Mercier Criterion Research Articles

The effect of pressure anisotropy on 3-D MHD stability for low magnetic field LHD equilibria

The magnetohydrodynamic stability of plasmas with an anisotropic pressure component is analysed for a low magnetic field configuration of the large helical device. Magnetic equilibria are calculated by the anisotropic Neumann inverse moments equilibrium code, an extension of the three-dimensional variational moments equilibrium code. A modified version of the bi-Maxwellian is used to model the anisotropic particle velocity distribution. Magnetohydrodynamic stability calculations for the $n=1$ mode family are carried out by TERPSICHORE, which has been expanded by the Kruskal–Oberman energy principle. For on-axis particle deposition, the growth rate and plasma displacement show that the parallel dominant plasmas are significantly more stable than isotropic or perpendicular dominant plasmas. For off-axis particle deposition, the growth rate and the Mercier criterion in the peripheral region $\rho =0.9$ , show that low field (LF) deposition perpendicular dominant plasmas are most unstable. For the most realistic off-axis deposition profile, it is found that parallel dominant plasmas are most stable for LF deposition, while perpendicular dominant plasmas are most stable for high field deposition. We conclude that, under low magnetic field conditions in the large helical device, tangential neutral beam injection heating has a stabilising influence on the plasma.

Generalized Mercier stability criterion for stellarators

The Mercier criterion is a well-known stability criterion for tokamaks. It is derived from a 2 × 2 matrix eigenvalue problem arising from the expansion of resonant solutions about a singular surface where m−nq=0, with m and n being the poloidal and toroidal mode numbers, respectively, and q being the safety factor. The stability criterion is that the eigenvalues must be real, otherwise, the solution oscillates, violating the Newcomb crossing criterion. Because of the non-axisymmetry of stellarators, different toroidal as well as poloidal harmonics couple to each other. It follows that each singular surface can have multiple resonant harmonics, with multiplicity M≥1. The corresponding matrix eigenvalue problem involves a 2M×2M matrix, resulting in M pairs of positive and negative eigenvalues. The generalized stability criterion is that all eigenvalues must be real. While the original Mercier criterion can be expressed in terms of quadratures of equilibrium quantities over the singular surface, which can be evaluated anywhere, the generalized Mercier criterion can only be evaluated on rational q surfaces with a given set of resonant harmonics.

Kinetic thermal ion effects on maintaining high beta plasmas above the Mercier criterion in the Large Helical Device

It is found that kinetic thermal ions have a significant stabilizing effect on Large Helical Device (LHD) plasmas from the kinetic magnetohydrodynamic (MHD) simulations. The simulation results for the LHD plasmas at high magnetic Reynolds number show that the high beta plasmas can be maintained since the saturation level of the pressure-driven MHD instabilities is significantly reduced by the kinetic thermal ions. This results from the fact that the response of the trapped ions to the instabilities is weakened by the precession drift motion in the three-dimensional magnetic field. Thus the supercritical stability of the LHD plasmas well above the Mercier criterion can be attributed to the precession drift motion of the trapped ions in the three-dimensional magnetic field.

The on-axis magnetic well and Mercier's criterion for arbitrary stellarator geometries

A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.

Magnetic well and Mercier stability of stellarators near the magnetic axis

We have recently demonstrated that by expanding in small distance from the magnetic axis compared with the major radius, stellarator shapes with low neoclassical transport can be generated efficiently. To extend the utility of this new design approach, here we evaluate measures of magnetohydrodynamic interchange stability within the same expansion. In particular, we evaluate the magnetic well, Mercier's criterion, and resistive interchange stability near a magnetic axis of arbitrary shape. In contrast to previous work on interchange stability near the magnetic axis, which used an expansion of the flux coordinates, here we use the ‘inverse expansion’ in which the flux coordinates are the independent variables. Reduced expressions are presented for the magnetic well and stability criterion in the case of quasisymmetry. The analytic results are shown to agree with calculations from the VMEC equilibrium code. Finally, we show that near the axis, Glasser, Greene and Johnson's stability criterion for resistive modes approximately coincides with Mercier's ideal condition.

