Resistive Wall Modes In Tokamaks Research Articles

General dispersion relations for resistive wall modes in tokamaks

The dispersion relation for the resistive wall modes (RWMs) is derived without the use of the trial function bHF proposed in S. W. Haney and J. P. Freidberg [Phys. Fluids B 1, 1637 (1989)] for the magnetic perturbation b outside the plasma. Another difference from the Haney–Freidberg (HF) approach is the incorporation of non-ideal effects in the plasma description. These enter the final result through the energy functional and affect the external solution for b through the boundary conditions only. This allows to perform the derivations in a general form without constraints on the dissipation mechanisms in the plasma. Then, the main mathematical difficulties are related to the description of the energy flow outside the plasma. This part of the task is presented with details allowing easy comparisons with the reference HF case. Being universally applicable, the resulting dispersion relation covers the existing variants, including those based on the so-called kinetic approaches. It shows that, because of its integral nature, the same predictions can be expected from various models for the plasma. Another conclusion is that, with a non-ideal contribution, just one or two free parameters would be enough to get agreement with experimental data on the plasma stability boundary. This, however, does not guarantee that the same choice of the fitting coefficients will be similarly efficient on other devices. The proposed relations provide a unified approach to the problem of plasma stability against RWMs.

Stabilization of resistive wall modes in tokamaks by drift kinetic effects combined with magnetic feedback

The synergetic effects of drift kinetic resonances, the resistive layer dissipation, the magnetic feedback, and the toroidal plasma flow on the stability of the resistive wall mode are numerically investigated using a full toroidal resistive magnetohydrodynamic-kinetic hybrid stability code MARS-K (Liu et al 2008 Phys. Plasmas 15 112503). It is found that the plasma resistivity, coupled with the favourable average curvature effect, can enlarge the stable domain predicted by the drift kinetic model. A synergy between the precessional drift resonance damping, the magnetic feedback and the plasma flow helps open two stability windows. The width of the inner stability window increases with the feedback gain, but decreases with the flow speed. In addition, optimization of the toroidal phase difference of the feedback gains between the upper and lower active coils can lead to a full suppression of the mode.

Rotational stabilization of the resistive wall modes in tokamaks with a ferritic wall

The dynamics of the rotating resistive wall modes (RWMs) is analyzed in the presence of a uniform ferromagnetic resistive wall with μ̂≡μ/μ0≤4 (μ is the wall magnetic permeability, and μ0 is the vacuum one). This mimics a possible arrangement in ITER with ferromagnetic steel in test blanket modules or in future experiments in JT-60SA tokamak [Y. Kamada, P. Barabaschi, S. Ishida, the JT-60SA Team, and JT-60SA Research Plan Contributors, Nucl. Fusion 53, 104010 (2013)]. The earlier studies predict that such a wall must provide a destabilizing influence on the plasma by reducing the beta limit and increasing the growth rates, compared to the reference case with μ̂=1. This is true for the locked modes, but the presented results show that the mode rotation changes the tendency to the opposite. At μ̂>1, the rotational stabilization related to the energy sink in the wall becomes even stronger than at μ̂=1, and this “external” effect develops at lower rotation frequency, estimated as several kHz at realistic conditions. The study is based on the cylindrical dispersion relation valid for arbitrary growth rates and frequencies. This relation is solved numerically, and the solutions are compared with analytical dependences obtained for slow (s/dw≫1) and fast (s/dw≪1) “ferromagnetic” rotating RWMs, where s is the skin depth and dw is the wall thickness. It is found that the standard thin-wall modeling becomes progressively less reliable at larger μ̂, and the wall should be treated as magnetically thick. The analysis is performed assuming only a linear plasma response to external perturbations without constraints on the plasma current and pressure profiles.

Ferromagnetic effects in the theory of slow and fast resistive wall modes in tokamaks

The traditional theory of the resistive wall modes (RWMs) in the toroidal fusion systems was developed assuming the magnetic permeability μ of the wall the same as the vacuum one μ0. Here, we analyze the dynamics of unstable RWMs at the presence of a ferromagnetic wall with μ≡μ/μ0≤4. This choice with μ=const corresponds to the saturated state of ferritic materials in a strong magnetic field, as it should be in a tokamak reactor. The study is based on the cylindrical dispersion relation valid for arbitrary s/dw, where s is the skin depth and dw is the wall thickness. This equation is solved numerically, and the solutions are compared with analytical asymptotes obtained for slow (s≫dw) and fast (s≪dw) RWMs. Within the model, only very slow RWMs are found insensitive to variations of μ, while slightly above the no-wall stability limit the growth rate of the modes increases with larger μ. It is shown that at s<dw this increase is roughly given by a factor of μ compared to a similar case with μ=1. The de...

