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

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

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)