緩和系高ベータプラズマの研究とQUEST計画への期待
政宗貞男
京都工繊大
共同研究者： 三瓶明希夫，比村治彦，田中裕之，上羽涼太，他 （京都工繊大）
R. Paccagnella (Consorzio RFX),K.J. McCollam, D.J. Den Hartog (University of Wisconsin-Madison
 RFP磁場配位について
 RELAX装置と運転領域
 Single Helical (SH) RFP配位とRELAXの特徴
 SH RFP緩和モデル
 2領域緩和モデルとRELAX SH の比較検討
 トカマクとの関連に関するコメント

第15回QUEST研究会
2018年7月20日 九州大学応用力学研究所

RFP磁場配位について

Reversed field pinch magnetic configuration• Magnetic fields are produced primarily by plasma current.
• Weak external magnetic field is sufficient.
－favorable to fusion reactor－strong magnetic shear, with (generally) weak toroidal effect→complementary to tokamaks and stellarator/heliotron• RFP plays important roles in self-organization and nonlinear MHD studies

Advantage of RFP as a fusion reactor：
Magnetic fields are primarily inside the plasma,with weak external field for confinement

Normal conducting coils could be usedbecause of the weak magnetic field at thecoil locations－magnetic pressure at the coils is 1/10of that of tokamak－high reliability of the system and easy
maintenanceVery high current density－Ohmic heating to the fusion reactor
regimes is possible－No need for plasma-facing auxiliary
heating system－High density limit （nG～Ip/a2）~1/r
towardmagnets

Advantage of weak external field

TITAN, Pwn =18 MW/m2

Pwn ≈ 6 MW/m2/m3/m3

Water-cooled Copper
coils will work
(Courtesy of John Sarff)

RELAX装置の概要

REversed field pinch of Low-Aspect-ratio eXperiment
R/a = A = 2(0.51 m/0.25 m)
Vacuum vessel without aconducting shell→resistive wall boundary
Operation region andplasma parameters:
Ip < 125 kA
ne = 1018 ~ 1019 m-3Te(0) ~ 100-200 eV
βpe0 ~ 5-15%
τD >3 ms

64 saddle coils
for active MHD Research objectives:- geometrical optimization of RFPcontrol- bootstrap current issues- MHD with resistive wall boundaryStable plasmas are realized over a wide operation region in ( Θ, F ) space(40 kA < Ip < 80kA)

Shallow reversal
and non-reversedregion

Deep reversal
regionBp (a)Bt

Bt (a)F
Bt
BFM(Ikezoe et al., PPCF 2012) In shallow reversal region,

 Quasi-periodic Quasi-Single Helicity (QSH) or Single Helical RFP state
tends to be realized In deep reversal, high-Θ region,

 Amplitudes of resonant modes are suppressed with broad spectrum
 SXR emission increases, indicating improved plasma performance

Single Helical (SH) RFP 配位

Single Helical (SH) RFP state
Lorenzini et al., Nature Phys. 5, 570 (2009)• Significant fraction of the plasma
volume involved• Theoretical explanation
Escande et al., PRL. 85, 3169 (2000)

Characteristics of SH states in
RFX-mod and MST

Periodic or intermittent transition to QSH(MST)

dominant mode

residual modesPurer QSH with higher persistence
for higher S (RFX-mod)
Formation of helical hot core with electron
thermal transport barrier (RFX-mod)Example of QSH in RELAX from edge magnetic fluctuation
Flat-topped phase

Toroidal plasma current

 QSH defined by Ns<2 can be
confirmed from edge magnetic
fluctuation spectrum.
 QSH appears quasi-periodically.

Spectral index Ns:
a quantitative measure for QSHQSH2
8 
1,n 2b
N S    8  1,n 2   n 3 b   n31(Oki et al, PFR 2012; FST 2013)

Dominant n numbers for the SH states at different aspect ratios
12XnExtrap T2R

10m=18Xx6RELAX

RFX modXMST

4X

R/a

2
12

34567

 Reversed field pinches operate in a wide range of aspect ratios (A=R/a),from 2 to almost 8.
 At all these different A’s single helical states are observed with different
toroidal periodicities (n numbers) and with poloidal mode number m=1.

Statistical characteristics of Single Helical RFP in RELAX

(a)

(b)

Dominant toroidal mode number n at different F values. Eachbox encloses 50% of the data with the median value (line).
 At shallow reversal the dominant modes are n=3,4 while at deep reversal n=5,6
could be dominant.
 When the probability of 3% is set as a threshold, only n=4 and marginally n=6
modes persist.
 The helical structure is not long-lived, and the most robust is the n=4 mode.

