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/
Recap Gyro-kinetics Numerics Results 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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Recap Gyro-kinetics Numerics Results www-users.york.ac.uk/~bd512
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
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Recap Gyro-kinetics Numerics Results www-users.york.ac.uk/~bd512
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
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Recap Gyro-kinetics Numerics Results www-users.york.ac.uk/~bd512
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
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Recap Gyro-kinetics Numerics Results www-users.york.ac.uk/~bd512
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
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Recap Gyro-kinetics Numerics Results www-users.york.ac.uk/~bd512
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?
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Recap Gyro-kinetics Numerics Results www-users.york.ac.uk/~bd512
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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Recap Gyro-kinetics Numerics Results www-users.york.ac.uk/~bd512
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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Recap Gyro-kinetics Numerics Results www-users.york.ac.uk/~bd512
DNS Scaling
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
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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
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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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DNS Scaling
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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Recap Gyro-kinetics Numerics Results www-users.york.ac.uk/~bd512
DNS Scaling
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
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Recap Gyro-kinetics Numerics Results www-users.york.ac.uk/~bd512
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
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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)
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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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Recap Gyro-kinetics Numerics Results www-users.york.ac.uk/~bd512
DNS Scaling
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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DNS Scaling
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● ...
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Recap Gyro-kinetics Numerics Results www-users.york.ac.uk/~bd512
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
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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
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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)
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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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DNS Scaling
Results - Turbulence
Waltz, Austin, Burrell, Candy, PoP 2006
GS2 (NSTX)GYRO (DIII-D)
Dorland (from hammett 2002 PPPL)
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DNS Scaling
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
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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”
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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
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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
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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
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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
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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.
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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
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Recap Gyro-kinetics Numerics Results www-users.york.ac.uk/~bd512
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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