Linear signatures in nonlinear gyrokinetics: interpreting turbulence with pseudospectra
Abstract
A notable feature of plasma turbulence is its propensity to retain features of the underlying linear eigenmodes in a strongly turbulent state—a property that can be exploited to predict various aspects of the turbulence using only linear information. In this context, this work examines gradientdriven gyrokinetic plasma turbulence through three lenses—linear eigenvalue spectra, pseudospectra, and singular value decomposition (SVD). We study a reduced gyrokinetic model whose linear eigenvalue spectra include ion temperature gradient driven modes, stable drift waves, and kinetic modes representing Landau damping. The goal is to characterize in which ways, if any, these familiar ingredients are manifest in the nonlinear turbulent state. This pursuit is aided by the use of pseudospectra, which provide a more nuanced view of the linear operator by characterizing its response to perturbations. We introduce a new technique whereby the nonlinearly evolved phase space structures extracted with SVD are linked to the linear operator using concepts motivated by pseudospectra. Using this technique, we identify nonlinear structures that have connections to not only the most unstable eigenmode but also subdominant modes that are nonlinearly excited. The general picture that emerges is a system in which signatures of the linear physics persist in the turbulence, albeitmore »
 Authors:
 Univ. of Texas, Austin, TX (United States). Inst. for Fusion Studies
 Univ. of California, Los Angeles, CA (United States). Dept. of Physics and Astronomy
 Max Planck fur Plasmaphysik, Garching (Germany)
 Univ. of WisconsinMadison, Madison, WI (United States)
 Publication Date:
 Research Org.:
 Univ. of Texas, Austin, TX (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC)
 OSTI Identifier:
 1272650
 Alternate Identifier(s):
 OSTI ID: 1272651; OSTI ID: 1326679
 Grant/Contract Number:
 FG0204ER54742
 Resource Type:
 Journal Article: Published Article
 Journal Name:
 New Journal of Physics
 Additional Journal Information:
 Journal Volume: 18; Journal Issue: 7; Journal ID: ISSN 13672630
 Publisher:
 IOP Publishing
 Country of Publication:
 United States
 Language:
 English
 Subject:
 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; gyrokinetics; pseudospectra; nonmodal; plasma turbulence; Landau damping; Hermite polynomials
Citation Formats
Hatch, D. R., Jenko, F., Navarro, A. Banon, Bratanov, V., Terry, P. W., and Pueschel, M. J.. Linear signatures in nonlinear gyrokinetics: interpreting turbulence with pseudospectra. United States: N. p., 2016.
Web. doi:10.1088/13672630/18/7/075018.
Hatch, D. R., Jenko, F., Navarro, A. Banon, Bratanov, V., Terry, P. W., & Pueschel, M. J.. Linear signatures in nonlinear gyrokinetics: interpreting turbulence with pseudospectra. United States. doi:10.1088/13672630/18/7/075018.
Hatch, D. R., Jenko, F., Navarro, A. Banon, Bratanov, V., Terry, P. W., and Pueschel, M. J.. Tue .
"Linear signatures in nonlinear gyrokinetics: interpreting turbulence with pseudospectra". United States.
doi:10.1088/13672630/18/7/075018.
@article{osti_1272650,
title = {Linear signatures in nonlinear gyrokinetics: interpreting turbulence with pseudospectra},
author = {Hatch, D. R. and Jenko, F. and Navarro, A. Banon and Bratanov, V. and Terry, P. W. and Pueschel, M. J.},
abstractNote = {A notable feature of plasma turbulence is its propensity to retain features of the underlying linear eigenmodes in a strongly turbulent state—a property that can be exploited to predict various aspects of the turbulence using only linear information. In this context, this work examines gradientdriven gyrokinetic plasma turbulence through three lenses—linear eigenvalue spectra, pseudospectra, and singular value decomposition (SVD). We study a reduced gyrokinetic model whose linear eigenvalue spectra include ion temperature gradient driven modes, stable drift waves, and kinetic modes representing Landau damping. The goal is to characterize in which ways, if any, these familiar ingredients are manifest in the nonlinear turbulent state. This pursuit is aided by the use of pseudospectra, which provide a more nuanced view of the linear operator by characterizing its response to perturbations. We introduce a new technique whereby the nonlinearly evolved phase space structures extracted with SVD are linked to the linear operator using concepts motivated by pseudospectra. Using this technique, we identify nonlinear structures that have connections to not only the most unstable eigenmode but also subdominant modes that are nonlinearly excited. The general picture that emerges is a system in which signatures of the linear physics persist in the turbulence, albeit in ways that cannot be fully explained by the linear eigenvalue approach; a nonmodal treatment is necessary to understand key features of the turbulence.},
doi = {10.1088/13672630/18/7/075018},
journal = {New Journal of Physics},
number = 7,
volume = 18,
place = {United States},
year = {Tue Jul 26 00:00:00 EDT 2016},
month = {Tue Jul 26 00:00:00 EDT 2016}
}
Web of Science

Linear signatures in nonlinear gyrokinetics: interpreting turbulence with pseudospectra
A notable feature of plasma turbulence is its propensity to retain features of the underlying linear eigenmodes in a strongly turbulent state—a property that can be exploited to predict various aspects of the turbulence using only linear information. In this context, this work examines gradientdriven gyrokinetic plasma turbulence through three lenses—linear eigenvalue spectra, pseudospectra, and singular value decomposition (SVD). We study a reduced gyrokinetic model whose linear eigenvalue spectra include ion temperature gradient driven modes, stable drift waves, and kinetic modes representing Landau damping. The goal is to characterize in which ways, if any, these familiar ingredients are manifest inmore »Cited by 2 
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The selfconsistency of the renormalized perturbation theory of Zhang and Mahajan (Phys. Rev. 132, 1759 (1985)) is demonstrated by applying it to the VlasovPoisson system and showing that the theory has the correct weak turbulence limit. Energy conservation is proved to arbitrary high order for the electrostatic drift waves. The theory is applied to derive renormalized equations for a lowbeta gyrokinetic system. Comparison of this theory with other current theories is presented. 
Freely decaying turbulence in twodimensional electrostatic gyrokinetics
In magnetized plasmas, a turbulent cascade occurs in phase space at scales smaller than the thermal Larmor radius ('subLarmor scales') [Tatsuno et al., Phys. Rev. Lett. 103, 015003 (2009)]. When the turbulence is restricted to two spatial dimensions perpendicular to the background magnetic field, two independent cascades may take place simultaneously because of the presence of two collisionless invariants. In the present work, freely decaying turbulence of twodimensional electrostatic gyrokinetics is investigated by means of phenomenological theory and direct numerical simulations. A dual cascade (forward and inverse cascades) is observed in velocity space as well as in position space, whichmore »