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Page 1: Entropy balance in electromagnetic gyrokinetic …m e/m i=5.669×10-4, T e=T i, β i=1%, ν i=2×10-3v ti/L n, ν e=ν i×√m i/m e. Numerical settings for linear runs: -11.9

Entropy balance in electromagnetic gyrokinetic simulations電磁的ジャイロ運動論的シミュレーションにおけるエントロピーバランス

Shinya Maeyama1), Akihiro Ishizawa2), Tomohiko Watanabe2), Noriyoshi Nakajima2), Shunji Tsuji-Iio1), Hiroaki Tsutsui1)

前山伸也1), 石澤明宏2), 渡邉智彦2), 中島徳嘉2), 飯尾俊二1), 筒井広明1)

1) Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku 152-8550, Japan2) National Institute for Fusion Science, 322-6 Oroshi-cho, Toki 509-5292, Japan

Motivation

We presented two methods for electromagnetic gyrokinetic simulations.

An electromagnetic gyrokinetic code is successfully developed.The outflow boundary condition reduces numerical oscillations in the

field-aligned coordinate.The entropy conservative difference substantially reduces numerical

errors in entropy balance.

Summary

Electromagnetic effects on turbulent transport play important roles in high beta plasmas. Rapid parallel motions of kinetic electrons bring some numerical difficulties to electromagnetic gyrokinetic simulation.

Recently, we extended the GKV code for electromagnetic gyrokinetic simulations. Here, we develop numerical methods to treat kinetic electrons and discuss benefits of the methods.

δf gyrokinetic equations Availability of the schemes for electromagnetic simulations

Linear simulations

17th Next MeetingMarch 15, 2012 Kashiwa, Japan

-0.4

-0.2

0

0.2

0.4

0.6

0 0.4 0.8 1.2 1.6 2

γGENE

ωGENE/4γGKV

ωGKV/4

Linear

gro

wth

rate

γL n

/vti,

Real f

requency

ωL n

/vti

Ion beta value βi [%]

Benchmark test with the GENE code:Beta dependence of linear instabilities

Zero-fixedOutflow

ote

ntial 1

10-1

Profile of the eigenfunction of electrostatic potential along the field file

Outflow boundary

First of all, the developed electro-magnetic GKV code is compared with the GENE code3.

The electromagnetic GKV code shows good agreements with the GENE code.

Following two tests are carried out to check the usefulness of the outflow boundary condition.

Plasma parameters: Ln/LTi=3.1, Ln/LTe=0, R/Ln=2.22, r/R=0.18, q=1.4, s=0.786,me/mi=5.669×10-4, Te=Ti, βi=1%, νi=2×10-3vti/Ln, νe=νi×√mi/me.

Numerical settings for linear runs: -11.9<kxρti<11.9, kyρti=0.2, -π<z<π, -4vts<v//<4vts, 0<µB/Ts<8,Nkx=25, Nky=1, Nz=32, Nv=64, Nµ=16, ∆t=0.002Ln/vti;

for nonlinear runs: -50.9ρti<x<50.9ρti, -62.8ρti<y<62.8ρti, -π<z<π, -4vts<v//<4vts, 0<µB/Ts<8,Nx=288, Ny=72, Nz=32, Nv=64, Nµ=16, adaptive time steps.

In a gyrokinetic framework, rapid gyrations are asymptotically decoupled from slow transport phenomena, and then reduced equations are,

kkkk ssss//s

//sd////

ˆ CSNfvm

Bivt

+=+

∂∂∇

−+∇+∂∂ µω

∑∑==

⊥ =

Γ−+

ei,sss

0ei,s0s

s

s2s

0

2 ˆ1ˆ)1(1kkk ne

Tnek

εφ

ε

∑=

⊥ =ei,s

//ss0//2 ˆˆ

kk ueAk µ

∑∑==

++=

++

ei,ss

s

As

s

Ass

MEei,s

s DLQ

LΓTWWS

dtd

Tp

Vlasov Eq.

Poisson Eq.

Ampere Eq.

We note that the gyrokinetic Eqs. satisfy the entropy balance relation,

which describes that the particle fluxes Γ and the heat flux Q act like sources

The GKV code1, originally developed for electrostatic turbulent simulations, solves time evolution of perturbations in a flux-tube simulation domain.

We extend the GKV code for electromagnetic gyrokinetic simulations with kinetic ions and electrons.

