Analysis of high β regimes for DEMO

Fusion Engineering and Design 86 (2011) 141–150

Contents lists available at ScienceDirect

Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes

Analysis of high ˇ regimes for DEMO I.T. Chapman ∗ , R. Kemp, D.J. Ward EURATOM/CCFE Fusion Association, Culham Science Centre, Abingdon, Oxfordshire OX14 3DB, United Kingdom

a r t i c l e

i n f o

Article history: Received 21 April 2010 Received in revised form 22 September 2010 Accepted 23 September 2010 Available online 29 October 2010 Keywords: Ideal MHD stability Fusion power plant

a b s t r a c t Stability limit calculations are presented for a range of tokamak power plant equilibria. The current drive requirements to sustain the optimised equilibrium profiles are confirmed by a transport code and the plasma shape is obtained from free-boundary equilibrium calculations. A pressure pedestal is included according to empirical scaling and ballooning mode stability limits. A terative optimisation of the profiles is undertaken to improve the baseline profiles in order to achieve the highest possible plasma performance and most favourable magnetohydrodynamic stability within conservative assumptions in order to increase confidence in the availability and control of the plasma. This results in a fully noninductive baseline operating scenario for a tokamak power plant design which has a broad low-shear q-profile which is meant to complement previous advanced tokamak design studies. © 2010 EURATOM/CCFE Fusion Association. Published by Elsevier BV. All rights reserved

1. Introduction It is desirable to operate a future power plant at the maximal normalised beta achievable in order to optimise fusion performance. Indeed, the EU Power Plant Conceptual Study (PPCS) [1] concluded that in order to operate a fusion power plant to produce electricity at economically attractive rates, plasma performance beyond the ITER baseline level [2] is required [3]. Previous studies [4] have shown that the cost of electricity scales as c.o.e. ∝

 DF 0.6 A

1 0.4 0.3 0.5 P 0.4 ˇN N th e

(1)

where D is the discount rate, F is the tenth of a kind factor, A is the availability, th is the thermodynamic efficiency, Pe is the unit size, N is the normalised density and ˇN is the normalised plasma beta, ˇN = 20  p  a/B0 Ip [MA], where p  is the volume averaged pressure, ˇ0 is the toroidal magnetic field, a is the minor radius and IP is the plasma current. Eq. (1) indicates that the cost of electricity will decrease as the normalised beta increases. Furthermore, enhancing the plasma beta leads to a virtuous feedback on plasma performance: high ˇP leads to a larger fraction of the plasma-generated non-inductive bootstrap current [5], which in turn lowers the internal inductance (li ) of the plasma. Operating at lower inductance permits stable operation at higher elongation. This in turn enhances the bootstrap fraction further. The increased bootstrap fraction means that more current is driven offaxis, which, for a fixed total current, raises the safety factor, q, at

∗ Corresponding author. Tel.: +44 1235 466243. E-mail address: [email protected] (I.T. Chapman).

the magnetic axis. By increasing the safety factor in the core, typically the plasma stability is improved, allowing operation at higher ˇN . Consequently, it is highly desirable to operate the plasma at the highest possible normalised beta to improve fusion performance. Indeed, high ˇN is often assumed in DEMO studies to alleviate the uncertainty regarding non-inductive current drive efficiency, for instance whether negative-ion neutral beam injection at energies greater than 1 MeV (1 MeV injection is planned for ITER [2]) is feasible. However, assuming high ˇN comes at the cost of highly tailored plasma profiles. Previous studies [6–12] of tokamak power plant stability have focussed on developing advanced tokamak (AT) scenarios [13–17] – which maximise ˇN performance by operating at high plasma pressure and low plasma current – without necessarily fully considering the self-consistency and realistic viability of the scenario. However, predicting plasma performance in DEMO, assessing the cost of electricity and optimising the plasma geometry parameters is heavily reliant upon an assessment of realistically achievable stability at the target ˇN . Thus, it is imprudent to predicate such extrapolations and design solely upon highly tailored profiles. In this study, less ambitious self-consistent current and pressure profiles are considered, including a pressure pedestal, and the ideal stability limits are computed, both with and without the presence of a close-fitting wall in order to complement the high performance scenarios developed in [8–12]. The focus of this work is on achievable plasma profiles; we employ self-consistent plasma profiles compatible with current drive assessments from transport calculations; we include a pressure pedestal according to empirical scalings and ballooning mode limits and we use a plasma shape achieved from a free-boundary equilibrium code. Naturally, this has the consequence that the bootstrap fraction is reduced and more non-inductively driven current must be supplied by external actuators than in advanced tokamak

0920-3796/$ – see front matter © 2010 EURATOM/CCFE Fusion Association. Published by Elsevier BV. All rights reserved doi:10.1016/j.fusengdes.2010.09.025

142

I.T. Chapman et al. / Fusion Engineering and Design 86 (2011) 141–150

scenarios. However, this provides a baseline operating scenario which inspires confidence in its operability. Whilst ˇN may be lower than in ATs, the availability, A, may be higher, which is even more important in determining the cost of electricity (see Eq. (1)). We proceed in a pedagogical fashion, first by developing the highest performance case in Section 2 to provide insight into how to optimally tailor the profiles, then proceeding to use this insight to optimise more conservative profiles in Section 3. Finally, these conservative plasma profiles are verified by a transport code, the optimised shape is confirmed as achievable by free-boundary equilibrium calculations, and the stability limit of the target DEMO plasma is assessed in Section 4 before the implications of this work are discussed in Section 5.

