A brief history of negative triangularity tokamak plasmas

Abstract

This review is devoted to tokamak plasmas with a cross sectional shape featuring negative triangularity, which appear to hold great promise as a candidate reactor configuration owing to their improved confinement. A brief historical perspective of its role in the worldwide magnetic fusion program is offered before reviewing theoretical predictions and experimental results on both magneto-hydrodynamic stability and turbulent transport. The material covers more prominently the confined plasma region, while limited work in the published literature is devoted to the scrape-off layer and plasma-wall interactions. A discussion on the suitability of this plasma shape in future reactors concludes the paper.

1 Introduction

The tokamak is presently the leading candidate for an economically viable magnetically confined nuclear fusion reactor. Tokamaks, conceptualized in the 1950s by Soviet physicists I. Tamm and A. Sakharov Shafranov (2001), make use of an intense magnetic field and an electric current to confine plasmas in the shape of a torus. The spatial coordinate around the axis of symmetry of the torus is called toroidal, while the plane orthogonal to the toroidal direction is referred to as the poloidal plane. The equilibrium, a cartoon of which is displayed in Fig. 1, is composed of nested surfaces over which a number of quantities, viz. pressure and current density, are approximately constant; the innermost surface degenerates into a toroidal line and is referred to as magnetic axis Kruskal and Kulsrud (1958). For any given flux surface, the radial distance of its centroid from the axis of symmetry of the device is called major radius, while its half-width on the mid-plane is referred to as minor radius.

Fig. 1

Cartoon illustrating the equilibrium of a plasma in a tokamak configuration. The toroidal and poloidal components of the confining magnetic field are denoted as $B_t$ and $B_p$, respectively, while the magnetic axis is represented by a dashed line. For illustration purposes, external coils are reduced to four toroidal field coils. Reproduced from F. Wagner, EPJ Web of Conferences 54, 01007 (2013)

Many of the early tokamaks, starting with the first-ever-built device T-1, employed a near-circular poloidal cross section, which was the natural choice one would make when folding the cylindrical plasma employed in linear devices into a torus. The fusion community began exploring non-circular cross sections in the late 1960s to early 1970s with the goal of either increasing the maximum limit to the current density Solovev et al. (1969); Laval et al. (1971); Coppi et al. (1972); Artsimovich and Shafranov (1972) above the Kruskal–Shafranov limit Shafranov and Sov. (1970), or to find configurations that would allow the confinement time to exceed the Bohm scaling Ohkawa and Voorhies (1969). It is instructive for the reader not familiar with plasma shaping to visualize a number of poloidal cross sections, some of which will be often referred to in the following, in Fig. 2.

Fig. 2

A few examples of poloidal cross sections used in tokamaks, in arbitrary units, with the axis of symmetry of the torus on the left of each figure. (a) circular, (b) elongated, (c) oblated, (d) dee or positive triangularity, (e) reverse-dee or negative triangularity, (f) bean

The highest quality of confinement steadily achievable in tokamaks is contingent on operation in scenarios free from the most virulent Magneto-Hydro-Dynamic (MHD) instabilities. As noted by H.P. Furth Furth (1986), the radial profile of the electron temperature is determined both by turbulence through the heat transport equation and by MHD stability via the shape of the current density radial profile entering Ohm’s law. The overall electron heat diffusivity can then be modelled as the sum of the MHD induced macroscopic contribution, $\chi _{mhd}$, and that due to microscopic fluctuations, $\chi _{turb}$, which generally dominates in the outer part of the poloidal cross section. In the late 1970s-early 1980s, while MHD theory matured to a level sufficient to quantitatively estimate the linear stability for given radial profiles of plasma pressure and current density, the theory of turbulent transport was still very preliminary. In particular, the observation that radial temperature profiles tend to fall within a narrow range due to the onset of strong turbulence driven transport beyond a given threshold had just been formulated as the Principle of Profile Consistency Coppi (1980), nowadays known as plasma stiffness, but no quantitative prediction could be made at that time.

It has to be noted that, even to this day, it is generally harder to predict anomalous transport than MHD stability due to the significantly larger computational resources needed. It was apparent, therefore, that a viable way to improve the overall plasma confinement was to minimize the core diffusivity via an optimization of the MHD configuration; additional improvements could be obtained if one was able to reduce turbulence near the plasma edge, thereby creating an insulating layer. The basic idea that guided the MHD optimization principle was to deform the poloidal cross section in a way as to allow one to increase the pressure content of a given plasma while maintaining MHD-quiescent operation. In particular, it was discovered that ballooning modes, a localized pressure driven instability, can be stabilized at higher pressure than that at which they first become unstable Mercier (1978); Lortz and Nührenberg (1978); Coppi et al. (1980): a phenomenon known as second stability. This resulted in the adoption of dee-shaped cross sections, also commonly referred to as positive triangularity (PT) shape; bean-shaped configurations Bell et al. (1990) were also considered (positive triangularity+indentation to provide 2nd ballooning stability conditions) before being later abandoned as their confinement properties proved to be inferior to those of dee shaped plasmas. Cross sectional shapes characterized by a reversed-dee configuration, also known as negative triangularity (NT), were examined and quickly dismissed on grounds of poor MHD stability.¹ Plasmas at positive triangularity were also attractive on technological grounds because, according to the Princeton D-coil design File et al. (1971), toroidal field (TF) coils are subject to pure tension from $J\wedge B$ forces when they are shaped as a dee: the coil is essentially the analogous to the catenary formed by a hanging chain for currents in a magnetic field whose strength is inversely proportional to the distance from the axis of symmetry of the torus.² The ability of the dee shape to sustain a stable plasma to higher performance was reported by numerous devices worldwide Troyon and A, W.A. Cooper, F. Yasseen, A. Turnbull, (1988); Neyatani (1996); Taylor (1997); Hender et al. (2007) thus providing overwhelmingly convincing validation of the MHD theory. The discovery of the H-mode in the early 1980s Wagner et al. (1982) provided a robust way of creating the edge insulating layer dubbed Edge Transport Barrier (ETB), where turbulence is stabilized by flow-shear Biglari et al. (1990); Burrell (1997), thereby paving the way to the achievement of the conditions necessary for self-sustained nuclear fusion.

