Power handling in a highly-radiative negative triangularity pilot plant

NT research has grown in popularity significantly in recent years. Three tokamaks, TCV [9, 10, 11, 12], DIII-D [13, 14, 15, 16, 17, 18, 19], and ASDEX Upgrade (AUG) [20] have successfully created NT-shaped plasmas. Many of these experiments have identified improvements in core performance. They also suggesting advantages for power handling when operating with a divertor designed for NT. The first of these is a geometric advantage in terms of a reduction in the heat flux density on the armored targets [21]. NT moves the divertor to larger radii, as shown in Figure 1, thereby increasing its surface area compared to a positive triangularity (PT) plant of equivalent major radius.

Results for plasmas in NT suggest that it may be possible to sustain good core confinement even without an edge transport barrier, a scenario more closely resembling an ELM-free L-mode edge [13, 22, 15, 16]. This is beneficial for power handling, as the absence of a pedestal and ELMs that accompany those in ELMy H-modes forgoes the need to survive large transients in heat and particle fluxes. Under the conditions expected in tokamaks at high plasma current, these will be particularly dangerous for machine survivability. Recent results on DIII-D bolster confidence in being able to retain this ELM-free edge for sufficiently negative triangularity, even at high heating powers [17, 23]. It was also observed that this ELM-free operation was compatible with non-seeded detachment, even with a divertor not fully optimized for NT [24, 25].

Other scenarios have also been proposed for reactors that emphasize the “power-handling first approach”. A recently proposed ARC-class reactor attempts to do this by remaining in L-mode while maintaining a high radiated core fraction, similarly to MANTA [26]. Many other regimes appear attractive for reactor concepts, including the EDA H-mode [27], the I-mode [28], the QH-mode [29], and the radiative X-point [30]. A radiative NT scenario, like that proposed in this work, offers similar ELM-free operation, with likely better confinement than the L-mode scenario and a likely broader heat flux width than the H/I-mode scenarios, as well as geometric advantages related to divertor location. Ability to access any of these regimes in a reactor remains uncertain, but if trends for benefits of NT extrapolate, these are perhaps easiest to access of all proposed regimes.

Power handling in tokamaks can be divided into three categories: plasma power exhausted into the scrape-off layer (SOL), radiated power, and power deposited by neutrons. The power transported from the core into the SOL, $P_{\mathrm{SOL}}$, flows along open field lines to the divertor targets, potentially creating significant heat loads on the target plasma-facing components (PFCs). Thus, the magnitude of $P_{\mathrm{SOL}}$ strongly influences divertor complexity and lifetime. $P_{\mathrm{SOL}}$ depends on heat sources and sinks, and can be expressed as

$P_{\alpha}$ comes from alpha particle heating, $P_{\mathrm{oh}}$ is the ohmic power generated by the tokamak’s central solenoid, $P_{\mathrm{aux}}$ is any auxiliary power used to heat the plasma, $dW/dt$ is the time rate of change of the plasma stored energy, $W$, and $P_{\mathrm{rad}}$ is any power that is radiated away in the core, either by the main ion species or by impurities.

Since NT FPPs like MANTA do not target H-mode operation, $P_{\mathrm{SOL}}$ need not exceed some threshold required for the transition. It is thought that H-mode access requires a certain critical level of ion heat flux to be driven through the edge. This is needed to build up a large enough ion pressure gradient to maintain $E\times B$ shear suppression and the development of a transport barrier in the typical H-mode edge [31]. Without such a requirement, an NT reactor with a highly radiating core could, in principle, achieve a large range in fusion power, $P_{\mathrm{fus}}$, simply by varying core density and without concern for radiating too much power such that $P_{\mathrm{SOL}}$ falls below the L-H power threshold, $P_{\mathrm{th}}^{\mathrm{L-H}}$. Furthermore, since both $P_{\mathrm{rad}}$ and $P_{\mathrm{fus}}$ increase with $n$, as $n$ increases, $P_{\mathrm{SOL}}$ can remain low as long as $P_{\mathrm{rad}}$ can be controlled to compensate for changes in $P_{\mathrm{fus}}$. Details of MANTA’s 0D analysis can be found in [1].

