Nonlinear Excitation of Damped Eigenmodes in Microturbulence Simulations

1
Nonlinear Excitation of Damped Eigenmodes in
Microturbulence Simulations

• D.R. Hatch, P.W. Terry, University of Wisconsin,
  Madison
• W.M. Nevins, Lawrence Livermore National
  Laboratory

13th EU-US TTF Workshop Copenhagen, Denmark
Sept. 2008
This research was performed under an appointment
to the U.S. Department of Energy, Fusion Energy
Sciences Fellowship Program, administered by the
Oak Ridge Institute for Science and Education
under contract number DE-AC05-06OR23100 between
the U.S. Department of Energy and Oak Ridge
Associated Universities.
2
Outline.

• Introduction to Damped Eigenmodes
• Damped Eigenmodes in Reduced Models
• Damped Eigenmodes in Gyrokinetics
• Zonal Damped Eigenmodes

3
Motivation.

• Nonlinear Energy Transfer to a Stable Manifold

4
Motivation

• Significance of Damped Eigenmodes

Fundamentally different way of looking at plasma
turbulence. Effect on energy transfer and
saturation Additional saturation
mechanism. Effect on Transport Contribute
s to an inward component of flux. Reduces
net transport.
5
Reduced Model.

• Three field ITG like model.
• Used to illustrate concepts of damped eigenmodes1.

1P.W. Terry, D.A. Baver, and S. Gupta, Phys.
Plasmas 13, 022307 (2006).
6
Reduced Model.

• Linear Eigenmode Basis

Linearize system of equations. Solve dispersion
relation for mode frequencies. Solve for
eigenvectors associated with each
frequency. Linear transformation changes to
eigenmode basis. Eigenmode basis diagonalizes
linear coupling and mixes nonlinearities.
7
Reduced Model.

• Effect of damped eigenmodes in reduced model.

All quantities (fluxes, conserved energy, etc.)
can be projected on eigenmode basis. Marginally
stable eigenmode has the largest
amplitude. Nonlinear frequency spectrum is
peaked at the marginal mode frequency Damped
eigenmode is the primary energy sink. Drastic
reduction in transport flux.
8
Gyrokinetics.

• Unable to construct eigenmode decomposition.
• Use of GLF23 gyro-Landau fluid equations
  eigenmode solver
• - several modes, representative of important
  gyrokinetic eigenmodes.
• Use of GYRO for nonlinear gyrokinetic
  simulations.
• CYCLONE base case parameters.
• Three tests
• Comparisons of nonlinear frequency spectra with
  damped mode frequencies.
• Examination of ?p phase angles (time histories
  and pdfs).
• Comparison of quasilinear flux and nonlinear flux.

9
Gyrokinetics Frequency Spectra.

• Frequency Spectrum in ky (kx0).
• Nonlinear spectrum is closely matched by the
  spread of linear eigenmode frequencies

Spectrum is wide several times linear growth
rate.
Solid lines frequencies Dashed lines growth
rates
10
Gyrokinetics Frequency Spectra.

• Frequency Spectrum in kx (ky0.2).
• Nonlinear spectrum is closely matched by the
  spread of linear eigenmode frequencies

11
Gyrokinetics Frequency Spectra.

• What can explain the
• width of the spectrum?
• -Linear growth rate?
• -Doppler shift from
• zonal flows?
• -Damped eigenmode
• frequencies?
• Width doesnt scale
• with linear growth rate.

R/LT6.0 ?0.065(cs/Ln)
R/LT6.9 ?0.093(cs/Ln)
R/LT11.0 ?0.19(cs/Ln)
R/LT8.0 ?0.12(cs/Ln)
12
Gyrokinetics Frequency Spectra.

• Width of spectrum doesnt scale with linear
  growth rate.
• Width doesnt scale with zonal flow velocity
  (Doppler shift).

13
Gyrokinetics Frequency Spectra.

• Width of spectrum doesnt scale with linear
  growth rate.
• Width doesnt scale with zonal flow velocity
  (Doppler shift).
• Width matches damped mode frequencies better than
  other possible explanations.

14
Gyrokinetics Frequency Spectra.

• Nonlinear spectrum width does not match spread in
  mode frequency variation in GLF23 magnetic shear
  scan.
• GLF23 narrower spread at low s.
• Nonlinear spectrum retains broad width.
• Possible explanations
• There are important eigenmodes not captured by
  GLF23?
• GLF23 poorly models shear dependence?
• Spectrum width determined by something other than
  damped mode frequencies?

15
Gyrokinetics Phase Angles

• Phase angle pdfs peaked near linear phase angle.
• Also exhibit nonlinear structure.

Secondary peaks Bumps
16
Gyrokinetics Phase Angles

• Phase angle pdfs exhibit nonlinear structure.

Significant width 0.5 to 1.5 radians.
17
Gyrokinetics Phase Angles.

• Phase angles.

Interaction of dominant mode ?1 with subdominant
mode, ?2.
Net phase angle to first order in ?2/?1
Interaction of two modes creates a pdf centered
at the dominant mode phase angle with a width
proportional to the relative amplitude of the
secondary mode.
18
Gyrokinetics Phase Angles.

• Phase angles.

Interaction of dominant mode ?1 with subdominant
mode, ?2.
Modeling exercise ?is modeled using segments
of a nonlinear simulation. ? reduces amplitude
of damped modes. Ri define correct phase angles
(from GLF23). Net phase angle calculated. Model
pdfs most closely match true pdfs when damped
modes dissipate 20-40 of energy input by
instability.
19
Gyrokinetics Transport Fluxes

• Quasilinear flux overestimates true flux.
• Consistent with damped mode effect of 35.

Fluxes are identical in linear phase.
In nonlinear phase, True flux 0.64 x QL Flux
20
Damped zonal modes.

• In ITG turbulence, saturation level is tied to
  fluctuations at ky0.
• The shear rate is on the order of linear growth
  rate and correlation time2.
• Fractional reduction in fluctuation amplitude,
  not order of magnitude reduction.
• More is needed to explain saturation.
• Zonal Damped eigenmodes?
• Fluctuations at ky0
• Rosenbluth Hinton weakly damped zonal flows.
• GAMS linearly damped, high frequency
• Other low frequency damped zonal modes.3,4

2Nevins et al., Phys. Plasmas 13, 122306, (2006).
3Z. Gao, K. Itoh, H. Sanuki, and J.Q. Dong, Phys.
Plasmas 13, 100702 (2006). 4K. Itoh, S.-I. Itoh,
P.H. Diamond, T.S. Hahm, A. Fujisawa, G.R. Tynan,
M. Yagi and Y. Nagashima, Physics of Plasmas 13,
055502 (2006).
21
Damped zonal modes.

• Phase angle behavior at ky0.
• Peak at ? (phase angle of zonal flows).
• Significant probability at all phase angles.

22
Damped zonal modes.

• Phase angle behavior at ky0.
• Phase angle fixes intermittently on ?.
• Rotation through all phase angles ? interaction
  of multiple modes (GAM frequency filtered out).

23
Damped zonal modes.

• Frequency spectrum at ky0.

24
Summary.

• Frequency spectra, phase angles, and quasilinear
  fluxes all show signs of damped eigenmode
  excitation.
• Frequency spectra closely match damped mode
  frequency spread.
• Phase angle pdfs exhibit structure and width
  consistent with damped mode excitation.
• True flux reduced from quasilinear estimate.
• Estimated magnitude of effect 35 at ky?0.
• ?p phase angle behavior of ky0 fluctuations
  indicate interaction of multiple modes.
• Main saturation mechanism excitation of damped
  modes at ky0?