Compact steady-state tokamak performance dependence on magnet and core physics limits

Compact tokamak fusion reactors using advanced high-temperature superconducting magnets for the toroidal field coils have received considerable recent attention due to the promise of more compact devices and more economical fusion energy development. Facilities with combined fusion nuclear science and Pilot Plant missions to provide both the nuclear environment needed to develop fusion materials and components while also potentially achieving sufficient fusion performance to generate modest net electrical power are considered. The performance of the tokamak fusion system is assessed using a range of core physics and toroidal field magnet performance constraints to better understand which parameters most strongly influence the achievable fusion performance. This article is part of a discussion meeting issue ‘Fusion energy using tokamaks: can development be accelerated?’.


Introduction
A fusion nuclear science facility (FNSF) is viewed as a potentially important element of the U.S. fusion development roadmap that would be complementary to the DEMO-based approach pursued by the majority of the world fusion program. In particular, a FNSF/ component test facility (CTF) [1][2][3][4][5][6][7] could make significant contributions to the development of fusion energy by providing the nuclear environment needed to develop fusion materials and components. The FNSF nuclear environment includes fusion-relevant neutron wall loading W n ≥ 1 MW m −2 , neutron fluence greater than or 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/ by/4.0/, which permits unrestricted use, provided the original author and source are credited.

Compact tokamak fusion performance scalings
The results described here are a direct extension of the 0D systems-code methodology used in §5 of [17]. The assumed normalized beta β N , elongation κ and vacuum toroidal magnetic field at the plasma geometric centre B T dependence on aspect ratio plays a very strong (approx. quartic) role in the projected fusion performance of high-bootstrap-fraction tokamak scenarios since P fusion ∝ (β N κB T ) 4 [17]. This scaling shows that the achievable β N , κ and B T influence the fusion power on a similar footing for steady-state tokamak scenarios. Furthermore, the inboard blanket and shield thickness assumptions play an important role in determining the space available for the inboard toroidal field magnet and associated support structure as described in the following subsection. In subsequent subsections, the plasma stability, elongation, magnet parameters and confinement scalings are varied to develop an understanding of which parameters most strongly influence fusion performance for R=3m pilot plants with 50 MW of 0.5 MeV deuterium NNBI auxiliary heating and current drive. Compact tokamak devices in this size and power range are projected to be capable of achieving net electricity production in steady-state if the toroidal magnetic field, energy confinement and MHD stability limits are sufficiently high.
(a) Model for shielding and tritium breeding blanket thickness Figure 1a shows the inboard breeding blanket and shield thickness and outboard breeding blanket thickness versus aspect ratio A used in this study and previous HTS-TF pilot-plant study [17]. Both inboard and outboard breeding blankets are assumed to be dual-cooled lead lithium (DCLL) [20]. First, for sufficiently low A (A = 1.7-1.8) and sufficiently large major radius (R ≥ 1.7 m) detailed neutronics analysis indicates no inboard breeding blanket is required to achieve TBR ≈ 1 [17,21] as long as breeding is provided at the top and bottom of the device, NBI heating aperture areas are sufficiently small and NBI ducts are tangential to the plasma. For HTS TF magnets, sufficient shielding is required to protect against both nuclear heating and/or neutron damage. Detailed calculations including ferritic steel structure (which has lower neutron attenuation) to support and contain a WC shield and water coolant indicate shieldaveraged attenuation decay lengths of 15-16 cm per decade, and this results in a requirement for approximately 60 cm of cooled WC shield (or equivalent shielding) to protect the HTS TF magnet in