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A gyrokinetic model for the tokamak periphery
Princeton Plasma Physics Laboratory, November 1, 2018
R. Jorge , B. Frei, P. Ricci, A. Baillod, N. F. Loureiro
1
The Tokamak Periphery = Edge + SOL
q Core boundary conditions
q Heat exhaust
q Plasma fueling and ashes removal
q Impurity controlCore
SOL
Edge
2
q Low-frequency
Properties of Periphery Turbulence
q Wide range of temperatures
and densities
q Large Scale Fluctuations
q Small Scale Fluctuations
3
Arbitrary Collisionality
q Low-frequency
Properties of Periphery Turbulence
q Wide range of temperatures
and densities
q Large Scale Fluctuations Gyrokinetic Theory
q Small Scale Fluctuations
4
First Heroic Steps On The Way To An Edge Model
1Qin et al., Physics of Plasmas 14, 056110 (2007)2Hahm et al., Physics of Plasmas 16, 022305 (2009)3Dimits, Physics of Plasmas 19, 022504 (2012)
Qin et al. 20071 Hahm et al. 20092 Dimits et al. 20123
Small Scale Fluctuations EM O(!2) EM O(!2) EM O(!2)Large Scale Fluctuations EM O(!) ES O(!2) ES O(!2)Poisson’s Eq. ∫(… ) "3# Long-Wavelength Limit "(… ) "3#
Collisions No No No
5
Our Model
Single Particle Dynamics
GyrokineticTheory
Moment Hierarchy
Retain
q Both large scale (full-F) and small scale fluctuations
q Second order
q Full Coulomb collisions
q Numerical efficiency
How we proceed
6
Arbitrary Collisionalities
(still magnetized)
Magnetized Plasma
Fluctuation Scales and Amplitude
Single Particle Dynamics - Ordering Assumptions
Electromagnetic
kkk?
⇠ ✏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
7
2-Step Phase-Space
Reduction
Single Particle Dynamics – Hamiltonian Perturbation Theory
Particle LagrangianL(x,v)
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
Lagrangian�(R, vk, µ)
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
(cyclic coordinate !, conserved !)
8
Single Particle Dynamics – First Step
Split between parallel and perpendicular velocity
Describe guiding center motion
Large Scales ,
Start from
9
Single Particle Dynamics – Second Step
Introduce small scale fluctuations
Describe gyro center motion
Small Scales ,
Start from
10
Single Particle Dynamics – Equations of Motion
Including large scale
and small scale fluctuations
Curvature Drift
Polarization Drift
Non-Linear Drifts
Other Drifts
11
Including large scale
and small scale fluctuations
Single Particle Dynamics – Equations of Motion
Other Drifts
Non-Linear Forces
12
Conserved Adiabatic Invariant
Other Drifts
Non-Linear Forces
Single Particle Dynamics – Equations of Motion
13
q 5-D + time
q Full Coulomb Collisions
q Coupling to Maxwell’s equations (integro-differential system)
Challenges
Gyrokinetic Equation
These challenges can be successfully approached by using a moment hierarchy
From Single Particle to Particle Distribution
14
Expand GK Equation into a Set of 3D
Moment Hierarchy Equations
Retain necessary kinetic effects and no more
Our Goal – Turn Gyrokinetic Eq. Into a Hierarchy of Fluid-Like Eqs.
Advantages of a Moment Hierarchy Model
Set of fluid-like equations with reasonable computational cost
Tune the number of moments according to the desired level of accuracy (function of collisionality)
15
Reduce to the fluid model at high collisionality
1. Choose an orthogonal polynomial basis for F
2. Project kinetic equation onto basis
3. Obtain evolution equation for the basis coefficients
16
3 Steps To Build a Moment Hierarchy Model
Z (GK Eq.)@Npj
@t= ...
