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Self organization of high p plasma equilibrium with an inboard poloidal magnetic field null in

QUEST

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2015 Nucl. Fusion 55 083009

(http://iopscience.iop.org/0029-5515/55/8/083009)

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1 © 2015 IAEA, Vienna Printed in the UK

1. Introduction

Operating Tokamak at a high poloidal beta β μ( = )p B2 /p 0 p2

value is usually attractive and this makes the spherical

tokamak an interesting choice for future fusion reactors [1–3]. Plasma pressure is expressed as ∫=p p V Vd / , where ... denotes volume averaged and Bp is the poloidal magnetic field averaged over the last closed flux surface. Operating a tokamak at high βp, utilizes a larger fraction of bootstrap cur-rent, which may reduce large external current drive require-ments and possibly produce better confinement properties due to modified equilibrium scenarios [4–7]. For attaining a high βp plasma, it is known to utilize an external source to heat the bulk Maxwellian component of the plasma [8, 9] or to create a confined anisotropic population of energetic particles [10, 11]. It is also proposed to operate in a shaped configuration

to increase βp [12]. High βp plasma has previously been pro-duced by injecting multi-megawatt NBI power to Ohmic plasmas in different machines [9, 13, 14]. The maximum achievable βp, however, is limited by a so called equilibrium limit, where an inboard poloidal magnetic field null (IPN) appears at the high field side of the vacuum vessel. Although it is predicted and studied theoretically in much detail [15–17], such an extreme situation is rarely achieved experimentally or it has been found extremely difficult to sustain such an equi-librium. There is also the concept of flux conserving tokamak (FCT) equilibrium [18, 19], where it is proposed that if the magnetic flux can be conserved during rapid enough heating of a low pressure plasma, it is possible to avoid any high βp limit. FCT theory also suggests that in such case no field null can appear inside the plasma. Nevertheless, such theoretical pre-diction has not been demonstrated experimentally and occur-rence of a βp limit and field null (IPN) formation continues to

Nuclear Fusion

Self organization of high βp plasma equilibrium with an inboard poloidal magnetic field null in QUEST

Kishore Mishra1,4, H. Zushi2, H. Idei2, M. Hasegawa2, T. Onchi2, S. Tashima1, S. Banerjee1,4, H. Hanada2, H. Togashi3, T. Yamaguchi3, A. Ejiri3, Y. Takase3, K. Nakamura2, A. Fujisawa2, Y. Nagashima2, A. Kuzmin2 and QUEST team

1 IGSES, Kyushu University, Kasuga, Fukuoka, 816–8580, Japan2 RIAM, Kyushu University, Kasuga, Fukuoka, 816–8580, Japan3 GSFS, University Tokyo, Kashiwa, 277–8561, Japan4 Institute for Plasma Research, Bhat, Gandhinagar, 382–478, India

E-mail: mishra@triam.kyushu-u.ac.jp

Received 12 January 2015, revised 21 May 2015Accepted for publication 28 May 2015Published 1 July 2015

AbstractSuccessful production of high βp plasmas (εβp ⩾ 1) fully non-inductively (NI) and their long pulse sustainment with the help of modest power (<100 kW) of electron cyclotron waves is demonstrated. High βp plasmas are found for the first time to be naturally self organized to form a stable natural inboard poloidal field null (IPN) equilibrium. A critical βp value is identified, which defines the transition boundary from inboard limiter (IL) to IPN equilibrium. A new feature of plasma self organization is evidenced, which enhances its negative triangular shape to sustain high βp. These results show a relatively simple method to produce and sustain high βp plasma close to the equilibrium limit in a stable configuration exploiting its self organization property.

Keywords: equilibrium limit, negative triangularity, plasma self organization, spherical tokamak

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K. Mishra et al

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doi:10.1088/0029-5515/55/8/083009Nucl. Fusion 55 (2015) 083009 (13pp)

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be a subject of discussion. The natural IPN configuration has been first observed transiently in CDX-U and DIII-D during the current ramp-up and closed flux formation phase [20], but detailed study of neither the equilibrium parameters nor the effect of external magnetic field on the IPN formation are per-formed. In TFTR [21–23], an IPN at the inboard side in high βp plasma formed by ramping down the plasma current (Ip) in a NBI heated plasma is considered to be the first experi-mental demonstration of a high βp limit sustained for 0.5 s. Such high βp plasma production and sustainment required very high NBI power. Recently transient null formation has also been reported during the current ramp up phase of EC driven plasma in TST-2 [24], however, the equilibrium is not sustained. In short, high βp induced IPN plasma formation in a tokamak is not a frequently observed phenomenon and it needs to be studied in detail. In QUEST, such an IPN configuration is easily achieved under a high magnetic mirror ratio [25] and high Bz/Bt values (≈10%) via electron cyclotron (EC) heating and current drive. Bz and Bt are externally applied vertical and toroidal magnetic fields, respectively. The typical IPN plasma is of rather low density (~1017 m−3) with significant domi-nance of energetic electrons generated by the EC waves and is far from the usual tokamak operational regime. Nevertheless, the present work is primarily focused on the investigation of equilibrium aspects of high βp IPN plasma close to the equi-librium limit and not it’s performance in comparison to the mainstream tokamaks, at least at present.

