# Presentation: 617228

## Gyrokinetic Theory and Simulation of Experiments

**Gyrokinetic Theory and Simulation of Experiments**

**G. W. Hammett**
**Princeton Plasma Physics Lab (PPPL)**
- _[http://w3.pppl.gov/~hammett](http://w3.pppl.gov/~hammett)_
**APS Div. Plasma Physics**
**Orlando, 11/12/2007**

- Candy, Waltz (General Atomics)

## Gyrokinetic Theory & Simulation of Experiments

- Intuitive pictures of gyrokinetic turbulence, & how to reduce it
- analogy with inverted pendulum / Rayleigh-Taylor instability
- reducing turbulence with sheared flows, magnetic shear, plasma shaping   advanced tokamak & advanced stellarator designs
- Development of & physics in gyrokinetic equations
- Development of nonlinear 5-D simulations of gyrokinetic turbulence
- Gyrokinetic simulations:  physics studies & comparisons w/ expts.
- Future challenges & opportunities
- more detailed comparisons w/ expts incl. fluctuation diagnostics
- directly couple turbulence simulations & long-time transport codes
- Edge Turbulence, ELMs, transport barriers

**Notes:**
Somewhere mention:  In this review talk I am primarily discussing other people’s work who deserve the credit for the development of gyrokinetic codes.  These are quite impressive as they are 5-Dimensional, Nonlinear, gyrokinetic Turbulence simulations.  My own background is as a user of gyrokinetic equations and codes, and a developer of gyrofluid codes based on gyrokinetics in the 1990’s, but it is other people who have developed fully gyrokinetic codes over the past 9 years.  (more below)

(later:) Mention turbulence simulations looked impossible in 1985.  Mention I interviewed with Ron Waltz for a job in 1985, and he said he was working on developing plasma turbulence simulations, and I thought he was crazy.   Show 1990 plot of theory/expt. chi. vs r showing factor of 1000 discrepancies.

## Gyrokinetic Invited & Contributed Talks at this meeting:

- Invited talks borrowed from in this talk:
**J E Kinsey, "First transport code simulations using the TGLF model", BI2.6, Monday, 12 noon **
**Anne White, "Electron temperature fluctuations in the core of high-performance DIII-D plasmas", NI1.2, Wednesday, 10 AM**
**G G Howes, "Turbulence in the solar wind: Theory, simulations and comparisons with observations", VI2.2 , Thursday 3:30 PM**
- More invited talks:
- David Mikkelsen, "A quantitative account of electron energy transport in an NSTX plasma", NI1.4, Wednesday, 11 AM
- Barrett Rogers, "Gyrokinetic simulations of plasma turbulence, transport and zonal flows in a closed field line geometry", Tuesday, 11 AM
- Z Lin, "Turbulent transport via wave-particle decorrelation in collisionless plasmas", CI1.3, Monday 3 PM
- J Lang, "Gyrokinetic delta f particle simulation of trapped electron mode driven turbulence", NI1.3, Wednesday 10:30 AM
- TS Hahm, "Turbulent equipartition theory of toroidal momentum pinch", YI1.6, Friday noon.
- Contributed Oral:
- Florian Merz, "Plasma microturbulence with dual drive", NO.00006, Wednesday 10:30 AM
- Ben McMillan, "Noise control in global gyrokinetic particle simulations", NO3.00010, Wednesday, 9:30 AM
- R V Budny, "Gyrokinetic simulations of electron density fluctuations and comparisons with measurement", NO3.00011, Wednesday 11:30 AM
- Yong Xiao, "Gyrokinetic simulation of trapped electron mode turbulence", NO3.00012, Wednesday 11:42 AM
- M Greenwald, "Particle transport and density peaking at low collisionality on Alcator C-Mod", PO3.00005, Wednesday, 2:48 PM
- Frank Jenko, "Decoupling of ion and electron heat transport via scale separation", TO3.00001, Thursday 9:30 AM
- Florian Merz, "Gyrokinetic turbulence simulations for stellarators", TO3.00002, Thursday 9:42 AM

## Motivation & Summary

## Fusion performance depends sensitively on confinement

- Normalized Confinement Time HH = E/Empirical

**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, ...

- Q = Fusion Power / Heating Power

## ↓ turbulence & ↑  could significantly improve fusion

**↓** turbulence & **↑**  could significantly improve fusion

- Galambos, Perkins, Haney, & Mandrekas 1995 Nucl. Fus.

- (Relative Cost of Electricity (COE) estimates in this study, see ARIES reactor studies for more detailed & lower costs estimates.)

## 5-slide executive summary

## Intuitive pictures of gyrokinetic turbulence, & how to reduce it
analogy w/ inverted pendulum / Rayleigh-Taylor instability
reduce turbulence with sheared flows, magnetic shear, ...

- Intuitive pictures of gyrokinetic turbulence, & how to reduce it
- analogy w/ inverted pendulum / Rayleigh-Taylor instability
- reduce turbulence with sheared flows, magnetic shear, ...

