The ablation of tokamak pellets and its influence on the tokamak plasma has been studied using several analytical and numerical approaches (see [2] and references therein). The inherent limitation of the previous ablation models has been the the absence of a rigorous inclusion of important details of physics processes in the vicinity of the pellet and in the ablation channel, and insufficient accuracy of numerical models due to extreme change of thermodynamics states on short length scales. The motivation of the present work is to improve both the accuracy of computational models by using the front tracking technology of the Interoperable Technologies for Advanced Petascale Simulations (ITAPS) Center [3], and physics modeling of the interaction of the pellet ablation channel with the tokamak magnetic field. The present work is a continuation of [5] which introduced sharp interface numerical MHD model for the pellet ablation, but omitted effects of charging and rotation of the pellet ablation channel. Both sharp interface numerical techniques and the resolution of complex physics processes in the pellet ablation channel are important not only for the calculation of pellet ablation rates and the fuelling efficiency, but also for the understanding of striation instabilities in tokamaks [6].

In this section, we will describe numerical ideas implemented in the FronTier-MHD code. In general, the system of MHD equations in the low magnetic Reynolds number

approximation is a coupled hyperbolic - elliptic system in a geometrically complex moving domain. We have developed numerical algorithms and parallel software for 3D simulations of such a system

\cite{SamDu07} based on the ITAPS front tracking technology \cite{FT_lite}.

The numerical method uses the operator splitting method. We decouple the hyperbolic and elliptic parts of the MHD system for every time step. The mass, momentum, and energy conservation equations are solved first without the electromagnetic terms

(Lorentz force). We use the front tracking hydro code FronTier with free interface support

for solving the hyperbolic subsystem. The electromagnetic terms are then found, in the general case,

from the solution of the Poisson equation for the electric potential in the conducting medium using

the embedded boundary method, as described in \cite{SamDu07}.

%In the current 2.5D axisymmetric model

%for the pellet ablation, the elliptic step is eliminated as the current

%density in each point of the ablation cloud is a function of the hydrodynamic state and some integrated

%quantities along the corresponding magnetic field line. In our current work in progress on 3D pellet

%ablation, solving of a modified 3D Poisson problem is required.

At the end of the time step, the fluid states are integrated along every grid line in the longitudinal

direction in order to obtain the electron heat deposition and internal hot currents.

The heat deposition changes the internal energy and temperature of fluid states, and therefore

the electrical conductivity. The Lorentz force and the centrifugal force are then added to the momentum equation.

FronTier represents interfaces as lower dimensional meshes moving through a volume filling grid (\Fig{FT_structures}).

The traditional volume filling finite difference grid supports smooth solutions located

in the region between interfaces. The location of the discontinuity and the jump in the solution

variables are defined on the interface. A computational stencil is constructed at every interface

point in the normal and tangential direction, and stencil states are obtained through interpolation.

Then Euler equations, projected on the normal and tangential directions, are solved. The normal

propagation of an interface point employs a predictor - corrector technique. We solve the Riemann problem for

left and right interface states to predict the location and states of the interface at the next time step.

Then a corrector technique is employed which accounts for fluid gradients on both sides of the interface.

Namely, we trace back characteristics from the predicted new interface location and solve Euler

equations along characteristics. After the propagation of the

interface points, the new interface is checked for consistency of intersections. The untangling of the

interface at this stage consists in removing unphysical intersections, and rebuilding a topologically

correct interface. The update of interior states using a second order conservative scheme is performed

in the next step. The tracked interface allows us to avoid the integration across large discontinuities

of fluid states, and thus eliminate the numerical diffusion. The FronTier code has been used for

large scale simulations on various platforms including the

IBM BlueGene supercomputer New York Blue located at Brookhaven National Laboratory.

[1] L. Baylor et al., Improved core fueling with high field pellet injection in the DIII-D tokamak, Phys. Plasmas, {\bf 7} (2000), 1878-1885.

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[2] Pegourie B Review: Pellet injection experiments and modelling, Plasma Phys. Control. Fusion 49 (2007) R87- R160.

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[3] http://www.tstt-scidac.org/

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[4] R. Samulyak, J. Du, J. Glimm, Z. Xu, A numerical algorithm for MHD of free surface flows at low magnetic Reynolds numbers, J. Comp. Phys., 226 (2007), 1532 - 1549.

\bibitem{SamLuParks07}

[5] R. Samulyak, T. Lu, P. Parks, A magnetohydrodynamic simulation of pellet ablation in the electrostatic approximation, Nuclear Fusion, 47 (2007), 103-118.

\bibitem{Parks96}

[6] P. Parks, Theory of pellet cloud oscillation striations, Plasma. Phys. Control. Fusion, 38 (1996) 571 - 591.

\bibitem{FT_lite}

[7] J. Du, B. Fix, J. Glimm, X. Li, Y. Li, L. Wu, A Simple Package for Front Tracking, J. Comp. Phys., 213, 613–628, 2006.

\bibitem{NGS}

[8] P. Parks, R. Turnbull, Effect of transonic flow in the ablation cloud on the lifetime of a solid hydrogen pellet in plasma, Phys. Fluids, 21 (1978), 1735 - 1741.

\bibitem{Ishizaki}

[9] R. Ishizaki, P. Parks, N. Nakajima, M. Okamoto, Two-dimensional simulation of pellet ablation with atomic processes, Phys. Plasmas, 11 (2004), 4064 - 4080.