Stability Of Ballooning Modes Research Articles

Overview of ASDEX upgrade results in view of ITER and DEMO

Experiments on ASDEX Upgrade (AUG) in 2021 and 2022 have addressed a number of critical issues for ITER and EU DEMO. A major objective of the AUG programme is to shed light on the underlying physics of confinement, stability, and plasma exhaust in order to allow reliable extrapolation of results obtained on present day machines to these reactor-grade devices. Concerning pedestal physics, the mitigation of edge localised modes (ELMs) using resonant magnetic perturbations (RMPs) was found to be consistent with a reduction of the linear peeling-ballooning stability threshold due to the helical deformation of the plasma. Conversely, ELM suppression by RMPs is ascribed to an increased pedestal transport that keeps the plasma away from this boundary. Candidates for this increased transport are locally enhanced turbulence and a locked magnetic island in the pedestal. The enhanced D-alpha (EDA) and quasi-continuous exhaust (QCE) regimes have been established as promising ELM-free scenarios. Here, the pressure gradient at the foot of the H-mode pedestal is reduced by a quasi-coherent mode, consistent with violation of the high-n ballooning mode stability limit there. This is suggestive that the EDA and QCE regimes have a common underlying physics origin. In the area of transport physics, full radius models for both L- and H-modes have been developed. These models predict energy confinement in AUG better than the commonly used global scaling laws, representing a large step towards the goal of predictive capability. A new momentum transport analysis framework has been developed that provides access to the intrinsic torque in the plasma core. In the field of exhaust, the X-Point Radiator (XPR), a cold and dense plasma region on closed flux surfaces close to the X-point, was described by an analytical model that provides an understanding of its formation as well as its stability, i.e., the conditions under which it transitions into a deleterious MARFE with the potential to result in a disruptive termination. With the XPR close to the divertor target, a new detached divertor concept, the compact radiative divertor, was developed. Here, the exhaust power is radiated before reaching the target, allowing close proximity of the X-point to the target. No limitations by the shallow field line angle due to the large flux expansion were observed, and sufficient compression of neutral density was demonstrated. With respect to the pumping of non-recycling impurities, the divertor enrichment was found to mainly depend on the ionisation energy of the impurity under consideration. In the area of MHD physics, analysis of the hot plasma core motion in sawtooth crashes showed good agreement with nonlinear 2-fluid simulations. This indicates that the fast reconnection observed in these events is adequately described including the pressure gradient and the electron inertia in the parallel Ohm’s law. Concerning disruption physics, a shattered pellet injection system was installed in collaboration with the ITER International Organisation. Thanks to the ability to vary the shard size distribution independently of the injection velocity, as well as its impurity admixture, it was possible to tailor the current quench rate, which is an important requirement for future large devices such as ITER. Progress was also made modelling the force reduction of VDEs induced by massive gas injection on AUG. The H-mode density limit was characterised in terms of safe operational space with a newly developed active feedback control method that allowed the stability boundary to be probed several times within a single discharge without inducing a disruptive termination. Regarding integrated operation scenarios, the role of density peaking in the confinement of the ITER baseline scenario (high plasma current) was clarified. The usual energy confinement scaling ITER98(p,y) does not capture this effect, but the more recent H20 scaling does, highlighting again the importance of developing adequate physics based models. Advanced tokamak scenarios, aiming at large non-inductive current fraction due to non-standard profiles of the safety factor in combination with high normalised plasma pressure were studied with a focus on their access conditions. A method to guide the approach of the targeted safety factor profiles was developed, and the conditions for achieving good confinement were clarified. Based on this, two types of advanced scenarios (‘hybrid’ and ‘elevated’ q-profile) were established on AUG and characterised concerning their plasma performance.