Parallel current density profiles in shallow- and deep-reversal regions
14



16



12(b)

(a)

10

128
8

6
4

4

2

r/ar/a000,20,40,60,81000,20,40,60,81

λ profile (in m-1) vs. normalized radius for shallow F (a) and deep F (b) values inRELAX. Dashed line represents a fit with λ=λ(0)[1-( r/a )4 ].
𝑱⋅𝑩 Parallel current density is defined by 𝜆 = 𝜇0 𝐵2
 Two 𝜆 profiles are deduced by the equilibrium reconstruction in RELAX atshallow and deep F values.
 The profile looks like almost the same for the two F values.

Single Helical (SH) RFP 配位に関する理論モデル

Single Helical, ideal wall, relaxation theories•

Mode Control has produced a quasi-ideal boundary and SH occurence
has increased (RFX-mod, Extrap T2R, RELAX)•It is known that Taylor’s relaxation requires an ideal wall forhelicity and flux conservation• Taylor’s theory is not able to predict the correct helicity:
because the on-axis modes are stable for perfectly flat λ profile• A theory with non-flat λ for a plasma dominated by a Single Mode alreadyexistsQuestions:i)

Is this theory predicting correctly the RFP states (F and Θ)?

ii)

Is this theory able to predict the observed scaling with R/a ? and with F?

A. Bhattacharjee, R. L. Dewar, and D. A. Monticello, Phys. Rev. Lett. 45 347 (1980).
A. Bhattacharjee and R. L. Dewar, Phys. Fluids 25, 887 (1982)
17
A. Bhattacharjee, R. L. Dewar, A. H. Glasser, M. S. Chance, and J. C. Wiley, Phys. Fluids 26, 526 (1983)

Relaxed states with a single mode
Beside Taylor’s invariants, 𝐾𝑜 and F (toroidal flux)
a new invariant related to SH relaxation is introduced:
where :K1 1 d A  B dV2V  qs  is the helical flux , 𝑞𝑠 =𝑚,𝑛and Ψ, Φ are the poloidal and toroidal magnetic fluxes→ The relaxed minimum energy states satisfy the simple relation:J0 (2  d )2dB• The wall is assumed to be IDEALLY CONDUCTING• d is a positive integer (related to the invariance request)
18

Two-region (TR) relaxation model with current sheets

Island formation in Single Helical stateschanges the topology of the magnetic fieldxx

2 magneticaxes

Two possible cases:i.

2 magnetic axes corresponding to
3 topological regions
ii. 1 magnetic axis corresponding to2 topological regions

Case ii corresponds to a mode saturated
at large amplitude and can be treatedwithin a two-region relaxation modelx

1 magneticaxis

R. Paccagnella, “Relaxation models for single helical reversed field pinch plasmas”, Phys. Plasmas 23, 092512 (2016)19
R. Paccagnella, “Aspect ratio scaling of single helical states in the RFP plasmas”, Nucl. Fusion 56, 046010 (2016)

TR relaxation model with current sheets (2)• The presence of a reconnecting mode can create current sheets in the plasma• We assume that in this two region a Taylor’s like relaxation is going on• We assume a given Taylor’s like expansion for the fields in each region and matchthe solutions at the location of the current sheet :𝑩 = 0, 𝐽1 𝛼1 𝑟 , 𝐽0 𝛼1 𝑟
𝑖𝑛 0 ≤ 𝑟 < 𝑟𝑐
𝑩 = (0, 𝑏1 𝐽1 𝛼2 𝑟 + 𝑏2 𝑌1 𝛼2 𝑟 ,𝑐1 𝐽0 𝛼2 r + c2 𝑌0 (𝛼2 𝑟)) 𝑖𝑛 𝑟𝑐 < 𝑟 ≤ 𝑎

𝛻×𝑩=𝛼 𝑟 𝑩

20

RELAXのSH RFP状態と緩和モデルとの比較検討
R. Paccagnella, S. Masamune, A. Sanpei, “Relaxation models forsingle helical reversed field pinch plasmas at low aspect ratio”,
Phys. Plasmas 25, 072507 (2018)

Comparison of the experiments with the relaxation model predictions
TR modelTR n=4TR n=6SHR n=4SHR n=6Relax data
SHR n=3SHR n=40

F-0,5xd=2-1

d=1

SHR model (2  d ) dJ 0
 B
2-1,5

-2
1

1,5

2

2,5

3

3,5F vs. Θ : experimental fit (crosses also labeled by exp); d=1 and d=2 SHRpredictions with n=3 and n=4 and n=4 and n=6 respectively; the 2 region
model predictions for n=4 and n=6.
 In the experiments the dominant modes are almost aligned along the same curve.
 The model curves are quite far from the experimental fit for the SHR predictions
with d=2, differently from the medium aspect ratio case.

