External Kink Modes Research Articles

Non-uniform rotation and the resistive wall mode

Advanced Tokamak Concepts [C. Kessel, J. Manickam, G. Rewoldt, and W. M. Tang, Phys. Rev. Lett. 72, 1212 (1994)] have been designed assuming that the ‘‘Resistive Wall Mode’’ (RWM) is stable. It has recently been shown that the RWM can be stabilized by a combination of strong uniform plasma rotation and visco-resistive dissipation. In this paper we examine the consequences of a sheared flow on the RWM, and contrast the results to the case of uniform flow. It is shown that, as for uniform flow, the rotation initially further destabilizes the resistive wall mode, but for higher rotation velocities the growth rate is reduced, and the presence of plasma dissipation may completely stabilize the mode. However, sheared rotation allows the possibility of the RWM coupling to and converting into a Kelvin-Helmholtz mode. It is shown that the position of the wall with respect to the critical position for stabilization of the external kink mode is of crucial importance.

The effect of partial poloidal wall sections on the wall stabilization of external kink modes

An analysis of the effect on the wall stabilization of external kink modes due to toroidally continuous gaps in the resistive wall is performed. The effects both with and without toroidal rotation are studied. For a high-β equilibrium, the mode structure is localized on the outboard side. Therefore, outboard gaps greatly increase the growth rate when there is no rotation. For resistive wall stabilization by toroidal rotation, the presence of gaps has the same effect as moving the wall farther away, i.e., destabilizing for the ideal plasma mode, and stabilizing for the resistive wall mode. The region of stability, in terms of wall position, is reduced in size and moved closer to the plasma. However, complete stabilization becomes possible at considerably reduced rotation frequencies. For a high-β, reverse-shear equilibrium both the resistive wall mode and the ideal plasma mode can be stabilized by close fitting discrete passive plates on the outboard side. The necessary toroidal rotation frequency to stabilize the resistive wall mode using these plates is reduced by a factor of three compared to that for a poloidally continuous and complete wall at the same plasma-wall separation.

Resistive wall mode stabilization by plasma rotation in advanced tokamaks

By combining previous results of Betti and Freidberg [Phys. Rev. Lett. 74, 2949 (1995)] and Eriksson [Phys. Plasmas 2, 3095 (1995)], a fully analytical description is obtained for the stabilizing effect of toroidal plasma rotation in a large aspect ratio tokamak surrounded by a resistive wall. As in advanced tokamak configurations with a large fraction of bootstrap current, it is assumed that the current gradient near the plasma edge is large. This assumption enables an analytical analysis of external kink modes with low poloidal mode numbers. An expression is obtained, showing explicitly how the window of stable wall distances depends on the current profile.

Toroidal coupling of ideal magnetohydrodynamic instabilities in tokamak plasmas

A theoretical framework is developed to describe the ideal magnetohydrodynamic (MHD) stability properties of axisymmetric toroidal plasmas. The mode structure is described by a set of poloidal harmonics in configuration space. The energy functional, δW, is then determined by a set of matrix elements that are computed from the interaction integrals between these harmonics. In particular, the formalism may be used to study the stability of finite-n ballooning modes. Using for illustration the s-α equilibrium, salient features of the n■∞ stability boundary can be deduced from an appropriate choice of test function for these harmonics. The analysis can be extended to include the toroidal coupling of a free-boundary kink eigenfunction to the finite-n ideal ballooning mode. A unified stability condition is derived that describes the external kink mode, a finite-n ballooning mode, and their interaction. The interaction term plays a destabilizing role that lowers the instability threshold of the toroidally coupled mode. These modes may play a role in understanding plasma edge phenomena, L–H physics and edge localized modes (ELMs).

External kink modes in a tokamak with a resistive wall and a high-beta rotating plasma

The effect of the toroidal plasma rotation on the external kink mode instability driven by the wall resistivity is studied theoretically. It is suggested that this effect is related to the presence of side-band poloidal harmonics with singular points lying inside the plasma. Near the singular points we allow for plasma inertia and related toroidal effects. The ion Landau damping and its weakening due to the toroidal trapping and taken into account. High-beta effects near the singular points are allowed for. They cause the finiteness of the Mercier index s. It is shown that the influence of the plasma rotation on the external kink modes is most significant for s<0, i.e., in the situation favouring the development of instability in a non-rotating plasma.

