Three-dimensional MHD Equilibria Research Articles

Unified nonlinear theory of spontaneous and forced helical resonant MHD states

It is shown that the theory of the nonlinearly saturated ideal internal kink mode by Rosenbluth et al. [Phys Fluids 16(11), 1894 (1973)] can be used to find the fully nonlinear ideal plasma response to an externally applied resonant magnetic perturbation. It is also demonstrated that the solution leads to a jump in the rotational transform across the resonant surface caused by a zonal current sheet. Its amplitude scales linearly with the plasma perturbation despite the nonlinearity of the solution. This confirms a recent conjecture that three-dimensional MHD equilibria with nested magnetic surfaces generally contain discontinuities in the rotational transform [J. Loizu et al., Phys Plasmas 22(9), 090704 (2015)]. It also lends support to Parker's long-standing suggestion that “almost all” MHD equilibria possess current sheets.

Three-dimensional distortions of the tokamak plasma boundary: boundary displacements in the presence of resonant magnetic perturbations

The three-dimensional plasma boundary displacements induced by applied non-axisymmetric magnetic perturbations have been measured in ASDEX Upgrade, DIII-D, JET, MAST and NSTX. The displacements arising from applied resonant magnetic perturbations (RMPs) are measured up to ±5% of the minor radius in present-day machines. Good agreement can be found between different experimental measurements and a range of models—be it vacuum field line tracing, ideal three-dimensional MHD equilibrium modelling, or nonlinear plasma amplification. The agreement of the various experimental measurements with the different predictions from these models is presented, and the regions of applicability of each discussed. The measured displacement of the outboard boundary from various machines is found to correlate approximately linearly with the applied resonant field predicted by vacuum modelling (though it should be emphasized that one should not infer that vacuum modelling accurately predicts the displacement inside the plasma). The RMP-induced displacements foreseen in ITER are expected to lie within the range of those predicted by the different models, meaning less than ±1.75% (±3.5 cm) of the minor radius in the H-mode baseline and less than ±2.5% (±5 cm) in a 9 MA plasma. Whilst a displacement of 7 cm peak-to-peak in the baseline scenario is marginally acceptable from both a plasma control and heat loading perspective, it is important that ITER adopts a plasma control system which can account for a three-dimensional boundary corrugation to avoid an n = 0 correction which would otherwise locally exacerbate the displacement caused by the applied fields.

The Finite Beta Effects on the Toroidal Field Ripple in a Tokamak Plasma

The efficiency of energetic ion confinement is reduced in a tokamak plasma by the non-axisymmetric field, namely the ripple field. The ripple field is produced by a finite number of toroidal field coils. It is affected by the non-axisymmetric finite beta effect. The three-dimensional MHD equilibrium calculation code VMEC is used to analyze the non-axisymmetric finite beta effect in a ripple tokamak. In the VMEC code, the flux coordinates are used, so the calculation region is limited to the area of plasma. To calculate the orbit outside the plasma, we develop a field calculation code, which is based on the Biot-Savart law. The details of the method and results are described in this paper.

Formation and healing of n = 1 magnetic islands in LHD equilibrium

Magnetic islands with the toroidal mode number n = 1, e.g. m/n = 1/1 and 2/1 islands, in a Large Helical Device (LHD) equilibrium are studied using the three-dimensional MHD equilibrium code, HINT. In order to accomplish this purpose, the HINT code has been improved. The equilibrium analysis, in particular an analysis of the LHD equilibrium with an m/n = 1/1 island, is required for the local island divertor experiment, in order to understand the magnetic structures of field lines, i.e. flux surfaces, islands and ergodic field lines. We find that the m/n = 2/1 island can be healed for a finite equilibrium beta, while the m/n = 1/1 island is not healed and is surrounded with ergodic field lines for finite-β. From the latter result, we can conjecture that the island divertor concept is effective even for finite equilibrium beta-values, but the performance of the island divertor is deteriorated for finite-β because of the existence of the ergodic zone between the closed surfaces (i.e. the core region) and the m/n = 1/1 island. We also find that the width of the m/n = 1/1 island depends on the equilibrium beta value and that the island located at the inside of the torus has the advantage of retaining its width.

