J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)504–507
000–000

Conceptual Approach to Steady State Reversed Field PinchReactor with Low Aspect Ratio
SHIINA Shoichi, NAGAMINE Yasuo, OSANAI Yukio, SAITO Katsunori, SAITO Katsuhiko,
AIZAWA Masamitsu, WATANABE Masayuki, TAGUCHI Masayoshi1 and KONDOH Yosiomi2
Institute of Quantum Science, College of Science and Technology,
Nihon University, Tokyo 101-8308, Japan
1
Nihon University, Narashino 275-0005, Japan
2
Gunma University, Kiryu 376-8515, Japan(Received: 9 December 2003 / Accepted: 3 April 2004)

Abstract
According to a simplified costing algorithm of steady state fusion reactor power plant, the relative attractivenessof advanced physics modes mainly depends on the stability and non-inductive current drive of the equilibrium. The
high stability beta and the good alignment of current profile with plasma self-induced current profile are compatible
for the low aspect ratio neoclassical RFP equilibrium solved self-consistently considering the bootstrap current. The
plasma stability is due to the hollow current profile making the magnetic shear increase locally and globally and theforce-free region wide. The good alignment of current profile reduces significantly the requisite power for non-inductive
current drive to generate the steady state magnetic configuration. As the result, the fusion power plant based on theneoclassical reversed field pinch (RFP) equilibrium with low aspect ratio offers the lowest cost of electricity.
Keywords:
RFP, low aspect ratio, high beta, plasma self-induced current, alpha particle finite banana-width effect,
steady state, reactor concept, D-T and D-3He Fusion, COE

1. Introduction

Two types of target magnetohydrodynamic (MHD)
equilibria for RFP with low aspect ratio are considered. The
first is the classical one described by partially relaxed statemodel (PRSM), which is reasonably close to dynamo-free,stable minimum energy state at finite beta. The second is theneoclassical one solved self-consistently considering thebootstrap current, which enhances the plasma stability owing
to the current profile making the magnetic shear increase
locally and globally and the force-free region wide although
it does not satisfy the PRSM condition [6]. The local andglobal magnetic shear in the neoclassical equilibrium issignificantly large compared with that in the classical (PRSM)
equilibrium where the Robinson’s stability criterion of ideal
kink modes is satisfied for the volume averaged beta value of<β > (≡ 2µ0 <p/B2>) = 66 % or the toroidal beta value of βt (≡2µ0 <p>/B02) = 59 % giving the stability beta limit against
Mercier’s localized mode.
In addition to the requisite rf power for the generationof steady state configuration, an additional rf power for thestabilization of resonant kink modes at distant conducting wallis required for the current profile control near the plasmaboundary to enhance the magnetic shear there and then todecrease the dependence of RFPs on a close fitting conductingwall.
The ideal externally nonresonant resistive wall kinkmodes (RWM) stability in the plasma surrounded by an
external shell of finite conductivity should be investigated todevelop the steady state reactor concept since it determines

Corresponding author’s e-mail: shiina@phys.cst.nihon-u.ac.jp©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

In our previous studies, the benefit of radiofrequency
wave current drive to widen the force-free field region had
been theoretically demonstrated by the significant reductionof nonlinearly turbulent level associated with the dynamo
activity in RFPs [1,2]. Recently the current profile
modification experiments have confirmed to enhance theenergy confinement time up to tokamak-like scaling at higher
beta and ten times smaller strength of surface magnetic field[3-5]. In order to realize its steady state with less requirementsof wave power, we are concerning with low aspect ratio
equilibrium where the neoclassical viscosity has a divergent
tendency for aspect ratio A ~ 1, which means the enhancementof bootstrap current.

2. Target equilibrium

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Shiina S. et al., Conceptual Approach to Steady State Reversed Field Pinch Reactor with Low Aspect Ratio

