nonlinear mhd modeling of tokamak plasmas on multiple time scales
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Nonlinear MHD Modeling of Tokamak Plasmas on Multiple Time Scales. S.E. Kruger 1 , D.B. Brennan 2 , T. A. Gianakon 3 , D.D. Schnack 1 , and C.Sovinec 4 and the NIMROD Team. 1)Science Applications International Corp. San Diego, CA USA. 2) General Atomics San Diego, CA, USA.

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nonlinear mhd modeling of tokamak plasmas on multiple time scales

Nonlinear MHD Modeling of Tokamak Plasmas on Multiple Time Scales

S.E. Kruger1, D.B. Brennan2, T. A. Gianakon3,

D.D. Schnack1, and C.Sovinec4and the NIMROD Team

1)Science Applications International Corp.

San Diego, CA USA

2) General Atomics

San Diego, CA, USA

3) Los Alamos National laboratory

Los Alamos, NM USA

4) University of Wisconsin-Madison

Madison, WI, USA

mhd instabilities often classified according to time scales
MHD Instabilities Often ClassifiedAccording to Time Scales
  • Ideal MHD instabilities grow on fastest time scales
    • g ~ tA
    • Large scale lengths: l/a ~ 1
    • Linear codes have been successful in predicting many aspects of experimental results
    • Often cause “hard disruptions”
  • Slower time scales often classified as “resistive modes”
    • g ~ SatA where 0 < a < 1
    • Small scale lengths: l/a << 1
    • Linear theory less successful:
      • Requires “extended MHD” to describe physics
    • Often causes “soft beta limits”
behavior of instabilities near marginal point important for understanding experiments
Behavior of Instabilities Near Marginal Point Important for Understanding Experiments


x0 = initial perturbation

xobs=perturbation size able to be


texp = duration of experiment

from onset of mode

Then there is a growth rate below which the experiment is

experimentally stable

Increased range in operating space depends on

mode passing through instability point has faster than exponential growth
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:

Perturbation growth:


mode does not grow because it is exactly at marginal point

nimrod studying behavior near marginal point for modes on two different time scales
NIMROD Studying Behavior Near Marginal Point For Modes on Two Different Time Scales
  • Shot 87009
    • Disruption when heated through b limit
      • Does simple analytic theory work?
      • What causes disruption?
  • Shot 86144 and 86166
    • ITER-like discharge
    • Sawteeth
      • Nonlinear generation of secondary islands
      • Destabilization of neoclassical tearing mode (NTM)?
    • Tests both resistive MHD and closure models for Extended MHD
nimrod uses advanced computational techniques to create flexible application framework
NIMROD Uses Advanced Computational Techniques To Create Flexible Application Framework
  • Pseudo-spectral discretization in toroidal direction
  • Finite-element discretization in poloidal plane:
    • Can use either structured rectangular cells or unstructured triangular cells
    • Uses Lagrangian elements of arbitrary polynomial degree in structured cells.
  • The non-dissipative semi-implicit method used for temporal discretization
  • MPI used for efficient scaling
nimrod solves the mhd form of the extended mhd equations
NIMROD Solves the MHD form of the “Extended MHD Equations”
  • Momentum Equation
  • Generalized Ohm’s law:
  • Temperature Equations:
diii d shot 87009 observes a mode on hybrid time scale as predicted by analytic theory
DIII-D SHOT #87009 Observes a Mode on Hybrid Time Scale As Predicted By Analytic Theory

High-b disruption slow heating

Growth is slower than ideal, but faster than resistive

Callen, Phys. Plasmas 6, 2963 (1999)

nimrod models discharge based on equilibrium profiles at t 1681 7 msec
NIMROD Models Discharge Based on Equilibrium Profiles at t = 1681.7 msec

Safety factor profile

  • Equilibrium reconstruction from experimental data
  • Negative central shear
  • Gridding based on equilibrium flux surfaces
    • Packed at rational surfaces
    • Bi-cubic finite elements

Poloidal gridding

Pressure contours

nimrod simulations begin near critical beta for ideal mhd linear stability
NIMROD Simulations Begin Near Critical Beta For Ideal MHD Linear Stability
  • Present version of NIMROD requires conducting wall at outer flux surface
    • Critical bN for ideal instability larger than in experiment
  • NIMROD gives slightly larger ideal growth rate than GATO
  • NIMROD finds resistive interchange mode below ideal stability boundary
nonlinear simulations with nimrod show faster than exponential growth
Nonlinear Simulations With NIMROD Show Faster-Than-Exponential Growth
  • Initial condition: equilibrium below ideal marginal bN
  • Use resistive MHD
  • 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