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.

Possibility of LHD Equilibrium with Zero Rotational Transform Surface

Possibility of a Large Helical Device equilibrium having a zero rotational transform surface is examined by using the three-dimensional MHD equilibrium code, HINT. We find the existence of the equilibrium but with the formation of two or three n = 0 islands composing the homoclinic-type structure near the center, where n is a toroidal mode number.

Magnetic Field Topology in Prominences

We present a study of the magnetic field lines of a prominence using MHD and thermodynamic/hydrodynamic (TH) models. Previous modeling of prominences has tended to emphasize either magnetic field modeling or TH modeling in isolation. In this paper, we combine these approaches to model a long-lived filament observed in 1996 August-September. In our new approach, we (1) use magnetograms to prescribe the boundary conditions for the magnetic flux in three-dimensional MHD simulations, (2) show that observed magnetic flux changes can produce a fluxrope and that the dipped (concave upward) portion of the field lines form in the approximate location of the observed prominence, and (3) show that TH computations, using the computed geometry of magnetic field lines that are in three-dimensional MHD equilibrium, have condensations forming in the dipped portions of the field lines.

Island Formation for Finite Pressure Equilibria in Non-axisymmetric Tori Obtained by the HINT Code

Confinement nature of an edge plasma is largely influenced by the structure of magnetic surfaces in the boundary region. This issue is of particular importance for non-axisymmetric tori. Three-dimensional (3D) MHD equilibrium codes, such as HINT and PIES, have been developed to answer the indeterminate question of whether or not a 3D finite-pressure equi-librium can really exist. One important discovery obtained is that magnetic surfaces are ergodized by the finite pressure effect in actual helical configurations, and the ergodization of surfaces in the core region often imposes severer limitation on the equilibrium β than the Shafranov shift. The breaking of surfaces often grows in the boundary region in particular. Also found by HINT is a remarkable property of ‘self-healing] of magnetic islands as β in-creases. We have extended the HINT code to include the effects of the net toroidal current on the equilibrium in a fully consistent manner, especially the neoclassical currents such as the bootstrap current. In addition, it has been modified to treat the full torus geometry, and to construct a nonlinear simulation code in a full 3D geometry. In this paper, a new series of computations is shown, which became achievable by a modification of the code to have existence of coil currents in the computation region. Usefulness of the code has largely increased. Based on those HINT computations, importance of the ‘global effect] as a physical mechanism of island formation is suggested, which is not described by the previous analytical predictions.

Island Formation for Finite Pressure Equilibria in Non-axisymmetric Tori Obtained by the HINT Code

Confinement nature of an edge plasma is largely influenced by the structure of magnetic surfaces in the boundary region. This issue is of particular importance for non-axisymmetric tori. Three-dimensional (3D) MHD equilibrium codes, such as HINT and PIES, have been developed to answer the indeterminate question of whether or not a 3D finite-pressure equilibrium can really exist. One important discovery obtained is that magnetic surfaces are ergodized by the finite pressure effect in actual helical configurations, and the ergodization of surfaces in the core region often imposes severer limitation on the equilibrium β than the Shafranov shift. The breaking of surfaces often grows in the boundary region in particular. Also found by HINT is a remarkable property of self-healing' of magnetic islands as β increases. We have extended the HINT code to include the effects of the net toroidal current on the equilibrium in a fully consistent manner, especially the neoclassical currents such as the bootstrap current. In addition, it has been modified to treat the full torus geometry, and to construct a nonlinear simulation code in a full 3D geometry. In this paper, a new series of computations is shown, which became achievable by a modification of the code to have existence of coil currents in the computation region. Usefulness of the code has largely increased. Based on those HINT computations, importance of the 'global effect' as a physical mechanism of island formation is suggested, which is not described by the previous analytical predictions.

