1 tokamak drift-wave instabilities （ itg ， etg ， ctem) ： nonlinear theory & simulations...
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Tokamak Drift-Wave Instabilities（ ITG ， ETG， CT
EM)：Nonlinear Theory & Simulations
X.Q.Xu
Lawrence Livermore National LaboratoryUSA
Institute of Fusion Theory and SimulationsZhejiang University
2009 Chinese Summer School on Plasma PhysicsHangzhou, China
July 18 – July 28, 2009
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Acknowledgments
• We thank Drs. J. Candy, L. Chen, P.H.Diamond, A.Dimits, D.Ernst,T. S. Hahm, G. Hammett, F. Jenko, Z.Lin, W. M. Nevins, H. Qin, and R.Waltz for fruitful physics discussions.
• This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory in part under Contract DE-AC52-07NA27344.
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Normalized Confinement Time HH = tE/tEmpirical
Fusion performance depends sensitively on confinement
Sensitive dependence on turbulent confinement causes some uncertainties, but also gives opportunities for significant improvements, if methods of reducing turbulence extrapolate to larger reactor scales.
Caveats: best if MHD pressure limits also improve with improved confinement. Other limits also: power load on divertor & wall, …
0
5
10
15
20
Q =
Fu
sion
Pow
er /
Hea
ting
Pow
er
Efusionext
EPP
dt
dE
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Overview of Mechanisms• “Islands”
– bootstrap etc.; magnetic flutter
• Ion Temperature Gradient Driven Mode (ITG), kri<1– electromagnetic with finite beta modes– Adiabatic electron model
• Trapped Electron Mode (TEM), kri≥1– Skin depth fluctuations– Non-adiabatic electron model
• Electron Temperature Gradient Driven Mode (ETG), kre<1 but kri>20– electrostatic + Streamer
• Critical theoretical issues:– ETG must couple to larger scales
a ri c/w
ce re
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Multiple scales in plasma microturbulence
• ITG/TEM: ion scale turbulence
• ITG/TEM and ETG scales separated by
• TEM/ETG: the same type mode with peak growth at different k┴
– TEM may smoothly connect to ETG at high k┴
trapped electron modes
ETG modes
ITG modesNot shown here:
- drift waves
- ballooning modes
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Nonlinear Saturation, Secondary Flows & Transport
• Nonlinear saturation due to wave coupling/cascades
• Turbulent Transport calculated from G=<dvEqxBdn>, fluxes
• Secondary flow generation via envelope modulation of drift-wave packet.– Zonal flow– GAM– Streamer
• Secondary flows in turn regulate the turbulence and transport
linear
nonlinear
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Diffusion
• Semi-empirical transport models – Generally diffusion can be modeled as a random walk of
steps of length d and time t. If the diffusion is collisional, then d is the mean free path and t is the inverse of the collision frequency. The diffusion coefficient D can be expressed variously as
c =d2/t
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Bohm Diffusion
• Bohm expressions for the transport coefficients– It was first observed in 1946 by David Bohm, et al, while studying magnetic arcs for
use in isotope separation. It has since been observed that many other plasmas follow this law.
• Bohm diffusion Scaling– turbulent transport is characterized by a radial correlation length Lr =(ria)1/2 and by a
decorrelation rate of the order of the diamagnetic frequency w*i =vti/a (vti being the ion thermal velocity and a the minor radius), as expected for the microinstabilities of the drift branch.
cB=ri2wci=T/eB
– the missing 1/16 in front is no cause for concern. Therefore, at least within a factor of order unity, Bohm diffusion is always greater than classical diffusion since nii<<wci.
eB
T
16
1
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Gyro-Bohm Diffusion
• gyro-Bohm expressions for the transport coefficients– turbulent transport is characterized by a radial correlation length Lr of the order of the
ion gyroradius ri and by a decorrelation rate of the order of the diamagnetic frequency w*i =vti/a (vti being the ion thermal velocity and a the minor radius), as expected for the microinstabilities of the drift branch.
cGB=ri2w*i
• For drift-wave, w*i=vTi/Lni
cGB=ri2vTi/Lni
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Mixing length theory
• Developed by Prandl in the early 20th century• only a rough approximation
• The mixing length is conceptually analogous to the concept of mean free path in thermodynamics: a fluid parcel will conserve its properties for a characteristic length d and time t, before mixing with the surrounding fluid.