the wall ratio. According to the linear theory on RWM [7],the stability windows increase with the velocity of toroidal
plasma rotation (v0) together with the viscosity and/or the ionsound wave damping effect for finite β plasma, depending onother equilibrium parameters so that the wall position
displaces further from the plasma for the shallowly reversed
RFPs and is closer to the plasma for higher beta obviously. A
further nonlinear study is necessary. It is noteworthy that theviscosity increases with lowering aspect ratio and theincreasing viscosity would contribute also to the stabilizationof neoclassical resistive g mode driven by perturbed PfirschSchluter (P-S) current in negative magnetic shear as in RFPs
[8,9], as well as the stabilization of RWM.
Even in MHD stable, dynamo-free RFPs, the energy
confinement time is determined by micro-instability. Thereexists a favorable stability for both micro- and macroinstabilities in low aspect ratio magnetic configuration with awide region of good curvature which results in high pitch
magnetic lines of force and enhances central q value [10].
Furthermore the equilibrium with a modest poloidal beta (βp≡ (2µ0 <p>/Bθ2(a)<1, Bθ (a) : poloidal magnetic field at plasmaboundary), which is realized in low aspect ratio RFPs andtokamaks, is essentially force-free, that is, a paramagnetism
with the current density nearly parallel to the magnetic lineof force [11]. Since the magnetic lines of force have a highpitch, a large poloidal current component is produced toenhance the toroidal magnetic field strength at the plasma axis
and then to lead the near-omnigeneity [11,12]. In theomnigenous systems the bounce-averaged drift remains withina flux surface. The maximum of longitudinal adiabatic
invariant (J) at the magnetic axis or ∇p∇J > 0 is attained in
low aspect ratio RFPs with strong paramagnetism, which
improves the stability for the ion mode of trapped particle
micro-instabilities, the toroidal ion temperature gradient (ITG)
mode and probably the confinement degradation of highenergy particle due to the non-closure magnetic field strength
resulting from a small Shafranov shift in tokamak withnegative magnetic shear [13]. The toroidicity-induced Alfven
eigenmodes (TAEs) can be expected to be reduced for theRFPs with small safety factor and strong poloidal magnetic
field at the plasma boundary.

balance along the magnetic field. Since the bootstrap current
is a parallel current, only the second term is kept for a total
bootstrap current, giving Ibs= 2π∫dψq(ψ )<j·B>bs /<B2>, whereq(ψ ) is the safety factor and < j·B>bs the bootstrap parallel
current density.
The bootstrap current obtained by the conventional
theory described above has been shown to be modified by
the finite banana-width effects around the magnetic axis
satisfying the condition ε1/2 < δa1/3, where ε is the inverse
aspect ratio and δa=2q0 ρa /κR0 with the central safety factor
q 0 , the Larmor radius ρa for species a, the elongationparameter κ and the major radius R0 [14]. The modification
is significant for high energy particle changing its trajectoryfrom the banana orbit to the potato one. The expression
obtained for the bootstrap current due to fusion-producedalpha particles by taking into account the finite banana-width
effect is applied for the low aspect ratio RFPs. For the RFPs
with a relatively small B0 , large ρa , small R0 and then large
δa , the ratio of the alpha particle-induced bootstrap current Iα
to the bulk bootstrap current Ibs becomes relatively large
depending on aspect ratio A and the profiles of pressure /temperature for a given beta value. Accordingly the plasma
self-induced current is increased in the low aspect ratio RFPequilibrium.
The “P-S current” flows in the boundary region ofplasma, then makes a good alignment with the equilibrium
current profile in conjunction with the bootstrap current
flowing in central and mid-radius region. The plasma selfinduced current ratio Fp is defined as the ratio of total toroidalbootstrap current and “P-S current” to equilibrium plasmacurrent Iϕeq, which determines the power required (PCD) for rf
current drive (RFCD) to generate the steady state
configuration. The RF power spectrum is selected in order tothat RFCD should create the current profile of equilibrium.
The RFCD equilibria calculation employs as many as three
types of wave: low frequency fast wave (LFFW), highfrequency fast wave (HFFW) and lower hybrid (LH) slow
wave. A low frequency fast wave (LFFW) provides the seedcurrent near the magnetic axis. The frequency of the LFFWis chosen to place the ω = 2Ωd (Ωd : deuteron cyclotronfrequency) resonance just outside the low field edge of theplasma; for A = 2.0, this places the tritium cyclotron harmonicω = 3Ωt (Ωt : tritium cyclotron frequency) well inboard of theaxis, with the result that LFFW deposit most of its power onelectrons in a single path through the plasma. An additional,
more strongly damped wave is usually needed to provide
RFCD at mid-minor radius. A high frequency fast wave(HFFW) at a high harmonic of the ion cyclotron (ω ~ 15Ωi)
is used for this role [15]. Finally, near the plasma surface thelower hybrid (LH) slow wave is ideal for the current drive.
Its depth of penetration is, however, strictly limited by theusual density accessibility constraints. The total RF power PCD
is given by the normalized current driving efficiency which
depends on the current driving system only, considering thecontributions to the power spectrum from each wave withdifferent current driving efficiency. Here is stipulated the3. Plasma self-induced current
The total toroidal current in an equilibrium is given by
Ip = –(2π) ∫(dp/dψ)(<Bθ2>/<B2>)dV + (1/2π) ∫F(<j·B>/<B2>)<1/R2>dV, where < > denotes the flux surface (ψ = const.)
average, V the volume of the region enclosed by a flux surface
and F the toroidal field function given by F = RBϕ . The firstterm in above equation represents contribution fromperpendicular current, that is the diamagnetic and the PfirschSchluter current, giving a substantial contribution to theequilibrium current for usual RFPs with Bθ (poloidal field)>Bϕ (toroidal field). Hereafter it is called “P-S current (Ips)”,which is the diamagnetic current and the P-S current. The
second term represents the contribution from parallel current
necessary to maintain the equilibrium parallel momentum
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Shiina S. et al., Conceptual Approach to Steady State Reversed Field Pinch Reactor with Low Aspect Ratio