- effective
- gravity

-     _**Inverted-density fluid**_
- Rayleigh-Taylor Instability

## 2.   Development of & physics in gyrokinetic equations

- if low frequencies *** <<* cyclotron frequency (***c*),   average over particle gyration, treat particles as rings of charge in spatially varying fields

- potential averaged
- around particle orbit,
- even if *k******i* large

- When calculating charge at point Q,
- have to sum over all particles whose
- guiding centers are on the dashed line,
- & have to include small variation of particle density around gyro-orbit ( polarization shielding)

- Development of nonlinear gyrokinetics
- was a major breakthrough

## Fairly Comprehensive 5-D Gyrokinetic Turbulence Codes Have Been Developed

- Solve for the particle distribution function *f(r,****,****,E,****,t)* (avg. over gyration: 6D  5D)
- 500 radii x 32 complex toroidal modes (96 binormal grid points) x 10 parallel points along half-orbitsx 8 energies x 16 v||/v12 hours on ORNL Cray X1E with 256 MSPs
- Realistic toroidal geometry, kinetic ions & electrons, finite- electro-magnetic fluctuations, collisions.  Sophisticated algorithms.

## 4. Gyrokinetic sims.: physics studies & comparisons w/ expts.

- Simulations often agree with core region of experiments within error bars on grad(T)

- Candy & Waltz, PRL 2003, Waltz et al. 2005

- TGLF transport model, based on gyrokinetic simulations, fits core of wide range of experiments

**J. **Kinsey  BI2.6, Monday 12:00 Noon

## 5.  Future challenges & opportunities:
more detailed comparisons w/ expts incl. synthetic fluctuation diagnostics
coupling turbulence simulations directly in long-time transport codes
Edge Turbulence, very challenging but critical problem
Edge important:  core depends on edge, ELMs, transport barriers 
present core codes don’t handle edge, need X-point separatrix, open & closed field lines, strong recycling, wide range of collisionality, ...

- 5.  Future challenges & opportunities:
- more detailed comparisons w/ expts incl. synthetic fluctuation diagnostics
- coupling turbulence simulations directly in long-time transport codes
- Edge Turbulence, very challenging but critical problem
- Edge important:  core depends on edge, ELMs, transport barriers
- present core codes don’t handle edge, need X-point separatrix, open & closed field lines, strong recycling, wide range of collisionality, ...

- From Kinsey, Staebler, Waltz, Sherwood 2002.
- Predictions for 2001 ITER-FEAT.

## (many of these insights developed with gyrofluid simulations in 1990’s, but gyrokinetics needed for better accuracy.)

**Intuitive pictures of gyrokinetic turbulence, & how to reduce it**

## Unstable Inverted Pendulum

_**Unstable Inverted Pendulum**_

- = (-g/|L|)1/2 = i(g/|L|)1/2 = i

- (rigid rod)

_**Density-stratified Fluid**_

- stable=(g/L)1/2

- =exp(-y/L)

- Max growth rate=(g/L)1/2

- =exp(y/L)

-     _**Inverted-density fluid**_
- Rayleigh-Taylor Instability

- Instability

## “Bad Curvature” instability in plasmas  Inverted Pendulum / Rayleigh-Taylor Instability

- Top view of toroidal plasma:

- plasma = heavy fluid

***B*** = “light fluid”

- geff =      centrifugal force

*R*

- Growth rate:

- Similar instability mechanism
- in MHD & drift/microinstabilities

- 1/L = p/p in MHD,
-   combination of n & T
- in microinstabilities.

## The Secret for Stabilizing Bad-Curvature Instabilities

- Twist in ***B*** carries plasma from bad curvature region
- to good curvature region:

- Unstable

- Stable

- Similar to how twirling a honey dipper can prevent honey from dripping.

## Spherical Torus has improved confinement and pressure limits (but less room in center for coils)

## These physical mechanisms can be seen

- in gyrokinetic simulations and movies

- Unstable bad-curvature
- side, eddies point out, direction of effective gravity

- particles quickly move along field lines, so density perturbations are very extended along fields lines, which twist to connect unstable to stable side

- Stable
- side, smaller eddies

## Sheared

- flows

## Slide 20

## Slide 21

## Rosenbluth-Longmire picture

## Rosenbluth-Longmire picture

- Can repeat this analysis on the good
- curvature side & find it is stable.
- (Leave as exercise.)

## Simple picture of reducing turbulence by negative magnetic shear

- Particles that produce an eddy tend to follow field lines.
- Reversed magnetic shear twists eddy in a short distance to point in the ``good curvature direction''.
- Locally reversed magnetic shear naturally produced by squeezing magnetic fields at high plasma pressure: ``Second stability'' Advanced Tokamak or Spherical Torus.
- Shaping the plasma (elongation and triangularity) can also change local shear

- Fig. from Antonsen, Drake, Guzdar et al. Phys. Plasmas 96
- Kessel, Manickam, Rewoldt, Tang Phys. Rev. Lett. 94

- (in std tokamaks)