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
17
Fluid Operator(density, velocity,
temperature)
Forces included at p>0
Spatial evolution ofMoments + Fields
Collisions
Z (GK Eq.)From Gyrokinetic Equation to Moment Hierarchy
Time Evolution
18
Collisions
Phase Mixing(coupling with other moments)
Time Evolution Electric Field Drive
Example – 1D Linear Gyrokinetic Moment Hierarchy
19
Example – 1D Linear Gyrokinetic Moment Hierarchy
Analytical Closed formula for the Gyroaverage operation
Full finite Larmor radius effects
Collisions
Phase Mixing(coupling with other moments)
Time Evolution Kernel
20Not immediate…
Projection of the Full Coulomb Collision Operator
Cpj =
ZhC(F )iHpLjdvkdµ
AAADBnicdVHPixMxFE7HX+v6q6tHL8FeulDKjCzowZXFguxB2BXs7kJThyR902Y3ycwmmUoJc/f/8C54EK9everJ/8aZ2SrMFB8h7+V93/cg32OZFNaF4e9OcO36jZu3tm5v37l77/6D7s7DE5vmhsOYpzI1Z4xakELD2Akn4SwzQBWTcMouRhV+ugRjRarfuVUGU0XnWiSCU1e24u7+6L3Pzgu8j4nQDhMJiXuBR/3Xu5gYMV+4l/gwzvCb+BzP8DImGTVUSpDli6g87vbCYVgH3iyiddFD6ziOdzpfyCzluQLtuKTWTqIwc1NPjRNcQrFNcgsZ5Rd0DpPcJc+nXugsd6B5A/NUWUXdYlBmu1KsysyumpxE5nYBetbsMgOXutmqbSxbGj7wVCmqZ54wKKprLrSHy7x2q2hRoKKU8/9LkAyXjNrSFmJq5MrgEjLQkP1VbSDmn6o1b+kWtPDL2Fe59V+mBuWp3DI2sYNVkmpnN0YfFZNo6klF41T6o4LIfi8iZrcolxy1V7pZnDwdRuEwervXO3i1XvcWeoyeoD6K0DN0gA7RMRojjj6hH+gn+hV8DD4HX4NvV9Sgs9Y8Qo0Ivv8BEwD9hw==AAADBnicdVHPixMxFE7HX+v6q6tHL8FeulDKjCzowZXFguxB2BXs7kJThyR902Y3ycwmmUoJc/f/8C54EK9everJ/8aZ2SrMFB8h7+V93/cg32OZFNaF4e9OcO36jZu3tm5v37l77/6D7s7DE5vmhsOYpzI1Z4xakELD2Akn4SwzQBWTcMouRhV+ugRjRarfuVUGU0XnWiSCU1e24u7+6L3Pzgu8j4nQDhMJiXuBR/3Xu5gYMV+4l/gwzvCb+BzP8DImGTVUSpDli6g87vbCYVgH3iyiddFD6ziOdzpfyCzluQLtuKTWTqIwc1NPjRNcQrFNcgsZ5Rd0DpPcJc+nXugsd6B5A/NUWUXdYlBmu1KsysyumpxE5nYBetbsMgOXutmqbSxbGj7wVCmqZ54wKKprLrSHy7x2q2hRoKKU8/9LkAyXjNrSFmJq5MrgEjLQkP1VbSDmn6o1b+kWtPDL2Fe59V+mBuWp3DI2sYNVkmpnN0YfFZNo6klF41T6o4LIfi8iZrcolxy1V7pZnDwdRuEwervXO3i1XvcWeoyeoD6K0DN0gA7RMRojjj6hH+gn+hV8DD4HX4NvV9Sgs9Y8Qo0Ivv8BEwD9hw==AAADBnicdVHPixMxFE7HX+v6q6tHL8FeulDKjCzowZXFguxB2BXs7kJThyR902Y3ycwmmUoJc/f/8C54EK9everJ/8aZ2SrMFB8h7+V93/cg32OZFNaF4e9OcO36jZu3tm5v37l77/6D7s7DE5vmhsOYpzI1Z4xakELD2Akn4SwzQBWTcMouRhV+ugRjRarfuVUGU0XnWiSCU1e24u7+6L3Pzgu8j4nQDhMJiXuBR/3Xu5gYMV+4l/gwzvCb+BzP8DImGTVUSpDli6g87vbCYVgH3iyiddFD6ziOdzpfyCzluQLtuKTWTqIwc1NPjRNcQrFNcgsZ5Rd0DpPcJc+nXugsd6B5A/NUWUXdYlBmu1KsysyumpxE5nYBetbsMgOXutmqbSxbGj7wVCmqZ54wKKprLrSHy7x2q2hRoKKU8/9LkAyXjNrSFmJq5MrgEjLQkP1VbSDmn6o1b+kWtPDL2Fe59V+mBuWp3DI2sYNVkmpnN0YfFZNo6klF41T6o4LIfi8iZrcolxy1V7pZnDwdRuEwervXO3i1XvcWeoyeoD6K0DN0gA7RMRojjj6hH+gn+hV8DD4HX4NvV9Sgs9Y8Qo0Ivv8BEwD9hw==AAADBnicdVHPixMxFE7HX+v6q6tHL8FeulDKjCzowZXFguxB2BXs7kJThyR902Y3ycwmmUoJc/f/8C54EK9everJ/8aZ2SrMFB8h7+V93/cg32OZFNaF4e9OcO36jZu3tm5v37l77/6D7s7DE5vmhsOYpzI1Z4xakELD2Akn4SwzQBWTcMouRhV+ugRjRarfuVUGU0XnWiSCU1e24u7+6L3Pzgu8j4nQDhMJiXuBR/3Xu5gYMV+4l/gwzvCb+BzP8DImGTVUSpDli6g87vbCYVgH3iyiddFD6ziOdzpfyCzluQLtuKTWTqIwc1NPjRNcQrFNcgsZ5Rd0DpPcJc+nXugsd6B5A/NUWUXdYlBmu1KsysyumpxE5nYBetbsMgOXutmqbSxbGj7wVCmqZ54wKKprLrSHy7x2q2hRoKKU8/9LkAyXjNrSFmJq5MrgEjLQkP1VbSDmn6o1b+kWtPDL2Fe59V+mBuWp3DI2sYNVkmpnN0YfFZNo6klF41T6o4LIfi8iZrcolxy1V7pZnDwdRuEwervXO3i1XvcWeoyeoD6K0DN0gA7RMRojjj6hH+gn+hV8DD4HX4NvV9Sgs9Y8Qo0Ivv8BEwD9hw==
