Edge Two Fluids and Gyrokinetic Continuum Simulations

Xueqiao Xu

Presented at ITER Fusion Simulation Workshop May 16, 2006, Peking University

Diverted tokamak magnetic fusion device and its poloidal Cross Section

After R. Waltz et al, 2002 SnowMass Mtg.

The Edge Transport Barrier isCritical to ITER’s Performance

• Transport barriers form spontaneously at plasma edge

• Studies of core turbulence show – Turbulent transport

constrains gradient scale lengths

Tcentral ~ proportional to Tped

• Tped is the largest source of uncertainty in projecting ITER’s performance – Fusion gain = Pfusion/Paux

Projection of ITER’s Fusion Gain

Edge codes (BOUT and TEMPEST… ) are aimed at reducing uncertainty in projections of ITER’s fusion gain

Edge simulation models/codes

• Two fluid model– Transport;

• UEDGE, B2– Turbulence

• BOUT• BDM• DALF3

• Gyrokinetic model– Continuum methods

• Tempest, FEFI– Particle-in-cell methods

• XGC, ASCOT, ELMFIRE

• Mont-Carlo Model– Degas, Eirene

BOUT is 3D EM Boundary Plasma Turbulence Code

• Braginskii --- collisional, two-fluids electromagnetic equations

• Realistic X-point geometry– open+closed flux surfaces

• BOUT is being applied to DIII-D, C-mod, NSTX, MAST, ITER (for Snowmass), ...

• LOTS of edge fluctuation data!– BES, GPI, PCI, Probe, and

Reflectometer– Provide excellent

opportunity for validating BOUT against experiments.

BOUT is a parallelized 3D nonlocal electromagnetic turbulence code using MPI

A suite of the codes work together to make BOUT simulation results similar to real experiments

A simple analytical neutral mode added

In BOUT simulations turbulence is found in divertor leg region

upperx-point

lowerx-point

ref=38

Cross-correlation plot shows that divertor turbulence is not correlated with upstream turbulence

Plot of spatial distribution of RMS fluctuations amplitude shows that fluctuations grow in regions of unfavorable curvature

BOUT simulations for C-Mod are consistent with experimental amplitude and spatial spectra of Ni

BOUT simulations yield filamentary structures as experiments for edge localized modes (ELMs)

• Ingredients of an ELM simulation– Nonlinear with “fast” explosive

nonlinearities– Produces fine scale fingers/blobs/ etc– Fast transport along the field line and

turbulent transport across the field line– Allow for magnetic reconnection

• BOUT simulations– Early structure & growth similar to linear

pressure-/current-gradient driven modes– Radially propagating filamentary

structures grow explosively (as seen in MAST, DIII-D)

– Filaments acting as conduits to pedestal, provide mechanisms for ELM losses

– Not able to simulate complete bursting event at present; mesh alignment is a problem

PRL 2004 A. Kirk et al.

BOUT simulations, Snyder et al, PoP 2005

Results show that strong spatial dependence of transport must be included

a) Typical previous model b) Our new coupled results

• Poloidal variation understood from curvature instability

Experiment (DIII-D, C-Mod)

0 2 4 Radial distance (cm)

0.1

Results consistent with expt.

Open - DIII-D Filled - C-Mod

A kinetic edge code is required to model

both today’s tokamaks and ITER

• Fluid approximation requires:

• Not satisfied on DIII-D todayWon’t be satisfied on ITER

• Need to move beyond fluid codes

DIII-D Edge Barrier

mfp T 2

n c ~ 2R

or

Describe each species with akinetic distribution function, F()(, , , E0, ,)

mfp

c

Orbit width orbit width plasma scale length

Tempest is a 5D Continuum Edge Gyrokinetic Plasma Code

• Gyrokinetic equations– Valid for edge ordering

• Nonlinear Fokker-Planck collision

• Realistic X-point geometry– open+closed flux

surfaces

Simulate neoclassical transport, turbulence and plasma-Surface interactions

Fully Nonlinear Ion gyrokinetic equations

We have designed and implemented a 4D edge simulation framework

pyMPI (parallel Python)pyMPI (parallel Python)

