and γ-ray telescopes, such as HESS and HESS2 have provided new observational constraints on the central regions of rotating black holes in AGN, suggesting that there is an inner- or spine-jet surrounded by a disk wind. This relativistic spine-jet is likely to be composed of electron-positron pairs extracting energy from the black hole and will be explored by the future γ-ray telescope CTA. Aims. In this article we present an extension to and generalization of relativistic jets in Kerr metric of the Newtonian meridional self-similar mechanism. We aim at modeling the inner spine-jet of AGN as a relativistic light outflow emerging from a spherical corona surrounding a Kerr black hole and its inner accretion disk. Methods. The model is built by expanding the metric and the forces with colatitude to first order in the magnetic flux function. As a result of the expansion, all colatitudinal variations of the physical quantities are quantified by a unique parameter. Unlike previous models, effects of the light cylinder are not neglected. Results. Solutions with high Lorentz factors are obtained and provide spine-jet models up to the polar axis. As in previous publications, we calculate the magnetic collimation efficiency parameter, which measures the variation of the available energy across the field lines. This collimation efficiency is an integral part of the model, generalizing the classical magnetic rotator efficiency criterion to Kerr metric. We study the variation of the magnetic efficiency and acceleration with the spin of the black hole and show their high sensitivity to this integral. Conclusions. These new solutions model collimated or radial, relativistic or ultra-relativistic outflows in AGN or γ-ray bursts. In particular, we discuss the relevance of our solutions to modeling the M 87 spine-jet. We study the efficiency of the central black hole spin to collimate a spine-jet and show that the jet power is of the same order as that determined by numerical simulations.

Effects of internal inductance on the Shafranov parameter by using the solution of equilibrium problem

2017 ◽  
Vol 31 (07) ◽  
pp. 1750078
Author(s):  
Muhammad Asif ◽  
Anila Asif
Keyword(s):  
Equilibrium Problem ◽  
Flux Function ◽  
Internal Inductance ◽  
Shafranov Equation ◽  
Plasma Internal Inductance

In this work, the dependence of Shafranov parameter on plasma internal inductance has been studied by using the solution of Grad–Shafranov equation (GSE) for Hefei Tokamak-7. The Shafranov parameter was obtained from the solution of GSE, using the expansion of free functions, which is quadratic in flux function. Then, we can find the dependence of Shafranov parameter on plasma internal inductance.

FRACTAL AND WADA EXIT BASIN BOUNDARIES IN TOKAMAKS

2007 ◽  
Vol 17 (11) ◽  
pp. 4067-4079 ◽  
Author(s):  
JEFFERSON S. E. PORTELA ◽  
IBERÊ L. CALDAS ◽  
RICARDO L. VIANA ◽  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Fractal Structure ◽  
Dynamical Structure ◽  
Stable And Unstable Manifolds ◽  
Chaotic Region ◽  
Chaotic Field ◽  
Unstable Manifolds ◽  
Field Lines ◽  
Basin Boundaries ◽  
Wada Property ◽  
Control Plasma

The creation of an outer layer of chaotic magnetic field lines in a tokamak is useful to control plasma-wall interactions. Chaotic field lines (in the Lagrangian sense) in this region eventually hit the tokamak wall and are considered lost. Due to the underlying dynamical structure of this chaotic region, namely a chaotic saddle formed by intersections of invariant stable and unstable manifolds, the exit patterns are far from being uniform, rather presenting an involved fractal structure. If three or more exit basins are considered, the respective basins exhibit an even stronger Wada property, for which a boundary point is arbitrarily close to points belonging to all exit basins. We describe such a structure for a tokamak with an ergodic limiter by means of an analytical Poincaré field line mapping.

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