In H-mode plasmas the confinement level is strongly correlated to the height of the edge pedestal which, as successfully explained by the theory of peeling-ballooning modes Wilson et al. (2002), increases with positive triangularity. Although it is beneficial to maximize the core confinement to reduce the capital cost of a power plant for a given power output, other constraints apply. More specifically, the main operational challenge intrinsic to the H-mode regime is that the power flow crossing the plasma edge must remain above the L$\rightarrow$H power threshold for the high confinement state to persist. However, parameters in future reactors are such that the power density that will be convected to the vessel will travel along a very narrow channel and will thus be much too large for Plasma Facing Components (PFC) to withstand. Therefore, intense research is being conducted worldwide to obtain stable H-mode regimes in which most of the power is radiated by impurities seeded near the plasma edge, thereby shielding PFCs in a configuration called detachment Krasheninnikov and Kukushkin (2017), in which a large fraction of the power crossing the plasma edge is radiated by seeded impurities Soukhanovskii (2017). This strategy is, however, complicated by the fact that impurities are advected inside the plasma through the main ion density gradient in the pedestal, thereby affecting its stability. The L$\rightarrow$H power threshold, therefore, gives rise to conflicting requirements dictated by the core and the edge of the plasma, which has recently resulted in a new line of research known as core-edge integration. In addition, detached regimes may not be easily controlled experimentally as their operating point is a sensitive function of parameters giving rise to transitions between attached and partially detached conditions known as detachment cliff McLean et al. (2015). It has been recently predicted that the detachment cliff is made more severe by steepening radial gradients near the separatrix, due to enhanced poloidally driven plasma flows, which typically happen in the H-mode regime Du et al. (2020). As a consequence, if reactors were to be maintained in a detached state, it would be easier to control the plasma if it was operated in the L-mode regime, as opposed to H-mode, provided that the global confinement was sufficient to sustain the desired pressure level.

Pedestals, due to their low turbulent levels, also give rise to other issues hampering the H-mode regime. The extremely low transport coefficients characterizing ETBs are such that, at fixed values of energy and particle fluxes, pedestals develop radial pressure gradients large enough to trigger bursting instabilities known as Edge Localized Modes (ELMs). Large and uncontrolled ELMs cannot be tolerated in future reactors because, in the absence of a large expansion of the heat flux footprint on the wall, the energy and particle fluxes they convect to the wall are projected to significantly decrease the lifetime of PFCs  Loarte et al. (2014). Therefore, unless stable ELM-free H-modes are found, if reactors are to operate in the standard H-mode regimes ELMs will have to be suppressed or mitigated by active techniques such as Resonant Magnetic Perturbations (RMPs) or ELM pacing via pellets or periodic vertical movement of the plasma column Lang et al. (2013); whereas, for obvious safety reasons, the use of passive techniques would be much preferred in reactors. Finally, the high particle confinement that characterizes the H-mode regime causes significant impurity retention, thereby lowering fusion performance due to excessive dilution of the main ion species.

All these issues have recently been made part of the fusion community question whether the H-mode regime is the optimal or, in the worst case scenario, even a viable candidate for operation in future fusion reactors, although it is commonly agreed that H-mode levels of confinement are necessary for a viable DEMO reactor. As a result, alternative magnetic configurations are being explored , or revisited, to look for optimal solutions using the more advanced numerical tools that are now available. One of the most prominent of such alternative configurations is the Negative Triangularity which, as will be explained in Sec. 3, modifies the poloidal cross sectional shape from standard to reversed-dee in an effort to improve confinement by reducing $\chi _{turb}$ rather than $\chi _{mhd}$.

This paper reviews past and recent experimental and theoretical work on plasmas with a Negative Triangularity shape: the MHD stability properties of this configuration are discussed in Sec. 2 while transport properties are the subject of Sec. 3; fast-ion and exhaust physics aspects are reviewed in Sec. 4 and Sec. 5, respectively, conclusions and future perspectives are offered at the end.

2 Magneto-hydro-dynamic stability

Most of the negative triangularity stability analyses performed until recently relate to the predictions of ideal MHD, which will be the focus of this Section. Ideal MHD is relevant to predicting the overall operational limit in terms of vertical stability for control and in terms of local and global modes for $\beta$-limits, i.e. the largest pressure the plasma can stably sustain for a given confining magnetic field. The paper by Medvedev Medvedev et al. (2015) is actually an excellent review of the expected ideal limits of a negative triangularity tokamak and we shall follow in large part this work for the recent results.

