Gyrokinetic simulation of electron turbulence

Gyrokinetic simulation of
electron turbulence spectrum
Chika Kawai,1),2) Shinya Maeyama,2)
Yasuhiro Idomura,2) Yuichi Ogawa1)
1)GSFS,Univ. Tokyo
2)JAEA
This work was carried out using the HELIOS supercomputer system at Computational
Simulation Centre of International Fusion Energy Research Centre (IFERC-CSC), Aomori,
Japan, under the Broader Approach collaboration between Euratom and Japan, implemented
by Fusion for Energy and JAEA.
Plasma in turbulent state 2
Zonal flow (ZF) can suppress turbulent transport[1],[2].
ZF formation through self-organization and its energy spectrum
structure in…
• fluid model (Hasegawa-Mima eq.)
well known.
• kinetic plasma (Gyrokinetic model) not well investigated.
(focused on this work)
𝐸(𝑘⊥ )
energy inverse cascade
enstrophy cascade
Zonal flow
Fig.1 schematic diagram of zonal flow structure.
[1]: Z. lin, et al., Science (1998)
[2]:A. Fujisawa, et al., Phys. Rev. Lett. (2004)
Energy source by
forcing
𝑘𝑠
𝑘𝜂
𝑘
Energy sink by
dissipation
Fig.2 schematic diagram of
plasma turbulence energy spectrum.
3
Gyrokinetics
Electron gyrocenter distribution function 𝐹𝑒 (𝑹, 𝜇, 𝑣∥ ; 𝑡) and electrostatic
potential 𝜙(𝒙; 𝑡) is solved.
•
Electrostatic approximation, adiabatic ion response 𝛿𝑛𝑖 ∼ 𝑞𝑖 𝜙 𝑇𝑖 × 𝑛0 ,
Long wavelength approximation 𝑘 𝜌𝑡𝑒 < 1 and shearless slab configuration
for background magnetic field 𝐵0 is assumed
•
Kinetic effect plays important roles
• Landau damping
• Finite Larmor radius (FLR) effect
𝜕𝐹𝑒
𝑐
+ 𝑣∥ 𝒃 +
𝒃 × 𝛻𝑹 𝜙
𝜕𝑡
𝐵0
−
𝛻2
𝛼
𝑒
⋅ 𝛻𝑹 𝐹𝑒 −
𝒃 ⋅ 𝛻𝑹 𝜙
𝑚𝑒
2
𝜌𝑡𝑒
1
+ 𝛁⊥ ⋅ 2 𝛁⊥ 𝜙 + 2 𝜙 = 4𝜋 𝑞𝑒
𝜆𝐷𝑒
𝜆𝐷𝑖
where 𝜌𝑡𝑒 =
𝑣𝑡𝑒
, 𝜆𝑑𝑠
Ω𝑒
=
4𝜋𝑒 2 𝑛𝑠
𝑚𝑠
𝜕𝐹𝑒
= 𝐶 𝐹𝑒
𝛼
𝜕𝑣∥
𝐹𝑒 𝛿 𝑹 + 𝝆 − 𝒙 𝐷d6 𝒁 + \𝑞𝑖 𝑛0
𝑠 = 𝑒, 𝑖 , ・ 𝛼 : gyroaverage, 𝐶 ・ : collision operator
Zonal flow formation through
self organization
4
Neglecting, 𝐶 𝐹 , 𝑘∥ ,and FLR effect yields Hasegawa-Mima (H-M) equation.[3]
𝜕 2 2
𝜌𝑠 𝛻⊥ 𝜙 − 𝜏𝜙 − 𝛻⊥ 𝜙 × 𝒛 ⋅ 𝛻⊥ 𝜌𝑠2 𝛻⊥2 𝜙 − ln 𝑛0 = 0
𝜕𝑡
𝑇𝑒 2
𝜆2𝐷𝑒
𝜏 = , 𝜌𝑠 = 1 + 2
𝑇𝑖
𝜌𝑡𝑒
2-D fluid eq. conserves
Energy inverse cascade:
Enstrophy cascade:
1
2𝑘2 𝜙2
𝜏
+
𝜌
𝑠
𝑘
𝒌
2
1
𝐄𝐧𝐬𝐭𝐫𝐨𝐩𝐡𝐲: 2 𝒌 𝜏𝑘 2 + 𝜌𝑠2 𝑘 4 𝜙𝑘2
𝐄𝐧𝐞𝐫𝐠𝐲:
energy inverse cascade
𝑬𝒇 𝒌 ∝ 𝒌−𝟓 𝟑 , 𝑘 < 𝑘𝑠
𝑬𝒇 𝒌 ∝ 𝒌−𝟑 ,
𝑘 > 𝑘𝑠
(dual cascade)
𝐸(𝑘⊥ )
enstrophy cascade
𝜔𝑒∗
𝜔𝑡
𝑘𝑠
Energy source by
forcing
[3]: A.Hasegawa, Adv. Physics.(1985)
[4]: R. H. Kraichnan, Phys. Fluids.(1967)
𝑘𝜇
𝑘
Energy damping by
dissipation
Fig.3 1-D energy spectrum in 2D fluid.