- “Normal” in stellarators

- Advanced Tokamaks

##  Development of & physics in Gyrokinetic Eqs.

- Development of gyrokinetic equations one of the triumphs of high-power theoretical plasma physics and asymptotic analysis
- Interesting pre-history and history of gyrokinetics...
- Key advance:  Frieman & Chen show nonlinear version of gyrokinetics possible
- Other advances:  Hamiltonian/Lagrangian derivations, insure conservation properties, easier to go to higher order

## Gyrokinetic Prehistory:

## Big Breakthrough: Nonlinear Gyrokinetics

- Long, interesting history of linear gyrokinetics, 1960’s, 1970’s.
- E. A. Frieman & L. Chen 79-82, showed it is possible to gyro-average nonlinear terms and keep full FLR-effects for arbitrary *k*****, & get rigorous solution w/o closure problem
- (usually, averaging nonlinear terms  closure problems, such as fluid equation closures, statistical turbulence theories,...Perhaps understood by some, or in restrospect:  J.B. Taylor ’67 demonstrated an adiabatic invariant still exists at arbitrary *k***** ...)
- GK ordering allows capture of drift/micro-instabilities & much of MHD at just order  & not 2

- Guided by expts., wave scattering, physics insights

## Slide 28

## Slide 29

## (borrowed  from B. D. Scott)

## alternative derivation:  next order correction to adiabatic invariant mu

## First Gyrokinetic PIC code

- Frieman & Chen had first derivation, but very complicated.
- Lee ’83 & ’87 derivations clearer, used Catto transformation to guiding center coordinates, & then asymptotic expansion.  Made clearer the role of the polarization density (higher order polarization drift can be dropped from gyrokinetic equation, but resulting polarization density contributes to the gyrokinetic Poisson equation (because even small charge densities lead to large forces in plasmas)).
- Lee made clearer that GK polarization density eliminates small Debye scale and high frequency plasma oscillations, making simulations much more tractable.  Demonstrates first GK PIC simulations (slab, electrostatic, 2-D on early 1980’s computers).

## Modern Lagrangian/Hamiltonian Lie-Perturbation methods

- Advantage:  *df/dt  = [H,f],* make approximations to Hamiltonian/Lagrangian, but preserve important Hamiltonian properties:  exact  conservation of an energy *H*, phase-space, symplectic etc., easier to extend to full f instead of breaking up *f=f**0**+f**1*, easier to extend to higher-order terms that may be important in some regimes (perhaps in edge turbulence where *f**1** << f**0* assumption weak), etc.
- Dubin, Krommes, Oberman, & Lee / borrowed from Littlejohn, Hamiltonian, slab, electrostatic
- Hahm:  Lagrangian approach better, extended to toroidal geometry & dB
- Brizard:  Lagrangian, extended to full dB  and dB|| , nonlinear properties
- Dimits & Lodestro generalization of ordering
- Sugama, others
- Brizard-Hahm RMP 2007
- Qin:  To ensure total energy conservation exactly, use variational field theory for full system of particles & fields, with a particle Lagrangian & a field Lagrangian.  Linear benchmarks with PEST MHD code, including kink mode.  Higher-order extensions that may be useful near edge.  Extensions to general frequency for RF resonant heating, etc.

## 3.  Development of Nonlinear 5-D Simulations of Gyrokinetic Turbulence

- The development of comprehensive gyrokinetic codes is one of the triumphs of computational/theoretical plasma physics (& of the modern explosion of computer power)

## Main Comprehensive Gyrokinetic Codes(Fully electromagnetic, being widely compared with expts.) 

- GS2 (Dorland & Kotschenreuther)  continuum, flux-tube
- GENE (Jenko, Garching) continuum, flux-tube
- GYRO (Candy & Waltz) continuum, global
- GEM  (Parker and Chen) F PIC, global

- All of these codes include: toroidal geometry, general axisymmetric plasma shapes (GS2 & GENE do stellarators), multiple species, trapped and passing non-adiabatic electrons, electromagnetic fluctuations, collision operators, equilibrium scale ExB shear flow,  GS2 & GENE use the *0 limit local flux-tube (equivalent to thin annulus).
- GYRO & GEM use  global/non-local thick annulus, can include radial shear of ***** profiles, turbulence spreading, etc. that can break gyro-Bohm scaling.
- These gyrokinetic codes use a number of advanced algorithms.

## Other Gyrokinetic Codes (1)

- Dimits PG3EQ:  First toroidal gyrokinetic code in high-resolution flux tube limit including zonal flows (electrostatic w/ adiabatic electrons).  Benchmark standard.  Discovered Dimits Nonlinear Shift in the temperature gradient threshold, at low *q* have to go somewhat beyond the linear stability point for significant turbulence
- Global F PIC codes (electrostatic, or fluid electron hybrid):
- Rick Sydora, LeBoeuf (UCLA, U. Alberta)
- S. E. Parker and Y. Chen (U.Col.)
- GTC (Z. Lin, U.C. I.)
- GTS (Weixing Wang PPPL), incl. plasma shaping, turbulence+neo
- M3D with gyrokinetic beam/thermal ions
- Edge/Global gyrokinetic codes (full f) in US:
- TEMPEST (Xu, LLNL)  continuum
- ESL Edge Simulation Lab (Xu, R. Cohen, Colella) continuum full f
- CPES Center for Plasma Edge Simulation (C.S. Chang) XGC edge PIC code