We first postulate an equilibrium with the plasma boundary typically defined as follows when designing new tokamaks Turnbull et al. (1988); Sauter (2016):

$$\begin{aligned} \begin{array}{r@{}l} R(r,\theta ) &{}{}= R_0(r) + r\cos \{\theta + \delta (r)\,\sin (\theta ) - \lambda (r)\,\sin (2\theta )\}\\ Z(r,\theta ) &{}{}= Z_0(r) + \kappa (r)\,r\,\sin (\theta ), \end{array} \end{aligned}$$

(1)

where $(R_0,Z_0)$ are the coordinates of the centroid of the surface, r is the half-width of the surface at the elevation of the centroid and $\theta$ is the angle around the poloidal direction, while $\kappa$, $\delta$ and $\lambda$ are shaping coefficients that describe the elongation, triangularity and squareness of the surface, respectively; higher order shaping coefficients can be used to further generalize the description. Other types of representations, such as Fourier expansion in the poloidal angle, are sometimes used depending on the application considered. From the usual definition of triangularity Sauter (2016), $\delta$ should be replaced by $\arcsin [\delta ]$ in Eq.1 as is performed in Miller’s equilibrium Miller et al. (1998), however this leads to less than 10$\%$ differences for the typical values $|\delta |<0.6$ used so far in experiments: the simplification $\arcsin (\delta )\simeq \delta$ will thus be adopted. A first question regarding a target equilibrium is the engineering difficulty and complexity of the toroidal-field and shaping coils. The design of a DEMO negative triangularity tokamak (NTT) has been actively pursued in recent years, thanks to the work of Kikuchi et al. (2019, 2015, 2014, March 2014). A positive triangularity (PT) naturally leads to TF coils with a curved outer shape which is best to reduce stresses. On the other hand, negative triangularity (NT) requires an over-sized design or a long straight TF portion at the LFS, hence potentially increased local curvature at the top/bottom LFS corners and reduced life cycle. This is in part why the early NTT designs used a large number of TF coils Kikuchi et al. (2019), however ongoing developments indicate that this may not be required. The remainder of this section describes the stability properties of axisymmetric ($n=0$) and non-axisymmetric ($n>0$) toroidal mode numbers in plasmas at $\delta <0$ and how they generally compare to more familiar configurations at $\delta >0$ .

2.1 The $n=0$ ideal stability

The next question relates to the control of such NT plasmas and in particular to the vertical growth rate of the ideal $n=0$ mode, with and without wall stabilization. It has been shown Medvedev et al. (2015) that a DEMO sized NTT can have $n=0$ growth rates in a range that can be controlled with modern feedback control systems. However Medvedev et al. (2015) also showed that NT plasmas have typically higher growth rates than PT and that a double-null up-down symmetric DEMO-NT has even higher growth rates. Vertical stability was first analyzed using a rigid displacement model, before full numerical codes were developed at the end of the 70’s-beginning of 80’s. Around the same time, non-circular shape tokamak plasmas were studied including positive and negative triangularity, also called “triangularization” or normal/standard and reversed/inverse-dee (D) shapes. For example the original work of Rebhan Rebhan (1975); Rebhan and Salat (1976, 1977, 1978), computing rigid displacement stability, analyzed elliptic and triangular Solovev-type equilibria Solov’ev (1968) ($p'=cst$ and $TT'=cst$). Rebhan scanned the parameter Q of the Solovev equilibria which essentially determines the triangular deformation leading to elliptical shapes for $Q\approx 0.5$ (for aspect ratio $A=R_0/a=3$), PT for $Q\approx 0$ or smaller and NT for $Q>1$. This work also showed that elongation is the main limiting factor, through vertical displacements, for the stability of axisymmetric equilibria and first found that “small aspect ratio and strong triangular deformation (both positive and negative) are favourable” Rebhan (1975), assuming rigid vertical displacements.

A subsequent work Rebhan and Salat (1977) found that “slip” modes ($m=1, n=0$) were destabilized by triangularity and that feedback control was necessary Rebhan and Salat (1978). The study was extended in Bernard et al. (1978), using the ERATO code Gruber (1978), and in Chu and Miller (1978), using AXISYM, and it was shown that triangularity is destabilizing for $n=0$ modes (Fig. 2 of Bernard et al. (1978) and Fig. 3 of Chu and Miller (1978)) when considering general $n=0$ deformations. This was also found in Yamazaki et al. (1983) using the PEST code Grimm et al. (1976). PEST calculations were also shown to compare well with experimental results of the Tokapole II experiments, with an increased $n=0$ growth rate with increasing $|\delta -0.1|$ Lipschultz et al. (1980).

The first experimental results on NT plasmas had shown that reverse-dee shapes are more unstable than “square” ($\delta =0$) shapes Lipschultz et al. (1979). Later, experimental results in PDX showed that NT $n=0$ growth rates were much larger (about three times) than for PT, but could be feedback controlled, confirming also the role of the magnetic decay n-index Takahashi et al. (1982). The difference between the rigid displacement model and full ideal MHD is particularly true for $Q<0$ and $Q>0.8$, that is strong PT or NT, including when a conducting wall is considered Bernard et al. (1978).

This also shows that present studies of NT should be careful when using a rigid displacement model for feedback control design in particular. The eigenfunction approach also explains well the differences between plasma shapes Bernard et al. (1978). Elliptical plasmas feature an essentially vertical displacement ($n=0$), PT have a maximum amplitude along the HFS straight side while NT plasmas tend to move preferentially towards the strong curvature of the plasma surface, towards the X-point(s). This may explain in part why diverted NT plasmas have a larger $n=0$ growth rate than PT, and why a PT wall can be stabilizing for NT plasmas if the wall can be close to the NT “X-point(s)” (see e.g. early studies like in Akkermans (1982)) .

