gyrokine tic simulations of tokamak microturbulencegs2.sourceforge.net/eps.pdfgyrokine tic...

Gyrokinetic Simulations ofTokamak Microturbulence

W Dorland, Imperial College, London

With key contributions from:

S C Cowley F Jenko G W HammettD Mikkelsen B N Rogers C BourdelleW M Nevins D W Ross K HallatschekR Budny E Belli M KotschenreutherJ W Connor E Quataert G D Kerbel

Gyrokinetics is Maturing Rapidly


Ë Mutually benchmarked, nonlinear codes exist



Ë Growing user base of experimentalists and theorists




Radial correlation functions

from three independentlydeveloped gyrokinetic codes,

run for identical physics

parameters.

W M Nevins, A Dimits, R E Waltz,J Candy, W Dorland




Ë Improving our understanding of experimental data




parameters.





Ë Improving our understanding of experimental data

Ë Providing guidance for theoretical advances




parameters.


First Principles Models Desirable

Ë Outliers valuable -- indicate missing physics



Ë Extrapolation less uncertain




GLF-23 model(Waltz, Staebler,Dorland, Konings, andKotschenreuther)

vs data from ITERprofile database

J Kinsey





vs data from ITERprofile database

J Kinsey

Ë Significant progress over the last nine years.





vs data from ITER

profile database

J Kinsey

Ë Worst outliers at high collisionality.

Nonlinear Physics BenchmarkedAgainst Theoretical Predictions

High b Alfvenic turbulence inhomogeneous, stirredplasma shows predictedperpendicular spectrum(and anisotropy, not shown).Here, b = 8 (i.e., 800%).

W Dorland, S C Cowley,G W Hammett and E Quataert

Nonlinear Physics BenchmarkedAgainst Theoretical Predictions

High b Alfvenic turbulence inhomogeneous, stirredplasma shows predictedperpendicular spectrum(and anisotropy, not shown).Here, b = 8 (i.e., 800%).

W Dorland, S C Cowley,G W Hammett and E Quataert

Need theory for inhomogeneous, unstable plasmas!

Parasitic Instability Model

Ë Equilibrium unstable to primary (linear) instabilitiesË Primaries unstable to secondary instabilities

-----------------------------------------------------------------------------------------------------

Ë Some secondary instabilities have zonal flow component

Ë Zonal flows unstable to tertiary instabilities

Key references: S C Cowley, R M Kulsrud, R Sudan, PF B, (3:2767:1991)J F Drake, et al., PF B, (4:488:1992)M N Rosenbluth, F Hinton, PRL (80:724:1998)B N Rogers, W Dorland, M Kotschenreuther, PRL, (85:5536:2000)W Dorland, F Jenko, M Kotschenreuther, B N Rogers, PRL, (85:5579:2000)


Ë Equilibrium unstable to primary (linear) instabilities Ë Primaries unstable to secondary instabilities

-----------------------------------------------------------------------------------------------------

Ë Some secondary instabilities have zonal flow component

Ë Zonal flows unstable to tertiary instabilities

Key references: S C Cowley, R M Kulsrud, R Sudan, PF B, (3:2767:1991)J F Drake, et al., PF B, (4:488:1992)M N Rosenbluth, F Hinton, PRL (80:724:1998)B N Rogers, W Dorland, M Kotschenreuther, PRL, (85:5536:2000)W Dorland, F Jenko, M Kotschenreuther, B N Rogers, PRL, (85:5579:2000)


Ë Amenable to analytical treatment

Ë Balances among primaries, secondaries and tertiariesexplain simulation results quantitatively and qualitatively

Ë For example, different kinds of secondaries have

different effects.l Toroidal ITG susceptible to strong secondary -> “weak” transport

l Toroidal ETG susceptible to weak secondary -> “strong” transportl Slab ETG/ITG susceptible to strong secondary -> “weak” transport

Nature of Secondary Instabilities

Contours of electrostatic

potential from simulation of

ETG turbulence

High-n microinstabilities

typically localised to low

field, bad curvature region;extended along field lines

G D Kerbel, W Dorland


Radially extended

structures clearly evident

on outboard midplane

Associated with existence

of high amplitude streamertransport

W Dorland, F Jenko


What is a secondaryinstability, and how is it

related to pictures like this?