a device that can achieve a peak outboard neutron fluence of 7 MW yr m −2 [22,23]. These results motivate the inboard WC shielding thickness of 60 cm at A = 1.8 as shown in figure 1a. Furthermore, at A = 2, there is insufficient outboard surface area to achieve TBR greater than or equal to 1 without inboard breeding. However, a 10 cm thick DCLL inboard breeding blanket can provide sufficient inboard breeding to achieve TBR greater than or equal to 1. This blanket is approximately half as effective for shielding as 10 cm of WC, so the inner WC shield can be reduced to 55cm thickness for a total shield plus breeding blanket thickness of 65 cm at A = 2. Dedicated neutronics calculations have not been carried out for higher A for an HTS-TFbased R=3m FNSF-Pilot. However, one can refer to the previous calculations finding that A = 4 advanced Tokamak (AT) pilot plants [12] with 40 cm of inboard breeding blanket (and 80 cm of outboard breeding blanket) can achieve TBR ≥ 1. These results correspond to the inboard shielding and breeding curves in figure 1a at A = 4 chosen to provide shielding equivalent to 60 cm of WC shielding. The inboard breeding blanket thickness for other aspect ratios is determined via interpolation while matching the aforementioned thickness values at A = 1.8, 2.0 and 4. Then, the WC shielding thickness is chosen so that all aspect ratios have an equivalent WC shielding thickness of 60 cm again assuming a DCLL blanket of a given thickness provides the same shielding as a WC shield half as thick. Clearly, the model curves in figure 1a are approximations and additional three-dimensional neutronics calculations are required to provide a more accurate dependence on A. As a cross-check, it is noted that the R = 2.7, A = 3.5 fusion development facility (FDF) [24,25] with copper magnets requires a 25 cm thick inboard DCLL blanket to achieve TBR ≥ 1. This implies a 73 cm inboard WC shield plus blanket total thickness is required to achieve a 60 cm WC-only shield equivalent, whereas figure 1a indicates 78 cm at A = 3.5. This indicates that figure 1a captures the leading order trends for required inboard shielding and breeding blanket thickness for TBR ≥ 1 for the range of aspect ratios considered here. Lastly, it is noted that small changes in inboard shielding thickness can have major impacts on device performance and lifetime. For example, an increase in effective inboard shielding thickness from 60 cm to 70 cm reduces the fusion and engineering gain by 25-35% for A = 2-3 but increases the peak outboard neutron fluence by a factor of 4 and increases HTS magnet lifetime by a factor of 5 in full-power-years [17].
(b) Aspect ratio dependence of normalized beta and elongation Figure 1b shows the assumed maximum β N (A) (n = 1 no-wall kink stability limit), κ(A), and shielding and blanket thickness versus aspect ratio A used in these studies as was done in [17]. The simplifying assumption is made that the β N (A) and κ(A) scalings are independent of absolute parameters such as device size, field or current and plasma parameters such as collisionality, toroidal rotation speed, poloidal beta or internal inductance. Holding the HTS winding pack thickness fixed at 0.24 m and varying the inboard toroidal field coil structure thickness versus A (increasing from 0.2 m to 0.45 m between A = 2 and 4) as was also done in [17], figure 2a shows the fusion gain Q DT , figure 2b the fusion power and figure 2c the net electric power versus A for an assumed effective inboard WC shield thickness of 0.6 m and varied assumptions of β N (A) and κ(A). As is evident from figure 2, fixing either β N or κ constant (at values of 3.5 and 2.0, respectively) significantly reduces the achievable fusion gain and power at lower aspect ratios and indicates an optimal aspect ratio near A = 2.5. Furthermore, holding both β N = 3.5 and κ = 2.0 fixed further reduces the fusion gain and power at low-A and makes the fusion gain and power nearly independent of aspect ratio for A ≥ 2.5. Given the importance of β N (A) and κ(A) on fusion performance evident in figure 2, determining whether the projected β N and κ values are sustainable in fully non-inductive low-aspect-ratio scenarios is a major research goal for the future NSTX-U operation [26].