In this paper, we present the experimental demonstration of a high βp induced IPN plasma configuration formed natu-rally and self sustained in steady-state in a fully non-induc-tively driven plasma current. Here we emphasize the term ‘naturally’ to distinguish the present configuration from the usual divertor configurations obtained by the use of shaping coils. The natural null formation at the inboard side is a con-sequence of the equilibrium βp limit, which is studied in this work. Because of a uniquely stable and sustained equilibrium configuration, its various equilibrium features, null forma-tion phases and response to external magnetic fields could be investigated. We found a critical value of βp, which defines a natural transition boundary from an inboard limiter (IL) to an IPN configuration. A very similar phenomenon of IL to IPN transition is also achieved in EC overdriven Ohmic plasma at controlled Ip to confirm that such transition is a consequence of a high βp equilibrium limit. It is further demonstrated that, during this transition at high βp, the plasma shape adjusts itself to become more negatively triangular to sustain a high βp equilibrium. Formation of the negative triangular shape is a new feature of plasma self organization in addition to the pre-viously observed [21] reduction in elongation of the plasma boundary in an IPN plasma. A simple analytical formalism as a solution to the Grad–Shafranov equation is adopted and the self organization features of IPN plasma equilibria are explained consistently.

The outline of the paper is as follows: device descrip-tion and experimental details are discussed next. Production and sustainment of IPN equilibrium in fully non-inductively driven plasma current and an EC overdriven Ohmic plasma are presented in section  3. Section  4 deals with the self

organization of this negative triangularity equilibrium. An analytical model description for an IPN plasma and different equilibrium parameters are discussed therein. Conclusions and future works are summarized in section 5.

2. Description of the QUEST device

QUEST is a medium sized spherical tokamak [26] with the major and minor radii of 0.68 and 0.4 m, respectively. The center stack (CS), which holds the Ohmic (OH) and toroidal field coils has an outer diameter of 0.4 m and the outer wall of the vacuum vessel is at Rvout = 1.4 m. There are flat divertor plates at z ~ ±1 m from the mid-plane. These plates are used for single null / double null configurations in QUEST. In the present experiment, these flat plates serve as a protector to terminate open field lines outside the confined plasma. The inboard plasma boundary is defined by a set of water cooled tungsten limiters on the CS at Rcs = 0.22 m and these lim-iters help to remove excess heat load caused due to separa-trix strike points in the IPN configuration. Working gas is either Hydrogen or Helium or both, supplied from electri-cally controlled piezo valves located on the CS or the out-board side. Plasma is initiated by fully non-inductive EC or EC assisted Ohmic discharge schemes. The EC system con-sists of two toroidally opposite 8.2 GHz Klystron systems (2 × 8 × 25 kW) corresponding to the fundamental resonance at Rfce = 0.29 to 0.55 m in off-axis or on-axis heating sce-narios. An additional 28 GHz 0.6 MW Gyrotron system is also available as a second harmonic off-axis start up and current drive system [27]. Four pairs of poloidal magnetic field (PF) coils generate different curvatures of field lines characterized by the magnetic mirror ratio M = Bt_end / Bt_start, along a field line, where Bt_start and Bt_end are Bt at starting (usually mid-plane z = 0) and terminating points on a particular Bz field line, respectively. In QUEST higher M is favorable [25, 28, 29] in confining energetic electrons generated at EC resonance layers in creating initial plasma current (Ip) before formation of the closed magnetic flux surfaces. This initial seed current is enhanced with suitable ramping up of the Bz strength, which enhances the trapped electron energy and fraction. Finally, closed flux surfaces are formed at Ip ⩾ 5 kA. Discharge fill pressure in the vessel is set in the range of 1–5 × 10−3 Pa with additional gas puffs injected from piezo valves to maintain density. Line averaged density usually remains of the order of 1017 m−3, which is less than the cut-off density (8.6 × 1017 m−3) for 8.2 GHz EC waves (ECW) and the elec-tron temperature is in the range of 100–600 eV depending on the injected RF power and gas puff. Various diagnostics are employed in QUEST, particularly 64 flux loops to determine plasma position and shape, Hard x-ray monitors for measuring energetic electrons, microwave interferometer for chord inte-grated density at the mid plane, a tangential visible camera and a 25 channel fiber based visible spectrometer for rotation measurement. Locations of these diagnostics are indicated in figure 1. Various geometrical parameters of the vacuum vessel and plasma shape are defined as follows. Aspect ratio of the vessel is defined as Av = [(Rvout + Rcs)/2]/[(Rvout − Rcs)/2] = 1.4. Plasma shape factors are defined as follows: horizontal

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minor radius a = (Rout − Rin)/2, vertical minor radius b = (Ztop − Zbottom)/2, geometrical center R0 = (Rout + Rin)/2, aspect ratio A = R0/a, inverse aspect ratio ε = 1/A = a/R0, elongation κ = b/a and triangularity δ = (R0 − Rztop)/a. Rztop is the radius corresponding to Ztop. Parameters Rin and Rout are defined as the inboard and outboard edges of plasma as defined by the last closed flux surface. In the case of IL plasma Rin = Rcs, whereas for IPN configuration Rin corresponds to poloidal magnetic field null defined by the separatrix. These definitions are shown in figure 1 and will now be dealt in the rest of the paper.

3. High βp equilibrium with IPN configuration

We report high βp plasma formed in two distinct experimental conditions. One is fully non-inductive start-up and sustain-ment with fundamental off-axis ECH/CD effect and using two pairs of PF coils, namely PF31-51 and PF32-52, where Ip is allowed to evolve self consistently without any external con-trol. The other is EC overdriven Ohmic plasma at constant Ip with the help of an external feedback circuit, where all four sets of PF coils are used to control the plasma equilibrium. These are discussed in the following subsections.