Effects of Magnetic Fields on Mass Entrainment of Supermagnetosonic Jets

We have performed three-dimensional MHD equilibrium jet simulations that have been designed to excite the Kelvin-Helmholtz (K-H) instability and allow us to investigate the spatial growth of the mixing layer between a magnetized jet and an initially unmagnetized external medium. These simulations differ in the magnetic field strength and orientation, in the jet-to-ambient density ratio, and in the amplitude of the initial (velocity) perturbation. We calculate as a direct measure of mass entrainment the mass of magnetized or high axial velocity material and compare the growth of entrained mass with dynamical and potentially observable properties of a jet. An equipartition toroidal field (whose magnetic pressure is approximately equal to the jet's thermal pressure) can inhibit the growth of the K-H instability and reduce mass entrainment significantly. An equipartition axial field has a slight stabilizing effect and reduces mass entrainment relative to a weak field for comparable magnetosonic Mach number. As the jet and ambient medium are mixed, the width of simulated total synchrotron intensity images increases and the fractional polarization of the jet decreases. In these simulations the jet evolves to a fast-moving magnetized spine surrounded by a slower moving, less magnetized sheath. These centralized spines are more easily disrupted in jets less dense than the surrounding medium. As a consequence of their greater instability, simulations with axial magnetic fields are more likely than simulations with toroidal magnetic fields to filament into axial (matter) streams. In both the width of the simulated intensity images and in the mass of magnetized material, we see evidence for a linear growth stage, a nonlinear growth stage, and a final saturation stage. Results from a normal-mode analysis suggest that the initial linear stage of mixing coincides with growth of the K-H-unstable normal surface modes and a spatial progression from higher to lower order modes. The nonlinear stage coincides with large-amplitude elliptical distortion to the jet cross section and filamentation of the jet. A subset of simulations was run at lower (9 zones/jet radius) and higher (25 zones/jet radius) spatial resolutions than the typical moderate spatial resolution of 15 zones/jet radius. We find that the instability saturates at approximately the same amount of entrained mass (relative to the jet mass) in the moderate- and high-resolution cases, although the transition between the linear and nonlinear stages is closer to the inlet in the higher resolution simulations.

Ultra Low Aspect Ratio Stellarator or Hybrid Configurations

Novel concepts of an ultra-low-aspect-ratio stellarator system, called a Spherical Stellarator (SS), or a stellarator-tokamak hybrid system, called SMARTH (Small Aspect Ratio Toroidal Hybrid) are discussed for high-{beta} regimes of operation. Self-consistent analysis of the three-dimensional MHD equilibria with inductively driven plasma current or bootstrap current is presented. It is stressed that the bootstrap current in SS can be large and is flowing in such a direction that the rotational transform produced enhances the vacuum rotational transform. This feature increases the equilibrium {beta}-limit. The high-{beta} equilibria in SS and SMARTH ({beta}(0) > 30%, {beta}O 8-9%) are demonstrated, correspondingly, with the bootstrap current and the ohmic current. Possible experiments and scalings, including those for reactor parameters, are outlined. 38 refs., 6 figs., 1 tab.

Existence of three-dimensional toroidal MHD equilibria with nonconstant pressure

We establish an existence result for the three-dimensional MHD equations with p ≠ const in tori T without symmetry. More precisely, our theorems insure the existence of sharp boundary solutions for tori whose departure from axisymmetry is sufficiently small; they allow for solutions to be constructed with an arbitrary number of pressure jumps. © 1996 John Wiley & Sons, Inc.

Three-dimensional MHD equilibrium in the presence of bootstrap current for the Large Helical Device

Three-dimensional MHD equilibria including the bootstrap current self-consistently are studied for the Large Helical Device (LHD). The magnitude of the bootstrap current is sensitive to the magnetic axis position or the Shafranov shift in LHD, which is proportional to the plasma pressure and inversely proportional to the rotational transform. Since the bootstrap current enhances the rotational transform and reduces the Shafranov shift, the self-consistent MHD equilibrium is crucial for estimating the bootstrap current. The total bootstrap current exceeds 100 kA for LHD plasmas with volume averaged beta ⟨β⟩ ≳ 1% if both electrons and ions belong to the almost collisionless 1/ν regime in the whole plasma region. Effects of the vertical field, the quadrupole field and the pressure profile on the bootstrap current are also investigated

Three-dimensional tokamak equilibria in the presence of resonant field errors

Numerical solutions are described for three-dimensional MHD equilibria in the presence of resonant magnetic field perturbations. The effects of a realistic spectrum of resonant field errors are calculated for a range of current profiles in cylindrical geometry, and the results are benchmarked against toroidal calculations for two different current profiles. It is found that field errors of the magnitude existing in present day devices, and contemplated for future devices, can produce a set of magnetic islands occupying a significant fraction of the plasma cross-section. The calculations are applicable to those situations where flow effects are unimportant. The conditions under which this is the case are discussed, and implications for several cases of interest are considered

Introduction to Confinement of Toroidal Plasma. 2. MHD Equilibrium of Toroidal Plasmas and Properties of Magnetic Surfaces.

