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Title: 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 gradient-driven 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 » in ways that cannot be fully explained by the linear eigenvalue approach; a non-modal treatment is necessary to understand key features of the turbulence.« less

Authors:
 [1];  [2];  [2];  [3];  [4];  [4]
  1. Univ. of Texas, Austin, TX (United States). Inst. for Fusion Studies
  2. Univ. of California, Los Angeles, CA (United States). Dept. of Physics and Astronomy
  3. Max Planck fur Plasmaphysik, Garching (Germany)
  4. Univ. of Wisconsin-Madison, 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:
FG02-04ER54742
Resource Type:
Journal Article: Published Article
Journal Name:
New Journal of Physics
Additional Journal Information:
Journal Volume: 18; Journal Issue: 7; Journal ID: ISSN 1367-2630
Publisher:
IOP Publishing
Country of Publication:
United States
Language:
English
Subject:
70 PLASMA PHYSICS AND FUSION TECHNOLOGY; gyrokinetics; pseudospectra; non-modal; 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:10.1088/1367-2630/18/7/075018.
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:10.1088/1367-2630/18/7/075018.
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:10.1088/1367-2630/18/7/075018.
@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 gradient-driven 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 non-modal treatment is necessary to understand key features of the turbulence.},
doi = {10.1088/1367-2630/18/7/075018},
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}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record at 10.1088/1367-2630/18/7/075018

Citation Metrics:
Cited by: 2works
Citation information provided by
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  • 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 gradient-driven 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 » 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 non-modal treatment is necessary to understand key features of the turbulence.« less
  • The self-consistency of the renormalized perturbation theory of Zhang and Mahajan (Phys. Rev. 132, 1759 (1985)) is demonstrated by applying it to the Vlasov--Poisson 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 low-beta gyrokinetic system. Comparison of this theory with other current theories is presented.
  • In magnetized plasmas, a turbulent cascade occurs in phase space at scales smaller than the thermal Larmor radius ('sub-Larmor 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 two-dimensional 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 » we diagnose by means of nonlinear transfer functions for the collisionless invariants. We find that the turbulence tends to a time-asymptotic state, dominated by a single scale that grows in time. A theory of this asymptotic state is derived in the form of decay laws. Each case that we study falls into one of three regimes (weakly collisional, marginal, and strongly collisional), determined by a dimensionless number D{sub *}, a quantity analogous to the Reynolds number. The marginal state is marked by a critical number D{sub *}=D{sub 0} that is preserved in time. Turbulence initialized above this value become increasingly inertial in time, evolving toward larger and larger D{sub *}; turbulence initialized below D{sub 0} become more and more collisional, decaying to progressively smaller D{sub *}.« less