The second exhaust channel is $P_{\mathrm{rad}}$. The highly-radiative core scenario considered here will radiate power throughout the plasma volume, including the core, common flux region (CFR), and divertor. The current analysis of the operating point does assume that the radiated fraction is a controllable parameter in NT plasmas, as it sets $P_{\mathrm{SOL}}$. The calculation of core radiation figures into transport analysis explained in more detail in [32], and the assertion of controllable radiation fraction is an active area of investigation. Radiation in the CFR and divertor is considered here, and it is included in the transport calculation described in Section 3. Success of a power handling solution requires both sufficiently low incident heat flux on the divertor plates to prevent damage as well as survivable radiation loading on the main chamber.

The final power exhaust channel is that of the 14.1 MeV neutrons, $P_{n}$, released via fusion reactions. While these neutrons do not impart substantial heat flux onto MANTA’s PFCs, avoiding substantial neutron flux to achieve reasonable magnet lifetime presents a strong constraint on the locations of the poloidal field (PF) coils. Thus, $P_{n}$ indirectly affects the achievable divertor configurations. The PF coil set used throughout this work balances the choice of long lifetimes or low currents, as detailed in Section 2.1.

While trends identified in this work are representative of a MANTA-like concept, the operating point scoped in this paper is that of an earlier iteration of MANTA, rather than the final MANTA design point [1]. However, this is still a self-consistent design and allows for representative scoping of divertor operation for negative triangularity tokamaks similar to MANTA. The remainder of the paper is organized as follows. Section 2 provides a recap of the design of MANTA from its 0D scoping, to the generation of its equilibrium using FreeGS. Section 3 details the SOL simulation workflow using the UEDGE code. Section 4 tests the robustness of the divertor solution to deviations from assumptions in the edge model. Sections 5 and 6 discuss how the outputs of the UEDGE solution have been utilized to design and determine power loading on the structural components of MANTA’s divertor. Section 7 presents initial scoping of advanced divertor concepts, namely the X-divertor. Finally, Section 8 will summarize the findings of the scoping work presented here.

The NT magnetic equilibrium used in this work differs from the final MANTA equilibrium in that a fourth poloidal field (PF) coil was employed. Beginning with an equilibrium generated by the Grad-Shafranov solver CHEASE[40] based off an early 0D scoping, the divertor geometry and PF coil locations/currents were developed with another Grad-Shafranov solver, FreeGS[41], which is able to model the X-point and divertor region. The resulting equilibrium is shown in Fig. 2, where parabolic fits were used for the current and pressure profiles. The over-arching goal while designing the PF coilset was to keep the divertor as simple as possible, meaning no complex magnetic topologies and minimal X-points, while still surviving $P_{\mathrm{SOL}}=25$ MW. This resulted in four pairs of PF coils, which were placed inside the toroidal field (TF) coils to reduce their cost. This was possible because MANTA’s TF coils are demountable, allowing access to their interior. The PF coil locations shown in Figure 2 were determined through moving the coils by hand and running neutronics simulations for each new coilset until the PF coil lifetimes were deemed acceptable from the perspective of reactor cost. The resulting PF locations and currents are given in Table 3.

Core transport modeling for MANTA found a fixed $P_{\mathrm{SOL}}$ = 25 MW to be compatible with a fusion core that satisfied the requirements of the NASEM report. The L-H transition power from the Martin scaling [44] for MANTA is $P_{\mathrm{th}}^{\mathrm{L-H}}\approx 100$ MW, meaning MANTA will operate at $f_{\mathrm{L-H}}=P_{\mathrm{SOL}}/P_{\mathrm{th}}^{\mathrm{L-H}}$ = 0.25. This value of $P_{\mathrm{SOL}}$ is thus used for the UEDGE point design. All 25 MW are applied at the core boundary, a few mm inside of the separatrix, and the power is assumed to be divided evenly between electrons and ions. The difference in power a few mm inside of the separatrix and at the separatrix is not expected to represent a significant fraction of $P_{\mathrm{SOL}}$. The exact partition of energy between electrons and ions, however, is a source of uncertainty, but is out of the scope of this paper. For the density boundary condition in MANTA’s design, we use the result of an integrated modeling workflow described in [1]. This determines a value of $n_{\mathrm{sep}}=9.6\times 10^{19}$ m^-3 yielded an edge solution compatible with that of the core. Particle balance on MANTA is achieved through cryopumping, with the objective of removing He ash produced in the core. This surface is designated at the PFR boundary on the outboard side, near the outer target where the highest neutral pressure and thus, pumping efficiency, is expected. A pump removal rate of $\sim$5$\times 10^{21}$s^-1 is chosen to meet ⁴He removal requirements, assuming a 2% ⁴He concentration in the plasma exhaust from the reactor core and an enrichment of $\sim$0.75 in the divertor like that of MANTA [1] (consistent with previous studies [45]).