• The diffusion coefficient c can be expressed variously as
c =d2/t
• Prandtl, L. (1926). "Proc. second int. Congr. appl. Mech.". Zurich.
.
Temperature is conserved for a certain distance as a parcel moves across a temperature gradient. The fluctuation in temperature that the parcel experienced throughout the process is .
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Simple mixing length estimates for transport
cs=rs2vTs/LTs
• Local balance between linear drive and nonlinear damping
Where g – linear growth rate, k┴ - perpendicular wavenumber No differentiation between particle and heat transport
• Mixing length model yields a gyro-Bohm scaling for drift-wave turbulence
• Where rs gyroradius for species s
• vTs thermal velocity for species s
• LTs equilibrium temperature gradient scale length for species s
D~cg/k┴2
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Simple mixing length estimates for transport from ITG, TEM, and ETG
cs=rs2vTs/LTs
• ITG, kri<1,
• ETG, kre<1 but kri>20– electrostatic
• The disparity in anomalous electron and ion thermal transport is
• This disparity is not typically observed in expt.– Suggesting different electron and ion scale physics– Electron transport is driven by ion-scale turbulence
• In ITB experiments– Electron transport remains high in the absence of
ion-scale turbulence
ci=ri2vTi/LTi
ce=re2vTe/LTe
ci ~ 60ce
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Transport fluxes from ExB drift
• Particle Flux
• Heat Flux
Gs=<dnsdvExB>
Qs=<dPsdvExB>
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BASIC INSTABILITY PICTURE
ITG-TEM-ETG
G.Hammett, APS 2007 review talk
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1. Intuitive pictures of gyrokinetic turbulence, & how to reduce it• analogy w/ inverted pendulum / Rayleigh-Taylor instability• reduce turbulence with sheared flows, magnetic shear, …
effectivegravity
Inverted-density fluidRayleigh-Taylor Instability
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Stable Pendulum
L
M
F=Mg w=(g/L)1/2
Unstable Inverted Pendulum
w= (-g/|L|)1/2 = i(g/|L|)1/2 = ig
gL
(rigid rod)
Density-stratified Fluid
stable w=(g/L)1/2
r=exp(-y/L)
Max growth rate g=(g/L)1/2
r=exp(y/L)
Inverted-density fluidRayleigh-Taylor Instability
Instability
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“Bad Curvature” instability in plasmas Inverted Pendulum / Rayleigh-Taylor Instability
Top view of toroidal plasma:
plasma = heavy fluid
B = “light fluid”
geff = centrifugal forceRv
2
R
Growth rate:
RLRLLtteffg vv
2
Similar instability mechanismin MHD & drift/microinstabilities
1/L = p/p in MHD, combination of n & T
in microinstabilities.
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The Secret for Stabilizing Bad-Curvature Instabilities
Twist in B carries plasma from bad curvature regionto good curvature region:
Unstable Stable
Similar to how twirling a honey dipper can prevent honey from dripping.
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21Rosenbluth-Longmire picture
22Rosenbluth-Longmire picture
Can repeat this analysis on the good curvature side & find it is stable?