with
q Bilinear
q Tensorial Nature
q Gyroaveraging Operation
q No parallel/perpendicular velocity symmetries
21
Collision Operator - Spherical Harmonic Decomposition
Expand the Rosenbluth potentials
in a Taylor series
with spherical harmonics as coefficients
Electrostatic potential of a charge
distribution F in velocity space
Spherical Harmonics
GyroangleBasis Tensors
!" #$
22
Collision Operator – Expansion in Spherical Harmonics
Large Scales
Small Scales
Gyroaveraging Procedure
Hp(vk)Lj(µ)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
23
After integration
Moments of the Collisional Operator
Cpj =
ZhC(F )iHpLjdvkdµ
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
Kernel × f(n) × moments "#$ of F
24
Physics That We Are Able To Capture – High Collisionality
Drift-Reduced Braginskii
equations retrievedSemi-collisional closure
with
Jorge et al., JPP 83, 6 (2018)
25
Towards a Numerical Implementation
Numerical and theoretical investigation of linear modes
GK non-linear model reduced to the drift-kinetic linear limit
q Compute collisional/hierarchy coefficients
q Solve time evolution problem
q Perform eigenvalue analysis
q Compare with collisionless result
26
Electron Plasma Waves –Damping at Arbitrary Collisionality
Collisionality
Ø One spatial and two velocity dimensions
Ø Electron perturbations only
Ø Time-evolution of !"" ∼ $
q Collisional and Landau Dampingcomputed for the first time with the full Coulomb collision operator
Parameterscan
Three collision operators
27
Electron Plasma Waves – Entropy Mode
q Electron-ion collisions known to yieldlong-time zero-frequency behaviour
q Full Coulomb collisions reduce the entropy mode damping rate
Zero-frequencyentropy mode
Jorge et al., to be submitted to JPP
28
First Study on Coulomb Collision Effects on the Drift-Wave Instability
Eigenmode Spectra
Jorge et al., PRL 121, 165001 (2018)
Ø Two spatial and two velocity dimensions in slab geometry
Ø Electron and ion perturbations
Ø Finite background density gradient
q Important deviations between full Coulomb collision operator and presently used collision operators
Instability Growth Rate
29
Summary
This Work Qin et al. 20071Hahm et al.
20092Dimits et al.
20123Madsen et al.
20134Mandell et al. 20185
Large Scales EM O(!2) EM O(!) ES O(!2) ES O(!2) ES O(!) ES O(!)Small Scales EM O(!2) EM O(!2) EM O(!2) EM O(!2) EM O(!) ES O(!)
Collisions Yes No No No No Simplified
Poisson’sEq. Moments !(… ) "3# Long-WavelengthLimit !(… ) "3# Long-WavelengthLimit, Padé Moments$∥∗ Exact Exact Exact Exact O(!) B
1 Qin et al., Physics of Plasmas 14, 056110 (2007)2 Hahm et al., Physics of Plasmas 16, 022305 (2009)3 Dimits, Physics of Plasmas 19, 022504 (2012)4 Madsen, Physics of Plasmas 20, 072301 (2013)5 Mandell et al., J. Plasma Phys 84, 905840108 (2018)