Gyrokinetic Simulations of Tokamak Microturbulence

Gyrokinetic Simulations of Tokamak Microturbulence W Dorland, Imperial College, London With key contributions from: S C Cowley F Jenko G W Hammett D Mikkelsen B N Rogers C Bourdelle W M Nevins D W Ross K Hallatschek R Budny E Belli M Kotschenreuther J W Connor E Quataert G D Kerbel

Gyrokinetics is Maturing Rapidly

Gyrokinetics is Maturing Rapidly Ë Mutually benchmarked, nonlinear codes exist

Gyrokinetics is Maturing Rapidly Ë Mutually benchmarked, nonlinear codes exist Ë Growing user base of experimentalists and theorists

Gyrokinetics is Maturing Rapidly Ë Mutually benchmarked, nonlinear codes exist Ë Growing user base of experimentalists and theorists Radial correlation functions from three independently developed gyrokinetic codes, run for identical physics parameters. W M Nevins, A Dimits, R E Waltz, J Candy, W Dorland

First Principles Models Desirable Ë Outliers valuable -- indicate missing physics

First Principles Models Desirable Ë Ë Outliers valuable -- indicate missing physics Extrapolation less uncertain

First Principles Models Desirable Ë Ë Outliers valuable -- indicate missing physics Extrapolation less uncertain GLF-23 model (Waltz, Staebler, Dorland, Konings, and Kotschenreuther) vs data from ITER profile database J Kinsey

Nonlinear Physics Benchmarked Against Theoretical Predictions High b Alfvenic turbulence in homogeneous, stirred plasma shows predicted perpendicular spectrum (and anisotropy, not shown). Here, b = 8 (i.e., 800%). W Dorland, S C Cowley, G W Hammett and E Quataert

Parasitic Instability Model 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)

Parasitic Instability Model Ë Amenable to analytical treatment Ë Balances among primaries, secondaries and tertiaries explain simulation results quantitatively and qualitatively Ë For example, different kinds of secondaries have different effects. Toroidal ITG susceptible to strong secondary -> weak transport Toroidal ETG susceptible to weak secondary -> strong transport 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

Nature of Secondary Instabilities Radially extended structures clearly evident on outboard midplane Associated with existence of high amplitude streamer transport W Dorland, F Jenko

Nature of Secondary Instabilities What is a secondary instability, and how is it related to pictures like this?

Nature of Secondary Instabilities What is a secondary instability, and how is it related to pictures like this? Let s go back in time, and consider the primary instabilities

Nature of Secondary Instabilities Primary instabilities have radial widths ~ 1/sqrt(n) and poloidal widths ~ 1/n

Nature of Secondary Instabilities Primary instabilities have radial widths ~ 1/sqrt(n) and poloidal widths ~ 1/n Since n >> 1, linear modes look like streamers

Nature of Secondary Instabilities Amplitude of linear perturbations increases exponentially in time

Nature of Secondary Instabilities Amplitude of linear perturbations increases exponentially in time Contours of potential are streamlines of ExB flows which are increasingly sheared and thus susceptible to Kelvin- Helmholtz-like instabilities Also, gradients in poloidal direction can be sqrt(n) stronger than radial gradients if eddies survive long enough

Nature of Secondary Instabilities ETG secondaries are complicated, so consider ITG secondary first. Same view as before. Secondary breaks up radial flows, tries to convert them to poloidal flows B N Rogers, W Dorland g, k x spectrum of secondary analytically tractable!

Secondary Instability of ITG Mode Selected Fourier harmonic amplitudes vs time in example GK ITG simulation: collisionless, adiabatic electrons, electrostatic

Secondary Instability of ITG Mode Selected Fourier harmonic amplitudes vs time in example GK ITG simulation: collisionless, adiabatic electrons, electrostatic Primary instability grows like exp[ g t]

Secondary Instability of ITG Mode Selected Fourier harmonic amplitudes vs time in example GK ITG simulation: collisionless, adiabatic electrons, electrostatic Primary instability grows like exp[ g t] Secondary instabilities grow like exp[exp[g t]] above a threshold

