Nested Flux Surfaces Research Articles

Ideal MHD spectrum calculations for the ARIES-CS configuration

Ideal MHD stability calculations for the ARIES compact stellarator (ARIES-CS) reactor design (Najmabadi et al 2008 Fusion Sci. Technol. 54 655) show a spectrum of instabilities. The ARIES design considered is a three field-period stellarator with engineering coil constraints optimized for magnetic well and alpha particle confinement. The reference design has high β ∼ 5%. The study is restricted to ideal modes and the calculations assume nested flux surfaces, with a limited plasma boundary surrounded by a vacuum. At β = 4%, with a conformal wall at twice the minor plasma radius, the equilibrium is slightly unstable to a periodicity-preserving, predominantly m/n = 9/6 mode peaked at the edge and a periodicity-breaking global m/n = 3/2 mode. At β ∼ 5%, these modes are destabilized but the growth rates are still moderate. At higher β, above the design value, several modes become unstable. Stabilization by a close fitting conducting wall is ineffective at β = 5% and below but becomes more effective at stabilizing external modes at higher β. The equilibrium at β ∼ 6% can be stabilized by a conformal wall at 1.1 times the minor plasma radius, although very weakly unstable internal modes remain at β > 6% with a wall on the plasma boundary. The sensitivity to the presence of the rational rotational transform ι = 2/3 surface at the edge of the plasma was also investigated. Generally, either the m/n = 3/2 mode is further destabilized or other modes are introduced. The stability calculations numerically impose a broadening of the singular perturbed current to eliminate spurious singularities. The effect of this is considered in detail and it is suggested that this numerical resonance detuning can model a physical broadening from non-ideal effects. Although the reference design with β ∼ 5% is above the strict ideal β limit, common experience in tokamaks indicates that weakly unstable internal modes and edge-localized modes result in relatively benign MHD activity. This is consistent with observations in large stellarator experiments that indicate some level of instability is tolerated and the results are discussed in this context and in relation to the numerical broadening of the singular perturbed currents.

Rapid Fourier space solution of linear partial integro-differential equations in toroidal magnetic confinement geometries

Fluctuating quantities in magnetic confinement geometries often inherit a strong anisotropy along the field lines. One technique for describing these structures is the use of a certain set of Fourier components on the tori of nested flux surfaces. We describe an implementation of this approach for solving partial differential equations, like Poisson's equation, where a different set of Fourier components may be chosen on each surface according to the changing safety factor profile. Allowing the resolved components to change to follow the anisotropy significantly reduces the total number of degrees of freedom in the description. This can permit large gains in computational performance. We describe, in particular, how this approach can be applied to rapidly solve the gyrokinetic Poisson equation in a particle code, ORB5 (Jolliet et al., (2007) [5]), with a regular (non-field-aligned) mesh.

Observation of Equilibria with a Double Magnetic Axis in LHD

Finite-beta equilibria with a double magnetic axis have been realized in the Large Helical Device (LHD). Since the rotational transform is weak in the central region of the LHD, the effect of an externally applied quadrupole field is more pronounced in the central region than in the periphery. Consequently, the magnetic axis splits due to a moderate elongation. In the case of vertical elongation, the figure-eight structure of the magnetic surfaces has been observed in a soft X-ray image. Degradation of confinement due to the appearance of the separatrix is suggested in the case of horizontal elongation, which is closely related to the equilibrium beta limit. The three-dimensional magnetohydrodynamic equilibrium code HINT, which does not assume the existence of nested flux surfaces, provides physical pictures consistent with the experimental observations.

New Calibration Method of Magnetic Measurements Based on the MHD Equilibrium with the Ergodic Region

A new calibration method for evaluating equilibrium parameters such as the beta value and its radial profile is introduced. The method is based on a real-coordinate equilibrium code (HINT), which does not assume nested flux surfaces, and on a signal-analysis code (JDIA), which evaluates the flux of magnetic measurements from the result of the HINT. Results of the present work are closer to experimental data than results of a conventional method which assumes nested flux surfaces.

Theoretical considerations of doublet-like configuration in LHD

In order to investigate the separatrix structure inside the plasma, the magnetohydrodynamic (MHD) equilibrium and the high-energy particle orbit are studied for the Large Helical Device plasmas with two magnetic axes located within the two lobes of a figure-eight magnetic separatrix. Magnetic axes are split for strongly elongated cross section by the external quadrupole field. MHD equilibria without assuming existence of nested flux surfaces are obtained from the HINT code. For the vertical elongation, the figure-eight structure evolves due to the finite beta effect and the structure of magnetic field lines outside the separatrix is ergodized. For the horizontal elongation, the Shafranov shift is very large Δ/a > 0.5. To see MHD equilibrium beta limit, high beta equilibrium (∼5%) is studied. The magnetic axis shifts extremely to the outside of the torus and a new figure-eight separatrix appears. The particle orbit on the separatrix is studied by solving the guiding-centre drift equation on the rectangular grid. If the reflecting point of the trapped particle is close to the separatrix, the banana width is very large implying large, nonlocal neoclassical transport.