3.1. Fully non-inductive IPN plasma

In fully non-inductive (NI) start-up, Hydrogen/Helium plasma is started up by the two 8.2 GHz EC systems from two phased array antennae (PAA) [30] located at the low field side. The toroidal magnetic field is set as 0.29 T at the major radius, Rfce = 0.33 m corresponding to the fundamental resonance for the 8.2 GHz frequency. It is earlier demonstrated [25] in QUEST that, under a high mirror ratio (curvature of Bz) and

high Bz/Bt ~ 10%, fast Ip ramp up and its sustainment is pos-sible. A high Bz of 20 mT at Rfce is thus applied 0.5 s prior to the EC breakdown phase and is kept constant throughout the plasma discharge. No inductive field is used to ramp up Ip. Owing to the magnetic field curvature at Rfce and R2fce (second harmonic), a better confinement of trapped energetic electrons produced at resonance layers due to multiple ECR interaction is anticipated. The current start-up is dominated by confined hot electrons generated, with energies up to several hundreds of keV as measured by Hard x-ray (HXR) diagnostics [31]. Ip is initiated instantaneously with EC injection and attained the −20 kA level in less than 0.14 s with dIp/dt ~ 0.14 MA s−1. The direction of Ip is defined as per the right handed cylin-drical coordinate system and negative Ip means in the direc-tion clockwise as seen from the top of the torus. Ip in this case is driven entirely by ECWs without any inductive field as evident from the measured loop voltage VL = 0 (figure 2).

We evaluate β β* = + l  /2ip p from the generalized Shafranov equation for radial force balance for the equilibrium vertical magnetic field [32]

⎡⎣⎢

⎤⎦⎥

μπ

β= + * −B

I R

R

a4ln

8 3

2,z

p

0

0

0p (1)

where li is the internal inductance of plasma and is assumed to be unity wherever βp is calculated in this work. The plasma boundary is computed from magnetic flux contours recon-structed from 64 flux loop measurements as per the proce-dure described in [33]. It may be noted that, the non-thermal

component of βp i.e. βphot is the dominant term in equation (1),

particularly due to well confined energetic electrons in high M configurations. It can be seen from figure 2 that, Rin, which indicates the inboard separatrix, reaches up to 0.45 m in the

Figure 1. (Left) Top view of the QUEST vessel indicating locations of the various auxiliary systems and diagnostics. (Right) Side view of the QUEST device showing positions of PF coils (square boxes) and positions of 64 flux loops (open circles). A typical plasma boundary is shown indicating the definition of Rcs, Rin, Rout, Ztop, and Rztop. The two vertical broken lines indicate fundamental and second harmonic resonance locations for 8.2 GHz. The top and bottom flat protector plates are shown by horizontal broken lines at z = +1 m and −1 m, respectively.

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very beginning of the discharge corresponding to the transient

peaking of β*.p Magnetic flux surfaces reconstructed for the same discharge at three different time slices are also shown in figure 2. It can be clearly seen that plasma is bounded by a nat-ural separatrix at the inboard side and the separatrix is moved to R = 0.45 m. This Rin position corresponds to the poloidal magnetic field null and this configuration is coined as IPN. As the Ip is increased further, a quick reduction in β*

p is observed and the plasma configuration is transformed from IPN to IL at

2.5 s. Nevertheless, after 6 s β*p again shows a steady increase

up to a value of 4 towards the end of the discharge as a result

of the steady decay in Ip. At 10 s, plasma is thus transformed again from IL to IPN. Figure 3 ensembles many such similar discharges at discrete time intervals. It can be seen that the

normalized Shafranov shift (Δ/a) increases rapidly with β*p

during the IL to IPN transition and remains constant during

the IPN regime. Figure 3(c) also shows that at β* ∼ 3,p IL to IPN transition occurs.

In the discharge shown in figure 2, Ip is not constant and is allowed to evolve self consistently, thereby the resulting βp is varied and an IPN-IL-IPN transition sequence is mani-fested. In this case, βp is seems to be determined mostly by

Figure 2. Typical high βp plasma discharge in a fully non-inductive current drive. Time traces of plasma current (Ip), one turn loop voltage

(VL), β β* = + l  /2ip p and inboard plasma position (Rin) are shown. ECW power and duration is indicated by a shaded bar. Magnetic flux surfaces reconstructed from the flux loops show the IPN → IL → IPN transition at three discrete times (stamped in each figure) of plasma evolution.

Figure 3. (a) β*p as a function of Bz/Ip, where Bz is the vertical field at R = R0, (b, c) Shafranov shift normalized to the minor radius (Δ/a)

and inboard edge of plasma (Rin) as a function of β*p shown for a number of fully non-inductive discharges. The open black squares

represent IL phases, whereas open circles show IPN configurations.

(a) (b) (c)

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the poloidal magnetic pressure μB /2p2

0 term as a reduction in

Ip increases βp. To independently verify the contribution of the plasma pressure term <p> in βp, we intend to investigate the high βp formation by regulating the Ip, described in the next subsection. <p> consists of thermal pressure contributed by bulk ne and Te and a non-thermal component contributed by energetic electrons confined in the QUEST magnetic geom-etry. If we consider the example of the discharge shown in figure 2, with values of plasma parameters at t = 1.95 s as a = 0.32 m, R0 = 0.62 m, b = 0.19 m and volume V = 2π2 R0ab = 0.74 m3. By taking the maximum electron pressure pe ~ 8 Pa as measured by the Thomson scattering system [34] in a similar

set of discharges, correspondingly β < 0.15pthermal is found to

be much below the measured βp. Hence a significant contribu-tion to βp due to non-thermal electron pressure generated by energetic electrons is evident. A similar non-thermal pressure contribution to βp has also been observed in LATE [35] EC driven non-inductive current drive experiments.