  • Follow nonlinear evolution through heating, destabilization, and saturation
nimrod scaling of growth rate of mode agrees with experiment
NIMROD Scaling of Growth Rate of Mode Agrees with Experiment
  • 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

  • Compare with theory:

Instability experiences no loss of confinement as it enters

nonlinear stage consistent with experimental observations.

slower growing mode also observe interesting behavior near marginal point
Slower Growing Mode Also Observe Interesting Behavior Near Marginal Point

DIII-D Shot 86166 observes loss of confinement when

m/n=3/2 island:

Why does 3/2 island appear at that particular sawtooth crash?

test linear theory using nimrod with two nearly identical equilibria
Test Linear Theory Using NIMROD With Two Nearly Identical Equilibria
  • Biggest difference is pressure near core
later case observes 3 2 mode that grows
Later Case Observes 3/2 Mode That Grows
  • Runs performed at S=106, Pr=1000
  • Runs at these parameters and higher S still ongoing.
  • Nonlinear runs confirm importance of beta in tearing mode instability
  • Linear stability useful in quantitative predictions? Not clear.
long pulse discharges on diii d observe modes operating on a longer time scale
“Long-Pulse” Discharges on DIII-D Observe Modes Operating on a Longer Time Scale

Shot 86144 similar to 86166

3/2 NTM

2/1 wall locking

1/1 sawteeth


  • Sawtoothing discharge
  • 3/2 NTM triggered at 2250 msec
  • 2/1 locks to the wall
model discharge 86144 at t 2250 msec
Model Discharge #86144 at t = 2250 msec

Grid (Flux Surfaces)

q - profile




  • ITER-like discharge
  • q(0) slightly below 1
discharge is unstable to resistive mhd
Discharge is Unstable To Resistive MHD

n = 1 linearly unstable


n = 2 linearly unstable

(D’ > 0)

n = 2 nonlinearly driven

by n = 1

Pressure and field lines

S = 107 Pr = 103

g= 4.58 X 103 / sec gexp ~ 1.68 X 104 / sec

resistive 3 2 islands driven by n 1 mode are too small to account for experimental observations
Resistive 3/2 Islands Driven by n=1 Mode Are Too Small To Account for Experimental Observations




  • Secondary islands are small in resistive MHD
    • Wexp ~ 6-10 cm
  • 3/2 island width decreases with increasing S
  • Need extended MHD to match experiment?
use heuristic closure to model neoclassical tearing mode
Use “Heuristic Closure” to Model Neoclassical Tearing Mode
  • Resistive MHD is insufficient to explain DIII-D shot 86144
  • Need “neo-classical” effects relevent for long mean-free-path regime:
    • Poloidal flow damping
    • Enhancement of polarization current
    • Bootstrap current
  • Simplified model captures most neo-classical effects
  • (T. A. Gianakon, S. E. Kruger, C. C. Hegna, Phys. Plasmas 9 (2002) 536)
  • Benchmarked against modified Rutherford Equation:

Hegna PoP 6 (1999) 3980

Lutjens PoP 8 (2001) 4267

closures agree with modified rutherford equation in determining the stability boundary
Closures Agree with Modified Rutherford Equation in Determining The Stability Boundary
  • TFTR-like equilibrium
  • Comparison with modified Rutherford equation
  • Anisotropic heat conduction leads to threshold island size (Wd)
  • Initialize NIMROD with various seed island sizes
  • Look for growth or damping
use analytic theory to help choose simulation plasma parameters of 86144 2250
Use Analytic Theory to Help Choose Simulation Plasma Parameters of 86144.2250

3/2 island



  • Use modified Rutherford equation
  • 3 values of anisotropic heat flux
  • 2 values of D¢
    • Vacuum
    • Reduced by factor of 10
  • Experimental island width ~ 0.1 m
plasma parameters severely constrained by ordering of length scales
Plasma Parameters Severely Constrained By Ordering of Length Scales
  • Need Wd >> dv High S
  • High S means smaller secondary island Need smaller threshold
  • To get smaller threshold, need higher anisotropy
  • Quickly leads into realistic plasma parameters

Nonlinear NTM calculations are extremely challenging!

  • NIMROD simulations of DIII-D discharges have several things in common:
    • Operation near marginal stability point
    • Self-consistent evolution to an unstable state
    • Simulations at realistic plasma parameters
  • Simulations on Fast Time Scale (87009-like discharge)
    • Heating through b limit
    • Super-exponential growth, in agreement with experiment and theory
  • Simulations on Slow Time Scale (shot #86144 and #86166)
    • Secondary islands driven by sawtooth crash, but must be near marginal stability point (high beta)
    • Resistive MHD insufficient, requires neo-classical closures
    • Must go to large S (~107) and anisotropy (kll/kprp ~ 109) to get proper length scales