MHD equilibrium of toroidal plasmas is discussed for both axisymmetric toroidal plasmas such as tokamak and reversed field pinch, and non-axisymmetric toroidal plasmas such as stellarator, heliotron and torsatron. The contents are; 1) Production of rotational transform, 2) Three-dimensional MHD equilibrium and Grad-Shafranov equation which includes (i) a generalized Grad-Shafranov equation, (ii) Grad-Shafranov equation under stellarator approximation, (iii) FCT equilibrium and currentless equilibrium, and (iv) Solove'v-Shafranov equilibrium equation, 3) Magnetic shear and magnetic well, 4) MHD equilibrium and magnetic islands, and 5) MHD equilibrium and equilibrium data analysis.It is emphasized that MHD equilibrium of stellarator/heliotron can be studied by generalizing Grad-Shafranov equation for tokamaks. MHD equilibrium of spatial axis stellarator is also discussed based on Solove'v-Shafranov equilibrium equation.

MHD instabilities in heliotron/torsatron

Recent theoretical results on MHD instabilities in heliotron/torsatron are reviewed. By comparing the results with experimental data in Heliotron E, Heliotron DR and ATF, it is pointed out that resistive interchange modes are the most crucial instabilities, since the magnetic hill occupies a substantial region of the plasma column.Development of three-dimensional MHD equilibrium codes has made significant progress. By applying the local stability criteria shown by DI (ideal MHD mode) and DR (resistive MHD mode) to the equilibria given by the three-dimensional codes such as BETA and VMEC, stability thresholds for the low n ideal modes or the low n resistive modes may be estimated with reasonable accuracy, where n is a toroidal mode number.

MHD instabilities in Heliotron/Torsatron.

Abstract Recent theoretical results on MHD instabilities in heliotron/torsatron are reviewed. By comparing the results with experimental data in Heliotron E, Heliotron DR and ATF, it is pointed out that resistive interchange modes are the most crucial instabilities, since the magnetic hill occupies a substantial region of the plasma column. Development of three-dimensional MHD equilibrium codes has made significant progress. By applying the local stability criteria shown by DI (ideal MHD mode) and DR (resistive MHD mode) to the equilibria given by the three-dimensional codes such as BETA and VMEC, stability thresholds for the low n ideal modes or the low n resistive modes may be estimated with reasonable accuracy, where n is a toroidal mode number.

Convergence properties of a nonvariational 3D MHD equilibrium code

Details are presented of the PIES code, which uses a nonvariational algorithm for calculating fully three-dimensional MHD equilibria. The convergence properties of the code are studied for several axisymmetric and nonaxisymmetric finite β equilibria that have magnetic surfaces.

Calculation of three-dimensional MHD equilibria with islands and stochastic regions

A three-dimensional MHD equilibrium code is described that does not assume the existence of good flux surfaces. Given an initial guess for the magnetic field, the code proceeds by calculating the pressure driven current and then by updating the field using Ampére's law. The numerical algorithm to solve the magnetic differential equation for the pressure driven current is described, and demonstrated for model fields having islands and stochastic regions. The numerical algorithm which solves Ampére's law in three dimensions is also described. Finally, the convergence of the code is illustrated for a particular stellarator equilibrium with no large islands.

Comparison of two-dimensional and three-dimensional MHD equilibrium and stability codes

Stability results obtained with the fully three-dimensional magnetohydrodynamic code BETA, the helically invariant code HERA, and the asymptotic stellarator expansion code STEP agree well for a straight l = 2, M = 5 stellarator model. This good agreement between the BETA and STEP codes persists as toroidal curvature is introduced. This validation provides justification for confidence in work with these models.