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Summary of Fusion Linear Stability Codes S.E. Kruger Tech-X Corporation ORNL SWIM Meet November 7, 2005 Long-wavelength, Low-frequency Instabilities in Fusion Plasmas Studied Using Fluid Equations • Take moments of kinetic equation and put into COM reference frame: MHD Form of Two-Fluid Equations: – Momentum Equation: V V V J B p t – Generalized Ohm’s Law: 1 1 J E V B J J B pe e VJ JV ne Whistler Dia magne tic Closures 0 pe t 2 Idea l MH D Re sistive Te rm MH D Te rm Ele ctron Inerita – Temperature Equations: T n nV T nT V 1 q 1Q t Linearization of Equations Give Eigenvalue Problem Which Determines Stability • Linearized equations -- neglecting equilibrium flow, closures, and two fluid terms for now: ˜ ˜ ˜ oiV J0 B J B0 p ˜ ˜ ˜ ˜ ˜J E V B0 0 J 0 Ideal MHD Resistive MHD ˜ ˜ ˜ ip V Tp0 p0 V 0 • Temperature equations have been combined with density equation to express things purely in terms of pressure. • Linear operators – Ideal MHD is purely Hermitian (I.e., can use variation approaches) => Real eigenvalues – Non-ideal MHD supports complex eigenvalues Ideal MHD codes exploit Hermitian properties • Write linear system as (assuming axisymmetery): i x F x ˆ ˆ 0 n n • Hermitian F for ideal operator allows variational form: 2K W • Can also formulate ideal MHD problem in an Euler-Lagrange equation form • For high-n, use analytic WKB-like analysis to derive ballooning equation. Also used for peeling-ballooning modes. • Boundary is also handled by extension to W and using the VACUUM code. 2K W plasma W vacuum W hot ... Resistive MHD Codes Use Several Techniques • Linear system as: ix F x • Only the MARS code solves this as an eigenvalue problem • Other codes use boundary layer theory to use “ideal region solution” from ideal code and match it to “inner layer solution”. • NIMROD and FAR nonlinear initial value codes separate the equilibrium fields from the perturbed solutions (NOT a linearization, but related). – NIMROD uses operator splitting. FAR creates 1 big matrix. – FAR has capability of adjust delta-t to find eigenfunctions and eigenvalues one at a time. – NIMROD just evolves the linear-only terms to find the most unstable eigenvalue and eigenfunction. Since NIMROD started using SuperLU, it’s fairly fast. • Resonances of hot particles with MHD modes are important. Can be done in either Brief Discussion of Coordinate Systems • All codes except NIMROD use flux coordinate systems. NIMROD is often “flux- aligned” in hot plasma region, but it is not required. • All codes discussed subsequently assume symmetry which allow for a decomposition by toroidal mode number. All codes discussed solve for the toroidal Fourier harmonic • Radial and poloidal discretizations vary widely. • Solver methods vary widely. Fusion research involves a tremendous amount of computing Major US fusion codes there is no “THE” fusion code (YET!) Nonlinear Gyrokinetic Vacuum & Linear Stability Gyrofluid Conductors MHD- + particles global Flux tube Flux tube VACUUM Ideal Non-Ideal low-n high-n low-n low-n high-n GYRO GS2 GRYFFIN VALEN PEST-I,II PEST-III NOVA-K HINST BALLOON Particle-in-cell global Flux tube Free Boundary NOVA MARS ORBIT CAMINO GTC SUMMIT Equilibrium DCON BALOOO Linear high-n EFIT TEQ gyrokinetic FP-Code GATO intermediate-n Antenna TSCEQ FULL CQL3D ELITE RANT3D 2D transport RF Heating Inverse Plasma Edge TSC & CD Equilibrium 3D Nonlinear MHD 2D plasma neutrals TRANSP JSOLVER Static Time -Dependent AORSA TORCH B2 DEGAS PIES M3D ORION LSC WHIST TOQ VMEC NIMROD UEDGE EIRENE ONETWO ESC TORIC TORAY FAR 3D plasma VMEC2D BOUT CURRAY CORSICA METS POLAR2D BALDUR CORSICA denotes parallel MPI code Courtesy S. Jardin – PPPL Linear Stability Codes Use Different Mathematical Formulations/Numerical Techniques Ideal Stability Codes Code Type Range Discretization Institution PEST-II W n=1-14? : FE/ : Spectral PPPL GATO W Low n Hybrid FE GA ERATO W Low n Hybrid FE Lausanne : Bicubic FE NOVA Eigenvalue Low n PPPL : Spectral DCON Newcomb Low n : FD/ : Spectral LANL BALLOON Ballooning High n Per flux surface PPPL BALOO Ballooning High n Per flux surface GA Camino Ballooning High n Per flux surface PPPL Peeling- ELITE ballooning Med.-hi n ? GA Linear Stability Codes Use Different Mathematical Formulations/Numerical Techniques Beyond-Ideal MHD Stability Codes Code Type Range Discretization Institution PEST-III Matching Low n Finite Element PPPL Resistive : FD/ : Spectral Matching Low n LANL DCON MARS Eigenvalue Low n : FD/ : Spectral GA Ideal+ : FE/ : Spectral NOVA-K Low n PPPL hot particle Linear Most unstable eigenvalue N=0-43 Lagrangian Elements NIMROD NIMROD All linear codes take input Grad-Shafranov Equations • Because of simple Ohm’s law, no information about it is needed for most codes. • Some work has been done for the case of purely toroidal flow because of trivial modifications required for GS solvers (GATO, MARS, NIMROD) • As CEMM pushes the two-fluid algorithms, a better understanding of steady-state solutions with drifts is needed (e.g., Steinhauer, PP 6, 2734 (1999)) • Mechanism for transfering codes from equilibrium solution to codes is file based and done as: (GS code) -> (Mapping code) -> (Stability Code) • In practice, this is very ugly: quality of mapping issues, difficult to swap out mapping codes, etc. • DCON and NIMROD use “mapping library approach” (forks of each other). Method could be easily adapted for something like XPLASMA. Output of linear codes is typically eigenvalues and eigenvectors • Typical usage: Worry about most unstable eigenvalue • Amplitude of eigenvector is arbitrary -- put into some model for what the amplitude should be. • Model is then for the effects of eigenfunction on the n=0 fields. – Example: ELM model takes eigenfunctions from ELITE. Scale them to some value, and set p=0 over width of eigenfunction. Personal comments about linear MHD stability and experiments • Low-n Ideal MHD codes have been very successful in guiding the design of tokamaks and experimental operating regimes (Manickam, Turnbull, and others) (Turnbull paraphrased: “I’ve never seen a fast disruption that was not ideal unstable within the error bars of equilibrium reconstruction”.) • Moderate-n Mercier criterion: Increased transport? Not relevant? • High-n modes: Give “soft beta limit” (?). Recently been more successful for peeling-ballooning modes. • Resistive MHD has been spectacularly less useful (Wesson paraphrased: “Useless”) • Problems: Lots of physics needed at that order, instabilities (NTMs) may be metastable, equilibrium reconstructions still may not be accurate enough, … • DIII-D results (Garafalo/Brennan) have recently shown that being close, but below the ideal stability boundary is still bad on longer time scales. Thus, linear resistive codes may not be “useless”, just require finesse to interpret.
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