Internal Kink Mode Research Articles

Dynamical analysis of the nonlinear growth of the m = n = 1 resistive internal mode.

A dynamical analysis is presented that self-consistently takes into account the motion of the critical layer, in which the magnetic field reconnects, to describe how the m=n=1 resistive internal kink mode develops in the nonlinear regime. The amplitude threshold marking the onset of strong nonlinearities due to a balance between convective and mode coupling terms is identified. We predict quantitatively the early nonlinear growth rate of the m=n=1 mode below this threshold.

Generation of radial electric field induced by collisionless internal kink mode with density gradient

Effects of density gradient on the collisionless m=1 (m/n=1/1) internal kink mode in a cylindrical tokamak plasma are studied by the gyrokinetic particle simulations. When the density gradient is not large enough to change the full reconnection process, the phenomena after the full reconnection, such as the secondary reconnection and the evolution of the safety factor profile, are changed considerably due to the self-generated radial electric field, i.e., the m/n=0/0 mode. The growing mechanism is explained by the difference of E×B drift motion between ions and electrons, which is caused by the fast parallel motion of electron. Once the radial electric field is triggered by the symmetrical flow induced by the 1/1 mode, the 0/0 mode grows up to the same level as the 1/1 mode, and drives an E×B plasma rotation in the ion diamagnetic direction, which breaks the symmetrical plasma flow induced by the 1/1 mode. The density and current distributions, and therefore minimum safety factor qmin after the full reconnection, are found to be affected by the asymmetrical flow driven by the 1/1 and 0/0 modes.

Bootstrap current destabilization of ideal MHD modes in three-dimensional reactor configurations

In current-free stellarators, the parallel current density is normally too weak to drive global external kink modes. However, at finite values of β, the bootstrap current (BC) can provide sufficient free energy to trigger this class of mode in some stellarator systems. The effect of the BC in the collisionless 1/ν regime has been investigated in several different types of stellarator reactor systems all with a volume V∼1000 m3. In quasiaxisymmetric and quasihelically symmetric stellarators, the BC is large at finite β and this can cause low order resonances to move into and emerge out of the plasma which in turn can destabilize global internal and external kink modes. In a six-field period system with poloidally closed contours of the magnetic field strength B, the BC is small and decreases the rotational transform only slightly. As a result, only intermediate to high n modes can become weakly destabilized. Furthermore, it is demonstrated in this system that the contours of the second adiabatic invariant \U0001d4a5|| close poloidally for all trapped particles at finite β*∼6%. This condition leads to the loss of a very small fraction of the collisionless α-particle orbits. In Sphellamak configurations with peaked toroidal currents required to generate nearly isodynamic maximum-B confining field structures, the BC accounts only for a small fraction of the total current. The loss of α-particles born within the inner quarter of the plasma volume is negligible while about (1/3) of those born at half volume escape the device within a slowing down time.

Internal Kink Instability in Shaped Tokamaks

A criterion of an ideal internal kink mode is derived for a shaped tokamak configuration in which q-profile is very flat in the core region. A combining criterion is obtained including the necessary criterion of Mercier and the sufficient criterion of Lortz. The new criterion makes progress compared with the necessary criterion of Mercier. In the elongated plasma, a poloidal beta can cause instability, while the triangularity has a stabilizing effect. The result is applicable for DIII-D and SUNIST.

Stabilization of sawteeth in tokamaks with toroidal flows

Sheared toroidal flows with a magnitude of the order of the sound speed are shown to stabilize sawteeth in tokamaks. In the absence of flows, tokamak equilibria in which the central safety factor q is less than unity are unstable to resistive tearing modes (resistive internal kink modes) with toroidal mode number n=1. As the ratio β of the plasma pressure to the magnetic field pressure increases, the growth rate of the n=1 mode rises because of the increasing pressure gradient. However, the addition of a toroidal flow to the equilibrium has a stabilizing effect. As the magnitude of the toroidal flow approaches the sound speed, the n=1 resistive tearing mode can be completely stabilized by the flow, eliminating sawteeth.

