Impact of ion diamagnetic drift effect on MHD stability at ...
Transcript of Impact of ion diamagnetic drift effect on MHD stability at ...
Impact of ion diamagnetic drift effect on MHD stability
at edge pedestal of rotating tokamaks
N. Aiba, M. Honda, K. Kamiya
Japan Atomic Energy Agency
21st NEXT Workshop 2016/Mar./10-11, Kyoto Terrsa, Kyoto
Introduction Peeling-ballooning mode (PBM) is the strongest candidate of
the trigger of the type-I edge localized mode (ELM). [Snyder PoP2002 etc.]
Usually, the toroidal mode number (๐๐) of the mode is intermediate (~30), but the results in JT-60U and JET-ILW imply higher-๐๐ modes sometimes trigger type-I ELM. [Aiba NF2011, Giroud PPCF2015]
What causes the discrepancy between theory and experiment?
Rotation is a candidate of the key parameter.
โข Theoretically, such high-๐๐ modes are stabilized by an ion diamagnetic drift (๐๐โ๐๐) effect.[Tang NF1980 etc.]
PBM stability diagram in JET-ILW (๐๐ is up to 50), [Giroud PPCF2015]
2/17
Drift MHD model To analyze the PBM stability with ๐๐โ๐๐ effect in rotating plasmas, we use the drift MHD model derived by Hazeltine and Meiss. [Hazeltine and Meiss, Plasma Confinement]
๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท
+ ๐ท๐ท๐ป๐ป โ ๐ฝ๐ฝ = 0, ๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐๐๐๐๐๐
+ ฮ๐ท๐ท๐ป๐ป โ ๐ฝ๐ฝ๐๐๐๐๐๐ = 0,
๐ฌ๐ฌ + ๐ฝ๐ฝ๐๐๐๐๐๐ ร ๐ฉ๐ฉ +1๐๐๐ท๐ท
๐ป๐ป๐ท๐ท = 0, ๐๐๐๐๐ท๐ท๐ท๐ท๐ฝ๐ฝ๐ธ๐ธ๐ท๐ท๐ท๐ท
+๐ท๐ท๐ท๐ท๐ท๐ท๐๐๐๐๐๐
(๐๐โฅ๐๐) = ๐ฑ๐ฑ ร ๐ฉ๐ฉ โ ๐ป๐ป๐ท๐ท.
๐ฝ๐ฝ = ๐ฝ๐ฝ๐๐๐๐๐๐ + ๐ฝ๐ฝ๐๐๐๐ = ๐ฝ๐ฝ๐ธ๐ธ + ๐๐โฅ๐๐, ๐ฝ๐ฝ๐๐๐๐ = ๐ฉ๐ฉร๐ป๐ป๐๐๐๐๐๐๐๐๐๐๐ต๐ต2
, ๐ฝ๐ฝ๐ธ๐ธ = ๐ฌ๐ฌร๐ฉ๐ฉ๐ต๐ต2
,
๐๐๐๐๐ท๐ท
= ๐๐๐๐๐ท๐ท
+ ๐ฝ๐ฝ โ ๐ป๐ป , ๐๐๐๐๐ท๐ท๐๐๐๐๐๐
= ๐๐๐๐๐ท๐ท
+ ๐ฝ๐ฝ๐๐๐๐๐๐ โ ๐ป๐ป
๐ท๐ท: number density, ๐ฝ๐ฝ: velocity, ๐ท๐ท: pressure, ฮ: heat capacity ratio,
๐ฌ๐ฌ: electric field, ๐ฉ๐ฉ: magnetic field, ๐ฑ๐ฑ: plasma current, ๐๐ โก ๐ฉ๐ฉ/๐ต๐ต, ๐๐: elementary charge, ๐๐๐๐: ion mass, ๐๐: effective charge, ๐๐๐๐: ion pressure
3/17
Approximations used for simplifying the drift MHD model
We simplify the model with Frieman-Rotenberg formalism. Approximations: 1. In Faradayโs law, non-ideal term is neglected.
