Kinetic simulation of heat pulse propagation through the

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and is supported by the U. S. Department of Energy, Office of Science,

Office of Fusion Energy Sciences and Office of Advanced Scientific Computing Research through the Scientific Discovery through Advanced Computing (SciDAC) project on Plasma-Surface Interactions

LLN L-PR ES-753272

Kinetic simulation of heat pulse propagation through the tokamak scrape-off layer

I. Joseph, M. A. Dorf, M. Dorr 1Lawrence Livermore National Laboratory

International Conference on Plasma Surface Interactionsin Controlled Fusion Devices

Princeton, New JerseyJune 22nd, 2018

I. Joseph, PSI 2018 2

Motivation: Prediction of particle and heat fluxes in future magnetic fusion devices such as ITER

• The PSI SciDAC is developing coupled models for the dynamic interaction between plasma & material surfaces at the edge of a magnetically confined fusion energy reactor

• Our goals are to – Determine importance of ELMs on

impurity production & material erosion– Understand dynamic recycling during

transient events

Plasmad ~ 10-100 mt ~ 10-7-10-3 s

Sheathd ~ 10-5-10-3 mt ~ 10-12-10-10 s

Materiald ~ 10-10-10-6 mt ~ 10-12-10-6 s


Goal: Simulate heat pulse propagation through the scrape-off layer (SOL) edge plasma• Goal: Determine the erosion rate of material

surfaces that are impacted by large transient events such as edge localized modes (ELMs)

• Study the transient behavior of a heat pulse as it travels along a flux tube using the 4D drift-kinetic COGENT code– Results will be benchmarked against heat pulse

test problems that have been used to compare physics models and numerical algorithms

• An important goal will be to determine under what conditions energetic particle tails are found to form and whether kinetic effects impact quantitative results

Plasma Phys. Control. Fusion 54 (2012) 045002 E Havlıckova et al

Figure 2. Parallel target energy flux for the reference case calculated by (left) the PIC code, (middle) the Vlasov code and (right) the fluidcode. In the fluid model, the parallel ion viscosity is limited and the total energy source due to the ELM is prescribed to be shared by ionsand electrons as 3 : 1, see the discussion below.

3. Results and comparison

3.1. Reference simulation

An ELM crash described by a set of pedestal parameters typicalfor JET type I ELMs is taken as a reference case (see [12], JETshot number 62221):

nped = 5 × 1019 m−3, Tped = 1.5 keV, tELM = 200 µs.

Geometric parameters are

L∥ ≈ 40 m, Ls ≈ 25 m.

From (14), we can obtain corresponding ELM power!WELM ≈ 0.4 MJ using the same values as in [3]: !RSOL =10 cm, R = 3 m and Lpol = 2.6 m (Lpol = Ls sin αu withαu = 6◦), but note that !RSOL is just a scaling parameterfor !WELM. In PIC simulations, the angle α is fixed (αu =αt = 6◦), while in this paper, we generally distinguish betweentarget and upstream angles.

Figure 2 shows the main analysed quantity, the totalparallel energy flux at the target Q∥, electron and ioncomponents Q∥,e and Q∥,i and in the case of the fluid codealso thermal fractions q∥,e and q∥,i due to heat conduction.Note that Q∥ is the energy flux deposited on the target in thedirection of the magnetic field, not the power load which isreferred to the flux perpendicular to the surface.

The ELM energy is transported along the magnetic fieldpreferentially by convection in all cases (low-collisionalitycase, no plasma–neutral interactions present). In the fluidmodel (figure 2 right and more details in the appendix),we observe two main time scales of the parallel transportcorresponding to conduction (the structure appearing betweent ≈ 10–80 µs) and convection (the main maximum, the timescale can be estimated as τ∥ ≈ L∥/cs,ped ≈ 104 µs). Thefirst structure does not appear so strongly in the PIC modeland could probably be modified or eliminated using heat fluxlimiters. On the other hand, we do not see any response atthe target before 10 µs in comparison with the rise of theenergy flux at the target in the PIC and Vlasov simulationsin figure 2 (left and middle), which appears due to fast ELMparticles and a reaction of the background plasma (the timescale is τ e

∥ ≈ L∥/veth,ped ≈ 2.4 µs, ve

th,ped is the electron thermal

speed in the pedestal). Such fast response is not observed inkinetic simulations if the transient propagates in the vacuum.These two features, clearly visible in log scale, define the maindifferences in fluid and kinetic results.