## Other Gyrokinetic Codes (2)

- ORB5 (T-M Tran, A. Bottino, S. Jolliet, Lausanne, Garching) delta-f PIC, global, adiabatic electrons
- GYSELA (V. Grandgirard, X. Garbet, Y. Sarazin, ..., Cadarache) semi-Lagrangian, global, adiabatic electrons
- FEFI, -FEFI (B.D. Scott, Garching) continuum, edge code, exploring algorithms
- ELMFIRE (J. Heikkinen, Finland) full F PIC, global (edge focus)
- G3D (Idomura, Japan) delta-f PIC, global
- G5D (Idomura, Japan) continuum, global (new, under development)
- GKV (Watanabe & Sugama, Japan) continuum, local
- gyrofluid codes based on moments of gyrokinetic equations, & fluid turbulence codes:  give a lot of useful insight into turbulence, recent versions improve comparison with gyrokinetics (Bruce Scott, Beer-Hammett-Dorland-Waltz, Klauss Hallatschek, Rogers & Drake, Xu...)

## Major Theoretical & Algorithmic Speedups

- Nonlinear gyrokinetic equation
- ion polarization shielding eliminates plasma freq. pe/ci ~ mi/me		x103
- ion polarization eliminates e & Debye scales   (i/e)3			x105
- average over fast ion gyration, ci / * ~ 1/*				x103
- Continuum or f PIC, reduces noise, (f0/f)2 ~ 1/*2 			x106
- Field-aligned coordinates (nonlinear extension of ballooning coord.)
- 			***||** / (****** q R / a)  ~ a / (q R *******)*				x70
- Flux-tube / Toroidal annulus wedge,  simulation volume
- ki = 0, 0.05, 0.1, ..., 1.0   n = 0,     15,  30, ...,  300  (i.e., 1/15 of toroidal direction)		x15
- Lr ~ a/5 ~ 140  ~ 10 correlation lengths				x5
- High-order / spectral algorithms in 5-D, 25 x 2				x64
- Implicit electrons							x5-50
- Total combined speedup of all algorithms				x1023
- Massively parallel computers (Moore’s law 1982-2007)		x105

- relative to simplest brute force, fully resolved, algorithm, for ITER 1/* = a/ ~ 700

## Major Theory/Algorithm Advances

-  f PIC, reduces PIC noise(Kotschenruether, Dimits, Parker, Hu & Krommes, Aydemir)	*f(x,v,t)* = smooth *f**0**(v)* + weighted particles ***f*
- Field-aligned coordinates, local flux-tube / toroidal annulus wedge  (Cowley, Beer, Hammett, Dimits)  (similar to shearing box simulations of accretion turbulence in astro)
- General toroidal annulus wedge (Waltz, Candy, Rosenbluth):

- Top view:

- Side view:

## Good convergence in velocity space

- Candy & Waltz, Phys. Plasmas 13, 032310 (2006)

- (for comparison typical PIC code uses 10’s of particles/cell)

- suppressed 0

- Std: 8 energies X
- 16 pitch angles

- Gauss-Legendre integration
- w/ 8 energies exact for Maxwellian X 8th order polynomial, super-exponential convergence.

## 5-D continuum codes show excellent scaling to large # of processors

- Candy, Dimits, Fahey, Mikkelsen

- Various US Supercomputers including the ORNL Cray XT4 (jaguar), X1E (phoenix) and LLNL IBM BlueGene/L (bgl).

- Strong scaling (fixed problem size) for GS2 & GYRO demon-strated over a factor of 1000 processors.

## Y. Chen & S. Parker breakthrough: working GK PIC algorithm for magnetic fluctuations 

- Long standing “high-” problem in GK PIC codes, first observed by Cummings (’94).  Chen & Parker JCP 03 fixed problem by careful treatment of two large terms (arising from canonical momentum transformation) that nearly cancel

- Y. Chen & S. Parker breakthrough: working GK PIC algorithm for magnetic fluctuations

- Benchmark 3 independent
- PIC & continuum codes

- theory

## Global code approaches local flux-tube limit as *  0

- Candy et al.  PoP 04

## Moderate amount of turbulence spreading occurs in some cases

- Waltz, Candy, Petty 2006 PoP 13, 072304

- DIII-D L-mode

- amount of spreading:   ~ 0.1 a
-    ~ 2-5 radial correlation lengths
-    ~ 20-50 ***i*
- see also Hahm et al. 2004, Lin et al. 2002, Garbet et al., Newman, Xu

## Successful Benchmarks of Independent Gyrokinetic Codes

- Good agreement in ** (+/- 10% on long time average *t > 1000 a/c**s*) between 3 continuum and 2 PIC codes

- Nevins et al.  2007

- Correlation functions agree well

## Continuum & PIC Gyrokinetic codes(Eulerian & Lagrangian/Monte-Carlo)

- Very useful to have both types of codes:  very different numerical properties,
- Very useful independent cross-check on each other
- “Continuum” codes use a range of advanced algorithms: pseudo-spectral,
- high-order upwind, conservative Arakawa methods, finite-element, etc.,
- not just simple grid.