3.2. EC overdriven ohmic plasma

An Ohmic target plasma is created, where Ip is fed back to the OH coil power supply in order to maintain it at −30 kA ± 10%. Bt is set similar to the NI case and a Bz ~ 26 mT is applied with the help of all four pairs of PF coils. At 2 s of the Ip flattop (see figure 4), an EC pulse is injected from the two antennae. This resulted an increase in Ip from −30 kA to −32 kA in less than 0.3 s. Due to the feedback circuit, the OH coil current

is reversed transiently to produce a retarding electric field (positive loop voltage VL) in order to bring back Ip to the feedback value. The recharging phenomenon of the OH cir-cuit in this case is transient only during ECW injection at t = 2 s, however, sustained recharging with a significant EC overdriven plasma has also been obtained in the later part of the discharge by suitable equilibrium control discussed later in this section. At present we focus our attention on the ini-tial phase (t = 2–2.2 s) of plasma transformation due to the ECW injection. The reconstructed magnetic flux contours just before (t = 1.95 s, OH) and after (t = 2.18 s) the EC injection are shown in figure 4. It can be seen that the plasma configu-ration is changed from IL during the OH phase to IPN in the EC phase, which is strikingly similar to the NI plasma dis-cussed in section 3.1. A tangentially viewing visible camera [36] image also confirms such transition in configuration, where the separatrix strike points on the inboard water cooled

limiters are clearly seen as depicted in figure 5. During this

transition phase, β*p shows an increment from 3 in the OH

phase to ~4.5 in the EC phase in <0.18 s. In order to maintain this high βp plasma in equilibrium, Bz is increased from 26 mT in the OH phase to 33 mT during the EC phase, or else Ip is not sustained. The Bz increment and ramp rate are carefully adjusted to sustain the plasma. The requirement of high Bz is also evident from the fact that, the increased pressure gra-dient (∇ = × )p j B  needs to be balanced by enhancing B at a fixed Ip.

The Rin distinctly shows that it has moved inside the vacuum vessel up to 0.4 m in the EC phase. The magnetic

Figure 4. ECW (8.2 GHZ ~ 100 kW) is injected to an OH target plasma with Ip feedback. Feedback level is indicated by a shaded region on

Ip trace. Traces of β*,p magnetic axis (Raxis) and inboard edge of plasma (Rin) shows IPN formation soon after ECW is injected at t = 2 s. Edge safety factor indicated by q* shows a sharp drop during IPN formation. Inboard limiter (IL) to IPN configuration transition is demarcated by double arrow lines. Both Hα (#19270) and line averaged density (ne) calculated over a few similar discharges (#18963–74) show a prompt rise with the ECW injection. Various plasma position and shape parameters like elongation (κ), inverse aspect ratio (ε), plasma minor radius (a) and normalized Shafranov shift (Δ/a) shows the IL to IPN transition process during the EC phase. Flux loop reconstructions shown during the OH phase (t = 1.95 s) and EC phase (t = 2.18 s) corroborate the IL to IPN formation. An analytic model reconstruction (discussed in section 4) of the boundary flux for an IL and IPN plasma configuration is superimposed.

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axis (Raxis) also shows an outward shift at the onset of the EC and remained outwardly shifted in the entire EC phase of the discharge. The outward shift of the magnetic axis is also confirmed from the density (ne) profile measured by Thomson scattering shown in figure 6. The radial pro-file shows the peak ne is shifted from 0.48 to 0.64 m as a result of high βp. This agrees reasonably well with the magnetic axis at 0.7 m deduced from the magnetic meas-urements within the experimental uncertainties. The kink safety factor (q* = πεa (1 + κ2) B0/μ0Ip) [12] shows a sharp drop from 4 during the OH phase to slightly above 1 in the EC phase. The sharp reduction of q* is primarily due to the reduction of a, larger R0 and reduced B0 due to the outward shift of the plasma column. Nevertheless, excluding the ini-tial minor disruption in Ip during the IL to IPN transition phase, the discharge remains stable for over 1 s from t = 2.5 s at low q*.

A number of plasma parameters like κ, ε, Rin and normal-ized Shafranov shift (Δ/a) are investigated as a function of

applied Bz and resulting β*p during the OH and EC phases.

We have taken a large number of discharges to find out the mean and standard deviation (σ) of these measurements and they are plotted in figure 7, which shows the high reproduc-ibility (thin shaded region) of these discharges. It can be seen

that β*p increases almost linearly with Bz, while at a very high

value of Bz, we have noticed frequent minor disruptions in Ip.

The β∼ *Bz p relationship remains very similar to that in the NI plasma case, however, it can be seen that εβp (li is assumed to be unity and not varying in the EC phase) is peaked at Bz = 26 mT corresponding to an optimum equilibrium configura-tion. It is also important here to mention that Ip is roughly kept constant at −30 kA ± 10% by the OH feedback system for the

entire β*p range obtained in all these discharges and hence we

can compare figures 7(a) and 3(a) easily in the range of Bz/Ip =

0.8 to 1 × 10−6 T A−1. As β*p is increased beyond 2.5, Rin is

moved from the limiter (R = 0.22 m) into the vacuum vessel allowing a natural separatrix formation, which further moved

radially outward linearly with β*.p With ECW injection, Δ/a increased from 0.5 during the OH phase to 0.57 in the EC

phase, which later remained constant at higher β*.p During the IL to IPN transition phase, κ is reduced dramatically (figure 7 (c)) from 0.68 to 0.6 at the onset of ECW to become an oblate shape and then saturated quickly prohibiting the plasma shape

from becoming further oblate at higher β*.p On the other hand,

ε does not vary too much before the transition phase but once

the IPN is formed, it declines linearly with β*.p These suggest that with ECW injection, the plasma first became oblate and

then this oblate plasma moved outward as β*p increased fur-

ther. If we notice figure 7(b), εβp initially increased until the

Figure 5. Visible camera image with superimposed magnetic flux contours shows separatrix strike points (bright spots) on CS formed during high βp plasma. On the left side, normalized intensity of the image is plotted along the broken line, and shows two distinct peaks at Z = ± 0.2 m. White dots appearing on the image are pixels affected by HXR bursts.