What is a secondary

instability, and how is it

related to pictures like this?

Let’s go back in time, and

consider the primaryinstabilities


Primary instabilities haveradial widths ~ 1/sqrt(n) and

poloidal widths ~ 1/n


Primary instabilities have

radial widths ~ 1/sqrt(n) and


Since n >> 1, linear modes look

like ‘streamers’


Amplitude of linear

perturbations increases

exponentially in time


Amplitude of linear



Contours of potential are

streamlines of ExB flows whichare increasingly sheared and

thus susceptible to Kelvin-

Helmholtz-like instabilities


Amplitude of linear




streamlines of ExB flows which

are increasingly sheared andthus susceptible to Kelvin-


Also, gradients in poloidal

direction can be sqrt(n)stronger than radial gradients

if eddies survive long enough


Amplitude of linear




streamlines of ExB flows which

are increasingly sheared andthus susceptible to Kelvin-


Also, gradients in poloidal

direction can be sqrt(n)stronger than radial gradients

if eddies survive long enough

Secondary growth rate is proportional to primary amplitude


ETG secondaries arecomplicated, so

consider ITG secondary

first.

Same view as before.

Secondary breaks upradial flows, tries to

convert them to

poloidal flows

B N Rogers, W Dorland

g, kx spectrum of secondary analytically tractable!

Secondary Instability of ITG Mode

Selected Fourier

harmonic amplitudes vstime in example GK ITG

simulation: collisionless,adiabatic electrons,

electrostatic


Selected Fourier


simulation: collisionless,

adiabatic electrons,electrostatic

Primary instability grows

like exp[ g t]


Selected Fourier


simulation: collisionless,adiabatic electrons,

electrostatic

Primary instability grows

like exp[ g t]

Secondary instabilities

grow like exp[exp[g t]]

above a threshold…


Growth rate of primaryis constant in time



Growth rate of

secondary increases intime



Growth rate of

secondary increases intime

Growth rate ofsecondary is

proportional to

amplitude of primary


Consider time whenprimary and secondary

growth rates are equal


Balance primary and

secondary growth rates to

estimate saturation

amplitude


Balance primary andsecondary growth rates to

estimate saturation

amplitude



estimate saturation

amplitude

Alternatively, view this as

the physics that

determines the radialmixing length (Cowley)



estimate saturation

amplitude

Alternatively, view this as

the physics that

determines the radialmixing length (Cowley)

This is not a modulational

instability -- amplitudes aretoo large, orderings

strongly violated


Is the secondaryphysics analytically

tractable?



tractable?

In the limit of highamplitude primary, low

amplitude secondary,

yes.



tractable?

In the limit of highamplitude primary, low

amplitude secondary,

yes.

Best satisfied slightly

before nonlinear break-

up of primary


Analytical treatment

tractable in limit of largeamplitude primary, small

amplitude secondaries

Fully turbulent regime toocomplicated



Analytical treatment

tractable in limit of largeamplitude primary, small

amplitude secondaries

Fully turbulent regime toocomplicated

Compare theoreticallypredicted secondary

growth rate spectrum with

simulation at t=62.3.



Secondary growth rate is

much larger than primarygrowth rate

Predicted spectrum in kx

remarkably independent ofprimary mode’s ky

Data taken fromcomplicated nonlinear

simulation



Agreement in the limit ofthe analytical treatment is

excellent (assumed

simplified primary mode

structure)



excellent (assumed


structure)

For each kx one must solve

a 2-D eigenvalue problem:in the y (~ poloidal)

direction and along the

field line



excellent (assumed


structure)

For each kx one must solve

a 2-D eigenvalue problem:in the y (~ poloidal)

direction and along the

field line

Component which is

constant in y and along

field line is special


With trapped particles, part

of the ky = 0 component of

the eigenmode is linearlyundamped in the

collisionless limit; this is

the Rosenbluth-Hinton

zonal flow.

Simulation: M A Beer, G D Kerbel,G W Hammett, W Dorland

Zonal Flows Can Quench Turbulence

Typical spectrum of zonalflows from gyrokinetic

simulation

Strongly peaked at longwavelengths

W M Nevins, W Dorland


Near but above the linear

threshold, Rosenbluth-Hinton zonal flows can

quench turbulence (Dimits)


Near but above the linear

threshold, Rosenbluth-Hinton zonal flows can

quench turbulence (Dimits)

Leads to important

question:Why doesn’t this alwayshappen?Equivalently, what limitsthe zonal flows well abovethe linear threshold?