## 4. Gyrokinetic Simulations:  Physics studies & comparisons with expts

## 2002: early detailed comparisons of gyrokinetic simulation and DIII-D experiment

- GS2 gyrokinetic simulation
- of fluctuation spectrum
- (Ross & Dorland, PoP 2002)

- Qualitatively similar to shape of measured spectrum
- (Ross, Bravenec, Dorland, et al. 2002)  (BES measurements by McKee, Fonck, et al.)

- Absolute amplitude of simulation fluctuations too large,
- but see next slide.
- Instrumental viewing volume contributes to roll off at high k,
- working on synthetic diagnostics to better account for instrumental effects

## Simulation gives heat flux 2x experiment at r/a=0.7
25%  T  outside error bars if applied everywhere, but may be within error bars since need to T for r/a < 0.55

- Simulation gives heat flux 2x experiment at r/a=0.7
*25% * ***T*  outside error bars if applied everywhere, but may be within error bars since need to ***T* for r/a < 0.55

- Ross, Dorland 2002: detailed comparisons of gyrokinetic simulation and DIII-D L-mode measurements

- ExB
- shear
- rate

- Growth
- rate

- Need to
- ***T*  here

- Need to
- ***T*  here

## Comparison of GYRO Code & Experiment

- Gyrokinetic turbulence codes now including enough physics (realistic geometry, sheared flows, magnetic fluctuations, trapped electrons, fully electromagnetic fluctuations) to explain observed trends in thermal conductivity, in many regimes.
- Big improvement over 15 years ago, when there were x10 – x100 disagreements between various analytic estimates of turbulence & expts.
- Now within experimental error on temperature gradient.  Importance of critical gradient effects emphasized in 1995 gyrofluid-based IFS-PPPL transport model.
- Caveats:  Remaining challenges: quantitative predictions of internal transport barriers, test wider range of parameters, & more complicated edge turbulence.

- Candy & Waltz, PRL03, Waltz Nucl. Fus. 05

- (-10%)

## ITG often within 5% of threshold in core

- Waltz, Candy, Petty 2006 PoP 13, 072304

- DIII-D H-mode

## 1980’s analytic turbulence theories had large disagreements (x10-1000) with experiments

- 1980’s analytic turbulence theories had large disagreements (x10-1000) with experiments

- Very smart people, but very hard problem
- Recent gyrokinetic simulations (and models based on them) now compare much better with experiments.  We’ve made a lot of progress.
- This plot made in 1990.  We and many theories didn’t appreciate at that time the importance of getting thresholds for marginal stability accurate, ... Much discussion about marginal stability in LM & SS, but pellet experiments apparently drive i > icrit (slab theory) without changing transport.  Proposed at the time: may not have been beyond marginal stability for toroidal modes (Rewoldt & Tang, 1990, Horton et al. 1992)
- see also S.D. Scott et al., Phys. Fluids B 1990

- Measured

## R/LTe dependence for ‘large’ density gradients

- Conventional (quasi-)linear models:
- 	no critical gradient (density gradient drive)
- Nonlinear simulations and new quasilinear model:
- 	effective critical gradient
- 	_electron heat flux_ has _offset-linear_ scaling

- R/LTe dependence for ‘large’ density gradients

-  similar as in adiabatic ITG case
-  implies Te profile stiffness
-  coupling of particle and electron heat flux

- Jenko, Angioni, et al. IAEA06

- http://www-pub.iaea.org/MTCD/Meetings/FEC2006Presentations/Presentations/20-oct-06/talk_EX8-5Ra.ppt

## Existence of a threshold in R/LTe

- AUG L-mode plasmas
- 	[0.8 MW ECRH, little OH)
- gradual reduction of central ECRH, balanced by increase
- 	of off-axis heating

- Existence of a threshold in R/LTe

- [F. Ryter et al., PRL 2005]

- ETG stable

- Threshold behavior is observed directly; _power balance_ and _transient_
- _transport_ consistent with both _linear gyrokinetics_ and _CG model_.

- Jenko, Angioni, et al. IAEA06

- http://www-pub.iaea.org/MTCD/Meetings/FEC2006Presentations/Presentations/20-oct-06/talk_EX8-5Ra.ppt

## Waltz, Austin, Burrell, Candy, PoP 2006

## Temperature corrugation and large sheared zonal flow

- Temperature corrugation and large sheared zonal flow
- occur near low-order rational surface

- Waltz, Austin, Burrell, Candy, PoP 2006

## Full-physics GYRO simulation of Negative Central Shear DIII-D case

- Full-physics GYRO simulation of Negative Central Shear DIII-D case

- DIII-D Shot 12717 w/ negative central shear, *q**min **= 1.925* (later time than cases in paper),  *q=2 @ r/a=0.2 & 0.54,  *******/a=0.003*
- experimental *grad(T)* used, but reduced ExB shearing rate by 20% to get finite turbulence
- 500 radii x 32 complex toroidal modes (96 binormal grid points)  x 10 parallel points along half-orbits x 8 energies x 16 v||/v, 12 hours on ORNL Cray X1E with 256 MSPs