Secondary Instability of ITG Mode Growth rate of primary is constant in time

Secondary Instability of ITG Mode Growth rate of primary is constant in time Growth rate of secondary increases in time

Secondary Instability of ITG Mode Growth rate of primary is constant in time Growth rate of secondary increases in time Growth rate of secondary is proportional to amplitude of primary

Secondary Instability of ITG Mode Consider time when primary and secondary growth rates are equal

Secondary Instability of ITG Mode Balance primary and secondary growth rates to estimate saturation amplitude

Secondary Instability of ITG Mode Balance primary and secondary growth rates to estimate saturation amplitude Alternatively, view this as the physics that determines the radial mixing length (Cowley)

Secondary Instability of ITG Mode Balance primary and secondary growth rates to estimate saturation amplitude Alternatively, view this as the physics that determines the radial mixing length (Cowley) This is not a modulational instability -- amplitudes are too large, orderings strongly violated

Secondary Instability of ITG Mode Is the secondary physics analytically tractable?

Secondary Instability of ITG Mode Is the secondary physics analytically tractable? In the limit of high amplitude primary, low amplitude secondary, yes.

Secondary Instability of ITG Mode Is the secondary physics analytically tractable? In the limit of high amplitude primary, low amplitude secondary, yes. Best satisfied slightly before nonlinear breakup of primary

Secondary Instability of ITG Mode Analytical treatment tractable in limit of large amplitude primary, small amplitude secondaries Fully turbulent regime too complicated B N Rogers, W Dorland

Secondary Instability of ITG Mode Analytical treatment tractable in limit of large amplitude primary, small amplitude secondaries Fully turbulent regime too complicated Compare theoretically predicted secondary growth rate spectrum with simulation at t=62.3. B N Rogers, W Dorland

Secondary Instability of ITG Mode Secondary growth rate is much larger than primary growth rate Predicted spectrum in k x remarkably independent of primary mode s k y Data taken from complicated nonlinear simulation B N Rogers, W Dorland

Secondary Instability of ITG Mode Agreement in the limit of the analytical treatment is excellent (assumed simplified primary mode structure)

Secondary Instability of ITG Mode Agreement in the limit of the analytical treatment is excellent (assumed simplified primary mode structure) For each k x one must solve a 2-D eigenvalue problem: in the y (~ poloidal) direction and along the field line

Secondary Instability of ITG Mode With trapped particles, part of the k y = 0 component of the eigenmode is linearly undamped 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 zonal flows from gyrokinetic simulation Strongly peaked at long wavelengths W M Nevins, W Dorland

Zonal Flows Can Quench Turbulence Near but above the linear threshold, Rosenbluth- Hinton zonal flows can quench turbulence (Dimits)

Zonal Flows Can Quench Turbulence Near but above the linear threshold, Rosenbluth- Hinton zonal flows can quench turbulence (Dimits) Leads to important question: Why doesn t this always happen? Equivalently, what limits the zonal flows well above the linear threshold?

Zonal Flow Amplitude Limited by Collisionless Tertiary Instability

Zonal Flow Amplitude Limited by Collisionless Tertiary Instability 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

Zonal Flow Amplitude Limited by Collisionless Tertiary Instability Increasing the temperature gradient slightly removes window of stability altogether

Zonal Flow Amplitude Limited by Collisionless Tertiary Instability Increasing the temperature gradient slightly removes window of stability altogether Although the Rosenbluth-Hinton zonal flows are linearly undamped, they are unstable to small perturbations above a threshold R/L T B N Rogers, W Dorland

Zonal Flow Amplitude Limited by Collisionless Tertiary Instability So far: 1. ITG modes linearly unstable: primary

Zonal Flow Amplitude Limited by Collisionless Tertiary Instability So far: 1. ITG modes linearly unstable: primary 2. Shear-flow instability limits growth of primary: secondary

Zonal Flow Amplitude Limited by Collisionless Tertiary Instability So far: 1. ITG modes linearly unstable: primary 2. Shear-flow instability limits growth of primary: secondary 3. Zonal flow component of secondary quenches primary: Dimits shift 4. Zonal flows unstable to tertiary: Stiff transport at higher R/L T