Plasma pressure can also influence vertical stability and it was first shown in Rebhan and Salat (1976) that PT was more favourable than NT in this respect. This was extended using NOVA-W Ward et al. (1993) in Cheng and Chance (1987); Ward et al. (1993) were a strong destabilization of NT plasmas with $\beta _p$ (ratio of the kinetic pressure to that of the poloidal magnetic field, volume averaged, $\beta _P\equiv 2\mu _0\langle P\rangle _V/\langle B_p^2\rangle _V$) was found despite the presence of a resistive conformal wall, contrary to PT (Fig. 4 in Ward et al. (1993)). This was linked to the Shafranov shift increasing with plasma pressure and squeezing flux surfaces near the LFS nose of PT shapes, effectively reducing elongation in the core. On the contrary, elongation is high in the core of NT plasmas (as shown in Fig. 3(e)) and can increase with $\beta _p$, leading to strong destabilization of the $n=0$ mode Ward et al. (1993). Note that this effect is important for other modes as well and for the magnetic well, as will be discussed below, and also for turbulence transport (Sec. 3).

Fig. 3

(a) Example TCV PT and NT shapes, with R flipped around $R_{geom}$. (b) to (f) using H-mode-like profiles (inspired by PT-H-mode profiles): $Istar=<j_\phi /R>/<1/R>$, q, magnetic shear, elongation and normalized Shafranov shift, $\Delta /a=(R_0(r)-R_0(a))/a$

In the more recent studies, motivated by the more clear confinement improvement observed in TCV L-mode plasmas, as described in Sec. 3, KINX Degtyarev et al. (1997) with a resistive wall was used to compare vertical growth rates between top/bottom positive and negative triangularity Medvedev et al. (2008). It was found that already a $\delta _{top}<0$ with $\delta _{bottom}>0$ yields much higher $\gamma _{n=0}$ ($\sim 300~s^{-1}$ vs $150~s^{-1}$ for $\delta _{top}>0$) and even more so with $\delta _{top}<0$ and $\delta _{bottom}<0$ ($\gamma _{n=0}\approx 1500~s^{-1}$). It was also found that $\gamma _{n=0}(NT)$ was very sensitive to the distance to the LFS wall, as well as to holes in the wall due to ports Medvedev et al. (2009). NT $n=0$ modes were also shown to be more sensitive to elongation Medvedev et al. (2009). Optimization can reduce the growth rates to controllable values but in some cases a LFS distance of less than $0.1a-0.2a$ ($\sim 2-5$cm in TCV) is required Medvedev et al. (2015). For example a double-null with strong $\delta <0$ was shown to be compatible with the TCV PF-coil limit and to feature a controllable $\gamma _{n=0}\sim 500~s^{-1}$ with a small LFS gap Medvedev et al. (2014). In Abate et al. (2020), $\delta <0$ plasmas in RFX-mod2 are considered and compared with PT DEMO-like shapes. They find a similar vertical growth rate, within controllable range of their control system and mainly sensitive to elongation. In this case, similar growth rates are found for single- and double-null. This shows that global trends are recovered and can be reviewed here. However, specifics of the assumed current density profiles and of the machine layout are important for quantitative analyses. Let us also mention another very recent study, using the DINA code Khayrutdinov and Lukash (1993), Xue et al. (2019) that analyzed hot vertical displacement events (VDE) based on HL-2M plasmas. They also find a higher growth rate for $\delta <0$ , with a faster VDE evolution and higher Halo currents in the LFS vacuum vessel, requiring special engineering attention, in particular near the X-point(s) Xue et al. (2019).

2.2 The stability of $n>0$ modes

Once we have equilibria that can be feedback controlled and kept vertically stable, the main operating limits are determined by $n>0$ ideal modes, and mainly by the global $n=1$ external kink. We will also discuss internal kink and local modes up to $n=\infty$ ballooning modes Connor et al. (1978), in particular related to edge stability limits in H-mode like plasmas. In all of these cases, it was recognized early on that NT plasmas are more unstable because of the absence of a magnetic well (favourable average curvature); which is why PT has been the main line of research for tokamaks from the mid-1980’s. However, viable solutions for demonstration fusion power reactors are projected to exist even at $\beta \simeq 3$ Federici et al. (2014); Kessel and Poli (2015); Sorbom et al. (2015) which, as it will be explained in Sec. 3, is a value that has already been achieved in $\delta <0$ plasmas. As such, the MHD stability of configurations at $\delta <0$ is deemed to be sufficient for a reactor, although the $\beta$ limit is still believed to be lower than the maximum value achievable in $\delta >0$ plasmas. The magnetic well is important for edge localized modes and the peeling-ballooning stability limit Connor et al. (1998). In Ohkawa (1969) analytical equilibria were obtained with a deep magnetic well and high shear with vertically elongated PT as well as with horizontally elongated NT ($\kappa <1$, comet shape). It was confirmed in Yamazaki (1980) that non-circular plasmas have a higher ballooning-$\beta$ limit with either PT and $\kappa >1$ or NT and $\kappa <1$. As discussed in Connor et al. (1998) the existence of a magnetic well ($d_M<0$) can be related to the ideal Mercier term Mercier (1960)

$$\begin{aligned} \begin{array}{r@{}l} d_M&{}= \frac{s^2}{\alpha }\;D_M, \end{array} \end{aligned}$$