Zonal Flow Amplitude Limited byCollisionless Tertiary Instability


Near

Increasing the amplitude of the zonal flows

increases the shear in the zonal flows, whichdecreases the growth rate of the primary


Near

Increasing the amplitude of the zonal flows

increases the shear in the zonal flows, which

decreases the growth rate of the primary

Further increases in the zonal flow amplitude lead to

collisionless tertiary instability B N Rogers, W Dorland


Increasing the temperature gradient slightly

removes window of stability altogether


Increasing the temperature gradient slightly

removes window of stability altogether

Although the Rosenbluth-Hinton zonal flows are linearly

undamped, they are unstable to small perturbationsabove a threshold R/LT B N Rogers, W Dorland


So far:1. ITG modes linearly

unstable: primary



unstable: primary

2. Shear-flow instability

limits growth of primary:secondary



unstable: primary



3. Zonal flow component

of secondary quenchesprimary: Dimits shift


So far:

1. ITG modes linearlyunstable: primary


limits growth of primary:

secondary3. Zonal flow component

of secondary quenches

primary: Dimits shift4. Zonal flows unstable to

tertiary: Stiff transport

at higher R/LT



unstable: primary



3. Zonal flow component

of secondary quenchesprimary: Dimits shift

4. Zonal flows unstable to

tertiary: Stiff transport

at higher R/LT

What happens with additional physics?

Ion-ion Collisions Damp Zonal Flows,Soften Threshold

Focus analysis on regionbetween linear critical

gradient and effective

nonlinear gradient


Ion-ion collisions dampzonal flows and thus

increase turbulent

transport(M N Rosenbluth, F Hinton,P Diamond, Z Lin, W W Lee,

W M Tang, T S Hahm)


Ion-ion collisions dampzonal flows and thus

increase turbulent

transport(M N Rosenbluth, F Hinton,P Diamond, Z Lin, W W Lee,

W M Tang, T S Hahm)

Corollary:

Confinement improvement

expected in reactor-sizedtokamaks

Non-adiabatic Electron DynamicsReverse Effect of Collisionality

Trapped electrons causelarge increase in transport

near marginal stability(Y Chen, S Parker;

D Mikkelsen, D W Ross, W Dorland)





Trapped electrons increasegrowth rate of primary, close

zonal flow stability window;

Dimits shift strongly reduced





Trapped electrons increasegrowth rate of primary, close

zonal flow stability window;

Dimits shift strongly reduced

Electron-ion collisions

reduce non-adiabatic

electron response, and thusreduce turbulent transport

Experimental Confirmation ofDimits Shift at High Collisionality

C-Mod H-mode 960116027 at

half radius

IFS-PPPL model overpredictstransport

NonlinearGS2

Simulations

High Collisionality Outlier from ITERProfile Database has Dimits Shift

C-Mod H-mode 960116027 athalf radius

IFS-PPPL model overpredicts

transport

Gyrokinetic simulations show

Dimits shift effect improvesagreement

General geometry, kineticelectrons, Lorentz collisions

D Mikkelsen, M KotschenreutherW Dorland

NonlinearGS2

Simulations

Lowering Collisionality IncreasesPredicted Heat Flux

Lowering overall collision-

ality by factor of 5 increases

predicted transport

IFS-PPPL model




predicted transport

Lowering only ion-ion

collisionality by factor of 5

has small effect, near kneeof Dimits shift

IFS-PPPL model




predicted transport

Lowering only ion-ion

collisionality by factor of 5has small effect, near knee

of Dimits shift

Results consistent with highcollisionality outliers from

profile database effort

D Mikkelsen, W Dorland

IFS-PPPL model

Gyrokinetic Simulations areStimulating and Guiding

Broad Theoretical Advances

Gyrokinetic Simulations areStimulating and Guiding

Broad Theoretical Advances

1. Ion-scale physics2. Electron-scale physics

Realistic Electron Dynamics AllowsSimulation of Particle Transport

Profiles from Tore-Supra

Electron Cyclotron Heating

Ion temperature profilesomewhat uncertain

No obvious central particlesource


Simulations near half-radius

Density gradient observed in

experiment

Focus on TEM+ITG

instabilities


Varied ion R/LT around

nominal value

Here, R/LTi increase (to

nominal value) att = 700 causes particle flux

to reverse (pinch)

Trapped electron effect?

