On the Use of High Magnetic Field in Reactor Grade Tokamaks
Abstract
A 0D model is developed to explore, from the physics point of view, the design options for future reactor grade tokamaks at values of the confining magnetic field exceeding the present technology. It is found that steady state devices with consistent exhaust parameters can indeed be designed at more compact geometry than presently envisaged, but the plasma performance, in particular the stability, is still at the upper end of what has been achieved in present day experiments, i.e. requires an ‘advanced tokamak’ approach.
Keywords
Fusion power plant Tokamak Magnetic confinementIntroduction
The performance of magnetically confined fusion plasmas usually increases with increasing magnetic field B for given size, expressed by the torus major radius R. Simple scalings [1] show that the fusion power scales as B^{4}R^{3}, and the power amplification Q increases with 1/(B^{−3.7}R^{−2.7}const) so that conversely, the major radius R can be decreased if B is increased according to these relations, keeping the plasma performance constant. This has even been interpreted misleadingly as ‘fusion performance not depending on size’ [2, 3]. Present designs for next step reactor grade experimental devices differ strongly in their assumptions about the technically achievable toroidal field. Relying either on ITER technology (Nb_{3}Sn), the value of B_{max} at the inner leg of the TF conductor of the order of 12 T [4]), or new High T_{c} (HTSC) superconductors of the REBCO type that could allow up to 24 T at the inner leg [5], although this value would, using present technology, require exceedingly large support structure to withstand the forces.
This contribution aims at analyzing the principal merits of high field tokamaks (where the term ‘high field’ means a field above the field possible using stateof theart ITER technology) and the new challenges arising, e.g. for exhaust of power through a poloidal divertor which can be more challenging in a compact device. We explicitly leave out a discussion of the technological challenges on the route to using HTSC in fusion, but remind the reader that solving these issues will be a prerequisite for using high field tokamaks as FPPs and remains a serious R&D task (at present, there is no convincing demonstration of a HTSC high field solution on the scale needed for a reactorgrade device).
In “The 0D Model” section, we describe the 0D model used for the analysis, an extension of the model used in [6] for application in a wider parameter range. In “Exploration of High Field Solutions” section, we discuss suitable figures of merit for quantifying the gain that high field operation may bring and analyze several routes to tokamak FPPs for their prospects on the high field path. In “Discussion and Conclusions” section, a concluding discussion is given.
The 0D Model
We use a 0D model based on the equations presented in [6], improved to explore a larger parameter space. This required to update the calculation of the fusion power for higher temperatures as well as a more detailed radiation model, including separately Bremsstrahlung, impurity radiation and synchrotron radiation (which specifically becomes important when exploring high field solutions).

normalized pressure β_{N}, related to ideal MHD stability,

safety factor q_{95}, related to the plasma current

normalized confinement time H = τ_{E}/τ_{E,ITER98p}, assuming that confinement scales similar to the ITER Hmode scaling

normalized density f_{GW} = n/n_{GW}.
Together with the machine geometry parameters, major radius R,^{1} aspect ratio A and toroidal magnetic field B, these allow the calculation of fusion power P_{fus}, radiation losses P_{rad} and auxiliary power needed to sustain the power balance or current drive power P_{CD} needed to drive the difference between total current and bootstrap current, i.e. for fully noninductive operation.
Input parameters used in the study of the 4 plasma scenarios
ITER Q = 10  AUG hybrid  Stepladder  DIIID/EAST  