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Gyrokinetic Theory developments
• Development of gyrokinetic equations one of the triumphs of high-power theoretical plasma physics and asymptotic analysis
• Key advance: Frieman & Chen show nonlinear version of gyrokinetics possible
• Other advances: Hamiltonian/Lagrangian derivations, insure conservation properties, easier to go to higher order
• GK ordering allows capture of drift/micro-instabilities & much of MHD at just order e & not e2
Guided by expts., mwave scattering, physics insights
)1(~,~~~~~||0
okk
k
T
e
F
f
L
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Two length scale separation, locally flatten gradients, quadratic nonlinearities
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Why gyroaverage?Reduce the dimensions to 5D
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Main Comprehensive Gyrokinetic Codes
• A partial list of dF codes– GTC (Lin & Lee) , GTS(Wang & Lee) PIC, global, USA– GS2 (Dorland & Kotschenreuther) continuum, flux-tube, USA– GENE (Jenko, Garching) continuum, flux-tube, Germany– GYRO (Candy & Waltz) continuum, global, USA– GEM (Parker and Chen) dF PIC, global, USA
– All of these codes include: toroidal geometry, general axisymmetric plasma shapes, multiple species, trapped and passing non-adiabatic electrons, electromagnetic fluctuations, collision operators, equilibrium scale ExB shear flow, GS2 & GENE use the r*0 limit local flux-tube (equivalent to thin annulus)
• A partial list of Full-F global, under developments & to be completed– TEMPEST (Xu &ESL) , continuum, USA– XGC (Chang & CPES), PIC, USA– GT5D (Idomura et al) , continuum, Japan– GYSELA (Garbet et al), semi-Lagrangian, France
• These gyrokinetic codes use a number of advanced algorithms
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ITG GYROKINETIC SIMULATION RESULTS
Dimits et al., Phys. Plasmas, 7 969 ( 2000).Z. Lin, T. S. Hahm, W. W. Lee, W. M. Tang, and R. B. White, Science 281, 1835 (1998).Z. Lin et al., Phys. Rev. Lett. 83, 3645 (1999).Z. Lin et al., Phys. Rev. Lett. 88, 195004 (2002).
28Dimits et al., Phys. Plasmas, 7 969 ( 2000).
Ion Temperature Gradient Driven Modes (ITG)
• In a toroidal system the instabilityis mainly driven by the magnetic field curvature and the ion temperature gradient
• ITG is of interest due to the successful interpretations of various experimental trends – related to the observed levels of turbulent transport in tokamak plasmas
• Theory/Simulations typically with adiabatic e- model
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Dimits et al., Phys. Plasmas, 7 969 ( 2000).
Parameterization dependence of the Dimits upshift?
ITG drives strong heat transport to force plasma near marginal stability
Explicit threshold of the Linear ITG instability, R/Lti,crit
Simulations find that the threshold of ci is different from that of linear growth. It is termed “Dimits upshift”
30Z. Lin, T. S. Hahm, W. W. Lee, W. M. Tang, and R. B. White, Science 281, 1835 (1998).
Zonal flow is found to regulate the turbulence and to play a key role in the Dimits upshift
31Z. Lin et al., Phys. Rev. Lett. 83, 3645 (1999).
Ion-ion collision is found to affect the ion heat transport via zonal flow damping
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Zonal flow
Zonal flow is a meteorological term
the general flow pattern is west to east along the Earth's latitude lines (as opposed to meridional flow).
Meridional flow
the general flow pattern is cross the latitude lines at a sharp angle
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Zonal flow• In toroidally confined fusion plasma experiments
the term zonal flow means a plasma flow within a magnetic surface primarily in the poloidal direction.
• Zonal flows in the toroidal plasma context are further characterized by
– being localized in their radial extent transverse to the magnetic surfaces (in contrast to global plasma rotation),
– having little or no variation in either the poloidal or toroidal direction -- they are m = n = 0 modes (where and m and n are the poloidal and toroidal mode numbers, respectively),
– having zero real frequency when analyzed by linearization around an unperturbed toroidal
equilibrium state (in contrast to the geodesic acoustic mode branch, which has finite frequency).
• They arise via a self-organization phenomenon driven by low-frequency drift-type modes, in which energy is transferred to longer wavelengths by modulational instability or turbulent inverse cascade.
From Wikipedia, the free encyclopedia
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c =d2/t
35Z. Lin et al., Phys. Rev. Lett. 88, 195004 (2002).
