Perhaps ironically, the confinement properties of tokamaks have become so good that various techniques are now being deployed to deliberately impair confinement locally in space and sometimes temporally, thereby either avoiding, or initiating controlled, destablization of MHD instabilities.
Not doing so can lead to delayed but consequently larger perturbations, which in the case of edge-localized modes 2 , 3 can damage the vessel wall, and in the case of core-localized sawteeth 4 can trigger 5 , 6 , 7 globally confinement-limiting, and potentially disruption-causing, neoclassical tearing modes NTMs 8. In tokamaks, it is well known that instabilities are primarily driven by steepening gradients in the current density or the pressure.
Additionally, minority energetic ion populations in ITER, including fusion born alpha particles, will have a considerable effect on plasma stability. In this case, the MHD activity can be seen by inspection of the periodic collapse of the core plasma temperature, thus tracing the characteristics of the so-called sawtooth oscillation. In this particular pulse, the energetic ion distributions arising from neutral beam injection NBI and ion cyclotron resonance heating ICRH were deliberately tuned in order to delay the onset of the sawtooth crash.
In Figure 1a it is seen that following the application of ICRH the longest sawtooth period exceeds one second, which is more than an order of magnitude longer than during the early phase of the pulse without auxiliary heating the Ohmic phase. Despite the plasma beta ratio of thermal to magnetic energy being half that of the expected high confinement mode H-mode standard operation in ITER, the magnetics signal indicates the growth of a secondary mode directly following the crash of the long sawtooth. The detection of the NTM by the machine protection system eventually leads to the auxiliary power, current, magnetic field and density being ramped down to avoid a plasma disruption.
This produces co-current propagating waves, thus creating a large tail in the co-passing ion phase space, and long sawteeth. The traditional method of controlling dangerous MHD activity, such as NTMs, has involved techniques that directly affect the current or pressure gradients in the vicinity of an instability. Such methods enable a degree of control over the timing and amplitude of the MHD oscillation, but are power intensive and would ultimately reduce the reactor's energy efficiency.
By contrast, it is shown in this article that instabilities can be controlled effectively without compromising plasma performance by careful tailoring of the phase space properties of auxiliary energetic ion populations. We specialize in the control of the sawtooth instability, and in particular, we demonstrate that the phase space properties of an energetic ion population can be deliberately tuned in the opposite way to those of the JET pulse shown in Figure 1 , so that sawteeth can be shortened in period, and consequential NTMs avoided.
This innovative technique, stemming from new theoretical insight, is demonstrated for the first time in high confinement mode H-mode plasmas, in which an actuator has created a high-performance reactor-like plasma in quiescent conditions that would otherwise suffer from potentially dangerous NTMs. Stabilization of MHD modes, by virtually collisionless energetic particles, is not unique to tokamak plasmas.
Figure 2 compares ion orbits confined in the magnetic field of the earths magnetosphere with ion orbits in the magnetic field lines of the tokamak. The rapid helical gyro motion has been averaged out in the tokamak, so that the guiding centres of particles moving in parallel with, and drifting across, the magnetic field lines can be seen clearly. Here, the magnetic moment is an adiabatic invariant over the single particle motion.
Such magnetically trapped particle orbits are shown in Figure 2a,b. In a tokamak, the toroidal coordinate is an angle of near symmetry, as it also is in the magnetosphere assuming a perfect dipole field. Here, a population is considered collisional if particles typically undergo many Coulomb collisions during one bounce cycle.
The energetic ions in Earth's magnetosphere a and the tokamak b are confined to spherical and toroidal surfaces, respectively. Three essential types of confined orbits are possible in the tokamak configuration b where field lines lie on a toroidal surface: trapped red , co-passing blue and counter-passing green. Particles that do not undergo bounce trapping are lost to the Earth's atmosphere in Figure 2a. However, in a torus, non-vanishing continuous vector magnetic fields 15 exist, and consequently collisionless untrapped particles can in principle follow magnetic field lines indefinitely without being lost.
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Such circulating ions are known as co- or counter-passing, with convention defining co-passing as those circulating toroidally in the direction of the Ohmically induced toroidal current. This article identifies the crucial effects of collisionless co- and counter-passing particles on general MHD disturbances, including the internal kink mode.