2. Advanced tokamak scenarios The ultimate goal of magnetic confinement fusion research is to produce steady-state burning plasmas in a fusion power plant. In order to achieve this goal it is necessary to develop a mode of operation in a tokamak which maximises ˇN whilst simultaneously minimising the amount of power required to supply the current non-inductively. Advanced tokamak scenarios represent an attractive way to operate future fusion power plant devices, since they aim to maximise the non-inductive bootstrap current by operating at high plasma pressure and low plasma current. At the same time, the safety factor is above one everywhere, so the plasma is more stable to core MHD, like sawteeth and neo-classical tearing modes (NTMs). ATs also have good transport properties, as they are often associated with internal transport barrier formation. However, the energy confinement degrades with decreased current, so in order to operate with economically attractive fusion performance, the energy confinement must be optimised. Furthermore, operating at increased pressure and lower current can result in MHD instabilities which are not unstable with conventional H-mode profiles. For instance, the broad current profiles and highly peaked pressure profiles mean that the plasma is more susceptible to external kink modes leading to resistive wall mode (RWM) instabilities. The RWM is a macroscopic pressure-driven kink mode, whose stability is mainly determined by damping arising from the relative rotation between the fast rotating plasma and the slowly rotating wall mode. In order to achieve a high fusion yield, it will probably be necessary to operate advanced tokamak scenarios with a pressure above the stability limit for RWM onset. Fortunately, recent experiments have shown that this can be achieved in the presence of rotation generated by unidirectional neutral beam injection (NBI) [18,19] or kinetic damping [20–24]. As well as increased susceptibility to RWMs, the advanced tokamak profiles are also often unstable to ideal n = 1 kink-ballooning modes, sometimes referred to as infernal modes [16,25–27]. Indeed, even if RWMs are not encountered, infernal modes or NTMs often inhibit plasma performance at high pressure [28]. Whilst these instabilities do not represent a hard performance limit since they can be avoided by specific tailoring of the plasma profiles, it is important to understand the domains of operation where such MHD instability will occur. Finally, AT plasmas are also more susceptible to Alfvénic fast ion driven instabilities than conventional q-profile plasmas [29]. Although Toroidal Alfvén Eigenmodes (TAEs) have been predicted to be stable in high ˇh reversed shear plasmas in the proposed FDF device [30], the 1.5 MeV neutral beams in DEMO (compared to 120 keV in FDF) will enhance the drive for Alfvén eigenmodes and even if TAEs are damped, RSAEs will remain unstable. Recent modelling for ITER has indicated that significant alpha particle loss can be expected in the reversed shear scenarios [31]. Despite these concerns, the AT has been considered as the basis for a fusion power plant [8,10,11,32,33] since it allows operation

Table 1 Table showing global plasma parameters for the proposed DEMO studied here derived from PROCESS systems code simulations, together with the Power Plant Conceptual Study Model C from Ref. [1] and the ITER-98 design [44].

R [m] a [m] V [m3 ] BT [T] Ip [MA] Pfus [GW] Paux [MW] H98(y,2 ) Target ˇNth n [1020 m3 ] ne /nGW Zeff fbs

DEMO

PPCS-C

ITER-98

8.5 2.83 2275 5.74 23 2.7 201 1.3 2.95 0.91 1.0 2.57 0.44

7.5 2.5 1750 6.4 20.1 3.4 112 1.3 3.4 1.2 1.5 2.2 0.63

8.14 2.8 2000 5.68 21 1.5 100–150 1 2.29 0.98 1.0 1.9

at high ˇN with a large fraction of bootstrap current, resulting in a lower predicted cost of electricity [6]. Whilst this provides the route to best performance for DEMO, it is a route fraught with challenges, most notably the reliability and accessibility of AT plasmas. Ref. [17] shows that reversed shear discharges which are stationary with respect to current diffusion time scales have yet to be reliably demonstrated on JET. However, high performance fully non-inductive plasmas have been demonstrated in DIII-D [34–36]. We have developed an AT scenario for the tokamak power plant proposed here in order to assess the optimum performance that one could expect from a power plant plasma. A series of equilibria were generated and the pressure profile was optimised in order to achieve good infinite-n ballooning stability and enhanced bootstrap current alignment simultaneously. Proposed DEMO power plant parameters were developed by using the PROCESS systems code [37], which contains simplified models of each of the systems in a power plant. Running PROCESS iteratively yields the necessary machine size required to produce a given net electrical output, here taken to be 1 GWe. The global parameters resultant from this systems study are given in Table 1. The first iteration of the plasma pressure and current profiles were taken from a transport calculation using the TRANSP code [38]. The plasma is heated with 1.5 MeV negative-ion neutral beam injection (N-NBI) directed on-axis. The equilibria are generated using the fixed-boundary HELENA code [39], which includes a selfconsistent assessment of the bootstrap current according to the formulae given in Ref. [40]. The plasma shape corresponds to the 99% flux surface of a free-boundary equilibrium calculation using the FIESTA code [41], with an elongation of  = 1.86 and triangularity of ı = 0.54. The pressure profile was then optimised for bootstrap current alignment and the resultant current profile was self-consistently calculated. A pressure pedestal is also retained in the profile, using the empirical scaling for the pedestal width 1/2 with ˇp,ped [42], and the corresponding pedestal height set by the infinite-n ballooning mode stability threshold. The resultant q-profile optimised for both stability and non-inductively driven bootstrap fraction is illustrated in Fig. 1. Such a reversed shear AT q-profile is reassuringly similar to those attained in similar studies with similar global machine parameters [8,10]. The location of the minimum in safety factor is deliberately as close to the plasma edge as possible in order to keep the magnetic shear large towards the edge, resulting in higher ballooning stability limits. This also has the practical benefit that regions requiring current profile tailoring are within external current drive limitations, for instance, the lower hybrid current penetration depth [11]. The stability of this AT equilibrium is tested using the MISHKA-1 linear MHD stability code [43]. The plasma is found to be unstable to low-n external kink modes. Fig. 2 shows the growth rate of the

I.T. Chapman et al. / Fusion Engineering and Design 86 (2011) 141–150

143

5

4.5

4 q 3.5

3

2.5

2 0

0.2

0.4

ψ

0.6

1

0.8

Fig. 1. The safety factor profile of an optimised advanced tokamak scenario for DEMO.