Influence of the shape of a conducting chamber on the stability of rigid ballooning modes in a mirror trap

MHD stabilization of flute and ballooning modes in an axisymmetric mirror trap is studied under the assumption of strong finite Larmor radius effect that suppresses all perturbations with azimuthal numbers m⩾2 and makes the m = 1 mode ‘rigid’. The rigid mode can be effectively suppressed using perfectly conducting lateral wall without any additional means of stabilization or in combination with end MHD anchors. Numerical calculations were carried out for an anisotropic plasma produced in the course of neutral beam injection into the minimum of the magnetic field at the right angle to the trap axis. The stabilizing effect of the conducting shell made of a straightened cylinder is compared with a proportional chamber, which, on an enlarged scale, repeats the shape of the plasma column. It is confirmed that for convincing wall stabilization of the rigid modes, the plasma beta (β, the ratio of the plasma pressure to the magnetic field pressure) must exceed some critical value βcr2 . When conducting lateral wall is combined with conducting end plates imitating MHD end anchors, there are two critical betas and, respectively, two stability zones β<βcr1 and β>βcr2 that can merge, making the entire range 0<β<1 of betas allowable for stable plasma confinement. The dependence of the critical betas on the plasma anisotropy, mirror ratio, width of the vacuum gap between the plasma column and the lateral wall, radial pressure profile and the axial magnetic field profile is examined.

Nonlinear second order electromagnetic gyrokinetic theory for a tokamak plasma

The steep plasma pressure gradient that forms at the edge of the high confinement, H-mode regime of tokamak operation provides free energy to drive electromagnetic micro-instabilities that are widely believed to influence the transport processes in this so-called pedestal region. This high pressure gradient also provides a high current density (bootstrap current), known to influence ballooning mode stability and to be important for driving kink modes in the ideal magneto-hydrodynamic plasma model (so-called peeling-ballooning modes). Furthermore, efficient, steady state future tokamak power plants must operate with a large bootstrap current in the core and especially concerning spherical tokamaks, confinement will be influenced by electromagnetic turbulence. To accommodate these important situations, conventional electromagnetic gyrokinetic theory is extended to incorporate neoclassical effects in the equilibrium drives, allowing (B 0 is the confining magnetic field, and B ϑ is its poloidal component). This provides a global gyrokinetic model that self-consistently captures the consequences of large bootstrap current fractions on the equilibrium distribution functions.

Finite magnetic well effects on resistive and drift-resistive ballooning modes in a shaped tokamak

The impact of plasma shaping through magnetic well modifications on the stability of resistive ballooning modes (RBMs) in tokamaks is analysed, also including finite diamagnetic flows. Various limiting cases of the dispersion relation, obtained by matching the ballooning equation across the ideal and resistive layers, are analysed. It is found that stability is generally improved by the combination of vertical elongation and positive triangularity, although, in some cases, the growth rate of the unstable mode can be enhanced by these effects. Usually, vertically elongated plasmas with no triangularity are prone to exhibit worse stability properties. A value for the critical β above which RBMs are driven unstable is identified, and a connection with type-III ELM activity is established.

On the stability of small-scale ballooning modes in axisymmetric mirror traps

It is shown that a steepening of the radial plasma pressure profile leads to a decrease in the critical value of beta, above which, small-scale balloon-type perturbations in a mirror trap become unstable. This may mean that small-scale ballooning instability leads to a smoothing of the radial plasma profile. The critical beta values for the real magnetic field of the gas-dynamic trap and various plasma pressure radial profiles was also calculated. For a plasma with a parabolic profile critical beta is evaluated at the level of 0.72. A previous theoretical prediction for this trap was almost two times lower than maximal beta 0.6 achieved experimentally.

Role of the pedestal current on the stability of non-ideal ballooning modes

On the basis of a three-field flute-reduced magnetohydrodynamic model, which mainly describes the edge instabilities by shielding a major part of the J × B force in the flute reduction, we study the stability of ballooning modes in the edge pedestal, highlighting the role of an equilibrium parallel current gradient. This effect, which is designated as the current gradient driven (CGD) term in this paper, is shown to have an influence on the stability of finite-n pedestal ballooning modes due to the existence of a highly localized bootstrap current. An analysis in the ideal limit shows that the CGD term destabilizes the ballooning modes regardless of the sign of its gradient, especially near the stability boundaries. An inclusion of the finite Larmor radius (FLR) effect via ion diamagnetic flow and finite resistivity results in a coupling of the FLR effect and the current gradient. In this particular regime where the deviation from the ideal stability is considerable, this coupling effect is shown to dominate stability in intermediate n (20&amp;lt;n≤40) modes. Here, n is the toroidal mode number. Stability analyses using a series of model pedestal equilibria indicate that an increase in a bootstrap current can move the most unstable position from the pedestal middle to the bottom and the negative gradient of the bootstrap current at the pedestal bottom leads to further destabilization of intermediate n modes.