(c) Fusion performance versus toroidal field magnet parameters
The results of figure 2 are representative of expected fusion performance for a particular choice of toroidal field coil winding pack thickness (0.24 m) and structural support width variation with aspect ratio that provides for sufficient space for at least a small central solenoid for A ≥ 2 and results in maximum magnetic field values ranging from 17 T to 19 T at the TF magnet. The results in figure 2 assume a maximum allowable TF structural support stress of 0.66 GPa typical of stainless steel, winding pack current density 70 MAm −2 , and winding pack stress limited to 0.4 GPa to ensure strains less than or equal to 0.3% to avoid any stress-related degradation in critical current [17]. To better understand the role of advanced superconductors on the achievable fusion performance versus aspect ratio, different combinations of maximum magnetic field at the magnet B max and winding pack current density are considered as shown in figure 3. An important reference case is the ITER toroidal field coil set [27][28][29] with B max ≈ 12 T and J WP = 20 MAm −2 . Figure 3a shows that for these magnet parameters the maximum allowable magnetic field cannot be reached for A ≤ 2.5 due to space constraints on the inboard TF magnet. Furthermore, as B max is increased to 19 Figure 3b shows the impact of using ITER LTS magnet assumptions versus advanced high J WP REBCO HTS magnet assumptions. In particular, the accessible vacuum toroidal field in the plasma increases by approximately a factor of 3 at A = 1.6, a factor of 2 at A = 2, a factor of 1.6 for A ≥ 2.5. Figure 4 shows that the maximum fusion and electricity gains are sensitive functions of both magnet parameters and aspect ratio. For ITER-like magnet parameters of B max ≈ 12 T and J WP = 20 MAm −2 , the fusion gain is constrained to be less than 3 while Q eng ≤ 0.5. Increasing the maximum field constraint from 12 T to 19 T increases Q DT to above 4 but Q eng remains less than unity. However, for B max ≈ 19 T and J WP = 30 MAm −2 , the fusion gain increases by a factor of 2 and Q eng ≥ 1 between A = 2.2 and 3.5. As shown in figure 5, as the winding pack current density is further increased the aspect ratio that produces the highest fusion power and net electric power approaches A ≈ 2 with P fusion up to 600 MW and net electric power greater than 100 MWe. These results highlight the importance of the simultaneous high winding pack current density and high  maximum field of HTS to leverage increased normalized plasma stability at reduced aspect ratio for maximizing fusion power and net electric power in a compact tokamak. Figure 6a shows that for J WP ≥ 30 MAm −2 nearly all aspect ratios have bootstrap fractions between 70% and 85% with average values near 80% for higher A. Figure 6b shows that the toroidal beta increases nearly inverse-quadratically with reduced aspect ratio (β T (%) ≈ 36A −1.8 ) and increases by roughly a factor of 5 between A = 4 and A = 1.6. Figure 6c shows that the kink safety factor q* is above 3 for all aspect ratios analysed and increases to up to 4.5 near A ≈ 2 for the highest winding pack current densities.
All of the results shown in figures 4 through 6 assume the energy confinement is sufficient to operate at the β N limit shown in figure 1b. To better understand the potential challenge in accessing these β N values in a high-field compact device, the required confinement is compared to several common confinement scalings. The most commonly used scaling is the ITER 98y2 H-mode confinement scaling [33,34] [34] when expressed in dimensionless parameters, several device-specific confinement scaling studies from JET, DIII-D and NSTX [35] have shown that the confinement scales weakly with β. Such weak β dependence is consistent with electrostatic turbulence with gyro-Bohm scaling represented in the 'Petty08' scaling expressed as: τ E−Petty08    In addition to the uncertainty in confinement scaling at low aspect ratio there is also uncertainty in how any scaling differences at low aspect ratio transition or connect to higher aspect ratio. Any such transitions/connections between low and higher aspect ratio confinement could have important implications for the ultimate choice of aspect ratio for compact pilot plants.