- Waltz, Austin, Burrell, Candy, PoP 2006

- Movie of density fluctuations from GYRO simulation _[http://fusion.gat.com/THEORY/images/0/0f/N32o6d0.8.mpg](http://fusion.gat.com/THEORY/images/0/0f/N32o6d0.8.mpg)_ from _[http://](http://fusion.gat.com/theory/Gyromovies)__[fusion.gat.com/theory/Gyromovies](http://fusion.gat.com/theory/Gyromovies)_

## Direct observation of TEM turbulence: Nonlinear GS2 simulations, with synthetic diagnostic, reproduce wavelength spectrum from phase contrast imaging in Alcator C-Mod ITB [Ernst-IAEA06].

*Direct observation of TEM turbulence: Nonlinear GS2 simulations, with synthetic diagnostic, reproduce wavelength spectrum from phase contrast imaging in Alcator C-Mod ITB [*_*Ernst-IAEA*_*06].*

*Synthetic phase contrast imaging diagnostic** in GS2 [*_*Ernst, IAEA06*_*].*

## Budny, Kramer, Mazzucatto, JET, et al., NO3.11, Wed. 11:30 am

## Anne White’s invited talk NI1.2, Wed. 9:30 AM 

- Anne White’s invited talk NI1.2, Wed. 9:30 AM

## Holland, White, GYRO synthetic diagnostics (UCSD, UCLA, General Atomics)

## Holland, White, GYRO synthetic diagnostics

## Largest GYRO simulations used to studyinteraction of ITG & ETG Turbulence

- Largest GYRO simulations used to studyinteraction of ITG & ETG Turbulence

- 1280 e x 1280 e 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/L**Ti**=0*

- ETG+ITG  *R/L**Ti**=6.9*

## ETG + kinetic ion GYRO simulation movie

- large box on right: full simulation domain, 1280 ***e* x 1280 ***e* = 64 i x 64 i
- small box on lower left:  zoom in on a 64 ***e* x 64 ***e* patch

- _[http://fusion.gat.com/THEORY/images/1/1f/ETG-ki.mpg](http://fusion.gat.com/THEORY/images/1/1f/ETG-ki.mpg)_ from http://fusion.gat.com/theory/Gyromovies

- Candy & Waltz

## ETG fluctuations (ki > 1) may account for significant fraction of transport in some plasmas

- ETG fluctuations (*k******i **> 1*) may account for significant fraction of transport in some plasmas

- Simple scaling from ITG to ETG:
-      ***itg**  ~ C**itg** ****i**2**  v**ti**/L*
***etg** ~ C**itg** ****e**2** v**te**/L ~ ****itg** /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.  TGLF transport model shows ETG / high-k TEM may still be important in some cases.

- DIII-D H-mode

**50%**

**J. Kinsey  BI2.6, Monday 12:00 Noon**

*** **(m**2**/s)*

## 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, NSTX

- Radial domain ~400 e.

## database of 400+ GYRO simulations available

can be used to test & fit theories

Used to develop improved transport model, TGLF, which fits experiments better than GLF23 over a wider range of parameters

- can be used to test & fit theories
- Used to develop improved transport model, TGLF, which fits experiments better than GLF23 over a wider range of parameters

## TGLF exhibits lower average global errors than GLF23 for a large L- and H-mode profile database of 96 discharges  

**TGLF exhibits lower average global errors than GLF23 for a ****large L- and H-mode profile database of 96 discharges  **

**Database: 25 DIII-D L-,33 DIII-D H-, 22 JET H-, 16 TFTR L-mode discharges **
**Avg RMS errors in W****inc**** is 19% for TGLF, 36% for GLF23**
**Avg RMS error in W****tot ****is ********R****Wtot****=10% for TGLF, 20% for GLF23**

- TGLF Model

- GLF23 Model

**J. Kinsey  BI2.6, Monday 12:00 Noon**

- Wider range of
- conditions than earlier
- GLF23 comparisons

## Gyrokinetic codes applied to astro/solar physics:

- Transition from MHD to Kinetic Alfven Wave Cascade

- AstroGK/GS2 simulation
- • Consistent with theory:
-    MHD: -5/3 spectra in E & B
-    KAW:  -1/3 E spectrum and
-               -7/3 B spectrum
- •See talk by Greg Howes for more:
-   Session VI2:  Thursday  3:30pm
-    Turbulence in the Solar Wind: Theory,
-         Simulations, and Comparisons with
-         Observations

- More than 500,000,000 computational meshpoints

- Howes, Dorland, Quataert, Cowley, et al.