Non-ideal effects on ballooning mode stability in the presence of resonant magnetic perturbations

The ideal ballooning mode model in the presence of the externally applied resonant magnetic perturbation (RMP), developed by Bird and Hegna [Nucl. Fusion 53, 013004 (2013)], is extended to include the non-ideal effects, i.e., the finite resistivity and diamagnetic drift effects. Using the eigenvalue equation with the non-ideal effects, a comprehensive analysis is carried out to elucidate the impact of the RMP on ballooning mode stability. Finite resistivity is shown to reduce the line bending stabilization in low to intermediate toroidal mode number (n), resulting in the more peaked growth rate spectrum with respect to n. It is shown that the combination of finite resistivity and the RMP-induced local shear modulation have a strong influence on line bending stabilization, leading to an interesting threshold behavior of the ballooning mode stability. This signifies the importance of the RMP to the line bending stabilization coupled with finite resistivity, as well as the local shear modulation.

Studies of the pedestal structure and inter-ELM pedestal evolution in JET with the ITER-like wall

The pedestal structure of type I ELMy H-modes has been analysed for JET with the ITER-like Wall (JET-ILW). The electron pressure pedestal width is independent of ρ * and increases proportionally to √β pol,PED. Additional broadening of the width is observed, at constant β pol, PED, with increasing ν * and/or neutral gas injection and the contribution of atomic physics effects in setting the pedestal width cannot as yet be ruled out. Neutral penetration alone does not determine the shape of the edge density profile in JET-ILW. The ratio of electron density to electron temperature scale lengths in the edge transport barrier region, η e, is of order 2–3 within experimental uncertainties. Existing understanding, represented in the stationary linear peeling–ballooning mode stability and the EPED pedestal structure models, is extended to the dynamic evolution between ELM crashes in JET-ILW, in order to test the assumptions underlying these two models. The inter-ELM temporal evolution of the pedestal structure in JET-ILW is not unique, but depends on discharge conditions, such as heating power and gas injection levels. The strong reduction in p e,PED with increasing D 2 gas injection at high power is primarily due to clamping of half way through the ELM cycle and is suggestive of turbulence limiting the T e pedestal growth. The inter-ELM pedestal pressure evolution in JET-ILW is consistent with the EPED model assumptions at low gas rates and only at low beta at high gas rates. At higher beta and high gas rate the inter-ELM pedestal pressure evolution is qualitatively consistent with the kinetic ballooning mode (KBM) constraint but the peeling–ballooning (P–B) constraint is not satisfied and the ELM trigger mechanism remains as yet unexplained.

A refined understanding of compressibility effects on the stability of drift ballooning modes

A study is conducted on the impact of plasma compressibility on the stability of drift ballooning modes. The two-fluid and four-field model developed by Hazeltine et al. [Phys. Fluids 28, 2466 (1985)] is employed in this study. Results of linear numerical simulations show that finite compressibility destabilizes ballooning modes which are otherwise stable due to the ion diamagnetic drift effect. A systematic study reveals that the parallel compressibility originating from the two-fluid effect, rather than the drift-acoustic wave coupling suggested by Hastie et al. [Phys. Plasmas 10, 4405 (2003)], plays the most important role in destabilizing the ballooning modes. An analytic evaluation of the dispersion relation underpins the strong sensitivity of this parallel compressibility term in the ballooning mode stability. The potential impact of this new understanding on the physics of small edge localized modes is discussed.

