Ideal Internal Kink Research Articles

MUSES: A nonlinear magnetohydrodynamics discontinuous Galerkin code for fusion plasmas

The article describes initial steps toward the development of our nonlinear magnetohydrodynamics code, MUSES. Unstructured grids and general coordinates are employed so that MUSES code is expected to have a capability to represent realistic geometries of fusion devices. Discontinuous Galerkin method is chosen as a code framework because of its high spatial accuracy and numerical stability on unstructured grids. The governing equation of the ideal magnetohydrodynamics in this work is Powell 8-wave model which can mitigate the divergence-free issue. Roe scheme is used as a numerical flux since the Powell 8-wave model is a nonconservative system. A cross code comparison is carried out with a linear magnetohydrodynamics code, MINERVA. Owing to discontinuous Galerkin method, a linear theory of an ideal internal kink mode is reproduced quantitatively although the upwind method is employed for numerical stability.

Numerical simulations of fishbones driven by fast ions in negative triangularity tokamak

The discharges with negative triangularity have lower turbulence induced transport and better energy confinement, so the tokamak with negative triangularity is recognized to be a better choice for future fusion device. In order to explore the features of the energetic particle driven instabilities with negative triangularity, the kinetic-magnetohydrodynamic hybrid code M3D-K is used to investigate the linear instability and nonlinear evolution of the fishbone driven by energetic ions with different triangularity. Based on EAST-like parameters, it is found that the negative triangularity destabilizes the ideal internal kink mode, but stabilizes the fishbone instability. Nonlinear simulations show that the fishbone instability with negative triangularity is hard to saturate without fluid nonlinearity. The possible explanation is that the orbits of fast ions are located more centrally with negative triagularity, so the energy exchange between energetic ions and the fishbone is more efficient than that with positive triangularity. These simulation results demonstrate that the negative triangularity does not have an obvious advantage over the positive triangularity, with the fishbone driven by energetic particles considered.

Ideal internal kink stability in presence of plasma flow and neoclassical toroidal viscosity due to energetic particles

The influence of energetic particles (EPs) on the ideal internal kink mode, in rotating tokamak plasmas, is numerically investigated by simultaneously solving MHD-kinetic hybrid equations together with a toroidal momentum balance equation utilizing the MARS-Q code (Liu et al 2013 Phys. Plasmas 20 042503). The neoclassical toroidal viscous (NTV) torque, induced by precessional drift resonances of trapped energetic particles, acts as the momentum sink term to damp the plasma flow. Quasi-linear initial value simulations show local reduction of the flow amplitude and enhancement of the flow shear near the q = 1 rational surface (q is the safety factor) due to EP induced NTV. Both effects in turn destabilize the internal kink mode. These numerical findings are robust against the initial linear stability of internal kink, the initial plasma flow profile, as well as the equilibrium distribution model for EPs.

Toroidal modelling of resistive internal kink and fishbone instabilities

The influence of energetic particles and plasma resistivity on the n=1 (n is the toroidal mode number) internal kink and fishbone modes in tokamak plasmas is numerically investigated, using the full toroidal, resistive magnetohydrodynamic-kinetic hybrid stability code MARS-K [Liu et al., Phys. Plasmas 15 112503 (2008)]. The results show that energetic particles can either stabilize or destabilize the ideal internal kink mode, depending on the radial profiles of the particles' density and pressure. Resistive fishbones with and without an ideal wall are investigated. It is found that, in the presence of energetic particles as well as plasma resistivity, two branches of unstable roots exist, for a plasma which is ideally stable to the internal kink instability. One is the resistive internal kink mode. The other is the resistive fishbone mode. These two-branch solutions show similar behaviors, independent of whether the initial ideal kink stability is due to an ideal wall stabilization for high-beta plasmas, or due to a stable equilibrium below the Bussac pressure limit. For a realistic toroidal plasma, the resistive internal kink is the dominant instability, which grows much faster than the resistive fishbone. The plasma resistivity destabilizes the resistive internal kink while stabilizes the resistive fishbone. Systematic comparison with an analytic model qualitatively confirms the MARS-K results. Compared to analytic models based on the perturbative approach, MARS-K offers an improved physics model via self-consistent treatment of coupling between the fluid and kinetic effects due to energetic particles.

