entropy balance in electromagnetic gyrokinetic …march 15, 2012 kashiwa, japan-0.4-0.2 0 0.2 0.4...
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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
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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
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0
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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.
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0
5
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0 20 40 60 80 100-10
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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
kφ
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).