Gyrokinetic Simulation of Multimode Plasma Turbulence Florian Merz, Frank Jenko, Pavlos Xanthopoulos, Matthias Kammerer IPP Theory Meeting Ringberg Castle 18. November 2008

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

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

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

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

Software library: SLEPc Rank( )~100.000 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)

Non-Hermitian degeneracies

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

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“

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!

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)

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)

TEM/ITG transitions

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

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

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

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

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

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

ITG turbulence in W7-X

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

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

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

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!

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

Additional material

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

High mirror equilibrium