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.

Trapped fast particle destabilization of internal kink mode for the locally flattened q-profile with an inflection point

The destabilization of ideal internal kink modes by trapped fast particles in tokamak plasmas with a “shoulder”-like equilibrium current is investigated. It is found that energetic particle branch of the mode is unstable with the driving of fast-particle precession drifts and corresponds to a precessional fishbone. The mode with a low stability threshold is also more easily excited than the conventional precessional fishbone. This is different from earlier studies for the same equilibrium in which the magnetohydrodynamic (MHD) branch of the mode is stable. Furthermore, the stability and characteristic frequency of the mode are analyzed by solving the dispersion relation and comparing with the conventional fishbone. The results suggest that an equilibrium with a locally flattened q-profile, may be modified by localized current drive (or bootstrap current, etc.), is prone to the onset of the precessional fishbone branch of the mode.

Bayesian soft x-ray tomography and MHD mode analysis on HL-2A

A Bayesian based tomography method using so-called Gaussian processes (GPs) for the emission model has been applied to the soft x-ray (SXR) diagnostics on HL-2A tokamak. To improve the accuracy of reconstructions, the standard GP is extended to a non-stationary version so that different smoothness between the plasma center and the edge can be taken into account in the algorithm. The uncertainty in the reconstruction arising from measurement errors and incapability can be fully analyzed by the usage of Bayesian probability theory. In this work, the SXR reconstructions by this non-stationary Gaussian processes tomography (NSGPT) method have been compared with the equilibrium magnetic flux surfaces, generally achieving a satisfactory agreement in terms of both shape and position. In addition, singular-value-decomposition (SVD) and Fast Fourier Transform (FFT) techniques have been applied for the analysis of SXR and magnetic diagnostics, in order to explore the spatial and temporal features of the saturated long-lived magnetohydrodynamics (MHD) instability induced by energetic particles during neutral beam injection (NBI) on HL-2A. The result shows that this ideal internal kink instability has a dominant m/n = 1/1 mode structure along with a harmonics m/n = 2/2, which are coupled near the q = 1 surface with a rotation frequency of 12 kHz.

Free boundary equilibrium in 3D tokamaks with toroidal rotation

The three-dimensional VMEC equilibrium solver has been adapted to numerically investigate the approximate toroidal rotation model we have derived. We concentrate our applications on the simulation of JET snakes and MAST long-lived modes under free boundary conditions. Helical core solutions are triggered when 〈β〉 exceeds a threshold value, typically 2.7% in JET-like plasmas. A large plasma current and edge bootstrap current can drive helical core formations at arbitrarily small 〈β〉 in which the ideal saturated internal kink coexists with an ideal saturated external kink structure of opposite phase. The centrifugal force linked with the rotation has the effect of displacing the plasma column away from the major axis, but does not alter significantly the magnitude of the edge corrugation of the plasma. Error field correction coil currents in JET-like configurations increase the outer midplane distortions by 2 cm. The edge bootstrap current enhances the edge modulation of the plasma driven by the core snake deformations in MAST.

Bifurcated helical core equilibrium states in tokamaks

Tokamaks with weak to moderate reversed central shear in which the minimum inverse rotational transform (safety factor) qmin is in the neighbourhood of unity can trigger bifurcated magnetohydrodynamic equilibrium states, one of which is similar to a saturated ideal internal kink mode. Peaked prescribed pressure profiles reproduce the ‘snake’ structures observed in many tokamaks which has led to a novel explanation of the snake as a bifurcated equilibrium state. Snake equilibrium structures are computed in simulations of the tokamak à configuration variable (TCV), DIII-D and mega amp spherical torus (MAST) tokamaks. The internal helical deformations only weakly modulate the plasma–vacuum interface which is more sensitive to ripple and resonant magnetic perturbations. On the other hand, the external perturbations do not alter the helical core deformation in a significant manner. The confinement of fast particles in MAST simulations deteriorate with the amplitude of the helical core distortion. These three-dimensional bifurcated solutions constitute a paradigm shift that motivates the applications of tools developed for stellarator research in tokamak physics investigations.