๐๐๐ฉ๐ฉ๐๐๐ท๐ท
= ๐ป๐ป ร ๐ฌ๐ฌ = โ๐ป๐ป ร ๐ฝ๐ฝ๐๐๐๐๐๐ ร ๐ฉ๐ฉ +1๐๐๐ท๐ท
๐ป๐ป๐ท๐ท
This approximation can be justified when a. Rotation is enough slow compared to ion thermal velocity. b. Density ๐ท๐ท or temperature ๐๐ is constant in a plasma c. Functional form of ๐ท๐ท is proportional to that of ๐๐. (Details are in Appendix)
2. Magnetic field varies slowly ๐ป๐ป ร (๐๐/๐ต๐ต) โช 1. This helps to change the continuity equation as follows.
๐ท๐ท๐ท๐ท๐ท๐ท๐ท๐ท
๐๐๐๐๐๐
+ ๐ท๐ท๐ป๐ป โ ๐ฝ๐ฝ๐๐๐๐๐๐ = 0
4/17
Simplified drift MHD equation Also, by assuming the incompressibility ๐ป๐ป โ ๐๐ = 0, the flute approximation ๐ฉ๐ฉ โ ๐ป๐ป ๐๐ โช 1 and ๐๐๐๐ = ๐๐๐๐, we can derive the following equation [Aiba submitted to PPCF]. (๐๐๐๐(๐๐๐๐): ion (electron) temperature).
๐๐0๐๐2๐๐๐๐๐ท๐ท2
+ 2๐๐0 ๐ฝ๐ฝ0,๐๐๐๐๐๐ โ ๐ป๐ป๐๐๐๐๐๐๐ท๐ท
+ ๐๐0 ๐ฝ๐ฝ0,๐๐๐๐ โ ๐ป๐ป๐๐๐๐โฅ๐๐๐ท๐ท
= ๐ญ๐ญ๐๐๐๐๐๐ + ๐ญ๐ญ๐๐๐๐, ๐ญ๐ญ๐๐๐๐๐๐ = ๐ฑ๐ฑ0 ร ๐ฉ๐ฉ1 + (๐ป๐ป ร ๐ฉ๐ฉ1) ร ๐ฉ๐ฉ0 โ ๐ป๐ป๐ท๐ท1 +๐ป๐ป โ ๐๐0๐๐โ ๐ฝ๐ฝ0 โ ๐ป๐ป ๐ฝ๐ฝ0,๐๐๐๐๐๐ โ ๐๐0๐ฝ๐ฝ0 โ ๐ฝ๐ฝ0,๐๐๐๐๐๐ โ ๐ป๐ป ๐๐ ,
๐ญ๐ญ๐๐๐๐ = ๐๐0๐ป๐ป โ ๐๐ ร ๐ป๐ป๐ท๐ท02๐๐๐๐0๐ต๐ต02
๐ฉ๐ฉ0 โ ๐ป๐ป ๐ฝ๐ฝ0,๐๐๐๐๐๐,โฅ,
๐ป๐ป โ ๐๐ = 0, ๐ฝ๐ฝ0 = ๐ฝ๐ฝ0,๐๐๐๐๐๐ + ๐ฝ๐ฝ0,๐๐๐๐ , ๐ฝ๐ฝ0,๐๐๐๐ =1
2๐๐๐๐๐๐0๐ต๐ต02๐ฉ๐ฉ0 ร ๐ป๐ป๐ท๐ท0
MINERVA code[Aiba CPC2009] is updated to solve this equation.
MINERVA-DI code 5/17
Benchmark test of MINERVA-DI In a static plasma, the growth rate ๐พ๐พ and the mode frequency ๐๐ can be estimated by the dispersion relation with ๐๐โ๐๐(= ๐๐ โ ๐ฝ๐ฝ0,๐๐๐๐) effect as follows.[Tang NF1980 etc.]
๐พ๐พ + ๐ค๐ค๐๐ = ๐ค๐ค ๐๐โ๐๐2
ยฑ ๐พ๐พ๐๐๐๐๐๐2 โ ๐๐โ๐๐2
4, ๐พ๐พ๐๐๐๐๐๐:๐พ๐พ of ideal MHD mode
MINERVA-DI shows good agreements with this when internal kink and high-๐๐ ballooning stability is analyzed.
Internal kink mode (๐๐ = 1) High-๐๐ ballooning mode
๐ด๐ด = 5.0
๐ด๐ด = 3.0
6/17
Ballooning mode stability with ๐๐โ๐๐ is analyzed in rotating tokamaks
Equlibrium has circular cross-section, and ๐ด๐ด = 3.0. Ideal ballooning mode is unstable for ๐๐ โฅ 12, but the mode is
suppressed by ๐๐โ๐๐ effect. Plasma (hydrogen) density is ๐ท๐ท = 2.0 ร 1019[1/๐๐3]. Rotation profile is determined as follow.
