1. Gyrokinetic Simulation of Multimode Plasma Turbulence
Florian Merz, Frank Jenko, Pavlos Xanthopoulos, Matthias Kammerer
IPP Theory Meeting
Ringberg Castle November 2008

2. Overview Introduction Results Summary GENE Eigenvalue solver
Non-Hermitian degeneracies
TEM-ITG transitions
ITG turbulence in W7-X
Summary

3. The GENE code
GENE is a Fortran 90/95 code to solve the gyrokinetic equations
Computes df on a fixed grid in 5D phase space
Massively parallel (MPI/OpenMP)
Electromagnetic perturbations
Arbitrary number of gyrokinetic species
Linearized Landau-Boltzmann collision operator
Realistic geometry via TRACER code
This talk: local flux tube limit

4. GENE equations GK equation for the modified distribution function g
Integrodifferential operators
Auxiliary fields
Electromagnetic fields

5. Eigenvalue solver in GENE
In addition to the initial value solver, an eigenvalue solver can be used to access arbitrary parts of the spectrum of Applications:
Computation the linear time step limit for explicit time stepping schemes
Investigations concerning subdominant unstable modes and mode transitions
beat of two competing modes

6. Software library: SLEPc
Rank( )~ for well resolved linear modes!
SLEPc - a Scalable Library for Eigenvalue Problems (Univ. Valencia)
MPI parallel
Only few EVs needed: iterative solver
RHS computation already optimized in GENE: matrix free solver
Before: spectral transforms to access arbitrary parts of the spectrum (e.g. shift-invert)
Collaboration with J. Roman: harmonic projection method to avoid inversion
(J. Roman, M. Kammerer, F. M., and F. Jenko, submitted to Parallel Matrix Algorithms and Applications)

7. Non-Hermitian degeneracies

8. Subdominant microinstabilities
For general plasma parameters, several microinstabilities are active simultaneously
Characterisation / naming according to drift direction / behaviour in parameter space
Trapped Electron Modes (TEM)
Ion Temperature Gradient (ITG)
Electron Temperature Gradient (ETG)
Traditional idea: if several instabilities are present, their identity can be determined by changing the plasma parameters to unambigous regimes
trapped electron modes
ETG modes
ITG modes

9. Linear mode transitions
Observation: The qualitative parameter dependence of the eigenmodes on one parameter can depend sensitively on the value of a second parameter.
Two unstable modes
can show
A clear transition
„Mixing“ or
„hybridization“

10. Linear mode transitions II
Connection of the two 1D scans in parameter space: closed scan along a rectangle in the 2D parameter space (R/LTe,R/Ln)
The modes are connected!

11. Topology of the transition
Full 2D scan: nontrivial topology of the eigenvalue surface in the two dimensional parameter plane
Explanation: a non-Hermitian degeneracy / Exceptional Point in the center
(Frequency analogous)

12. Non-Hermitian degeneracies
The non-Hermiticity of the linear operator creates the instabilities and also allows for degeneracies
Non-Hermitian degeneracy / Exceptional Point
creation of a non-trivial Jordan block
eigenvectors coincide
codimension 2 in the gyrokinetic parameter space (encountered frequently)
Further scans: pairwise connections between ITG, ETG, KBM, temperature/density gradient driven TEMs, i.e. all microinstabilities can be transformed into each other by continuous parameter variations!
A strict distinction of microinstabilities is impossible, traditional naming should be avoided close to exceptional points
(M. Kammerer, F. M., and F. Jenko, PoP 15, 2008)

13. TEM/ITG transitions

14. Nonlinear TEM-ITG transition
Parameter scan far from non- Hermitian degeneracies
With the additional instability, heat and particle fluxes are suppressed instead of increased
ITG branch: Nonlinear upshift of critical R/LTi
Transition at
R/LTi=4.63
linear modes at ky=0.25

15. Zonal flows Shearing rate shows a jump at the transition
Zonal flow suppression shows almost no effect in TEM regime but large increase of transport for ITG regime
Nonlinear upshift of critical R/LTi also observable for ITG saturation mechanism (linearly R/LTi=3.3 to R/LTi=4.5 nonlinearly)
Zonal flows suppressed
With zonal flows

16. Spectral decomposition
Nonlinear frequency distribution shows coexistence of TEM and ITG at different wave numbers and even at the same wave number
linear values
TEM
ITG
Dominant mode is determined by linear growth rate
Spectral dependence is confirmed by other TEM/ITG specific properties like the flux ratios

17. Particle transport at the transition
Zero crossings of the particle transport important for quasistationary state in actual experiments
Mechanisms: spectral balance of ITG pinch and TEM- or ITG induced outward transport

18. Quasilinear flux ratios
Flux ratios can be determined from linear computations and are enough to determine G=0
Only one ‘central’ ky=0.25 to describe the whole spectrum
Average of the two microinstabilities via
with , p=10
nonlinear
quasilinear
TEM
Model
ITG

19. Application of the model
The simplicity of the model allows for higher dimensional investigations of the G=0 hypersurface. The most important parameters are the gradients
Example: dependence of the G=0 surface in gradient space on the collisionality
nC=0.0005
nC=0
G=0
gTEM=gITG

20. ITG turbulence in W7-X

21. Stellarator simulations
TRACER code extracts metric quantities for the GENE flux tube from 3D MHD equilibria
Complicated structure of the metric coefficients, many points in parallel direction (expensive simulations!)
In the following: W7-X high mirror case, a/Ln=a/LTe=0
First simulations with gyrokinetic electrons

22. Microinstabilities
Even with pure ITG drive (here: a/LTi=3), many instabilities are present (SLEPc limit was set to 10)
Adiabatic electrons
Gyrokinetic electrons

23. Nonlinear results I Spectra of the ion heat transport
Parallel structure dominated by the three most unstable modes
gyrokinetic electrons
adiabatic electrons

24. Nonlinear results II
Heat diffusivity is offset-linear (tokamaks: heat flux)
adiabatic electrons
gyrokinetic electrons
Linear threshold
Linear threshold
Adiabatic electrons: zonal flows are overestimated!

25. Summary
Combination of initial value + eigenvalue solver as implemented in GENE is very useful
Linear gyrokinetics: non-Hermitian degeneracies connect different microinstabilities, so that the traditional naming can be misleading
TEM/ITG turbulence:
remnants of both modes are observed in the same simulation and even at the same wavenumber
ITG turbulence shows a significant upshift of critical R/LTi of due to nonlinear TEM-ITG interaction
Zero transport due to spectral cancellation can be modelled using linear eigenvalue computations
W7-X simulations of ITG turbulence show
Many instabilities
Adiabatic approximation overestimates zonal flows
Potential for optimization of anomalous transport in stellarators

26. Additional material

27. Spectral dependence
Dominant microinstability depends nontrivially on gradient and wave number
Nonlinearly, the normalized sum of amplitudes of positive/negative frequencies (ITG/TEM contributions) shows a steep transition
The average of the nonlinear frequency distribution can be used to determine the nature of the turbulence
Linear dominance of ITG
TEM
nonlinear frequency
Amplitudes of positive/negative frequencies for
ky=0.25
zero frequency

28. High mirror equilibrium