Free-boundary Equilibrium Research Articles

Finite beta plasma equilibrium in toroidally linked mirrors

The problem of finite pressure plasma equilibrium in a system with closed magnetic field lines consisting of quadrupole mirrors linked by simple toroidal cells with elliptical cross sections is analyzed. An appropriate analytical procedure is developed, that uses conformal mapping techniques, which enables one to obtain the magnetic field structure for the free boundary equilibrium problem. This method has general applicability for finding analytic solutions of the two-dimensional Dirichlet problem outside an arbitrary closed contour. Using this method, the deformations of the plasma equilibrium configuration due to finite plasma pressure in the toroidal cell are calculated analytically to the second order in a λ expansion, where λ∼β/εE, β is the ratio of plasma pressure to the magnetic field pressure, ε is the inverse aspect ratio, and E is the ellipticity of the plasma cross section. The outer displacement of the plasma column is shown to depend nonlinearly on the increase of plasma pressure, and does not prevent the achievement of substantial β∼10% in the toroidal cells.

Faraday rotation as a density diagnostic for helical axis stellarators

The Faraday rotation of a linearly polarized electromagnetic wave probing a helical-axis-stellarator plasma is a useful electron density diagnostic. The twisted magnetic geometry results in a large magnetic held component in the vertical plane that is assumed to contain the probe beam(s). As a result, significant polarization rotations result for far-infrared beams traversing plasmas of modest densities and magnetic fields. Linear-birefringence-induced beam elliptization is small and the sensitivity to secondary (bootstrap) currents is weak. A new technique for measurement of the polarization state and its application to densitometry of plasma confined by the H-1 Heliac is discussed and modelled using the three-dimensional, free-boundary MHD equilibrium code, VMEC.

Axisymmetric Magnetic Control Design in Tokamaks Using Perturbed Equilibrium Plasma Response Modeling

A model of linearized tokamak plasma response to variations in external toroidal currents is developed for purposes of axisymmetric control analysis. The plasma model is based on free-boundary equilibria with perturbations in the poloidal field coils. Plasma response to currents in arbitrary passive conductors such as the vacuum vessel is approximated by mapping their effect to equivalent poloidal field coil currents. Ideal magnetohydrodynamic flux conservation is satisfied approximately by allowing total plasma current and current-density profile to vary. The resulting flux-conserving plasma response model including vacuum vessel effects is used along with multivariable control system techniques to design high-performance axisymmetric control systems for the Alcator C-Mod tokamak to stabilize the vertical instability and to provide stable shape control.

Original Communications: The Parabola as an Equilibrium Plasma-Vaccum Interface

Abstract The free-boundary MHD equilibrium is considered in the case of plane symmetry and constant current profile. We present a global solution of the problem, given in terms of elementary functions, which shows a parabolic plasma-vacuum interface

Plane free-boundary equilibria

The free-boundary MHD equilibrium is considered for plane symmetry and a constant current profile. The field produced by external currents is prescribed and the plasma current is determined such that the plasma extends up to the separatrix.

Modelling the behaviour of the magnetic field diagnostic coils on the W VII-AS stellarator using a three-dimensional equilibrium code

Three-dimensional equilibrium solvers have been extensively used to investigate stellarator equilibria, but it was only recently that they have been adapted to model the magnetic diagnostics on a functioning experiment. The paper presents the new computer code NEMEC-DIAGNO which was written to model the diagnostic coils on the W VII-AS stellarator. This code extends the free-boundary equilibrium solver NEMEC to calculate the magnetic field and the vector potential due to the plasma currents alone and incorporates detailed models of the coils themselves. In its application to W VII-AS over a range of its operation, the code yields a set of simple curves which provide a prescription for finding the average plasma beta under varying conditions of pressure profile and net longitudinal ‘bootstrap’ current.

Natural elongation and triangularity of tokamak equilibria

Quasianalytic formulas are calculated for the elongation κ and triangularity δ of the plasma surface of a free-boundary tokamak equilibrium. The final results give κ and δ as functions of five quantities: the inverse aspect ratio ε, the poloidal beta βp, the internal inductance li, and the quadrupole and hexapole moments of the externally applied field. The agreement with numerically computed equilibria is found to be quite good when A≥3, κ≤1.5, and δ≤0.2 and when the plasma is limited by the vacuum vessel wall and not diverted by the presence of a separatrix on the plasma surface.