Ponderomotive feedback stabilization of external kinks and disruptions in tokamaks

A novel mechanism for feedback stabilization of external kink modes and disruptions is proposed in which modulated radio-frequency (RF) antennas apply a stabilizing ponderomotive force (PF) to the plasma. The RF power required for n=1 kink stabilization in the Princeton Beta Experiment-Modified (PBX-M) [Phys. Fluids B 2, 1271 (1990)] is estimated, and it is shown that the idea can be tested with the existing antenna system. A simulation of a preliminary n=0 modulation experiment on PBX-M also shows that the PF can balance applied vertical field oscillations.

Pressure and conducting wall effects on kink modes in large aspect ratio tokamaks with large current gradients

The recently developed analytical theory of external kink modes in large aspect ratio tokamaks with large current gradients is extended to include effects due to locally large pressure gradients and a conducting wall. The results are presented in terms of simple integrals over the equilibrium profiles.

Reversed-field pinch discharge improvement with parallel-connected auxiliary toroidal field coils

Improvements in discharge duration and peak current have been obtained on the Reversatron reversed-held pinch by the addition of auxiliary parallel-connected toroidal field coils. Parallel connection forces the toroidal flux in each coil to be equal. These coils were used without removing or disconnecting the normal series-connected coils. In discharges with the auxiliary coils, F and Theta are unchanged, the plasma resistance is reduced and there is less shot-to-shot variation in the discharges. Diagnostic coils show that field errors with poloidal mode number m=0, and toroidal mode numbers n=6 and n=12 arising from asymmetries in the vacuum vessel are reduced in the setting-up phase. In the sustainment phase, there is a significant reduction in the external kink modes with m=1 and n=3, 4, 5, and 6.

Analytical stability boundaries for ideal kink modes in tokamaks with large edge currents.

The ideal magnetohydrodynamic stability analysis of the m\ensuremath{\ge}2 external kink modes in cylindrical tokamaks is reconsidered. Except for very small edge currents, the instability bands of these models are found to be accurately described by m-(m+1)${\mathit{I}}_{\mathit{m}}$/${\mathit{I}}_{0}$${\mathit{nq}}_{\mathit{a}}$m, where the quantities ${\mathit{I}}_{\mathit{m}}$ are defined as ${\mathcal{F}}_{0}^{\mathit{a}}$J(r)(r/a${)}^{2\mathit{m}}$2\ensuremath{\pi}r dr and J(r) is the current density in the plasma. This result may help to clarify the role played by the bootstrap current for the edge localized modes observed in high-confinement tokamak discharges.

Effect of toroidal flow on the stability of a diffuse high-beta tokamak

The effect of toroidal flow on the stability of a diffuse current model tokamak to external kink modes is examined. For a high-β tokamak it is shown that, to leading order in the inverse aspect ratio, the effect of toroidal flow is simply to modify the kinetic pressure by adding to it the flow pressure. Then the limit on the plasma beta in the presence of toroidal flow can be obtained from the results for a static plasma. It is shown that, for a given safety factor profile and shape of the plasma boundary, the maximum plasma beta for stability to external kink modes is always lower in the presence of flow compared to a static configuration. This is true for any arbitrary profile of the toroidal flow velocity.

Improved plasma performance in tokamaks with negative magnetic shear.

A tokamak plasma configuration is reported that simultaneously improves on the maximum stable plasma pressure, the bootstrap current contribution, and kinetic stability to temperature and density gradient driven modes in toroidal geometry. It is characterized by negative magnetic shear in the plasma interior and a peaked pressure profile. Stability to the ideal low-n external kink modes requires a conducting shell at 1.3 times the plasma minor radius. This novel plasma configuration is promising for improved plasma performance in advanced tokamak experiments.

