s - Fusion

Gyrokinetic Theory Working Group Meeting
CIEMAT
June 30, 2014,
Madrid
Electromagnetic gyrokinetic turbulence
in finite-beta helical plasmas
A. Ishizawa, T.-H. Watanabe, H. Sugama,
S. Maeyama and N. Nakajima
National Institute for Fusion Science
Reference:
Ishizawa et al., Phys. Plasmas 21, 055905 (2014).
GK simulations of LHD plasmas
• 
Regulation of ITG turbulent transport due to zonal flows occurs more clearly in the inward
shifted configuration than in the standard configuration.
T.-H. Watanabe,
PRL (2008)
• 
Comparison between results from GK ITG turbulence simulation and high-Ti LHD experiment
• 
Previous GK simulations of LHD plasmas assumed adiabatic electron response. Turbulent
transport problem in finite beta helical plasma has not been previously explored because of
numerical difficulties.
Problem about saturation of electromagnetic
GK turbulence in finite beta plasmas
Failure of the transport levels to saturate at finite beta
in gyrokinetic simulations in flux tube geometry
Cyclone base case (tokamak)
Finite beta (ITG)
Runaway above β e = 0.75%
R. E. Waltz, Phys. Plasmas, (2010)
Finite beta (ITG) β = 0.9%
M. J. Pueschel, Phys. Rev. Lett., (2013)
Zonal flows are weak in electromagnetic turbulence.
ITG Mode (low beta) vs Ballooning Mode (high beta)
tokamak
ηe = 0
Structure formation that affects saturation of
instabilities shows difference between ITG mode
and KBM.
•  Low beta: ITG mode
ITG
KBM
–  Stabilized by zonal flow structure.
A. Ishizawa, et.al.,
Nuclear Fusion, 053007
(2013)
•  High beta: Ballooning (MHD) instability
Acceleration by finger-like structure
Linear mode structures of ITG
and KBM are similar. Both of
them have ballooning structure
in the linear growth phase.
This work
•  Electromagnetic gyrokinetic (KBM)
turbulence simulation is done for a finitebeta helical plasma.
•  Saturation of KBM turbulence is found to
result from nonlinear interaction of
dominant unstable modes (inclined
modes) which have finite radial
wavenumbers.
EM δ f gyrokinetic equations
Dδf sk
∂δf sk
qF
+ vTs v// b* ⋅ ∇δf sk − vTs µb ⋅ ∇B
= −iv ds ⋅ k ⊥ (δf sk + s sM φk J 0 s )
Dt
∂v//
Ts
+ iv * s ⋅ k ⊥
qs FsM
qF
(φk − vTs v// A// k ) J 0 s + vTs v// s sM E// k + C (δf sk )
Ts
Ts
'
$
qs2
λ k φ = ∑ %% qsδnsk − [1 − Γ0 s ]φk ""
Ts
s &
#
2
2
Di ⊥ k
k ⊥2 A// k = β i ∑ qsδu sk
D ∂ 1
= + [φJ 0 s , ]k
Dt ∂t B
1
b* ⋅ ∇ = b ⋅ ∇ − [ A// J 0 s , ]k
B
s
Flux tube geometry
∂A
E// k = −b ⋅ ∇φk J 0 s − // k J 0 s
∂t
δnsk = ∫ dv 3δf sk J 0 s
*
δu sk = ∫ dv 3v//δf sk J 0 s
(s = e, i )
( x, y, z , v|| , µ )
y (field line label)
x (radial)
z
(along the magnetic
field line)
Linear instability analysis
using equilibrium magnetic field of
LHD #88343
High Ti discharge
LHD #88343
KBM
ITG
•  Unstable against ITG mode
–  B_0=2.75T, Rax=3.6m
(shifted to 3.75m)
–  Beta(r/a=0.65)=0.3%
–  banana regime
•  KBM is unstable at high beta with
the same configuration
•  KBM with finite radial wavenumber
(red) is more unstable than that
without it (blue).
The profile of electrostatic potential along the
field line (ITG mode at β = 0.2 % )
Turbulence simulation for
finite beta LHD plasmas
using a model LHD configuration and
n & T profiles (different from LHD #88343)
ηe = 0
enables saturation of KBM turbulence.
Most unstable KBM has finite radial wavenumber
ηe = 0
KBM in LHD
ITG in LHD
KBM
ITG
poloidal wave number
ηe = 0
KBM
ITG
radial wave number
A. Ishizawa, et.al., Nuclear Fusion,
053007 (2013)
The most unstable KBM has
finite radial wavenumber, which
corresponds to finite θ k in the
ballooning representation.
For CBC tokamak
KBM is most
unstable for kx=0.
The most unstable KBM has a finite radial
wavenumber (finite θk)
ITG
φ
KBM
Nonlinear simulation results
Efficiency of zonal flow
generation by KBM turbulence
is much lower than that by ITG
turbulence.