Finite-beta equilibrium magnetic field perturbations in stellarator plasmas

Magnetic field perturbations due to finite-beta operation in stellarators have been simulated by using the three-dimensional free-boundary equilibrium code VMEC to overcome the limitations imposed by averaged equilibrium and fixed-boundary methods. Results of these computations have been compared with analytic predictions for cylindrical stellarator models and confirm a linear relationship between the average beta and the plasma dipole moment. Only a weak sensitivity of the computations to details of the pressure profile is found. The distortion of the magnetic surfaces can be significant even at moderate beta, so that careful modelling is required when analysing the data.

Experimental identification of Tokamak equilibrium using magnetic and diamagnetic signals

In the experimental identification of Tokamak equilibria from external magnetic measurements, the addition of integral constraints to the full equilibrium analysis provides extra information on the toroidal current distribution, improving the stability of the solution against measurement errors, and increasing the accuracy of its determination. It is shown that the use of the toroidal flux value measured by the diamagnetic loop as an additional constraint to the free-boundary equilibrium solver increases to four the number of independent parameters that can be identified in the current density profile, providing a more precise evaluation of the relative weight of the pressure and paramagnetic terms.

Tokamak equilibrium reconstruction using Faraday rotation measurements

A new equilibrium reconstruction procedure using magnetic, line integrated electron density and Faraday rotation measurements has been developed. The method has been applied to a number of elongated tokamak equilibria which were computed by using a free-boundary MHD equilibrium code. Typical errors in four global plasma parameters (βp, ℓi, μ and q0) are evaluated as functions of the measurement errors and the number of source function parameters. Assuming realistic random perturbations in the measurements, the method allows source functions with up to six independent parameters to be reconstructed. It is shown that, when electron density and Faraday rotation measurements are included, the accuracy in the determination of q0 is increased by at least a factor of two, compared with cases without Faraday rotation data. The effects of adding a diamagnetic probe and of varying the plasma elongation are also investigated.

FBT - a free-boundary tokamak equilibrium code for highly elongated and shaped plasmas

A new axisymmetric, free-boundary equilibrium code, FBT, has been developed and tested. The code allows the computation of arbitrarily shaped tokamak equilibria, with external or internal separatrices and multiple magnetic axes. It is shown that bifurcations can be controlled by an iterative procedure using shape feedback. Three specific applications are discussed: (1) The problem of shape accuracy, (2) the calculation of a startup evolution for a highly elongated tokamak, and (3) methods to control saddle points.

The free-boundary equilibrium problem for helically symmetric plasmas

An iterative technique for solving the ideal MHD equilibrium equations for a helically symmetric plasma with a free boundary is described. The method involves an application of Green's theorem and has been formulated for the geometry of a heliac. It is used to determine a stability diagram for the SHEILA heliac as a function of the plasma pressure and the current in one of the external coils.

Studies of pressure profile effects on MHD stability in Heliotron-E using the stellarator expansion

The effects of varying the width of the pressure profile for MHD stability in the Heliotron-E configuration are studied. The calculation employs free-boundary stellarator expansion MHD equilibrium and stability codes, using a model for the Heliotron-E vacuum magnetic field which was constructed by means of a Biot-Savart integration of the exact coil configuration. The calculated critical beta values are in the same range as those observed in Heliotron-E experiments.

Adaptive triangular mesh generation

A general adaptive grid algorithm is developed on triangular grids. The adaptivity is provided by a combination of node addition, dynamic node connectivity and a simple node movement strategy. While the local restructuring process and the node addition mechanism take place in the physical plane, the nodes are displaced on a monitor surface, constructed from the salient features of the physical problem. An approximation to mean curvature detects changes in the direction of the monitor surface, and provides the pulling force on the nodes. Solutions to the axisymmetric Grad-Shafranov equation demonstrate the capturing, by triangles, of the plasma-vacuum interface in a free-boundary equilibrium configuration.