Observation of 1/1 impurity-related ideal internal kink mode locking in the EAST tokamak

m/n = 1/1 pressure gradient driven impurity-related ideal internal kink mode locking is observed in EAST limit-cycle state H-mode plasmas. This 1/1 mode has a radial structure with no inversion around the resonance, which implies that the mode has an ideal internal kink structure. The rotation direction of the 1/1 internal kink mode, as obtained by tomography of a high-resolution multi-array soft x-ray system, is the ion diamagnetic drift velocity direction. Impurities accumulated in the central region play a vital role in mode excitation. Remarkable loss of the yield of neutron rate due to 1/1 mode is also observed. A notch at the location in the vicinity of q = 1 is also observed in the local toroidal rotation profile. Furthermore, the frequency chirping behavior of the 1/1 kink mode is related to the toroidal rotation velocity of the plasma.

MHD properties in the core of ITER-like hybrid scenarios

MHD stability is studied in the frame of ITER like hybrid scenarios, characterized by weak reversed shear, with particular attention on ideal internal kink modes and infernal mode stability. Numerical simulations for ITER like equilibria, with hollow q profiles with an off axis minimum close to unity, were carried out using the 3-D equilibrium code ANIMEC, which showed the presence of a 3-D helical core [1] with the characteristics of a saturated internal kink mode. The internal kink perturbation has been investigated non-linearly using the XTOR code, in the ideal frame. A scan in the current was performed and it has been found that when the minimum of q is above the unity (low currents), the helical distorsion shows good agreement with the results provided by the 3-D ANIMEC simulations while for qmin below 1 (high currents), XTOR gives a residual distorsion in contrast with the ANIMEC results. Moreover infernal mode stability in hybrid scenarios has been studied analytically extending the quasi-interchange model with the inclusion of resistive and both electron and ion diamagnetic effects. This enables us to investigate kinetic effects on plasma scenarios susceptible to saturated kink structures exhibited into ANIMEC simulations.

Magnetohydrodynamic equilibrium and the stability of tokamak and reversed-field pinch systems with 3D helical cores

Bifurcated magnetohydrodynamic (MHD) equilibrium states are computed for ITER hybrid scenario and RFX-mod SHAx configurations with very flat or reversed core magnetic shear conditions. In the ITER studies, the minimum inverse rotational transform qmin is near unity, while for RFX-mod it is 1/8. Two equilibrium states are obtained: one is axisymmetric, the other displays a 3D helical core. In tokamak devices, the structure resembles a saturated ideal MHD internal kink mode. In the reversed-field pinch, the structure is seven-fold toroidally periodic. The equilibrium magnetic field spectrum in the Boozer coordinate frame is calculated in both the ITER and RFX-mod configurations and the implications are discussed. The RFX-mod equilibria are strongly unstable to external ideal MHD kink modes, which become stabilized with a closely fitting conducting shell when the equilibrium state has a weak reversed core shear. It is marginally unstable with a monotonic q-profile. Unstable modes are driven by the Ohmic current, with pressure and Pfirsch–Schlüter currents having a very weak effect. The external kink mode spectrum is dominated by coupled m = 1, n = 6 and m = 2, n = 13 Fourier components, which revert to m = 1, n = 8 and m = 2, n = 15 terms with a conducting wall in proximity to the plasma–vacuum interface.

Helical ITER hybrid scenario equilibria

Numerical computations of ITER equilibria in the hybrid scenario using a three-dimensional (3D) magnetohydrodynamic equilibrium code with nested magnetic flux surfaces demonstrate the formation of internal 3D helical cores similar to saturated ideal internal kink modes under reversed magnetic shear conditions when the minimum value of the inverse rotational transform is in the neighbourhood of unity.

A new sawtooth control mechanism relying on toroidally propagating ion cyclotron resonance frequency waves: Theory and Joint European Torus tokamak experimental evidence

The sawtooth control mechanism in plasmas employing toroidally propagating ion cyclotron resonance waves is extended. The asymmetrically distributed energetic passing ions are shown to modify the ideal internal kink mode when the position of the minority ion cyclotron resonance resides within a narrow region close to the q=1 surface. An analytical treatment of the internal kink mode in the presence of model distribution function with parallel velocity asymmetry is developed. The fast ion mechanism explains the strong sensitivity of sawteeth to resonance position, and moreover is consistent with dedicated Joint European Torus [F. Romanelli, Nucl. Fusion 49, 104006 (2009)] experiments which controlled sawteeth despite negligible current drive.

