Gyro-kinetics and tokamak turbulence simulations

Recap Gyro-kinetics Numerics Results www-users.york.ac.uk/~bd512

DNS Scaling

Gyro-kinetics and tokamak turbulence simulations

Image: General Atomics

Dr Ben Dudson, University of York. http://www-users.york.ac.uk/~bd512/


DNS Scaling

Plasma turbulenceSummary from last lecture:

● Turbulence is chaotic (apparently random) fluid motion which results in enhanced diffusion / transport

● A key feature is spreading of energy across spatial scales● In neutral fluids, large scale 'stirring' spreads to small-scale

fluctuations (energy cascade)● In 2D fluids and plasmas, this can be reversed, with energy

moving from small to large scales (inverse cascade)

kk=0 (flow)

cascade

Inverse

cascade

'Stirring' / instabilityTurbulence and mean flows interact through Reynolds stress. This looks like a viscosity, but just transfers energy & momentum between flow and fluctuations.

NB: Different from (inverse) cascade. Cascade is between eddies of similar size; Reynolds stress is direct.DissipationReynolds stress

Small scalesLarge scales

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DNS Scaling

Solving turbulence

Though the governing equations are known (e.g. Navier-stokes), analytic progress has been very slow

Some of the most successful “theories” are based on simple pictures of turbulence, and dimensional arguments.

If you give up trying to predict details, and settle for statistical averages● The variables are well-behaved and reproducible ● The equations governing these variables cannot be written down!

This is due to the closure problem, similar to deriving fluid equations from the Vlasov equation

This non-linear term means each velocity moment depends on the next higher one



DNS Scaling

Direct calculation of turbulence● We need a way to actually calculate turbulence in current and future

situations.

● Since ~1970s, computer simulations have played an increasing role (Fluid turbulence Orszag & Patterson 1972)

● Computational Fluid Dynamics (CFD) is now a vast field of study

● Plasma simulation of turbulence is getting to the stage of being able to reproduce experimental results.

● For core turbulence, Gyro-kinetics is the model most often used



DNS Scaling

Direct calculation of turbulence

NB: Analytic turbulence theory is still vital because● Clever techniques are needed to efficiently simulate turbulence● It provides a framework to help make sense of complicated data

from experiment and simulations

“The danger from computers is not that they will eventually get as smart as men, but we will meanwhile agree to meet them halfway” – Bernard Avishai

● We need a way to actually calculate turbulence in current and future situations.

● Since ~1970s, computer simulations have played an increasing role (Fluid turbulence Orszag & Patterson 1972)

● Computational Fluid Dynamics (CFD) is now a vast field of study

● Plasma simulation of turbulence is getting to the stage of being able to reproduce experimental results.

● For core turbulence, Gyro-kinetics is the model most often used



DNS Scaling

Plasma simulation models

Length scales

Tim

e s

cale

s

Gyro-radius

Cyclotron frequency

Fluid modelse.g. MHD

On large length and time-scales, fluid models can be

used

KineticVlasov

The kinetic equations (e.g. Vlasov) are valid at all scales, but are impractical to solve for all except small scales and times

What about the middle bit?



DNS Scaling

Plasma simulation models

Length scales

Tim

e s

cale

s

Gyro-radius

Cyclotron frequency

Fluid modelse.g. MHD

On large length and time-scales, fluid models can be

used

KineticVlasov

The kinetic equations (e.g. Vlasov) are valid at all scales, but are impractical to solve for all except small scales and times

What about the middle bit?

Gyro-kinetics!

Gyro-kinetics

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DNS Scaling

Distributions and F-P equation

Collision terms

Describes a set of particles with velocities v and locations x

Recall the Fokker-Planck equation:

6D (+ time) problem. Too hard to simulate most of the time



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Gyro-averaging

Not interested in phenomena faster than cyclotron frequency

In ITER,

Need 6 dimensions, but we're free to choose what those

dimensions are.Parallel v Perp. v

gyro-phase

Average over gyro-phase



DNS Scaling

Gyro-kinetics

Distribution of particles Distribution of loops

Gyro-kinetics is the description of plasma as a collection of charged current loops

Average over Larmor orbit

c.f. Fokker-Planck, which is a description of a collection of point particles

Derived by gyro-averaging the Vlasov equation

In real spaceIn Fourier space becomes

Bessel functions

This is much quicker to calculate, so often solve G-K equations in Fourier space



DNS Scaling

Gyro-kinetics II● What properties does a current loop have?