Effect of combined triangularity and ellipticity on the stability limit of the ideal internal kink mode in a tokamak

The effect of combined triangular and elliptical shaping of the plasma cross section on the stability limit of the ideal, internal kink mode in a tokamak is analyzed by means of a computer algebra expansion of the magnetohydrodynamic equations. The work extends the result of a previous investigation of the effect of ellipticity alone on this mode in toroidal plasmas [Wahlberg, Phys. Plasmas 5, 1387 (1998)]. It is shown that, in contrast to the strongly destabilizing effect of (positive) ellipticity alone, the effect of combined positive ellipticity and positive triangularity is stabilizing. The effect is of the same importance as the effect of ellipticity alone if the triangularity of the q=1 surface is of the same order of magnitude as the inverse aspect ratio. When the safety factor at the magnetic axis is close to (but below) unity, the geometrical factor governing the stability of the ideal, internal kink mode is found to be the same as the corresponding factor in the Mercier criterion for shaped tokamaks.

Alpha particle physics in a tokamak burning plasma experiment

Much is known about the behavior of energetic ions in tokamak devices but much remains to be understood. Single-particle effects are well understood and provide a firm basis for extrapolation to a burning plasma. In contrast, collective effects involving fast ions are more poorly understood and extrapolations are unreliable. Collective modes of concern include toroidicity-induced and ellipticity-induced Alfvén eigenmodes, kinetic ballooning modes, and internal kink modes. In addition to these magnetohydrodynamic normal modes, there are also energetic particle modes characterized by strong dependence on the fast-ion distribution function. Although many issues are important areas of study in current experiments, five issues distinguish burning plasma experiments. First, the energetic alphas are not the dominant source of free energy for the instabilities unless the fusion power exceeds the heating power by a factor of 10. Second, the damping of the instabilities depends sensitively on mode coupling to other heavily-damped waves. The magnitude of this coupling is expected to depend on the normalized thermal gyroradius, which is much smaller in a reactor. Third, in a reactor, both the radial extent of the instabilities and the fast-ion orbit contract relative to current experiments, so the fast-ion transport will change. Fourth, when instability occurs, a larger number of modes are unstable, so the mechanism of nonlinear saturation could shift from fast-ion transport to mode coupling. Fifth, because of the extreme sensitivity of energetic particle modes to the distribution function, an isotropic alpha particle distribution function differs from anisotropic fast-ion populations.

Effects of circulating energetic ions on sawtooth oscillations.

In contrast to the well-known result that the effects of the trapped energetic ions (TEI) on the internal kink mode are due to the toroidal precession of the TEI, it is found that the effects of the circulating energetic ions (CEI) on sawtooth are due to the toroidal circulation of the CEI. The effects of the CEI on sawtooth oscillations are found to be different from the well-known purely stabilizing effects of the TEI on sawtooth oscillations; the toroidal circulation of the co-CEI provides an additional sink of free energy and stabilizes the mode; the toroidal circulation of the counter-CEI provides an additional source of free energy and destabilizes the mode.

Modeling of diamagnetic stabilization of ideal magnetohydrodynamic instabilities associated with the transport barrier