Strongly driven surface-global kinetic ballooning modes in general toroidal geometry

Kinetic ballooning modes in magnetically confined toroidal plasmas are investigated putting emphasis on specific stellarator features. In particular, we propose a Mercier criterion which is purposely designed to allow for direct comparison with local flux-tube gyrokinetics simulations. We investigate the influence on the marginal frequency of the mode of a magnetic curvature which is inhomogeneous on the magnetic flux surface due to the fieldline-label dependence. This is a typical (surface) global effect present in non-axisymmetry. Finally, we propose an artificial equilibrium model that explicitly retains the fieldline-label dependence in the magnetic drift, and analyse the stability of the system by introducing a representation of the perturbations similar to the flux-bundle model of Sugamaet al. (Plasma Fusion Res., vol. 7, 2012, 2403094). The coupling of flux bundles is shown to have a stabilising effect on the most unstable local flux-tube mode.

Ideal magnetohydrodynamic theory for localized interchange modes in toroidal anisotropic plasmas

Ideal magnetohydrodynamic theory for localized interchange modes is developed for toroidal plasmas with anisotropic pressure. The work extends the existing theories of Johnson and Hastie [Phys. Fluids 31, 1609 (1988)], etc., to the low n mode case, where n is the toroidal mode number. Also, the plasma compressibility is included, so that the coupling of the parallel motion to perpendicular one, i.e., the so-called apparent mass effect, is investigated in the anisotropic pressure case. The singular layer equation is obtained, and the generalized Mercier's criterion is derived.

Global magnetohydrodynamic instabilities in the L-2M stellarator

Analysis of global magnetohydrodynamic (MHD) instabilities in the L-2M stellarator (Prokhorov General Physics Institute, Russian Academy of Sciences) is presented. The properties of free-boundary equilibria states are outlined, the stability conditions for small-scale modes are briefly discussed, and the number of trapped particles is estimated. All the magnetic configurations under study are stable against ballooning modes. It is shown that global ideal internal MHD modes can be found reliably only in Mercier unstable plasmas. In plasma that is stable with respect to the Mercier criterion, global unstable modes that are localized in the vicinity of the free plasma boundary and are not associated with any rational magnetic surface inside the plasma (the so-called peeling modes) can be found. The radial structure of all perturbations under study is almost entirely determined by the poloidal coupling of harmonics. The results of calculations are compared with the available experimental data.

The negative triangularity tokamak: stability limits and prospects as a fusion energy system

The paper discusses edge stability, beta limits and power handling issues for negative triangularity tokamaks. The edge magnetohydrodynamic stability is the most crucial item for power handling. For the case of negative triangularity the edge stability picture is quite different from that for conventional positive triangularity tokamaks: the second stability access is closed for localized Mercier/ballooning modes due to the absence of a magnetic well, and nearly internal kink modes set the pedestal height limit to be weakly sensitive to diamagnetic stabilization just above the margin of the localized mode Mercier criterion violation. While a negative triangularity tokamak is thought to have a low beta limit with its magnetic hill property, it is found that plasmas with reactor-relevant values of normalized beta βN > 3 can be stable to global kink modes without wall stabilization with appropriate core pressure profile optimization against localized mode stability, and also with increased magnetic shear in the outer half-radius. The beta limit is set by the n = 1 mode for the resulting flat pressure profile. The wall stabilization is very inefficient due to strong coupling between external and internal modes. The n > 1 modes are increasingly internal when approaching the localized mode limit, and set a lower beta in the case of the peaked pressure profile leading to a Mercier unstable core. With the theoretical predictions supported by experiments, a negative triangularity tokamak would become a prospective fusion energy system with other advantages including a larger separatrix wetted area, more flexible divertor configuration design, wider trapped particle-free scrape-off layer, lower background magnetic field for internal poloidal field coils, and larger pumping conductance from the divertor room.

Ballooning Modes Instabilities in Outward LHD Configurations

The aim of this 3D ballooning mode growth rates study is to point out the existence of ballooning activity in outward LHD configurations near the edge, and to find out how these instabilities are triggered in equilibria that simulate plasmas of high density core operations with internal transport barrier. 3D ballooning mode growth rates and Mercier stability were studied for several magnetic configurations with different beta values, and the research reveals an intense activity in outward configurations near the plasma edge, where Mercier criterion predicts stability for interchange modes and experimental data situates the density collapse events [S. Ohdachi et al., Contrib. Plasma Phys. 50, 552 (2010)].