K Hallatschek, W DorlandF Jenko, T Hoang

Passing Electrons Dominate Pinch

Integrated particle flux vspitch angle shows relativecontributions to total flux

Hybrid PIC-fluid models thatassume passing electrons

are adiabatic will miss effect

Direct comparison withexperiment possible

K Hallatschek, W Dorland

At re Scales, New Physics Emerges:

Return to Parasitic Instability Theoryfor Guidance

Secondary physics reduces to

2D eigenvalue problem: alongthe field line, and in the y

direction (perp to B and —y)

Secondary Structure at ri Scales





Shown here are the dominantFourier harmonics of the

solution in the long wavelength

(ITG) limit





solution in the long wavelength

(ITG) limit

Note presence of significant

zonal flow component

(constant along field line, ky=0)


At re Scales, Secondaries Change

Ë Quickly establish terminology:

Secondary driven by perpendicular shear of

perpendicular flows that are associated with the

primary instability will be the “Rogers” secondary; thishas been the main secondary so far

At re Scales, Secondaries Change

Ë Quickly establish terminology:

Secondary driven by perpendicular shear of

perpendicular flows that are associated with the

primary instability will be the “Rogers” secondary; thishas been the main secondary so far

The secondary driven by perpendicular shear of parallelflows that are associated with the primary instability

will be the “Cowley” secondary; this was the first

secondary identified as potentially important in ITG/ETG

turbulence

Shown here are the dominant

Fourier harmonics of theRogers secondary in the short

wavelength (ETG) limit

Rogers Secondary at re Scales


Fourier harmonics of theRogers secondary in the short


At small scales, adiabatic ionresponse comes from gyration,

not streaming along field line;

weakens Rogers secondary



Fourier harmonics of the

Rogers secondary in the short


At small scales, adiabatic ion

response comes from gyration,not streaming along field line;

weakens Rogers secondary

Note absence of zonal flowcomponent (constant along

field line, ky=0)

W Dorland, B N Rogers, F Jenko


Cowley Secondary at re Scales

If primary instability requiressignificant parallel compress-ibility (e.g., the sheared-slab hemode) the Cowley secondary is

excited


excited



Cowley secondary in the shortwavelength (ETG) limit


If primary instability requires

significant parallel compress-ibility (e.g., the sheared-slab hemode) the Cowley secondary isexcited


Cowley secondary in the short


Again, note absence of zonal

flow component (constant

along field line, ky=0)S C Cowley, W Dorland



excited



Cowley secondary in the shortwavelength (ETG) limit

Again, note absence of zonalflow component (constant

along field line, ky=0)S C Cowley, W Dorland


Cowley Secondary Strong on re Scales

¸ Theoretically predicted:

growth rate; parallel

wavenumber increasing with

the amplitude of the primary





Cowley secondary not

weakened like Rogers at small

scales; breaks up streamerswhen excited






Cowley secondary not

weakened like Rogers at small

scales; breaks up streamerswhen excited

Zonal flows irrelevant on rescales because drive is weak

and tertiary is strong


Balance of Primaries, Secondariesand Tertiaries Explains Simulations

¸ Foregoing predicts slab

ITG/ETG should be similar (innormalized units) because slab

primary requires parallel

compressibility

Balance of Primaries, Secondariesand Tertiaries Explains Simulations

¸ Foregoing predicts slab

ITG/ETG should be similar (in

normalized units) because slab

primary requires parallelcompressibility

¸ In the limit of constant

curvature, parallel compress-ibility irrelevant to primary, so

normalized ETG saturation

level should be much higher

Balance of Primaries, Secondariesand Tertiaries Predicts Simulations

Major result:

Balance of primary and

secondary growth rates

predicts when highamplitude streamer

transport is found with

simulations

Balance of Primaries, Secondariesand Tertiaries Predicts Simulations

Major result:

Balance of primary and

secondary growth rates

predicts when highamplitude streamer

transport is found with

simulations

Toroidal ETG branch most

dangerous

F Jenko, W Dorland

Experimental Confirmation of Theory?

