THE NIMROD HIGH-S
CAMPAIGN
D. D. Schnack
S. E. Kruger
Center for Energy and Space Science
Science Applications International Corp.
San Diego, CA 92121
and
The NIMROD Team
MOTIVATION
- NIMROD is designed
for high Lundquist number calculations:
- Grid aligned with flux surfaces
- Grid packing at selected rational
surfaces
- Push the limits of the NIMROD code for linear and nonlinear calculations
at large S
- Modern tokamaks operate at
- Present nonlinear calculations limited
to
- Linear calculations are
"easier", but accuracy and efficiency at high S is difficult to
achieve
- Non-circular poloidal cross-sections are
difficult
How high in S can
NIMRODgo?
APPROACH
- Remain within the resistive MHD
model
- "Easy" problems => difficult problems
- "Easy" problems (linear):
- Linear resistive tearing mode
- Geometry:
- Circular cross-section, doubly periodic
cylinder
- Circular cross-section torus
- Shaped cross-section torus
- Difficult problems (nonlinear):
- Single resistive tearing mode in circular
cylindrical geometry
- Multiple resistive tearing modes in circular
cylindrical geometry
- Tearing modes in a circular cross-section
torus
- Tearing modes in a shaped cross-section
torus
- Neo-classical tearing modes and two-fluid
physics
-
THE NIMROD CODE
- Contains full resistive,
two-fluid, and neo-classical physics,
including:
- Anisotropic non-linear heat flux
- 2 choices for neo-classical
closure
Use only resistive MHD model for this study
- Can be run in linear or nonlinear
mode
- Toroidal, with arbitrarily
shaped poloidal cross-section, including R= 0
- Linear, doubly periodic circular
cylinder
- Slab
- Finite elements for poloidal
plane, FFTs for toroidal (or axial) direction
- Semi-implicit time advance
- MPI
- Nearly ideal scaling demonstrated
- Machine independent:
- Serial Linux PCs to the T3E!
NIMROD is available
from www.nimrodteam.org
- Linear resistive tearing mode in doubly
periodic circular cylinder
- Well-known test cases:
J. A. Holmes, B. A. Carreras, T. C. Hender, H.
R. Hicks, V. E. Lynch, and B. F. Masden, Phys. Fluids 26, 2569
(1983)
Case I:
- Linearly unstable to 2/1 and 3/2 tearing
modes
- Flat q-profile makes interior Suydam
unstable to many modes with m/n slightly
greater than 1, eg., 8/7, 9/8, 10/9, 11/10,
…..
- Extremely difficult nonlinear
case
Case III:
- Linearly unstable to 2/1 tearing
mode
- Advertised to be relatively benign
nonlinear case
- Concentrate on Case I linear 2/1 tearing
mode
- Obtain gas a function of S for
S > 105
This is an ongoing project!
CASE I
Saftey Factor vs. Radius
CASE III
Safety Factor vs. Radius
GRID PACKING
- Essential for efficient computation at high
S
- Algorithm
gives Dr/Dr0= f(r), with
Drsmall
near rational surface:
Here,
Dr0 = 1/Nradial = 1/520
gives Drat rational surface equivalent to 2763
equally spaced radial grid points!
Radial Velocity Eigenfunction
2/1 Tearing Mode
S=107
Radial Velocity Eigenfunction
2/1 Tearing Mode S=107
(Zoomed)
Radial Velocity Eigenfunction
2/1 Tearing Mode S = 107
(Zoomed) with Grid Points
STATUS
- Obtained converged linear growth rates
for Case I 2/1 tearing mode for
NIMROD scaling:
Compare with analytic scaling:
- Grid packing essential for efficient
calculations, but:
- Linear calculations have required up to 128
processors
- Computational requirement will only increase
as S becomes larger
- Large time steps
(Dt~ 10tA)are possible,
but:
- Accuracy (as measured by linear growth rate)
requires
to
decrease with S, not 
Time step seems tied to the Alfvén time, not the tearing
mode growth time
- Nonlinear calculations have been less
successful
- Case I calculations are dominated by
m/n ~ 1 Suydam modes in the interior. Spectrum
always fills and code crashes
- Case III calculations with
2/1 mode at S = 105
have begun
FUTURE DIRECTIONS
- Continue to push Case I linear calculations
to higher S
- S = 108 calculations are
underway
- Pursue nonlinear Case III
calculations
Are there more "interesting" nonlinear cylindrical cases?
- Repeat in more interesting geometry
- Circular cross-section toroidal
- Shaped cross-section toroidal
- Document the parameter space available to the
NIMROD code