The Vlasov and PIC simulations are in fair agreementas far as the total energy flux is concerned, though the peakvalue is slightly underestimated by the Vlasov model. In bothcases, the input energies are equally shared between ions andelectrons (unlike the fluid code where the energy source isredistributed as SE,i/SE,e = 3, see section 3.3). In general, theVlasov code displays a lower energy flux for ions but a higherone for electrons compared with PIC. It must be noted that theVlasov code is completely collisionless, whereas collisions areincluded in the PIC simulation. The effect of collisions maythus be to enhance the energy transfer from the electron to theion population. A simulation where collisions are removedfrom the PIC code indeed produces results very similar to thoseobtained with the Vlasov code, for both the ion and electronenergy fluxes [23].

It is worth mentioning that the power to the targetcan also be described well by a free-streaming model [24](no collisions, no background plasma). Earlier, it wasdemonstrated that the heat pulse shape at the target calculatedanalytically agrees with the shape typically observed inexperiment [25]. Results from the free-streaming model (withan ad hoc assumption that electrons transfer all of their parallelenergy to the ions so that both species are quickly acceleratedto the sound speed) show good agreement with both the Vlasovand PIC codes. These comparisons will be published in [26].To properly compare the free-streaming model with the kineticcodes, the analytic impulse response must be numericallyconvolved with the temporally and spatially distributed sourcesthat were used in the kinetic codes. However, in the case whenthe source is localized in space at the distance L∥ from thetarget and distributed in time as a step function with durationtELM, an analytic solution for the parallel energy flux to onetarget can be found as follows:

Q∥

ϵinp= 1

3τ∥

tELM√

2π

1t

exp(

−12

(τ∥

t

)2)

+1

2tELM

(1 − erf

(τ∥√2t

)), if 0 < t ! tELM,

5

E. Havlikoca, Plasma Phys. Control. Fusion 54 (2012) 045002

DWELM = 0.4 MJtELM = 200 µs


Plan: develop 4D COGENT model for ELM-relevant simulation of transient heat loads

• Develop a 4D COGENT model for simulation of transient heat loads relevant to edge-localized mode instabilities (ELMs)− Implement ELM-relevant heat pulse model within the COGENT code in simplified

geometry & perform verification studies − Predict the plasma fluxes impinging on the sheath− Compare heat pulse simulations to fluid models and experimental data− Develop models for nonlinear sheath BCs & for accurately exchanging data with

sheath code

• As new capabilities are developed jointly with the ESL team & AToM-SciDAC, we will develop new capabilities− Unlike species collisions: electron-ion− Neutral physics models including ionization, charge-exchange, radiation− Implicit treatment of both collisionless and collisional kinetic transport− 5D simulation of drift-kinetic plasma instabilities


Overview of COGENT

• COGENT is a full-F continuum gyrokinetic (GK) code– Multi-species gyro-kinetic equations and gyro-Poisson field equations– Fokker-Planck collision operators– At present, the code handles the long-wavelength drift-kinetic regime kr<<1, but

extension to short wavelength is planned for the future– Reduced physics models are also available: model collision operators, vorticity

equation, fluid electrons, etc.

• To date, the main research thrust has been focused on obtaining axisymmetric solutions in realistic tokamak geometry, in isolation from wall physics– A non-axisymmetric (5D) version of the code has recently become operational and has

been successfully verified in the simulations of the collisionless drift (universal) instability in simplified slab geometry

• We will focus on improving the capabilities necessary for modeling plasma-wall interactions in the divertor region


The COGENT team includes both physics & math developers

• COGENT is part of the Edge Simulation Laboratory (ESL), the integrated modeling (AToM) SciDAC, and the PSI SciDAC– ESL: FES physics team at LLNL & ASCR applied math team at LBNL & LLNL

• Algorithmic Capabilities– 4th Order Finite Volume Discretization and interpolation

• Discretization errors are bounded, even near the X-point of a separatrix– Mapped multiblock grid technology

• Flux surfaces in different topological regions are mapped from the physical toroidal geometry onto topologically rectangular grid blocks

• High-order interpolation is used to provide data communication in the region where grid blocks overlap, e.g. near the X-point

– Implicit algorithms & IMEX capability• IMEX capability has been successfully used to treat the Fokker-Planck

collision operator implicitly for like-species collisions • Improves simulation efficiency well into the collisional regime, a regime that

is notoriously difficult to treat using kinetic codes


Example: COGENT has recently obtained self-consistent results for the tokamak pedestal*

• Results of an axisymmetric (4D) COGENT simulation of cross-separatrix plasma transport in a DIII-D discharge using full Fokker-Plank ion-ion collisions and self-consistent 2D electrostatic potential variations with the reduced vorticity model for isothermal electrons.

*M. Dorf and M. Dorr, Contrib. Plasma Phys. (2018) DOI: 10.1002/ctpp.201700137.