ITG
ZF
KBM
ITG
KBM
ZF
KBM turbulence is less efficient for transport
than ITG turbulence
2
2
ITG Qi ≈ 5n0Ti vTi ρ i / Ln
Qi (ITG)
KBM Qi ≈ 3n0Ti vTi ρ i2 / L2n
Qi (KBM)
flux = electrostatic + electromagnetic
Γ = Γ ES + Γ EM
Qe (ITG)
Q = QES + QEM
Qe (KBM)
ITG at β = 0.2 %
ΓES = 0.82 n0vTi (ρi2/Ln)2
ΓEM = - 0.02 n0vTi (ρi2/Ln)2
Q iES = 4.98 n0TivTi (ρi2/Ln)2
Q iEM = - 0.05 n0TivTi (ρi2/Ln)2
Q eES = 1.0 n0TivTi (ρi2/Ln)2
Q eEM = - 0.03 n0TivTi (ρi2/Ln)2
KBM at β = 1.7 %
ΓES = 0.37 n0vTi (ρi2/Ln)2
ΓEM = 0.00 n0vTi (ρi2/Ln)2
Q iES = 2.86 n0TivTi (ρi2/Ln)2
Q iEM = - 0.02 n0TivTi (ρi2/Ln)2
Q eES = 0.47 n0TivTi (ρi2/Ln)2
Q eEM = 0.23 n0TivTi (ρi2/Ln)2
Structures of potential in (x, y)-plane
ITG
turbulence
β = 0.2%
Linear
growth
Nonlinear
saturation
KBM
turbulence
β = 1.7%
Linear
growth
φ
Nonlinear
saturation
φ
ηe = 0
Radial direction
Radial direction
•  Most unstable KBM has an inclined mode structure.
•  Saturation of the KBM turbulence is caused by
nonlinear interactions between inclined modes.
Structures of potential in (x, y, z)-space
•  Saturation of the KBM turbulence is caused by mutual
shearing between convection cells of inclined modes.
•  In order to quantitatively investigate saturation process of
KBM turbulence, entropy transfer analysis is done.
Entropy balance equation
Γs = Γes , s + Γem, s
"T Γ
%
%
d"
qs
s s
+
+ Ds ''
$ ∑δ Ss + δWes + δWem ' = ∑$$
dt # s
& s # L ps LTs
&
Ts | δf sk |2 3
δS s = ∑ ∫
d v
2 FMs
k
k ⊥2 | δA// k |2
δWem = ∑
2β i
s
Γes , s = Re
∑ δns
k
Γem, s = Re
∑ δu
k
ik yδφ
*
k
B
ik yδA//* k
s
B
qs = qes,s + qem,s
qs FsM
ϕ k J 0s )* Cs d 3v
Ts
k
' 2 2
$ | δφk |2
qs2
δWes = ∑ %% λDi k ⊥ + ∑ (1 − Γ0 s ) ""
k &
s Ts
# 2
Ds = ν ss
∑ ∫ (δ fsk +
# δ p//s
& ikyδϕ k*
5
qes,s = Re ∑%
+ δ p⊥s − Tsδ ns (
$ 2
' B
2
k
*
" δ q//s
% −ikyδ A//k
qem,s = Re ∑$
+ δ q⊥s '
#
&
2
B
k
Sugama et al., Phys. Plasmas 1996 & 2009
Nonlinear turbulent entropy transfer in k space
Spectral transfer function [Nakata et al. PoP (2012)]
Ts g sk*
T (k ; k ' , k ' ' ) = ∫ dv
δ k ,k '+ k ''b ⋅ k '×k ' ' ( χ sk ' g sk '' − g sk ' χ sk '' )
2 FMs
3
gsk = fsk +
qs
ϕ k J 0s FMs
Ts
χ sk = (ϕ k − vTs v// A//k )J 0s
an element of
the diagram
In saturated KBM turbulence, nonlinear interaction between
dominant unstable modes with finite kx produces turbulent
entropy spectral transfer to ZFs and other stable modes.
In (kx, ky) space, the dominant
KBMs transfer turbulent
entropy in the inclined direction
subsequently from unstable
KBMs to higher wavenumber
stable modes.
ZFs produced by KBMs
causes spectral transfer
in the horizontal (kx-)
direction.
Linear growth rates
in (kx, ky) space
Turbulent entropy transfer
in (kx, ky) space
Summary
•  GK analysis of finite-beta LHD plasmas
•  Saturation of KBM turbulence in LHD
configuration
–  Most unstable inclined modes (with finite kx)
are dominant in (kx, ky)-space.
–  Weaker zonal flow generation but lower
efficiency of transport than ITG turbulence.
–  Nonlinear interaction between the dominant
inclined modes produces turbulent entropy
spectral transfer to ZFs and other stable
modes.
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