Non-local effects on pedestal kinetic ballooning mode stability

The H-mode pedestal height plays an important role in determining the global confinement of the tokamak plasma. In type I ELMy H-mode the ultimate limit for the pedestal pressure at constant width is set by the ideal MHD peeling-ballooning modes (PBMs) that are thought to be the trigger for the ELMs. However, the PBM criterion does not uniquely determine the pedestal. Increasing the width of the pedestal, the marginally peeling-ballooning stable pedestal height increases as well. The second criterion for the pedestal is set by the transport processes in the pedestal that limit the gradient between the ELMs. One candidate for driving this transport is the kinetic ballooning mode (KBM) that is driven by the pressure gradient (Snyder et al 2009 Nucl. Fusion 49 085035). The KBM growth rate increases very rapidly after the critical pressure gradient is exceeded leading to very stiff profiles with the pressure gradient clamped near to the stability limit. In the local linear gyrokinetic analysis of experimental MAST and JET plasmas we have found that, like the ideal MHD ballooning modes, the KBMs can access locally so called second stability if the magnetic shear becomes low enough (Dickinson et al 2011 Plasma Phys. Control. Fusion 53 115010; Saarelma et al 2013 Nucl. Fusion 53 123012). However, in the pedestal region the local assumption that the equilibrium can be considered radially constant for the investigated modes is no longer justified. In this paper we revisit the KBM analysis using a global code ORB5 to investigate whether second stability access exists for KBMs. We find that counter to the local analysis, the global KBM stability is not sensitive to the magnetic shear in the pedestal region. At sufficiently high β (but still below the ideal peeling-ballooning limit) the pedestal region becomes KBM unstable regardless of the amount of bootstrap current assumed in the equilibrium reconstruction. However, just as in local analysis, the mode is stabilised by reducing the pressure gradient. This suggests that KBMs can regulate the pedestal pressure gradient during the ELM cycle even when local analysis finds them stable due to high bootstrap current.

Stationary QH-mode plasmas with high and wide pedestal at low rotation on DIII-D

A stationary, quiescent H-mode (QH-mode) regime with a wide pedestal and improved confinement at low rotation has been discovered on DIII-D with reactor relevant edge parameters and no ELMs. As the injected neutral beam torque is ramped down and the edge E × B rotation shear reduces, the transition from standard QH to the wide pedestal QH-mode occurs. At the transition, the coherent edge harmonic oscillations (EHO) that usually regulate the standard QH edge cease and broadband edge MHD modes appear along with a rapid increase in the pedestal pressure height (by ⩽60%) and width (by ⩽50%). We posit that the enhanced edge turbulence-driven transport, enabled by the lower edge E × B flow shear due to lower torque reduces the pedestal gradient and, combined with the high edge instability limit provided by the balanced double-null plasma shape, permits the development of a broader and thus higher pedestal that is turbulence-transport-limited. Even with the significantly enhanced pedestal pressure, the edge operating point is below the peeling ballooning mode stability boundary and thus without ELMs. Improved transport in the outer core region (0.8 ⩽ ρ ⩽0.9) owing to increased E × B flow shear in that region and the enhanced pedestal boost the overall confinement by up to 45%. These findings advance the physics basis for developing stationary ELM-free high-confinement operation at low rotation for future burning plasma where similar collisionality and rotation levels are expected.

Impact of ion diamagnetic drift on ideal ballooning mode stability in rotating tokamak plasmas

Drift magnetohydrodynamic (MHD) equations have been derived in order to investigate the ion diamagnetic drift effect on the stability to ideal MHD modes in rotating plasmas. These drift MHD equations have been simplified with the Frieman-Rotenberg formalism under the incompressible assumption, and a new code, MINERVA-DI, has been developed to solve the derived extended Frieman-Rotenberg equation. Benchmark results of the MINERVA-DI code show good agreements with the analytic theory discussing the stability to an internal kink mode and that to a ballooning mode in static plasmas. The stability analyses of the ballooning mode with respect to toroidal rotation with the ion diamagnetic drift effect have been performed using MINERVA-DI. The stabilizing effect by the ion diamagnetic drift is found to be negligible when the rotation frequency is large compared to the ion diamagnetic drift frequency. The direction of plasma rotation affects the ballooning mode stability when the ion diamagnetic drift effect is taken into account. It is identified that there are two physics mechanisms responsible for the dependence of MHD stability on the rotation direction. One is the correction of the dynamic pressure effect on MHD stability by the ion diamagnetic drift, and the other is the change of the MHD eigenmode structure by the combined effect of plasma rotation and ion diamagnetic drift.

