Method for Loading Marker Particles forArbitrary Distribution Functions and

Application for Simulation ofHigh-Energy Ion Dynamics in Tokamak Plasmas

Andreas Bierwage1, Claudio Di Troia2,

Sergio Briguglio2, Giuliana Fogaccia2,Gregorio Vlad2 and Fulvio Zonca2

1: Naka Fusion Institute, JAEA, Ibaraki, Japan2: Frascati Research Center, ENEA, Rome, Italy

7th Supercomputing in Nuclear Application and 3rd Monte CarloJoint International Conference, Tokyo, Japan, October 17-21, 2010

This work has been partially supported by EFDA co-funded GOTIT fellowship.SNA 2010, Oct. 20 — 1/20

Motivation & backgroundSituation: In modern tokamaks, energetic ion orbits deviatesignificantly from magnetic flux surfaces due to magnetic drift motion.This results in a complicated kinetic phase-space distribution function.

X / aZ

/ a

large drift (eps=0.3, ρH0

/a=3×10−2)

−0.5 0 0.5 1

−0.5

0

0.5

X / a

Z /

a

small drift (ε=3×10−3, ρH0

/a=3×10−4)

−0.5 0 0.5 1

−0.5

0

0.5

v|| < 0

v|| > 0

Task:Given an approximateanalytical model andexperimental data as areference ...

→... develop a methodto construct an exactkinetic equilibrium ...

→... for initial-value particlesimulations &benchmarking.

SNA 2010, Oct. 20 — 2/20

Outline1 Background2 Method3 Software tool VSTART4 Properties of the kinetic distribution function5 Summary6 Outlook7 References and acknowledgments

About the software tool “VisualStart” (VSTART):

A new software tool, VSTART, has been developed for the purpose ofinitializing particle simulations with an accurate kinetic equilibrium. Itprovides an interactive graphical user interface (GUI), which aids in thedesign of the distribution function. The resulting initial snapshot is saved ina portable format (NetCDF) to be read by a high-performance particle code.

Here, VSTART is used to prepare the initial snapshot for a simulation of asparse population of drift-kinetic energetic ions that interact with the bulkplasma represented by a reduced MHD model in a shifted-circle tokamakequilibrium. This is the physics model underlying the hybrid MHDdrift-kinetic code HMGC [1,2].

SNA 2010, Oct. 20 — 3/20

1. BackgroundPhase-space is sampled by marker particles (randomly and uniformly),weighted so as to represent a given initial distribution functionF0 = F (t = 0) for physical particles. Examples for F0:

Local Maxwellian: F loc ∝ n(ψ)[T (ψ)]−3/2 exp[−E/T (ψ)]

Canonical Maxwellian: F can ∝ n(Pϕ) exp[−E/T0]

Conventionally, markers are loaded uniformly in space, and F0 is takento be the same in the entire plasma.

Problems:

F loc is not a true equilibrium since ψ is not a constant of motion⇒ F after phase-mixing differs from input distribution F0Due to loss orbits, no tractable formula for F0 holds for the entireplasma exactly (unless such orbits are artificially refilled in the simulation)

⇒ even F can (fct. of const. of motion) is not a true equilibriumVarying velocity leads to periodic bunching and spreading of markers⇒ noise and signal are not statistically independentF can mixes spatial and velocity coord. in Pϕ ≈ (R0ωc/B)ψ + Rv‖⇒ not a readily measurable quantity

SNA 2010, Oct. 20 — 4/20

2-1. Method: Basic concept (See [3] for a similar procedure.)

Construct exact kinetic equilibrium F0 = Feq from particle orbits.

1. Sample phase-space in Z = 0 mid-plane; uniformly and randomly.2. Compute unperturbed guiding center (GC) orbits for all samples. Orbits

with v‖ > 0 (v‖ < 0) are initialized on low-(high-)field side of Raxis.3. Fill volume by placing markers along orbits, taking into account ∇ · v.4. Assign weights so as to match the desired reference profiles.

Formally, Feq is defined asa boundary-value problem,given the GC equations ofmotion and a boundary con-dition F bc(E , µ, ψ(Z = 0)).

SNA 2010, Oct. 20 — 5/20

2-2. Method: Sampling and weighting (example)Phase-space sampling: (j = marker label, λ = λj = orbit label)

1. Define radial grid in Z = 0 plane: Nρ cells one each side of themagnetic axis X = Xaxis.Here: Let ∆X (ρ) such that ∆ρ = const. (with ρ2 = 1− ψ/ψaxis).

