Solution Of Grad-Shafranov Equation Research Articles

Retraction Note to: Low Dissipation with Normalized Flux Surfaces in 2-Dimensional Coordinate for IR-T1 Tokamak Using Grad–Shafranov Equation Solution

Retraction Note to: Low Dissipation with Normalized Flux Surfaces in 2-Dimensional Coordinate for IR-T1 Tokamak Using Grad–Shafranov Equation Solution

Solution of Grad-Shafranov equation with linear triangular finite elements providing magnetic field continuity with a second order accuracy

The formulation presented here is applied to the Grad-Shafranov equation, which describes the equilibrium of two-dimensional axisymmetric fusion plasmas. The equilibrium configurations, obtained via standard finite element formulations in terms of poloidal flux using linear triangles, provide the magnetic flux with an accuracy O(h2). However, the poloidal magnetic field is obtained with a lower accuracy O(h) and is discontinuous across adjacent elements, with consequent problems in the derivation of linearized models for Magneto-Hydro-Dynamic stability analysis. This requires the calculation of terms related to magnetic field derivatives. The method presented here, based on Helmholtz's theorem, achieves continuity and convergence rate of order O(h2) for the magnetic field by using three-node linear triangular finite elements. It can be implemented as a postprocessor of the magnetic flux solution obtained with linear triangles, in such a way to provide the magnetic field with an accuracy O(h2). The continuity properties of the magnetic field provide a high level of accuracy and the possibility of deriving reliable linearized models with a limited computational effort. The accuracy is highly reliable for some applications requiring a high precision in the vicinity of magnetic poloidal field null points, like for instance breakdown analysis, or control of the X-point of diverted configurations in a tokamak. The effectiveness of the method is shown in linear and nonlinear cases for which analytical solutions are available.

Axisymmetric Force-Free Magnetospheres and Theoretical Models Beyond General Relativity: Magnetic Moment of Axion Stars

The magnetosphere structure of a magnetar is considered in the context of a theory of gravity with dynamical torsion field beyond the standard General Relativity (GR) with the purpose of determining in a theoretical way the corresponding electromagnetic fields of the compact star, as well as being able to express phenomenologically the magnetic moment of an axion star. To this end, the axially symmetric version of the Grad–Shafranov equation (GSE) is obtained in this theoretical framework where the resulting GSE solution in the case of the magnetosphere corresponds to a stream function containing also a pseudoscalar part. As we shown before, this function solution under axisymmetry presents a complex character that could be associated with an axidilaton.

Non-Riemmanian geometry, force-free magnetospheres and the generalized Grad-Shafranov equation

The magnetosphere structure of a magnetar is considered in the context of a theory of gravity with dynamical torsion field beyond the standard General Relativity (GR). To this end, the axially symmetric version of the Grad-Shafranov equation (GSE) is obtained in this theoretical framework. The resulting GSE solution in the case of the magnetosphere corresponds to a stream function containing also a pseudoscalar part. This function solution under axisymmetry presents a complex character that (as in the quantum field theoretical case) could be associated with an axidilaton field. Magnetar–pulsar mechanism is suggested and the conjecture about the origin of the excess energy due the GSE describing the magnetosphere dynamics is claimed. We also show that two main parameters of the electrodynamic processes (as described in GR framework by Goldreich and Julian (GJ) [Astrophys. J. 157 (1969) 869]) are modified but the electron-positron pair rate [Formula: see text] remains invariant. The possible application of our generalized equation (defined in a non-Riemannian geometry) to astrophysical scenarios involving emission of energy by gravitational waves, as described in the context of GR in [S. Capozziello, M. De Laurentis, I. De Martino, M. Formisano and D. Vernieri, Astrophys. Space Sci. 333 (2011) 29–35], is briefly discussed.

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

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.

Application of poloidal beta and plasma internal inductance in determination of input power time of Damavand tokamak

In this study, magnetic measurement of poloidal fields were used to determine poloidal beta and plasma internal inductance of Damavand tokamak combination of poloidal beta and plasma internal inductance ( $$\beta _p+\dfrac{l_i}{2}$$ ), known as Shafranov parameter, was obtained experimentally in terms of normal and tangential components of the magnetic field. Plasma internal inductance and poloidal beta were obtained using parametrization method based on analytical solution of Grad-Shafranov equation (GSE) and compared with parabolic-like profile of toroidal current density approach for determination of the plasma internal inductance. Finding evolution of $$\beta _p+\dfrac{l_i}{2}$$ and plasma internal inductance. Finding poloidal beta (Shafranov parameter and internal inductance) and using energy balance equation, thermal energy and energy confinement were determined qualitatively in terms of poloidal beta during a regular discharge of Damavand tokamak.