q_{95}  3.1  5.4  4.5  6.5 
H  1.0  1.1  1.2  1.5 
β_{N}  1.8  2.8  3.5  3.0 
f_{GW}  0.85  0.7  1.2  1.0 
In Eqs. (1) and (2), we have accounted for the dilution due to Helium ash and the Xenon seed impurities which are assumed to be injected to increase the radiation from inside the separatrix (no other impurities are considered in this work). Both densities are normalized to the electron density, i.e. f_{He} = n_{He}/n_{e} and f_{Xe} = n_{Xe}/n_{e}. The quantity Z_{Xe} is the average charge state of Xenon in the core plasma (He is fully ionized under these conditions). We note that we do not correct the heating power for the radiation losses, motivated by the finding that for stiff temperature profiles, confinement is hardly affected by radiation losses as long as P_{rad}(r) does not overlap with P_{fus}(r) [8], which is the case for Xenon for typical profiles in reactor grade devices that tend to be peaked offaxis [8].
Exploration of High Field Solutions
In this section, we apply the model described in the previous section to study the possible parameter space of future reactorgrade tokamaks allowing high toroidal field and neglecting, for the moment, the present technological limitations to the increase of the field. The study will analyse different routes from present day experiments to reactor grade plasmas, namely the ITER Q = 10 scenario at low q_{95} [10], a ‘hybrid’type steady state scenario demonstrated on ASDEX Upgrade [11] and DIIID [12] at intermediate q_{95} as well as a lower q_{95} version proposed for the ‘stepladder’ in [6] and a high q_{95} variant demonstrated on DIIID and EAST [13] for use on CFETR [14]. In choosing these cases, the study is limited to conventional aspect ratio of A = 3.1 (the ITER value is used throughout). A study of compact solutions at low aspect ratio is subject to further work.
Table 1 shows the parameters q_{95}, H, β_{N} and f_{GW} for these 4 cases, noting that f_{He} = 0.05, f_{LH} = 1.1 and A = 3.1 are kept constant in the study. The values for ITER Q = 10, AUG Hybrid and Stepladder have been taken directly from refs [6, 10, 11], respectively. For the DIIID/EAST scenario, we took the values for the discharge discussed in [13], and chose q_{95} such that the value of the plasma current is matched. Due to the slightly differing aspect ratio and shape, this leads to a lower value than quoted in [13].

Fusion power P_{fus}: even though high field may allow smaller unit sizes, we still anticipate that an FPP will generate several GW of fusion power in order to arrive at reasonable recirculating power fraction due to the relatively large auxiliary power needed for a tokamak. Hence, we explore P_{fus} up to 3.5 GW, aiming at around 1 GW of electrical power at conventional efficiency.
 Power amplification w.r.t. power balance, Q_{PB} = P_{fus}/P_{AUX}: this shows how close the plasma is to ignition, and how effective it can be in generating electrical power. Assuming that P_{AUX} is the dominant electrical power needed to sustain the plant, the recirculating power iswhere we have accounted for the thermal power generated in the blanket by nuclear reactions by the factor 1.18 and introduced the efficiencies η_{AUX} (wall plug efficiency of the auxiliary heating system) and η_{TD} (thermodynamic efficiency to generate electricity from heat). For example, for η_{AUX} = 0.4 and η_{TD} = 0.35, we obtain f_{rec} = 6/Q_{PB} and a reasonable value of f_{rec} < 10% will require Q_{PB} > 60. We note here that in our approach, for an ignited plasma, there is no attempt to fulfill exactly the power balance, i.e. these cases would strictly not be stationary, but for the scoping studies shown here this is not considered to be too important.$$ f_{rec} = \frac{{P_{el,AUX} }}{{P_{el,tot} }} = \frac{1}{{1.18Q_{PB} \eta_{AUX} \eta_{TD} }} $$(9)

Power amplification w.r.t. current drive power Q_{CD} = P_{fus}/P_{CD}: while Q_{PB} is calculated from the power balance, the requirement of steady state will often lead to values of P_{CD} [see Eq. (8)] that exceed P_{AUX} and in this case Q_{CD} will determine the recirculating power fraction, calculated by using Q_{CD} and η_{CD} in Eq. (9), where η_{CD} is the wall plug efficiency of the CD system (the current drive efficiency in the plasma is already taken into account by calculating P_{CD} according to (8)). Hence, we also map out this quantity in the R–B space, noting that for a pulsed tokamak FPP, this constraint will not exist. In principle, P_{CD} and P_{AUX} should have the same value for a stationary solution, but usually, P_{CD} > P_{AUX} is found. In these cases, the fusion performance might be higher than estimated by our model, since there is excess heating, but as for the ignited cases discussed above, this is not considered to be too important for the scoping studies shown here.