To assess this in a highly approximate way, a hybrid 'NSTX-Petty08' scaling is used. This ad hoc hybrid scaling accounts for the fact that the NSTX ST scaling has thus far been developed for plasmas with A ≤ 1.7 ( ≥ 0.6). Similarly, the Petty08 scaling has been developed using data primarily from conventional aspect ratio plasmas with A ≥ 2.5 ( ≤ 0.4). Assuming a linear interpolation in between the two scalings is justified, a combined/hybrid scaling can be defined as the weighted sum of the NSTX and Petty08 scalings according to: Figure 7a shows that in order to reach the β T values shown in figure 6b elevated normalized confinement is required for all aspect ratios. In particular, the required H-factor relative to the ITER 98y2 scaling is as high as 1.7-1.8 at lower A and is 1.5-1.6 for high A = 3-4. Similarly, figure 7b shows that even for the Petty08 confinement scaling with weak β dependence, the H-factor relative to the scaling is between 1.2 and 1.4 for nearly all aspect ratios and is nearly independent of A with H Petty08 ≈ 1.25 for A ≥ 2 for the highest winding pack current densities. Figure 7c shows the NSTX ST confinement scaling H-factor needed to achieve the assumed β T is a rapidly decreasing function of increasing aspect ratio. Again, this ST scaling has not yet been extended to higher field, current or plasma temperature (i.e. reduced collisionality) and is obviously not applicable to higher aspect ratios A > 2.5. Figure 7d shows the NSTX-Petty08 confinement scaling H-factor needed to achieve the assumed β T is near or below 1 for A in the  This result opens up the interesting (albeit speculative) possibility of the optimal aspect ratio from an integrated magnet, shielding, confinement, stability and non-inductive sustainment standpoint being between A = 1.8 and 2.3-a range of aspect ratios not previously studied experimentally. A further consideration in the aspect ratio optimization of compact tokamaks is the impact of the maximum magnetic field achievable or allowed at the magnet. To gain further insight into this dependence, a set of three maximum fields is considered (17,19 and 23T) in combination with a set of two winding pack current densities (70 and 160 MAm −2 ) consistent with conductor on round core (CORC) magnet cables under development [30][31][32]. Figure 8 shows the achievable fields for this combination of magnet parameters. Figure 8a shows the maximum achievable toroidal magnetic field for these parameters and indicates that nearly all aspect ratios (A = 1.6 is the exception) can achieve 17T at the magnet for both J WP assumptions. Figure 8a also shows that A ≥ 2.2 is required to achieve 19 T at the magnet for both J WP assumptions and that A ≥ 1.8 can achieve 19 T for J WP = 160 MAm −2 . None of the aspect ratios considered here can achieve 23 T at the magnet due to inboard structural space limitations, but A = 4 and J WP = 160 MAm −2 does exceed 22T. Figure 8b shows the achievable toroidal magnetic field at the plasma geometric centre and shows that in comparison to the results of figure 3b, the field at the geometric centre at lower aspect ratios is more strongly influenced by the ability of the TF magnets to access J WP ≥ 40 MAm −2 than by the ability to access B max ≥ 19 T.  For the same magnet parameters as figure 8, figure 9 shows that for B max = 19 T the highest fusion and engineering gains occur near A = 2. However, as the maximum field is relaxed to the highest achievable (i.e. constrained by space for inboard TF coil structural support), there is a broad range of aspect ratio A = 2 to 3 with similar gains. Figure 10a shows that the plasma current versus aspect ratio depends relatively weakly on J WP and B max for the cases considered. However, figure 10b shows that the achievable ramp-up plasma current depends more strongly on the combination of J WP and B max with the lowest B max and highest J WP combination providing the most space for a central solenoid and highest inductive ramp-up plasma current. These calculations assume a double-swing OH solenoid is located inside the bore of the inner toroidal field coil support structure using a solenoid with 20T maximum field and J WP = 70 MAm −2 [17]. Figure 10c shows the ratio of ramp-up plasma current to steady-state flat-top current is small or    aspect ratio are largely similar to those shown in figure 7, but figure 11d shows a clearer minimum H-factor requirement near 1 for A ≈ 2 for the hypothesized ad-hoc NSTX-Petty08 confinement scaling.