MHD Instabilities in Tokamaks

A general introduction to ideal magnetohydrodynamic (MHD) stability of tokamak plasmas is given, using linear perturbations of the ideal MHD equations. Subsequently the Energy Principle for ideal MHD instabilities is derived. The specific instabilities which are then discussed are loosely divided into two categories. Under the name “current driven instabilities”, external and internal kink modes, which are modes with a large radial extent, are discussed. The internal m = 1 kink mode is responsible for sawtooth collapses and fishbone oscillations in tokamaks. Under the header “pressure driven instabilities”, more localized modes are presented. These modes may limit the pressure gradient in the plasma without causing sizeable disruptions. The ballooning limit and the Mercier criterion are presented. The Troyon limit is mentioned as a synthesis of several of these stability boundaries.

Improved-confinement plasmas at high temperature and high beta in the MST RFP

We have increased substantially the electron and ion temperatures, the electron density, and the total beta in plasmas with improved energy confinement in the Madison Symmetric Torus (MST). The improved confinement is achieved with a well-established current profile control technique for reduction of magnetic tearing and reconnection. A sustained ion temperature >1 keV is achieved with intensified reconnection-based ion heating followed immediately by current profile control. In the same plasmas, the electron temperature reaches 2 keV, and the electron thermal diffusivity drops to about 2 m2 s−1. The global energy confinement time is 12 ms. This and the reported temperatures are the largest values yet achieved in the reversed-field pinch (RFP). These results were attained at a density ∼1019 m−3. By combining pellet injection with current profile control, the density has been quadrupled, and total beta has nearly doubled to a record value of about 26%. The Mercier criterion is exceeded in the plasma core, and both pressure-driven interchange and pressure-driven tearing modes are calculated to be linearly unstable, yet energy confinement is still improved. Transient momentum injection with biased probes reveals that global momentum transport is reduced with current profile control. Magnetic reconnection events drive rapid momentum transport related to large Maxwell and Reynolds stresses. Ion heating during reconnection events occurs globally, locally, or not at all, depending on which tearing modes are involved in the reconnection. To potentially augment inductive current profile control, we are conducting initial tests of current drive with lower-hybrid and electron-Bernstein waves.

MHD Instabilities in Tokamaks

A general introduction to ideal magnetohydrodynamic (MHD) stability of tokamak plasmas is given, using linear perturbations of the ideal MHD equations. Subsequently the Energy Principle for ideal MHD instabilities is derived. The specific instabilities which are then discussed are loosely divided into two categories. Under the name “current driven instabilities”, external and internal kink modes, which are modes with a large radial extent, are discussed. The internal m = 1 kink mode is responsible for sawtooth collapses and fishbone oscillations in tokamaks. Under the header “pressure driven instabilities”, more localized modes are presented. These modes may limit the pressure gradient in the plasma without causing sizeable disruptions. The ballooning limit and the Mercier criterion are presented. The Troyon limit is mentioned as a synthesis of several of these stability boundaries.

Mercier Stability Improvement in Nonlinear Development of LHD Plasma

A stable path to high-beta plasmas is investigated for an inward-shifted LHD configuration. An improvement of Mercier stability is observed due to the nonlinear saturation of the resistive interchange modes. For this study, a multi-scale numerical scheme is used. In this scheme, the beta value is increased by adding a small pressure increment to the background pressure. We focus on the dependence of the Mercier stability on the profile of the pressure increment. It is found that the total pressure profile approaches a profile marginally stable to the Mercier criterion when the pressure increment has a fixed profile.

MHD Instabilities in Tokamaks

A general introduction to ideal magnetohydrodynamic (MHD) stability of tokamak plasmas is given, using linear perturbations of the ideal MHD equations. Subsequently the Energy Principle for ideal MHD instabilities is derived. The specific instabilities which are then discussed are loosely divided into two categories. Under the name “current driven instabilities”, external and internal kink modes, which are modes with a large radial extent, are discussed. The internal m = 1 kink mode is responsible for sawtooth collapses and fishbone oscillations in tokamaks. Under the header “pressure driven instabilities”, more localized modes are presented. These modes may limit the pressure gradient in the plasma without causing sizeable disruptions. The ballooning limit and the Mercier criterion are presented. The Troyon limit is mentioned as a synthesis of several of these stability boundaries.