2. Inside each cell, initialize Nv orbits at random pos. X . The Nv orbitssample the velocity space with probability distribution fct. (PDF) P.Here: Sample (E , α)-plane, let PE ,α = const. (with cosα = v‖/v).→ volume element V v

j ∝ JE ,α/PE ,α =√2µB (GC Jacobian/PDF)

3. Along each orbit with trajectory Rλ, length Lλ and period τλ, place Mλ

markers at pos. Rj = Rλ(tj) for uniformly random times, tj ∈ [0, τλ).→ volume element V x

j ∝ Lλ 〈R〉τ ∆X/Mλ (with orb. average 〈·〉τ)

Marker weight: (i = 1, 2, ... iteration #)

w(i)j =

V xj

︷ ︸︸ ︷

Lλ 〈R〉τ ∆X︸ ︷︷ ︸

orbit volume

/ Mλ︸︷︷︸

markerson orbit

×

V vj =J /P×Fbc

︷ ︸︸ ︷√

2µB︸ ︷︷ ︸

Jacobian /marker PDF

×F bc(i)(ψ(X ,Z = 0), µ,E )︸ ︷︷ ︸

boundary condition fordistrib. function F

eq

Here: Let Fbc(i) = C(i)λ

exp(−Eλ/T(i)λ) = iteratively matched boundary condition

SNA 2010, Oct. 20 — 6/20

2-3. Method: Iterative matching (example)Iteratively determined boundary condition: (i = 1, 2, ... iteration #)

C(i)λ = C

(i)λ (Pϕ,λ, µλ,Eλ) = C

(i)0 C

(i−1)λ (ψλ(t))

1

τλ

∫ τλ

0dt

nref(ψλ(t))

n(i−1)num (ψλ(t))

C(i)0 =

[∫ 1

0dψ q(ψ)nref(ψ)

]/[∫ 1

0dψ q(ψ)n(i)num(ψ)

]

, C(0)λ = n(0)num = 1

X / a

Z /

a

flux surfaces andtrapped orbit in poloidal plane

−0.5 0 0.5 1

−0.5

0

0.5

0 0.5 1

reference density profile andeffective "orbit density"

ρ

0 0.5 1ρ

time

spen

t in

a ra

dial

cel

l

0 500 10000.5

0.6

0.7

0.8

t / τA0

r / a

nref

(ρ)

orbitsamples

(a)

(b)

(c)

(d)

Cλ(1)=nλ

C(1)λ = nλ is computed as the

effective contribution of an orbitto the reference profile nref(ψ)by averaging nref over the orbittrajectory.

C(i)λ is iteratively adjusted by

the ratio between the referenceprofile and the numericallycomputed flux-surface averaged

0th moment n(i)num(ψ) of Feq(i).

SNA 2010, Oct. 20 — 7/20

3-1. Software tool VisualStart: XHMGC version overviewProcedure:

1. Create new case or load existing case.

2. Set up MHD equilibrium.

3. Set up reference distribution function and profiles for energetic ions.

4. Build orbit database, check accuracy, and analyze orbits if needed.

5. Load markers and adjust weights iteratively to match reference profile.

6. Analyze qualitative properties and quantitative accuracy of constructedequilibrium distribution.

7. Save snapshot as initial condition for particle code.

Main control panel:

SNA 2010, Oct. 20 — 8/20

3-2. VSTART: MHD equilibriumSet parameters for ananalytical model to fit anexperimental data; then,solve Grad-Shafranovequation directly.Here: Zero-beta shifted-circle model equilibriumfrom HMGC simulation ofDIII-D shot 122117 [5].

Feature preview:Import equilibrium from anexternal equilibrium solver.Right screenshot showsVSTART for MEGA code[4,5] with ITER equilibriumcomputed by MEUDAS.

SNA 2010, Oct. 20 — 9/20

3-3. VSTART: Reference distribution function and profiles

Define phase-space sampling grid.

Import/design reference distribution F bc andreference profiles nref , Tref , etc.

Here: F bc has the formof a local isotropicMaxwellian with con-stant temperatureTH = 1. All otherparameters and nH(ψ)are based on HMGCsimulation of DIII-Dshot 122117 [6].

SNA 2010, Oct. 20 — 10/20

3-4. VSTART: Orbit database

Analyze individual orbits

Build orbit database

Verify accuracy of all orbits in the database

Here:Parameters for large-drift scenario basedon HMGC simulationof DIII-D shot 122117[5].

SNA 2010, Oct. 20 — 11/20

3-5. VSTART: Marker loadingLoad markers using orbit database.

Analyze properties of resultingdistribution Feq.

Adjust weights iteratively, untilmoments match reference profiles.

Here: Parameters for large-driftscenario based on HMGC simu-lation of DIII-D shot 122117 [5].