ETG threshold formulaobtained from GS2

Tore-Supra finds Te profile

is stiff above a criticalgradient

Experimental and theoretical thresholds similar and GK ETG

simulations predict experimental stiffness beyond threshold

F Jenko, G W Hammett W Dorland

NSTX Confinement Consistent withGyrokinetic Predictions

Gyrokinetic analysis of NSTX

discharges indicates long

wavelength instabilities weakor non-existent, but ETG

modes unstable

C Bourdelle, W Dorland, NSTX team





modes unstable

TRANSP analysis of NSTX

discharges indicates electrons

are dominant energy loss

channel (not shown)






modes unstable

TRANSP analysis of NSTX

discharges indicates electrons

are dominant energy loss

channel (not shown)

Detailed results in press


Can Higher b and Higher b GradientImprove ST Confinement?

GK analysis of NSTX data suggestsconfirmation of long wavelength

“second microstability” predictions

for STC Bourdelle, G W Hammett

W Dorland, et al.

-db/dr

Can Higher b and Higher b GradientImprove ST Confinement?

GK analysis of NSTX data suggestsconfirmation of long wavelength

“second microstability” predictions

for STC Bourdelle, G W Hammett

W Dorland, et al.

GK simulations of ETG turbulence

indicate 1/b scaling of electron

energy diffusion coefficient in

some regimesF Jenko, W Dorland

-db/dr

Conclusions

Ë First-principles simulation of turbulence in fusion plasmas is

a rapidly maturing area. Gyrokinetic simulations areexplaining experimental data. For example:

1. ETG turbulence identified in NSTX2. ITG turbulence identified in C-Mod

3. TEM-induced particle transport identified in Tore Supra?

Ë Parasitic instability model is a useful theoretical frameworkfor understanding nonlinear simulation results. Competition

among primary, secondary and tertiary instabilities explains

simulation results.

Nonlinear Physics BenchmarkedAmong Independent Codes

GS2 and GENE,

benchmark of heat flux

for toroidal ETGturbulence

F Jenko, W Dorland


Primary instabilities haveradial widths ~ 1/sqrt(n) and


Since n >> 1, linear modes looklike ‘streamers’

Note: Flux tube simulationstypically ignore radial envelope

because nonlinear coupling

dominates: turbulent radial

correlation length ~ 1/n

Alpha Heating + NeoclassicalTransport + ST = Stable Profiles

Simple alpha power depositionmodel + self-consistent

bootstrap current + small seed

current on axis + neoclassicaltransport + electron energy

transport (possibly strong) in

ST configuration predicted to

yield MHD- and micro-stableprofiles.

W Dorland, M Kotschenreuther

Alpha Heating + NeoclassicalTransport + ST = Stable Profiles

Simple alpha power depositionmodel + self-consistent

bootstrap current + small seed

current on axis + neoclassicaltransport + electron energy

transport (possibly strong) in

ST configuration predicted to

yield MHD- and micro-stableprofiles. Need higher b !

But particle transport problem?

W Dorland, M Kotschenreuther


ETG threshold formula

obtained from approx.

3000 GS2 runs (F Jenko, W

Dorland, G W Hammett)






Smoothly interpolates

analytical results of

Romanelli (toroidal) and

Hahm-Tang (slab)






Smoothly interpolates

analytical results of

Romanelli (toroidal) and

Hahm-Tang (slab)

Tore-Supra finds Te profile

is stiff above a criticalgradient

Linear Physics Benchmarked forWide Range of Problems

Linear micro-

stability calcu-lations for NCSX

with GS2 and

FULL agree

E Belli, G Rewoldt,G W Hammett,

W Dorland


Adequate resolution ofradial structures is

challenging, but achievable


Adequate resolution of

radial structures is


In normalized units, ETG

transport is much larger

than ITG transport whenstreamers are observed


Adequate resolution ofradial structures is


In normalized units, ETGtransport is much larger

than ITG transport when

streamers are observed

The difference can be

traced to differences insecondary and tertiary

physics

W Dorland, F Jenko, B N Rogers

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