The internal kink is a core-localized pressure and current driven mode, whose stability determines the onset of the sawtooth reconnection event, and thus potentially the triggering of an NTM. The novel theoretical advances described here are exploited in the creation of experiments that produce an imbalance in the number, or energy density, of co- and counter circulating ions, and in so doing, control the sawtooth instability in high-performance tokamak conditions.
In order to assess the impact of collisionless energetic ions on stability, the hybrid kinetic MHD model 16 , 17 is applicable. The linearized equations follow MHD, except that the closure of the system departs from the equation of state, which is only applicable to collisional populations.
Breaking the populations into a core ' c ' collisional population and a hot ' h ' , essentially collisionless ion population, the linearized equation of motion for components perpendicular to the equilibrium magnetic field B is:. The perturbed distribution function about equilibrium f h can be written 18 in terms of perturbations of the three quantities that are conserved in the equilibrium state:. Shown in Figure 3a is the radial r component of the normalized unstable internal kink displacement 19 , or the equivalent poloidal electric field in accordance with Ohm's law, as a function of r.
The radial component of the perturbed electric field is also important, but for clarity, its effect is not illustrated in what follows. The latter are keV 3 He ions orbiting in a 3T tokamak magnetic field. Owing to the fact that trapped ions do not complete a full poloidal circuit, the convolution is dominated by a positive poloidal drift velocity that is, counter-clockwise in Fig.
Integrating over velocity space, and assuming that f h is a Maxwellian with 3 He tail temperature keV, parabolically distributed in radius, it is seen in Figure 3b that the radial component of the normalized force points inwards. In the context of the momentum equation 1 this states that the reaction of trapped ions to a radially outward fluid displacement is inwards, that is, the kinetic-trapped ion response attempts to damp the initial perturbation.
This effect is responsible for the trapped ion stabilization of low frequency modes in the magnetosphere 14 and the tokamak 12 , This occurs especially for particles confined in the region overlapping a resonant MHD surface, and it is in this respect that the mechanism outlined here is relevant to a broad class of modes, including interchange modes 20 , toroidal Alfven eigenmodes 21 and resistive wall modes For the internal kink mode, counter-passing ions will observe a larger poloidal electric field in the region where the poloidal drift velocity is positive where the poloidal drift is anti-clockwise in Fig.
This variation in the electric field over the orbit occurs because of the radial drift motion of the single particle across magnetic field lines, as shown in Figure 3a. The radial drift excursion is enhanced as the energy of the single particle is increased relative to a fixed cyclotron frequency. Moreover, circulating particles that are almost trapped have the largest radial drift excursion 23 , and it is these particles that yield the largest contribution to.
Finally, it should be noted that there are additional finite orbit width modifications 23 associated with the adiabatic response the toroidal magnetic potential term on the right hand side of equation 3 , and it is the sum of all the finite orbit width effects that are ultimately evaluated for passing ions. Plotted in Figure 3c,d are the contributions to the radial component of due to the radial excursion of, respectively, co- and counter-passing ions.
It is seen that for co-passing particles the force points radially inwards close to r 1 , and thus acts so as to diminish the initial outward plasma displacement. In contrast, counter-passing particles create a force that points radially outward close to r 1. In this sense, the counter-passing ions have the opposite dynamics to energetic trapped ions, and attempt to amplify the initial outward radial fluid displacement. We note that the forces plotted in Figure 3 have the same normalization, and hence it is seen that passing ion kinetic effects compete with trapped ion kinetic effects.
Simulations undertaken with effective tail temperature larger than keV for the same cyclotron frequency show 23 , 24 that passing ion kinetic effects are dominant by virtue of the enhanced radial guiding centre motion. Nevertheless, crucially, we point out here that the effects of co- and counter-passing ions shown in Figure 3c,d almost cancel if the distribution of fast ions is symmetric in the parallel velocity note that a resonance effect not of concern here can remain under certain conditions In a , the radial profile of the normalized poloidal electric field , or equivalently the normalized radial fluid displacement , is shown for an internal kink displacement.
