self-collision algorithms for fokker-planck operator ... · the operator can be linearized with...

Self-collision algorithms for Fokker-Planck operator simulation in full-f PIC codes and

direct verification of classical transport

8th IAEA technical meeting on Theory of Plasma Instabilities

T. Nicolas1, H. Lütjens2, J.-F. Luciani2, J. P. Graves1

EPFL, Swiss Plasma Center, SwitzerlandEcole Polytechnique, Centre de Physique Théorique, Palaiseau, France

Outline

Motivation : long hybrid kinetic/MHD simulations

A solution to the background reaction problem using effective Maxwellian distributions

Results in local 3D-in velocity space

Classical transport (4D or 5D)

Sawteeth are almost inevitable in tokamaks, particularly preoccupying for ITER

Sawteeth are almost inevitable in tokamaks, particularly preoccupying for ITER

Frequent sawteethFrequent sawteeth

Long sawteeth (>10 s)

Long sawteeth (>10 s)

DisruptionDisruption

Confinement degradation by NTM

Confinement degradation by NTM

3

fast α particles

fast α particles

Motivation: Sawteeth, a serious problem for ITER

[Porcelli PPCF 1996]

(modelling)

[Chapman N

F 2010]

(empirical s

caling)

Impurity penetration

Impurity penetration

[Wesso

n N

F 1989]

The hybrid approach to model energetic particle effects

The hybrid kinetic/fuid model : the natural tool to understand fast particles impact

The hybrid kinetic/fuid model : the natural tool to understand fast particles impact

4

XTOR-K code:● 3D Extended MHD implicit solver● Hybrid kinetic/MHD coupling● Kinetic PIC ions with full cyclotronic

orbits● Ohm’s law electric field

Allows studies of: ● Sawtooth stabilization by fast

particles (FLR effects on MHD)● Self-consistent ion neoclassical

transport (with collisions)● Kinetic Alfven modes excitation

Particle PushParticle Push

MHD evolutionMHD evolution

Converged ? Converged ?

Next time stepNext time step

∇⋅Πi ,k

YES

NO

E(t

), B

(t)

Fie

lds

alon

gT

raje

ctor

y

Why are collisions needed

5

Alpha particles slow down on background plasma (mainly electrons)

Radial transport of high Z impurities [cf. M. Raghunathan’s talk I-8]

When background is also kinetic:

● Relaxation toward Maxwellian after MHD event

● Self-consistent Ion neoclassical transport

Prevent accumulation points in distribution function

Requires self-collision algorithmRequires self-collision algorithm

The background reaction problem

6

dfdt

=C [ f , f ] C [ f , f ]=− ∂∂ v

⋅(F f −12

∂∂ v

⋅(D̄ f ))

F and d can be computed numerically for an arbitrary distribution (via Rosenbluth potentials) but this is a very expensive operation.

They are known analytically for Maxwellians

The operator can be linearized with respect to a neighboring Maxwellian:

The Landau Fokker-Planck Collision OperatorThe Landau Fokker-Planck Collision Operator

C [ f , f ]≃C [ f M ,δ f ]+ C [δ f , f M ]⏟Background reaction

(C [ f M , f M ]=0 )

How can we handle the background reaction ? How can we handle the background reaction ?

Δ v=F(v )Δ t+d ( v)N (1,0)√Δ tIto-Rule SDE:

The issue with standard collision methods

7

The two-weight schemeThe two-weight scheme The binary collisionsThe binary collisions

dfdt

=C [ f , f ] C [ f , f ]=− ∂∂ v

⋅(F f −12

∂∂ v

⋅(D̄ f ))

[Brunner et al. PoP 1999, Vernay et al. Varenna 2010, Sonnendrücker et al. JCP 2015]

[Takizuka Abe JCP 1977, Nanbu PRE 1997, Bobylev Nanbu PRE 2000]

Weighted markers representing δf collide on Maxwellian distributions. Background reaction is approximated

Weights are evolved with sophisticated techniques to limit ‘weight spreading’

Issue: XTOR-K formulation is full-f, heavy reformulation needed

Neighbour particles are paired randomly to undergo collision

Reproduces the Landau integral exactly in the limit N → ∞, Δt → 0

Issue: In 6D, requires a very expensive sort accross processes in a parallelized framework (manageable in 5D or less)

Outline





The principle of the new approach

9

C [ f , f ]≃C [ f M (T ,V ) ,δ f ]+C [δ f , f M (T ,V )]f =f M (T ,V )+δ f

f M (T eff ,V eff )=f M (T ,V )+δ f M

C [ f M (T eff ,V eff) , f ] = C [ f M (T ,V ) , f ] + C [δ f M , f ]≃ C [ f M (T ,V ) ,δ f ] + C [δ f M , f M (T ,V )]

Distribution function and collision operator are decomposed around a Maxwellian distribution

A Maxwellian with different temperature and velocity can be related to fM(T , V)

Collisions of test particles of f let appear a background-reaction like term

Can we choose Teff and Veff so that the background reaction term is emulated?

Can we choose Teff and Veff so that the background reaction term is emulated?