Zonal flow formation through
self organization
𝑘2𝜙
,
𝜏+𝜌𝑠2 𝑘 2
𝐿−1
𝑛 𝑘𝑦
𝜔𝑡 ∼
Low-k : drift wave dispersion:
𝜔𝑒∗ = − 𝜏+𝜌2𝑘2 ,
𝜔𝑒∗
𝑘𝜇
𝑘
Energy damping
by dissipation
Fig.3 1-D energy spectrum in 2D fluid.
𝑘𝑥
1
𝐿𝑛 = 𝛻 ln 𝑛0 , 𝜖 =
𝑉
𝜔𝑡
𝑘𝑠
Energy source by
forcing
𝟏
1
−𝟐 −𝟏
k 𝑐,x = 𝑳𝒏 𝝐 𝟒 cos 𝜃 sin2 𝜃 ,
𝟏
3
−𝟐 −𝟏
𝟒
k 𝑐,𝑦 = 𝑳𝒏 𝝐 sin2 𝜃
𝑘
−1 𝑦
[5]:P. B. Rhines, J. Fluids Mech.(1975)
[6]:G. F. Vallis, and M. E. Maltrud, J. Phys. Oceanogr(1993)
enstrophy cascade
𝑠
Energy inverse cascade is suppressed due to
linear dispersion.
Energy spectrum stagnate around
𝑘𝑐 𝜔𝑡 𝑘𝑐 ∼ 𝜔𝑒∗ 𝑘𝑐 :Rhines scale:[5]
𝜃 = tan
energy inverse cascade
𝐸(𝑘⊥ )
High-k: nonlinear transfer rate:
5
𝒌𝒚
Zonal flow
Energy inverse
cascade
Linear
mode
𝒌𝒙
𝛻𝜙 2 d𝑉
Fig.4 2-D turbulence energy spectrum:
zonal flow structure(𝑘𝑦 = 0, 𝑘𝑥 ≠ 0) is induced
Summary of results
1. Relevance between gyrokinetic plasma turbulence
simulation and H-M eq. (fluid) theory.
2. Spectrum anisotropy modified by key parameters: 𝜏, 𝜌𝑠
6
Summary of results
1. Relevance between gyrokinetic plasma turbulence
simulation and H-M eq. (fluid) theory.
2. Spectrum anisotropy modified by key parameters: 𝜏, 𝜌𝑠
7
8
Simulation condition
Electron scale plasma turbulence simulation is conducted
Simulation code:
Equilibrium configuration
G5D[7]
single helicity, shearless slab configuration
Density(temperature) profile :
𝑛 𝑥 , 𝑇 𝑥 ∝ exp − 𝐿
0.3 𝐿𝑥
(𝑛,𝑇𝑒 )
Grid size:
tanh
𝑘𝑥 , 𝑘𝑦 𝜌𝑡𝑒 = 0.01 ∼ 2.6, 0.02 ∼ 2.6
𝑁𝑥 , 𝑁𝑦 , 𝑁𝑣∥ , 𝑁𝑣⊥ = (512,256,64,8)
Reference value of plasma parameter:
𝑇
𝜏 ≡ 𝑇𝑒 ∼ 0.3
𝑖
𝜌𝑠2
∼ 300𝜌𝑡𝑒
∼ 600𝜌𝑡𝑒
𝑩𝟎
𝑥−0.5𝐿𝑥
0.3 𝐿𝑥
𝑦
𝒗∗𝒆
𝒏𝟎 (𝒙)
≡1+
𝑘 𝐿
𝑩𝟎
𝜆2𝐷𝑒
2
𝜌𝑡𝑒
∼ 11
Θ ≡ 𝑘 ∥ 𝜌 𝑛 = 0.75
𝑦
𝑡𝑒
𝑧
𝑥
Fig.5 schematic diagram of shearless slab configulation.