## 5. Future challenges & opportunities

- more detailed comparisons w/ expts incl. synthetic fluctuation diagnostics
- move from flux prediction to profile prediction mode:
- makes experimental comparisons easier, more direct
- a step to  coupling short-time turbulence simulations & long-time transport codes
- transport code coupling: study ITB formation, heat/cold pulse perturbative studies
- Many multiscale problems here, incl. Neoclassical Tearing Mode interaction with turbulence
- Most important & difficult problem:  **Edge Turbulence**, ELMs, H-mode transport barrier

## Areas of possible improvements for core gyrokinetic codes

- Dominant terms that can break gyro-Bohm scaling have been included (shear in profiles, turbulence spreading)
- However, there are some small ***** terms and small *k**||**/k*** terms that have been dropped for convenience.  Could be put in.
- Collision operators simplified to various degrees, improve
- Can’t handle separatrix, not efficient for high collisionality regimes (like edge), and so need new edge gyrokinetic codes...

## Fusion performance depends sensitively on Edge

- 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, ...

- From Kinsey, Staebler, Waltz, Sherwood 2002.
- Predictions for 2001 ITER-FEAT.

- Lithium may help  edge T

## Edge boundary layer very important & uncertain

- Edge pedestal** **temperature  profile near the edge of an H-mode discharge in the DIII-D tokamak. [Porter*2000*]*. *Pedestal is shaded region.

- Schematic views of divertor tokamak and edge-plasma region (magnetic separatrix is the red line and the black boundaries indicate the shape of magnetic flux surfaces)

- R (cm)

- Temp. (eV)

- 0

- 500

- 1000

-  Marginal stability: Tcore  Tped
-  Periodic instabilites in edge region
- can dump outer ~5-10% of plasma onto
- divertor plates.  Might be manageable, or
- divertor erodes, melts?

## Beginning Work: Edge Gyrokinetic Turbulence

- Crucial:  Need large H-mode pedestal & small ELMs,  fusion *Q* depends on *T**ped*
- Complicated:
- Character of edge turbulence different: not ITG/TEM but nonlinear / drift resistive ballooning, strong non-adiabatic electrons, significant magnetic fluctuation...
- Open & closed field lines, X-point, H-mode forms near separatrix
- Strong sources & sinks, neutral recycling, radiation, particle fuelling, Debye sheath boundary conditions
- Large variation in density and & temperature over scale of simulation
- need algorithms that can handle high and low collisionality regimes
- Not a large separation of equilibrium and fluctuation scales, need accurate conservative full-F code
- (edge fluid work: B.D. Scott, Rogers & Drake, Hallatschek, Xu)
- Two initial gyrokinetic efforts:
- ESL/TEMPEST, continuum approach (R. Cohen, LLNL, LBNL, GA, PPPL, ...)
- CPES, PIC approach (C.S. Chang, NYU, Colorado, PPPL, ...)

## Other Unfinished Gyrokinetic Work:

- Sometimes GLF23/TGLF transport models predict too little transport near the magnetic axis.  Something missing?  Turbulence spreading?  Microtearing modes?  ETG?
- While gyrokinetic simulations & transport models often predict temperature profiles within ~10% experimental uncertainties, there are some outliers which need further study.  This relatively good accuracy is in part a consequence of stiff transport with critical gradients, which makes the prediction of temperature profiles less sensitive to uncertainties in turbulence saturation levels, but which can also make it more difficult to quantify when other transport mechanisms (like ETG, microtearing, gyro-Bohm breaking effects, or turbulence spreading) might be playing some role.
- Turbulent transport in ST’s:  Long-wavelength ITG/TEM stabilized .  Microtearing modes?, ETG?, ?, nonlinear saturation?
- Transport barrier formation, transition threshholds, etc.
- Interaction of low-n MHD (NTM) and high-n turbulence...
- Gyrokinetics in stellarators, alternate concepts like RFPs, ...

## Gyrokinetic Theory & Simulation of Experiments

- Intuitive pictures of gyrokinetic turbulence, & how to reduce it
- analogy with inverted pendulum / Rayleigh-Taylor instability
- Development of & physics in gyrokinetic equations
- Development of nonlinear 5-D simulations of gyrokinetic turbulence
- Gyrokinetic simulations:  physics studies & comparisons w/ expts.
- Future challenges & opportunities
- more detailed comparisons w/ expts incl. fluctuation diagnostics
- coupling fast-time turbulence simulations & long-time transport codes
- Edge Turbulence, ELMs, transport barriers