Choosing Teff and Veff

10

The background reaction ensures conservation properties:

● momentum and energy lost by the δf part of the distribution must be given to the bulk

There may exist Teff and Veff such that energy/momentum is conserved

⟨ΔΕ⟩=⟨2 vF+2d ⊥+d∥⟩Δ t+⟨F2⟩Δ t 2

Δ v=F(v )Δ t+d (v)N (1,0)√Δ t

⟨ΔP ⟩=⟨vv

F ⟩Δ t

Teff and Veff are obtained by setting the first order terms to zero

Teff and Veff are obtained by setting the first order terms to zero

Parallelization is straightforward and efficient [Nicolas et al. PPCF 2017]

Outline





Two maxwellians without velocity shift (1)

12

Distribution f = fa + fb , where fa and fb are both Maxwellian with different temperature and density

Compares very well with Takizuka-Abe

Deviation with theoretical prediction as expected

The agreement remains very good even when na ~ nb and significant ΔT

The error in the energy transfer rate is of order

∼nb

na(ΔTT a

)2

Two maxwellians with velocity shift

13

Distribution f = fa + fb , where fa and fb are both Maxwellian with different temperature, density and average velocity (beam-like study)

Good agreement in the small density case

The agreement becomes less good when density is increased, but the order of magnitude remains good

Parameter study in nb/na, |Vb|/Vth,a space

14

Distribution f = fa + fb , where fa and fb are both Maxwellian with different temperature, velocity and density

Ratio between the momentum transfer rates in the two algorithms

Clearly, two regimes where the error is small (overestimation by less than one third) are visible :

● Beam case: Low density, arbitrary velocity

● Cold/hot slow bubble case: low velocity, arbitrary density

The algorithm tends to overestimate the transfer rates

More than 25 % error

Fast particles slowing down

15

Fast ion slowing down is mainly on electrons, but we must verify that self-collisions don’t introduce spurious effects

The agreement with binary collisions is very bad unless the second order term is dealt with

With this δ F correction the agreement becomes again very good

⟨ΔΕ⟩=⟨2 vF+2d ⊥+d∥⟩Δ t+⟨F2⟩Δ t 2

( F+δF )2Δ t2+2δF v Δ t=0

Outline





Direct verification of classical transport

17

A lot of literature on Braginskii equations for Classical transport. In particular● Original works by Braginskii: [Braginskii Sov. Phys. JETP 1958, Braginskii Reviews of Plasma

Physics 1965]

● Corrections in moderate collisionality regimes, using cartesian or spherical harmonics expansions and numerical techniques: [Epperlein and Haines Physics of Fluids 1986, Bendib et al. PoP 2002]

Although some of this work is numerical, it is difficult to find a direct verification of the Braginskii classical ion transport equations in the literature.

Direct verification of classical transport equations: Simulate directly the kinetic equation in presence of gradients and check that the plasma quantities

evolve accordingly

Direct verification of classical transport equations: Simulate directly the kinetic equation in presence of gradients and check that the plasma quantities

evolve accordingly

Teff Algorithm versus BraginskiiAnd Binary Collisions

18

∂T∂ t

=2√29√π

ρth2∇⋅( 1

√T∇T )

Braginskii normalized heat equation for ion classical transport :

Teff Algorithm overestimates classical transport by a factor of approximately 1.5

Binary collisions seem to underestimate classical transport (by a similar factor)

It does not seem to be a convergence problem

Same phenomenon in 4D or 5D

The study of [Ma et al., Computer Physics Communications, 1993] using binary collisions

19

[Ma et al. CPC 1993]

Finite differenceFinite difference PIC SimulationPIC Simulation

This study argues that binary collisions are in check with classical transport. Braginskii is not cited and the formula does not have the correct dependency…

∂T∂ t

=2√29√π

ρth2∇⋅( 1

√T∇T ) ∂T

∂ t=

4 ν0

3mωc2 ∇⋅(T ∇ T )

[Ma et al. CPC 1993][Braginskii RPP 1965]

Conclusion

The new method yields good agreement with binary collisions in 3D in velocity space configuration

It is easily parallelized for parallel large scale PIC codes

It can simulate classical transport, although it is too fast by a factor of 1.5

It is not understood why binary collisions do not agree with Braginskii classical ion transport

Parallel implementation

21

Setting the red terms (sums on the distribution) to zero is done by varying the Teff and Veff parameters using a Newton algorithm

In a parallel environment, particles in a given cell of the PIC algorithm live on different parallel tasks

To parallelize the algorithm, it is only necessary to compute the sums locally and then use MPI_Reduce with operator sum

The obtained Teff and Veff parameters are finally broadcasted to all the tasks

⟨ΔΕ⟩=⟨2 vF+2d ⊥+d∥⟩Δ t+⟨F2⟩Δ t 2

⟨ΔP ⟩=⟨vv

F ⟩Δ t

[Nicolas et al. PPCF 2017]

Two maxwellians without velocity shift (2)

22

The algorithm also captures the deviation δf with respect to the Maxwellian

Why are collisions needed

23

Prevent accumulation points in distribution function

Noise reduction

Alpha particles slow down on background plasma (mainly electrons)

When background is also kinetic

● Relaxation toward Maxwellian after MHD event

● Self-consistent Ion neoclassical transport

Requires self-collision algorithmRequires self-collision algorithm

Numerical reasonsNumerical reasons Physical reasonsPhysical reasons

self-collision algorithms for fokker-planck operator ... · the operator can be linearized with...

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