Impact of plasma rotation on the linear physics of resistive wall modes in tokamaks

Rotation effects on the linear stability of the resistive wall mode (RWM) are investigated numerically based on the ideal magnetohydrodynamics (MHD) model. One topic is impact of poloidal rotation in a toroidally rotating plasma. Since the difference between mode frequency and rotation frequency is important to change the MHD stability by rotation, modulation of the Doppler-shifted frequency by poloidal rotation can play an important role even when the poloidal rotation velocity is significantly lower than the toroidal one. In addition, when poloidal rotation is fixed, the absolute value of the Doppler-shifted frequency relies on the direction of toroidal rotation, and hence, linear MHD stability can depend on this rotation direction. Such a dependence is sometimes observed experimentally. The other topic is the re-destabilization of the RWM by toroidal rotation in a reversed shear tokamak. Several numerical results imply that a candidate of this re-destabilized mode is a stable MHD eigenmode, which becomes unstable due to coupling with the RWM when rotation frequency is similar to the frequency of this stable eigenmode in a static equilibrium.

Analysis of the dispersion relation for resistive wall modes in tokamaks with account of the skin effect

The accuracy of the dispersion relations derived earlier analytically for the resistive wall modes (RWMs) in tokamaks is evaluated. Existing models use the expansions in the ratio of the wall thickness d w to the skin depth s. This parameter is small in the standard theory of “slow” RWMs, but it is large in the recently developed approach for the “fast” RWMs. Here, a dispersion relation applicable not only to these extreme cases, but also to the intermediate range is derived in the single-mode cylindrical approximation without restrictions on s/d w . The derived equation is solved numerically, the result is compared with the predictions of the earlier analytical models with either s ≫ d w or s ≪ d w . The applicability limits of the asymptotic expressions for the RWM growth rate are clarified. In particular, the limits are specified in which their error does not exceed 10%. It is shown that these expressions always underestimate the growth rate. The study confirms the necessity of the skin effect incorporation in the description of RWM dynamics.

Numerical study of physical properties of resistive wall modes in tokamaks

The effect of plasma with toroidal rotation on the resistive wall modes in tokamaks is studied numerically. An eigenvalue method is adopted to calculate the growth rate of the modes for changing plasma resistivity and plasma density distribution, as well as the diffusion time of magnetic field through the resistive wall. It is found that the resistive wall mode can be suppressed by the toroidal rotation of the plasma. Also, the growth rate of the resistive wall mode decreases when the edge plasma density is the same as the core plasma density, but it only changes slightly with the plasma resistivity.

Energy approach to stability analysis of the locked and rotating resistive wall modes in tokamaks

A method is proposed for stability analysis of the locked and rotating resistive wall modes (RWMs) in tokamaks. The method is based on the relations describing the balance of energy permeating through the vessel wall. This is a natural extension of the traditional energy approach to the plasma stability tasks which allows incorporation of the energy outflow (absent in the classical energy principle) and its dissipation in the wall. The proposed method covers the locked and rotating modes with a complex growth rate. Its efficiency is proved by derivation of a general dispersion relation for such modes with further reduction to particular consequences for slow and fast RWMs. It is shown that in the latter case, when the skin depth becomes smaller than the wall thickness, the mode rotation essentially amplifies its damping, weakening and even suppressing the instability. This effect was earlier found in the frame of the slab model [V. D. Pustovitov, Phys. Plasmas 19, 062503 (2012)]. Here, it is confirmed with equations valid for toroidal geometry, which are obtained as a supplement to the standard energy principle. The presented results predict strong rotational stabilization of the fast RWMs, which occurs at the mode rotation frequency above a critical level. The estimates are given to allow comparison of these predictions with experimental results.

Thick-wall effects on the rotational stabilization of resistive wall modes in tokamaks

This paper is devoted to studying the combined effect of mode rotation and energy dissipation in the resistive wall on plasma stability. The problem is analysed on the basis of the energy approach complementing the standard methods of the traditional MHD theory of plasma stability. The key element that makes our model different from this theory and commonly used thin-wall approaches to the stability analysis of resistive wall modes (RWMs) is the incorporation of the skin effect. In the ideal MHD theory of plasma stability, the skin depth is, formally, zero. In contrast, the conventional thin-wall theory of RWM stability assumes a skin depth much larger than the wall thickness. The presented model considers the intermediate case with a finite skin depth compared with the wall thickness. This covers the modes in between the typical RWMs and the ideal MHD modes when wall resistivity still affects the mode dynamics. It is shown that, in this region, the growth rate of the locked modes must be substantially larger than that calculated in the thin-wall models. On the other hand, the fast RWMs can be completely stabilized by mode rotation above some critical level. Qualitatively, this corresponds to the rotational stabilization observed in the DIII-D tokamak and allowing the plasma operation above the no-wall stability limit (Strait et al 2003 Nucl. Fusion 43 430). This is the main result of this study, which is completely analytical with all dependences explicitly shown. In particular, the dispersion relations for the fast RWMs and the critical frequency of mode rotation necessary for rotational stabilization are expressed through quantities that depend on the plasma parameters or can be experimentally found by magnetic measurements outside the plasma.

