Loss of beam ions to the inside of the PDX tokamak during a fishbone instability

Using data from two vertical charge exchange detectors on the Poloidal Divertor Experiment (PDX), a set of conditions have been identified for which loss of beam ions inwards in major radius is observed during a fishbone instability. Previously, it was reported that beam ions were lost only to the outside of the PDX tokamak. The losses to the inside are two orders of magnitude smaller than the losses to the outside and are consistent with numerical predictions based on the mode particle pumping theory.

ABSTRACT. Using data from two vertical charge exchange detectors on the Poloidal Divertor Experiment (PDX), a set of conditions have been identified for which loss of beam ions inwards in major radius is observed during a fishbone instability. Previously, it was reported that beam ions were lost only to the outside of the PDX tokamak. The losses to the inside are two orders of magnitude smaller than the losses to the outside and are consistent with numerical predictions based on the mode particle pumping theory.
In high /3-j-q discharges with near-perpendicular neutral beam injection on the Poloidal Divertor Experiment (PDX) an MHD instability called the fishbone instability ejected beam ions from the plasma [ 1 ]. In theoretical studies, White et al. [2] found that rotating MHD modes similar to those observed in the experiment resonate with the toroidal precession of the beam ion orbits, resulting in the loss of many of the injected beam ions. The mode resonance was predicted to produce a beacon of outwardly exiting beam particles correlated with the mode rotation [2]. In studies of the charge exchange efflux with toroidally displaced detectors the predicted n = 1 modulation of the beam ion loss was found [3]. The mode particle resonance theory also correctly predicted linear scaling of the beam ion loss with the amplitude of the * Present address: GA Technologies Inc., San Diego, CA 92138, USA. mode, as observed experimentally [4]. The loss of nonresonant fusion products also could be explained using White's fishbone model [5]. Further support for the model was given by Beiersdorfer et al. who reported in Ref. [3] that measurements with two vertically viewing neutral particle analysers indicated that beam ions were lost to the outside of the plasma but not to the inside. This letter presents data from the inner vertical charge exchange detector (IDE) [3] which indicate that 50 keV beam ions were lost to the inside of the PDX tokamak during a fishbone instability when the ions measured by the IDE were on confined orbits. Figure 1 shows bursts of charge exchange flux on the IDE that correlate with bursts of fishbone MHD activity. In this discharge, the plasma underwent a transition to the high mode [6] at 480 ms. Before the transition, large variations G>10%) in H« emission were not observed, but, after the transition, sudden changes in neutral density associated with edge relaxation phenomena (ERP) [7] were correlated with charge exchange bursts and fishbones. Conceivably, variations in neutral density may account for the bursts on the IDE in the high mode, but in the low mode the charge exchange bursts are almost certainly due to an increase in the number of fast ions in the IDE sightline. Because of the relatively gradual decay of the MHD bursts (about 1 ms) the possibility that sawteeth were coincident with the fishbones can be excluded [4,8] and therefore the bursts are not due to the transport of fast ions when sawteeth occurred [9]. Since the IDE sightline skims the inner radius of the plasma, these bursts indicate that fishbones transport beam ions inwards in major radius as well as outwards.
Usually, the signal-to-noise ratio on the IDE was inadequate to ascertain whether the charge exchange flux was modulated at the mode frequency by the fishbone oscillations (as it was on the outer edge of the plasma [1,3]). Figure 2 shows one of the few events LETTERS where internal structure seems to be present. This burst was synchronous with a large fishbone (fractional change in neutron emission AI n /I n = 25%) in a discharge with relatively large neutral density (Pb =4.7 MW). It is estimated that the peak signal level  corresponds to less than one hundred counts at the IDE channeltron detector, so even in this case errors associated with counting statistics are appreciable. A peak in the Fourier spectrum occurs near the frequency of peak amplitude of the Mirnov oscillations, suggesting that the spikes are due to beam ion motion induced by the mode.