A new code, MISHKA-D (Drift MHD), has been developed as an extension of the ideal magnetohydrodynamics (MHD) code MISHKA-1 in order to investigate the finite gyroradius stabilizing effect of ion diamagnetic drift frequency, ω*i, on linear ideal MHD eigenmodes in tokamaks in general toroidal geometry. The MISHKA-D code gives a self-consistent computation of both stable and unstable eigenmodes with eigenvalues |γ|≅ω*i in plasmas with strong radial variation in the ion diamagnetic frequency. Test results of the MISHKA-D code show good agreement with the analytically obtained ω*i spectrum and stability limits of the internal kink mode, n/m=1/1, used as a benchmark case. Finite-n ballooning and low-n kink (peeling) modes in the edge transport barrier just inside the separatrix are studied for high confinement mode (H-mode) plasmas with the ω*i effect included. The ion diamagnetic stabilization of the ballooning modes is found to be most effective for narrow edge pedestals. For low enough plasma density the ω*i stabilization can lead to a second zone of ballooning stability, in which all the ballooning modes are stable for any value of the pressure gradient. For internal transport barriers typical of the Joint European Torus [JET, P. H. Rebut et al., Proceedings of the 10th International Conference, Plasma Physics and Controlled Nuclear Fusion, London (International Atomic Energy Agency, Vienna, 1985), Vol. I, p. 11] optimized shear discharges, the stabilizing influence of ion diamagnetic frequency on the n=1 global pressure driven disruptive mode is studied. A strong radial variation of ω*i is found to significantly decrease the stabilizing ω*i effect on the n=1 mode, in comparison with the case of constant ω*i estimated at the foot of the internal transport barrier.

Stabilization of the Mercier modes in a tokamak by toroidal plasma rotation

The stability of localized modes (Mercier modes) in a tokamak with a toroidally rotating plasma is analyzed within the framework of compressible, ideal magnetohydrodynamics. For equilibria with large aspect ratio, poloidal beta value of order unity, and isothermal magnetic surfaces, it is found that sonic, toroidal rotation provides a strongly stabilizing effect for the Mercier modes, similar to the stabilization recently found for the internal kink mode in a rotating plasma [Wahlberg and Bondeson, Phys. Plasmas 7, 923 (2000)]. A finite oscillation frequency (Brunt–Väisälä frequency), of the order of the sound frequency, is shown to be associated with each magnetic surface. If Γ>1, where Γ is the exponent in the equation of state, the rotation transforms the Mercier instabilities to stable oscillations at the local Brunt–Väisälä frequency associated with the magnetic surface where the mode is located. If the plasma satisfies an isothermal equation of state (Γ=1), however, the stability of the Mercier modes becomes sensitive to the profile of the toroidal flow. In this case, the rotation is found to be stabilizing if the kinetic energy density of the rotation is an increasing function of the minor radius. In the opposite case, the rotation is destabilizing unless the pressure profile is much more peaked than the kinetic energy density profile.

Destabilization of internal kink modes at high frequency by energetic circulating ions.

A theoretical model is proposed to interpret the high-frequency fishbone instability observed in tangential neutral-beam-injection discharges in a tokamak. It is shown that, when the beam ion beta exceeds a critical value, energetic circulating ions can indeed destabilize the internal kink mode through circulation resonance at a high frequency comparable to the circulation frequency of the energetic ions. The critical beta value of the energetic ions, the real frequency, and the growth rate of the mode are in general agreement with the high-frequency fishbone instability observed in experiments.

Generation of radial electric field in the process of full reconnection by kinetic kink mode

Effects of density gradient on the kinetic m = 1 (m/n = 1/1) internal kink mode in a cylindrical tokamak plasma are studied by the gyro-kinetic particle simulations. When the density gradient is not large enough to change the full reconnection process, the phenomena after the full reconnection, such as the plasma flow and the secondary reconnection, are changed considerably due to the self-generated radial electric field, i.e. the m/n = 0/0 mode. The growing mechanism is explained by the difference of E × B drift motion between ions and electrons, which is caused by the fast parallel motion of electron.