Effect of rigid toroidal rotation on the stability of a high-beta tokamak to external kink modes

The effect of rigid toroidal rotation on the stability of a tokamak to external kink modes is examined. For simplicity a surface current model is assumed. For a high-β tokamak it is shown that to leading order the equations governing stability in the presence of rotation are identical to those for a static plasma with β replaced by β+S, where β is the plasma beta and S is a parameter which provides a measure of the rotation rate. For a circular cross-section tokamak the critical beta for stability to external kink modes in the presence of rigid toroidal rotation is given, to leading order in the inverse aspect ratio ε, by β=0.21ε−S. For an elliptical cross-section tokamak the largest critical beta is obtained for a vertical elongation of 2.2 and is given, to leading order in ε, by β=0.37ε−S. The lower limit on the kink safety factor q* or the Mercier q increases with increase in S. Quantitative estimates of this increase can be obtained from the stability boundaries for the static problem.

Effect of rigid rotation on the stability to external kink modes

The effect of rigid toroidal rotation on the stability of a tokamak to external kink modes is examined. The effect of rigid rotation is modeled by a centrifugal force field. The problem is studied in the cylindrical limit for which this reduces to a uniform force field acting in a direction transverse to the plasma column. The effect of a transverse force field on the limit on the safety factor q for stability to external kink modes is studied. For values of q above this, limits on the plasma beta, denoted β, are obtained. A study of the equilibrium of a toroidal plasma undergoing rigid toroidal rotation yields another limit on β as a function of q. Combining the results of cylindrical stability with that of toroidal equilibrium provides approximate estimates of the maximum β that could be obtained. The effect of rigid rotation on this maximum β is also studied.

The role of edge current density on kink mode stability and its implication for magnetohydrodynamic activity associated with edge localized modes

The stability of a plasma equilibrium to the external kink mode is shown to be dependent on the details of the current density near the plasma edge and the value of the safety factor, q, at the edge. The buildup of the current density near the plasma edge is shown to decrease the shear in the safety-factor profile and lead to destabilization of the kink mode. Tokamak plasmas in the high confinement mode of operation have high current density near the edge and are known to exhibit activity referred to as edge localized modes. An important feature of this activity is intermittent bursts of magnetohydrodynamic (MHD) activity near the plasma edge. A model is proposed for the MHD activity in terms of the external kink mode. The role of the plasma geometry and equilibrium profiles is discussed.

Analytical study of plasma heating by resonant absorption of the modified external kink mode

A simplified analytic description is used to understand recent results of large-scale numerical simulations of resonant absorption and to disentangle the basic physics. It is shown that very efficient absorption takes place at frequencies where a discrete external kink and an Alfvén continuum mode merge into a modified external kink mode. The relation between this ‘hybrid’ mode and ‘pure’ continuum or discrete spectrum modes is discussed.

Interactions between the plasma and the stabilizing shell during disruptions in PBX-M

To stabilize vertical and external kink modes in the elongated, indented PBX-M plasma, a close-fitting stainless-steel-clad aluminum conducting shell surrounds over 85% of the plasma surface. During normal operation, the currents induced in the shell by the plasma are small, usually reflecting changes in plasma shape initiated by the shaping control system. However, voltage spikes of less than 2 kV were observed across gaps in the plates during both low- q and density limit disruptions. After the thermal quench, the plasma became unstable to the n = 0 vertical mode. The resulting rapid (20 to 5000 m/s, a resistive timescale) plasma motion induced large poloidal currents through segments of the shell to the vacuum vessel, despite the fact that each segment was electrically isolated except for a common single-point ground. Through impact loading, these large currents caused the failure of insulators and mechanical members in the shell support structure. These events have been numerically simulated with a resistive force-free halo plasma surrounding the main plasma, which provides a current path between the plates along open field lines. This model is consistent with measured plasma current decay times of 1–3 ms, characteristic of a resistive 10 eV plasma. Confirmation that the duration of the post-disruption plasma is about a resistive tearing time was provided from bolometers viewing both along the mid-plane and into the divertors. About 30–60% of the total energy and 30% of the magnetic energy was radiated during the thermal and current quench phases. This model led to improvements in the electrical configuration and insulator design which eliminated all failures. Disruptions have been prevented even when radiating over 100% of the input power during current profile control experiments with counter beam injection.