Fig. 2. The growth rate of n = 1 − 4 modes with respect to ˇN in the absence of a wall. It is clear that the n = 1 mode has the lowest marginal stability limit.

n = 1 . . . 4 modes as a function of ˇN in the absence of an ideal wall. In all the ˇ scans in this paper, as ˇN is increased, the parameter I/aB is held constant, which means that the q-profile does vary slightly. It is clear that the n = 1 external kink mode is the most unstable low-n mode. The pressure profile, including the pressure pedestal, is optimised to be stable to infinite-n ballooning modes to pressures well above the no-wall low-n stability limits. Fig. 3 shows the effect of moving the ideal wall closer to the edge of the plasma. The shape of the wall used is the same as

0.4

No Wall r w =1.6a r w =1.5a r w =1.3a

γ / ω A (0)

0.3

0.1 0.0 3.5

4.0

that designed for the ITER-98 physics basis [44], which has similar machine size parameters, as seen in Table 1. It is seen that the presence of a perfectly conducting wall at rw = 1.3a, where a is the plasma minor radius, results in an ideal n = 1 with-wall stability b = 5.5. A realistic position for the wall [1,10,45] would limit of ˇN be in the range of rw = [1.2a, 1.3a]. Naturally, such axisymmetric modelling does not take into account the detailed 3d structure associated with port holes and vessel structures, but such toroidal discontinuity is not expected to have a significant effect on lown stability. The ˇ-limit achievable in this optimised AT scenario is slightly less than that achieved in the ARIES-AT design [8], presumably due to the inclusion of a pressure pedestal. Indeed, neglecting b = 6.4. It a pedestal increases the rw = 1.3a stability limit to ˇN is worth noting that the no-wall stability limit is rather low, at ∞ = 3.1. Whilst the ideal wall can stabilise the external kink mode, ˇN in practice the surrounding vessel wall is resistive, which means the plasma is susceptible to resistive wall modes. Consequently, operation beyond the no-wall limit will require active control of RWMs from an externally applied field which opposes the change in the radial magnetic field at the wall [46,47]. Such an external coil system, located outside of the vacuum vessel, would also need to address n > 1 RWM control, since Fig. 4 shows that in the presence of a close-fitting wall, the higher n modes become more unstable than the n = 1 external kink mode. For ˇN = 4.0, it is clear that the proximity of the ideal wall is more important for stability to higher-n modes. Whilst the n = 1 mode has the largest growth rate at rw /a = 2, it is strongly stabilised by the presence of a close-fitting ideal wall. In contrast, for higher-n, the wall only begins to stabilise the mode when it is very close. This susceptibility to higher-n modes will be heightened when full 3d structures are considered, and this will be the subject of future investigation. 3. Baseline scenario optimisation

0.2

3.0

Fig. 4. The growth rate of n = 1 − 4 modes with respect to the position of the ideal wall at ˇN = 4. It is evident that the wall is more stabilising for lower n, but in all cases, can stabilise the mode at realistic wall positions.

4.5

5.0

5.5

6.0

6.5

βN Fig. 3. The growth rate of n = 1 modes with respect to ˇN for different wall positions. Whilst the no-wall ˇ-limit is low, the presence of a close-fitting ideal wall at rw /a = 1.5 increases the ˇ-limit to 5.5.

Whilst steady-state advanced tokamak scenarios with strongly reversed shear and a high bootstrap fraction, as explored in Section 2, are clearly more economically attractive in a tokamak power plant, we consider here a conventional q-profile as a baseline operating scenario. The monotonic or low-shear scenario developed here is intended to complement advanced tokamak scenarios such as those in Section 2 or Refs. [8,10,11,32] in order to demonstrate viability of DEMO scenarios for a range of availability and neutral beam current drive assumptions. Although such non-bootstrap optimised equilibria necessarily require a larger auxiliary noninductive current drive power, they are presently well-understood

144

I.T. Chapman et al. / Fusion Engineering and Design 86 (2011) 141–150

6

0.07

TRANSP PPCS Mild ITB Peaked Broad

Pressure [au]

0.06 0.05

5 4

0.04

q3

0.03

2

0.02

1

0.01 0 0

TRANSP PPCS Mild ITB Peaked Broad

0 0

0.2

0.4

ψ

0.6

0.8

0.2

0.4

1

Fig. 5. The pressure profiles for five iterations of the baseline operating scenario. A pressure pedestal has been included in accordance with the empirical scaling in Ref. [42].

and regularly controlled in present-day devices, increasing confidence in the operability and availability of such an operating mode. It is worth noting that the so-called ‘hybrid’ regime [17,48], which has a broad low shear region with the safety factor above unity and can operate for long pulse duration in a quiescent state, has achieved better performance than advanced scenarios. Ref. [17] shows that hybrid plasmas have reached a higher confinement factor H89 ˇN /q295 , been sustained for longer in terms of resistive diffusion times and are more easily reproduced in present day devices than comparison reversed shear plasmas. Furthermore, high values of ˇN are potentially realisable without the need to incur RWM feedback control systems, and high confinement can be sustained even at high densities approaching the Greenwald density limit, which are required to allow efficient use of the plasma ˇ and radiation of exhaust power to the reactor walls. It is worth emphasising for clarity that in this paper we consider fully non-inductive scenarios, in contrast to the hybrid scenario planned for ITER which is long-pulse but still inductively driven. In that sense, it is just the q-profile which is similar. Once more, we begin from equilibrium profiles derived using the Transp code for a DEMO power plant with the global parameters given in Table 1. The equilibrium is generated with the fixed boundary HELENA code with plasma geometry given by  = 1.85 and ı = 0.55. The linear ideal stability of low-n modes and the bootstrap current fractions are then assessed for a sequence of equilibria with slightly varied pressure profiles in order to optimise the stability of such a baseline operating scenario. The pressure profiles are illustrated in Fig. 5. The TRANSP profile is very similar to that used in Ref. [10] with the added inclusion of a small pressure pedestal. The PPCS-C model [1] gives a more peaked pressure profile. Three further profiles are considered which include a significant pedestal assumed by following the empirical observation that the pedestal 1/2 width scales with ˇp,ped [42] and the pedestal height is set by the infinite-n ballooning stability limit. It has been shown that reducing the pressure gradient near the plasma edge naturally increases the achievable ˇN [8]. An optimisation can be reached whereby the ideal pressure gradient is that which allows a high ˇN for sufficient bootstrap driven current to allay requirements of auxiliary heating power, whilst being simultaneously stable to ballooning modes. Since H-mode confinement is assumed in the power plant specification, it is important to include the pedestal. Of course, the empirical scaling employed here is inevitably flawed, but its inclusion is intended to enhance the realistic nature of the profiles and result in more conservative and readily achievable performance limits. The inclusion of the pedestal also results in a high plasma