Gyrokineticmodule

Gyrokineticmodule

pyUEDGEmodule

pyUEDGEmodule

Visualization module (pyGist)

Visualization module (pyGist)

Co

llision

sC

ollisio

ns

Ad

vection

AccelerationS

treaming

Radial D

rift

Ad

vection

AccelerationS

treaming

Radial D

rift

Field

solve

Field

solve

SUNDIALSSUNDIALS

Distribution Function moduleDistribution Function module

Data ManagerData Manager

SAMRAISAMRAI

HypreHypre

We have implemented a gyrokinetic Poisson equation field solver

2

2

22

2

2

ln22

NDD

p

eZNnNZe

De

22

2 1

24

• Discretized in - coordinates using standard finite differencing• Uses Hypre library of parallel linear algebra solvers and

preconditioners– Solvers:

• Conjugate Gradient (CG)• Generalized Minimum Residual (GMRES)• Stabilized BiConjugate Gradient (BiCGSTAB)

– Preconditioners• Diagonal scaling• Block Gauss-Seidel with PFMG or SMG in each block• BoomerAMG

• Currently implemented with Boltzmann electron model

Ne=<Ni0 >ee/Te /< ee/Te>

Simulation results agree very well with neoclassical theory in Ring geometry

Radial Position R(m)

Ion distribution function F(R,Z,E0 in DIII-D geometrywith endloss at plates in the SOL looks as expected

V||

V||

V||

V||

Tempest recovers theoretical U|| inside separatrix and increases as expected in SOL

Tempest exhibits collisionless damping of GAMs and zonal Flow

• Axis-symmetric mode (no toroidal variation)– Parallel ion dynamics– Magnetic curvature TEMPEST should see GAMs

• Tempest model– Drift kinetic ions with radial drift, streaming,

and acceleration– Boltzmann electron– Gyrokinetic Poisson equation in limit small

s/Lx

– Dirichlet radial boundary conditions• GAMs provide opportunity to “verify” TEMPEST

physics– Rosenbluth-Hinton residual

– Frequency

2

( ) 1

( 0) 1 1.6

t

t q

7

8Ti

GAM Rv

Time(vti/R0)

Rosenbluth-Hinton Residual zonal flowCollisionless damping of zonal flow and GAM

(t)/t)

2

( ) 10.02, 2.23, 0.02

( 0) 1 1.6

tq

t q

GAMsim/GAM

th=1.06

Tempest exhibits collisionless damping of GAMs and zonal Flow

Time(vti/R0)

Rosenbluth-Hinton Residual zonal flowCollisionless damping of zonal flow and GAM

(t)/t)

2

( ) 10.02, 2.23, 0.02

( 0) 1 1.6

tq

t q

GAMsim/GAM

th=1.06

GAMs simulations converge with nv, n, and KEmax

KEmax=15rtol=10-7, atol=10-12

r/R=0.02q=2.23

n=30, n=50, nE=30, n=15

n=30, n=50, nE=60, n=30

n=30, n=50, nE=100, n=50

n=30, n=100, nE=60, n=30

n=30, n=50, nE=30, n=15, KEmax=10

Rosenbluth-Hinton Residual zonal flow

Maximum kinetic energy has to be 10x thermal energy

n=30, n=50, nE=30, n=15rtol=10-7, atol=10-12

r/R=0.02,q=2.2

KEmax=15

KEmax=10

KEmax=5

Contour plot of distribution function

Time=0 Time=75

resonance

Summary

• Edge simulation and modeling are critical to ITER’s Performance

• Two fluid turbulence code BOUT yields simulation results consistent with experiments for present day tokamaks

• Edge gyrokinetic continuum code TEMPEST is under development

• A lot of scientific phenomenon remain to be discovered via advanced computing!