● Location

● Perpendicular velocity

● Parallel velocity

A useful property is magnetic moment because it is (approximately) conserved

Replace perpendicular velocity

We can also replace the parallel velocity by the kinetic energy

Replace parallel velocity

Now have a distribution function

NB: Several different choices of coordinates are commonly used



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Gyro-kinetics IIIWhat is the evolution equation for this distribution function?

Follow F-P equation:

= 0 since Mag. Moment ~ conservedVelocity of a current loop

Kinetic energy of a current loop



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Velocity of current loop

Parallel velocity

Curvature drift Grad-B driftExB drift

Guiding-centre velocity

Velocity is just the sum of all the particle drifts

Parallel flow along

perturbed field

One you may not have come across before...

Sometimes called the magnetic “flutter” term



DNS Scaling

Drift-kinetic equation

Drift-kinetic equation

Just need the change in kinetic energy.

The drift-kinetic equation has made an important assumption:

Magnetic and electric fields do not change over the gyro orbit

Electric fieldMagnetic induction

(Electric field around gyro-orbit)

This is now a 5-D kinetic equation, which has eliminated the fast cyclotron-frequency gyro motion

However

Need to average particle drifts over the gyro orbit



DNS Scaling

Gyro-kinetic equationWhat about small-scale fluctuations?

Expected to be produced by turbulent cascade, and observed in experiments

At small scales, E and B change over the orbit

Same for the magnetic perturbations

Average EM potential fields over gyro-orbit

Reminder: Gyro-average operator(Bessel function in Fourier space)



DNS Scaling

Gyro-kinetic equation II

Still need equations for EM potential fields and

● Like MHD, quasi-neutrality approximation used● Fast time-scales neglected, so remove displacement current

Need to solve a Poisson equation for the electrostatic potential

The magnetic potential is given by:

Sum over species Species charge



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Gyro-kinetic Poisson equation

To calculate charge density at Q, need to add contributions from all particles with guiding centres on dashed line

Applied electric field polarises charge ring → polarisation shielding



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Summary of Gyro-kinetics

● Averages over fast cyclotron time-scales, but retains small length-scales (gyro-radius or smaller)

● Kinetic description, with self-consistent evolution of fields

Requires the evolution of a 5-D distribution function in time, and inverting differential equations for the EM fields

This is still challenging to simulate, so extra tricks are commonly used, including:

● Ordering methods (e.g. Delta-f) to simplify the equations● Coordinate systems which take advantage of anisotropy● ...



DNS Scaling

Delta-f method

Densi

t y,

Tem

pera

ture

Minor radius

Fluctuations in tokamaks are generally found to be small-amplitude (~1%), at least in the core. Doesn't work for edge plasma.

Can simplify the equations by expanding the distribution function

Background

However, non-linear fluctuations must be able to “turn off” their drive (i.e. gradients). Therefore,

Note:

Equilibrium length-scale



DNS Scaling

Small parallel variation

Perpendicular to the magnetic field, fluctuations are of the order of the gyro radius

Due to fast transport along field-lines, variation parallel to B tends to be on equilibrium length-scales

This allows simplifications such as

Neglect parallel derivatives in many cases



DNS Scaling

Coordinate systemsTrying to simulate structures which are very elongated along fields, and small across fields.

Using a “simple” coordinate system, would need to use high resolution (~gyro-radius) in all directions

By aligning one of the coordinates with the magnetic field, fewer points can be used in this direction

Leads to orders of magnitude savings in memory and CPU timeCoordinates are now non-orthogonal, curvilinear (differential geometry)



DNS Scaling

Simulation codes

Main electromagnetic codes● GS2 (Dorland & Kotschenreuther) cont. flux.● GENE (Jenko, Garching) continuum, flux tube● GYRO (Candy & waltz) continuum, global● GEM (Parker & Chen) df PIC, global

Notable mentions● PG3EQ (Dimits) continuum, flux● GTC (Z.Lin) df PIC, global

Edge codes (new)● TEMPEST (Xu, LLNL)● ESL Edge Simulation Lab (Xu, Cohen, Colella)● XGC (C.S.Chang)

Together, algorithmic and theoretical advances over the past ~25 years

have sped up G-K simulations by ~1023

Compare this with Moore's law over same period ~105 speedup[G.Hammet, APS 2007]