ψ

0.6

0.8

1

Fig. 6. The self-consistent safety factor profiles for the five pressure profiles shown in Fig. 5.

density at the separatrix, which increases the current drive requirements. After each iteration of the pressure profile, a self-consistent equilibrium is generated using the HELENA code. The ne , Te ( = Ti ), Zeff and j profiles are supplied to HELENA, which produces j · B . The bootstrap current is then found by solving the equations given in Ref. [40]. Finally, the remaining non-inductive current is calculated and adjusted in order to keep Ip as requested. This iteration continues until a fully self-consistent equilibrium is reached for each of the pressure profiles. Fig. 6 shows the resulting safety factor profiles for each of the pressure profiles illustrated in Fig. 5. The more peaked pressure profiles result in enhanced bootstrap current drive near the plasma core, which reduces the safety factor in the core. Conversely, the broader pressure profile case results in a strong bootstrap current drive outside the mid-radius, raising q0 . The three profiles that include a strong pressure pedestal all give rise to significant bootstrap current near the edge of the plasma, which produces regions of low magnetic shear towards the edge. The coupling of this low shear with a strong pressure gradient in the unfavourable curvature region makes the plasma unstable to high-n ballooning modes. However, the ultimate stability limit is still determined by the global low-n kink modes, which remain more unstable than high-n pressure driven modes. This is in contrast to scoping studies in Refs. [32,33] where a close conformal superconducting wall at rw /a = 1.1 assures wall stabilisation of low-n kink modes. Since the tungsten armour and blanket are not toroidally continuous, we assume an effective conducting wall at rw /a > 1.25. It should be noted that these profiles are themselves the result of a significant optimisation process, whereby many profile nuances, for instance the pressure peaking factors or the ITB width and height, have been varied to reach the best performance in each case. The linear ideal MHD stability of each of these equilibria is assessed using the MISHKA-1 code. The ˇ-limits for the equilibrium profiles illustrated in Figs. 5 and 6 are given in Table 2. The original equilibrium derived using the TRANSP code profiles is unstable to n = 1, 2 global kink-ballooning modes, or infernal modes, but stable

Table 2 Table showing ˇ-limits for rw = 1.3a and the corresponding limiting instabilities for the equilibria profiles given in Figs. 5 and 6. Model

ˇN -limit

Instability

TRANSP PPCS-C Peaked Broad ITB

3.7 4.0 2.9 2.7 3.5

Infernal Infernal External kink peeling External kink Infernal

I.T. Chapman et al. / Fusion Engineering and Design 86 (2011) 141–150

Fig. 7. The growth rate of n = 1 − 9 modes for a peaked pressure profile with ˇN = 3.0 and rw /a = 2.0. It is clear that the plasma is more susceptible to higher n modes.

for n ≥ 3. When an ideal wall is placed at rw = 1.3a, the stability b = 3.7. However, the peaked pressure profile results in limit is ˇN a strong external component to the eigenfunction, such that the wall plays a strongly stabilising role. Without a wall, the stabil∞ = 1.9. Such an equilibrium would inevitably ity limit is only ˇN require RWM feedback control methods, which is an unattractive proposition given the rather poor no-wall performance. Whilst the more peaked PPCS profiles result in a much improved no-wall sta∞ = 3.3, the absence of an edge pressure pedestal bility limit of ˇN means that the required confinement is unlikely to be achieved with this profile. Consequently, only the profiles that include an edge pedestal are considered as potentially viable baseline scenarios.

145

Fig. 9. The growth rate of n = 1 − 4 modes with respect to ˇN for rw /a = 1.7 for a broad pressure profile. The n > 1 modes are the most unstable since the strongest pressure gradient is localised near higher rational surfaces.

radial extent means that they are unlikely to be disruptive. That said, peeling modes are expected to result in Edge Localised Modes (ELMs) [49,50] which result in divertor heat loads that are unlikely to be tolerable in a tokamak power plant. Whilst the presence of the wall within a realistic position range (i.e., rw > 1.27a) does stabilise the external kink mode, this scenario would also incur the requirement for RWM control. 3.2. Broad pressure profile

Fig. 7 shows the growth rate of n = 1 . . . 9 instabilities for the peaked pressure profile when ˇN = 3.0 and the wall is at rw = 2a. The plasma is unstable to n = 1, 2 external kink modes and n ≥ 3 peeling modes. The external kink modes can be stabilised by the presence of an ideal wall. Fig. 8 shows the growth rate of the n = 1 . . . 4 eigenmodes with respect to the position of a perfectly conducting wall when ˇN = 3.0. The wall does not affect the peeling mode stability until it is unrealistically close. However, the peeling modes are not considered as a hard-limit to plasma operation, since their small

The broad pressure profile case is even more unstable to the external kink mode. Fig. 9 shows the growth rate of the n = 1 . . . 4 ideal modes as a function of ˇN when an ideal wall is located at rw = 1.7a. It is evident, that the plasma is more unstable to higher toroidal mode numbers. The n > 1 modes are more unstable in this case, since the strong pressure gradients that drive the instability are located near the higher order rational surfaces. Whilst an ideal wall can stabilise the plasma, feedback control for RWM stabilisation would once again be required in this case. However, since it is higher-n modes that are most unstable, this would require a more sophisticated feedback control system involving a larger number of external coils. Fig. 10 shows the influence of the perfectly conducting wall on the n = 1 . . . 4 modes for the broad pressure profile case when

Fig. 8. The growth rate of n = 1 − 4 modes with respect to the position of an ideal wall for a peaked pressure profile and ˇN = 3.0. The n > 1 unstable modes are edgelocalised peeling modes, which would probably result in ELMs, but not be disruptive.

Fig. 10. The growth rate of n = 1 − 4 modes with respect to the position of an ideal wall for ˇN = 3.0 and a broad pressure profile. Even with a wall in a realistic position, the plasma is unstable for ˇN = 3 forn > 1 modes.

