MHD Simulation of Pellet Ablation in Tokamak CSC Seminar April 19, 2007, BNL MHD Simulation of Pellet Ablation in Tokamak Tianshi Lu Computational Science Center Brookhaven National Laboratory Roman Samulyak, CSC/BNL Paul Parks, General Atomics Jian Du, Stony Brook/AMS

Talk outline Motivation Main models Numerical algorithms for MHD equations Simulations result of pellet ablation

Pellet Ablation in the Process of Tokamak Fueling Detailed studies of the pellet ablation physics (local models) Studies of the tokamak plasma in the presence of an ablating pellet (global models) ITER schematic Problems Pellet ablation (rate and structure) Striation instabilities Plasma disruption mitigation

Main Models MHD with low magnetic Reynolds number approximation Equation of state with atomic processes Kinetic model for the interaction of hot electrons with the ablated gas Surface ablation model Cloud charging and rotation models New conductivity model (ionization by electron impact) Penetration of the pellet through the plasma pedestal region Finite shielding length due to the curvature of B field Schematic of processes in the ablation cloud

2.5D MHD with Low ReM Approximation Full system of MHD equations Low magnetic Re approximation Finite time spin-up has been implemented

Equation of State with Atomic Processes Saha equation for the dissociation (ionization) fraction

Equation of State with Atomic Processes Second law of thermodynamics: Compatibility with the second law of thermodynamics requires:

Equation of State with Atomic Processes Energy Sinks Conductivity

Equation of State with Atomic Processes High resolution solvers (based on the Riemann problem) require the sound speed and integrals of Riemann invariant type expressions along isentropes. Therefore the complete EOS is needed. For better numerical efficiency, FronTier operates with three pairs of independent thermodynamic variables: (r,E), (r,p) and (r,T). For the first two pairs of variables, solve nonlinear equation for T by iteration. Such an approach is prohibitively slow for the calculation of Riemann integrals. To speedup calculations, we precompute and store values of Riemann integral as functions of pressure along isentropes. Two dimensional table lookup and bi-linear interpolation are used.

Hot Electron Energy Deposition In the cloud: On the pellet surface:

Surface Ablation Model Some facts: The pellet is effectively shielded from incoming electrons by its ablation cloud Processes in the ablation cloud define the ablation rate, not details of the phase transition on the pellet surface No need to couple to acoustic waves in the solid/liquid pellet The pellet surface is in the super-critical state As a result, there is not even well defined phase boundary, vapor pressure etc. This justifies the use of a simplified model: Mass flux is given by the energy balance (incoming electron flux) at constant temperature Pressure on the surface is defined through the connection to interior states by the Riemann wave curve Density is found from the EOS.

Cloud Charging Model Sheath potential depends on the line-by-line cloud opacity.

Conductivity Model Ionization by Impact

Warm-up Time in Plasma Pedestal

Finite Shielding Length Without MHD, the cloud expands in three dimensions, so that the ablation rate reaches a finite value in the steady state. With MHD, the cloud expansion is one dimensional, so that the ablation rate would goes to zero by the ever increasing shielding, unless a finite shielding length in introduced. The steady state ablation rate is smaller with a longer shielding length.

Talk outline Motivation Main models Numerical algorithms for MHD equations Simulations result of pellet ablation

FronTier-MHD numerical scheme Elliptic step Hyperbolic step Point Shift (top) or Embedded Boundary (bottom) Propagate interface Untangle interface Update interface states Apply hyperbolic solvers Update interior hydro states Calculate electromagnetic fields Update front and interior states Generate finite element grid Perform mixed finite element discretization or Perform finite volume discretization Solve linear system using fast Poisson solvers

Numerical Implementation: Front Tracking based MHD Code for Free Surface Flows Interior and interface states for front tracking Explicitly tracked interfaces: resolution of material properties and multiple scales Equations are discretized separately in each domain: no numerical diffusion. Interfaces are propagated according to solutions of the Riemann problem. FronTier-MHD is a 3D code for free surface MHD: solves the coupled hyperbilic – elliptic problem in geometrically complex evolving domains. Supports topological changes in 2D and 3D (formation of droplets). In the axisymmetric pellet problem, we avoid solving the elliptic problem as the current density is a known function of the velocity and magnetic field.