ฮฉ๐๐ = ๐ถ๐ถโ๐๐โ๐๐,๐๐๐๐๐๐๐๐ 0 โค ๐๐ โค ๐๐๐๐๐๐๐๐๐๐ = ๐ถ๐ถโ๐๐โ๐๐ ๐๐,๐ ๐ 0 ๐๐๐๐๐๐๐๐๐๐ โค ๐๐ โค 1
Here ๐๐โ๐๐,๐๐๐๐๐๐๐๐ = 9.51 ร 10โ3๐๐๐ด๐ด0 at ๐๐ = ๐๐๐๐๐๐๐๐๐๐, ๐๐๐ด๐ด0 is the toroidal Alfven frequency on axis.
๐๐ and ๐๐โ๐๐ โจ๐๐ โ ๐ฉ๐ฉโฉ and ๐๐ Growth rate Eigen function (ideal mode)
7/17
Plasma rotation can cancel ๐๐โ๐๐ stabilizing effect on ballooning mode
Plasma rotation changes ballooning mode stability as follows. โข Intermediate-๐๐ modes become unstable, though high-๐๐
modes are stabilized. โข Ballooning mode becomes unstable even when ๐๐โ๐๐ effect
is taken into account. โข Maximum ๐พ๐พ of ballooning mode with ๐๐โ๐๐ effect is
converged to that of ideal ballooning mode.
Dependence of ๐พ๐พ on ๐๐ (left: ๐ถ๐ถโ = 3.0, right: ๐ถ๐ถโ = 5.0) Dependence of ๐พ๐พ on ๐ถ๐ถโ
8/17
PBM stability with ๐๐โ๐๐ is analyzed in a rotating shaped tokamak
๐๐ and ๐๐๐๐ โจ๐๐ โ ๐ฉ๐ฉโฉ and ๐๐ ฮฉ๐๐ and ๐๐โ๐๐ Contours of ๐๐
PBM stability is analyzed in D-shaped (double-null) plasma. Density is assumed as ๐ท๐ท = 1.0 ร 1020[1/๐๐3]. Rotation profile is determined as follow.
ฮฉ๐๐[krad/s]= 49.5 1 โ ๐๐48 4 + 0.5. By changing bulk ion species from hydrogen ๐ป๐ป to deuterium ๐ท๐ท, ฮฉ๐๐/๐๐๐ด๐ด0 and ๐๐โ๐๐/๐๐๐ด๐ด0 is increased by factor of โ2.
PBM stability is analyzed in both ๐ป๐ป and ๐ท๐ท plasmas. 9/17
Rotation of โheavyโ plasma has large impact on PBM stability
Results in ๐ป๐ป plasma
Results in ๐ท๐ท plasma
Large ฮฉ๐๐/๐๐๐ด๐ด0 has impact on PBM stability, but impact of ๐๐โ๐๐ on PBM is not affected by ion species so much,
Plasma effective mass potentially changes (ideal) PBM stability. 10/17
Large ๐๐๐๐๐๐๐๐ reduces impact of ๐๐โ๐๐ on PBM stability
Results in ๐ท๐ท plasma (๐๐๐๐๐๐๐๐ = 1.0)
Results in ๐ท๐ท plasma (๐๐๐๐๐๐๐๐ = 2.0)
Plasma effective charge changes ๐๐โ๐๐ effect on PBM stability.
Impact of ๐๐โ๐๐ on PBM stability depends on ๐๐๐๐๐๐๐๐ due to ๐๐โ๐๐ โ ๐๐๐๐๐๐๐๐โ1 , but ideal MHD stability is independent from ๐๐๐๐๐๐๐๐.
11/17
The equilibira analyzed numerically are E49228 and E49229; these are type-I ELMy H-mode plasmas [A. Kojima et al., NF, (2009)].
Rotation effect on type-I ELM in JT-60U
E49228 (CO.) fELM~37Hz E49229 (CTR.) fELM~45Hz
We compare the results of the numerical stability analysis of these equilibria just before ELM.
0
Profiles of p, j.B, ฮฉ
12/17
Stability boundary is shifted by ๐๐โ๐๐ effect in static plasmas ๐๐๐๐๐๐๐๐ โ ๐ผ๐ผ94 diagram (๐๐eff = 2.0) (left:CO., right CTR.)