Transition from Bohm to gyro-Bohm Scaling
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Global code approaches local flux-tube limit as r* 0
Candy et al. PoP 04
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Moderate amount of turbulence spreading occurs in some cases
Waltz, Candy, Petty 2006 PoP 13, 072304DIII-D L-mode
amount of spreading: ~ 0.1 a ~ 2-5 radial correlation lengths ~ 20-50 ri
see also Hahm et al. 2004, Lin et al. 2002, Garbet et al., Newman, Xu
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Successful Benchmarks of Independent Gyrokinetic Codes
Good agreement in c (+/- 10% on long time average t > 1000 a/cs) between 3 continuum and 2 PIC codes
Nevins et al. 2007
Correlation functions agree well
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Gyro movie
Evolution of potential fluctuations in a plasma very similar to DIII-D 101381/101391. Simulation is centered at r/a=0.6. Note the strong equilibrium sheared rotation, which leads to a strong reduction in transport. This landmark simulation from 2002 includes kinetic electrons at finite-beta, along with the equilibrium ExB variation.
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TEM GYROKINETIC SIMULATION RESULTS
Ernst et al., APS invited talk 2008
Ernst et al., PoP 2004, 2007, 2009
Lang, Chen, Parker, PoP 2007, 2008
Lin et al, TTF talk 2009
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Trapped Electron Modes (TEM)• In typical ITG simulations, e- is
adiabatic• Adding trapped electron effects
introduce a new unstable root, the TEM root, – Due to resonances between the
mode eigenfrequency and the orbit-time-average magnetic drift frequency (precession frequency)
– in addition to increasing the ITG growth rate
• ITG has a real frequency in the ion diamagnetic drift direction (negative)
• TEM has a real frequency in the electron diamagnetic drift direction (positive)
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At Very Steep Density Gradients, Mode Structure Changes
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At Very Steep Density Gradients, Mode Structure Changes
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GTC simulations show CTEM saturation & eddy size regulated by Zonal flow
CTEM: collisionless trapped electron modes, nei=0
• Linear streamers mostly broken by zonal flows
• Transport much higher if zonal flows removed
• Significant tails in correlation function indicate existence of meso-scale eddies, evidences of non-diffusive transport
Xiao and Lin, submitted to PRL 2009
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ETG GYROKINETIC SIMULATION RESULTS
T.Görler and F. Jenko, PRL 100, 185002 (2008)
T. Görler, 1st September 2008, TTF
Waltz, Candy and Fahey, PHYSICS OF PLASMAS 14, 056116 2007
Lin et al, PoP 2005
Lin et al, PRL 2007
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ETG fluctuations (kri > 1) may account for significant fraction of transport in some plasmas
Simple scaling from ITG to ETG:
citg ~ Citg ri2 vti/L
cetg ~ Citg re2 vte/L ~ citg /60
But Dorland & Jenko (2000) showed ETG turbulence larger because:
perpendicular adiabatic ions for ETG gives more shielding of zonal electric fields than does parallel adiabatic electrons for ITG.
Candy showed ETG will be reduced by kinetic ions, more so if strong ITG turbulence
ITG can be weak near marginal stability w/ ExB shear. Görler and Jenko shows ETG / high-k TEM may still be important in some cases.
Dorland et al., Phys. Rev. Lett. 85, 5579 (2000).
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Electron Temperature Gradient Mode (ETG)
• Electron temperature gradient modes are driven by the circulating electrons
• TEM mode are driven by trapped electrons
• ETG and TEM mode are the same family of linear electron modes with growth rates peak at different k
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Waltz, Candy, and Fahey, Phys. Plasmas 14, 056116 2007
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Waltz, Candy, and Fahey, Phys. Plasmas 14, 056116 2007
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Largest GYRO simulations used to studyinteraction of ITG & ETG Turbulence
• 1280 re x 1280 re x 20 parallel pts/orbit x 8 energies x 16 v||/v
• electrons + kinetic ions, mi/me = 202 - 302
• 5 days on DOE/ORNL Cray X1E w/ 720 Multi-Streaming Processors
Candy, Waltz, et al. JPSC 2007
ETG w/ kinetic ions R/LTi=0 ETG+ITG R/LTi=6.9
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Gyro movie
Electron density fluctuations in a simulation coupling ITG/TEM instabilities with ETG instabilities. In this case, ITG modes are linearly unstable.