(2)

with $D_M$ the ideal Mercier term (criteria: stable if $D_M<1/4$ or also $-D_I=-D_M+1/4\,>\,0$), s the magnetic shear and $\alpha$ the ballooning parameter proportional to the pressure gradient Connor et al. (1978). In this context it is useful to use the dominant triangularity and elongation dependence of $D_M$Lutjens et al. (1992). Let us first write the relation obtained by Lutjens et al. (1992) (Eq. (7)) in terms of $\kappa$, $\delta$ and $\Lambda =d\Delta /dr\approx \beta _p+l_i/2$:

$$\begin{aligned} \begin{array}{r@{}l} D_M = -\,\frac{2r\,p'}{s^2\,B_0^2} \biggl \{ 1\,-\,q^2\,+\,\frac{3\,q^2}{4}\; \biggl [ \kappa -1\, + \frac{\kappa 'r}{2} \, (1-2\Lambda ) \\ - \,\frac{(\kappa -1)\,\delta }{3\,\epsilon }(5+\frac{\delta 'r}{\delta } \,+\,\frac{\kappa 'r}{4(\kappa -1)}(9-3\frac{\delta 'r}{\delta }) \biggr ] \biggr \} \end{array} \end{aligned}$$

(3)

Assuming $\kappa ' = 0$ and $(\delta /r)'=0$ (which allows to write $r \delta '=\delta$, we obtain (without assuming $q=1$ as in Lutjens et al. (1992)):

$$\begin{aligned} \begin{array}{r@{}l} D_M \approx -\,\frac{2r\,p'}{s^2\,B_0^2}\;\bigl \{ 1\,-\,q^2\,+\,\frac{3\,q^2\,(\kappa -1)}{4}\,(1\,-\,\frac{2\,\delta }{\epsilon }) \bigr \}, \end{array} \end{aligned}$$

(4)

with r an equivalent minor radius and $'=d/dr$. It should be noted that $\delta /\epsilon$ is about $2\delta _a$ (cst) up to mid-radius and increases to $4\delta _a$ at the edge, with $\delta _a=\delta (LCFS)$ and for TCV aspect ratio, thus it is not small over the whole radius. Note that early analytical expressions obtained by expanding with respect to elongation and triangularity led to analysis showing favourable dependence on negative triangularity Galvão (1975). We see from Eq. 4 that $D_M$ changes sign if triangularity is $\lesssim -0.2$ and $\kappa >1$ or if $\kappa <1$.

As will appear within this short review, $D_M$ is related to the stability of the peeling-ballooning modes Connor et al. (1998); Wilson and Miller (1999), of the local modes in general but also of the $n=1$ internal kink Lutjens et al. (1992); Reimerdes et al. (2000) and of the global modes through the link with the magnetic well. This explains why, after the early studies up to $\sim$1980, two main types of plasma shapes were studied: D-shaped PT (most MHD stability research from mid 1980s) and comet-shape NT with $\kappa <1$ (e.g: Ohkawa et al. (1989); Yamazaki (1980), as well as Wootton (1978) for “horizontal” stability), directly linked to the main dependence of $D_M$, Eq. (4). The work of Peng et al. (1978) showed that the form of the pressure profile also plays a role in the Mercier stability, in addition to PT. The overall proportionality of plasma $\beta$ with plasma current $I_p$Troyon et al. (1984); Troyon and Gruber (1985), yielding the normalized pressure performance $\beta _N=\beta [\%]/(I_p[MA]/a[m]B_0[T])$, led naturally to the focusing on D-shape plasmas with high elongation and high current, and in large part to the design of the TCV tokamak in particular Hofmann et al. (1986); Turnbull et al. (1988). On the other hand, recent studies of DEMO-like NTT plasmas Medvedev et al. (2015) show that $\beta _N\sim 3$ can be stable and these values have also been reached experimentally in DIII-D experiments Austin et al. (2019); Marinoni et al. (2019). This is related to the difference between the Mercier-Ballooning pressure profile optimization and the limit related to low-n external kink modes. Mercier modes are easily stabilized by non-ideal effects Porcelli and Rosenbluth (1998), so are not considered for stability boundaries (although $D_M$ influences the stability and global extent of low and high n modes as mentioned above Lutjens et al. (1992); Connor et al. (1998); Medvedev et al. (2015)). The optimized pressure profile obtained with $p'$ near the ballooning limit is usually at significantly higher values than the ones limited by the $n=1$ external kink. This is why the effective operating boundary is obtained from the $n=1$ stability limit, see e.g. Turnbull et al. (1988) for the TCV design, especially for PT D-shaped plasmas. The reduced ballooning stability of NT plasmas mainly reduces the difference with the $n=1$ limit but not necessarily reduce the operating boundary significantly. On the other hand, the absence of a magnetic well influences the access to the 2nd ballooning stability region at low shear, high $\alpha$ which is important for ELM physics Wilson and Miller (1999).

2.2.1 Ballooning modes

The ballooning stability limit ($n=\infty$ modes) Connor et al. (1978) and related high n modes have been extensively studied and we shall not attempt to review these results here. We will focus on the studies related to negative triangularity. Present-day tokamak studies focus on the access to the 2nd stability region and its effect on the edge stability of H-mode plasmas with large edge pressure gradient and finite edge current density Wilson and Miller (1999); Connor et al. (1998). This access was shown to exist with a sufficiently deep magnetic well ($d_M$ sufficiently negative, $d_M\lesssim -0.6$) Wilson and Miller (1999).