AS UEDGE does not resolve turbulent temporal and spatial scales, radial fluxes are calculated using a set of user-supplied cross-field transport coefficients. Since there is a lack of radiative L-mode negative triangularity experiments to compare to and current physics models for edge transport are incomplete, it is necessary to use existing scalings to choose a transport model. To do so, three main quantities are used to guide transport model selection: the e-folding length of the parallel heat flux at the separatrix, $\lambda_{q}$, the e-folding length of the plasma density at the separatrix, $\lambda_{n}$, and the ratio of inboard to outboard power, $P_{\mathrm{in-out}}$. Predictions for these are made for their values using empirical scalings and data from high-field devices.

The first of these, $\lambda_{q}$, is very influential for power handling. It dictates the size of the divertor wetted area that must handle the bulk of the power crossing the separatrix. The decay length in the near-SOL for the heat flux in an NT edge is likely to be similar to that in H-mode, though perhaps not as small. A variety of scalings for $\lambda_{q}$ scalings[46, 47, 48] serve to provide bounds that MANTA’s $\lambda_{q}$ is apt to fall in. The results for these scalings for MANTA’s primary operating parameter set are tabulated in Table 4. Interestingly enough (and perhaps unfortunately), Eich and Brunner’s scalings both predict heat flux widths possibly as low as 0.3 - 0.4 mm. Eich’s scaling is a unified scaling, and succeeds in predicting $\lambda_{q}$ in many current devices spanning a large range of operational paremters. It is, however, purely an H-mode scaling. Brunner’s on the other hand is multi-regime and might fare well in an NT edge, but it was constructed solely on data from Alcator C-Mod. Other scalings of C-Mod data by Brunner, but using $B_{p}$ could suggest $\lambda_{q}$ three times as high if the NT edge is to resemble the L-mode. Further, a multi-machine scaling by Horacek would put MANTA’s $\lambda_{q}>$ 10 mm. Considering this variation in possible values of $\lambda_{q}$ and in the interest of computational efficiency associated with grid size resolution, a middling value of $\lambda_{q}\approx$ 0.75 mm at the divertor throat is chosen. Figure 4 shows the parallel heat flux profiles at the X-point, with fits to the near-SOL $q_{\parallel}$, from which $\lambda_{q}$ is extracted.

For $\lambda_{n}$ and $P_{\mathrm{in-out}}$, there is even less predictive capability than for $\lambda_{q}$ and no empirical scalings to extrapolate from. Instead, these are chosen by considering measurements from Thomson Scattering and Langmuir probes on current experiments [49, 43]. Using these data, it is expected that $\lambda_{n}=\frac{n}{\nabla n}$ at MANTA’s high $B_{p}$ may be somewhere around 6 - 10 mm near the separatrix for L-mode plasmas. Finally, to determine $P_{\mathrm{in-out}}$, data for power flow to the inner and outer divertor of current devices is used. Ballooning-like transport on the LFS is thought to carry significantly more particles to the outer divertor than the inner divertor [50, 51, 52]. This effect is due in part to a large volume of “bad curvature” on the outboard side, making it more susceptible to interchange-driven turbulence [53]. Because of the flipped cross-section, this type of turbulence may not be as detrimental in NT because of the decreased surface area of the bad-curvature LFS separatrix. On the other hand, while DN configurations enable power-sharing between the top and bottom targets, the inner and outer SOLs are disconnected, leading to a larger in-out power asymmetry than witnessed in single null (SN) cases. Thus, MANTA is assumed to have a 30-70 power split between the inner and outer divertor targets, respectively, corresponding to $P_{\mathrm{in-out}}=P_{\mathrm{in}}/(P_{\mathrm{in}}+P_{\mathrm{out}})=0.3$.