-1.5

-1

-0.5

0

0.5

1

1.5

-6 -4 -2 0 2

(a)

V|| (co-Ip)

D

ni

Er

Ti

R-Rsep (cm)

ni (5x1019m-3)

Ti (300 eV)

Er (20 kv/m)

D (3.4 m2/s)

Vi,|| (20 km/s)

-40

-30

-20

-10

0

10

20

0 1 2 3

(b)

Time (ms)

Er (kv/m)

Vi,|| (km/s)

!" !#⁄ %Φ !#⁄

Temperature Potential Outer midplane @ 3 ms


ELM Benchmark (*): determine heat flux due to ELMs for JET-like pedestal parameters

• JET-like SOL parameters mi = 2 mp

– Bt = 3 T, R = 3 m, Lpol = 8.3 m nped = 5x1019 m3

– angle = 6o, Bp/Bt = 0.11 Tped = 1.5 keV– 2 L|| = 80 m, Lsrc = 25 m Cs,ped = 3.8x105 m/s

• Maxwelian Source– Source parameters are set to the pedestal parameters (A=1.2)

• (*) References1. E. Havlickova, W. Fundamenski, D. Tskhakaya, et al., Plasma Phys. Control. Fusion 54,

045002 (2012).2. A. Chankin and D. P. Coster, Contrib. Plasma Phys. 54, 493 (2014).3. E. L. Shi, A. H. Hakim, and G. Hammett, Phys. Plasmas 22, 022504 (2015).4. T. D. Rognlien, R. H. Cohen, D. D. Ryutov, et al., J. Nucl. Mater. 438, S418 (2013).

! = #$%&'()$%&*

+/- ./ 01-/'234 /)$%& !567 = 89:;< =5,:;</?567


We can develop understanding using kinetic ions and an adiabatic electron model

• Collisionless kinetic ion model

• Adiabatic = Boltzmann electron model (fixed Te)

• Heat flux at target plate (Assume Te= Ti at target plate)

!" = !"$%&'(% + *& log .//.1 !"$%&'(% = 23*& log 4/||3 6&/28*&

9& = 2*&Γ/

9/ = 9/,<=>? +9/,<=>@ +9/,$%&'(%

9/,$%&'(% = !"$%&'(%Γ/

9/,<=>? = A3*/Γ/

9(=( = 9/+ 9&


COGENT results: The heat is on! • Parameters for these cases

– Resolution: RxZ=8x32, v|| x µ = 32x32– nSOL = 2x1019 m3

– TSOL = 175 eV, Te=210 eV– Tsrc = 1500 eV– Ssrc = 9x1023 /sm3

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

7

time (ms)

Q|| (G

W/m

2 )

Heat Flux

QpariQsheathQiQeQtot

0 10 20 30 40 50 60 700

2

4

6

8

10

length (m)

Ni (

1019

m−3

), Ti

(100

eV)

Plasma Parameters

Ni,initNi,finalTi,initTi,final

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

SOL Plasma

time (ms)

Ni (

1019

m−3

), Ti

(100

eV)

Ni,upNi,dnTi,upTi,dn


COGENT results: tELM = 200µsMoments• Tup rises to 1 keV• Tdn rises to 0.9 keV• Heat flux Q||=5 HW/m2 peaks after

tELM

• Temperature profile inverts after tELM

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

time (ms)

Q|| (G

W/m

2 )

Heat Flux

QpariQsheathQiQeQtot

0 10 20 30 40 50 60 700

2

4

6

8

10

length (m)

Ni (

1019

m−3

), Ti

(100

eV)

Plasma Parameters

Ni,initNi,finalNi,peakTi,initTi,finalTi,peak

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

2

4

6

8

10

SOL Plasma

time (ms)

Ni (

1019

m−3

), Ti

(100

eV)

Ni,upNi,dnTi,upTi,dn


COGENT results: tELM = 200µs Particle distribution function• PDF is not Maxwellian

– Sonic outflows at target plates – T||<Tperp=1.5keV at midplane– Transition to ½ Maxwellian in v||

near target plates

−10 −5 0 5 100

2

4

6

8

10

12

14

16

v||/v0

(µ/µ0)1/2

log10(fi)

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

0 0.2 0.4 0.6 0.8

−10

−5

0

5

10

z/L0

v ||/v0

log10(fi)

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−10 −5 0 5 100

2

4

6

8

10

12

14

16

v||/v0

(µ/µ0)1/2

log10(fi)

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

−10 −5 0 5 100

2

4

6

8

10

12

14

16

v||/v0

(µ/µ0)1/2

log10(fi)

