1. NIMROD Simulations of a DIII-D Plasma Disruption
S. Kruger, D. Schnack (SAIC)
20th IAEA Fusion Energy Conference
November 2004
Vilamoura, Portugal

2. Outline Acknowledgements: D. Whyte, C. Sovinec, J. Callen (UW-M),
Motivation
High-beta disruption discharge: #87009
Simple analytic theory
NIMROD Modeling
Fixed boundary
Free-boundary using equilibria above marginal limit
Acknowledgements:
D. Whyte, C. Sovinec, J. Callen (UW-M),
A. Turnbull, M. Chu (GA)
E. Held (USU)

3. DIII-D SHOT #87009 Observes a Plasma Disruption During Neutral Beam Heating At High Plasma Beta
DIII-D shot #87009 is a high beta discharge that exhibits a disruption as it is slowly heated through the ideal MHD stability boundary. The disruption occurred at approximately 1682 msec.
The top figure on the left shows the plasma current and neutral beam power as functions of time. The bottom figure on the left shows beta-normal and (SXR??) flux as functions of time. Beta-normal rises as the neutral beam power is increased.
The magnetic signal of the perturbation associated with the disruption is observed to grow with a rate faster than a simple exponential in time. In the top figure on the right the data points are the measured signals from the perturbation. Note that these do NOT lie on a straight line in this semi-log plot. The solid blue line represents simple exponential growth on the idea MHD time scale. The dashed blue line represents simple exponential growth on the resistive diffusion time scale. The green line represents simple exponential growth of an ideal MHD instability with a growth rate determined by GATO. The red line represents a best fit to the data assuming exp(t**3/2) time dependence (as predicted by theory; see slide #4).
The bottom figure on the right shows the magnetic perturbation signal vs. poloidal angle. It is largest on the outboard side.
We are studying the onset of the disruption with NIMROD.
Callen et.al, Phys. Plasmas 6, 2963 (1999)

4. Mode Passing Through Instability Point Has Faster-Than-Exponential Growth
In experiment mode grows faster than exponential
Theory of ideal growth in response to slow heating (Callen, Hegna, Rice, Strait, and Turnbull, Phys. Plasmas 6, 2963 (1999)):
Heat slowly through critical b:
Ideal MHD: 
This slide shows Callen, et al’s linear theory of mode growth when heated slowly through an ideal marginal stability point. Nomenclature:
* beta_c is the critical beta_normal
* gamma_h is the heating rate
* gamma(t) is the instantaneous growth rate
* gamma_hat_MHD is the incremental MHD growth rate at the marginal point.ie., d(gamma)/d(beta) evaluated at beta=beta_c.
The theory assumes gamma_heat << gamma_hat_MHD. In the experiment the ratio is about 10**(-5).
The first line is the assumed time dependence of beta. The second line is the ideal MHD energy principle; the assumed time dependence of beta yields the expression for the instantaneous growth rate. The third line is the equation for growth of the perturbation. With the given expression for gamma(t) it can be easily integrated to give the indicated time dependence; it is super-exponential and fits well with experiment. The scaling of the time constant with gamma-heat and gamma_hat_MHD will be determined by NIMROD and compared with this theory.
Perturbation growth: 

5. DIII-D SHOT #87009 Observes a Mode on Hybrid Time Scale As Predicted By Analytic Theory
Growth is slower than ideal, but faster than resistive
DIII-D shot #87009 is a high beta discharge that exhibits a disruption as it is slowly heated through the ideal MHD stability boundary. The disruption occurred at approximately 1682 msec.
The top figure on the left shows the plasma current and neutral beam power as functions of time. The bottom figure on the left shows beta-normal and (SXR??) flux as functions of time. Beta-normal rises as the neutral beam power is increased.
The magnetic signal of the perturbation associated with the disruption is observed to grow with a rate faster than a simple exponential in time. In the top figure on the right the data points are the measured signals from the perturbation. Note that these do NOT lie on a straight line in this semi-log plot. The solid blue line represents simple exponential growth on the idea MHD time scale. The dashed blue line represents simple exponential growth on the resistive diffusion time scale. The green line represents simple exponential growth of an ideal MHD instability with a growth rate determined by GATO. The red line represents a best fit to the data assuming exp(t**3/2) time dependence (as predicted by theory; see slide #4).
The bottom figure on the right shows the magnetic perturbation signal vs. poloidal angle. It is largest on the outboard side.
We are studying the onset of the disruption with NIMROD.
Callen et.al, Phys. Plasmas 6, 2963 (1999)
Turnbull, IAEA, Sorrento, Italy (2000)

6. Resistive MHD Equations Used to Numerically Model Disruption
MHD Equations Solved:
Density Equation:
Momentum Equation
Resistive MHD Ohm’s Law:
Temperature Equations:

7. NIMROD’s Spatial Discretization Advantageous For Disruption Simulations
Finite-Elements in Poloidal Plane, Pseudo-spectral in Toroidal Angle
Can parallelize by FE blocks and by toroidal mode number
Lagrangian elements of arbitrary polynomial degree (specified at runtime)
Spectral convergence needed for realistic conditions: Error ~ hp+1
High S:
Use polynomial degree ≥3
High k||/kperp: Use polynomial degree ≥4
See Sovinec et.al. JCP 195 (2004) 355

8. Initial Simulations Performed Using Fixed Boundary Equilibria
Use q and pressure profile from experiment
Negative central shear
This slide shows the equilibrium used in the study. It was reconstructed from experimental data just before the disruption using the TOQ code. It is highly shaped and has negative central shear. The example of poloidal gridding (lower right) is based on the equilibrium flux surfaces, and is just one of several grids that have been used in this study. For most runs reported, the poloidal grid had 128 radial (psi) points and 64 poloidal angle (theta) points. Bicubic finite elements were used, so there is actually 9 times as much poloidal information as indicated by the grid. The packed grid was used for the linear studies, but a uniform grid was used for most nonlinear runs.

9. Fixed Boundary Simulations Require Going to Higher Beta
Conducting wall raises ideal stability limit
Need to run near critical bN for ideal instability NIMROD gives slightly larger ideal growth rate than GATO
NIMROD finds resistive interchange mode below ideal stability boundary
NIMROD was run in the linear mode to determine growth as a function of beta-normal near the marginal point. The present version of NIMROD requires that the last closed flux surface be a solid conducting wall. Thus the marginal value of beta-normal is larger (~ 4.5) than is the case with a free boundary. (Equilibria with different beta-normal were produced by retaining the q(psi) and P(psi) functional profiles and changing the magnitude of the pressure at the magnetic axis.)
All growth rates are converged values. Above the marginal point, NIMROD gives consistently higher growth rtes than GATO. The eigenfunctions (not shown) are in excellent agreement. (The NIMROD eigenfunctions, which all have n=1, actually look “better” than those from GATO, which appears to have a numerical problem at the magnetic axis.) The reason for the difference in the growth rates is not well understood. (NIMROD has agreed well with other codes, eg., M3D.)
The marginal point as determined by DCON is also shown. Determination of marginal stability boundaries with an initial value code such as NIMROD is notoriously difficult because you are looking for a limit as the growth rate goes to zero. In principle this would require an infinite amount of computer time. Therefore a definitive NIMROD marginal point is not shown.
NIMROD finds localized resistive interchange modes below the ideal marginal points.

10. Log of magnetic energy in n = 1 mode vs. time
Nonlinear Simulations Find Faster-Than-Exponential Growth As Predicted By Theory
Impose heating source proportional to equilibrium pressure profile
Log of magnetic energy in n = 1 mode vs. time
S = 106 Pr = 200 gH = 103 sec-1
Nomlinear resistive MHD simulations with NIMROD were begun with an equilibrium below the marginal point. A heating source was added to the energy equation to give an (approximately) linear increase in beta. The flux and pressure surfaces thus slowly shift outward during the run. As the marginal point is exceeded an ideal mode starts to grow.
The figure on the right shows the log of the magnetic energy in the n=1 mode as a function of time. This run had S=10**6, Pr(=nu/eta)=200, and gamma_heat/gamma_hat_MHD ~ 10**(-3). (Compare with 10**(-5) in the experiment.) Simple exponential growth would be a straight line on this plot. Clearly, “super-exponential” growth is observed as beta increases in response to the heating. The mode saturates nonlinearly.
This run had n=0 and n=1 modes only. Because of the large value of the critical beta_normal (due to the conducting wall), both GATO and NIMROD find modes with higher n increasingly unstable. These are not seen in the experiment, nor with GATO when run with lower beta-normal and free boundary. We will repeat these nonlinear runs with more modes when the free-boundary/vacuum capability is available in NIMROD.
The grid was decomposed in on 154 processors. The time step corresponded to a CFL that approached 10**4. The total run took a little over 7 CPU years on the T3E! This clearly points up the need for faster computers.
Follow nonlinear evolution through heating, destabilization, and saturation

11. Scaling With Heating Rate Gives Good Agreement With Theory
NIMROD simulations also display super-exponential growth
Simulation results with different heating rates are well fit by x ~ exp[(t-t0)/t] 3/2
Time constant scales as
Log of magnetic energy vs. (t - t0)3/2
for 2 different heating rates
Nonlinear runs were done with 2 different heating rates. They both showed super-exponential growth that was well fit by exp[((t-t_0)/tau)**3/2]. Here t_0 is the time for the onset of the mode. The figure on the right shows the log of the amplitude of the n=1 perturbation for these 2 cases plotted as a function of (t-t_0)**3/2; a straight line thus indicates exp[((t-t_0)/tau)**3/2] dependence, as indicated by theory. The fit is excellent. Knowing gamma_hat_MHD from the linear stability studies allows us to determine the scaling of tau with gamma_heat from these data. The result is shown in the first equation on the left. It is to be compared with the theoretical prediction shown in the second equation. The quality of the agreement is likely in the eye of the beholder, but I don’t think it’s bad. Discrepancies may arise because NIMROD, having resitivity and viscosity, is NOT ideal MHD.
Compare with theory:
Discrepancy possibly due to non-ideal effects

