SCIDAC Center for Extended MHD Modeling
Dalton D. Schnack Center for Energy and Space Science Science Applications International Corp. San Diego, CA
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 The Extended MHD model • The computational challenges: –  Extreme separation of time scales –  Extreme separation of spatial scales – Extreme anisotropy –  Importance of geometry, boundary conditions –  Causality: can’t parallelize over time! At least as challenging as hydrodynamic turbulence! . Present computational approaches: – Implicit time differencing –  Specialized spatial grids . Status of present models . Vision for integrated modeling
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 Hydrodynamics - A mathematical model that describes the motion of a continuous, isotropic fluid . Magnetohydrodynamics (MHD) - A mathematical model that describes the motion of a continuous, electrically conducting fluid in a magnetic field –  Hydrodynamics and Maxwell equations coupled through Lorentz body force and Ohm’s law . Ideal MHD - the fluid has infinite electrical conductivity (zero resistivity) . Resistive MHD - The fluid has finite conductivity and resistivity . Extended MHD - additional effects of electron dynamics and/or non-Maxwellian species
Example: DIII-D shot 86144 •Sawtoothing discharge •3/2 NTM triggered at 2250 msec •2/1 locks to the wall
• Slow evolution Nonlinear fluid model required • Plasma shaping Realistic geometry required • High temperature Large “Reynolds’ numbers” • Low collisionality Extensions to resistive MHD required • Strong magnetic field Highly anisotropic transport required • Resistive wall Non-ideal boundary conditions required
•  Quasi-equilibrium: ∇p = J×B  +   constraints -  ““Magnetohydrostatics” (MHS?) -  Eliminates all waves -  Basis for 1-1/2 dimensional transport models -  Extension to 3-D? •  Time dependent -  Solve 2-fluid equations -  Retain all normal modes -  Focus of present SciDAC efforts
. Kinetic models of plasmas based on distribution function for each charge species . Satisfies kinetic equation - six dimensions plus time - computationally impractical for time scales of interest . Fluid models derived by taking successive velocity moments of kinetic equation –  Reduce dimensionality by 3 • . Hierarchy of equations for n, v, p, Π, q, ....... Equations truncated by closure relations –  Express high order moments in terms of low order moments –  Capture kinetic effects in these moments . Result is Extended MHD
• Maxwell (no displacement current):
• Current and quasi-neutrality:
• Add electron and ion momentum equations:
All effects beyond resistivity constitute Extended MHD
● Extreme separation of time scales –  Realistic “Reynolds’ numbers” –  Implicit methods ● Extreme separation of spatial scales –  Important physics occurs in internal boundary layers –  Small dissipation cannot be ignored –  Requires grid packing or adaption ● Extreme anisotropy –  Special direction determined by magnetic field Accurate treatment of  B  ∇  operator is important -   Requires specialized gridding
v Affvén transit time t s Sound transit time V
V Resistive diffusion time
Explicit time step impractical:
Require implicit methods
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 Partially implicit methods –  Treat fastest time scales implicitly –  Time step still limited by waves • Semi-implicit methods –  Treat linearized ideal MHD operator implicitly –  Time step limited by advection –  Many iterations .
 Fully implicit methods –  Newton-Krylov treatment of full nonlinear equations –  Arbitrary time step –  Still a research project
. Extremely large condition number :  >  10 10 !! –  Specialized pre-conditioners – Anisotropy . Ideal MHD is self-adjoint –  Symmetric matrices - CG • Advection and some 2-fluid effects (whistler waves) are not self- adjoint –  Need for efficient non-symmetric solvers • Everything must be efficient and scalable in parallel . Should interface easily with F90
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 Important dynamics occurs in internal boundary layers –  Structure is determined by plasma resistivity or other dissipation –  Small dissipation cannot be ignored • Long wavelength along magnetic field • Extremely localized across magnetic field: δIL  ~   S ª  << 1 for S >>  1 . It is these long, thin structures that evolve nonlinearly on the slow evolutionary time scale
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 Magnetic field locally defines special direction in space Important dynamics are extended along field direction, very narrow across it Propagation of normal modes (waves) depends strongly on local field direction .