Tokamak Fusion Reactor Start-Up Simulation

A simulation code TSEC (Time-dependent Spectral Equilibrium Code) has been developed to model the axisymmetric evolution of a tokamak on the resistive (L/R) time scale of the external coils, conductors, or shell. The electromagnetic interaction between the plasma and the external circuit is taken into account in a self-consistent manner. TSEC is Lagrangian and utilizes magnetic flux coordinates with spectral decomposition in the angle variable theta. The plasma is modeled as a finite-size, zero-inertia, finite-pressure fluid which adjusts its position and shape to remain in free-boundary equilibrium consistent with the currents in the external circuits. At the heart of TSEC is a fast method of calculating the self-consistent free-boundary plasma equilibrium at each time step which is based on the minimization of a certain mean-square error. 3 refs., 6 figs., 3 tabs.

Volterra’s series solutions of free boundary plasma equilibria

In this paper the nonlinear problem of the free boundary equilibrium of a plasma inside a conducting circular shell has been solved analytically in the high beta limit. The unknown function describing the plasma shape has been expanded in a Volterra functional series, the functional analogous to the Taylor series. The hierarchy of the linear integral equations obtained from the expansion is, at least in principle, analytically solvable, so that the solution of each equation can be given in a closed form. The analytical computations have been carried out up to the fourth order and the results compared with numerical computations.

Three-dimensional free boundary calculations using a spectral Green's function method

The plasma energy W p = ∫ ω p ( 1 2 B 2 +p dV is minimized over a toroidal domain ω p using an inverse representation for the cylindrical coordinates R = ΣR mn ( S) cos( mθ − nζ) and Z = ΣZ mn ( s) sin( mθ − nζ), where ( s, θ, ζ) are radial, poloidal and toroidal flux coordinates, respectively. The radial resolution of the MHD equations is significantly improved by separating R and Z into contributions from even and odd poloidal harmonics which are individually analytic near the magnetic axis. A free boundary equilibrium results when ω p is varied to make the total pressure 1 2 B 2 + p continuous at the plasma surface Σ p and when the vacuum magnetic field B v satisfies the Neumann condition B v ·d Σ p = 0 . The vacuum field is decomposed as B v = B 0 + ∇φ , where B 0 is the field arising from plasma currents and external coils and φ is an single-valued potential necessary to satisfy B v d Σ p = 0 when p ≠ 0. A Green's function method is used to obtain an integral equation over Σ p for the scalar magnetic potential φ = Σφ mn sin( mθ − nζ). A linear matrix equation is solved for φ mn to determine 1 2 B 2 v on the boundary. Real experimental conditions are simulated by keeping the external and net plasma currents constant during the iteration. Applications to l = 2 stellarator equilibria are presented.

An integral equation technique for the exterior and interior neumann problem in toroidal regions

An integral equation technique for the Neumann problem of finding a function Φ satisfying ΔΦ = 0 with prescribed values of ∂Φ∂n on the boundary is described. Fourier representation of the potential Φ on the boundary with respect to two angle-like variables transforms the integral equation to an infinite set of linear equations for the Fourier coefficients of Φ. The singularity of the Green's function is treated by a regularization method: a function with the same singularity is subtracted and its analytically calculated Fourier-transform is added to the Fourier transformed integral equation. A computer code named NESTOR is developed. Applications include studies of toroidal magnetic vacuum fields and calculation of the vacuum field contribution for the 3D free-boundary equilibrium problem.

Axisymmetry stability of highly-elongated Tokamak plasma-a new method and initial results

A new method is presented for computing global axisymmetric instabilities of Tokamak plasmas in a conducting shell. The method uses a free-boundary equilibrium code to find displacment vectors compatible with ideal MHD. It is applied to the computation of vertical instability growth rates of highly-elongated, race-track shaped plasmas. Results obtained for up-down symmetric configurations are compared with ERATO calculations. The effects of plasma-wall distance, plasma pressure and current profile are analyzed. In addition, the gain in stability by asymmetric positioning of a given plasma in a given shell is evaluated.

Torsatron equilibrium and stability studies

Equilibrium and stability results are presented for the Advanced Toroidal Facility (ATF) device. The results of three-dimensional equilibrium calculations and free-boundary average method calculations are shown to be in good agreement with previous fixed-boundary average method results. These favourable comparisons serve as a valuable validation of the simple and computationally efficient fixed-boundary average method. Stability calculations for the free-boundary average method equilibria are also in good agreement with fixed-boundary calculations, showing instability only when the plasma is shifted inwards with an applied vertical field.