1. 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

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

3. 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

4. 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

5. 2.5D MHD with Low ReM Approximation
Full system of MHD equations
Low magnetic Re approximation
Finite time spin-up has been implemented

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

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

8. Equation of State with Atomic Processes
Energy Sinks
Conductivity

9. 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.

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

11. 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.

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

13. Conductivity Model
Ionization by Impact

14. Warm-up Time in Plasma Pedestal

15. 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.

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

17. 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

18. 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.

19. 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

20. 3D implementation
Parallel 3D implementation has been completed and fully tested
Same principle as 2D
Bilinear interpolation of flux

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

22. 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

23. 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

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

25. 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

26. Axially symmetric MHD simulation
Mach number distribution at

27. 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.

28. 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

29. 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

30. 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?

31. 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