3.1. Peaked pressure profile

146

I.T. Chapman et al. / Fusion Engineering and Design 86 (2011) 141–150

Fig. 11. The growth rate of n = 1, 10 modes with respect to ˇN for a pressure profile with a mild internal transport barrier. The presence of an ideal wall has only a small effect on the n = 1 mode stability since the eigenfunction is primarily internal associated with the pressure gradient at the ITB. n = 10 ballooning instabilities do become unstable, but at higher ˇN than the n = 1 mode.

Fig. 12. The ˇ-limit for the optimised pressure profile with respect to the triangularity, ı at a fixed elongation  = 1.85. For reference, the original value used in the previous ˇ scans was ı = 0.55.

3.4. Effect of plasma shaping ˇN = 3.0. Whilst the wall can completely stabilise the n = 1 mode at rw = 1.5a, the n > 3 modes remain unstable even with the wall in a realistic position. The n = 4 mode is found to be the most unstable external kink mode of those tested, and the corresponding withwall (rw = 1.27a) stability limit for this equilibrium is found to be b = 2.7. ˇN 3.3. Optimised pressure profile Consequently, the pressure profile must not be too peaked or else the plasma is limited by the n = 1 external kink mode to a with-wall ˇN -limit below 3, nor too broad, or else the n > 1 external modes limit plasma performance to ˇN < 3. Furthermore, in both cases, the no-wall limits are even lower and RWM feedback control would be essential to maintain the equilibrium at economically viable plasma pressures. In order to balance these two extremes, it is necessary to optimise the pressure profile so that the n = 1 mode remains the most dangerous, but the associated no-wall limit is increased. This can be achieved by invoking a small increase in the pressure gradient towards the plasma mid-radius, as associated with a very weak internal transport barrier (ITB). Fig. 11 shows the growth rate of both the n = 1 and n = 10 modes with respect to the normalised pressure for different ideal wall locations. The plasma is most unstable to the n = 1 infernal mode which is driven by the enhanced local pressure gradient at the ITB location. The mode eigenfunction is primarily internal, meaning that the effect of the wall is not as strong as for the peaked or broad pressure profiles, which were unstable to predominantly external kink modes. ∞ = 3.05 Furthermore, the no-wall limit is significantly higher, at ˇN b and the realistic with-wall limit is higher still at ˇN = 3.5. Whilst these equilibrium profiles are clearly beneficial for n = 1 stability, the enhanced pressure gradient does also mean that the plasma is unstable to mid-n ballooning modes at the ITB location. However, the ˇ-limit of the n = 10 mode shown in Fig. 11 is clearly higher than the n = 1 mode, and the stability criterion is set by the n = 1 kink-ballooning mode. The stability to n = 2–4 modes has also been tested, and shown to have higher ˇ-limits than the n = 1 infernal mode. It should be noted that whilst some optimisation has been required in order to achieve this no-wall limit above 3, these profiles are regularly achieved in many modern-day tokamaks and the navigation of parameter space necessary to attain such profiles is well understood.

Having varied the plasma pressure profile in order to optimise the stability of the power plant baseline scenario in the presence of a strong edge pressure pedestal, it is now of interest to consider how the plasma shaping affects the stability of the mild-ITB equilibrium (which was developed with plasma triangularity, ı = 0.55 and plasma elongation,  = 1.85). It is known that both the triangularity and elongation can have a significant effect on ideal MHD stability. Increasing the triangularity is typically stabilising to highn ballooning modes since it enhances the good curvature along a magnetic field line, and is similarly beneficial for low-n external kink stability due to the increased magnetic shear at the plasma edge. Fig. 12 shows the ˇ-limit for the optimised pressure profile with respect to the triangularity at a fixed elongation,  = 1.85. The plasma boundary is parameterised in the form: x = a cos( +  sin() +  sin(2))

(2)

y = b sin()

(3)

where the poloidal angle  is found from  by inversion, () = arctan(

b/a sin() ) cos( + ı sin() +  sin(2))

(4)

In these scans, the parameter I/aB is held fixed when generating the equilibria. In the stability analysis, q0 is scaled slightly so that  = m − nqa is constant, since the stability of the external kink is sensitive to , and can be stabilised when the rational surface is sufficiently far from the edge of the plasma. The small change to q0 is then applied to the ˇ-limit (since raising q0 is achieved by scaling the toroidal field). It is clear that increasing the triangularity does allow access to higher normalised beta, though a rollover occurs at ı = 0.75. Meanwhile, reducing the triangularity has a strongly deleterious effect on stability. Furthermore, increasing the triangularity also increases the distance the ideal wall must be from the plasma edge in order to stabilise the n = 1 external kink mode for a given ˇN . Fig. 13 shows the distance of an ideal wall normalised to the minor radius required to marginally stabilise the n = 1 mode with respect to the plasma triangularity at a fixed elongation  = 1.85. Whilst it is clear that at higher triangularities, the position of the wall required to marginally stabilise the n = 1 mode moves outwards, it is also worth noting that eventually, the n = 2 mode becomes more unstable and the position of the ideal wall is constrained by the higher-n modes.

I.T. Chapman et al. / Fusion Engineering and Design 86 (2011) 141–150

147

Fig. 13. The position of an ideal wall required for marginal stability to the n = 1 mode at ˇN = 3.5 with respect to triangularity. It is clear that as triangularity increases, the wall does not need to be as close to stabilise the n = 1 mode.

Fig. 15. The position of an ideal wall required for marginal stability to the n = 1 mode at ˇN = 3.3 with respect to elongation for different triangularities. When  > 2.1, the wall needs to be moved closer to the plasma in order to stabilise the n = 1 mode.