Common ways to classify codes● Continuum / PIC● Global / flux-tube (thin annulus)● Delta-f / full-f



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Results - Turbulence

Waltz, Austin, Burrell, Candy, PoP 2006

GS2 (NSTX)GYRO (DIII-D)

Dorland (from hammett 2002 PPPL)



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Results - Scaling

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

From W.Dorland, 2002 EPS

High beta, homogeneous stirred plasma shows kolmogorov scaling. Here beta = 8



DNS Scaling

Results – Dimits shiftFirst observed by A.Dimits using PG3EQ code

Upwards shift in critical temperature gradient for ITG turbulence

From GS2 [gs2.sourceforge.net]

● Simulations show an initial burst of turbulence, which then dies down to a low level

● Found to be due to generation of zonal flows through Reynolds stress

● Resulting low-turbulence state called a “Rosenbluth-Hinton equilibrium”



DNS Scaling

Results - Transport

0.010

0.10

1.0

10

0.01 0.1 1 10

DIII-D L-DIII-D H-JET H-TFTR L-

TG

LF P

redi

cted

Win

c (M

J)Experimental W

inc (MJ)

∆RWinc

= 19%

<RWinc

> - 1 = 2%

G.Hammet, APS 2007

For core turbulence, Gyro-kinetic codes can now be very close. c.f. 15 years ago, fluxes were out by factors of 10 - 100!

Waltz, Candy, Petty.2006 PoP 13, 072304

ITG threshold within 5% in core

GYRO simulation results fitted with simpler transport model, then this used to compare with experiment



DNS Scaling

Summary of Gyro-kineticsStandard G-K ordering uses a small parameter

But small perpendicular fluctuations

● Evolve a 5-D distribution function in time with self-consistent EM fields● Removes high-frequency phenomena from the Fokker-Planck equation,

making the problem solvable● Captures the essential physics needed to calculate turbulent transport in

tokamak core plasma● Huge progress made over past ~25 years● Can now reproduce core turbulence with reasonable accuracy

● Work currently ongoing to extend this success to the plasma edge, which is not understood nearly so well

● Calculations are still far from “routine”, and a few more orders of magnitude improvement is needed



DNS Scaling

Transport scaling laws

● Theoretical understanding of turbulence is still poor

● Gyro-kinetic simulations are improving quickly, but it's still too expensive to simulate the full tokamak

● Understanding of turbulence at the edge of the plasma is even less well understood

● Transport barriers (ETB / ITB) are not well understood, and have a large effect on the overall confinement

● Engineers still need to predict performance of tokamak designs



DNS Scaling

Bohm scaling

Taylor (1961) proved that this is the largest value that cross-field diffusion can attain

Bohm (1949)

Cross-field advection dominated by ExB motion

Boltzmann-like response (equipartition of energy)

In this scaling, eddy size scales with the system, which is terrible news for confinement



DNS Scaling

Gyro-Bohm scalingIn this scaling, eddy size is fixed to the gyro-radius scale. Diffusion coefficients acquire a factor of rho / L

Since the electron gyro-radius is much smaller than the ion gyro-radius

This is what tends to be observed in experiment, and the values are typically within an order of magnitude (or two) of the observed value.



DNS Scaling

Empirical scalingAs a necessary (but desperate) measure, transport levels are fitted to data from existing machines in order to extrapolate to future devices

These generally take the form of power laws, and come in two varieties: those in engineering parameters such as

And those in normalised “physics” parameters such as nu-star & rho-star

NB: These methods have very little physics basis, and are purely statistical, ad-hoc models. They must be fitted extremely carefully to avoid introducing bias, and should be treated with extreme caution.

Warning signs:● No error bars● Data points which can be counted on one hand● No separation into training and testing datasets



DNS Scaling

Summary● Averaging the Vlasov equation over gyro-orbits gives a 5D

equation for the dynamics of charged current loops - Gyro-kinetics

● Advances in analytic theory, numerical methods, and computer power mean GK codes can now reproduce core turbulence quite accurately

● Edge turbulence, and transport barriers are still not well understood or reproducible.

● Empirical scaling laws, and simple scaling such as Bohm or Gyro-Bohm are often used to extrapolate to future machines. Need to be very careful when extrapolating trends.

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Gyro-kinetics and tokamak turbulence simulations

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