[7]: Y.Idomura, et al., J. Compt. Phys. (2006)
ETG mode instability dependence
on 𝚯 and 𝜼𝒆
9
Linear instability problem for ETG(electron temperature gradient) mode is described by
following eigenvalue equation
−𝑘 2
1
+ 2
𝜆𝐷𝑒
exp −𝑏 𝐼0 𝑏 − 𝜂𝑒
1
+ 𝑏 exp −𝑏 𝐼0 𝑏 + 𝜂𝑒 𝑏 exp −𝑏 𝐼1 𝑏
2
× 𝜉∗𝑍
Electron temperature gradient(ETG) mode is unstable for certain value of
Θ≡
𝑘∥ 𝐿𝑛
𝑘𝑦 𝜌𝑡𝑒
=
𝜔𝑒∗
𝑘∥ 𝑣𝑡𝑒
,𝜂 ≡
𝐿𝑛
𝐿𝑇𝑒
Fig.6 Θ scan for 𝛾(Θ), 𝑘𝑦 𝜌𝑡𝑒 fixed at ∼ 0.5
Fig.7 𝜂𝑒,𝑖 scan for 𝛾(Θ), Θ is fixed at ∼ 0.75
Decaying turbulence
simulation
10
ETG mode is stable for 𝜂𝑒 ∼ 0 .
• Fluid model
energy source: artificial forcing in low-k
sink: artificial viscosity in high-k
energy source and sink is well separated: 𝑘𝑠 ≪ 𝑘𝜇
• Kinetic mode
energy source: ETG mode
(dependent on Θ, 𝜂)
sink: Landau damping
(dependent on Θ)
energy source and sink separation is not obvious
Fig.8 initial perturbation
(a): real space e𝜙(𝒙)/𝑇𝑒 , (b): energy spectrum: E k x , k y =
𝑘+Δ𝑘
𝑘
𝜏 + 𝜌𝑠2 𝑘 2 𝜙𝑘2
Zonal flow formation in
11
electron decaying turbulence
Set of simulations for various 𝐿𝑛 value
was conducted to check Rhines scale:[8]
𝐿𝑛 0 = 1500𝜌𝑡𝑒
Fig.9 saturated state for case
𝐿𝑛
𝜌𝑡𝑒
(a): real space e𝜙(𝒙)/𝑇𝑒 ,
(b): energy spectrum:
E k x , k y = 𝑘+Δ𝑘
𝜏 + 𝜌𝑠2 𝑘 2 𝜙𝑘2
𝑘
= 1500
Fig.10 Rhines scale for 𝑘𝑍𝐹 ≡
Blue: Numerical simulation,
−𝟏 𝟐
Red: 𝒌𝒁𝑭 = 𝝐−𝟏 𝟒 𝑳𝒏
𝑘𝑥 𝐸𝑓 𝑘𝑥 ,𝑘𝑦 =0 d𝑘𝑥
𝐸𝑓 𝑘𝑥 ,𝑘𝑦 =0 d𝑘𝑥
[8]:Y. Idomura, Phys. Plamsam(2006)
Turbulence energy spectrum:
inertial range for dual cascade
Isotropic 1-D turbulent energy spectrum:
2
2
𝐸𝑓 𝑘⊥ = 𝑛Δ𝑘
𝑘⊥ = 𝑛−1 Δk 𝐸𝑓 𝑘𝑥 , 𝑘𝑦 , 𝑘⊥ = 𝑘𝑥 + 𝑘𝑦
12
−𝟓 𝟑
Blue line: ∝ 𝒌⊥ (energy inverse cascade),
Green line:∝ 𝒌−𝟑
⊥ (enstrophy cascade)
Fig.11 1-D energy spectrum 𝐸𝑓 𝑘⊥
for decaying turbulence
Decaying turbulence (transient response to initial perturbation):
Kolmogorov-like power law is not reproduced
 Steady state simulation (energy source and sink) is needed
Turbulence energy spectrum:
inertial range for dual cascade
Isotropic 1-D turbulent energy spectrum:
𝐸𝑓 𝑘⊥
=
𝑛Δ𝑘
𝑘⊥ = 𝑛−1 Δk
𝐸𝑓 𝑘𝑥 , 𝑘𝑦 , 𝑘⊥ =
𝑘𝑥2
+
𝑘𝑦2
13
−𝟓 𝟑
Blue line: ∝ 𝒌⊥ (energy inverse cascade),
Green line:∝ 𝒌−𝟑
⊥ (enstrophy cascade)
Fig.11 1-D energy spectrum 𝐸𝑓 𝑘⊥
(left). ETG turbulence:
𝑞𝑒 𝜙
𝑇𝑒
∼ 𝟎. 𝟐%
(right). ETG turbulence:
𝑞𝑒 𝜙
𝑇𝑒
∼ 𝟓%
ETG turbulence (steady state: energy sink and source balanced):
left: energy inverse cascade 𝑘⊥−5 3 is not clear
(𝜖 is too small, Rhines scale 𝒌𝒁𝑭 is too close to energy source range 𝒌𝒔 )
right: dual cascade of energy and enstrophy is clarified.