Understanding sawtooth activity during intense electroncyclotron heating experiments on TCV

Different types of central relaxation oscillations areobserved in the presence of ECH dependingon the location of the deposited power. In the TCV tokamak, normal sawteeth, i.e. triangular sawteeth similar toohmic sawteeth, and saturated sawteeth are observed with central ECHpower deposition, while giant sawteeth and `humpback'oscillations occur when heating close to the sawtooth inversionsurface of the local soft X ray emissivity. New measurements with hightemporal resolution show that the crash phase of these sawtooth typesis accompanied by a reconnection process associated with an m/n = 1 resistiveinternal kink mode. After the sawtooth crash, full magneticreconnection is observed in normal and in saturated sawteeth, whilefor giant and humpback sawteeth the reconnection process is incompleteand poloidally asymmetric temperature profiles persist after the crash. The detailed dynamics of the magnetic island associatedwith the resistive internal kink mode are described by a displacementfunction which is inferred from the experimental data. In normalsawteeth, the kink mode is destabilized just before the crash, whilein all other sawtooth types a magnetic island exists for a significantfraction of the sawtooth period. The different types of sawteeth havebeen simulated using a numerical code based on a theoretical modelwhich describes the evolution of the electron temperature in thepresence of localized heat sources and of a magnetic m/n = 1/1 island.

The effects of sheared toroidal plasma rotation on the internal kinkmode in the banana regime

The stability of the ideal internal kink mode is calculated, taking intoaccount the kinetic response of thermal ions in the external region and thesingular layer. By extending the collisionless dispersion relation to includethe equilibrium radial electric field it is found that the stability of theinternal kink mode depends sensitively on sheared toroidal plasma rotation.The sheared toroidal plasma rotation can increase the critical pressure forinternal kink mode displacements typically by a factor of two.

Internal kink stability of large aspect ratio high-β tokamaks

Theoretically, high-β (β∼1) tokamaks offer a large fusion power efficiency advantage over low-β devices. However, if high-β tokamaks are inherently unstable then such devices will never be realized. In particular, kink modes are thought to be the most serious obstacle to high-β operations. High-β tokamaks are characterized by a very large Shafranov shift with a thin “boundary layer” on the outboard side of the device and a large “core” region of vertical flux surfaces comprising most of the central volume. In this paper, the energy principle is used to compute the magnetohydrodynamic internal kink stability of such devices in the large aspect ratio limit with a low toroidal mode number. A class of internal kink mode similar to the usual low-β kink is present; the stability against these modes is computed. A set of parameters describing a kink stable high-β equilibrium is given. Stability is shown to be dependent on the shape of the plasma boundary.

Critical magnetic shear for the tokamak ideal internal kink mode in the ion kinetic regime

It is shown that for the ideal internal kink mode of tokamak plasma, there exists a critical magnetic shear s0c for its excitation when the finite ion Larmor radius effect is considered in the mode singular layer. This can be related to the sudden onset of the sawtooth and also to the stabilization of the sawtooth by the trapped hot ions observed experimentally. The alpha particle effect on this mode is revisited and some new results are reported.

Isothermal internal kink instability in a toroidally rotating tokamak plasma

The properties of the internal kink instability in a tokamak with a toroidally rotating plasma is analysed from the ideal magnetohydrodynamic (MHD) equations, for plasmas satisfying an isothermal equation of state. The plasma equilibrium is assumed to have large aspect ratio, circular cross section, poloidal beta value ( β p) of order unity, and a rotational frequency comparable to the sound frequency. For an isothermal plasma, the rotational effect on the internal kink mode can be expressed in terms of a `rotational beta value' ( β Ω ) of the plasma. This quantity is defined in the same way as β p, with the kinetic energy density of the rotation ( u Ω ) replacing the plasma pressure. The rotation is found to be strongly stabilizing if u Ω is an increasing function of the minor radius ( β Ω<0 ). In the opposite case, the rotation is destabilizing unless the pressure profile is much more peaked than the kinetic energy density profile.