Study of electromagnetic microinstabilities in helical systems with the stellarator expansion method

Electromagnetic microinstabilities in helical systems are studied by numerically solving integral eigenmode equations, which are derived from the ion gyrokinetic equation, the quasineutrality equation, the Ampère’s law, and the massless electron approximation. The stellarator expansion technique is used to evaluate finite-beta effects on the guiding-center drift in the helical configuration, where the toroidal plasma shift and the magnetic shear strongly influence the magnetic curvature and accordingly the stability of both magnetohydrodynamics (MHD) and kinetic modes. The kinetic integral equations are shown to reduce to the ideal MHD ballooning mode equation in the fluid limit, from which the Mercier criterion is obtained. For helical geometry like the Large Helical Device (LHD) [Motojima, et al., Nucl. Fusion 43, 1674 (2003)], it is confirmed that, when increasing the beta value, the ion temperature gradient mode is stabilized while the kinetic ballooning mode (KBM) is destabilized due to the unfavorable geodesic curvature resulting from the negative magnetic shear combined with the toroidal plasma shift. Also, dependencies of these kinetic-mode properties on the poloidal wave number and the magnetic shear are investigated. It is found that the KBM-unstable parameter region is narrower than the Mercier-unstable region in the LHD-like configuration.

MHD equilibrium and pressure driven instability in L = 1 heliotron plasmas

Free boundary MHD equilibrium properties of Heliotron J are investigated by VMEC, HINT and PIES codes, and ideal MHD stability properties are studied by the Mercier criterion, the ballooning mode equation and the CAS3D global stability code. It is shown by the equilibrium calculations that the change of the plasma boundary shape is substantial in a low shear helical system even if β is relatively low. Preliminary comparison between PIES results and HINT results shows that the β value at which the magnetic island begins to be perceptible is almost the same in both codes, but the island width seems to be different. It is considered that the difference comes from a difference of pressure distribution in real space which is determined through each numerical algorithm. In the stability analysis, the effect of the global magnetic shear on the three-dimensional or helical ballooning mode whose mode structure shows strong poloidal and toroidal mode (helical mode) coupling is investigated. It is found that the positive shear of the rotational transform is favourable for the three-dimensional ballooning mode stability in a low shear helical system.

The effect of hexapole and vertical fields on α-particle confinement in heliotron configurations

Collisionless confinement of monoenergetic α particles in three-dimensional magnetic fields produced by the magnetic coils of the Large Helical Device is calculated. It is found that the inward shift of the magnetic axis due to the vertical field improves the α-particle confinement. In contrast to the vertical field, both large positive and negative hexapole fields do not improve the confinement. The study of the β effect and Mercier criterion calculations for different hexapole fields are also presented.

Violating Suydam criterion produces feeble instabilities

Eigenmode analysis of a magnetic shear localized ideal interchange mode indicates a marginal stability condition consistent with the Suydam (and Mercier) criterion. However, the linear eigenmode analysis shows that it is not a very restrictive condition because the mode growth rate is exponentially small just above the marginal stability condition and the modes are easily stabilized by finite ion Larmor radius (FLR) effects. This suggests that for practical applications to toroidal confinement experiments, the Suydam–Mercier criterion does not introduce a restrictive condition for operational limits. The criterion for robust growth appears to be almost a factor of two greater than the Suydam criterion. It is also shown that even the presence of usual nonideal effects (FLR, electron diamagnetic flow and resistivity) does not alter this higher criterion for robust growth.

Effects of net toroidal current profile on the Mercier criterion in heliotron plasmas

The effects of variations in net toroidal currenton the Mercier criterion are studied in a heliotron configurationunder the free boundary condition.The plasma column and the magnetic axis are shifted horizontally by the net toroidal current because the free boundary condition changes the effective vertical field.The direction of the shift depends on the direction of the current. In the case of a subtractive current density peaked at the axis, which decreases the rotational transform,the equilibrium is more stable than the no net current equilibrium,although the plasma axis is shifted inwards by the current itself. This is because the Shafranov shift is greater than the inward axis shift caused by the current itself, and the magnetic well is enhanced.In the case of additive currents which increase the rotational transform,there is a tendency for the stability to be improvedas the peak position of the current densityvaries from the axis to the peripheral region.This improvement is due to the enhancement of the magnetic shear at low betaand the magnetic well at high beta. As in the no net current case,the equilibrium with an additive hollow currentunder the free boundary condition is more stable than that under the fixed boundary conditionbecause magnetic well stabilization is enhanced. When the subtractive current peaked at the axis and the additive hollow current flow simultaneously so that the total current is zero or slightly additive,the interchange mode can be stabilized even if the corresponding equilibrium with no net currentis strongly Mercier unstable.