Before reviewing early studies on the effect of shape on ballooning modes, it is instructive to look at the profiles of the ideal Mercier $D_M$ and the magnetic well $d_M$ for two examples, inspired by TCV experiments, with a positive and negative triangularity SND and a current density profile with an edge bootstrap current density and large edge pressure gradients typical of H-modes. The two shapes are shown in Fig. 3(a), with $|\delta |\approx 0.5$ and the LCFS chosen symmetric in R around $R=R_0(LCFS)$. The equilibria have been computed with the CHEASE code Lütjens et al. (1996) with $q_0=0.95, \beta _N\approx 1.6$ and the same H-mode like pressure and $I*=<j_\phi /R>/<1/R>$ profiles. The latter is shown in the top panel of 3(b). The resulting q profiles (c) show the typical difference with $q_{NT}<q_{PT}$ according to the value of triangularity (Eq. (36) Sauter (2016)). The (average) magnetic shear s (d) is not very different, in particular the dip towards $s\approx 0$ is also present in the NT case. Note a stronger increase in shear near the LCFS for the PT case. We show also the profiles of elongation (e) and Shafranov shift (f) with differences between PT and NT which are typical as mentioned above for the $n=0$ stability and the role of elongation. Increasing $\beta$ pushes the flux surfaces to the low field side. With PT, this requires that the flux surfaces be compressed vertically, which explains why the elongation on axis is smaller in PT than in NT, as is the Shafranov shift ($\Delta /a<5\%$ vs $>10\%$ in NT). To show that this is a generic feature of NT’s, we also plot the profiles obtained with zero pressure in NT (dashed red line). We see that the Shafranov shift is still much larger than the high $\beta$ PT case. In addition the elongation remains high up to the magnetic axis. As mentioned above, this has an effect on the $n=0$ growth rate and mode structure. It also increases $D_M\propto \kappa -1$ and leads to more unstable intermediate n internal modes Medvedev et al. (2015) and $n=1$ internal kink Lutjens et al. (1992); Reimerdes et al. (2000); Eriksson and Wahlberg (2002); Martynov et al. (2005); Graves et al. (2005). On the other hand, increased elongation is favourable for reducing radial turbulent transport and could play a role in the global confinement improvement of NT, as well as the larger Shafranov shift (see Sec. 3).

Using these two equilibria, CHEASE also computes the Mercier term $D_M$, plotted in Fig. 4 (top panel) along with the magnetic well term $d_M$ (bottom panel). It clearly shows that $D_M(PT) < D_M(NT)$ everywhere, yielding the wide unstable core region with $D_M>1/4$ related to the destabilization of the internal kink (see below). We also clearly see the opposite peak near $\rho _\psi \sim 0.9$ in the dip of the magnetic shear. $D_M$ is positive, destabilizing, in NT and negative, stabilizing, in PT. This leads to an overall positive $d_M$, unfavourable average curvature, for NT and the presence of a magnetic well reaching the “deep well” values $d_M<-0.6$ at the edge in PT. The simplified analytic formula, Eq. (4), assumes $\kappa '=0$ and $(\delta /r)'=0$, which is mainly valid in the core and for internal kink studies, but not near the edge. To check the direct effect of triangularity, we computed the full analytic formula given in Eq. (3) which is shown as dashed lines in Fig. 4(b), which follow well the exact calculations (solid lines, using Eq. 19 of Lütjens et al. (1996)). We also show the result using the NT parameters inside Eq. (3), but reversing the sign of the terms proportional to $\delta$. This is the dotted red line in Fig. 4(b) which uses the same parameters as the red dashed line (NT case) but with $(-\delta )$ instead of $(\delta )$ in the analytic formula. We see that it yields a profile very close to $d_M(PT)$, demonstrating that the flip from positive to negative $d_M$ is only due to the sign of $\delta$ over the whole radius. Note that the term related to the Shafranov shift in Eq. (3) is not significant because of the factor cancelling the $\kappa -1$ term leaving only the small contribution from $\kappa '$.

Fig. 4

(a) Mercier term $D_M$ Lutjens et al. (1992), (b) magnetic-well term $d_M$

The bad curvature on the LFS is not the only drive for ballooning modes. It was shown in Greene and Chance (1981) that the local shear also plays an important role. A low or negative local shear is stabilizing Manickam (1986). This was confirmed in a series of studies Bishop et al. (1984); Bishop (1986, 1987) where the ballooning stability near the separatrix was calculated analytically in the case of a divertor, or more precisely of a “bulge” on the LCFS with a much stronger local radial curvature as near the X-point of a single-null (SN) diverted shape. In Greene and Chance (1981) it is shown that the first stability region for ballooning modes is influenced mainly by the length along the flux surface with bad curvature. However the 2nd stability region is obtained if the local shear is small or negative over the whole region of bad curvature. The work in Bishop et al. (1984); Bishop (1986, 1987) showed that this can be obtained if the bulge is on the inside (HFS) of the LCFS ($R_{bulge}<R_0$, i.e. $\theta _{bulge}$ in the 2nd or 3rd quadrant). However, when the X-point is on the LFS equatorial plane ($\theta _{bulge}=0$), then the negative local shear cannot extend to the good curvature region and no access to the 2nd stability region is found, even with finite edge current. It was proposed Bishop et al. (1984); Bishop (1986, 1987) that this was the main cause of the difficulty experienced by JT-60 in achieving H-mode JT-60 TEAM (1987), since it had such a LFS X-point on the outer mid-plane. These results were confirmed by numerical calculations Roy and Troyon (1988); Roy (1990) which showed that for $|\theta _{bulge}|<3\pi /8\approx 70^o$ there was always a positive local shear region inside the bad curvature region, along the relevant flux surfaces, explaining the lack of 2nd stability access.