Stabilization of the internal kink mode in a tokamak by toroidal plasma rotation

The stability of the internal m=n=1 kink mode is analyzed for a tokamak with a toroidally rotating plasma, by a large aspect ratio expansion of the compressible magnetohydrodynamic equations. Assuming that the central poloidal beta is of order unity, it is found that the internal kink mode is stabilized by rotational frequencies of order Ω/ωA∼ε, where ωA is the Alfvén frequency and ε is the inverse aspect ratio. The internal kink then turns into a stable oscillation with a Doppler-shifted frequency ∼ΩM(1−1/Γ)1/2, where Γ is the adiabatic index and ℳ is the sonic Mach number. The stabilization comes from the centrifugal force which gives a stable density (or entropy) distribution within each magnetic surface. The parallel motion associated with the internal kink mode then behaves as the Brunt–Väisälä oscillations of a stably stratified fluid in a gravitational field. At lower rotational frequencies, Ω/ωA∼ε2, the only effect of the rotation is a co-rotation of the usual (nonrotating) m=n=1 instability, whereas the ordering Ω/ωA∼ε3/2 represents a transition regime where the stabilizing effect of the rotation competes with the drive from the internal kink instability. Kinetic behavior along the field lines is expected to influence this stabilization mechanism, as it depends on the adiabatic index Γ.

Structure of the compressible MHD equations for the internal kink mode in a toroidal plasma with large aspect ratio

The equations for the ideal, internal m = n = 1 kink mode in a toroidal plasma are derived from a direct, large-aspect-ratio perturbation expansion of the compressible magnetohydrodynamic (MHD) equations. The derivation complements earlier investigations of the internal kink mode based either on the energy principle or on direct expansions of the incompressible MHD equations. It is shown that five poloidal harmonics (m = −1, 0, 1, 2 and 3) have to be retained in a direct expansion of the compressible MHD equations, as compared with the three poloidal harmonics m = 0, 1 and 2 needed in the case of an incompressible plasma, or when working from the energy principle. Furthermore, the sound velocity is found to replace the Alfvén velocity in the generalized Pfirsch–Schlüter factor (the kinetic energy enhancement factor in a toroidal plasma) previously derived for an incompressible plasma. Taking this factor fully into account in the calculation of the growth rate of the m = n = 1 mode, it is shown that, while the Bussac result γB is recovered near marginal stability, growth rates of the order of 30% larger than γB are obtained when γB becomes of the order of the sound frequency.

Linear gyrokinetic theory for kinetic magnetohydrodynamic eigenmodes in tokamak plasmas

A two-dimensional (2D) numerical solution method is developed for the recently derived linear gyrokinetic system which describes arbitrary wavelength electromagnetic perturbations in tokamak plasmas. The system consists of the gyrokinetic equation, the gyrokinetic Poisson equation, and the gyrokinetic moment equation. Since familiar magnetohydrodynamic (MHD) results can be recovered entirely from this gyrokinetic model, and all interesting kinetic effects are intrinsically included, this gyrokinetic system offers an approach for kinetic MHD phenomena which is more rigorous, self-consistent, and comprehensive than the previous hybrid models. Meanwhile, drift type microinstabilities can be also investigated systematically in this theoretical framework. The linear gyrokinetic equation is solved for the distribution function in terms of the perturbed fields by integrating along unperturbed particle orbits. The solution is substituted back into the gyrokinetic moment equation and the gyrokinetic Poisson equation. When the boundary conditions are incorporated, an eigenvalue problem is formed. The resulting numerical code, KIN-2DEM, is applied to kinetic ballooning modes, internal kink modes, and toroidal Alfvén eigenmodes (TAEs). The numerical results are benchmarked against the well-established FULL code [G. Rewoldt, W. M. Tang, and M. S. Chance, Phys. Fluids 25, 480 (1982)], the PEST code [J. Manickam, Nucl. Fusion 24, 595 (1984)], and the NOVA-K code [C. Z. Cheng, Phys. Rep. 211, No. 1 (1992)]. More importantly, kinetic effects on MHD modes can be investigated nonperturbatively. In particular, the kinetic effects of the background plasma on internal kink modes and the hot particle destabilization of TAEs are studied numerically.