[1] T.-H. Watanabe and H. Sugama, Nucl. Fusion 46, 24 (2006).

[3] F Jenko, et al., Phys. Plasmas 7, 1904 (2000).

Flux-tube simulation domain

Magnetic field B

L//L⊥

Charged-particle gyromotion

Outflow boundary improves the convergence with the length along the field line.

Nonlinear simulations

Numerical methods

Outflow boundary condition along the field-aligned coordinate

Entropy conservative difference for parallel motions

Entropy conservative difference significantly reduces entropy balance error.

1−znf

znf

zz nn ff =+1

12 2 −+ −=zzz nnn fff

Schematic picture of the outflow boundary condition

zzLzLz ∆− zLz ∆+ zLz ∆+ 2

0// >v

-3 -2 -1 0 1 2 3

Trajectories of particle parallel motions

Para

llel ve

loci

ty v

||/v

ti

Field-aligned coordinate z

3

2

1

0

-1

-2

-3

Outflow

Inflow

Instabilities drive perturbations at the bad curvature region (z=0), and the electron perturbation is rapidly advected along the field-aligned coordinate z.

Outflow boundary condition is employed to reduce numerical oscillations at the edge of the field-aligned coordinate.

Parallel dynamics can be written as an incompressible form,

where . Application of Morinishi’s scheme2 guarantees the entropy conservative property,

where <・> denotes the flux-surface average. Es is non-zero when the outflow boundary condition is employed.

//

sMs0ssM

ss

ss0////

s

sMs

//

s

s

//s//// ,ˆˆ1ˆˆˆ

+=∇−∂∂∇

+∇− FJeFfT

mJv

TFe

vf

mBfv kk

kkk

kk φφµ

-0.1

0

0.1

0.2

0.3

0.4

0 5 10 15 20 25 30 35

-0.12

-0.08

-0.04

0

0.04

0 5 10 15 20 25 30 35Err

or

of

gro

wth

rate

(γ-γ

ref)

/γre

f

Length along a field line Lz/π

Convergence of the growthrate and frequency with the length along the field line

Zero-fixedOutflow

Err

or

of

freq

uen

cy (ω-ω

ref)

/ωre

f

Length along a field line Lz/π

Zero-fixedOutflow

-15 -10 -5 0 5 10 15Ele

ctro

stat

ic p

o|φ

k(z)

/φk(

0)|

Field-aligned coodinate z/π

10

10-2

10-3

yreduces numerical oscillations in the field-aligned coordinate.

Numerical errors from the parallel dynamics act like an artificial source term, and enhance the collisional dissipation.

Using the outflow boundary condition, two nonlinear turbulent simulations are carried out: the one employs the 4th-order central finite difference, and the other employs the 4th-order entropy conservative difference.

-10

-5

0

5

10

0 20 40 60 80 100-10

-5

0

5

10

0 20 40 60 80 100

Time evolution of entropy variables by using (Left) central finite and (Right) entropy conservative differences

Entr

opy

variable

s [n

0T

ivtiρ t

i2/L

n3]

Time t vti/Ln

Entr

opy

variable

s [n

0T

ivtiρ t

i2/L

n3]

Time t vti/Ln

d(Si+Se+WE+WM)/dtTiΓi/Lpi+Qi/LTi

TeΓe/Lpe+Qe/LTe

DiDe

(LHS)-(RHS)

z

z

Lz

Lz

dvJeFfT

TFv

dzgE

=

−=

+−= ∫∑

∫3

2

s0ssM

ss

s

sM//s

ˆˆ

21

kkk

The outflow boundary introduces artificial sinks in the entropy balance,

We employs the numerical methods same with the original GKV code,Discretization in x and y: Fourier expansionDiscretization in z, v// and µ: Finite differenceTime integration: 4th-order Runge-Kutta-Gill method

and develops special methods to treat kinetic electrons.

s//

//

sMs0ssM

ss

s

*s0s

sM

*ss ,ˆˆ1ˆˆ

EdvFJeFfT

mJe

FfT

=

+

+∑ ∫

kkk

kkk

k φφ

{ } gfgfgf vv ////// ////, ∇∂−∂∇=

which describes that the particle fluxes Γs and the heat flux Qs act like sources of the perturbed energy (the entropy variables Ss, the electric energy WE and magnetic energy WM) and the collisional dissipation Ds act like sinks.

[2] Y. Morinishi, et al., J. Comp. Phys. 143, 90 (1998).