The primary actuators used to achieve these values for $\lambda_{q}$, $\lambda_{n}$, and $P_{\mathrm{in-out}}$ are the particle and electron/ion heat diffusivities, $D$ and $\chi_{e,i}$ respectively. As a result of difficulty in measuring the ion temperature, it is hard to constrain $T_{i}$, although it is known that ion to electron temperature ratio, $\tau_{i}=T_{i}/T_{e}$ can be anywhere between 2 - 6 in the edge [54]. For simplicity, $\chi_{i}$ is chosen equal to $\chi_{e}$, a common assumption made in the SOL [55]. Further, these transport coefficients are applied uniformly in the radial direction, yielding an entirely flat transport profile without a “transport well” characteristic of an H-mode pedestal. Modeling with radially invariant transport profiles in the open field line region is seen to reproduce gradients typical of L-mode confinement for the near-SOL. Though this may miss some features of the far-SOL (e.g. density shoulders, filament transport, etc), the focus of this study is reproducing the expected near-SOL heat flux width properties and power exhaust, so these far-SOL features are not considered.

In order to reach sub-mm values for $\lambda_{q}$ consistent with the $B_{p}$ and $<p>$ scalings, $\chi_{e,i}$ had to be reduced to below 10^-2. It was found that a value of $\chi_{e,i}=7.5\times 10^{-3}$ m²s^-1 was sufficient to achieve $\lambda_{q}\approx 0.76$ mm at the entrance to the outer divertor. In order to yield larger density gradient scale lengths, the particle diffusivity was slightly increased, relative the thermal diffusivities. $D=2.4\times 10^{-2}$ m²s^-1 yielded a $\lambda_{n}$ at the OMP separatrix of 7.1 mm. $D$ also influences $\lambda_{q}$ through the convective term in the heat equation, $\propto D\nabla nT$, so $D$ was first used to match $\lambda_{n}$ and then $\chi_{e,i}$ was adjusted to yield the desired $\lambda_{q}$. Finally, in order to replicate the expected in-out power asymmetry, poloidal variation was introduced in $\chi_{i,e}$. The grid was split along a radial cut connecting the center of the plasma to the X-point, and $\chi_{i,e}$ on the inboard side was dropped further, to $8\times 10^{-4}$ m²s^-1. These transport coefficients were applied in the common flux region (CFR) of both the inboard and outboard side individually. It was found that for numerical stability reasons, the transport coefficients had to be increased in the PFR. Plasma in this region is generally cold and rather tenuous, so differences in transport in this region are not expected to have a large impact on the overall solution. Figure 5 shows the spatial variation of transport coefficients required to reproduce the gradient scale lengths outlined in this section.

Table 5 records the three primary plasma parameters used as inputs for UEDGE, discussed in Sections 3.1 and 3.2. Figure 6 shows the resulting upstream profiles for these parameters as well as the transport coefficients shown in Figure 5. The plasma density, $n$, clearly decays more slowly into the SOL than $T_{e}$ and $T_{i}$, which have small $\lambda_{T}$ consistent with the small $\lambda_{q}$ expected from high $B_{p}$. $T_{i}$ is higher than $T_{e}$ across most of the SOL. At the separatrix, $T_{i}$ falls off more slowly than $T_{e}$, but the gradient scale lengths resemble each other more closely in the far-SOL. The left panel of Figure 7 shows the 2D distribution of $T_{e}$, with a hot core plasma inside the LCFS that cools towards the walls and divertor targets. Regardless, the parallel temperature gradient is weak, an indication of an attached plasma with little power dissipation. In-out differences in $T_{e}$ are not obvious without impurity seeding, but much larger temperatures remain in the outer SOL than in the inner SOL for the seeded case, consistent with detachment beginning in the inner target before the outer. The left panel of Figure 8 shows a very peaked heat flux density profile both at the outer and inner targets. This is also characteristic of an attached divertor, in which the heat flux on the target is dominated by the thermal plasma component. The peak heat flux density for this plasma is $q_{\mathrm{surf}}\approx$ 35 MW/m², significantly above the limit for the divertor’s material tolerances.