Embedded Boundary Elliptic Solver Main Ideas Based on the finite volume discretization Potential is treated as cell centered value, even if the center is outside the computational domain Domain boundary is embedded in the rectangular Cartesian grid, and the solution is treated as a cell-centered quantity Using finite difference for full cell and linear interpolation for cut cell flux calculation

3D implementation Parallel 3D implementation has been completed and fully tested Same principle as 2D Bilinear interpolation of flux

Talk outline Motivation Main models Numerical algorithms for MHD equations Simulations result of pellet ablation

Previous Studies Transonic Flow (TF) (or Neutral Gas Shielding) model, P. Parks & R. Turnbull, 1978 Scaling of the ablation rate with the pellet radius and the plasma temperature and density 1D steady state spherical hydrodynamics model Neglected effects: Maxwellian hot electron distribution, geometric effects, atomic effects (dissociation, ionization), MHD, cloud charging and rotation Theoretical model by B. Kuteev et al., 1985 Maxwellian electron distribution An attempt to account for the magnetic field induced heating asymmetry Theoretical studies of MHD effects, P. Parks et al. P2D code, A. K. MacAulay, 1994; CAP code R. Ishizaki, P. Parks, 2004 Maxwellian hot electron distribution, axisymmetric ablation flow, atomic processes MHD effects not considered

Spherically symmetric simulation No Atomic Processes (Polytropic EOS) EOS with Atomic Processes Normalized ablation gas profiles at 10 microseconds Excellent agreement with TF model and Ishizaki. Verified scaling laws of the TF model Poly EOS Plasma EOS Sonic radius 0.66 cm 0.45 cm Temperature 5.51 eV 1.07 eV Pressure 20.0 bar 26.9 bar Ablation rate 112 g/s 106 g/s

Axially Symmetric Hydrodynamic Simulation Temperature, pressure, and Mach number of the steady-state ablation flow Mach number Temperature, eV Pressure, bar

Axially symmetric MHD simulation Main simulation parameters: Plasma electron temperature Te 2 keV Plasma electron density ne 1014 cm-3(standard) 1.6x1013 cm-3(el. shielding) Warm-up time tw 5 – 20 microseconds Magnetic field B 2 – 6 Tesla Velocity distribution. Channeling along magnetic field lines occurs at

Axially symmetric MHD simulation Mach number distribution at

Properties of the steady state ablation channel. Solid line: 2 Tesla, dashed line: 4 Tesla, dotted line: 6 Tesla. Warm up time is 10 microseconds.

Radius of the ablation channel Solid line: tw =10 microseconds, ne = 1.0e14 cm-3 Dashed line: tw = 10 microseconds, ne = 1.6e13 cm-3 Dotted line: tw = 5 microseconds, ne = 10e14 cm-3

Density along the axis of symmetry and the ablation rate Solid line: MHD model, B = 6 Tesla, ne = 1.0e14 cm-3 Dashed line: MHD model, B = 2 Tesla, ne = 1.6e13 cm-3 Dotted line: 1D spherically symmetric model Solid line: tw = 10, ne = 1.0e14 cm-3 Dashed line: tw = 10, ne = 1.6e13 cm-3 Dotted line: tw = 5, ne = 10e14 cm-3

Factors Affecting Ablation Rate Maxwellian hot electron distribution vs. Mono-energetic electrons Increase by 2.75 (Ishizaki, 1D) Atomic processes in ablation cloud vs. Polytropic gas Reduce by 0.95 (both Ishizaki and our work) Axisymmetric flow vs. Spherical flow Reduce by 0.82 (our work) MHD (Lorentz force) vs. 2D hydrodynamic model Reduce by 0.3 (B = 6 Tesla) ~ 0.4 (B = 2 Tesla) (our work) Cloud charging and rotation vs. MHD without rotation Increase by 2~3 ? (our ongoing work) Striation instability Increase by ??? TF Model has none of these factors, but it is claimed to agree with experiments. What will our sophisticated models and simulations predict?

Conclusions and future work Developed MHD pellet ablation model based on front tracking Performed numerical simulation of the deuterium pellet ablation 1D spherical model: excellent agreement with TF model and Ishizaki 2D pure hydro model: explained the factor of 2 reduction of the ablation rate Performed first systematic studies of the ablation in magnetic fields Subsonic ablation flow everywhere in the channel Lower and uniform pressure on the pellet surface compared to hydro model Extended plasma shield reduces the ablation rate Channel radius and ablation rate strongly depend on the warm-up time In ITER, a fast pellet injection will result in a small ablation rate Ongoing and future work Benchmark against DIII-D experiment Simulation using ITER parameters 3D simulations of the pellet ablation and studies of striation instabilities Coupling our pellet ablation model as a subgrid model with a tokamak plasma simulation code