0.6 0.8 1Intensity [a.u.]

Z [m

]R [m]

Shot no # 21944

0.4 0.6 0.8 1

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Figure 6. Normalized density profile measured in similar shots by Thomson scattering at 1.8 s (OH) and 2.1 s (ECW) shows that the peak is shifted radially outward during the EC phase.

0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

R [m]

n e [no

rmal

ized

]

1.8 s (OH)2.1 s (ECW)

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IL to IPN transition phase (broken vertical lines) and encoun-tered some kind of limit. After the transition, it started to decrease and allowed the plasma equilibrium to be sustained by modifying its shape by decreasing ε (figure 7(d)). In addi-tion to this, these plasmas also exhibit one more self adjust-ment feature described in section 4.

As mentioned earlier in this section, under the specific conditions of equilibrium control, significant recharging of the OH circuit and an overdriven ECCD plasma is observed in this experiment. Figure 8 compares two such discharges in OH feedback experiments with similar experimental conditions. At the onset of the ECW power to the OH target plasma, a transient recharging and IL to IPN transition is observed at t = 2 s, which is a usual feature in all these dis-charges. However, if βp increased up to β ε≅ +1/  0.5c

p i.e.

εβ ε( = + )1 /2 cp [37], then Ip does not sustain and starts to

reduce with the reduction of plasma a and q* as shown in figures 8(b) and (g). In this particular scenario, the plasma switches to an IPN configuration at t = 2.5 s and does not exhibit strong recharging as can be seen in the other dis-

charge, where βp remains below β .cp In the second case when βp remains below β ,cp a does not shrink and q* remains above 3. This set of discharges shows strong OH circuit recharging at t ~ 3 s with a steady increase in Ics to control increasing Ip beyond the feedback level. Nevertheless, Ip continues to increase and ultimately Ics reaches an upper limit and then the discharge is terminated as per a pre-programmed safety interlock. In the discharge, where the recharging phenomenon has occurred, the plasma remains in the IL configuration from t = 2.3 s until the end of the discharge.

3.3. Summary of high βp IPN plasma

Now we summarize the high βp discharges from both the NI and OH experiments discussed in the previous sections. First,

we investigate the dynamics of high βp plasmas based on a

simple force balance relation. In figure 9(a), β*p contours are

plotted in the Ip–R0Bz plane as defined by equation (1), where

14 equispaced lines are drawn from β*p = 2 to 15. Time trajecto-

ries in NI and EC driven OH discharges are plotted for different time segments. Similarly, pressure evolution can be depicted

by rewriting equation  (1) as π≅ ≅ − μp V p I B R I R Gz op

24 p

20

0

with ⎡⎣ ⎤⎦= + −G ln .R

a

l8

2

3

2i0 V is assumed constant and not

varying. Thus in figure  9(b) p contours are plotted in the

−I R Bzp 02 plane. Though the value of G and R0 are not constant

in our case, we take the averaged value of G = 1.9 and R0 = 0.6 m for the two trajectories of p. For the NI case, p increased very fast during the Ip ramp-up phase due to the contribution of energetic electron driven non-thermal pressure and quickly saturated at ~210 Pa. This fast ramp up in p is much faster com-

pared to similar observations in LATE NI discharges [35]. β*p

evolves to ~15 during this ramp up phase and then is reduced

to ~2 as seen from figure 9(a). This reduction in β*p is simply

ascribed to the enhancement in the magnetic pressure term. For the case of an ECW injection in OH plasma (red trace in figure 9(b)), the rise in p is much higher than the NI case. Thus with external Ip control, the plasma pressure could be increased 3 times compared to that without the control at a similar injected EC power. This additional increase in p is due to the contribution of the non-thermal pressure (<p> hot) component

Figure 7. Equilibrium properties of high βp plasma formed during EC injection into Ohmic plasma as a function of external Bz and

resulting β β* ( ) *a b. ,p p and εβp as a function of applied Bz at Rfce. (c)–( f ) Plasma elongation (κ), inverse aspect ratio (ε), normalized Shafranov shift (Δ/a) and the position of inboard plasma edge (Rin) as a function of β*

p during OH phase (red) and in the EC phase (black). All data points are the mean values sampled over a large number of discharges with two standard deviation spread in data points shown as the gray region. The broken vertical lines demarcate IL and IPN plasma transition phase.

(a)

(b)

(c)

(d)

(e)

(f)

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Figure 8. Recharging of the OH circuit at t = 3 s in EC overdriven plasma (#19548) shown with a similar discharge (#19542), where recharging does not occur. Time traces of plasma current (Ip), OH coil current (Ics) and loop voltage (VL) show that Ip is driven against a large opposite VL induced by Ics reversal at t ~ 3 s. Plasma minor radius (a) and edge safety factor (q*) show that recharging occurs only when both are kept high up to t ~ 3 s. εβp are shown for both discharges along with their respective εβ ε= +1 /2 c

p values shown in broken traces. They show that for the discharge, where recharging has not occurred (#19542, black trace) εβp encounters the εβc

p limit (shown by black dotted trace) at t = 2.5 s. For the discharge #19548 (blue trace) εβp is kept below εβc

p and subsequently recharging is observed at t = 3 s. Change in line averaged density with respect to the OH phase (Δne), visible light emission (Hα) and ECW power and duration are shown in (h) and (d). The shaded region in Ip traces (a) indicates the OH feedback level.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 9. (a) β*p and (b) pressure (p [Pa]) equi-spaced contours (black) in Ip–R0Bz and −I R Bzp 0

2 planes respectively for QUEST discharges.