To dissipate some of this power, MANTA’s divertor uses impurity seeding, a technique routinely used for removing power to the targets via impurity radiation [56, 57, 58]. This technique is the same used in MANTA’s core to achieve a high radiated fraction. To model extrinsic impurity injection, a fixed-fraction impurity model is used in UEDGE, with neon (Ne) as the radiative species. This model sets the neon fraction of the main ion density, $f_{\mathrm{Ne}}=n_{\mathrm{Ne}}/n$, where $n_{\mathrm{Ne}}$ is the density of Ne. The value for $f_{\mathrm{Ne}}$ was increased until enough power was radiated in the divertor to reduce $q_{\mathrm{surf}}$ within material tolerances. It was found that for $f_{\mathrm{Ne}}$ = 0.13%, $q_{\mathrm{surf}}$ on the more heavily loaded outer target dropped well below 10 MW/m².

The right panels of Figures 7 and 8 show the effects of additional impurity injection on the plasma most clearly. As the impurity fraction is increased, the plasma at the divertor targets, in particular at the inner target, gets colder. The heat flux density at the inner target broadens and drops to below 1 MW/m², indicative of detachment. This is largely a result of the lower power sharing to the inner divertor, driven by reduced turbulence in the inner SOL, and facilitating detachment at that target. Regardless, Figure 8 shows that the injection of impurities does also affect the outer divertor leg. The peak heat flux density on the outer target drops more than four-fold, to a more manageable value, $q_{\mathrm{surf}}$ = 7.8 MW/m².

Figures 9 and 10 highlight some of the details of this power dissipation. Figure 9 shows the maximum temperatures, both $T_{e}$ and $T_{i}$ along each of the divertor legs. Not unsurprisingly, temperatures on the inner leg are lower and decay to within 10 eV at a distance of just over 15 cm from the X-point. At the same distance, temperatures along the outer leg remain above 30 eV. Local flattening of the parallel temperature gradient only occurs right in front of the target ($\sim$49 cm from the X-point along the outer leg). While some difference between $T_{e}$ and $T_{i}$ exists close to the X-point, halfway to the target, temperatures have equilibrated, likely a result of increased collisions as a result of larger plasma density. By the time plasma arrives at each of the targets, however, it is below 5 eV, the typically assumed limit for target sputtering [59]. Figure 10 shows target temperature profiles. At the inner target, both $T_{e}$ and $T_{i}$ are relatively flat, while at the outer target, some structure remains in the profiles. The peak temperature is $\sim$10 - 15 mm from the strike point, indicating a significant reduction in heat flux resulting from plasma convection.

The first scan takes the impurity fraction of neon to be fixed at $f_{Ne}=0.35\%$. This impurity fraction is somewhat higher than that required to just barely meet divertor tolerances as shown in Section 6, which provides considerable margin for the basecase input parameters parameters listed in Table 5. With this fixed, scans in $n_{\mathrm{sep}}$ and $P_{\mathrm{SOL}}$ are performed. While neither of these can be controlled with absolute certainty, they represent fairly good proxies for two of the main control room actuators, fueling and heating, respectively. The scan in density is particularly useful because uncertainty in predictive capabilities for $n_{\mathrm{sep}}$ given a particular fueling scheme renders it vital to scan about a value extrapolated from current experiments. On the other hand, even if predictions for the point value of $n_{\mathrm{sep}}$ are correct, it may still be desirable from a performance point of view to increase the density in an attempt to access larger fusion gain. Core modeling of MANTA shows that for a fixed value of $P_{\mathrm{SOL}}$, MANTA can generate more fusion power simply by increasing the line-averaged density, $\bar{n}$, [1], which might also raise $n_{\mathrm{sep}}$. Varying the density also opens the possibility of a larger upper limit for $P_{\mathrm{SOL}}$ because of the ability to dissipate more power at higher $n_{\mathrm{sep}}$. Furthermore, heating in a burning plasma will be dominated by $P_{\alpha}$, and so fueling is expected to be an even more important actuator in a reactor than $P_{\mathrm{aux}}$. Higher $n_{\mathrm{sep}}$ may then offer a higher margin for error for $P_{\mathrm{SOL}}$, inspiring more confidence in an integrated core-edge solution.

