Medium to high n resistive ballooning modes are driven nonlinearly unstable by growth of the internal kink (q=1/1) instability in TFTR geometry. The equilibrium was used in prior XTOR simulations and is closely related to M3D simulations of the destabilization of kink ballooning modes by the internal kink. (H. Lutjens and J.F. Luciani, "Stability thresholds for ballooning modes drvien by high beta internal kinks," Phys. Plasmas 4 (4192) 1997.

The equilibrium code used is CHEASE which produces an output file INP1 normally used by PEST3. Fluxgrid can process this PEST3 input file. The equilibrium is essentially a TFTR equilibria with normalized minor radius and was communicated to me by Hinrich Lutjens. The equilbrium was also processed with DCON and a region of infinite n-ballooning instability in a region of positive shear for 0.84807 < q < 0.8537 was indicated.

In the Lutjens' simulation the authors comparared across a range of beta values and aspect ratios, but in this comparision the focus will be on an aspect ratio A=4 result in what the Lutjens paper refers to as LS or low shear. The mesh used by Lutjens was 11 toroidal harmonics and 17 polodial harmonics and 300 equidistant radial mesh points. In comparision, two sets of NIMROD simulations have been completed: The first uses n=0 through 10 toroidal harmonics, 64 poloidal cells, and 64 radial cells and the second uses n=0 through n=21, 128 poloidal cells, and 128 radial cells. Even though the radial mesh is coarser in the NIMROD simulations, the salient features of the kink destabilization mechanism are reproduced. As with the Lutjens paper, the resistivity is set to zero in these simulations, the paper indicates that they have used "minute" viscosity. In the NIMROD simulations the viscosity was set to kin_visc = 190 m²s^-1.

Grid Case 1: n=0 to 11, 64x64.

Comparison of linear growth rates (gamma tau_A) between NIMROD and XTOR indicate relatively good agreement between the two codes. The observed growth rate difference between the two codes can be attiributable to different viscosities and also possibly different definitions of the Alfven time. XTOR predicts linear instability for n=2 and n=3 and this is not observed in NIMROD. Likewise an n=10 ballooning instability is indicated by NIMROD but is not observed by XTOR. Except for the instability associated with the internal kink, these instabilities do not seem to matter once the simulation has reached the nonlinear stage and the physics becomes dominated by mode coupling effects.

The nonlinear evolution is first characterized by mode coupling that leads to well known behaviors for the growth rates, e.g, n=2 is twice n=1, because 1/1 * 1/1 --> 2/2, etc. This describes most of the physics up to appoximately t=1.60e-04 s. At approximately t=1.62e-04 s, the n=10 growth rate is first to observed to increase and shortly after that at t=1.64e-04 s the lower n harmonics also begin to grow faster. This is then intrepreted as the kink destabilization of the kink ballooning mode (see for example Figure 3b in the Lutjens behavior for similiar behavior. The ballooning behavior as evidenced in the following animation of the poincare section spanning the set of times 1.50e-04, 1.55e-04, 1.56e-04, 1.57e-04, 1.58e-04,1.59e-04,1.60e-04,1.61e-04,1.62e-04,1.63e-04,1.64e-04,1.65e-04, 1.70e-04. Note that the first and last frames are different time steps then the remaining steps.

In this second case, with the finer grid, the prior simulation is restarted from the perturbations at t=1.4e-04 s, which is at a state where the internal kink is still in the linear phase. The time evolution of the kinetic energy is presented in the following four figures. Early in the simulations, the internal kink decays a bit, presumbaly as energy increases in the higher order harmonics. The n=10 is also apparently stabilized in this case, though continues to grow due to mode coupling. Once the growth of the internal kink is re-established and the mode coupling growth rates seem to be established, an interesting observation can be made that first n=20 changes slope, then slightly later in time n=19 changes slope, and this effect seems to continue at least through n=10. Eventually, the simulation becomes very nonlinear, and a strong grouping of harmonics becomes established. The nonlinear evolution of the pressure contours illustrates the establishment of some very long figures, but without the resistivity a topology change is not observed.

The grid case 1 the simulations were performed on the T3E with 11 layers in the Fourier direction and a single 64x64 block on each layer. A block size of 64x64 is roughly the largest block size that fits on a T3E processor. The mode of operation, since I wasn't in a hurry, was to let the simulations run 500 timesteps a night for a total of 17500 time steps. The total CPU time from start to finish after totaling results from nimrod.out files was 183163.74 CPU seconds for grid case 1. For grid case 2 the problem was divided into 22 layers and four rblocks (nxbl=4 and nybl=1). The bl_diaga solver was also used. The total number of timesteps was 5458 and a total of 184773.8 CPU seconds was used. Additional timing information is available in the file timing1.dat for grid case 1 and timing2.dat for grid case 2.