(d) Fusion performance versus scaled normalized beta
As discussed in the introduction to §2, the strong dependence of the fusion power on β N where P fusion ∝ β 4 N is a strong motivation for accessing improved stability to pressure-driven modes for all aspect ratios. Figure 12 shows the fusion and engineering gains versus aspect ratio for a scan of total (thermal + fast-particle) β N scaled by factors ranging from 0.7 to 1.  As is evident from figure 12, both gains increase by approximately a factor of 8 as the no-wall limit β N is scaled from 0.7 to 1.2 consistent with the β 4 N scaling. Strictly speaking, the thermal β N and bootstrap fraction should be considered in these scaling comparisons, but similar scaling trends are also observed when these parameters are considered. Figure 13 shows fusion power and net electric power for the same scan. Figure 13a shows that fusion powers exceeding 1 GW are projected for β N values 20% above the assumed no-wall limit. Furthermore, figure 13b shows that net-electric powers above 200 MWe are projected for the same elevated β N assumption. Thus, accessing plasma pressures above the no-wall stability limit relying on either passive/kinetic stabilization [47-58] and/or active RWM feedback control [59][60][61][62] can have a substantial impact on fusion performance and is potentially a requirement for modular approaches to steady-state power generation utilizing compact tokamaks [18]. Figure 14 shows that the power densities can be very high for these compact configurations operating at elevated β N . Figure 14a shows that the surface-average heat flux exceeds 1 MWm −2 for aspect ratios A ≥ 2.5. Figure 14b shows that the surface-average neutron wall loading exceeds 3 MWm −2 for aspect ratios A ≥ 2.2. Assuming an outboard mid-plane neutron wall loading peaking factor of 1.5 [23], figure 14c shows that the peak outboard neutron wall loading exceeds 5 MWm −2 for aspect ratios A ≥ 2.5. Reducing the power exhaust and neutron wall loading parameters tends to favour lower aspect ratio due to larger plasma boundary (and first-wall) surface area when the plasma major radius is fixed. Figure 15 shows trends for the plasma current and beta parameters for this normalized beta scan. Figure 15a shows that the plasma current scales approximately linearly with the scaled β N , and figure 15b shows the scaled β N itself for reference. Since the toroidal beta is proportional to the normalized beta and plasma current, figure 15c shows the expected result that the toroidal beta β T scales approximately quadratically with the scaled β N . Figure 16a shows that the bootstrap current fraction is highest at the highest β N and is between 80 and 85% for nearly all aspect ratios considered. As the β N is lowered, the bootstrap current (which is proportional to the thermal plasma pressure) also decreases and the non-inductive current drive from the 50 MW of 0.5 MeV NNBI contributes an increasingly larger fraction of the total current. Figure 16b shows that the  highest β N cases have the lowest kink safety factor q* consistent with these cases also having the highest plasma current. A ≈ 2 has the highest q* and nearly all cases have q * ≥ 3 as a metric for stability against the external current-driven kink mode [63]. Figure 17 shows that for the set of confinement scalings treated here (ITER 98y2, Petty08, NSTX-Petty08) there is a non-monotic dependence on β N values used. In particular, for the lowest values of β N (A) scaling factor considered (0.7 and 0.8), increasing H is required for increasing β N (A) scaling factor. However, as the scaling factor is further increased above values in the range of 0.9 to 1, the required H progressively decreases. This nonlinear result arises because the increased fusion power and alpha-driven plasma heating at higher assumed β N scaling reduces the required confinement multiplier. Such results may point to the complexity of controlling potentially highly nonlinear plasma states in which increasing fusion gain increases the plasma pressure, bootstrap and total current, and thus the confinement time (through the increased plasma current) thereby further increasing the fusion gain until other effects intervene to saturate or limit the fusion gain. Such saturating effects include reduced fusion reactivity and/or increased plasma radiation power loss at very high plasma temperature, and limiting effects include pressure-driven disruptions.