A unified approach to description of the fast and slow resistive wall modes in tokamaks

The formulation of the boundary conditions is considered in the problem of the resistive wall mode (RWM) stability in tokamaks. The mode-wall interaction, usually modeled in the thin-wall approximation, is described here with account of the finite thickness of the wall and skin effect. It allows one to step beyond the standard restrictions into the area of faster RWMs than the usual “slow” RWMs near the stability threshold. The analysis is carried out with the energy balance equations incorporating the dissipation in the wall. The approach is equally applicable to the modes of any kind and allows natural matching of the exterior problem with the models for the inner region. For example, it allows one to connect the outer task to the classical energy principle for the inner area. It is shown how to calculate the RWM growth rates within this model. A general algorithm with equations applicable to arbitrary toroidal systems and its full realization in the conventional cylindrical model are described. In the latter case, it is shown that the growth rate of the “fast” RWMs essentially differs from the estimates of the standard theory of slow RWMs. The analysis proves that the RWM theory has to be complemented by the additional block of calculations for more correct formulation of the boundary conditions on the inner side of the wall than that in the theory with an ideal or thin resistive wall.

Neural Network Control of Resistive Wall Modes in Tokamaks

A neural network (NN) implementation of an adaptive optimal stochastic output feedback control is developed to stabilize the resistive wall mode (RWM), a critically important instability in fusion machines like tokamaks. The design of an adaptive optimal stochastic output feedback control was discussed and reported earlier by Sun, Sen, and Longman. The system dynamics is experimentally determined via the extended least square method with an exponential forgetting factor and covariance resetting. The optimal output feedback controller is redesigned periodically online based on the system identification. The output measurements and past control inputs are used to construct new control inputs. These are achieved by an architecture of NNs consisting of a pool of sequentially linked Hopfield networks and implemented in hardware with digital NN processors made by the Accurate Automation Corporation. The simulations have shown that the NN adaptive output controller can stabilize the time-dependent RWM in a slowly evolving tokamak discharge. This is accomplished within a time delay of the inverse of the nonlinear growth rate, which is the time scale of the development of the possibly dangerous levels of fluctuations. This stabilization is similar to that of the simulation with a C++ implementation. It is expected that significant gains can be achieved for the systems of a higher order, which are to be found in future fusion devices.

Adaptive Optimal Stochastic State Feedback Control of Resistive Wall Modes in Tokamaks

An adaptive optimal stochastic state feedback control is developed to stabilize the resistive wall mode (RWM) instability in tokamaks. The extended least square method with exponential forgetting factor and covariance resetting is used to identify the time-varying stochastic system model. A Kalman filter is used to estimate the system states. The estimated system states are passed on to an optimal state feedback controller to construct control inputs. The Kalman filter and the optimal state feedback controller are periodically redesigned online based on the identified system model. This adaptive controller can stabilize the time dependent RWM in a slowly evolving tokamak discharge. This is accomplished within a time delay of roughly four times the inverse of the growth rate for the time-invariant model used.

Adaptive stochastic output feedback control of resistive wall modes in tokamaks

An adaptive optimal stochastic output feedback control is developed to stabilize the resistive wall mode (RWM) instability in tokamaks. The system dynamics is experimentally determined via the extended least square method with an exponential forgetting factor and covariance resetting. The optimal output feedback controller is redesigned online periodically based on the system identification. The output measurements and past control inputs are used to construct new control inputs. The adaptive output controller can stabilize the time dependent RWM in a slowly evolving tokamak discharge. This is accomplished within a time delay of roughly three times the inverse of the growth rate. The design procedure is simpler and the computation time is shorter than the state feedback method reported earlier in Sun, Sen, and Longman [Phys. Plasmas13, 012512 (2006)].

Adaptive optimal stochastic state feedback control of resistive wall modes in tokamaks

An adaptive optimal stochastic state feedback control is developed to stabilize the resistive wall mode (RWM) instability in tokamaks. The extended least-square method with exponential forgetting factor and covariance resetting is used to identify (experimentally determine) the time-varying stochastic system model. A Kalman filter is used to estimate the system states. The estimated system states are passed on to an optimal state feedback controller to construct control inputs. The Kalman filter and the optimal state feedback controller are periodically redesigned online based on the identified system model. This adaptive controller can stabilize the time-dependent RWM in a slowly evolving tokamak discharge. This is accomplished within a time delay of roughly four times the inverse of the growth rate for the time-invariant model used.

