Resistive Modes in CDX-U

1
Resistive Modes in CDX-U

• J. Breslau, W. Park. S. Jardin, R. Kaita PPPL
• D. Schnack, S. Kruger SAIC
• APS-DPP Annual Meeting
• Albuquerque, NM
• October 30, 2003

2
Abstract

• CDX-U is an attractive device to model to
  benchmark resistive MHD codes. Its small size and
  low temperature (S lt 105) make it possible to
  simulate MHD events in it using actual
  experimental parameters in a reasonable time on
  present-day computers. The dominant MHD activity
  during normal operation is the sawtooth
  oscillation, a resistive internal kink mode with
  toroidal mode number n1 and dominant poloidal
  mode number m1 1. We model both the linear
  growth of the instability and the nonlinear
  reconnection event at the q1 rational surface
  (the sawtooth crash) as a test problem for both
  the M3D 2 and NIMROD 3 codes. Depending on
  the resistivity model employed and on the initial
  value of q0, the crash can either lead to a
  disruption or to a quiescent state with q on axis
  above 1.
• 1 D. Stutman et al., Plasma Phys. Control.
  Fusion 41, 867 (1999).
• 2 W. Park et al., Phys. Plasmas 6, 1796 (1999).
  
• 3 C.R. Sovinec et al., Phys. Plasmas 10, 1727
  (2003).

3
Characteristics of the Current Drive Experiment
Upgrade (CDX-U)

• Low aspect ratio tokamak (R0/a 1.4 1.5)
• Small (R0 33.5 cm)
• Elongation ? 1.6
• BT 2300 gauss
• ne 4?1013 cm-3
• Te 100 eV
• Ip 70 kA
• Soft X-ray signals from typical discharges
  indicate two predominant types of low-n MHD
  activity
• sawteeth
• snakes

4
The TSC Sequence
TSC follows 2D (axisymmetric) evolution of
typical CDX-U discharge
Equilibrium at t12.25ms (as q0 drops to 0.95) is
used to initialize 3D runs
5
Typical Case

• Equilibrium taken from a TSC sequence (Jsolver
  file).
• R/a 1.4
• q(0) ? 0.955
• q(a) 10

toroidal current density

• Questions to investigate
• Linear growth rate and eigenfunctions
• Nonlinear evolution
• disruption?
• stagnation?
• repeated reconnections?

6
Typical Baseline Parameters
Lundquist Number S 2?104 on axis.
Resistivity ? Spitzer profile ?Teq-3/2.
Prandtl Number Pr 10 - 100 on axis.
Viscosity ? Constant in space and time.
Perpendicular thermal conduction ?? 200 m2/sec (CDX-U energy confinement time)
Parallel thermal conduction ? ?? for main (isotropic) case. 108 m2/sec for anisotropic NIMROD case.
Plasma ? 10-2 (low-beta).
Density Evolution Turned on for nonlinear phase.
7
Comparing the Codes

• M3D and NIMROD are both parallel 3D nonlinear
  extended MHD codes in toroidal geometry
  maintained by multi-institutional collaborations,
  and comprise the two members of the Center for
  Extended MHD Modeling (CEMM) SciDAC.

M3D
NIMROD

• Uses linear finite elements in-plane.
• Uses finite differences between planes.
• Partially implicit treatment allows efficient
  time advance but requires small time steps.
• Linear operation full nonlinear filtering,
  active equilibrium maintenance.
• Nonlinear operation all components of all
  quantities evolve nonlinearly.

• Uses high-order finite elements in-plane.
• Uses Fourier decomposition in toroidal direction.
• Fully implicit treatment requires costly matrix
  inversions but allows large time steps.
• Linear operation evolve perturbations to
  particular modes only.
• Nonlinear operation perturbations to fixed
  equilibrium are evolved, with nonlinear couplings
  between modes.

8
Poloidal Meshes for the CDX-U Case
M3D

• 40 ? 24 structured grid
• 4th order basis functions on quadrilateral
  elements
• Conducting wall
• Fourier decomposition toroidally 10 or more
  modes retained

• 70 radial zones, up to 210 in ? in unstructured
  mesh
• Linear basis functions on triangular elements
• Conducting wall
• Finite differences toroidally 16 or more planes

9

• M3D RESULTS

10
Linear n1 Eigenfunctions, Pr 10
incompressible poloidal velocity stream function
B?
temperature
J? isosurface

• Large m 1 mode at two minor radii
• Higher m components are insignificant
• ??A 3.76 ? 10-3

11
The Nonlinear Phase
17,367
Total kinetic energy
Kinetic energy by mode number
linear prediction

• n 1 first decays, then exceeds linear
  prediction.
• Higher n modes appear to couple nonlinearly,
  grow to disruption.

12
Current Flattening in the Nonlinear Phase
q(r)
J? (surface plot)
Before
minor radius
After
minor radius
13
Magnetic Field Poincaré Section Series for
Nonlinear Phase
Initial state
1,1 mode dominant
High-m islands growing
Stochastic
The narrowing of the current channel leads to a
disruption.
This result applies to a case with a current
density that peaks in time. A more realistic
resistivity model that more closely resembles the
CDX-U experiment is now under study. It is not
expected to yield the same current profile
behavior, so the present result should not be
taken as a prediction of disruption in the
experiment.
14

• NIMROD RESULTS

15
Linear Eigenfunctions
B?
J?

• n 1
• Pr 10
• ? const.
• ? Teq-3/2 (fixed)
• Isotropic heat flux
• ?? 200 m2/sec
• Small m 1
• Higher m components at inboard edge
• Density evolution (no effect on linear behavior)
• ??A 3.7 ? 10-4

n
P
16
Mode Structure

• Low aspect ratio
• Low-n field lines make more turns on inboard side
• Mode localized along equilibrium field line will
  have more structure on inboard side
• Higher-n ?

Equilibrium field line with pitch
m 3, n 1
17
Effect of toroidal mode number n

• Modes occur at edge of discharge
• Move slightly outboard with increasing n
• Growth rate increases with n

18
Nonlinear Phase

• Linear n 1 is dominantly 1/1 (sawtooth)
• Nonlinear n 1 changes from 1/1 to 2/1
• Finite-sized 2/1 mode grows nonlinearly in
  Rutherford regime
• No indication of high-n instability

• n 10, 9, most unstable
• Lower n (e.g., n 1) now nonlinearly driven
• ?n1 ?n9 ?n10
• Different from linear picture

19
n 1 mode Changes After Reconnection
1/1 with harmonics
2/1 with harmonics
20
Conclusions

• There is a qualitative agreement between the
  linear eigenfunctions found by M3D and the 1,1
  mode seen during the NIMROD run labeled Case 2.
• The M3D nonlinear result shows a possible route
  to disruption for tokamaks in which narrow
  current channels can form.
• NIMRODs Case 1 eigenfunctions, with q0 0.95,
  are anomalous. This seems likely to be a result
  of the quality of the initial equilibrium, which
  is now under investigation.

21
Future Work

• Both codes should run converged linear and
  nonlinear studies on the same case for
  comparison.
• M3D will switch on current drive to maintain
  equilibrium q profile.
• More physically accessible and well-behaved
  equilibria, e.g. q00.97, will be considered.
• Several effects have been seen that merit
  follow-up study
• Presence and stability of high-m modes on inboard
  side in spherical tori, and their sensitivity to
  q0.
• Nonlinear triggering of higher-m modes off of 1,1
  sawtooth crashes.
• Current channel narrowing as a cause of
  disruptions.