12. Free-Boundary Simulations Models “Halo” Plasma as Cold, Low Density Plasma
Typical DIII-D Parameters: Tcore~10 keV Tsep~1-10 eV ncore~5x1019 m nsep~ 1018 m-3
Spitzer resistivity: h~T-3/2
Suppresses currents on open field lines
Large gradients 3 dimensionally
Requires accurate calculation of anisotropic thermal conduction to distinguish between open and closed field lines

13. Goal of Simulation is to Model Power Distribution On Limiter during Disruption
Plasma-wall interactions are complex and beyond the scope of this simulation
Boundary conditions are applied at the vacuum vessel, NOT the limiter.
Vacuum vessel is conductor
Limiter is an insulator
This is accurate for magnetic field:
Bn=constant at conducting wall
Bn can evolve at graphite limiter
No boundary conditions are applied at limiter for velocity or temperatures.
This allows fluxes of mass and heat through limiter
Normal heat flux is computed at limiter boundary

14. Free-Boundary Simulations Based on EFIT Reconstruction
Pressure raised 8.7% above best fit EFIT
Above ideal MHD marginal stability limit
Simulation includes:
n = 0, 1, 2
Anisotropic heat conduction (with no T dependence) kpar/kperp=108
Ideal modes grow with finite resistivity (S = 105)

15. First Macroscopic Feature is 2/1 Helical Temperature Perturbation Due to Magnetic Island
The 1/1 and 3/1 perturbations do not cause islands as one would expect from an ideal perturbation. At this point, there are already a lot of islands out in the divertor region that one can see in a poincare puncture plot.
Plotted here: two field lines started near each other. Color indicates temperature along field line and brightness of nodes indicates distance along field lines.
Isosurfaces are of temperature

16. Magnetic Field Rapidly Goes Stochastic with Field Lines Filling Large Volume of Plasma
Region near divertor goes stochastic first
Islands interact and cause stochasticity
Rapid loss of thermal energy results. Heat flux on divertor rises
Note the temperature variation along the magnetic field and that the two field lines do not end near each other (sensitivity to initial conditions). Shown here in addition to the field-lines and temperature isosurfaces are the heat flux incident on the limiter.

17. Maximum Heat Flux in Calculation Shows Poloidal And Toroidal Localization
Heat localized to divertor regions and outboard midplane
Toroidal localization presents engineering challenges - divertors typically designed for steady-state symmetric heat fluxes
Qualitatively agrees with many observed disruptions on DIII-D
This is time of maximum heat flux. The bright white spots are where the maximum heat flux is located. Unfortunately, the divertor temperature diagnostic wasn’t working for this shot so I can’t compare the heat temperature.

18. Investigate Topology At Time of Maximum Heat Flux
Regions of hottest heat flux are connected topologically
Single field line passes through region of large perpendicular heat flux. Rapid equilibration carries it to divertor
Complete topology complicated due to differences of open field lines and closed field lines
It is difficult to convey the complete topological picture in a single powerpoint slide. There are field lines that connect to the top (bottom) and terminate in the plasma, fieldlines that connect both top and bottom (as pictured and as what the original topology is), and field-lines that are completely contained in the plasma. Visualization tools are currently inadequate for easily visualizing this.
I changed the heat flux colormap here to show up better on white.

19. Initial Simulations Above Ideal Marginal Stability Point Look Promising
Qualitative agreement with experiment: ~200 microsecond time scale, heat lost preferentially at divertor.
Plasma current increases due to rapid reconnection events changing internal inductance
Wall interactions are not a dominant force in obtaining qualitative agreement for these types of disruptions.
I need to add vertical lines to show where the previous slides were taken from.
Qualitatively, we’re doing pretty well. To move into quantitative comparisons is hard because we need to push up in Lundquist number. Tough.

20. Future Directions
Direct comparison of code against experimental diagnostics
Increased accuracy of MHD model
Temperature-dependent thermal diffusivities
More aggressive parameters
Resistive wall B.C. and external circuit modeling
Extension of fluid models
Two-fluid modeling
Electron heat flux using integral closures
Energetic particles
Simulations of different devices to understand how magnetic configuration affects the wall power loading

21. Conclusions
NIMROD’s Advanced Computational Techniques Allows Simulations Never Before Possible
Heating through b limit shows super-exponential growth, in agreement with experiment and theory in fixed boundary cases.
Simulation of disruption event shows qualitative agreement with experiment.
Loss of internal energy is due to rapid stochastization of the field, and not a violent shift of the plasma into the wall.
Heat flux is localized poloidally and toroidally as plasma localizes the perpendicular heat flux, and the parallel heat flux transports it to the wall.