Dynamo-free plasma in the reversed-field pinch: Advances in understanding the reversed-field pinch improved confinement mode

Generation and sustainment of the reversed field pinch (RFP) magnetic configuration normally relies on dynamo activity. The externally applied electric field tends to drive the equilibrium away from the relaxed, minimum energy state which is roughly described by a flat normalized parallel current density profile and is at marginal stability to tearing modes. Correlated fluctuations of magnetic field and velocity create a dynamo electric field which broadens the parallel current density profile, supplying the necessary edge current drive. These pervasive magnetic fluctuations are also responsible for destruction of flux surfaces, relegating the standard RFP to a stochastic-magnetic transport-limited device. Application of a tailored electric field profile (which matches the relaxed current density profile) allows sustainment of the RFP configuration without dynamo-driven edge current. The method used to ascertain that a dynamo-free RFP plasma has been created is reported here in detail. Several confinement improvements during the accompanying periods of low magnetic fluctuations are observed. Namely, the magnetic fluctuation level is reduced to the point where stochastic-magnetic transport is no longer the dominant process in the core and nested flux surfaces are restored in the core of the dynamo-free RFP.

Magnetohydrodynamic Equilibrium of Heliotron J Plasmas

Magnetohydrodynamic (MHD) equilibria of Heliotron J plasma are investigated by using the HINT code. HINT calculates an MHD equilibrium by using the relaxation method, which calculates the time evolution of dissipative MHD equations. Since HINT uses Eulerian rectangular grids, it does not assume nested flux surfaces. In the standard configuration, magnetic islands appear inside the plasma in spite of low β (<β> ~ 0.4%). The width of the islands depends on the pressure distribution. To reduce island width, feedback control of the external vertical field is introduced. Because of the effect of the external vertical field, magnetic islands are suppressed. The effects of the net toroidal currents on MHD equilibrium are also investigated. The rotational transform is changed by the currents, but the Shafranov shift is almost unaffected. The width of the islands changes because of the change in the rotational transform.

Non-existence of normal tokamak equilibria with negative central current

Recent tokamak experiments employing off-axis, non-inductive current drive have found that a large central current hole can be produced. The current density is measured to be approximately zero in this region, though in principle there was sufficient current drive power for the central current density to have gone significantly negative. Recent papers have used a large aspect-ratio expansion to show that normal magnetohydrodynamic equilibria (with axisymmetric nested flux surfaces, non-singular fields, and monotonic peaked pressure profiles) cannot exist with negative central current. That proof is extended here to arbitrary aspect ratio, using a variant of the virial theorem to derive a relatively simple integral constraint on the equilibrium. However, this constraint does not, by itself, exclude equilibria with non-nested flux surfaces, or equilibria with singular fields and/or hollow pressure profiles that may be spontaneously generated.

Constructing integrable high-pressure full-current free-boundary stellarator magnetohydrodynamic equilibrium solutions

For the (non-axisymmetric) stellarator class of plasma confinement devices to be feasible candidates for fusion power stations it is essential that, to a good approximation, the magnetic field lines lie on nested flux surfaces; however, the inherent lack of a continuous symmetry implies that magnetic islands responsible for breaking the smooth topology of the flux surfaces are guaranteed to exist. Thus, the suppression of magnetic islands is a critical issue for stellarator design, particularly for small aspect ratio devices. Pfirsch–Schlüter currents, diamagnetic currents and resonant coil fields contribute to the formation of magnetic islands, and the challenge is to design the plasma and coils such that these effects cancel.Magnetic islands in free-boundary high-pressure full-current stellarator magnetohydrodynamic equilibria are suppressed using a procedure based on the Princeton Iterative Equilibrium Solver (Reiman and Greenside 1986 Comput. Phys. Commun. 43 157) which iterates the equilibrium equations to obtain the plasma equilibrium. At each iteration, changes to a Fourier representation of the coil geometry are made to cancel resonant fields produced by the plasma. The changes are constrained to preserve certain measures of engineering acceptability and to preserve the stability of ideal kink modes. As the iterations continue, the coil geometry and the plasma simultaneously converge to an equilibrium in which the island content is negligible, the plasma is stable to ideal kink modes, and the coils satisfy engineering constraints. The method is applied to a candidate plasma and coil design for the National Compact Stellarator eXperiment (Reiman et al 2001 Phys. Plasma 8 2083).