(e) Fusion performance sensitivity to confinement and stability limits The parametric scans addressed in previous sections dealt primarily with constraints on the maximum β N and/or magnetic field or magnet current density while leaving the confinement unconstrained. To investigate the impact of constrained confinement using the NSTX-Petty08 model, figure 18 shows the confinement multiplier, achievable β N and fusion gain assuming H NSTX-Petty08 ≤ 1 as shown in figure 18a (with the exception of a reference unconstrained case indicated by the blue diamond symbols). Figure 18b shows that the assumed β N constraint is only accessible for narrow range of aspect ratios determined by the value of 2 from the ad hoc NSTX-Petty08 confinement scaling model. In particular, as the aspect ratio for which ST confinement transitions to Petty08 confinement is reduced (i.e. 2 is increased), an increasingly narrow window of sufficient confinement is accessible to reach the β N constraint. If the transition aspect ratio is as low as A = 2 (i.e. 2 is as large as 0.5) then no aspect ratio can reach the β N constraint for H NSTX-Petty08 ≤ 1. It is also evident that for H NSTX-Pett08 ≤ 1 many aspect ratios have lower β N ≈ 2 and much lower fusion gain Q DT ≈ 1 than the β N constraint would allow. This is due to the strong dependence of the gain on H as described in more detail in appendix A. These results also indicate that if ST scaling does in fact extend to lower collisionality and higher field and current in NSTX-U and MAST-U, important questions may still remain on how rapidly the transition from ST to conventional-A confinement occurs for intermediate A = 1.8 to 2.5. Finally, figure 19 shows that for B max = 19 T and J WP = 70 MAm −2 , the assumed aspect ratio dependence of β N and κ has a significant impact on the achievable fusion gain and net electrical power. In particular, the fusion gain decreases by nearly a factor of 2 and net electrical power becomes negative if the β N and κ are held constant at the values applicable to A = 4 (β N = 3.45 and κ = 2.06). These results are similar to those shown in figure 2 indicating that the aspect ratio dependence of β N and κ is important for a range of magnet parameters.

Conclusion
For tokamaks with plasma major radius as small as R = 3 m, net electric power production becomes accessible from a plasma stability standpoint if the toroidal field coil winding pack current density and maximum field at the coil are sufficiently high (17-20 T). As the maximum magnetic field at the TF coil is increased above approximately 19-20 T, magnet structural support requirements leave progressively less space available in the central bore until only small or no space remains for a central solenoid. In the limit of small or no central solenoid for plasma current formation, aspect ratios near 2 become increasingly attractive for maximizing fusion power and electrical power production. For such lower-A solutions to be viable, high toroidal field magnet current density is required and appears feasible if (for example) CORC HTS cables can be developed into large-bore TF magnets. Furthermore, the aspect ratio dependence of β N and κ is important to include since such dependence can potentially increase the fusion gain by up to a factor of 2 relative to fixed β N and κ. Lower-A configurations would require some fraction of the current ramp-up to be provided through non-inductive current overdrive or via other methods of current formation [64]. All of the compact tokamak scenarios studied here require elevated confinement relative to conventional aspect ratio scalings in order to access n = 1 nowall beta values. Interestingly, for an ad hoc confinement scaling connecting the NSTX and Petty08 scalings, A = 2 − 2.3 (a range of aspect ratios not previously studied experimentally) lowers the required confinement multiplier to H ≈ 1. Operation near and above the n = 1 no-wall limit could be particularly advantageous since P fusion ∝ β 3−4 N for high bootstrap current fraction scenarios for which kink safety factor q* ≥ 3 is intrinsically met and is not a stability constraint. Net electric powers of 200-300 MWe are projected at elevated β N values ≈ 1.2× the no-wall limit in a compact device with R = 3 m, but such configurations would have challenging exhaust heat fluxes and high neutron wall loading. These high power fluxes would lead to central HTS toroidal field magnet lifetimes of 1-2 full-power years from neutron irradiation damage potentially requiring blanket and central TF magnet change-out every 1-2 years if very high duty factor was ultimately achieved. HTS TF magnet lifetime could be increased by up to a factor of 5 by increasing the effective inboard shielding thickness from 60 cm to 70 cm [17]. Increased inboard shielding at fixed plasma major radius does however result in decreased plasma and engineering gains. Appendix A. Fusion gain scaling for steady-state tokamaks In this Appendix, fusion gain scalings are derived for high-field steady-state tokamaks with highbootstrap-fraction configurations for which limits on the external current drive are typically more constraining than kink stability limits on the minimum edge safety factor. Such scalings can be expressed as a combination of both dimensional and dimensionless parameters most relevant to reactor design to aid understanding of the trade-offs between different parameters.