The effect on MHD stability of varying the plasma elongation has also been investigated numerically. Once again, the normalisation factor I/aB is held fixed as the plasma shaping is changed. Fig. 14 shows the ˇ-limit for the optimised pressure profile equilibrium with respect to the plasma elongation for ı = 0.55, 0.7, 0.85. For a relatively mild shape (ı = 0.55, the baseline reference used in pressure profile optimisation scans), varying the elongation does not dramatically affect stability to the n = 1 infernal mode. For ı = 0.55, the highest achievable plasma pressure occurs for  = 2.1, both for n = 1 global modes and infinite-n ballooning modes. At higher triangularities, increasing the elongation has a more favourable effect on the stability limit. However, the increase in elongation comes at a price. Firstly, the position of an ideal wall required to marginally stabilise the n = 1 infernal mode is moved significantly closer to the plasma edge for very high elongations. Fig. 15 shows rw /a for marginal stability with respect to  for three different triangularities. It is apparent that when  > 2.1, the wall must be moved much closer to the plasma to facilitate stability. Indeed, when  > 2.2, the wall must be unrealistically close, so stable operation (here, at ˇN = 3.3) would be prohibited. The second ramification of increasing the elongation is an increased demand upon the vertical position feedback system to control the n = 0 vertical instability. The vertical stability of the

system is not discussed in detail here, though initial calculations suggest that adoption of an elongation  < 2 is prudent. We have developed baseline scenario equilibria for a DEMO tokamak power plant which exhibit global stability above ˇN = 3 despite making conservative assumptions about the plasma profiles. Whilst attaining fully non-inductive operation using this approach puts a greater demand on the external current drive actuators since the bootstrap fraction is reduced, the reliability of such monotonic, or broad low-shear q-profiles is likely to be considerably higher. Adoption of such an equilibrium does, of course, rely on the temperature scaling of current drive actuator efficiency more strongly than the advanced tokamak scenarios. Similarly, it is likely that active control of neo-classical tearing modes will be required at the q = 3/2, 2 rational surfaces in order to access high ˇN performance. That said, ECCD control of NTMs has been robustly demonstrated on many present-day devices [51–53], though establishing an empirical prediction for the current drive power required for NTM control will require experiments in ITER.

Fig. 14. The ˇ-limit for the optimised pressure profile with respect to the elongation,  at fixed triangularities. For reference, the original value used in the previous ˇ scans was  = 1.85. Higher elongation increases the ˇ-limit for high triangularity, but infinite-n stability is optimised at  = 2.1.

4. Self-consistent equilibrium verification and final stability assessments The final stage in the development of a baseline operating scenario for a tokamak power plant is to generate steady-state profiles using a transport code. The pressure and safety factor profiles that have been optimised for stability in Section 3 are supplied as input to the TRANSP code, which solves the poloidal field diffusion equation at each time step until steady-state conditions are reached, typically after 1000 s. We then assess the MHD stability of the resultant equilibrium which is naturally self-consistent from a current drive point of view with fully relaxed profiles. The TRANSP simulations include anomalous fast ion diffusion in the core region to represent the effects of turbulence or fast-ion driven instabilities which are thought to account for the observed neutral beam driven current in present devices being less than predicted by transport codes [54,55]. An anomalous diffusion coefficient of D = 0.1 m2 s−1 is employed in the region N ∈ [0, 0.15]. Firstly, the optimal deposition for the NBI power must be ascertained. Fig. 16 shows the q-profile for a transport simulation after 1000 s (i.e., fully relaxed) when all the beam power is injected onaxis. For the 100% on-axis case, q drops below unity for N < 0.1 on a time-scale of 100–200 s, which has deleterious implications for stability, as indicated in Fig. 17. Indeed, q0 is unrealistically low as sawteeth would almost certainly mediate a q-profile with q0 > 0.7. This plasma is extremely unstable to n = 1 internal kink modes. If

148

I.T. Chapman et al. / Fusion Engineering and Design 86 (2011) 141–150

1

5

4

0.6 3

0.4

q

2

0.2 0 0

Safety Factor

Plasma Pressure [MPa]

Pressure

0.8

0.2

0.6

0.4

0.8

1 1

r/a

Fig. 16. The q-profiles for Transp equilibrium after 1000 s when all the injected NBI power is aimed on-axis, and when one-third of the NBI power is injected at a tangency radius of 10 m, one-third at 9.5 m and one-third on-axis.

we fix q0 = 0.9 in order to find external kink modes as the limit∞ = 2.55 and ing instability, then the n = 1 no-wall ˇ limit is only ˇN when an ideal wall is placed at rw /a = 1.3 the stability limit for the b = 2.9. In order to achieve a higher q n = 1 mode is ˇN min some of the NBI power is required further off-axis. Fig. 16 also shows the q-profile achieved when one-third of the NBI power is injected at a tangency radius of 10 m, one-third at 9.5 m and one-third on-axis. The off-axis beams are at an angle of inclination of 10◦ and produce a beam shine-through of only 1.2 MW (less than 0.5%). In this ∞ = 30 case, qmin = 153, and the stability limits are enhanced to ˇN b and ˇN = 3.45, as illustrated in Fig. 17. The injection of some NBI power counter to the plasma current was also considered, but this had a significantly smaller impact on qmin than the off-axis NBI, whilst also resulting in a far larger auxiliary power requirement. Having optimised the NBI tangency radii for current drive requirements, the stability of the fully relaxed steady-state equilibrium is assessed. Fig. 18 shows the relaxed q and pressure profiles whilst Fig. 19 shows the current density profile for the baseline steady-state equilibrium. In order to achieve this safety factor profile and meet the 1 GW net electrical power design criterion, 207 MW of NBI power is required, which is slightly above the PROCESS target value given in Table 1. The calculated no-wall ∞ = 3.0 and when an ideal stability limit for this equilibrium is ˇN

Fig. 17. The ˇ-limit for the transport-consistent q-profiles given in Fig. 16. Injecting some of the NBI power off-axis raises qmin sufficiently to significantly improve n = 1 mode stability.

Fig. 18. The plasma pressure and q-profiles for the optimised steady-state baseline equilibrium.

wall is placed at rw /a = 1.3a the stability limit for the n = 1 mode b = 3.45. The n = 1 mode is found to be more unstable than the is ˇN n > 1 modes, and so represents the ˇ-limiting instability. The total neutral beam driven current is not dissimilar when half of the NBI power is injected off-axis since the drop in temperature that occurs when depositing the NBI power further from the core (resulting in reduced NBCD efficiency) is compensated by the drop in density (increased NBCD efficiency). It is anticipated that the 207 MW of NBI power would be provided by NBI boxes capable of supplying 16 MW from 10 ports, some of which will have both on- and off-axis injectors. For comparison, the PPCS-C scoping study [1] assumed 24 ports would be required for NBI power provision. Furthermore, the IPB98(y,2) guidelines for the scaling of engineering parameters [44] suggests that a significant fraction of the power required for neutral beam current drive will be required for H-mode access anyway. High plasma shaping in DEMO is complicated by the need for the poloidal field coils to be outside the shielding and far from the plasma. The FIESTA free-boundary equilibrium code [41] was used to find a viable single-null equilibrium with  = 1.91 and ı = 0.57, which is predicted to be vertically stable to n = 0 displacements. This equilibrium is shown in Fig. 20. Coil currents are within reasonable engineering limits.