Fig. 5

Separation between good vs bad curvature regions and contours of local magnetic shear in (a) PT (b) NT

To demonstrate this important difference between PT and NT, Fig. 5 shows the results from CHEASE of the location of zero curvature (good curvature on the HFS of the blue stars line, bad curvature on the LFS) for the same PT and NT cases shown earlier, as well as the contour of the local shear S. We see that for PT the local maximum of S near the LCFS is in the good curvature region, while it is in the bad curvature region in NT, preventing $S<0$ over the whole bad curvature region of a given flux surface. Note that this is consistent with the difficulty observed in TCV Porte et al. (2020) and DIII-D Marinoni et al. (2020) in entering H-mode in NT plasmas as predicted by Bishop et al. (1984); Bishop (1986, 1987).

Fig. 5 explains why the 2nd stability region cannot be obtained in NT. This is also consistent with the pedestal height decreasing significantly as triangularity is decreased Merle et al. (2017) and with the H-L transition occurring when the top $\delta$ decreases below -0.2 in TCV after the ELMs becoming smaller Pochelon et al. (2012). Note that -0.2 is also consistent with the value of $\delta$ mentioned above below which $d_M$ becomes always positive. This confirms the implications of the results of Merle et al. (2017), showing an order of magnitude smaller pedestal height for $\delta <-0.2$ as compared to $\delta >0.4$ (see Fig. 10 of Merle et al. (2017) and Fig. 3 of Kikuchi et al. (2019) for a normalized form).

Let us mention that another shape was studied as a result of the quest for 2nd ballooning stability effects. This is the bean shape, which can lead to a fully accessible 2nd stability region with sufficient indentation Chance et al. (1983); Grimm et al. (1985); Manickam et al. (1983); Bol et al. (1986); Bell et al. (1990) – the opposite case with respect to NT plasmas, for which the access to 2nd stability is closed instead. The bean shape was also shown to be related to extending the negative local shear outside the bad curvature region. The negative local shear at the LFS is due to the local increase of the poloidal magnetic field, itself required to equilibrate the increased pressure gradient when squeezing flux surfaces to the LFS. Using a bean shape, the idea is to squeeze these surfaces even further, increasing $B_p$ in the outer LFS region and changing the local shear further. This was already discussed in Mercier (1978), when analyzing the local criteria of the 2nd kind. This also showed that the criterion of the first kind (ideal Mercier criterion) is a limiting case of the criterion of the 2nd kind (ballooning mode criterion). Finally, Yamazaki et al. (1985) studied the 1st region of ballooning stability, comparing triangularity, bean shape and squareness. This work found that the $\beta _N$ limit can be compared with the Troyon limit, even with only the 1st stability region, adding a favourable triangularity dependence. It also found that squareness was not favourable.

2.2.2 The $n=1$ ideal internal kink

TCV experiments demonstrated a striking difference in the behaviour of sawteeth to additional central ECH (electron cyclotron heating) Reimerdes et al. (2000); Pochelon et al. (2001, 1999); Holger (2001). For small $\kappa$ and/or large $\delta$, the sawtooth period $\tau _{saw}$ increases with ECH power (solid triangles in Fig. 6), while for large $\kappa$ and/or small/negative $\delta$ $\tau _{saw}$ decreases with increasing central ECH (open triangles in Fig. 6).

Fig. 6

Sawtooth period response to additional central ECH in terms of elongation and triangularity and related marginal ideal Mercier criteria for two $\beta _p$ values at $q=1$ Reimerdes et al. (2000)

We also see that these different behaviors are consistent with the discharges being stable or unstable with respect to the ideal Mercier criterion which depends on both $\kappa$ and $\delta$ as seen in Eq. (4). Another characteristic difference is the fact that the first group (on the stable side of the Mercier term) has peaked pressure profiles in between sawtooth crashes with $\beta _{p1}$ increasing with ECH power (with $\beta _{p1}=(<p>_1-p_1)/(B_p^2/2\mu _0)$), while $\beta _{p1}$ stays small and constant, with flat pressure profiles inside $q=1$, for the 2nd group in the unstable Mercier zone (Fig. 7 of Reimerdes et al. (2000)). This was shown to be consistent with a significant change of the ideal internal kink, using KINX calculations. At small elongation and/or large PT, the ideal internal kink is stable up to $\beta _{p1,crit} >0.3-0.35$ consistent with standard expected values Bussac et al. (1975). However for high $\kappa$ and/or negative $\delta$, the internal kink can be unstable even at $\beta _{p1}=0$, that is p should remain constant inside $q=1$ Reimerdes et al. (2000); Martynov et al. (2005); Martynov (2001). This confirms the link between the Mercier term and the ideal $n=1$ internal kink growth rates Lutjens et al. (1992). This change of behavior is associated with sawteeth being triggered by the ideal trigger condition at low/negative $\delta$ instead of the resistive one Porcelli et al. (1996); Sauter et al. (1999); Martynov et al. (2005). Extending these experiments to more negative $\delta$ values, the sawtooth period was seen to exhibit a minimum at slightly negative $\delta$, consistent with a maximum in the $n=1$ ideal growth rate Martynov et al. (2005); Martynov (2001). This non-monotonic behavior was predicted by the derivation of the dominant dependencies of the potential energy of the ideal internal kink with respect to $\epsilon , \kappa$ and $\delta$ Eriksson and Wahlberg (2002), which shows terms proportional to $\delta$ and $\delta ^2$ Eriksson and Wahlberg (2002); Martynov et al. (2005). The derivation of Eriksson and Wahlberg (2002) also shows explicitly that the Mercier term contributes directly to the internal kink $\delta W$. Using many KINX simulations, Martynov et al. (2005) modified the formula used for the ideal kink growth rate to be used for sawtooth modeling (Eq. (16) of Martynov et al. (2005), for $\epsilon (LCFS)\sim 1/3$):

$$\begin{aligned} \begin{array}{r@{}l} \gamma \,\tau _A &{}{}= 0.44\;\frac{\epsilon _1\,\kappa _1}{1\,+\,7\epsilon _1\,s_1}\;(\beta _{p1}-\beta _{p1}^{c}),\\ \beta _{p1}^{c} &{}{}= 0.9 - (0.6 + 0.1\,s_1)\,\kappa _1, \end{array} \end{aligned}$$