Color traces are time trajectories from the two discharges #19408 (NI, figure 2) and #19270 (OH + EC, figure 4). For the NI discharge

(indicated by arrow mark in figure (a)) Ip ramp up phase is shown from t = 1.3 s–1.7 s indicating rapid increase in p and β*p and trajectories

thereafter up to t = 4.5 s are marked by the blue patch. Time trajectories for OH + EC discharge are shown for t = 1.95–2 s, with the OH

phase (magenta) and EC phase (red). Equispaced β*p and p contour levels are marked in the figures.

(a) (b)

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of energetic electrons during the ECW injection. During this

time <p> hot increases by a factor of >4 and μB /2p2

0 increases

by a factor of 2 albeit the fixed Ip, primarily due to the reduc-

tion of the plasma boundary during the IPN formation. Thus β*p

is increased by a factor >2 as seen in figure 9(a).Secondly, to demonstrate the role of β*

p on the IL to IPN transition, all the high βp plasma discharges from NI and EC overdriven OH plasmas are summarized in figure 10. In the

case of NI plasmas, β*p scales as −I ,p

1 independent of the con-figuration and a transition discriminates IPN and IL at Ip ≅ 25 kA. Such a transition has not been observed in a similar high βp LHCD plasma in Versator-II [10], where an analogous

relationship between β*p and Ip has been observed. However,

we found that when Ip is kept below this critical value of 25 kA, the IPN plasma is self sustained for more than 600 s [38, 39] without any natural transition to IL, corresponding to only region-I shown in figure 10(a). It may be noted that, the IPN formation is limited to the region-I, while the region-II consists of the IL configuration and accessibility to the region-III at higher Ip is not achieved under the present NI plasma conditions. Meanwhile, accessibility to the region-III from II (figure 10(a)), which was otherwise not possible in NI plasma, is demonstrated by keeping Ip constant in the OH

target plasma. In this scenario, β*p is promptly increased in

the EC phase and causes a transition from IL (region II) to IPN (region III) at Ip = 30 kA (figure 10(a), shown by a solid

arrow). This increase in β*p is consistent with the increase in

p shown in figure 9(b). Although ECW injection into the OH target plasma with Ip > 30 kA has been carried out, IPN equi-librium is only formed by reducing Ip to the 30 kA level by itself as shown by a broken arrow in figure 10(a). One of the possible reasons for this observation may be due to the fact that ECW injection for the OH target plasma at Ip = 40 and 45 kA is performed at a higher Bt and correspondingly Rfce is set at 0.54 m. Additionally, in QUEST a strong inverse ECW driven Ip dependence on Rfce is generally observed. Any effect of Rfce on equilibrium configuration remains to be investigated further.

Now let us examine the role of β*p in the IL to IPN tran-

sition. In an NI discharge, where there is no control on Ip, the IL to IPN transition seems to be determined by Ip as may appear from figure 10(a). However, this transition in fact is

associated with the increase in β*p > 3 at decreasing Ip. The β*

p value directly and Ip indirectly (through Bp

2 term) determines the transition in this case. This fact is again confirmed with the EC overdriven OH discharge at fixed Ip. It is already dem-onstrated that by fixing Ip in the OH discharge, additional EC

power has resulted in a concurrent increase in p and β*p at an

approximately constant Ip ~ 30 kA, which is higher than the apparent critical value of Ip ~ 25 kA in the NI case at the tran-

sition boundary. Furthermore, summarizing β*p and Rin values

obtained from NI and OH discharges in figure 10(b), it is seen

rather distinctly that a critical boundary at β*p ~ 3 demarcates

IL and IPN regions. No such relationship is observed between

β*p and Ip in figure  10(a). Hence it is concluded that there

exists the critical value of β*p that dictates the IL to IPN transi-

tion in all these experiments.

4. Discussion on IPN equilibrium

With high βp formation, circular plasma naturally self orga-nizes itself to reduce κ as first observed in TFTR [21, 22], where Ip is reduced in NBI heated plasma. In the present case of EC overdriven OH plasma, where βp is dominated by ener-getic electron pressure, similar reduction in κ is also observed. However, we found a new additional self organization feature, where the self adjustment mechanisms work on the plasma shape so as to become more negatively triangular (δ < 0). This new feature overcompensates the diminution of βp due to the reduction in κ. A simple analytic solution of the Grad–Shafranov equation  (GSE) [40, 41] is applied to investigate such aspect and this is discussed next.

For toroidal axisymmetric equilibria (independent of toroidal angle φ) magnetic field can be represented by

μ φ π φ π ψ= ∇ ( ) + ∇( ) × ∇B G /2 /2 ,0 where ψ is the total poloidal magnetic flux and not the poloidal flux per unit radian. R, Z and φ are as defined in the usual cylindrical coor-dinate system. Defining ψ( )G as the poloidal current function

Figure 10. β* − Ip p relation for NI and OH plasma. (a) Region-I is IPN, II is IL and III is high βp IPN at higher Ip accessed through Ip

control. (b) A critical β* ∼ 3p defines the IL to IPN (shaded region) transition boundary.