Diagnostics:

number density, temperature;

flows v‖, vϕ, vX , vZ ;

markers per cell.

Summation performed over a window

(a) along ρ direction,

(b) in (X ,Z ) plane, or

(c) in velocity space.SNA 2010, Oct. 20 — 12/20

4-1. Properties of kinetic distribution F eq: Profile fitting

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ρ

n H/n

H0

small driftε=3×10−3, ρ

H0/a=3×10−4

16,359 orbits, 1,402×103 markers, Nρ=32 cells

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

large driftε=0.3, ρ

H0/a=3×10−2

14,642 orbits, 813×103 markers, Nρ=32 cells

ρ

n H/n

H0

0 0.2 0.4 0.6 0.8 10

0.5

1

ρ

TH

/TH

0

0 0.2 0.4 0.6 0.8 10

0.5

1

ρ

TH

/TH

0

ref.

i=1

i=2 (match n)

i=3 (match n)

i=4 (match T)

ref.i=1i=2 (match n)

(a) (b)

(d)(c)

Successful matching of number density profiles demonstrated for small andlarge orbit width. SNA 2010, Oct. 20 — 13/20

4-2. F eq: density and temperature field

−0.5 0 0.5 1

−0.5

0

0.5

Z /

a

large orbit width

−0.5 0 0.5 1

−0.5

0

0.5

−0.5 0 0.5 1

−0.5

0

0.5

X / a

Z /

a

−0.5 0 0.5 1

−0.5

0

0.5

−0.5 0 0.5 1

−0.5

0

0.5

Z /

a

small orbit width

−0.5 0 0.5 1

−0.5

0

0.5

−0.5 0 0.5 1

−0.5

0

0.5

X / a

Z /

a

−0.5 0 0.5 1

−0.5

0

0.5

0 0.5 10

0.2

0.4

0.6

0.8

1

ρ

n H/n

H0

number density

0 0.5 10

0.5

1

ρ

TH

/TH

0

temperature

reference

small orbit width

large orbit widthn

H(X,Z)

TH

(X,Z)

Despite similar radial profiles, density and temperature fields in the casewith large drifts exhibit two separate peaks in the (X ,Z )-plane, one on thelow-field side (LFS, X > Xaxis) and one on the high-field side (HFS,X < Xaxis). The fields vary significantly on a flux surface. SNA 2010, Oct. 20 — 14/20

4-3. F eq: spatial dependence of (µ, v‖) distribution

0 0.5 1 1.5 2−2

0

2

0 0.5 1 1.5 2−2

0

2

0 0.5 1 1.5 2−2

0

2small orbit width

v ||

0 0.5 1 1.5−2

0

2

0 0.5 1 1.5 2 2.5−2

0

2

0 0.5 1 1.5 2 2.5

−2

0

2large orbit width

0 0.5 1 1.5 2 2.5−2

0

2

0 0.5 1 1.5 2 2.5−2

0

2

0 0.5 1 1.5 2−2

0

2

0 0.5 1 1.5 2

−2

0

2

µ

X / a

Z /

a

−0.5 0 0.5 1

−0.5

0

0.5

−0.5 0 0.5 1

−0.5

0

0.5

X / a

Z /

a

−0.5 0 0.5 1

−0.5

0

0.5

0 0.5 1 1.5 2−2

0

2

v ||

0 0.5 1 1.5 2−2

0

2

µ

v ||

LFS

HFS

v|| > 0

v|| < 0

With large drifts, the velocity distribution varies significantly betweendifferent regions in configuration space. The above example shows thedifference between the low-field side (LFS, X > Xaxis) and high-field side(HFS, X < Xaxis), and shows orbits in the loss-cone region.SNA 2010, Oct. 20 — 15/20

4-4. F eq: flow fields and profiles (entire v-space)

large orbit width

Z

/ a

−0.5 0 0.5 1

−0.5

0

0.5

−0.5 0 0.5 1

−0.5

0

0.5

0

0.2

0.4toroidal flow

v φ / v H

0

Z /

a

−0.5 0 0.5 1

−0.5

0

0.5

−4

−2

0

x 10−5

horizontal flow

v X /

v H0

X / a

Z /

a

−0.5 0 0.5 1

−0.5

0

0.5

0 0.5 1−10

−5

0

x 10−3

vertical flow

ρ

v Z /

v H0

small orbit width

Z /

a

−0.5 0 0.5 1

−0.5

0

0.5

−0.5 0 0.5 1

−0.5

0

0.5

Z /

a

−0.5 0 0.5 1

−0.5

0

0.5

X / a

Z /

a

−0.5 0 0.5 1

−0.5

0

0.5

small orbit width

large orbit widthvφ

vZ

vX

With large drift, two oppositely directed toroidal beams form (vφ); one onthe low-field side (LFS, X > Xaxis), the other on the high-field side (HFS,X < Xaxis). Each has its respective curvature drift field (vX , vZ ).

