1. Brookhaven Science Associates U.S. Department of Energy 1 Axisymmetric MHD Simulation of Pellet Ablation Roman Samulyak Computational Science Center Brookhaven National Laboratory CPPG Seminar, Princeton Plasma Physics Laboratory July 26, 2006, Princeton, NJ Tianshi Lu CSC/BNL, modeling, software development, fusion applications Paul Parks General Atomics, MHD theory, fusion applications Hydrodynamic and MHD Multiphase Flows: James Glimm, Stony Brook University / BNL, modeling, numerical algorithms Jian Du, Stony Brook University, software development, accelerator applications Zhiliang Xu, Xiaolin Li, CSC/SBU, front tracking methods

2. Brookhaven Science Associates U.S. Department of Energy 2 Pellet Ablation in the Process of Tokamak Fueling Detailed studies of the pellet ablation physics (local models) ITER schematic Studies of the tokamak plasma in the presence of an ablating pellet (global models)

3. Brookhaven Science Associates U.S. Department of Energy 3 Previous Studies: Local Models Transonic Flow (TF) (or Neutral gas shielding) model, P. Parks & R. Turnbull, 1978 Provides the scaling of the ablation rate with the pellet radius and the plasma temperature and density 1D steady state hydrodynamics model, monoenergetic electron distribution Neglected effects: MHD, geometric effects, atomic effects (dissociation, ionization) 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 Non-spherical ablation flow (axial symmetry), proper treatment of scattering Kinetic calculation of the electron heat deposition, atomic physics processes MHD effects not considered

4. Brookhaven Science Associates U.S. Department of Energy 4 Simulations using MH3D code, H. Strauss & W. Park, 1998 Finite element version of the MH3D full MHD code Details of the ablation are not considered Pellet is given as a density perturbation of initial conditions Smaller values of density and larger pellet radius (numerical constraints) Simulations using MHD code based on CHOMBO AMR package, R. Samtaney, S. Jardin, P. Colella, D. Martin, 2004 Analytical model for the pellet ablation: moving density source 8-wave upwinding unsplit method for MHD AMR package – significant improvement of numerical resolution Previous Studies: Global Models (examples)

5. Brookhaven Science Associates U.S. Department of Energy 5 Improved model is needed: Studies of the local pellet ablation physics were still missing MHD 3D effects Charging models for the ablation cloud Global plasma simulations in the presence of an ablating pellet need a better local model as input Studies of striation instabilities, observed in all experiments, are not possible without a 3D detailed physics model We are working on building and validations of such models

6. Brookhaven Science Associates U.S. Department of Energy 6 Pellet Ablation Model Based on Front Tracking Explicitly tracked interfaces: resolution of material properties and multiple scales MHD equations in the low magnetic Reynolds number approximation and numerical methods for free surface flows Equation of state with atomic processes Kinetic model for the interaction of hot electrons with the ablated gas Surface ablation model

7. Brookhaven Science Associates U.S. Department of Energy 7 MHD equations and approximations Full system of MHD equationsLow magnetic Re approximation Assumptions that will be verified later. Near the pellet Downstream in the ablation channel

8. Brookhaven Science Associates U.S. Department of Energy 8 Simplification of the elliptic equation for the pellet problem Axial symmetry: Pellet cloud charging, polarization currents, and the axial rotation of the ablation channel are neglected in this study (implementation of the charging and rotation of the channel is in progress and will be discussed later): Magnetic field is constant: B = (0,0,B z ) Therefore, and the only non-zero component of the current is

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

10. Brookhaven Science Associates U.S. Department of Energy 10 Stencil and equations for the interface point propagate

11. Brookhaven Science Associates U.S. Department of Energy 11 Schematic of the interface point propagate algorithm

12. Brookhaven Science Associates U.S. Department of Energy 12 Embedded Boundary Elliptic Solver Main Ideas Based on the finite volume discretization Domain boundary is embedded in the rectangular Cartesian grid, and the solution is treated as a cell-centered quantity The discretized operator is centered in centroids of partial cells Using finite difference for full cell and linear interpolation for cut cell flux calculation

13. Brookhaven Science Associates U.S. Department of Energy 13 Validation of the Elliptical Problem 2D problem 3D problem Letbe a solution of Derive the R.H.S. and B.C., solve numerically and compare with the exact solution. Mesh sizeErrorConv. RateCPU timeIterations 64x649.09e-05N/A0.08744 128x1282.01e-052.1750.38998 256x2564.80e-062.1222.223264 512x5121.78e-061.89315.445500 Mesh sizeErrorConv. RateIterations 32x32x321.32e-03N/A42 64x64x643.18e-042.05076 128x128x1288.05e-052.016144

14. Brookhaven Science Associates U.S. Department of Energy 14 Validation of the MHD Code A free mercury jet travels longitudinally along the z axis in a magnetic field (0,By,0) A satisfactory perturbation theory and experimental data are available (S. Oshima et al., JCME Int. J., 30 (1987), No. 261. Solid line: theory Dots: simulations

15. Brookhaven Science Associates U.S. Department of Energy 15 Muon Collider target: a brief summary of modeling and simulation Jet surface instabilities Cavitation in the mercury jet and thimble n Simulation of the mercury jet target interacting with a proton pulse in a magnetic field Studies of surface instabilities, jet breakup, and cavitation MHD forces reduce both jet expansion, instabilities, and cavitation

16. Brookhaven Science Associates U.S. Department of Energy 16 Saha equation for the dissociation (ionization) fraction Equation of State with Atomic Processes.