MHD equilibrium of a low-shear helical axis Heliotron

The HINT code is a three-dimensional MHD equilibrium calculation code which does not assume nested flux surfaces. This code has been applied to the study of helical system plasma and it has reproduced the phenomenon of magnetic islands in finite pressure equilibria. We modify the numerical scheme of HINT to make it more efficient. In particular, the calculation of the plasma pressure distribution is improved for the case of a low-shear stellarator like Heliotron J. Using the modified HINT code, we study the finite plasma pressure equilibria of Heliotron J plasmas. In the standard configuration of Heliotron J plasmas, the equilibrium with a peaked distribution of the plasma pressure has magnetic islands appearing at β∼0.8%. When β∼3%, the edge region of plasma is ergodized by the finite pressure effect and flux surfaces are broken. On the other hand, for a broad initial pressure distribution the island width is decreased, but the structure of magnetic field remains stochastic.

Hamiltonian guiding center equations in toroidal magnetic configurations

Guiding center equations for particle motion in a toroidal magnetic configuration are derived using general magnetic coordinates. For the case of axisymmetry, the explicit transformation to exact Hamiltonian canonical variables is presented for the first time. Approximate canonical coordinates are introduced also for three-dimensional configurations with strong toroidal magnetic field. Previous derivations made use of so-called Boozer equilibrium coordinates, which are highly nonuniform and are canonical only in the exceptional case of low beta, up–down symmetric configurations. The present formalism is valid for arbitrary, spatially well distributed magnetic coordinates, greatly increasing the accuracy of calculations. Magnetostatic equilibrium is not assumed in the present formalism, the analysis holds for any configuration with nested flux surfaces.

Explicit spectrally optimized Fourier series for nested magnetic surfaces

The nonuniqueness of the poloidal angle θ in the parametric representation of a space curve x(θ)=[R(θ),Z(θ)] can be exploited to condense the Fourier spectra of R and Z. The nonlinear equation describing this spectral condensation was previously derived and solved numerically using Lagrange multipliers. Here a special case of the condensation equation is shown to be exactly solvable, leading to an explicit representation for x. A family of such representations is generated that possesses increasingly condensed spectra as a parameter is varied. Applications to a variety of curves are considered as models for three-dimensional magnetohydrodynamic (MHD) equilibria with nested flux surfaces. A substantial improvement occurs in spectral convergence compared with a polar representation, while retaining the numerical simplicity of the polar constraint. The asymptotic behavior for the R and Z spectral coefficients near a magnetic axis is analyzed. Implications for improvements of MHD equilibrium calculations are discussed.

The KINX ideal MHD stability code for axisymmetric plasmas with separatrix

The paper presents the KINX code for computing linear ideal MHD growth rates and eigenvectors of axisymmetric plasmas surrounded by a vacuum layer and a conducting wall. Plasma equilibrium magnetic surfaces are assumed to be nested either in the whole plasma domain (separatrix at the plasma boundary is possible) or in the domains separated by an internal separatrix (doublet and divertor configurations). The computational domain is decomposed into subdomains with nested flux surfaces. In each subdomain finite hybrid elements are used on an equilibrium grid adapted to magnetic surfaces. Numerical destabilization is eliminated; this results in better convergence properties and makes possible efficient stability index calculation (δW-code). An inverse vector iteration method and a vectorizable matrix solver are applied to the matrix eigenvalue problem. The stability studies of external kink modes for doublet and single null configurations are given as application examples of the KINX code. Another version of the code, KINX-W, computing resistive wall n = 0 mode growth rates, is also presented for single null, doublet and divertor plasma configurations.

Single-particle motion in nonaxisymmetric static electric and magnetic fields

The technique of iteration of canonical transformations for a Hamiltonian system is applied to study the motion of a single particle of small Larmor radius in static, nonaxisymmetric electric and magnetic fields. For use in applications in kinetic theory, it is asked when is μ(r,v), where μ is the magnetic moment adiabatic invariant, single valued in r, with v held fixed? Even when μ(r,v) is a good adiabatic invariant, it may fail this additional condition, so that f[1/2v2+Φ(r), μ(r,v)] is not an acceptable particle distribution function, where Φ(r) is a single-valued electrostatic potential. It is shown that in a toroidal domain the condition is satisfied only for axisymmetric fields, or for magnetic fields near a zero shear field with nested flux surfaces that fill a toroidal domain. The appropriate transformations to guiding center coordinates and Hamiltonians are given for these cases.