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ETG + kinetic ion GYRO simulation movie
• large box on right: full simulation domain, 1280 re x 1280 re = 64 ri x 64 ri
• small box on lower left: zoom in on a 64 re x 64 re patchhttp://fusion.gat.com/THEORY/images/1/1f/ETG-ki.mpg from http://fusion.gat.com/theory/Gyromovies
Candy & Waltz
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Gyro movie
Electron density fluctuations in a simulation coupling ITG/TEM instabilities with ETG instabilities. In this case, ITG modes are linearly stable.
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ExB shear can affect even ETG
GYRO ETG-ki sim. turbulent e flux ~ 1 MW (NSTX expt ~ 2 MW)ExB shearing rate varied from 2X to 1/4 experimental rate.Eddies grow longer (and wider) as shearing rate is reduced.
2X
experimental ExB rate
1/2X
1/4X
Mikkelsen, NSTXRadial domain ~400 e.
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• GENE: a nonlinear gyrokinetic Vlasov code– fully gyrokinetic electrons and ions– fully electromagnetic fluctuations– realistic collision operators– flux tube volume in general toroidal geometry
• here: electrostatic, collisionless, s-α-model equilibrium
• reduced mass ratio: from mi/me = 1836 to mi/me = 400
– computational effort lowered by one order of magnitude:TCPU ~ (mi/me)3/2
– still more than 100,000 CPUh / simulation
Background: Nonlinear gyrokinetic simulations
T. Görler, 1st September 2008
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• Physical parameters Cyclone-like – except for profile gradients
• Numerical parameters: perpendicular box size:
64 x 64 ion gyroradii perpendicular resolution:
1.5 x 3 electron gyroradii
• Unconventional way of plotting transport spectra conventional plot
‘area preserving’ plot
Simulation parameters and plotting conventions
64 ρi
64 ρ
i
T. Görler, 1st September 2008
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A.) R/LTi = 6.9, R/LTe
= 6.9, R/Ln = 2.2B.) R/LTi
= 5.5, R/LTe = 6.9, R/Ln = 0.0
C.) R/LTi = 0.0, R/LTe
= 6.9, R/Ln = 0.0
T. Görler, 1st September 2008
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Dominant and subdominant modes for case A
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Case A: R/LTi=6.9,R/LTe
= 6.9, R/Ln = 2.2 – strong ITG
T. Görler, 1st September 2008
ions
electrons
• ion heat transport– localized at low-k– unrealistically high:
• Electron heat transport– ETG nonlinearly present – ‘high-k’ contribution to
electron heat transport ~10%
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Case A: R/LTi=6.9,R/LTe
= 6.9, R/Ln = 2.2 – strong ITG
T. Görler, 1st September 2008
small-scale streamers are subject
to large-scale vortex shearing
Isotropic spectrum
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A.) R/LTi = 6.9, R/LTe
= 6.9, R/Ln = 2.2
B.) R/LTi = 5.5, R/LTe
= 6.9, R/Ln = 0.0C.) R/LTi
= 0.0, R/LTe = 6.9, R/Ln = 0.0
T. Görler, 1st September 2008
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ions
• ion heat transport still dominat but reduced by one order of magnitude
• more than 40% of the electron heat transport is in the ‘high-k’ regime!
Scale separation between ion and electron heat transport
electrons
T. Görler, 1st September 2008
Case B: R/LTi=5.5,R/LTe
= 6.9, R/Ln = 0.0 – weak ITG
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T. Görler, 1st September 2008
Case B: R/LTi=5.5,R/LTe
= 6.9, R/Ln = 0.0 – weak ITG
Thermal fluxes (2D):
Particle flux (2D):
ions electrons
Particle flux
• similar to case A (not shown)
• negative (pinch)
• restricted to low-k due to increasing ion “adiabaticity” at higher wavenumbers
→ high-k transport
anisotropy which has been absent in case A
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A.) R/LTi = 6.9, R/LTe
= 6.9, R/Ln = 2.2
B.) R/LTi = 5.5, R/LTe
= 6.9, R/Ln = 0.0
C.) R/LTi = 0.0, R/LTe
= 6.9, R/Ln = 0.0
T. Görler, 1st September 2008
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T. Görler, 1st September 2008
Case C: R/LTi=0.0,R/LTe
= 6.9, R/Ln = 0.0 – no ITG
• Only TEM and ETG modes unstable (dominant electron heating, high beta values, substantial equilibrium ExB shear, ITBs)
• More than 50% of the electron heat transport is in the ‘high-k’ regime!