β

10 20 30 401

2

3

4

5

6

1 2 3 4 5 60.2

0.3

0.4

0.5

NI-IPNNI-ILOH-IPNOH-IL

β

I (IPN) III(IPN)

II (IL)

NI-IPNNI-ILOH-IPNOH-IL

IPNIL (b)

(a)

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so as to generate μ π=φB G R/20 within a particular flux sur-face, the GSE for such equilibrium can be expressed as,

⎜ ⎟⎛⎝

⎞⎠

ψ ψ μψ

μ πψ

∂∂

∂∂

+ ∂∂

= − − ( )RR R R Z

GG

Rp1 d

d2

d

d.

2

2 02

02 (2)

Here, p is plasma pressure, which is a flux function. The GSE, in general, needs to be solved numerically because of its non-linearity. However, a number of powerful yet simpler models to solve GSE analytically are described in [42–48] and they have their inherent ability to choose geometrical param-eters independently. We have here used the analytical solution discussed in [47] for the spherical torus plasma equilibrium in particular, given as

⎡

⎣⎢⎢

⎛

⎝⎜

⎞

⎠⎟⎤

⎦⎥⎥

ψψ

σω

τ

( ) = ( − ) + ( − )

− − +

R ZR

R RZ

R R

R RR

RR R

, 4

2 ln ,

b

ba a

aa

a

42 2 2

2

22 2

2 22

22 2

(3)

where ψ = σσ( + )Rb

cb8 140

2

2 is the plasma boundary value of

poloidal flux, π= ( )c B R C2 /0 0 02 is related to pressure gradient

with a dimensionless pressure constant C, Ra represents the magnetic axis and Rb is a constant. ω and σ are functions of plasma geometrical parameters through some complex functions of the plasma diamagnetism factor τ. The factor τ is particularly important in spherical torus high beta plasma equilibria. For τ set to zero, equation  (3) is reduced to the Solov’ev’s plasma equilibrium solution. The constants in the above equation can be expressed as functions of plasma geo-metrical parameters ε, κ, δ [47].

ε τ ω τ ε δ σ κ ε δ τ= ( ) = ( ) = ( )R

Rf R g R R h R, , , , , , , , , , , ,a

ba b a0

(4)

Here τ and c0 are chosen iteratively so that the imposed boundary condition is satisfied while constraining the parallel current distribution λ ψ μ( ) = ( − ) ( − )′ ′GI IG qG I/ ,0 at ψ ψ= b to zero. Here, both I (toroidal current) and q (safety factor profile) are flux functions and the prime denotes differentia-tion with respect to ψ. All these functions are obtained from the two basic surface functions U(ψ) and V(ψ) evaluated by numerical integration method.

In [47], no constraint on Ip is imposed and it is calculated self consistently. However in the present case we have forced the total Ip bounded by ψ = ψb to be fixed and is set to the measured Ip in addition to the λ(ψb) = 0 boundary condition. By doing so we could choose the constant c0 corresponding to the actual Ip. This simplified the iteration procedure and only the optimum τ is to be found to satisfy the boundary condition. The plasma geometrical parameters ε, κ, δ, R0 inferred from the magnetic measurements, Ip measured from the Rogowsky coil and the applied toroidal magnetic field B0 are given as the input parameters to the model. ψ(R, Z) are

the output of the model, which are used to find the boundary flux contour. Such boundary flux contours for the IL and IPN configuration are computed from the model and are superim-posed on the LCFS boundaries inferred from the magnetic measurements (figure 4). The model input parameters for the IL and IPN examples shown in figure 4 are given in table 1. It should be noted that, while the analytic solution agrees excellently with the flux loop reconstruction at the boundary, they do not coincide well inside the plasma region. This is one of the caveats of using a simple analytic expression for an equilibrium calculation. It is also important to choose τ judiciously as it has a direct bearing on the resulting βp and plasma boundary. For a critical value of τ, a poloidal field null just appears inside the vessel, which corresponds to a critical βp for IL to IPN transition as discussed later in this section.

Once a convergence of the solution is found at the boundary, various flux functions like magnetic well W (ψ), magnetic shear, safety factor q (ψ), etc are evaluated. A mag-

netic well defined [47, 49] as ( )ψ μ( ) ≅ +W V B p2 / ,V

B2 d

d 2 0

2

which is closely related to the average curvature of the mag-netic field line, is believed to have an important role in the stability of the plasma. A positive W signifies that the average field curvature is in the favorable region of the equilibrium. In figure 11, W and q for the two configurations (IL and IPN) are shown. The computed W(ψ) profile shows the presence of a finite magnetic well in the IPN plasma, which is expected due to the finite diamagnetic effect in high βp plasma. From the q(ψ) profile it can be seen that q95 is reduced from 4.3 in IL to 1.5 in the IPN configuration and such a reduction of q during the IPN formation is consistent with the similar reduction of the measured q* during the IL to IPN transi-tion as shown in figure  4. Reduction of q* is observed in almost every high βp discharge with an IL to IPN transition. Nevertheless, IPN plasma remains stable for >1 s without any disruptions or MHD instabilities. The computed central q (see figure 11) for the IPN plasma goes below one during the transition. The absence of any confirmed severe MHD oscillations in these discharges, however, suggests that the central q may not be below unity. Considering ECW heating in low density plasma (ne_ave < 6 × 1017 m−3), strong pres-sure anisotropy ( ≫ )⊥ ∥p p is expected. The analytic solution

Table 1. Analytical model parameters for the IL and IPN configurations shown in figure 4.

Model Parameters IL IPN

κ 0.61 0.56ε 0.53 0.36δ 0.029 −0.26C 1.2 2.93ω −0.0176 −0.3726σ 0.9239 1.4183τ 0.37 0.605Ra 0.5936 0.6887Rb 0.44 0.3672

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presented here however, assumes an isotropic pressure profile. It has been shown earlier [50] that the estimation of central q can be significantly underestimated by such assumptions. The role of pressure anisotropy in the present equilibrium is left as future work.