electric-to RF power conversion efficiencies to be ηCD = 0.62
in view of the small variation in ηCD computed for advanced
steady state tokamak power plants [16].

(PCD).
As the result, PE ~ 778.5 MWe, MPD ~ 310.5 kWe/tonne, PCD ~ 11.0 MW and QE ~ 14.1, then the lowest cost ofelectricity of COE ~ 47 mill/kWe·h for the D-T-fueled steadystate fusion power plant based on the neoclassical RFP
equilibrium with the low aspect ratio of A = 2.0 under thewall ratio of rw /a = 1.35 as predicted by the linear theory on
RWM while r w /a = 1.55 for the perfect conducting shell.
Generally the smaller the wall ratio is, the higher the MPD
is, and the COE is the lower but not so sensitive to the wallratio for a given QE . However the surface average neutron
wall loading achieves Wn ~ 7.55 MW/m2. For the reductionof Wn , the D-3He-fueled reactor design operating at a highertemperature (volume average electron temperature of Te = 50keV) and a stronger toroidal field (B0 = 6 T) for a given betavalue might be desired. In this case, PE ~ 385 MWe, MPD~ 381 kWe/tonne, PCD ~ 3.5 MW and QE ~ 15.7, then COE~ 60 mill/kWe·h with the lunar-3He fusion supply contributing~ 20 % to the COE. The net electric power decreases by afactor of ~ 2.0 without changing significantly QE and COE,
but the neutron wall loading is reduced to Wn = 0.18 MW/m2.
For the D-T-fueled fusion power plant based on the classical(PRSM) equilibrium, PE ~ 1215 MWe, MPD ~ 339 kWe/tonne, PCD ~ 421 MW and QE ~ 3.0, then COE ~ 170 mill/
kWe·h and Wn ~ 9.0 MW/m2. Comparing with the D-T-fueled
fusion power plant based on the neoclassical equilibrium, thenet electric power and the neutron wall loading become large
by a factor of ~ 1.6 with a smaller QE and a higher COE.

4. Economic analysis
The calculation of the cost of electricity (COE) at each
design point is achieved by using the costing algorithms
described in detail in ref. [17]. This economic model findsthat the engineering power gain (QE ; the ratio of plant grosselectric power PET to circulation power PC for sustaining
reactor operation) and the mass power density (MPD; the ratioof net electric power (PE) to the grid divided by the total massof the fusion power core MFPC) are the primary variables
determining COE. The MPD is evaluated by the size scalingof MFPC . We desire to achieve the compact power plants withhigh MPD as well as low circulation power (high QE). This
goal is achieved with the low toroidal current equilibria andby optimizing the plasma self-induced current contribution,
thereby minimizing the non-inductive seed current.
The design points in economic analysis for the classicaland the neoclassical equilibrium are summarized in Table 1.
For the RFP reactor, a factor H is defined by the ratio ofenergy confinement time for burning τ ERe against theempirical scaling results for the global energy confinement
time τ EES , depending on the inverse of Greenwald normalized
density IN (≡ Iϕeq / πa2<ne>20) [18]. For the D-T-fueled plasma,IN ~ 2.7, then H ~ 2.6 and τ EES ~ 1 sec. For the classical(PRSM) equilibrium, the alignment of bootstrap current
profile with equilibrium one becomes poor although thestability beta limit increases, with flattening plasma pressure
profile. The neoclassical equilibrium with a relatively flatpressure profile has a dominant (~ 100 %) plasma selfinduced current, simultaneously the high stability beta of
toroidal beta βt ~ 63 % owing to hollow current profile
making the magnetic shear stronger and the force-free region
wide. The compatibility of the high stability beta and the goodalignment of plasma self-induced current profile withequilibrium one gives an economical attractiveness ofenhancing the plant gross electric power (PET) and reducing
significantly the non-inductive current driving power required
for the generation of steady state magnetic configuration