A New Perspective on Protection of Nuclear Reactor Surfaces From High Energy Plasma Irradiation by Equilibrium Reconstruction

Ignition is the point at which a nuclear fusion of hydrogen isotopes reaction becomes self-sustaining. This occurs when the energy being given off by the fusion reactions heats the fuel mass more rapidly than various loss mechanisms cool it. At this point, the external energy needed to heat the fuel to fusion temperatures is no longer needed. As the rate of fusion varies with temperature, the point of ignition for any given machine is typically expressed as a temperature. On the other hand, energy confinement time is expressed by an equilibrium state. Analytical solutions of Grad-Shafranov equation (GSE) can be used for theoretical studies of plasma equilibrium, transport and Magnetohydrodynamic stability of tokamaks. In this paper we have presented two families of exact solutions. With applying these solutions to IR-T1 tokamak, a small, air core, low beta and large aspect ratio tokamak with a circular cross section, we calculated poloidal magnetic flux. Due to the generality and high accuracy of the second exact solution for all of the magnetic configurations of interest, the result of this solution for IR-T1 tokamak is good (tokamak-relevant) equilibrium compare to the first exact solution for this tokamak. We intend to use these analytical solutions as benchmark of numerical equilibrium codes.

RETRACTED: Feedback controller input design for ignition of deuterium–tritium in NSTX tokamak

Nuclear fusion is a nuclear reaction in which two or more atomic nuclei (such as a deuterium–tritium) come very close and then collide at a very high speed and join to form a new high energy nucleus (Helium). Determination of accurate plasma horizontal position during plasma discharge is essential to transport it to a control system based on feedback. The solutions of Grad-Shafranov equation (GSE) analytically can be used for theoretical studies of plasma equilibrium, transport and magneto hydrodynamic stability. Here we have presented specific choices for source functions, kinetic pressure and poloidal plasma current, to be quadratic in poloidal magnetic flux and derive an analytical solution for Grad-Shafranov equation. With applying this solution to NSTX tokamak, we calculated poloidal magnetic flux, toroidal current density and normalized pressure profiles for this tokamak. Toroidal and poloidal flows can considerably change the equilibrium parameters of tokamak. These effects on the equilibrium of tokamak plasmas are numerically investigated using the code FLOW. As a comparative approach to equilibrium problem, the code is used to model equilibrium of NSTX tokamak for case pure toroidal flow. Comparison of the results of these two methods for NSTX tokamak shows good agreement between two and that our analytical solution can be served as good benchmark against the equilibrium code FLOW.

Effects of internal inductance on the energy confinement time by using the solution of equilibrium problem

In this work, dependence of energy confinement time on plasma internal inductance has been studied by using the solution of Grad–Shafranov equation (GSE) for circular cross-section HT-7 tokamak. For this, the Shafranov parameter (asymmetry factor) and poloidal beta were obtained from solution of GSE. Then we can find the dependence of energy confinement time, on plasma internal inductance. It is observed that the maximum energy confinement time is related to the low values of internal inductance [Formula: see text].

RETRACTED ARTICLE: Low Dissipation with Normalized Flux Surfaces in 2-Dimensional Coordinate for IR-T1 Tokamak Using Grad–Shafranov Equation Solution

Together with the problem of confinement, plasma–wall interactions present the major constraints toward a magnetic fusion reactor. The solutions of Grad–Shafranov equation (GSE) analytically can be used for theoretical studies of plasma equilibrium, transport and magneto-hydrodynamic stability. Here we introduce specific choices for source functions, kinetic pressure and poloidal plasma current, to be quadratic in poloidal magnetic flux and derive an analytical solution for GS equation. With applying this solution to IR-T1 tokamak, we have calculated the poloidal magnetic flux, toroidal current density and normalized pressure profiles for this tokamak. Toroidal and poloidal flows can considerably change the equilibrium parameters of tokamak. These effects on the equilibrium of tokamak plasmas are numerically investigated using a code FLOW. As a comparative approach to equilibrium problem, the code is used to model equilibrium of IR-T1 tokamak for case pure toroidal flow.