Neoclassical dissipation and resistive wall modes in tokamaks

It is shown that the critical toroidal plasma flow speed that is required to stabilize the resistive wall mode in tokamaks is reduced by a factor of the order of B/Bθ or of 1.265ε3/4B/Bθ depending on the plasma parameters when the perturbed neoclassical viscosity driven current is taken into account. Here, B is the magnetic field strength, Bθ is the poloidal magnetic field strength, and ε is the inverse aspect ratio. This effect is illustrated using an existing model for the resistive wall modes by including the neoclassical dissipation in the derivation of the dispersion relation. The derivation is based on fluid equations with the plasma viscosity, calculated using kinetic equation, as the closure. The reduction of the critical toroidal speed is a consequence of the parallel (to the magnetic field B) momentum equation when neoclassical viscosity becomes important. The results are compared with experimental observations in tokamaks.

Physics and stabilization of resistive wall modes in tokamaks

The theory of resistive wall modes (RWMs) is discussed and compared with experimental results. Special attention is given to the possibilities of stabilizing the RWM by plasma rotation and active feedback. A simple cylindrical model is used to illustrate various aspects of active control. Fully toroidal computations are also presented, including predictions for RWM stabilization in ITER. According to ideal MHD, robust control of RWM should be straightforward. Theory for stabilization by rotation is discussed, and a semikinetic model is introduced, which compares favourably with experiment. The semikinetic model produces somewhat lower rotation thresholds than previous models.

Optimal control of tokamak resistive wall modes in the presence of noise

A scheme of optimal control based on “state” feedback for the resistive wall modes in tokamaks is described. The important effects of ever present magnetohydrodynamic noises, which have been neglected in the past, are included in the stochastic formulation of the basic equations. The optimality of the system is obtained both in terms of minimization of the stabilized fluctuation levels of the modes and the control energy. It is found that the stabilized fluctuation level and control signal level are of the order of the noise level of the system in the steady state.

Drift stabilization of internal resistive-wall modes in tokamaks

The problem of drift stabilization of the internal resistive-wall modes (RWMs) in tokamaks is theoretically investigated. The basic assumption of the model is that, when the drift effects are neglected, these modes are unstable in the absence of a conducting wall and stable in the presence of a close-fitting perfectly conducting wall. In the former case, the instability condition is expressed as Δ′∞>0, where Δ′∞ is the matching parameter calculated under the assumption that the wall is removed to infinity. In the latter case, one has ΔW′<0, where ΔW′ is the external matching parameter of tearing modes calculated assuming a perfectly conducting wall at the plasma boundary. In the case with a resistive wall, the relevant parameter can be either Δ′∞ or ΔW′, depending on whether the value of the dimensionless parameter ωτs/2m is small or large, respectively (here ω is the mode frequency, τs is the resistive time constant of the wall, and m is the poloidal mode number). In the presence of drift effects, the mode frequency ω is approximately equal to the electron drift frequency, ω≈ω*e. The value of the parameter ω*eτs/2m, which therefore determines the behavior of internal RWMs, is estimated for several existing tokamaks, namely, AUG (ASDEX-Upgrade), DIII-D, JET, TFTR, and JT-60U, as well as for the projected ITER-FEAT. It is shown that, although drift effects do not stabilize internal RWMs in current devices, they should be efficient in suppressing these modes in reactor-grade tokamaks.

Active control of resistive wall modesin the large-aspect-ratio tokamak

The large-aspect-ratio model for current-driven external kinks is applied to study control of non-axisymmetric resistive wall modes in tokamaks. Comparison with toroidal computations indicates that the cylindrical instabilities react in similar ways to feedback as the pressure-driven toroidal modes, when the feedback and sensor coils are placed on the low-field side of the torus. However, higher gain is required in the cylindrical case. The cylindrical model is used to gain insights into design issues concerning a feedback system for ITER, with a double wall and superconducting coils. Good control performance and acceptable coilvoltages are found in an initial value problem, where the initial conditions correspond to expected noise levels, when sensors for the poloidal field are placed inside the first wall. This can be accomplished with a PI (proportional plus integral) controller from the voltages of the sensor loops to the voltage over the active coils. The two-wall structure of ITER makes control somewhat more demanding than for tokamaks with a single wall. Adequate control can be achieved also when poloidal sensors are placed outside a single wall, but this requires additional derivative action (PID) and the resulting voltages are significantly higher.

Physics and control of resistive wall modes

Active feedback stabilization of resistive wall modes in tokamaks is studied both analytically, using large aspect ratio theory, and by means of toroidal computations. Extensive studies show that robust stabilization, with respect to variations in plasma current, pressure and flow velocity, can be achieved with a simple control system using poloidal sensors inside the first wall. The required coil voltages are modest, even for the two-wall structure of a tokamak reactor.