−6

−5.5

−5

−4.5

−4

−3.5

−3

−2.5

−2

Midplane

Targetplate


Dependence of maximum heat flux on tELM is sublinear

• Maximum heat flux achieved ~ t|| = 60 µs after tELM

• Magnitude of heat flux Q|| has a power law dependence on tELM

with exponent < 1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

τELM (ms)

τ max

(ms)

Time to Peak Heat Flux

τmaxτelm+τs||

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

1

2

3

4

5

6

7

8

τELM (ms)

Qm

ax (G

W/m

2 ), Ti

(100

eV)

Maximum Heat Flux

QmaxQi,maxQe,max


We are working towards simulations with 2 kinetic species: electrons and ions

• Example: Solution with a self-consistent sheath

• Results of a simulation of the sheath using kinetic electrons with BGK

collisions, kinetic hydrogen and a full 2D electrostatic potential model

– Resolution: 8X x 48Y x 32V|| x 24µ

• Low density ne ~ 2x1016/m3 allows one to resolve the sheath ld= 0.2 mm

in a domain of 1 cm length for temperature Te ~ 13.5 eV, Ti = 4.5 eV

2 4 6 8

5

10

15

20

25

30

35

40

45

x index

y in

dex

phi (eV)

0.5

1

1.5

1 cm0 500 1000 1500 2000 2500 3000 3500 4000 4500

0

1

2

3

4

5

6

time

heat

flux

(W/m

2 )

qiqeqi+qe


Implementation of “gyrokinetic sheath” BCs* allows for efficient quasineutral simulation of a large domain

0 0.5 10

0.5

1

x (cm)z

(m)

ni

0.5

0.6

0.7

0.8

0.9

0 0.5 10

0.5

1

x (cm)

z (m

)

Ti

0.84

0.86

0.88

0.9

0.92

0 0.5 10

0.5

1

x (cm)

z (m

)

miniV||i

−0.5

0

0.5

0 0.5 10

0.5

1

x (cm)

z (m

)

V||i

−1

−0.5

0

0.5

1

0 0.5 10

0.5

1

x (cm)

z (m

)ne

0.3

0.4

0.5

0.6

0.7

0 0.5 10

0.5

1

x (cm)z

(m)

Te

0.9995

1

1.0005

0 0.5 10

0.5

1

x (cm)

z (m

)

meneV||e

−0.01

0

0.01

0 0.5 10

0.5

1

x (cm)

z (m

)

V||e

−5

0

5

0 0.5 10

0.5

1

x (cm)

z (m

)

ρ

0.1

0.2

0.3

0 0.5 10

0.5

1

x (cm)

z (m

)Φ

0

0.5

1

1.5

2

0 0.5 10

0.5

1

x (cm)z

(m)

Ez

−4−20246

0 0.5 10

0.5

1

x (cm)

z (m

)

J||

−1

0

1

• Gyrokinetic Poisson: retain polarization current but eliminate vacuum polarization

• “Sheath BC” = electrons reflected if mev||2 /2 < f in last grid cell in domain before boundary

• Parameters: 8R x 16Z x 32v|| x 24µninit = 1019 m-3

Tinit = 100 eVme/mi = 0.01Snapshot at t = 3 µs

*P. Collela, M. R. Dorr, D. D. Wake, J. Comp. Phys. 149, 168 (1999)E. L. Shi, Ph.D. Dissertation, Princeton University (2017)


Future Work: explore the importance of kinetic plasma effects

• Kinetic effects have the potential to strongly impact surface evolution– Heat fluxes are strongly flux-limited during the early phase of an ELM– Energetic ions can alter the

• ion saturation current• charge exchange rate• rates for implantation, sputtering, and defect formation

– Energetic electrons can alter the• heat flux directed on PFCs• sheath potential which determines ion energy on impact• rates for threshold processes such as ionization, recombination, and radiation

– It is known that kinetic effects alters the quantitative ratio of electron to ion heat flux 1:1 vs. 3:1 (Havlikova PPCF 2012)

• Kinetic processes are potentially important for interpretation of experimental data and for validation exercises – Non-Maxwellian distributions can change the interpretation of standard

diagnostic techniques based on Langmuir probes and impurity radiation


Conclusions

• PSI SciDAC is developing dynamically coupled plasma-wall models– Ultimate goal is to determine the erosion rate of material surfaces that

are impacted by large transient events such as ELMs

• ELM heat pulse benchmark has been simulated using an adiabatic Boltzmann electron model– Results appear to match reasonably well– Still need to perform numerical convergence study

• Future work will focus on two kinetic species: both electrons and ions

Documents

Kinetic simulation of heat pulse propagation through the