Numerical solutions of Grad-Shafranov equation in a field-reversed configuration

The solution of Grad-Shafranov equation in field-reversed configuration (FRC) is a basic problem. The solution of Grad-Shafranov equation can help to understand most of physical processes in FRC plasma, such as magnetohydrodynamic (MHD) instabilities and plasma transport. In the present paper, based on the FRC asymptotic theory by Barnes D C, the code for solving the two-dimensional Grad-Shafranov equation in FRC is developed. By using the code, the equilibriums of FRC with different elongations and separatrix radii are investigated in the present paper. The one-dimensional numerical results show that the plasma density gradient increases linearly with magnetic flux increasing in the FRC center, while, it steepens due to the high magnetic field distribution at the separatrix. The results also show that the plasma density in the closed field region increases with the density at the separatrix increasing, which implies that FRC embodies the strong confinement ability. It is a key problem to choose equations determining the shape of the separatrix in a two-dimensional numerical investigation. In the present paper, the shape equation is described as rs = rs max (1 - z2a), in which a is the shaping parameter. When a=1, the separatrix shape is elliptical, and when a1, the separatrix shape is like a racetrack. The geometry character of the separatrix appears in the one-order equations (in one-order equations: (0)/(z) = (0)/(rs)(rs)/(z), where (0)/(rs) is determined by lead equations and (rs)/(z) is given by separatrix equation). The two-dimensional numerical results show that O-point moves outward as the sparatrix radius increases. The curvature radius of magnetic flux surface increases with the separatrix radius increasing. The O-point of magnetic flux surface is just at the curvature center. Thus O-point moves outward as the sparatrix radius increases.

Study of Energy Confinement Time by the Analytical Solution of Grad–Shafranov Equation with Lithium Limiter for Circular Cross-Section Tokamak

In this work we calculated the energy confinement time by analytical solution of Grad–Shafranov equation (GSE) with Lithium limiter for circular cross-section HT-7 tokamak. A generalized Grad–Shafranov-type equation has been used. Specific functional forms of plasma internal energy and current are used. For this, the Shafranov parameter (asymmetry factor) and poloidal beta were obtained from by analytical solution of GSE. Than we can find the plasma energy confinement time. It is observed, the energy confinement time obtained from the analytical solution of GSE by using liquid lithium limiter is longer than that using graphite limiter, which shows that the plasma performance was improved.

Study of Effective Edge Safety Factor Using Analytical Solution of Grad-Shafranov Equation for Circular Cross Section Tokamak

In this work, we present effective edge safety factor using analytical solution of the Grad-Shafranov equation based on expansion of free functions of first order and magnetic probes for circular cross section HT-7 tokamak.

Plasma Position Determination by Using Fixed Boundary Conditions in GSE Solution Versus Experimental Technique in IR-T1 Tokamak

Plasma position displacement is presented by considering linear source functions in the Grad-Shafranov equation (GSE) solution and by assuming circular fixed boundary conditions and a constant plasma minor radius in the IR-T1 Tokamak. A new method for the GSE analytical solution is demonstrated by imposing constraints on the plasma current measured by the Rogowski coil and βp + li/2 measured by four discrete magnetic coils. Plasma position displacement is also measured by four discrete magnetic coils as an experimental technique, and the result is compared with our analytical technique. The result comparison shows good agreement around the steady-state equilibrium.

Study of Shafranov shift by the simplest Grad-Shafranov equation solution for HT-7 superconducting Tokamak

Study of Shafranov shift by the simplest Grad-Shafranov equation solution for HT-7 superconducting Tokamak

Calculation of plasma horizontal displacement by using a simulated plasma column in the IR-T1 tokamak

The calculation of plasma horizontal displacement by using the special solution of the Grad–Shafranov equation (GSE) and by using the measurements of several magnetic coils is presented. The constants of Zheng's solution could be determined by four magnetic probes and three poloidal flux loops. The calculated plasma position obtained by the special solution of GSE for the IR-T1 tokamak is compared with the measured plasma position determined by using four discrete magnetic probes. In the equilibrium region, our theoretical results are in good agreement with experimental results.

Theoretical model of steady-state magnetic reconnection in collisionless incompressible plasma based on the Grad–Shafranov equation solution

The problem of steady-state magnetic reconnection in an infinite current layer in collisionless, incompressible, nonresistive plasma, except of the electron diffusion region, is examined analytically using the electron Hall magnetohydrodynamics approach. It is found that this approach allows reducing the problem to the magnetic field potential finding, while last one has to satisfy the Grad–Shafranov equation. The obtained solution demonstrates all essential Hall reconnection features, namely proton acceleration up to Alfvén velocities, the forming of Hall current systems and the magnetic field structure expected. It turns out that the necessary condition of steady-state reconnection to exist is an electric field potential jump across the electron diffusion region and the separatrices. Besides, the powerful mechanism of electron acceleration in X-line direction is required. It must accelerate electrons up to the electron Alfvén velocity inside the diffusion region and on the separatrixes. This is a necessary condition for steady-state reconnection as well.

On the existence of a free boundary solution of the Grad-Shafranov equation

The existence of a non-trivial free boundary solution of the nonlinear Grad-Shafranov equation is studied. In the case of axisymmetric toroids, the existence of the solution is proved by the standard variational approach using assumptions which are not physically restrictive. Also, the existence of a cylindrically symmetric solution is proved in the case of straight cylinders with circular cross-section.