17. Brookhaven Science Associates U.S. Department of Energy 17 EOS with Atomic Processes Incomplete EOS (known from literature): 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. Using the second law of thermodynamics we found the complete EOS and showed that the compatibility with the second law of thermodynamics requires:

18. Brookhaven Science Associates U.S. Department of Energy 18 Complete EOS with Atomic Processes Notations: We will define the sound speed in a form typical for the polytropic gas: where the effective gamma is the Gruneisen coefficient is and the entropy is

19. Brookhaven Science Associates U.S. Department of Energy 19 For better numerical efficiency, FronTier operates with three pairs of independent thermodynamic variables: Numerical Algorithms for EOS For the first two pairs of variables, solve numerically nonlinear algebraic equation, and find T. Using, find the remaining state. Such an approach is prohibitively slow for the calculation of Riemann integrals (involves nested nonlinear equations). To speedup calculations, we precompute and store values on Riemann integrals as functions of the density and entropy. Two dimensional table lookup and bi-linear interpolation are used.

20. Brookhaven Science Associates U.S. Department of Energy 20 Influence of Atomic Processes on Temperature and Conductivity

21. Brookhaven Science Associates U.S. Department of Energy 21 Redlich-Kwong EOS for the cold and dense gas Real gas EOS (work in progress) We have derived an extension of the Redlich-Kwong EOS to include atomic processes (dissociation and ionization) The EOS contains three terms; the partial pressure/energy of the molecular gas is written in the Redlich-Kwong form, and the partial pressure/energies of the dissociated and ionized components is written in the ideal EOS form. Complete EOS has been derived (expressions for entropy, sound speed etc.) The numerical implementation is in progress

22. Brookhaven Science Associates U.S. Department of Energy 22 Electron Energy Deposition In the cloud:On the pellet surface:

23. Brookhaven Science Associates U.S. Department of Energy 23 Physics Models for Pellet Studies : Surface Ablation Features of the pellet ablation: 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

24. Brookhaven Science Associates U.S. Department of Energy 24 Main Simulation Parameters Pellet radiusrprp 2 mm Pellet density0.2 g/cm 3 Plasma electron temperature TeTe 2 keV Plasma electron density nene 10 14 cm -3 (standard) 1.6x10 13 cm -3 (electrostatic shielding) Length of the ablation channel LcLc 15 cm Warm-up time5 – 20 microseconds Magnetic fieldB2 – 6 Tesla

25. Brookhaven Science Associates U.S. Department of Energy 25 Three types of simulation 1D hydrodynamic model: spherically symmetric, no JxB force 2D hydrodynamic model: axially symmetric, directional heating along magnetic field lines, no JxB force 2D MHD model: axially symmetric, directional heating along magnetic field lines, JxB force is applied

26. Brookhaven Science Associates U.S. Department of Energy 26 We will compare results with TF model: P. Parks and R. Turnbull, Phys. Fluids, 21 (1978), 1735. Kuteev: B. V. Kuteev, Sov. J. Plasma Phys, 11 (1985), 236. MacAulay: A. K. MacAulay, Nucl. Fusion, 34 (1994), 43. Parks 2000: P. Parks, W. Sessions, L. Baylor, Phys. Plasmas., 5 (2000), 1968 Ishizaki: R. Ishizaki, P. Parks, N. Nakajiama, M. Okamoto, Phys. Plasmas, 11 (2004), 4064

27. Brookhaven Science Associates U.S. Department of Energy 27 Spherically symmetric simulation Normalized ablation gas profiles at 10 microseconds Polytropic EOSPlasma EOS Poly EOSPlasma EOS Sonic radius0.66 cm0.45 cm Temperature5.51 eV1.07 eV Pressure20.0 bar26.9 bar Ablation rate112 g/s106 g/s Excellent agreement with TF model and Ishizaki. Verified scaling laws of the TF model

28. Brookhaven Science Associates U.S. Department of Energy 28 Axially Symmetric Hydrodynamic Simulation Temperature, eVPressure, barMach number Distributions of temperature, pressure, and Mach number of the ablation flow near the pellet at 20 microseconds.

29. Brookhaven Science Associates U.S. Department of Energy 29 Axially Symmetric Hydrodynamic Simulation Temperature, pressure, and Mach number of the ablation flow near the pellet in the longitudinal (solid line) and radial (dashed line) directions at 20 microseconds.

