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Adaptive Mesh Refinement MHD for Magnetic Fusion Applications

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### Adaptive Mesh Refinement MHD for Magnetic Fusion Applications

Ravi Samtaney and Stephen C. Jardin

Computational Plasma Physics Group

Princeton Plasma Physics LaboratoryPrinceton University

Phillip Colella, Daniel F. Martin, Terry J. Ligocki

Applied Numerical Algorithms Group

Lawrence Berkeley National Laboratory

SciDAC PI Meeting, March 22-24, Charleston, SC

Pellet Injection: Objective and Motivation

- Motivation
- Injection of frozen hydrogen pellets is a viable method of fueling a tokamak
- Presently there is no satisfactory predictive model for use in the design of a system for ITER
- Pellet-plasma interactions:
- Ablation: Considered well-understood
- Mass deposition: Large scale MHD driven but not-so-well understood
- Objectives
- Identify the mechanisms for mass distribution during pellet injection in tokamaks
- Quantify the differences between “inside launch” and “outside launch”

Pellet injection in TFTR

Approach/ Model

- Detailed 3D AMR simulations of pellet injection using the MHD equations– pellet treated as moving density source
- Ratio of pellet size to device size is ~O(10-3)
- Phased approach to understand the basic physics of mass redistribution with varying degrees of complexity
- Simple Cartesian geometry (Samtaney, Jardin, Colella and Martin, Sherwood Fusion Theory Conference 2003)
- Toroidal geometry (ICNSP 2003 Invited talk. To appear in Comput. Phys. Comm.)
- Physical assumptions
- Pellet ablates with an analytic ablation model
- Instantaneous heating of ablated mass by electrons
- Single fluid MHD equations describe plasma

Godunov SymmetrizationMathematical Model I

- Equations in conservation form + source terms
- Flux vector
- BT is the toroidal component of the equilibrium magnetic field
- Equation of state

Mathematical Model II

- Mass source is given by an ablation model :
- Delta function in source term approximated as a Gaussian function of width ten times the pellet radius
- Energy source terms
- Solve “Classical” 1D model and storeenergy sink as a function of and t

Source Sink

Current: Reconnection S= 104

Time sequence of current (Jz)Thin current layer “clumps” followedby high pressure plasma ejections6 Level AMR run. Effective unigrid: 4096x2048.(Samtaney et al. APS DPP 2002)

1.669

1.713

1.732

1.760

1.876

1.942

2.253

2.593

1.978

MHD Richtmyer-Meshkov Instability

t=t1

- RMI occurs
- at a fluid interface accelerated by a shock
- Linear stability analysis by Richtmyer (1960)
- Experimental confirmation by Meshkov (1970)
- in inertial confinement fusion where it is the main inhibiting hydrodynamic mechanism
- in astrophysical situations
- RMI stabilization by a magnetic field (Samtaney, Phys. Fluids 2003)
- Results are shown for an effective mesh of 16384x2048 points which took approximately 150 hours on 64 processors on NERSC.
- Speedup of over a factor of 25 compared to a non-AMR code

t=t2

t=t3

Comparison of Numerical and Analytic Results

- For regular refraction at the contact discontinuity, in a small neighborhood of the point where all discontinuities meet, the MHD PDEs can be reduced to algebraic equations.
- TF and RF are fast shocks
- Local analysis shows that the RS is a slow shock, while shock TS is a 2-4 intermediate shock (Vincent Wheatley, Dale Pullin, GALCIT, Caltech, Private communication)
- Example of code verification

Transitions in Solution Type with Increasing

- Two solution branches with different wave structures
- Singular approach to hydrodynamic limit
- The width of the inner layer decreases with . It scales like -1/2.
- Both solutions have the same leading order behavior

Ranges of validity of each solution type and comparison of inner layer width to values from simulations.

Wheatley, Pullin, Samtaney (APS DFD 2003). Also submitted to Journal of Fluid Mechanics

Summary of 2003 Applications/Accomplishments

- Implicit treatment of resistive, viscous, conductivity terms
- Implemented and tested for constant properties
- 3D AMR simulations of pellet injection
- Includes model for pellet ablation and prescribed motion of pellet
- Mass redistribution dominantly along magnetic field lines
- “Anomalous” mass redistribution, i.e., outward radial displacement of pellet mass.
- HFS more efficient than LFS
- Pellet injection: Estimated speed up ranged from 16-237
- AMR is a viable approach to efficiently resolve the relatively small pellet
- Magnetic Reconnection in 2D
- Current layer “bunching” and “plasmoid” ejections lead to a reconnection rate which is faster than that predicted by theory.
- Richtmyer-Meshkov Instablity
- Stabilization and lack of “mixing” if magnetic field is present
- MHD Hydrodynamic limit is singular with non-unique solutions

MHD Simulation Codes

*Exploratory project together with APDEC

Where are we?