Fig. 19. The current density profile for the optimised steady-state baseline equilibrium. The solid line is the total current density; the dotted line is the beam driven current (73% of total); the dashed line is the bootstrap current (36% of total) and the dash-dot line is other currents (e.g., diamagnetic and fusion fast-ion currents).

I.T. Chapman et al. / Fusion Engineering and Design 86 (2011) 141–150

Fig. 20. A free-boundary equilibrium for DEMO, showing the plasma shape and coil positions. ˇN = 3.0,  = 1.91, ı = 0.57, q0 = 1.5q0 = 1.5, q95 = 3.8.

Consequently, we have produced a tokamak power plant operating scenario based on a free-boundary equilibrium, transport simulation and MHD stability assessment. The bootstrap fraction in this baseline monotonic q-profile plasma is less than 40%, which is significantly smaller than that achieved in AT scenarios [8]. However, whilst the non-inductive current drive requirements are much larger and the normalised beta is much lower, the nature of these profiles, including a pressure pedestal and making deliberately conservative assumptions, means that the likelihood of achieving such a scenario is greatly improved. This is designed to complement more aggressive advanced tokamak scenarios previously designed for DEMO [8,10,11,32] in order to demonstrate viability of DEMO scenarios for a range of performance assumptions. 5. Discussion and conclusions Many previous studies of operating scenarios for a tokamak power plant have focussed on the development of advanced tokamak plasmas [8–11], despite the uncertainty inherent in operating ATs and concerns about the effect of RWMs and fast ion-driven instabilities. In this work we have developed a baseline operating scenario with more conservative parameters resulting in a safety factor profile readily achieved in present-day machines. In order to enhance the credibility of this baseline scenario, the plasma shape is verified with free-boundary equilibrium calculations, the optimised pressure profile is then iterated using a transport code to calculate the expected non-inductive driven current and a pressure pedestal is included according to empirical scaling. The resulting broad low-shear q-profile is found to have a no-wall limit above ˇN = 3.0, which is the performance assumed in complementary systems studies. This work will naturally form an integral part of an optimisation process to refine the plant specification to achieve the highest performing plasmas in DEMO. Whilst this forms an initial scoping study of the stability in power plant plasmas, there are many issues which need further investigation. The role of fast particles in a burning plasma like DEMO is likely to be significant. Future work will consider how these energetic ions affect plasma stability. Whilst one of the reasons for adopting

149

the weak shear q-profile was that it is predicted to be significantly more stable to fast ion-driven instabilities than strongly reversed shear plasmas with higher q0 [31], it is nonetheless important to consider the implications of energetic particle modes for plasma confinement. It is still important to keep q > 1 though, since the significant fast ion population in DEMO would probably result in very long sawtooth periods [56,57], which are more likely to trigger deleterious NTMs [58–60] which in turn degrade confinement and can disrupt the plasma. Similarly, it has been shown recently that kinetic effects can substantially alter RWM stability [20,22–24]. Although operation in the wall-stabilised regime is not required, such kinetic damping could permit operation at higher ˇN , thus increasing the bootstrap fraction and relaxing the external current drive requirements. The stability to RWMs above the no-wall limit can only be properly assessed by including a full three dimensional resistive wall. Whilst the q-profile optimised for stability necessarily operates in a regime which avoids sawtooth oscillations, the absence of such MHD reconnection could result in a deleterious accumulation of helium ash in the plasma core. Consequently, it is important to perform nonlinear calculations with different transport models and impurity mixes to assess the expected impurity transport in DEMO plasmas, and consider actuators for the amelioration of impurity accumulation. For instance, central ECRH deposition has suppressed impurity accumulation by increasing the anomalous diffusion and by flattening the profile of the main plasma density which reduces neoclassical inward convection for the impurities [61,62]. Furthermore, full gyrokinetic predictions are ultimately required in order to predict the temperature and density profiles to assess whether the optimised pressure profile is realistic. Whilst every effort has been made to employ a realistic pressure profile including an edge pedestal, the model for the pedestal width and height would benefit from improved studies. Similarly, whilst the equilibrium shape has been verified as achievable with a free boundary equilibrium code, a comprehensive study of the demands upon the vertical position feedback system, especially with regards to feedback requirements to control the n = 0 vertical instability will be conducted in the future. Finally, the assumptions made for NBI provision and the corresponding vessel and port structures should be scrutinised. The 207 MW assumed for NBCD in order to facilitate fully non-inductive plasma operation is more conservative than previous studies [1], but the assumption of 10 ports, a fraction of which will have offaxis tangential injectors, needs to be assessed from an engineering perspective. Further, neutronics studies to assess how much blanket coverage is needed for tritium breeding self-sufficiency are also underway, though initial results suggest that a loss of first wall area of 0.5–1% necessary to accommodate 10 ports will not decisively affect the tritium breeding ratio. Acknowledgements This work was funded by the United Kingdom Engineering and Physical Sciences Research Council under grant EP/G003955 and by the European Communities under the Contract of Association between EURATOM and CCFE. The views and opinions expressed herein do not necessarily reflect those of the European Commission. References [1] D. Maisonnier, et al., Fus. Eng. Des. 81 (2006) 1123. [2] M. Shimada, et al., Nucl. Fusion 47 (2007) S1. [3] D.J. Campbell, et al., 21st IAEA Fusion Energy Conference, Chengdu, 2006, FT/11. [4] D.J. Ward, I. Cook, P.J. Knight, 18th IAEA Fusion Energy Conference, Sorrento, 2000, IAEA-CN-77-FT/P2-20. [5] R.J. Bickerton, et al., Nat. Phys. Sci. 229 (1971) 110.

