Summary of Fusion Linear Stability Codes by N6w9v83

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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   1Q
t


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
MHD

˜ ˜                   ˜
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
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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