• ETG transport level( ) is in line with pure ETG simulations
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ETG streamers and
TEM vortices coexist
Anisotropic spectrum
TEM streamer
ETG streamer
T. Görler, 1st September 2008
Case C: R/LTi=0.0,R/LTe
= 6.9, R/Ln = 0.0 – no ITG
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T. Görler, 1st September 2008
Case C: R/LTi=0.0,R/LTe
= 6.9, R/Ln = 0.0 – no ITG
Thermal fluxes (2D):
Particle flux (2D):
• small & mainly positive
• still restricted to low-k
• Ion heat flux negligible
• Transport fraction:
– kyρs >1: ~50%
– kxρs >1: ~30%
→ high-k transport anisotropic
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Comparison: Density spectra ((x,z,t) average)
T. Görler, 1st September 2008
(Linearly) Pure ITG, TEM and ETG
simulation resultsMultiscale simulation results
• A second ‘knee’ is observed in multiscale simulations
• Power laws in pure and mixed turbulence sims: <|ne|>2~ky-α with α~3.5
• Disagreement with experimental results (α~6)?
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Comparison: Density spectra (kx=0, (z,t) avg.)
T. Görler, 1st September 2008
Some diagnostics detect e.g. kx≈0 contribution; asymmetry important!
Exp. Results (P.Hennequin et al., PPCF 46, B121)
• Power laws steeper at kx=0, closer to experiment
• Similarities between blue (TEM/ETG) curve and exp. results
• High power law exponent (case C: α~5) does not imply negligible transport contribution
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Comparison: Frequency spectra
T. Görler, 1st September 2008
Case A Case B
Case C • Nonlinear and linear frequencies match over a wide range except for transition regime from dominant ITG to TEM/ETG
• Phase velocity on the order ofvph < 5 csρs/R~
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Linear and nonlinear cross phases
R/LTi = 0.0 R/LTi = 5.5 R/LTi = 6.9
Linear
Nonlinear
nonlinear spread of
ITG features
T. Görler, 1st September 2008
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• If ETG modes are unstable,– there tends to be a scale separation between ion and electron
heat transport (the latter can exhibit substantial or even dominant high-k contributions) [T. Görler and F. Jenko, PRL 100, 185002 (2008)]
• discharges with dominant electron heating, high beta, large equilibrium ExB shear
• residual electron heat fluxes in transport barriers
– density spectra tend to be anisotropic at higher k and may exhibit a flat region or modified power laws at kyρe~0.15-0.25 (kyρs~9-15 for D plasmas)
• Linear features (like cross phases or frequencies) tend to survive in the nonlinear simulations; deviations most pronounced in mode-transitional regimes
Summary
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Streamers• Streamers: Radially extended eddies
– kr=0, n=m=0, or kr << kq
– Increased correlation length can lead to increased transport
• Origin– remnants of linear modes (Jenko,
Dorland,…)– Radial streamers, which represent ExB
velocity fields, are generated by nonlinear toroidal coupling (Lin).
– Nonlinear driven convective cells
(Champeaux, Diamond, …)
• Stability– Typical scale of streamers?
• ITG turbulence acting like Zonal flow for
ETG to break up the radial streamers and reduce/suppress the ETG transport (Candy &Waltz) Dorland et al., Phys. Rev. Lett. 85, 5579 (2000).
Lin et al, PoP 2005
c =d2/t
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Differences between ITG and ETG
• ITG– Electrons are Boltzmann due
to k||v||>>w
• ETG– Ions are Boltzmann due to
k┴ri>>1
• ITG turbulence acting like Zonal flow for ETG to break up the radial streamers and reduce/suppress the ETG transport.
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Problems
• Using mixing length model, derive ETG thermal transport scaling and compare with simulation results in Dorland et al., Phys. Rev. Lett. 85, 5579 (2000).