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Thermal and magnetic energy content of the plasma have to be rapidly and uniformly removed during a disruption mitigation attempt so as to prevent damage to the plasma-facing components. Injection of high-Z impurities, either through massive gas injection at the plasma periphery (MGI), or by direct shattered pellet injection (SPI), aims to accomplish this goal by uniformly radiating away the plasma stored energy to the wall$^1$. However, due to geometric limitations on injection sites, nonuniformities in the radiation patterns and the resulting plasma response are inevitable (see, for example, Eidietis$^2$ and the references therein). The goal of this work is to report on the preliminary results of a systematic numerical study of the effects of injection asymmetries on the expected radiation patterns and the generated magnetohydrodynamic (MHD) activity.

Initially certain simplifying assumptions are made, and ablation and ionization physics are not directly addressed. Instead the injected impurities are assumed to lead to a localized cooling of the plasma with a three dimensional Gaussian profile at one or more toroidal locations. The resulting nonlinear perturbation with $\delta p < 0$ propagates as a rarefaction wave, mostly in the parallel direction, forming well-defined flux tubes on nearby low-order rational surfaces. The initial dynamics of this expansion wave confirms the poloidal torque analysis presented in Aydemir$^3$ and is consistent with the radiation patterns observed on DIII-D and elsewhere$^{1,2}$.

Figure 1 shows a flux surface poloidally and toroidally cut opened near the $q=3$ surface. In Fig. 1 (a), a single injection at $\theta=1.5,~\zeta=0$ (poloidal and toroidal angles, respectively) and the resulting $m=3, n=1$ low-pressure impurity flux tube is shown. Two perfectly balanced injections at the same radial and poloidal locations but toroidally separated by $\pi$ radians result in the $m=6, n=2$ flux tube seen in Fig. 1 (b). Clearly this symmetric double injection leads to a radiation pattern more uniformly distributed around the flux surface. Figures 1(c,d) show the kinetic energies in various toroidal mode numbers for these two cases. In steady-state, the single-injection case in (a,c) has higher energies in both the $n=1$ and $n=2$ modes: $W_K(n=1)=1.55\times 10^{-4}, W_K(n=2)=1.05\times 10^{-4}.$ For the symmetric double injection in (b,d),we have $W_K(n=2)=6.62\times 10^{-5}$, with negligible energies in all the odd mode numbers. Thus, the symmetric injection produces a more uniform radiation pattern and lower-amplitude MHD activity.

However, the perfectly-balanced double injection results shown above represent a very unrealistic case. Even if the two injections can be synchronized perfectly, it would be difficult to ensure that the impurity payload in the same in both locations; one is likely to be of different size than the other. Figure~2 shows the consequences of two synchronous injections $\pi$-radians apart where one ``pellet'' size is only half of the other.

Because the injections are geometrically symmetric but have different payloads, the nonlinear perturbation on the plasma pressure is not consistent with the $q=3$ symmetry any longer and generates odd-n modes also, as seen in Fig. 2(c). Ballooning-like modes that develop at the plasma edge (Fig. 2(a) eventually lead to a significant shrinking of the plasma column, mostly due to convective ELM-like losses, as seen in Fig. 2(b). These computational results will be extended and compared with experimental observations where available.

**References**

$^1$ E.M. Hollmann, P.B. Aleynikov, T.Fulop, *et al.,* Phys. Plasmas, 22:021802, 2015.

$^2$ N.W. Eidietis, V.A. Izzo, N.Commaux, *et al.*, Phys. Plasmas, 24:102504, 2017.

$^3$ A.Y. Aydemir. Phys. Plasmas, 25:050702, 2018.

Affiliation | National Fusion Research Institute, Daejeon, Korea |
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Country or International Organization | Korea, Republic of |