The deuterium-tritium (D-T) fusion power P f in a magnetic confinement system scales as P f ∝ n 2 σ v DT V [65] which for ion temperatures in the range of 8-30keV is approximately proportional to , p is the volume-averaged plasma pressure, B T0 is the vacuum toroidal field at the plasma geometric centre and V is the plasma volume. The maximum β T is limited by plasma magnetohydrodynamic instabilities and the maximum toroidal magnetic field in the plasma is limited by engineering constraints including maximum field, force, current density and/or cooling of the magnet. Furthermore, p ∝ P h τ E /V where P h is the plasma heating (loss) power and τ E is the energy confinement time. For the scalings here, core radiation losses and divertor power exhaust challenges are not considered and are topics for future study. The energy confinement time can be expressed as a scaling in terms of dimensional parameters as τ E ∝ HI α I P B α B T n α n e P −α P h R α R κ α κ α where H is the confinement multiplier, I P is the plasma current, n e is the plasma electron density, R is the plasma major radius, κ is the plasma boundary elongation and is the plasma inverse aspect ratio, where ≡ A −1 and A ≡ R/a and a is the plasma minor radius. From these definitions is also follows that V ∝ κRa 2 ∝ κR 3 2 . The fusion power therefore scales as P f ∝ H 2 I 2α I P B 2α B T n 2α n e P 2(1−α P ) h R 2α R −3 κ 2α κ −1 2α −2 . The plasma heating power is the sum of the auxiliary heating and current drive power P aux and the self-heating power from alpha-particles P α = λ DT P f : P h = P aux + λ DT P f = P aux (1 + λ DT Q DT ) where Q DT ≡ P f /P aux and λ DT ≈ 0.2, i.e. 20% of the DT fusion power is in confined α particles that can heat the plasma. Thus, P f = Q DT P aux and Q DT are determined from the solution of the equation Q DT /(1 + λ DT Q DT ) 2(1−α P ) = Q * DT where Q * DT ∝ H 2 I 2α I P B 2α B T n 2α n e P 2(1−α P )−1 aux R 2α R −3 κ 2α κ −1 2α −2 . It is noted that for λ DT Q DT 1 it follows that Q DT ≈ Q * DT . For λ DT Q DT 1 it follows that Q DT ∝ (Q * DT ) 1 2α P −1 which is a sensitive function of α P and implies α P > 0.5 is required for physically meaningful results using this simple zero-dimensional scaling. Thus, for typical values of α P ≈ 0.7 and all other parameters held fixed, P f ∝ H 2 at low fusion gain and P f ∝ H 5 at high fusion gain indicating the fusion power is a very strong function of the confinement multiplier as the plasma transitions from the burning plasma regime (Q DT = 5-10) towards ignition.
The density and MHD pressure limits are important parameters for reactor design. The empirical density limit for tokamaks [66,67] can be expressed as Greenwald density fraction f gw ≡n e /n gw ≤ 1 where n gw [10 20 m −3 ] ≡ I P [MA]/π a 2 and thus n e ∝ f gw I P /a 2 . From this scaling it follows that the normalized fusion gain scales as: Similarly, the plasma current I P can be recast as I P = (I P /aB T )aB T and since I P /aB T ∝ β T /β N then I P ∝ (β T /β N )aB T . From this relation it follows that: and that the aspect ratio dependence of β N , B T and κ are important factors in determining the optimal aspect ratio for maximizing fusion gain when other parameters are held fixed.