To vary $n_{\mathrm{sep}}$, the density boundary condition at the innermost radial flux surface is varied. At differing values of this boundary condition, the power entering the simulation domain at this same boundary is varied. Robust divertor operation is found for $P_{\mathrm{SOL}}>25$ MW, so long as $n_{\mathrm{sep}}$ remains sufficiently high. Figure 11 shows a scan in $n_{\mathrm{sep}}$ from $0.7-1.2\times 10^{20}$m^-3 and a scan in $P_{\mathrm{SOL}}$ from 25 - 50 MW. Regardless, it is found that past $\sim$40 MW, increases in $n_{\mathrm{sep}}$ do not help considerably with power dissipation. Higher values of $P_{\mathrm{SOL}}$ may require additional impurity seeding, as described below.

A second set of scans explores how MANTA’s operators may need to adjust the level of injected neon in order to maintain acceptable heat fluxes in the divertor in the event that the upstream value for $P_{\mathrm{SOL}}$ varies. Or conversely, it may inform leniency in the allowable level of $P_{\mathrm{SOL}}$ if MANTA can reliably regulate the Ne content in its divertor. Since $n_{\mathrm{sep}}$ remains uncertain, scans are performed at both a lower density and a higher density, $n_{\mathrm{sep}}=0.8\times 10^{20}$m^-3 and $n_{\mathrm{sep}}=1.1\times 10^{20}$m^-3. These correspond to separatrix Greenwald fractions, $n_{\mathrm{sep}}/n_{G}$, of 0.36 and 0.5 respectively. Taking $\bar{n}=2n_{\mathrm{sep}}$ would give overall Greenwald fractions for these two operating points of $f_{G}\approx 0.72$ and $f_{G}\approx 1$ respectively. Of course $f_{G}$ at unity is nominally an unstable operating point. Recent results in DIII-D’s NT campaign, however, show operation even at $f_{G}\approx 2$, even though its edge remained below $n_{G}$, implying greater density peaking than the assumption made here of $\bar{n}=2n_{\mathrm{sep}}$. If MANTA can achieve this level of peaking, it is conceivable that MANTA may also be able to operate at this density.

This high density scan presents a potential, if optimistic, expansion to MANTA’s operational space if robust high density operation in NT is successful.

These scans vary $P_{\mathrm{SOL}}$ between $25-50$ MW and $f_{\mathrm{Ne}}$ between $0-1\%$. As with the previous set, they demonstrate a very flexible operational space for MANTA’s divertor. The left panel of Figure 12 shows that even at low density, it remains within 10 MW/m² for $P_{\mathrm{SOL}}$ up to 50 MW with $f_{\mathrm{Ne}}<1\%$. The $q_{\mathrm{surf}}$ line is linear throughout most of the scan and perhaps becomes quadratic only at high $P_{\mathrm{SOL}}$. It is a good rule of thumb then, that the amount of additional impurity needed should scale as the amount of additional power crossing the separatrix.

At high density, shown in the panel on the right of Figure 12 the operational space is somewhat reduced. This is a result of the imposition that the state of detachment remain the same as in the standard operating scenario, i.e. inner leg detached, outer leg attached. Whether or not this might be an actual requirement of this plasma or is a physical result is outside the scope of this paper. In some sense, the fact that partially detached solutions exist only at fairly moderate impurity fractions is a testament to the relative ease with which an NT plasma may detach. If full detachment does not compromise core performance, then this regime might be easily reached. Regardless, even with the requirement of partial detachment it is found that very little impurity seeding is required. At $P_{\mathrm{net}}$ up to $\sim$35 MW, $f_{\mathrm{Ne}}<0.2$ is sufficient to keep $q_{\mathrm{surf}}<10$ MW/m².