 Transport (heat and momentum flux) is also highly anisotropic ==>  Requires accurate treatment of operator B · ∇ Inaccuracies lead to “spectral pollution”’ and anomalous perpendicular transport
. Spatial stiffness and anisotropy require special gridding – Toroidal and poloidal dimensions treated differently .
 Toroidal (φ, primarily along field) –  Long wavelengths, periodicity => FFTs (finite differences also used) . Poloidal plane (R,Z) –  Fine structure across field direction –  Grids aligned with flux surfaces (~ field lines) –  Unstructured triangular grids –  Extreme packing near internal boundary layers . Finite elements –  High order elements essential for resolving anisotropies Dynamic mesh adaption in research phase .
DIII-D poloidal cross-section with flux aligned grid (NIMROD) Circular poloidal cross-section with triangles and grid packing (M3D) Poloidal grids from SciDAC development projects

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2-fluid effects
–  Whistler waves (Hall term) require implicit advance with non-symmetric solver –  Electron inertia treated implicitly –  Diamagnetic rotation may cause accuracy, stability problems . Kinetic effects - influence of non-Maxwellian populations –  Analytic closures •  Seek  local expressions for Π, q, etc. – Particle closures •  Subcycle gyrokinetic  δf calculation •  Minority ion species - beam or α-particles
2 major SciDAC development projects for time-dependent models . –  M3D - multi-level, 3-D, parallel plasma simulation code •  Partially implicit •  Toroidal geometry - suitable for stellarators •  2-fluid model •  Neo-classical and particle closures –  NIMROD - 3-D nonlinear extended MHD •  Semi-implicit •  Slab, cylindrical, or axisymmetric toroidal geometry •  2-fluid model (evolving computationally) •  Neo-classical closures •  Particle closures being de-bugged Both codes have exhibited good parallel performance scaling Other algorithms are being developed in the fusion program .
Sawtooth in NSTX computed by M3D Stellarator ballooning mode computed by M3D
Secondary magnetic islands generated during sawtooth crash in DIII-D shot 86144 by NIMROD
. Effect of energetic particle population on MHD mode . Subcycling of energetic particle module within MHD codes . M3D agrees well with NOVA2 in the linear regime . Energetic particles are being incorporated into NIMROD
Computed NTM stability diagram Sawtooth and secondary islands Secondary island growth Varying S
. Non-local kinetic physics, MHD, and profile evolution are all inter- related –  Kinetic physics determines transport coefficients –  Transport coefficients affect profile evolution –  Profile evolution can destabilize of MHD modes –  Kinetic physics can affect nonlinear MHD evolution (NTMs, TAEs) –  MHD relaxation affects profile evolution –  Profiles affect kinetic physics . Effects of kinetic (sub grid scale) physics must be synthesized into MHD models –  Extensions to Ohm’s law (2-fluid models) –  Subcycling/code coupling –  Theoretical models (closures), possibly heuristic Effects of MHD must be synthesized into transport models . . Predictions must be validated with experimental data
. Largest, fastest computers! –  But intermediate computational resources often neglected, and… – The computers will never be large or fast enough! . Algorithms –  Parallel linear algebra –  Gridding, adaptive and otherwise . Data structure and storage –  Adequate storage devices –  Common treatment of experimental and simulation data –  Common tools for data analysis • Communication and networking –  Fast data transfer between simulation site and storage site –  Efficient worldwide access to data –  Collaborative tools –  Dealing with firewalls • Advanced graphics and animation
. Predictive simulation capability has 3 components –  Code and algorithm development – Tightly coupled theoretical effort –  Validation of models by comparison with experiment . Integration required for: –  Coupling algorithms for disparate physical problems –  Theoretical synthesis of results from different models – Efficient communication and data manipulation . Progress is being made with Extended MHD – Integration of energetic ion modules into 3-D MHD –   Computationally tractable closures –  Resistive wall modules Need to bring a broader range of algorithms and codes to bear for overall fusion problem