# Presentation: 084107

## 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

## Abstract

-    CDX-U is an attractive device to model to benchmark resistive MHD codes. Its small size and low temperature (*S* < 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 *n*=1 and dominant poloidal mode number *m*=1 [1]. We model both the linear growth of the instability and the nonlinear reconnection event at the *q*=1 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 *q**0*, 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).

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

- Low aspect ratio tokamak (*R**0*/*a* = 1.4 – 1.5)
- Small (*R**0* = 33.5 cm)
- Elongation ** ~ 1.6
*B**T* ~ 2300 gauss
*n**e* ~ 41013 cm-3
*T**e** *~ 100 eV
*I**p** *~ 70 kA
- Soft X-ray signals from typical discharges indicate two predominant types of low-*n* MHD activity:
- sawteeth
- “snakes”

## The TSC Sequence

- Equilibrium at t=12.25ms (as q0 drops to 0.95) is used to initialize 3D runs

- TSC follows 2D (axisymmetric) evolution of typical CDX-U discharge

- _The TSC Sequence_

## Typical Case

- _Typical Case_

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

- Equilibrium taken from a TSC sequence (Jsolver file).
*R*/*a* = 1.4
*q*(0)  0.955
*q*(*a*) ~ 10

- toroidal current density

## 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. |

## Comparing the Codes

- 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.

- 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.

- 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.

- _M3D_

- _NIMROD_

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

- 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

- M3D

- _Poloidal Meshes for the CDX-U Case_

## M3D RESULTS

## Linear n=1 Eigenfunctions, Pr = 10

- Large *m* = 1 mode at two minor radii
- Higher *m* components are insignificant
**A = 3.76  10-3

*B***

- temperature

- incompressible poloidal velocity
- stream function

***J*********** isosurface**

## The Nonlinear Phase

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

- linear prediction

- 17,367

- Total kinetic energy

- Kinetic energy by mode number

## Current Flattening in the Nonlinear Phase

- _Current Flattening in the Nonlinear Phase_

- Before

- After

*J*** (surface plot)

*q*(*r*)

- minor radius

- minor radius

## 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.

## NIMROD RESULTS

## Linear Eigenfunctions

*n* = 1
- Pr = 10
** = const.
** ~ *T*eq-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

*B***

*J***

*n*

*P*

## 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

## Effect of toroidal mode number n

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

## 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*:
*    ****n*=1 ~ ***n*=9 + ***n*=10
- Different from linear picture

## n = 1 mode Changes After Reconnection

- 1/1 with harmonics

- 2/1 with harmonics

## 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.
- NIMROD’s “Case 1” eigenfunctions, with *q**0* = 0.95, are anomalous. This seems likely to be a result of the quality of the initial equilibrium, which is now under investigation.

## 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. *q**0*=0.97, 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 *q**0*.
- Nonlinear triggering of higher-*m* modes off of 1,1 sawtooth crashes.
- Current channel narrowing as a cause of disruptions.