3D torus

3D slab

Geometry

M3D and NIMROD

2D slab

Physics Model

single fluid MHD

two fluid MHD

two fluid MHD/hybrid

Required Resources for future studies

*Possible today

Estimate P ~ S1/2 (a/e)4 for uniform grid explicit calculation. Adaptive grid refinement, implicit time stepping, and improved algorithms will reduce this.

Necessary (but not sufficient) Requirements for Fusion MHD

- Geometry
- Anisotropic transport
- Two-fluid MHD Model
- Implicit treatment of fast compressive wave
- Other application specific requirements (e.g. Pellet ablation model)

Geometry

- In the AMR MHD code source terms are employed to treat toroidal geometry
- Approaches
- Generalized non-orthogonal curvilinear mesh
- Embedded boundary approach
- Hybrid approach: Treat toroidal direction explicitly and employ EB approach in RZ planes
- In general, toroidal variation is “small” compared with poloidal variations
- Implies anisotropic refinement

Anisotropic Transport

- Parallel conduction of heat is orders of magnitude larger than perpendicular heat conduction (||/? = O(108))
- M3D treats this by solving an auxiliary equation (the “artificial sound” method)
- Numerical studies by the NIMROD group suggests that large anisotropy in heat conduction can be effectively handled by higher order accurate finite elements (O(h5))
- Recent studies (S. Jardin) utilizing higher-order finite elements with C1 (C2) continuity at edges (vertices) show large anisotropy can be effectively handled with higher-order accuracy
- Similar conclusion by P. Fischer (Argonne) for spectral elements
- Higher order of accuracy (fourth) with Chombo

Two-fluid MHD

- Presently the AMR MHD code solves single-fluid resistive MHD equations
- Separate treatment of ions and electrons with Hall term in Ohm’s law leads to the “two-fluid” MHD equations
- Nature of equations is dispersive
- quadratic dependence on wave number in dispersion relation. Whistler waves
- Investigation of relative importance of r pe term (Kinetic Alfven wave) is yet unknown

Explicit vs. Implicit Time Stepping

- Consider Pellet Injection in CDXU
- Pellet radius rp=0.3 mm with injection velocity vp=450 m/s, Minor radius a=0.3 m, Alfven speed VA¼ 1500 km/s
- If rp resolved by 3 points, we need about 2 x 107 time steps on a 30003 uniform mesh with t = 3.3 x 10-11 s,
- Adaptivity is a practical necessity
- 5 refinement levels (nref=2) or 3 refinement levels (nref=4)
- 4 x 105 time steps on coarsest mesh
- Resolution/time step requirements more stringent for larger devices
- Full implicit treatment is desirable but as a practical first step implicit treatment of the fast compressive wave would reduce the large number of time steps needed to complete the simulation

Application specific wish list

- Pellet injection
- Better treatment of anisotropic heat conduction
- More accurate treatment of toroidal geometry
- Realistic parameters
- Equilibrium: Presently the code results in a quasi-periodic solution when initialized with an analytical equilibrium state
- Due to truncation errors
- Subtracting the toroidal equilibrium component of B increases robustness
- Investigation of other pellet ablation models
- Magnetic Reconnection in 2D
- Inclusion of Hall-term
- Effect of r pe term in Ohm’s law. Model studies indicate that r pe dominates over the JxB term when a strong background is present
- Good parallel scaling is important and needs to be verified systematically for the AMR MHD code

Future Plan (2005-2006)

- Implicit treatment of fast compressive wave
- Higher-spatial order (fourth order)
- For anisotropic heat conduction
- Investigate different pellet ablation models
- Comparison of pellet injection with analytic pellet injection models and experimental data (TFTR, JET or DIIID)
- Include the r pe term in Ohm’s law and investigate the importance of the Kinetic Alfven wave on 2D magnetic reconnection

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