Rf stabilization of external kink modes in the presence of a resistive wall

The problem of controlling external kink instabilities that arise on the slow time scale characteristic of magnetic field diffusion through a conducting wall with finite resistivity is considered. An expression for the growth rate of these ‘‘resistive wall’’ kink modes is derived in a low-beta cylindrical tokamak model assuming a thin wall and small resistivity, including the additional stabilizing effect of a spatially localized, externally applied surface force. It is shown that the marginal stability condition is identical to that of the ideal kink in the absence of a conducting wall. The model is applied to ponderomotive stabilization arising from the near field of an ion-Bernstein-wave (IBW) antenna system. It is found that radio-frequency (rf) fields below the marginal stability threshold can significantly reduce the growth rate of the m=2 instability. The required rf fields can be attained using present IBW antennas with Ωi≲ω&amp;lt;2Ωi in tokamaks with relatively low magnetic fields and may therefore be useful for high-beta and second-stability studies.

Quenching of external kink modes by localized radio-frequency fields

The quenching of external kink modes by a spatially localized, externally applied surface force is studied in a low-beta cylindrical tokamak model. It is shown that m=2 modes may be stabilized by the application of a radio frequency (rf) field in the ion cyclotron range of frequencies (ICRF) excited by either a toroidal or a poloidal antenna structure surrounding the plasma. Whereas linear mode coupling induced by toroidal localization is negligible, that caused by poloidal localization may degrade the stabilization and, thus, require increased power levels for stability. The model is directly applicable to the ponderomotive force exerted by an evanescent slow wave in the ion cyclotron range of frequencies. Quantitative estimates are made for ponderomotive stabilization of kink modes by means of ion-Bernstein-wave (IBW) antennas. This method of stabilization has the advantage that the antenna system does not have to match the spatial or temporal structure of the unstable modes. For typical IBW antenna systems, it is found that the required rf fields for kink stabilization are reasonable if the current profile is not too flat.

Transition to the second region of ideal MHD stability

Transition to the second region of MHD stability has been examined in circular cross-section boundary tokamak geometry. Values of ⟨β⟩ and ∊βp, corresponding to equilibria which are marginally stable in the second region on each flux surface, are presented. The aspect ratio scaling of these functions is shown to be sensitive to the value of the on-axis safety factor qo. Equilibria evolving from the first to the second region of stability have been generated by a transport code which calculates the two-dimensional evolution of the magnetic configuration self-consistently with the thermal and particle diffusion. The transport model enhances the diffusion coefficients on flux surfaces which are unstable to high-n modes, thereby altering the pressure profile locally. The q-profile, modified by neutral beam driven current so that q0 > 1, reduces the size of the unstable region sufficiently to allow transition to the second region. Auxiliary power requirements for this transition are estimated. Stability of the transition equilibria to low-n external and internal modes is also examined. All ideal modes restabilize as ∊βp approaches and exceeds unity. A close fitting conducting wall is needed to stabilize low-n external kink modes.

Ponderomotive stabilization of external kink modes in tokamaks

It is shown that external kink modes in tokamaks can be stabilized by applied radio frequency (rf) waves in the ion cyclotron range of frequencies. This stabilization results from the work done by the plasma against the ponderomotive force exerted by the rf field. For a strongly evanescent surface field (such as the near field of an ion Bernstein wave coupler), the ponderomotive layer is confined to the vicinity of the plasma surface and, if sufficiently strong, influences kink stability in the same way as a close-fitting conducting wall. This effect is illustrated by deriving an analytic condition for stabilizing a single (m,n) mode in the low-beta straight tokamak model by means of a cylindrically symmetric E∥ rf field. The effects of poloidally or toroidally nonuniform rf fields are examined in a two-mode coupling perturbation analysis. This stabilization mechanism may be useful in extending the beta limit of ion-Bernstein-wave-heated tokamaks against both ideal and resistive modes.