Ion-ion Collisions Damp Zonal Flows, Soften Threshold Focus analysis on region between linear critical gradient and effective nonlinear gradient

Ion-ion Collisions Damp Zonal Flows, Soften Threshold Ion-ion collisions damp zonal flows and thus increase turbulent transport (M N Rosenbluth, F Hinton, P Diamond, Z Lin, W W Lee, W M Tang, T S Hahm)

Non-adiabatic Electron Dynamics Reverse Effect of Collisionality Trapped electrons cause large increase in transport near marginal stability (Y Chen, S Parker; D Mikkelsen, D W Ross, W Dorland)

Non-adiabatic Electron Dynamics Reverse Effect of Collisionality Trapped electrons cause large increase in transport near marginal stability (Y Chen, S Parker; D Mikkelsen, D W Ross, W Dorland) Trapped electrons increase growth rate of primary, close zonal flow stability window; Dimits shift strongly reduced

Experimental Confirmation of Dimits Shift at High Collisionality C-Mod H-mode 960116027 at half radius IFS-PPPL model overpredicts transport Nonlinear GS2 Simulations

High Collisionality Outlier from ITER Profile Database has Dimits Shift C-Mod H-mode 960116027 at half radius IFS-PPPL model overpredicts transport Nonlinear GS2 Simulations Gyrokinetic simulations show Dimits shift effect improves agreement General geometry, kinetic electrons, Lorentz collisions D Mikkelsen, M Kotschenreuther W Dorland

Lowering Collisionality Increases Predicted Heat Flux IFS-PPPL model Lowering overall collisionality by factor of 5 increases predicted transport

Lowering Collisionality Increases Predicted Heat Flux IFS-PPPL model Lowering overall collisionality by factor of 5 increases predicted transport Lowering only ion-ion collisionality by factor of 5 has small effect, near knee of Dimits shift Results consistent with high collisionality outliers from profile database effort D Mikkelsen, W Dorland

Gyrokinetic Simulations are Stimulating and Guiding Broad Theoretical Advances

Gyrokinetic Simulations are Stimulating and Guiding Broad Theoretical Advances 1. Ion-scale physics 2. Electron-scale physics

Realistic Electron Dynamics Allows Simulation of Particle Transport Profiles from Tore-Supra Electron Cyclotron Heating Ion temperature profile somewhat uncertain No obvious central particle source

Realistic Electron Dynamics Allows Simulation of Particle Transport Simulations near half-radius Density gradient observed in experiment Focus on TEM+ITG instabilities

Realistic Electron Dynamics Allows Simulation of Particle Transport Varied ion R/L T around nominal value Here, R/L Ti increase (to nominal value) at t = 700 causes particle flux to reverse (pinch) Trapped electron effect? K Hallatschek, W Dorland F Jenko, T Hoang

Passing Electrons Dominate Pinch Integrated particle flux vs pitch angle shows relative contributions to total flux Hybrid PIC-fluid models that assume passing electrons are adiabatic will miss effect Direct comparison with experiment possible K Hallatschek, W Dorland

At r e Scales, New Physics Emerges: Return to Parasitic Instability Theory for Guidance

Secondary Structure at r i Scales Secondary physics reduces to 2D eigenvalue problem: along the field line, and in the y direction (perp to B and y)

Secondary Structure at r i Scales Secondary physics reduces to 2D eigenvalue problem: along the field line, and in the y direction (perp to B and y) Shown here are the dominant Fourier harmonics of the solution in the long wavelength (ITG) limit

At r e 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; this has been the main secondary so far The secondary driven by perpendicular shear of parallel flows 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

Rogers Secondary at r e Scales Shown here are the dominant Fourier harmonics of the Rogers secondary in the short wavelength (ETG) limit

Rogers Secondary at r e Scales Shown here are the dominant Fourier harmonics of the Rogers secondary in the short wavelength (ETG) limit At small scales, adiabatic ion response comes from gyration, not streaming along field line; weakens Rogers secondary