โข When the plasma is assumed as static, PBM stability boundary is shifted to larger-๐ผ๐ผ side by ๐๐โ๐๐ effect as expected.
โข Difference between operation point and boundary increases from ~10% to >20% of ๐ผ๐ผ94 on operation point in E49228, and from ~20% to ~40% in E49229.
๐๐ = 10 (no ๐๐โ๐๐)
๐๐ = 10 (with ๐๐โ๐๐)
๐๐ = 14 (no ๐๐โ๐๐)
๐๐ = 7,12 (with ๐๐โ๐๐)
13/17
Plasma rotation can counteract ๐๐โ๐๐ effect on PBM stability boundary ฮฉ๐๐ of ๐ถ๐ถ and ๐ท๐ท
โข Rotation profile of deuterium (๐ท๐ท) is estimated with TOPICS integrated simulation code. [M. Honda NF2013]
โข Rotation has little impact on pedestal stability in E49228. โข Rotation cancels ๐๐โ๐๐ effect on pedestal stability in E49229,
but the stability boundary is still far from operation point.
๐๐๐๐๐๐๐๐ โ ๐ผ๐ผ94 diagram (๐๐eff = 2.0) (left:CO., right CTR.)
๐๐ = 12 (no ๐๐โ๐๐) ๐๐ = 10
(with ๐๐โ๐๐)
๐๐ = 16 (no ๐๐โ๐๐) ๐๐ = 14
(with ๐๐โ๐๐)
14/17
Poloidal rotation can affect edge MHD stability Difference of the mode frequency nฯ from the Doppler-shifted frequency k.v is essential for destabilizing edge MHD modes.[Aiba NF2011].
: toroidal rotation freq., : poloidal rotation freq.: toroidal mode number, : poloidal mode numbern m
ฯ ฮธฮฉฮฉ
For the Fourier harmonics which satisfy m-nq=0, mฮฉฮธ ~ nฮฉฯ even when vฮธ=(nr/mR)vฯ ~0.1vฯ if q=3 and R/r ~3.3.
: freq. parallel to magnetic field,: freq. in toroidal directiont
ฮฉ
ฮฉ
Near rational surfaces, k.B~0. โ ฮฉt will be important.
= ๐ค๐คฮฉโฅ ๐๐ โ ๐ฉ๐ฉ ๐ต๐ตโ โ ๐ค๐ค๐๐ฮฉ๐ท๐ท
๐๐ โ ๐๐ = โ๐ค๐ค๐๐ฮฉ๐๐ + ๐ค๐ค๐๐ฮฉ๐๐
MINERVA(-DI) can identify the poloidal rotation effect on MHD stability (at present, poloidal rotation effect on equilibrium is neglected.)
Re-evaluate stability diagram with not only ฮฉ๐๐ but also ฮฉ๐๐. 15/17
PBM stability boundary can be predicted by counting ๐๐๐๐ and ๐๐๐ฝ๐ฝ
โข In E49228, poloidal (+toroidal) rotation moves the ideal MHD boundary near operation point, but ๐๐โ๐๐ effect shifts the boundary far from the point.
โข In E49229, ๐๐โ๐๐ effect is negligible as the toroidally rotating case, and the boundary becomes closer operation point.
ฮฉ๐๐ and ฮฉ๐๐ of ๐ท๐ท ๐๐๐๐๐๐๐๐ โ ๐ผ๐ผ94 diagram (๐๐eff = 2.0) (left:CO., right CTR.)
๐๐ = 18 (no ๐๐โ๐๐) ๐๐ = 18,22 (with ๐๐โ๐๐)
๐๐ = 12 (no ๐๐โ๐๐) ๐๐ = 10
(with ๐๐โ๐๐)
16/17
Summary Linearized drift MHD equation was derived with Frieman-
Rotenberg formalism. MINERVA-DI code was developed to solve the drift
MHD equation. It is found that ion diamagnetic drift (๐๐โ๐๐) effect on
ballooning/peeling-ballooning(PBM) stability can be canceled by plasma rotation.
Effective mass/charge have potential to change PBM stability when rotation and ๐๐โ๐๐ are counted.
Cancellation of ๐๐โ๐๐ effect by rotation plays an important role on PBM stability boundary in JT-60U.
PBM stability analysis in JET ITER-like metallic wall is on going, and the result will be reported in near future.
17/17