(𝜖 is large enough that 𝒌𝒁𝑭 and 𝒌𝒔 is well separated.)
Summary of results
14
1. Relevance between gyrokinetic plasma turbulence
simulation and H-M eq. (fluid) theory.
2. Spectrum anisotropy modified by key parameters:
𝝉, 𝝆𝒔
2-D energy spectrum anisotropy
dependence on 𝝉, 𝝆𝒔
15
Linear dispersion relation for H-M equation:
𝐿−1
𝑛 𝑘𝑦
𝜔𝑘 =
2 2,
𝜏 + 𝜌𝑠 𝑘⊥
𝑇𝑒 2
𝜆2𝐷𝑒
𝜏 ≡ , 𝜌𝑠 = 1 + 2
𝑇𝑖
𝜌𝑡𝑒
𝝉 and 𝝆𝒔 affects linear dispersion property.
Linear dispersion is
• Rossby wave like
• No dispersion
𝜔𝑘
𝑘𝑦
if 𝜏 ≪ 𝜌𝑠2 𝑘𝑐2
anisotropic structure[8]
= const.
if 𝜏 ≫ 𝜌𝑠2 𝑘𝑐2
(𝑘𝑐 = 𝜖 −1 4 𝐿−𝑛1 2 : Rhines scale)
[8] T. Nozawa and S. Yoden, Phys. Fluids, 9, 3834(1997).
2-D energy spectrum anisotropy
dependence on 𝝉, 𝝆𝒔
16
Parameter scan for 𝜏 and 𝜌𝑠 is performed for decaying turbulence simulation.
𝐿
(𝜌 𝑛 = 1500, Θ ∼ 0.117, 𝑘𝑠 𝜌𝑡𝑒 ∼ 0.45)
𝑡𝑒
Case 1. 𝝉: 0.3 → 1
Case 2. 𝝆𝟐𝒔 : 11 → 2
(𝑇𝑖 decreased)
(𝜆𝐷𝑒 decreased)
Fig.12 2-D turbulent energy spectrum 𝐸 𝑘𝑥 , 𝑘𝑦 for saturated state.
(left) 𝝉 increased
(center) Reference case
(left) 𝝆𝒔 decreased
Saturated energy spectrum 𝐸 𝑘𝑥 , 𝑘𝑦 deformed to isotropic structure
when (a) 𝝉 =
𝑻𝒆
𝑻𝒊
is increased, or (b)
𝝆𝟐𝒔
=𝟏+
𝝀𝟐𝑫𝒆
𝝆𝟐𝒕𝒆
is decreased
Conclusion
17
Gyrokinetic simulation of electron turbulence is conducted
and followings are clarified:
• Rhines scale
• Dual cascade
• Anisotropic structure in
2-D turbulent energy spectrum
due to energy inverse cascade (similar to H-M eq.)
• 𝝉, 𝝆𝒔 𝝐 dependence for anisotropy of
2-D turbulent energy spectrum
Followings will be investigated
• Evaluation of heat transport coefficient for
ETG turbulent simulations.
Cascading in spectrum structure
18
Nonlinear cascade induces power law for turbulence spectrum:
Fluid theory:
Energy cascade  -5/3 power law (Kolmogorov)
Kinetic theory:
Entropy cascade  -10/3 power law
energy inverse cascade
Turbulent
energy
enstrophy
cascade
entropy
cascade
Zonal flow
ETG mode
instability
𝑘⊥ 𝜌
𝑘𝑠
Fig. Kolmogorov’s -5/3 power law for
navier-stokes turbulence in various
experiment[2]
Fig. possible structure for
ETG driven turbulence
Fig. spectrum of entropy and
field energy[3]
[2]: H. Woo., «Computational fluid dynamics» (2010)
[3]: T.Tatsuno, et al., Phys. Rev. Lett. (2009)
19
Energy balance
Fig. energy balance for decaying
turbulence simulation
Fig. set of decaying turbulence
simulation: Landau damping
weakened as Θ =
𝜔𝑒∗
𝑘∥ 𝑣𝑡ℎ
−1
decrease
Fig. enstrophy selectively dissipates
in decaying turbulence simulation
Fig. energy balance for ETG
turbulence simulation

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