Contribution of synchrotron radiation: a particularity of high field tokamaks is the possibility of synchrotron radiation dominating the power balance. We define as a rough indicator of this the quantity f_{sync} = P_{sync}/P_{rad,required}, where P_{rad,required} is defined by Eq. (7). If f_{sync} > 1, the radiation power exceeds the required power even at f_{Xe} = 0, i.e. the synchrotron losses are intolerable. Operation above this line will not be possible.
 Similarity in exhaust: in principle, an exhaust solution should be modelled using more sophisticated codes than the one discussed here to find the seed impurity concentration needed in the SOL and divertor to provide a detached solution. This concentration would then have to be fed back to the core plasma, assuming a certain compression ratio. Such a model has been recently developed and applied in [15]. However, this is beyond the scope of the present study. Hence, we rather adopt a similarity criterion put forward in [16], that states that the impurity concentration in the SOL and divertor needed to obtained a detached divertor solution is expected to scale likeat constant f_{LH}, A and shape. Note that fixing f_{LH} means that f_{Xe} varies with fusion power, feeding back into the power balance via Eq. (1). This scaling has also been found in [15]. In the following, we use it to connect the existing model points from Table 1 to other points in parameter space, arguing that the exhaust problem will be similar along this line and the solution developed for the points in Table 1 will apply for all points on the line. For scenarios where no exhaust scenario has been studied, we plot an ‘ITER Q = 10 exhaust similarity line’$$ f_{z} \propto \frac{{B^{0.88} R^{1.33} q^{0.32} }}{{f_{GW}^{1.18} }} $$(10)$$ B = B_{ITER} \left( {\frac{{f_{GW} }}{{f_{GW;ITER} }}} \right)^{1.34} \left( {\frac{{q_{95,ITER} }}{{q_{95} }}} \right)^{0.37} \left( {\frac{{R_{ITER} }}{R}} \right)^{1.51} $$(11)
These parameters will now be analysed for the 4 cases from Table 1.
The conclusion for the ITER Q = 10 point is of course in line with many previous studies and steady state solutions are usually explored at higher q_{95}, which will increase the bootstrap fraction, but at the same time, reduce the fusion power at given β_{N} and also make ignition harder due to the reduced current.
It can be seen that the approach is indeed successful in avoiding the synchrotron limit and providing reasonable Q_{CD}. If one wants to reduce the size of the FPP point along the exhaust similarity line, one quickly enters the region of too high P_{fus}, meaning that this process should start from a smaller machine, e.g. the DEMO point. One notes, however, that this leads into the region of relatively high f_{Xe}, which has the effect that Q_{CD} does not rise too strongly along this line.
Obviously, a problem in our approach is that the exhaust similarity line does not deviate too strongly from the lines of constant Q_{CD}, making it hard to profit from the smaller size along this line w.r.t. steady state. This is an inherent problem, because from a combination of (1) and (8), one can see that Q_{CD} roughly scales like B^{3}R^{3}, and on top of this, the decease of size leads to an increase in f_{Xe} which decreases the CD efficiency. Inserting the exhaust similarity, Eq. (10), leads to Q_{CD}~ 1/R^{1.5}, i.e. a decrease of size by a factor of 2 will lead to a gain in Q_{CD} of 2.8 at best. On the contrary, the fusion power will increase by up to a factor of 1/R^{3} = 8 along the same route, slightly diminished if the temperature becomes so high that the correction shown in Fig. 1 applies.
From the plots in Fig. 5, one can see that this indeed represents a step in a direction where Q_{CD} becomes higher, synchrotron radiation is not a problem, and fusion power stays below the 3.5 GW limit. We have highlighted a point at R = 6 m, B = 10 T, which sits at Q_{CD}= 43 and P_{fus} = 2.35 GW and might be an attractive steady state scenario. We note from the plot of Q_{PB} that from the view of power balance, the point already sits deeply in the ignited regime, meaning that the assumption H = 1.5 could even be relaxed (Q_{CD} = Q_{PB} for H = 1.17 in this case). We also note that no studies of a consistent exhaust scenario exist for this point. The ITER Q = 10 exhaust scenario line according to (11) is indicated in blue in the diagram, showing that finding a consistent exhaust scenario for this approach may also be challenging.
Discussion and Conclusions

a low q_{95}, low β_{N} approach, which is applied to maximize fusion power in pulsed ITER discharges, does not extrapolate to steady state at higher B since the gain in QCD is relatively small when moving on the exhaust similarity line