The assumed 3 He Maxwellian ion population is distributed parabolically in radius. It is crucial that sawtooth control is demonstrated in reactor-relevant high-performance conditions. Consequently, the experiments addressed by this paper make the leap to conditions akin to those in ITER, the design of which has been made possible because of our growing understanding of fast ion interaction with sawteeth, as reviewed in refs 26 , The JET experiments reported here have been designed with large auxiliary heating power, where enhanced co-current NBI power stabilizes sawteeth, and in this respect takes the place of fusion alpha particles in ITER.
In Figure 4, we quantify the degree of asymmetry for auxiliary heating methods used in the JET sawtooth control experiments described below.
An illustration of the co-current NBI is shown in Figure 4a , while Figure 4b plots the radial deposition of the current associated with the fast ions. The current is essentially the parallel velocity moment of f h , and is therefore a measure of the degree of parallel velocity asymmetry through the plasma. As expected for mirror-trapped particles, the distribution is close to symmetric in the trapped cone, while there is a significant asymmetry in the distribution of co- and counter-passing particles. The co-passing particles 23 , 29 and the trapped particles 12 , 13 stabilize the internal kink mode, and therefore sawteeth, by virtue of the mechanism visualized, respectively, in Figure 3b,c.
As will be seen, sawteeth are subsequently controlled, or shortened, by depositing 3 He minority ICRH tangent to the internal kink mode resonant surface r 1 on the inboard side of the device, as indicated in the SCENIC code 31 simulations shown in Figure 4d. Counter current propagating waves are employed in order to preferentially heat counter-passing ions.
The reason for the extremely effective control on sawteeth is indicated by inspecting the ICRH-driven current density in Figure 4e , where it is seen that very localized asymmetries in the parallel velocity are produced. Figure 4f plots the distribution function of ICRH 3 He ions close to the resonant surface r 1 , where it is seen that a large tail is generated for negative counter Ohmic current parallel velocity. In e the ICRH-driven current is plotted with respect to the minor radius normalized to the plasma edge, and in f the distribution of ICRH ions at the same location, with colour map scaling the same as the NBI distribution in c —see also Supplementary Movie 3.
Figure 5 shows JET pulse in which 4. With toroidal field 2. In contrast, the long sawtooth JET pulse , shown in the introduction Fig. Meanwhile, in an otherwise similar pulse to , JET pulse Fig. These 3 He minority ICRH experiments demonstrate for the first time that sawteeth in H-mode can be efficiently controlled by phase space engineering, where the sawtooth period has been reduced by a factor of two in this case. Such surgical but sustained instability control, by just tailoring the orbits of the energetic ion population, is an optimum solution for ITER or fusion reactors, where fusion energy production would ideally not be compromised by MHD control techniques.
Showing JET pulse in blue with 4. The fast ion mechanism described, and experimentally demonstrated, in this article is quantified in Figure 6 in terms of the potential energy integrated over the plasma volume. These are plotted against the difference between ICRH resonance position r res and the internal kink mode resonance radius r 1 , each normalized to the edge radius.
In the simulations, r res was held constant on the low field side of the device, and r 1 varied. The potential energy is obtained in terms of the perturbed force for each fast ion species, as calculated in Figure 6.
ogadospen.tk | Magnetohydrodynamic Stability of Tokamaks, Hartmut Zohm | | Boeken
The total fast ion potential energy is the sum of these terms. Furthermore, additional control techniques 26 , 27 , such as electron cyclotron current drive 34 , could be much more effective in plasmas where ICRH has nullified the stabilizing effects of alpha particles. Minority populations of energetic ions in ITER, including fusion born alpha particles, are expected to have a strong impact on plasma stability. The deployment of techniques that directly affect the current or pressure gradients in the vicinity of the instability can control the timing and amplitude of MHD oscillations, but such methods are power intensive and would ultimately reduce the reactor's energy efficiency.
By contrast, careful tailoring of the phase space properties of auxiliary energetic ion populations will enable MHD instabilities to be controlled effectively without compromising plasma performance. In this article, a mechanism has been identified capable of controlling MHD instabilities by velocity phase space engineering in tokamaks. The mechanism is unique to a toroidally confined plasma, in which energetic particles that are not magnetically bounce trapped can be distributed asymmetrically in the velocity parallel to the magnetic field.