Computation of βp from the analytic model is quite intriguing, particularly in IPN plasma. The hunting for an optimum τ corresponding to a λ(ψb) minimum is often cum-bersome in the method described in the previous section. Choice of τ is also severely constrained for arbitrary choice of input plasma shape parameters. Additionally from figure 4

and 7( f ) it is seen that as β*p increases beyond ~3 with the

application of ECW, Rnull (=Rin) moves into the vacuum vessel

and has a very good correlation with β*.p So to study the IPN plasma and relation of β*

p with Rnull more easily, we dwelt upon the treatment described in the [48], where the IPN equi-libria can be studied more flexibly by introducing the fol-lowing transformations

ωω

σω

τ= −−

=−

= −D E H1

,1

, and 2 (5)

where D represents plasma triangularity, E relates to plasma elongation and H is related to diamagnetism. The analytical

solutions described in [47, 48] are exactly similar. However, the latter does not introduce any boundary constraint on par-allel current density numerically other than a conducting limiter at the high field side, which defines the boundary in a divertor equilibrium. No parameter hunting is required in this case and the diamagnetism factor can be chosen flexibly to study different equilibria. The diamagnetic factor (H) in this case is important to describe high βp plasma equilibria. As H is increased at constant D and E values, a poloidal field null appears at the high field side. Correspondingly the critical H is defined as a function of Rx = Rin/R0 as = ( − )H R R2 1 /ln ,x xc

2 2 where the null point is just about to appear in the vacuum vessel.

In figure 12, we have computed β*p as per the procedure

described in [48] as a function of Rin and D from the analytic

model for various IPN equilibria corresponding to β= *H H .c p shows an increasing trend both with larger Rin position and higher negative values of D for the range of E = 0.85–1.43.

This prediction is consistent with the measured values of β*p

as shown earlier in figure  4. Figure  12(a) shows a critical β*

p ~ 3 at which the IL-IPN transition occurs and is supported by the model. The model also agrees with the experimental observation of negative δ formation at high βp. This also

Figure 11. Safety factor (q) and magnetic well (W) as a function of normalized flux to boundary flux value, computed from the analytic model: (a) limiter plasma; (b) IPN plasma.

(a)

(b)

Figure 12. Poloidal beta (βp) computed from the analytic solution (circles) and magnetic measurements (cross) as a function of null position (Rin) and triangularity. D is the triangularity parameter in the analytic model and δ is the measured plasma triangularity. The values of model parameters for different points are chosen to best match the LCFS, E = 0.85 − 1.43, D = 0 to −0.56, H = Hc.

(a)

(b)

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predicts that βp can be raised for negatively triangular shaped plasma. Negative δ has also been shown to increase εβp for various ranges of κ discussed in [48]. It may be noted that in this model it is also difficult to find a suitable combina-tion of parameters so as to satisfy the measured boundary flux. Equilibria calculated for βp (shown in figure  12) also overestimates κ as compared to the measured one. These dis-crepancies also arise from the fact that the model solution does not agree appreciably with the measurements inside the plasma region. Inclusion of realistic and measured pressure profiles, pressure anisotropy and plasma flow (substantially seen in IPN plasma) in the model may improve its accuracy and it is a future work. However, with these limitations such a simple analytic model qualitatively explains the experimental results.

5. Conclusion and future work

A high βp natural divertor configuration (IPN) is success-fully formed under a high magnetic mirror ratio (M ~ 2) and high Bz/Bt ratio (~10%) both in fully non-inductive plasma and Ohmic plasma over driven by EC waves. The dominant contribution of non-thermal pressure generated by confined energetic electrons in generating high βp is confirmed. βp is seen to be linearly increasing with applied Bz strength. Better confinement of energetic electrons at higher Bz contributes to the higher non-thermal pressure. Additionally, high Bz keeps the Shafranov shifted plasma in equilibrium and helps in the manifestation of a stable IPN configuration. The IPN equilibrium is a consequence of the equilibrium poloidal beta limit and the steadystate sustain-ment of this equilibrium limited plasma is demonstrated for the first time. The stable IPN equilibrium is possible due to self-organization features of the plasma by suitably modifying its boundary near the equilibrium βp limit and loss of equilibrium is avoided by self regulating its ε and δ. At high βp, the plasma shows enhancement of its negative triangularity, which plays an essential role to maintain high βp regardless of decreasing κ. The critical βp from the IL to IPN transition and negative δ formation has been explained well by the simple analytical equilibrium solution. The experiment supports the theoretical prediction of achieving a higher βp with negative triangularity. It should be noted that self-sustainment of the high βp IPN configuration is ascribed to self-adjusting mechanisms for plasma shape with increasing negative δ. The unique stable IPN configu-ration in QUEST allowed us to investigate all such sce-narios. We believe that this experiment of IPN equilibrium near the βp limit may recreate interest in further work in the tokamak community for this long forgotten rare plasma equilibrium.

Acknowledgments

This work is supported by Grant-in-aid for Scientific Research (S24226020, A21246139). This work is also performed with the support and under the auspices of the NIFS Collaboration

Research Program (NIFS13KUTR085). One of the authors (KM) was also partly supported by the JSPS A3 Foresight Program ‘Innovative TokamakPlasma Startup and Current Drive in Spherical Torus’. He also gratefully acknowledges the inspiring discussions with Professor P. Kaw, G. Navratil and S. Sabbagh at the IAEA conference, St. Petersburg .

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