Magnetic Configuration Scans in H-1 Heliac

The results of the experimental study of the magnetic configurations in the H-1 heliac are presented. The shape of the flux surfaces and the rotational transform in H-1 can be controlled by varying external coil currents. Electron beam magnetic mapping has been performed to show the existence of closed nested flux surfaces and to observe the effect of small errors in coil alignment on the vacuum magnetic structure in H-1. Langmuir probes have been used to study the electron density profiles in a current-free collisional RF-sustained plasma (n{sub e}<=4 x 10{sup 12} cm{sup -3}, T{sub e}<= 15 eV). In standard magnetic configuration and for the present moderate RF power levels, the highest central density is achieved at rather low magnetic field (0.07 T). This regime is characterized by peaked density profiles that appear to have a maximum coincident with the position of the vacuum magnetic axis. When a lowest-order m=1, n=1 resonance is introduced inside the outermost magnetic surface a strong asymmetry in both the vacuum magnetic structure and the plasma density profiles is observed. We observed low frequency (2-3 kHz) density fluctuations having low radial mode numbers and internal parallel plasma current localized in the regions of highest densitymore » gradient. These fluctuations are effectively suppressed by an increase of the magnetic field. 4 refs., 6 figs.« less

Experimental investigation of the magnetic structure in the H-1 heliac

The results of an experimental study of the magnetic structure in the H-1 heliac are presented. Electron beam magnetic mapping has confirmed the existence of closed nested flux surfaces, in good agreement with a computer model. Measurements over a wide range of helical winding currents demonstrated a variety of attainable magnetic configurations within a rotational transform range of 0.6 ⩽ t0 ⩽ 1.8. The observed islands can be attributed to deduced small errors in the coil alignment in H-1. A magnetic island study and a correction in the model to fit the experimental observations revealed the error sources in the magnetic field

Detailed investigation of the vacuum magnetic surfaces on the W7-AS stellarator

The fluorescent rod technique is a rather new and sensitive method of experimentally mapping vacuum magnetic surfaces in toroidal confinement devices. It was applied to the modular low shear W7-AS stellarator before starting plasma experiments. Magnetic island widths down to less than 5% of the plasma radius could be recorded, these being caused by relative perturbation field amplitudes of down to 10-5. The almost complete Fourier spectrum of the resonant perturbation field components is derived for values of the rotational transform iota between 1/4 and 2/3. The results obtained on W7-AS demonstrate once more that the existence of magnetic islands is a common phenomenon in toroidal confinement devices, and that the assumption of closed, nested flux surfaces in transport theories may not always be justified. On the basis of the measured perturbation field spectrum, the enhancement of the radial transport caused by the vacuum field islands is estimated with a simple transport model. The predictions of the model are compared with the experimental dependence of the diamagnetic energy content on iota in W7-AS

Numerical simulation of fluctuation suppression via DC helicity injection in a reversed field pinch

The sustainment of a reversed field pinch using electrostatic helicity injection in conjunction with the usually applied axial electric field is investigated. Non-linear magnetohydrodynamic simulations indicate that it may be possible to sustain the plasma in a resistive steady state free from current driven modes and the associated dynamo relaxation. Moreover, the core region is well confined by nested flux surfaces. Practical considerations such as the impedance and efficiency of the injection circuit are discussed.

The beta limit in the Advanced Toroidal Facility due to local ideal MHD instabilities

The marginal pressure profile that is imposed either by ideal ballooning modes or by interchange modes on every nested flux surface of a three-dimensional equilibrium modelling the ATF device is determined for conditions of vanishing net toroidal plasma current. The corresponding limiting values are β = 3.75% and β(0) = 9.89%. Mercier modes dominate in regions of strong stellarator-like magnetic shear (ι′/ι > 0), and non-symmetric ballooning modes (non-symmetric eigenstructures in the extended poloidal angle domain) dominate in regions with strong tokamaklike magnetic shear (ι′/ι < 0).

Stochastic stability of a plasma torus

The hamiltonian equations for magnetic field lines in a toroidal plasma are derived from a variational principle; we find an equation for critical fluctuations ▪ n by assuming a marginal overlap of adjacent resonances throughout the small cross section for the torus. Comparison with the flux surfaces of JET as obtained from a static equilibrium code leads to critical fluctuations of the order of δB crit ≲ B p x10 −4 for n ≳ 10, where B p is the poloidal equilibrium field. Finally, we solve the equations for magnetic field lines in the case of a set of nested flux surfaces perturbed by critical fluctuations; the picture of a completely destroyed torus has been verified with a stochasticity parameter √3, but no stable integration scheme for a stochastic magnetic line has been found.