Effects of line-tying on magnetohydrodynamic instabilities and current sheet formation

An overview of some recent progress on magnetohydrodynamic stability and current sheet formation in a line-tied system is given. Key results on the linear stability of the ideal internal kink mode and resistive tearing mode are summarized. For nonlinear problems, a counterexample to the recent demonstration of current sheet formation by Low and Janse [Astrophys. J. 696, 821 (2009)] is presented, and the governing equations for quasistatic evolution of a boundary driven, line-tied magnetic field are derived. Some open questions and possible strategies to resolve them are discussed.

Ideal internal kink modes in a differentially rotating cylindrical plasma

The Velikhov effect leading to magnetorotational instability (MRI) is incorporated into the theory of ideal internal kink modes in a differentially rotating cylindrical plasma column. It is shown that this effect can play a stabilizing role for suitably organized plasma rotation profiles, leading to suppression of MHD (magnetohydrodynamic) instabilities in magnetic confinement systems. The role of this effect in the problem of the Suydam and the m = 1 internal kink modes is elucidated, where m is the poloidal mode number.

Safety factor corrections to the magnetohydrodynamic internal kink mode in a tokamak

It has long been acknowledged that the well known and frequently used stability criterion βp<0.3 for the toroidal ideal magnetohydrodynamic internal kink mode in a tokamak is inaccurate for an empirically relevant safety factor. The present paper outlines the severity of the usual approximation, and presents improved analytical approximations of the general solution in M. N. Bussac, R. Pellat, D. Edery, and J. L. Soulé, Phys. Rev. Lett. 35, 1638 (1975), thus providing new insights into the nature of the instability, together with simple formulas that can be incorporated into transport codes with sawtooth cycle algorithms.

Sawtooth behaviour in highly elongated TCV plasmas

The effect of plasma elongation κ on the sawtooth behaviour is studied in ohmically heated TCV plasmas up to values of κ = 2.8. Above a certain value of κ, which varies from 2.2 to 2.6 and depends on the current profile, the sawtooth oscillations disappear completely. The sawteeth are then replaced by continuous oscillations with frequencies in the range from 5 to 20 kHz, which are resonant at the q = 1 surface and seem to prevent a peaking of the temperature and pressure profiles in the plasma core. It is suggested that the transition is a consequence of an ideal internal kink mode becoming unstable, even with a flat pressure profile inside the q = 1 surface, if κ is sufficiently high.

M = 1 ideal internal kink modes in a line-tied screw pinch

It is well known that the radial displacement of the m=1 internal kink mode in a periodic screw pinch has a steep jump at the resonant surface where k∙B=0 [Rosenbluth, Dagazian, and Rutherford, Phys. Fluids 16, 1894 (1973)]. In a line-tied system, relevant to solar and astrophysical plasmas, the resonant surface is no longer a valid concept. It is then of interest to see how line-tying alters the aforementioned result for a periodic system. If the line-tied kink also produces a steep gradient, corresponding to a thin current layer, it may lead to strong resistive effects even with weak dissipation. Numerical solution of the eigenmode equations shows that the fastest growing kink mode in a line-tied system still possesses a jump in the radial displacement at the location coincident with the resonant surface of the fastest growing mode in the periodic counterpart. However, line-tying thickens the inner layer and slows down the growth rate. As the system length L approaches infinity, both the inner layer thickness and the growth rate approach the periodic values. In the limit of small ϵ∼Bϕ∕Bz, the critical length for instability Lc∼ϵ−3. The relative increase in the inner layer thickness due to line-tying scales as ϵ−1(Lc∕L)2.5.

High frequency fishbones excited by near perpendicular neutral beam injection

The high frequency fishbone instability observed in experiments with near perpendicular neutral beam injection is interpreted as the ideal internal kink mode destabilized by circulating energetic ions. The mode frequency is close to the transit frequency of circulating ions. The beta value of the circulating ions is required to peak on the magnetic axis and the average value within the q=1 magnetic surface must exceed a critical value for the mode to grow up.