Summary of Fusion Linear Stability Codes by N6w9v83


									 Summary of Fusion
Linear Stability Codes

      S.E. Kruger
   Tech-X Corporation

   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  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
        Idea l MH D Re sistive
                                   Te rm                                                           
                     MH D                       Te rm                              Ele ctron Inerita 

          – Temperature Equations:

                n        nV  T  nT   V   1  q   1Q

  Linearization of Equations Give Eigenvalue
     Problem Which Determines Stability
• Linearized equations -- neglecting equilibrium flow, closures,
  and two fluid terms for now:

                   ˜        ˜ ˜
               oiV  J0  B  J  B0  p
                   ˜    ˜           ˜ ˜J
                   E  V  B0  0 J   0
                        Ideal MHD   Resistive
                 ˜ ˜                   ˜
               ip  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:      2K  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.
        2K  W plasma  W vacuum  W hot  ...
Resistive MHD Codes Use Several Techniques

•   Linear system as:
                           ix  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
     – 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
• All codes discussed subsequently assume
  symmetry which allow for a decomposition
  by toroidal mode number. All codes
  discussed solve for the toroidal Fourier
• Radial and poloidal discretizations vary
• 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
  global       Flux tube                Free Boundary       NOVA                          MARS        ORBIT
  GTC       SUMMIT                      Equilibrium
                                                            DCON             BALOOO                             Linear high-n
                                         EFIT    TEQ
                FP-Code                                     GATO         intermediate-n
 Antenna                                         TSCEQ                                                            FULL
                 CQL3D                                                       ELITE

                             2D transport
   RF Heating                                    Inverse                                              Plasma Edge
   & 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
                                                               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
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
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
            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,
• 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
      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
• 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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