Looking into the model predictions: importance in parallel current density profile

r/ar/ar/ar/a

𝜆 profiles vs. normalized radius (r/a).
Dotted lines: experimental fit; plain lines: SHR; dashed lines: TR curves.
Top curves refer to shallow and bottom curves to deeper F cases.
 For the SHR model, there is a tendency to develop a bump in 𝜆 profile by decreasing
the F value.
 Apart a small region near the edge, when d is set to 2, the TR model is very close to
the experimental fit for both reversal ratios, with assumption that 𝜆0 be equal to theaverage of the 𝜆 profile in the region from the maximum to the axis.

Appropriate choice of parallel current density and resonant surface
location improves the agreement between the experiment and TR model
0,2FTR model

TR n=4 r = 0.390cTR n=6 r = 0.39
cexperiment
TR n=4,6 r = 0.5-0,2xc-0,4-0,6-0,8-1-1,21,522,53



3,5F vs. Θ : experimental fit with error bars (red long dashed), TR model with
rc=0.4 (circles and squares), TR model with rc=0.5 (light blue short dashed).
 There is a good agreement between the TR model and the experimental data within
their typical error bars.
 It is only by using the SHR and the TR model that we could obtain a good fit. The SHeigenvalue 𝜆(0) and the position of the resonance of the dominant mode are thetwo essential ingredients used within the TR model.

Predictive capability of the model:
preferred n minimizes the integrated “error” E at different F’s
8[a.u]

F =0

7

F=-1

6E

54
32
1n01234

56

7Integral error E vs. n for shallow and deep F values.
2𝜆0,1𝜒
Integrated error: 𝐸 𝑞𝑠 =𝐽|| − 𝐽|| 𝑑𝑟𝜆𝜒
𝐽|| 0,1 : deduced by the TR model; 𝐽|| : deduced directly from the SHR modelSummary and Conclusions : The SH-relaxation model seems NOT ABLE to predict theexperimental F-Θ trend (including n=4 and n=6 alignement) at R/a=2
 The two-region Taylor’s model can fit the experimental data withanappropriate choice of the truncation radius, i.e. position of thecurrent sheet and assuming a flat core λ as the AVERAGE obtained
from the SH model Dominant n modes at various F predicted correctly by theCLOSENESS between the two models (SH and 2 regions) ofrelaxation
Possible explanations: A residual stochasticity is flattening the λ profile in the core (theparallel current bump is not observed experimentally) Need for a toroidal theory? (but difficult) (the relaxed SH model isobtained assuming cylindrical geometry)

Comments on the relation of the TR model with tokamak research:
Importance of helical relaxed state in fusion plasmas- Hybrid operation scenario for high-beta tokamak core
- Spontaneous formation of the helical core
- A possible explanation for the helical core is given by Piovesan et al.
“Impact of ideal MHD stability limits on high-beta hybrid operation”,
Plasma Phys. Control. Fusion 59, 014027 (2017)“Role of a continuous MHD dynamo in the formation of a 3D equilibria
in fusion plasmas”, Nucl. Fusion 57, 076014 (2017)
where similarities to the electrostatic dynamo for helical RFP is emphasizedAppendix: Some examples of dust particles in the RELAX vacuum vesselP. Piovesan et al., Nucl. Fusion 57 076014 (2017)

Helical flux surface reconstruction andelectrostatic dynamo emf in DIII-D (#164661)

saturation of 1/1 mode leads to helicalcore in presence of external 1/1 field.
(Specyl simulation)

The assumption that 𝜆0 be equal to the average of the 𝜆 profile in the region
from the maximum to the axis

rmax

r/a

𝜆 vs. normalized radius (r/a): the dotted line is the experimental fit, the plain line
the SHR model (d=2) and the dashed line is the TR model. Note that the TR 𝜆(0) is
set to the average of the plain line (SHR model) between rmax and r=0.

q

n=4
n=6SHR model

TR modelr/aq profile vs. normalized radius (r/a):
Plain line: SHR model; crosses: TR model
Resonances of n=4,6 modes are shown.
Careful analysis of the matching radius in the TR model is important at low aspect ratio