## Selected Gyrokinetic References

- This talk available at  _[w3.pppl.gov/~hammett/talks](http://w3.pppl.gov/~hammett/talks)_
- 3 GYRO movies shown (d3d.n16.2x_06_fly, n32o6d0.8, & ETG-ki) from _[http://](http://fusion.gat.com/theory/Gyromovies)__[fusion.gat.com/theory/Gyromovies](http://fusion.gat.com/theory/Gyromovies)_
- Web sites for 4 main gyrokinetic codes discussed here (incl. refs., documentation):
- GYRO (Waltz & Candy, GA):  _[fusion.gat.com](http://fusion.gat.com/theory/Gyro)__[/theory/Gyro](http://fusion.gat.com/theory/Gyro)_
- GS2 (Dorland & Kotschenreuther, U. Maryland/Texas): _[gs2.sourceforge.net](http://gs2.sourceforge.net/)_
- GENE (Jenko, Garching):  _[www.ipp.mpg.de/~fsj](http://www.ipp.mpg.de/~fsj)_
- GEM (Parker & Chen, U. Colorado):  _[cips.colorado.edu/simulation/gem.htm](http://cips.colorado.edu/simulation/gem.htm)_
- “Anomalous Transport Scaling in the DIII-D Tokamak Matched by Supercomputer Simulation”, J. Candy & R. E. Waltz, Phys. Rev. Lett. 2003
- “Burning plasma projections using drift-wave transport models and scalings for the H-mode pedestal”, Kinsey et al., Nucl. Fusion 2003
- “Electron Temperature Gradient Turbulence”, W. Dorland, F. Jenko, M. Kotschenreuther, B.N. Rogers,  Phys. Rev. Lett. 2000
- “Generation & Stability of Zonal Flows in Ion-Temperature-Gradient Mode Turbulence”, Rogers, Dorland, Kotschenreuther, Phys. Rev. Lett. 2000
- "Comparisons and Physics Basis of Tokamak Transport Models and Turbulence Simulations",  Dimits et al., Phys. Plasmas 2000.
- “Simulations of turbulent transport with kinetic electrons and electromagnetic effects”,** **Y. Chen, S.E. Parker, B.I. Cohen, A.M. Dimits et al., Nucl. Fus. 43, 1121 (2003)

## Selected Gyrokinetic References (cont.)

- Brizard & Hahm, Reviews of  Modern Physics 2007
*“*_[A Short Introduction to General Gyrokinetic Theory](http://www.pppl.gov/~hongqin/QinPapers/Qin-Fields.pdf)_*”*, H. Qin,  in Fields Institute Communications **46**, Topics in Kinetic Theory, American Mathematical Society, 171 (2005).  see also _[http://](http://www.pppl.gov/~hongqin/QinPapers.php)__[www.pppl.gov/~hongqin/QinPapers.php](http://www.pppl.gov/~hongqin/QinPapers.php)_
*“*_[Geometric Gyrokinetic Theory for Edge Plasmas”](http://www.pppl.gov/~hongqin/QinPapers/QinGeometric%20Gyrokinetic%20Theory.pdf)__[,](http://www.pppl.gov/~hongqin/QinPapers/QinGeometric%20Gyrokinetic%20Theory.pdf)_ H. Qin, R. H. Cohen, W. M. Nevins, and X. Q. Xu, Physics of Plasmas **14**, 056110 (2007)
- “_[Theory and Computation in Full-F Gyrokinetics](http://www.ipp.mpg.de/~bds/presentations/sem0506.pdf)_” B. D. Scott, Princeton PPL Theory seminar, June 2005, and other useful presentations at _[http://](http://www.ipp.mpg.de/~bds/)__[www.ipp.mpg.de/~bds](http://www.ipp.mpg.de/~bds/)__[/](http://www.ipp.mpg.de/~bds/)_
- E. A. Frieman and L. Chen, Phys. Fluids **25**, 502 1982
- T. M. Antonsen and B. Lane, Phys. Fluids **23**, 1205 1980
- P. J. Catto, W. M. Tang, and D. E. Baldwin, Plasma Phys. **23**, 639 (1981)
- D. H. E. Dubin, J. A. Krommes, C. Oberman, & W. W. Lee, Phys. Fluids **26**, 3524 (1983)
- T. S. Hahm, Phys. Fluids **31**, 2670 (1988)
- A. Brizard, J. Plasma Phys. **41**, 541  (1989)
- A. M. Dimits, L. L. Lodestro, and D. H. E. Dubin, Phys. Fluids B **4**, 274 (1992)
- W.W. Lee, Phys. Fluids 26, 556 (1983)
- "Astrophysical Gyrokinetics: Basic Equations and Linear Theory," Gregory G. Howes, Steven C. Cowley, William Dorland, Gregory W. Hammett, Eliot Quataert, Alexander A. Schekochihin, Ap.J 651, 590 (2006), astro-ph/0511812

## Acknowledgements:

- J. Candy & R. E. Waltz (GA)  (GYRO results & movies)
- W. Dorland (Univ. Maryland), M. Kotschenreuther (U. Texas)
- F. Jenko (Garching)
- S. Cowley (UCLA)
- W. Nevins, B.I. Cohen, A.M. Dimits, R. Cohen (LLNL)
- S.E. Parker and Y. Chen (U. Colorado)
- B. D. Scott (Garching)
- Hong Qin (PPPL)
- T.S. Hahm, A. Brizard, W.W. Lee, W. Tang, J. Krommes, T. Stoltzfus-Dueck
- Chris Holland, Anne White, Jon Kinsey, Gary Staebler
- D. Ernst, D. Mikkelsen, R. Budny
- Center for Multiscale Plasma Dynamics
- DOE Scientific Discovery Through Advanced Computing (SciDAC)
- Center for the Study of Plasma Microturbulence
- Edge Simulation Laboratory
- Earlier DOE SciDAC & Computational Grand Challenge projects, including Plasma Microturbulence Project & Numerical Tokamak Project
- DOE National Energy Research Supercomputing Center (NERSC)
- Many others...