SNA 2010, Oct. 20 — 16/20

4-5. F eq: phase-space marker distribution

−0.5 0 0.5 1

−0.5

0

0.5

Z /

a

small orbit width

−0.5 0 0.5 1

−0.5

0

0.5

0

2

4

6

x 104 entire v−space

# m

arke

rs

−0.5 0 0.5 1

−0.5

0

0.5

Z /

a

large orbit width

−0.5 0 0.5 1

−0.5

0

0.5

0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

µ

v ||

Z /

a

−0.5 0 0.5 1

−0.5

0

0.5

0

500

1000

v|| > 0 window

# m

arke

rs

X / a

Z /

a

−0.5 0 0.5 1

−0.5

0

0.5

0 0.5 10

500

1000

v|| < 0 window

ρ

# m

arke

rs

Z /

a

−0.5 0 0.5 1

−0.5

0

0.5

X / a

Z /

a

−0.5 0 0.5 1

−0.5

0

0.5

−1 −0.5 0 0.5 1

marker loading cells ∆X(∆ρ=const.)

small orbit width

large orbit width

Emax

entire v−space

With small drifts, markers are distributed uniformly in ρ. In (X ,Z )-plane,modulation of marker density reflects nonuniformity of cells ∆X (Z = 0)for uniform ∆ρ. When drifts are large, marker density in (X ,Z ) revealsloss and confinement regions for given window in v-space. SNA 2010, Oct. 20 — 17/20

5. SummaryProblem and method

When magnetic drifts are significant, the phase-space distribution of theparticles concerned is too complex to be expressed by analytical modelsand exp. data. Model and measurements may only serve as a reference.

We define the equilibrium distribution function Feq as a boundary-valueproblem: Given a reference distribution serving as a boundary conditionF bc in the mid-plane, the unperturbed guiding center equations of motiondetermine Feq in the entire plasma volume.

Status

Developed new GUI-aided software tool, VSTART, which constructs anaccurate true equilibrium distribution given an arbitrary reference distrib.

Implemented simple case for the simulation of a sparse energetic ionpopulation in an MHD bulk plasma with shifted-circle equilibium.

Demonstrated proper functioning of the loading algorithm. Iterativeprofile matching performed successfully. Developed diagnostics tools fordetailed analysis of the distribution in velocity and configuration spaces.

SNA 2010, Oct. 20 — 18/20

6. OutlookNext steps

Include a greater variety of analytical reference distributions such asICRH+NBI (currently, Maxwellian and slowing-down).

Import numerical reference distributions obtained from transport codes.

Extension to initialize δf simulations (currently full-f ),

Possibly even apply to gyrofluid (“Eulerian”) codes.

Perspectives

The ultimate goal is to construct self-consistent “kinetic equilibria” toreplace MHD Grad-Shafranov equilibria; new elements include:

◮ energetic ion pressure contribution (anisotropic, not a fct. of ψ),◮ intrinsic flows,◮ boundary effects (in particular, loss orbits).

Interface with multiple codes; currently (X)HMGC [1,2] and MEGA [3,4].

Facilitate meaningful benchmark studies by providing an accuratestandard initial snapshot of the plasma. Comparisons may then focus onthe solver algorithm (numerical schemes, physics models).

SNA 2010, Oct. 20 — 19/20

7. References & Acknowledgements

References

[1] S. Briguglio et al, Phys. Plasmas 2 (1995), 3711.

[2] X. Wang et al, 2010 Proc. 23rd Int. Fusion Energy Conf. (Daejon,Korea 2010) THW/2-4Ra.

[3] J.A. Heikkinen et al, J. Comput. Phys. 173 (2001) 527.

[4] Y. Todo and T. Sato, Phys. Plasmas 5 (1998) 1321.

[5] Y. Todo, Phys. Plasmas 13 (2006) 082503.

[6] G. Vlad et al, Nucl. Fusion 49 (2009) 075024.

Acknowledgments

This work was partially supported by EFDA Co-funded GOTIT programmeand A.B. thanks the Frascati Research Institute ENEA for its hospitalityfor the duration of the fellowship. A.B. would also like to thank ZhihongLin (UC Irvine) for stimulating discussions on the subject and for providingaccess to the dragon cluster at UC Irvine for testing the new markerloading method with the HMGC code.

SNA 2010, Oct. 20 — 20/20