in the usual advanced tokamak approach of increasing q_{95} and β_{N} for higher bootstrap fraction and, at the same time, H to compensate for the lower confinement and higher q_{95}, the Greenwald fraction also has to be increased because otherwise, the absolute density gets so low that no exhaust scenario compatible with present day approaches exists. In addition, high q_{95} at low f_{GW} also leads very high temperatures at which, together with the high field, synchrotron radiation becomes important. On the other hand, high f_{GW} will also help solving the exhaust problem.
Hence, the use of high field in an advanced tokamak approach will require high f_{GW}, an ingredient which is presently not integrated with this approach.
To study a possible target for such a scenario, we show the R–B space for an advanced tokamak (β_{N} = 4.0, H = 1.3, f_{GW} = 1.2, q_{95} = 5.7). It can be seen that this choice of parameters leads to an attractive operational point at R = 5 m, B = 9.3 T, producing 2.7 GW of fusion power at Q_{CD} = 80, i.e. with the prospect to reach low recirculating power fraction below 10%. Due to the high f_{GW}, the average electron density is 1.6 × 10^{20} m^{−3} and the average temperature is 20 keV, leading to a large margin against synchrotron radiation and compatibility with the ITER exhaust requirements. This point, by no means optimized, can be taken as a start for optimization studies to find how to best exploit the high field.
In conclusion, we have shown that the possibility to build tokamaks at higher field than is presently possible calls for an optimization procedure that is not necessarily similar to that applied for present designs of reactorgrade devices. We have proposed a procedure to obtain design points which are steady state and fulfill an exhaust similarity criterion with ITER, meaning that the exhaust scheme could be validated there. An important finding is, however, that attractive points in operational space, especially if they should be steady state, still require quite optimistic assumptions about the plasma performance in terms of H and/or β_{N}. This is also evident from previous studies of high field devices [2, 3, 5].
Finally, we remind the reader that a rigorous technology R&D programme will be needed to solve the presently unresolved issues, such as high mechanical forces or neutron shielding at reduced radial build, if the high field approach should be pursued in future.
Footnotes
 1.
Plasma shape is assumed to be that of ITER so that the volume can be calculated as V = V_{ITER} (R/6.2 m)^{3}/(A/3.1)^{2}.
Notes
Acknowledgements
Open access funding provided by Max Planck Society. The author wants to acknowledge enlightening discussions with V. Chan, E. Fable, T. Pütterich and M. Siccinio as well as the programming support by A. Bock. The help of P. Bonoli with the comparison to ARC is also gratefully acknowledged. This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014–2018 under Grant Agreement Number 633053. The views and opinions expressed herein do not reflect those of the European Commission.
References
 1.H. Zohm, Fusion Sci. Technol. 58, 613 (2010)CrossRefGoogle Scholar
 2.A. Costley et al., Nucl. Fusion 56, 066003 (2017)ADSCrossRefGoogle Scholar
 3.W. Biel et al., Nucl. Fusion 57, 038001 (2017)ADSCrossRefGoogle Scholar
 4.G.F. Federici et al., Fusion Eng. Des. 89, 882 (2014)CrossRefGoogle Scholar
 5.B. Sorbom et al., Fusion Eng. Des. 100, 378 (2015)CrossRefGoogle Scholar
 6.H. Zohm et al., Nucl. Fusion 57, 086002 (2017)ADSCrossRefGoogle Scholar
 7.H.S. Bosch, M. Hale, Nucl. Fusion 32, 611 (1992)ADSCrossRefGoogle Scholar
 8.E. Fable et al., Nucl. Fusion 57, 022015 (2017)ADSCrossRefGoogle Scholar
 9.N. Uckan et al ITER physics design guidelines: 1989, ITER documentation series 10 (Vienna: IAEA), based on B.A. Trubnikov, Rev. Plasma Physics 7 (1979), ed. M. Leontovich (New York: Plenum Press) p. 345Google Scholar
 10.E. Dyole et al., Nucl. Fusion 47, S18 (2010)Google Scholar
 11.A. Bock et al., Nucl. Fusion 57, 126041 (2017)ADSCrossRefGoogle Scholar
 12.F. Turco et al., Phys. Plasmas 22, 056113 (2015)ADSCrossRefGoogle Scholar
 13.A. Garofalo et al., Nucl. Fusion 55, 123025 (2015)ADSCrossRefGoogle Scholar
 14.Y.X. Wan, Nucl. Fusion 57, 102009 (2017)ADSCrossRefGoogle Scholar
 15.M. Siccinio et al., Nucl. Fusion 58, 016032 (2018)ADSCrossRefGoogle Scholar
 16.M.L. Reinke, Nucl. Fusion 57, 034004 (2017)ADSCrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.