Gyro-fluid and two-fluid theory and simulations of edge-localized-modes

This paper reports on the theoretical and simulation results of a gyro-Landau-fluid extension of the BOUT++ code, which contributes to increasing the physics understanding of edge-localized-modes (ELMs). Large ELMs with low-to-intermediate-n peeling-ballooning (P-B) modes are significantly suppressed due to finite Larmor radius (FLR) effects when the ion temperature increases. For type-I ELMs, it is found from linear simulations that retaining complete first order FLR corrections as resulting from the incomplete “gyroviscous cancellation” in Braginskii's two-fluid model is necessary to obtain good agreement with gyro-fluid results for high ion temperature cases (Ti≽3 keV) when the ion density has a strong radial variation, which goes beyond the simple local model of ion diamagnetic stabilization of ideal ballooning modes. The maximum growth rate is inversely proportional to Ti because the FLR effect is proportional to Ti. The FLR effect is also proportional to toroidal mode number n, so for high n cases, the P-B mode is stabilized by FLR effects. Nonlinear gyro-fluid simulations show results that are similar to those from the two-fluid model, namely that the P-B modes trigger magnetic reconnection, which drives the collapse of the pedestal pressure. Due to the additional FLR-corrected nonlinear E × B convection of the ion gyro-center density, for a ballooning-dominated equilibrium the gyro-fluid model further limits the radial spreading of ELMs. In six-field two fluid simulations, the parallel thermal diffusivity is found to prevent the ELM encroachment further into core plasmas and therefore leads to steady state L-mode profiles. The simulation results show that most energy is lost via ion channel during an ELM event, followed by particle loss and electron energy loss. Because edge plasmas have significant spatial inhomogeneities and complicated boundary conditions, we have developed a fast non-Fourier method for the computation of Landau-fluid closure terms based on an accurate and tunable approximation. The accuracy and the fast computational scaling of the method have been demonstrated.

Ideal ballooning modes, shear flow and the stable continuum

There is a well-established theory of ballooning modes in a toroidal plasma. The cornerstone of this is a local eigenvalue λ on each magnetic surface—which also depends on the ballooning phase angle k. In stationary plasmas, λ(k) is required only near its maximum, but in rotating plasmas its average over k is required. Unfortunately in many cases λ(k) does not exist for some range of k, because the spectrum there contains only a stable continuum. This limits the application of the theory, and raises the important question of whether this ‘stable interval’ gives rise to significant damping. This question is re-examined using a new, simplified, model—which leads to the conclusion that there is no appreciable damping at small shear flow. In particular, therefore, a small shear flow should not affect ballooning mode stability boundaries.

Numerical analysis of key factors for the appearance of grassy ELMs in tokamak plasmas

Numerical stability analysis of edge-localized MHD mode is performed to identify the origin of small-amplitude ‘grassy ELMs’ on the basis of current understanding of kinetic effects on ballooning mode stability. These qualitative and quantitative analyses show that short wavelength ballooning mode can play an important role in a grassy ELM stability even when kinetic effects are taken into account. After showing the importance of kinetic effects for discussing grassy ELM physics, impacts of plasma parameters important for realizing a grassy ELM plasma experimentally are investigated numerically from the viewpoint of the edge-localized MHD stability including these kinetic effects. These analyses show that low plasma ellipticity is preferable to realize a grassy ELM plasma due to destabilizing ballooning mode by preventing access to the second stability region of the ballooning mode.

Analysis of high β regimes for DEMO

Stability limit calculations are presented for a range of tokamak power plant equilibria. The current drive requirements to sustain the optimised equilibrium profiles are confirmed by a transport code and the plasma shape is obtained from free-boundary equilibrium calculations. A pressure pedestal is included according to empirical scaling and ballooning mode stability limits. A terative optimisation of the profiles is undertaken to improve the baseline profiles in order to achieve the highest possible plasma performance and most favourable magnetohydrodynamic stability within conservative assumptions in order to increase confidence in the availability and control of the plasma. This results in a fully noninductive baseline operating scenario for a tokamak power plant design which has a broad low-shear q-profile which is meant to complement previous advanced tokamak design studies.