30. Brookhaven Science Associates U.S. Department of Energy 30 The ablation rate, 90 g/s, agrees with Kuteev and MacAulay, disagrees with Ishizaki The reduction of the ablation compared to the spherically symmetric case is ~ 18%. This disagrees with prevailing expectations of the factor of 2 reduction (Kuteev and Ishizaki) However Kuteev did not calculate the 1D ablation rate: he compared with the FT model that used the monoenergetic electron distribution, and found a 2.2 reduction Our explanation of the factor of 2.2 reduction: Maxwellian heat flux increases the 1D ablation rate by 2.75 (Ishizaki) Directional heat flux in 2D reduces the ablation rate by 0.82 (this work) 2.75 x 0.82 = 2.25 Reduction of the ablation rate in 2D

31. Brookhaven Science Associates U.S. Department of Energy 31

32. Brookhaven Science Associates U.S. Department of Energy 32 Mach number distribution of the ablation flow near the pellet in 6 Tesla magnetic field. Warm up time is 20 microreconds. 3 microseconds 5 microseconds 9 microseconds

33. Brookhaven Science Associates U.S. Department of Energy 33 Temperature (eV) distribution of the ablation flow near the pellet in 6 Tesla magnetic field. Warm up time is 20 microseconds. 3 microseconds 5 microseconds 9 microseconds

34. Brookhaven Science Associates U.S. Department of Energy 34 Pressure (bar) distribution of the ablation flow near the pellet in 6 Tesla magnetic field. Warm up time is 20 microseconds. 3 microseconds 5 microseconds

35. Brookhaven Science Associates U.S. Department of Energy 35 Pressure along the z-axis of steady state ablation channel. No shieldingElectrostatic shielding Solid line: 2 Tesla, dashed line: 4 Tesla, dotted line: 6 Tesla. Warm up time is 10 microseconds.

36. Brookhaven Science Associates U.S. Department of Energy 36 Temperature along the z-axis of steady state ablation channel. Solid line: 2 Tesla, dashed line: 4 Tesla, dotted line: 6 Tesla. Warm up time is 10 microseconds. No shieldingElectrostatic shielding

37. Brookhaven Science Associates U.S. Department of Energy 37 Mach number along the z-axis of steady state ablation channel No shieldingElectrostatic shielding Solid line: 2 Tesla, dashed line: 4 Tesla, dotted line: 6 Tesla. Warm up time is 10 microseconds.

38. Brookhaven Science Associates U.S. Department of Energy 38 Normalized temperature and pressure in the ablation channel Solid lines: Simulation in 2 Tesla field with electrostatic shielding. Dashed lines: Theoretical parallel flow model for the ablation channel (Parks 2000)

39. Brookhaven Science Associates U.S. Department of Energy 39 Radius of the ablation channel Solid line: 10 microseconds warm up time, n e = 1.0e14 cm -1 Dashed line: 10 microseconds warm up time, n e = 1.6e13 cm -1 Dotted line: 5 microseconds warm up time, n e = 10e14 cm -1

40. Brookhaven Science Associates U.S. Department of Energy 40 Density along the axis of symmetry and the ablation rate Solid line: t w = 10, n e = 1.0e14 cm -1 Dashed line: t w = 10, n e = 1.6e13 cm -1 Dotted line: t w = 5, n e = 10e14 cm -1 Solid line: MHD model, B = 6 Tesla, n e = 1.0e14 cm -1 Dashed line: MHD model, B = 2 Tesla, n e = 1.6e13 cm -1 Dotted line: 1D spherically symmetric model

41. Brookhaven Science Associates U.S. Department of Energy 41 Justification of the Electrostatic Approximation B n e 2 Tesla4 Tesla6 Tesla 10 14 cm -3 0.5300.8220.1280.2230.0510.100 1.6x10 13 cm -3 0.1100.1800.0290.0550.0150.026 The ratio of induced magnetic field to the toroidal magnetic field.

42. Brookhaven Science Associates U.S. Department of Energy 42 Current work: implementation of the pellet charging model

43. Brookhaven Science Associates U.S. Department of Energy 43 Other Low Magnetic Reynolds Number Flows in Tokamaks Laser driven pellet acceleration Gyrotron driven pellet acceleration Liquid jet for plasma disruption mitigation High density, low temperature gas jets for tokamak fueling (inefficient method?) Laser driven pellet accelerationFueling using a high speed gaseous jet

44. Brookhaven Science Associates U.S. Department of Energy 44 Conclusions and future work Developed MHD code for free surface low magnetic Re number flows Front tracking method multiphase/multimaterial flows Elliptic problems in geometrically complex domains Phase transition models Performed numerical simulation of tokamak fueling through the injection of frozen deuterium - tritium pellets Computed ablation rates in hydro and MHD case Explained the factor of 2 reduction of the ablation rate Performed first systematic studies of the ablation in magnetic fields Future work 3D simulations of the pellet ablation Studies of striation instabilities Coupling our pellet ablation model as a subgrid model with a tokamak plasma simulation code Laser -- plasma interaction model with sharp absorption front, laser acceleration of pellets (possible)