Cowley Secondary at r e Scales If primary instability requires significant parallel compressibility (e.g., the sheared-slab h e mode) the Cowley secondary is excited

Cowley Secondary at r e Scales If primary instability requires significant parallel compressibility (e.g., the sheared-slab h e mode) the Cowley secondary is excited Shown here are the dominant Fourier harmonics of the Cowley secondary in the short wavelength (ETG) limit

Cowley Secondary Strong on r e Scales Theoretically predicted: growth rate; parallel wavenumber increasing with the amplitude of the primary

Cowley Secondary Strong on r e 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 streamers when excited

Balance of Primaries, Secondaries and Tertiaries Explains Simulations Foregoing predicts slab ITG/ETG should be similar (in normalized units) because slab primary requires parallel compressibility

Balance of Primaries, Secondaries and Tertiaries Explains Simulations Foregoing predicts slab ITG/ETG should be similar (in normalized units) because slab primary requires parallel compressibility In the limit of constant curvature, parallel compressibility irrelevant to primary, so normalized ETG saturation level should be much higher

Balance of Primaries, Secondaries and Tertiaries Predicts Simulations Major result: Balance of primary and secondary growth rates predicts when high amplitude streamer transport is found with simulations

Experimental Confirmation of Theory? ETG threshold formula obtained from GS2 Tore-Supra finds T e profile is stiff above a critical gradient F Jenko, G W Hammett W Dorland Experimental and theoretical thresholds similar and GK ETG simulations predict experimental stiffness beyond threshold

NSTX Confinement Consistent with Gyrokinetic Predictions Gyrokinetic analysis of NSTX discharges indicates long wavelength instabilities weak or non-existent, but ETG modes unstable TRANSP analysis of NSTX discharges indicates electrons are dominant energy loss channel (not shown) C Bourdelle, W Dorland, NSTX team

Can Higher b and Higher b Gradient Improve ST Confinement? GK analysis of NSTX data suggests confirmation of long wavelength second microstability predictions for ST C Bourdelle, G W Hammett W Dorland, et al. -db/dr GK simulations of ETG turbulence indicate 1/b scaling of electron energy diffusion coefficient in some regimes F Jenko, W Dorland

Conclusions Ë First-principles simulation of turbulence in fusion plasmas is a rapidly maturing area. Gyrokinetic simulations are explaining experimental data. For example: 1. ETG turbulence identified in NSTX 2. ITG turbulence identified in C-Mod 3. TEM-induced particle transport identified in Tore Supra? Ë Parasitic instability model is a useful theoretical framework for understanding nonlinear simulation results. Competition among primary, secondary and tertiary instabilities explains simulation results.

Nonlinear Physics Benchmarked Among Independent Codes GS2 and GENE, benchmark of heat flux for toroidal ETG turbulence F Jenko, W Dorland

Nature of Secondary Instabilities Primary instabilities have radial widths ~ 1/sqrt(n) and poloidal widths ~ 1/n Since n >> 1, linear modes look like streamers Note: Flux tube simulations typically ignore radial envelope because nonlinear coupling dominates: turbulent radial correlation length ~ 1/n

Alpha Heating + Neoclassical Transport + ST = Stable Profiles Simple alpha power deposition model + self-consistent bootstrap current + small seed current on axis + neoclassical transport + electron energy transport (possibly strong) in ST configuration predicted to yield MHD- and micro-stable profiles. W Dorland, M Kotschenreuther

Experimental Confirmation of Theory? ETG threshold formula obtained from approx. 3000 GS2 runs (F Jenko, W Dorland, G W Hammett)

Experimental Confirmation of Theory? 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)

Linear Physics Benchmarked for Wide Range of Problems Linear microstability calculations for NCSX with GS2 and FULL agree E Belli, G Rewoldt, G W Hammett, W Dorland

Nature of Secondary Instabilities Adequate resolution of radial structures is challenging, but achievable

Nature of Secondary Instabilities Adequate resolution of radial structures is challenging, but achievable In normalized units, ETG transport is much larger than ITG transport when streamers are observed The difference can be traced to differences in secondary and tertiary physics W Dorland, F Jenko, B N Rogers