Unified theory of resistive and inertial ballooning modes in three-dimensional configurations

Analytic results for the stability of resistive ballooning modes (RBMs) and electron inertial ballooning modes are obtained using a two-scale analysis. This work generalizes previous calculations used for axisymmetric ŝ−α geometry [R. H. Hastie, J. J. Ramos, and F. Porcelli, Phys. Plasmas 10, 4405 (2003)] to general three-dimensional geometry. A unified theory is developed for RBMs and inertial ballooning modes, in which the effects of both ideal magnetohydrodynamic free energy (as measured by the asymptotic matching parameter Δ′) and geodesic curvature drives in the nonideal layer are included in the dispersion relation. This unified theory can be applied to determine the stability of drift-resistive-inertial ballooning modes in the low temperature edge regions of tokamak and stellarator plasmas where steep density gradients exist.

Effects of a sheared toroidal rotation on the stability boundary of the MHD modes in the tokamak edge pedestal

Effects of a sheared toroidal rotation are investigated numerically on the stability of the MHD modes in the tokamak edge pedestal, which relate to the type-I edge-localized mode. A linear MHD stability code MINERVA is newly developed for solving the Frieman–Rotenberg equation that is the linear ideal MHD equation with flow. Numerical stability analyses with this code reveal that the sheared toroidal rotation destabilizes edge localized MHD modes for rotation frequencies which are experimentally achievable, though the ballooning mode stability changes little by rotation. This rotation effect on the edge MHD stability becomes stronger as the toroidal mode number of the unstable MHD mode increases when the stability analysis was performed for MHD modes with toroidal mode numbers smaller than 40. The toroidal mode number of the unstable MHD mode depends on the stabilization of the current-driven mode and the ballooning mode by increasing the safety factor. This dependence of the toroidal mode number of the unstable mode on the safety factor is considered to be the reason that the destabilization by toroidal rotation is stronger for smaller edge safety factors.

MHD stability analysis of small ELM regimes in JET

We have analysed the edge stability of JET discharges with small edge localized modes (ELMs) using the high resolution Thomson scattering system for accurate edge profiles in the equilibrium reconstruction. For the reference plasmas with large Type I ELMs we confirm the results from earlier analyses that the edge stability is limited by intermediate-n peeling–ballooning modes with a relatively large radial extent. The double null configuration needed to replace Type I ELMs by smaller Type II ELMs considerably increases the stability against these modes while the stability against n = ∞ ballooning modes is not affected. When this is combined with high collisionality (which is the other requirement for Type II ELMs), we find that the plasma cannot reach the Type I ELM triggering peeling–ballooning mode stability boundary before it is destabilized by high-n ballooning modes resulting in more benign ELMs. The ELM mitigation by magnetic perturbation causes the edge stability to be limited by pure peeling modes with a narrow radial extent. This explains the smaller ELM size and also why the ELMs are not fully suppressed. The transition from Type I ELMs to Type III ELMs by increasing the edge radiation fully stabilizes the edge plasma against ideal MHD modes. Therefore, the Type III ELMs are due to be triggered by some other mechanism than an ideal MHD instability.

Effect of equilibrium properties on the structure of the edge MHD modes in tokamaks

Effects of the pressure profile and the current density profile inside the top of the pedestal and that of the plasma shape on the expansion of the structure of the unstable edge MHD mode are investigated numerically. The structure of the edge MHD mode is expanded by spreading the envelope of the edge ballooning mode structure due to the increase of the pressure gradient inside the top of the pedestal, and by decreasing the toroidal mode number of the most unstable mode due to the increase of the current density inside the top of the pedestal. In strongly shaped plasmas, the pressure gradient inside the top of the pedestal can approach to the ballooning mode stability boundary and the current density increases enough to reduce the toroidal mode number of the most unstable mode. These increases of the pressure gradient and the current density destabilize the edge MHD mode and expand the structure of this mode.