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W. Park, E. Belova, G.Y. Fu, X.Tang, PPPL
L.E. Sugiyama, MIT


Nonlinear MHD simulations of pellet injection [1] are in qualitative agreement with recent ASDEX results [2]. MHD forces can accelerate large pellets, injected on the high field side of a tokamak, to the plasma center. More recently we have studied pellet - driven disruptions, which can occur if the pressure perturbation of the ablated pellet cloud is sufficiently large [3]. Nonlinearly, we find that a pellet - driven disruption can cause anomalous penetration of the pellet cloud. This may explain the results of ``killer pellet" experiments [4] in which the pellet material penetrated and cooled the plasma center.

It is assumed that pellet ablation is rapid and produces a `` tex2html_wrap_inline67 bubble," which is not in MHD equilibrium. The ablated pellet and background plasma are modeled with dissipative MHD equations. A single temperature, the electron temperature, is assumed for simplicity. The temperature transport parallel to the magnetic field, which tends to make the temperature approximately constant on magnetic field lines, is modeled with the ``artificial sound" method [3, 5].


Inboard pellet injection confers advantages with regards to MHD stability of the background plus pellet system. Because of the high local tex2html_wrap_inline67 at the pellet cloud, the system may be unstable to pressure driven modes, even though the background equilibrium is stable [3, 6]. Pellets on the outboard side tend to be more destabilizing, because the pellet pressure gradient and the equilibrium gradient add on the large R side of the equilibrium. With inboard injection, the pressure gradients oppose one another. In addition, on the low field side, the velocity perturbations resemble typical moderate wavelength ballooning modes. They produce disruptions in nonlinear simulations. On the high field side, the velocity perturbations are much more localized.

Figure 1: (a) pressure including pellet. (b) electrostatic potential.

A high tex2html_wrap_inline67 equilibrium was produced, with peak tex2html_wrap_inline75 . The equilibrium becomes unstable if tex2html_wrap_inline67 is increased to .30 a / R . The q profile varied from 1.7 on axis to 3.7 at the wall, and R/a = 3. The D shaped boundary was the same as in the previous section.

This was modified by the presence of a pellet perturbation, but now varied as tex2html_wrap_inline89 for improved numerical resolution. The low toroidal mode number part of the MHD equations were not allowed to evolve, only modes with mode number n ;SPMgt; 4. Inboard and outboard pellet perturbations were centered on approximately the same flux surface, so that a given density perturbation produced the same amplitude pressure perturbation. The density perturbation is tex2html_wrap_inline93

Figure 2: Growth rate of pellet driven ballooning modes for outboard (upper curve) and inboard pellet.

Inboard injection is more favorable, because the instability threshold tex2html_wrap_inline95 is higher, and because the unstable modes are more localized.


Nonlinear simulations of disruptions induced by pellet perturbations indicate anomalous penetration of the pellet material. The instability causes fragmentation of the major major radius side of the pellet cloud, while the inner part is relatively less disturbed.

Nonlinear simulations were performed with similar initial states as the linear cases, with tex2html_wrap_inline97 The pellet positions are somewhat different from the linear cases. The nonlinear runs were initialized by starting with an equilibrium (the same as in the previous section), adding a pellet perturbation, as well as a small velocity perturbation. The qualitative result, that there is a net density pinch, is independent of the sign of the velocity perturbation.

The density at first evolves without instability. At a later time, tex2html_wrap_inline99 ripples appear in the density contours. Later, at time tex2html_wrap_inline101 the density perturbations become more pronounced, as well as more extended poloidally.

Figure 3: (a) Initial density at tex2html_wrap_inline103 (b) density at tex2html_wrap_inline105

The motion of the density distribution can also be seen by considering the flux surface averaged density, performing the flux surface average in the toroidally averaged magnetic field. The initial flux surface averaged density tex2html_wrap_inline107 is labeled (a). The density at time tex2html_wrap_inline111 is shown in curve (b). As expected, the peak of the curve has shifted to the right, corresponding to a shift towards the outer boundary at tex2html_wrap_inline115 The density at tex2html_wrap_inline117 is shown in curve (c). Now the peak of the density has shifted inwards, to the left of the initial density peak. The density has shifted inwards towards the magnetic axis.

Figure 4: flux averaged density tex2html_wrap_inline121 at a - t = 0, b - tex2html_wrap_inline111 c - tex2html_wrap_inline127

A plausible cause for this inward transport is the change in the density distribution as the density perturbation flows along the magnetic field. An isoplot shows the initial density and at tex2html_wrap_inline127 At this point significant density perturbations appear halfway around the torus.

Figure 5: (a) Isoplot of density at t = 0, (b) tex2html_wrap_inline127

Because of the rotational transform (about 2) there is now a substantial density perturbation on the inside of the torus. It appears that the density perturbation is larger on the inside of the torus than on the outside. This can be attributed to the mixing caused by the ballooning like modes, which are localized to the outside, bad curvature part of the torus. The density distribution now resembles a pellet perturbation centered on the inside of the torus. As such, it tends to move radially outward, carrying the pellet material towards the magnetic axis.

Acknowledgments This work was supported by U.S. DoE grant DE-FG02-86ER53223 and contract DE-AC02-76-CHO-3073.

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Next: References

Hank Strauss
Mon Nov 22 16:49:15 EST 1999