150

I.T. Chapman et al. / Fusion Engineering and Design 86 (2011) 141–150

[6] D.A. Ehst, et al., Nucl. Fusion 38 (1998) 13. [7] K. Okano, Y. Asaoka, R. Hiwatari, Y. Ogawa, 31st EPS Conference on Plasma Physics, London, 2004, P2.135. [8] C.E. Kessel, T.K. Mau, S.C. Jardin, F. Najmabadi, Fus. Eng. Des. 80 (2006) 63. [9] G.V. Pereverzev, E. Strumberger, S. Guenter, K. Lackner, 21st IAEA Fusion Energy Conference, Chengdu, 2006, T-23. [10] E. Strumberger, S. Guenter, K. Lackner, G. Pereverzev, 33rd EPS Conference on Plasma Physics, Rome, 2006, P4.165. [11] S.C. Jardin, et al., Fus. Eng. Des. 80 (2006) 25. [12] S.C. Jardin, C.E. Kessel, C.G. Bathke, D.A. Ehst, T.K. Mau, F. Najmabadi, T.W. Petrie, Fus. Eng. Des. 38 (1997) 27. [13] C.E. Kessel, J. Manickam, G. Rewoldt, W. Tang, Phys. Rev. Lett. 72 (1994) 1212. [14] A. Becoulet, G.T. Hoang, Plasma Phys. Control. Fusion 50 (2008) 124055. [15] E. Joffrin, Plasma Phys. Control. Fusion 49 (2007) B629. [16] T.S. Taylor, Plasma Phys. Control. Fusion 39 (1997) B47. [17] A.C.C. Sips, Nucl. Fusion 47 (2005) A19. [18] E.J. Strait, et al., Phys. Rev. Lett. 74 (1995) 2483. [19] T.C. Hender, et al., 21st IAEA Fusion Energy Conference, Chengdu, 2006, EX/P818. [20] B. Hu, R. Betti, Phys. Rev. Lett. 93 (2004) 105002. [21] H. Reimerdes, et al., Phys. Rev. Lett. 98 (2007) 055001. [22] I.T. Chapman, et al., Plasma Phys. Control. Fusion 51 (2009) 055015. [23] Y.Q. Liu, et al., Phys. Plasmas 16 (2009) 056113. [24] Y.Q. Liu, et al., Phys. Plasmas 15 (2008) 092505. [25] J. Manickam, et al., Phys. Plasmas 1 (1994) 1601. [26] J. Manickam, et al., Nucl. Fusion 27 (1987) 1461. [27] I.T. Chapman, et al., Nucl. Fusion 50 (2010) 045007. [28] C. Challis, et al., 36th EPS Conference on Plasma Physics, Sofia, 2009, O5.057. [29] F. Zonca, et al., Plasma Phys. Control. Fusion 48 (2006) B15. [30] V.S. Chan, et al., Fusion Sci. Technol. 57 (2010) 66. [31] G. Vlad, et al., Nucl. Fusion 46 (2006) 1. [32] Y.R. LinLiu, R.D. Stambaugh, Nucl. Fusion 44 (2004) 548. [33] J.E. Menard et al., “Unified Ideal Stability Limitd for Advanced Tokamak and Spherical Torus Plasmas” PPPL Report 3779 (2003).

[34] M. Murakami, et al., Phys. Plasmas 13 (2006) 056106. [35] A.M. Gerofalo, et al., Phys. Plasmas 13 (2006) 056110. [36] J.R. Ferron, et al., 22nd IAEA Fusion Energy Conference, Geneva, 2008, EX/P4-27. [37] T.C. Hender, P.J. Knight, I. Cook, Fusion Technol. 30 (1996) 1605. [38] R.V. Budny, et al., Nucl. Fusion 32 (1992) 429. [39] G.T.A. Huysmans, J. Goedbloed, W. Kerner, Proceedings of the CP90 Conference on Computer Physics, World Scientific, Singapore, 1991, p. 371. [40] O. Sauter, C. Angioni, Y.R. Lin-Liu, Phys. Plasmas 6 (1999) 2834. [41] G. Cunningham, Private Communication. [42] P.B. Snyder, et al., Nucl. Fusion 49 (2009) 085035. [43] A.B. Mikhailovskii, G.T.A. Huysmans, S.E. Sharapov, W.O. Kerner, Plasma Phys. Rep. 23 (1997) 844. [44] ITER Physics Basis, Nucl. Fusion 39 (1999) 2137. [45] J. Pamela, A. Becoulet, D. Borba, J.-L. Boutard, L. Horton, D. Maisonnier, Fus. Eng. Des. 84 (2009) 194. [46] A.H. Boozer, Phys. Plasmas 4 (1997) 3180. [47] Y.Q. Liu, et al., Phys. Plasmas 7 (2000) 3681. [48] E. Joffrin, Nucl. Fusion 45 (2005) 626. [49] J.W. Connor, Plasma Phys. Control. Fusion 40 (1998) 191. [50] J. Manickam, Phys. Fluids B 4 (1992) 1901. [51] R. Prater, et al., Nucl. Fusion 43 (2003) 1128. [52] R.J. LaHaye, Phys. Plasmas 13 (2006) 055501. [53] H. Zohm, et al., Nucl. Fusion 47 (2007) 228. [54] S. Günter, et al., Nucl. Fusion 45 (2005) S98. [55] M.R. Turnyanskiy, et al., Nucl. Fusion 49 (2009) 065002. [56] F. Porcelli, D. Boucher, M. Rosenbluth, Plasma Phys. Control. Fusion 38 (1996) 2163. [57] I.T. Chapman, et al., Plasma Phys. Control. Fusion 49 (2007) B385. [58] R.J. Buttery, et al., Nucl. Fusion 44 (2004) 678. [59] O. Sauter, et al., Phys. Rev. Lett. 88 (2002) 105001. [60] I.T. Chapman, et al., Nucl. Fusion 50 (2010) 102001. [61] R. Dux, et al., J. Nucl. Mater. 313 (2003) 1150. [62] R. Neu, et al., Plasma Phys. Control. Fusion 44 (2002) 811.