A further indication that ions were lost to the inside of the tokamak is that the change in flux at a fishbone scales approximately linearly with the severity of the fishbone (Fig. 3). The quantity AI n /I n in Fig. 3 is approximately the fraction of energetic ions lost at the fishbone event [4]. Figure 3 shows that the flux depends much more strongly on AI n /I n than on the existence of a coincident ERP instability.
All of the above data are for neutrals with energy E approximately equal to the neutral beam injection energy Ei n j = 45 keV and I p ;> 350 kA. For energies well above Ejnj (E -E^j ^ 10 keV) or slightly below E^j (E ~ Einj ^ ~3 keV), bursts of beam ions were not LETTERS §10"   clearly observed on the IDE unless an ERP or a sawtooth instability was coincident with the fishbone (Fig. 4). For E -E^j <^ -10 keV, a fishbone combined with an ERP sometimes increased the flux by an order of magnitude relative to the flux at either a fishbone or an ERP alone. It appears that ions with E = 25-35 keV were not transported into the IDE sightline unless the fishbone instability was assisted by transport from another instability.
Bursts on the IDE during the fishbone instability were not observed unless the plasma current was sufficiently large to confine beam ions in the IDE sightline (Fig. 5). For deuterons with energy E = 45 keV, most of the ions in the IDE sightline hit vacuum vessel hardware when the plasma current was below I p <J275 kA. The corresponding cutoff for the outer detector (ODE) at 45 keV was I p <, 225 kA. The PDX rarely operated below 225 kA, however, and therefore a reduction in the amplitude of the burst at low plasma current was not observed on the ODE for E <, E^j. In fact, the ODE measured bursts during the fishbone instability even if the detected ions were on unconfined orbits; for example, large bursts were observed with the ODE at E = 80 keV, I p s 225 kA.
In conclusion, for I p J> 350 kA and E ^ E^j, fast ions were lost to the inside of the tokamak at the fishbone instability. However, these losses were still two orders of magnitude less than the losses to the outside of the tokamak observed on the ODE. Detailed Monte Carlo simulations of fishbones [10] predict that some ions should move inwards in major radius because of a combination of mode particle pumping [2] and classical Coulomb scattering. Thus, the observation of some transport to the inside of the tokamak is not inconsistent with either the mode particle pumping theory [2] or the major conclusions of the PDX paper that first reported results from the IDE [3]. ABSTRACT. The Grad-Shafranov equation for axisymmetric plasma equilibria has been solved in the ellipsoidal co-ordinate system (p, f, 0) with quasi-uniform current density profile up to the sixth order in f. Bean shaped and elongated elliptic equilibria and dee shaped equilibria can be described in this co-ordinate system. The basic characteristics of these equilibria are determined by the size, aspect ratio, elongation, poloidal beta, triangularity, and separatrix location.
The Grad-Shafranov equation is known to have separable solutions in cylindrical [9] and toroidal [10] co-ordinate systems using multipolar series expansion at moderate plasma elongation. To describe equilibria with large elongation, superposition of a large number of these multipolar solutions is necessary. An ellipsoidal co-ordinate system has co-ordinate surfaces approximating the desired external plasma shapes at high elongation. Thus it is natural to seek solutions of the equilibria in the ellipsoidal co-ordinate system. The purpose of this letter is to show how equilibria are obtained in an ellipsoidal co-ordinate system in terms of the basic characteristics of the equilibria.

INTRODUCTION
In a large number of experimental [ 1 ] and theoretical [2][3][4][5] analyses it has been established that the maximum beta attainable in a toroidal configuration is given by the scaling law |3(%) = C {I(MA)/[a(m)B(T)]}, with C being approximately 3.5. Plasma surface elongation permits an increased total plasma current I at a fixed edge safety factor qg. Triangulation [6] increases the magnetic well depth for beta optimization. Advanced triangulation of the plasma results in shapes with indentation of the plasma inside edge (bean shapes) [7]. Depending on the elongation and triangularity, these configurations resemble an ellipsoidal shell or a spherical shell [8].