5. Conclusion
The plasma self-induced current in the neoclassical RFP
equilibrium with low aspect ratio has a hollow current profile,which enhances the stability beta making the magnetic shear
increase locally and globally and the force-free region wide,
and a good alignment with equilibrium current profile, which
reduces the requisite power of non-inductive current drive togenerate the steady state magnetic configuration. The
compatibility of high stability beta and small amount of RF
current driving power allows the economical design ofcompact fusion power plant with the lowest cost of electricity.
The D-3He-fueled reactor designed has a net electric power

Table 1 Design points for RFP equilibria in the economic analysis; A is the aspect ratio, κ the cross sectional ellipticity, βt thetoroidal beta, F the field reversal parameter, R0 the major radius, B0 the magnetic field strength, ne0 and Te0 the electron
density and temperature on magnetic axis, Iϕeq the total toroidal current in equilibrium, over bar denotes volume average,ne20 the electron density in unit of 1020 /m3, Fp the plasma self-induced toroidal current fraction to Iϕeq, Z the effective charge
number. CL and NEC denote operating modes for the classical and the neoclassical equilibrium, respectively; (1) D-Tfueled fusion, (2) D-3He-fueled fusion.
Aκ

βt[%]

βp[%]

q0

F

R0[m]

B0[T]

Iϕeq
[MA]

ne0 /<ne>Te0 /<Te><ne>20[m-3]

<Te>
[keV]

FpZ

CL

2

2.0

59

72

0.68

-0.21

2.8

3.0

24.2

1.04

1.63

2.42

22.9

1.29

2.0

NEC(1)

21.4

63

45~1.0

-0.10

2.8

3.0

30.8

1.10

1.26

1.88

29.6

0.96

2.0

NEC(2)

21.463178~1.0-0.10

2.8

6.0

30.8

1.10

1.26

4.40

50.0

0.96

2.0

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Shiina S. et al., Conceptual Approach to Steady State Reversed Field Pinch Reactor with Low Aspect Ratio

less by a factor of ~ 2.0 without changing significantly an
economical attractiveness but with reducing the neutron wallloading, comparing with D-T-fueled reactor designed. The
problem remained to be solved is to optimize the design point
parameters meeting to the stability window of wall ratio on
RWM and the engineering technology of reactor.

[8] S.E. Kruger, C.C. Henda and J.D. Callen, Phys. Plasmas
5, 455 (1998).
[9] G. Rewoldt, W.M. Tang and J. Menard, Phys. Plasmas
3, 1667 (1996).
[10] Y. Nagamine, S. Shiina et al., Proc. 11th ICPP (Sydney,2002).
[11] S. Shiina, Y. Nagamine et al., Proc.10th ICPP and 42nd
Annual Meeting of DPP (Quebec, 2000).
[12] S. Shiina, Proc. 9th IEA-RFP/WS (Tsukuba, 2003).
[13] M. Furukawa, Y. Nakamura and M. Wakatani, J. Plasma
Fusion Res. 74, 598 (1998).
[14] M. Taguchi, J. Plasma Fusion Res. 77, 153 (2001).
[15] S. Shiina, Y. Kondoh and H. Ishii, Nucl. Fusion 34, 1473
(1994).
[16] D.A. Ehst et al., Nucl. Fusion 38, 13 (1998).
[17] R.A. Krakowski, Fusion Technol. 27, 135 (1995).
[18] K. Miyamoto, J. Plasma Fusion Res. 75, 238 (1999).

References[1] T. Nagai, K. Kusao, K. Kondoh et al., Proc. 8th Intr.
Conf. Plasma Phys. (Nagoya, 1996) Vol.I p.1042[2] E. Uchimoto et al., Phys. Plasmas 1, 3226 (1994).
[3] J.S. Sarff et al., Nucl. Fusion 43, 1684 (2003).
[4] T. Bolzonellaa et al., Phys. Rev. Lett. 87, 195001 (2001).
[5] Y. Yagi et al., Plasma Phys. Control. Fusion 44 (2002).
[6] Y. Nagamine, S. Shiina et al., Proc. 30th EPS (Montreau),27A, P-1.204 (2003).
[7] S.C. Guo et al., Phys. Plasmas 6, 3868 (1999).

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