(5)

with the growth rate normalized by the Alfven time $\tau _A$. This shows the significant dependence of $\beta _{p1,crit}$ on elongation and to a lesser extent on shear. An interesting result of these analyses is that the above formula reproduces well the non-monotonic behavior with respect to $\delta$ even without an explicit $\delta$ dependence in the fit. As discussed in Martynov et al. (2005) this is due to the different penetration of elongation with triangular shapes. Therefore the $\kappa _1$ term in Eq. (5) includes the dependence on $\delta$ in a self-consistent equilibrium. It is also noted that the current profile alters $\kappa (\rho )$ as well, which is why $s_1$ enters in $\beta _{p1}^{c}$, although as a smaller effect. It is important in the case of high elongation, positive $\delta$, which naturally leads to low $l_i$ and where sawteeth are observed to disappear Pochelon et al. (2001); Reimerdes et al. (2006) and be replaced by a continuous internal mode  Reimerdes et al. (2006). The sawteeth were already observed to “disappear” or change behavior at NT, leading then to more unstable q profiles with respect to tearing modes Weisen et al. (1997). In Martynov et al. (2005) it was found that the elongation profile $\kappa (\rho )$ changes again at much stronger negative triangularity, contributing to the overall non-monotonic behavior, i.e. $\kappa (q=1)/\kappa (edge)$ has a maximum near $\delta \sim 0$ (yielding a maximum ideal n=1 growth rate at slightly NT, see Fig. 17 of Martynov et al. (2005)). This also shows that full consistent equilibria are required to understand these dependencies, even of an internal mode like the n=1 internal kink, which is nevertheless relatively global. A similar change of sawtooth behavior has been observed between bean and elliptical shapes Lazarus et al. (2006).

We shall discuss later the link between MHD and transport, but let us note here the results of Kirneva et al. (2012) which found a similar degradation of the confinement time with respect to ECH power in positive and negative triangularity plasmas. In the same paper, they compare the effect of sweeping the ECH deposition from off-axis to on-axis in a NT and a PT plasma. In this case the confinement is much higher in NT than in PT, but also the relative increase of the scaling factor when heating is deposited inside $q=1$ is higher for NT. In addition, it shows that the $q=1$ radius is more central in NT than PT plasmas Kirneva et al. (2012). Since the $q=1$ radius can vary when changing the input power, and its radius can be quite central, one has to be careful to not change the relative deposition with respect to the inversion (or near $q=1$) radius, since it can change significantly the plasma thermal energy and bias the transport studies. The difference in confinement between the low shear region just outside $q=1$ in NT and PT has not been studied yet.

2.2.3 Resistive modes

Most of the “old” studies were related to the study of the resistive interchange criterion Glasser et al. (1975), stable if $D_R <0$ with:

$$\begin{aligned} D_R = D_I + (H- \frac{1}{2})^2 = D_M + H^2 - H \end{aligned}$$

(6)

Combining the ideal Mercier (Sec. 2.2) and resistive interchange criteria:

$$\begin{aligned} D_M - \frac{1}{4}< D_R < 0, \end{aligned}$$

(7)

which defines the overall local stability conditions (Ballooning modes have been discussed in Sec. 2.2.1). We see that one needs to be significantly stable to the ideal criterion in order to be stable to the resistive interchange criterion. The resistive interchange is also a local criterion and, if violated, does not necessarily mean that a finite island [neoclassical] tearing mode ([N]TM) is unstable. However it remains a good indication of the local stability. Therefore, since we have seen that $D_M$ is more unstable for NT plasmas (higher than in PT), $D_R$ is also more unstable. We can look at the triangularity dependence as we did for $D_M$ and $d_M$ in Sec. 2.2 using the same two equilibria, as well as the analytical expansion. Lütjens also derived the main shaping dependencies of $D_R$ in Eq. 5.24 of Lütjens (1993) (transposed here in terms of $\kappa$ and $\delta$):

$$\begin{aligned} \begin{array}{r@{}l} D_R = D_M(\text {Eq. 3}) - \,\frac{2r\,p'}{s^2\,B_0^2}\;s\,q^2\, \biggl \{\frac{1}{8}\,(\kappa -1+r\kappa ')\,-\,\Lambda \,[1-\frac{3}{4}(\kappa -1+r\kappa ')] \\ +\,\frac{\kappa -1}{16}\frac{\delta }{\epsilon }\,[3-\frac{r\delta '}{\delta }\,-\,\frac{r\kappa '}{\kappa -1}\,(1+\frac{3\,r\delta '}{\delta })] \biggr \}. \end{array} \end{aligned}$$

(8)

For completeness, if we use the same approximation as for Eq. 4, $\kappa =\text {cst}$ and $\delta /r=\text {cst}$, we get (Eq. 5.25 of Lütjens (1993)):

$$\begin{aligned} \begin{array}{r@{}l} D_R = D_M(\text {Eq. 4}) - \,\frac{2r\,p'}{s^2\,B_0^2}\;s\,q^2\, \biggl [\frac{\kappa -1}{8}\,(1+\frac{\delta }{\epsilon })\,-\,\Lambda \,(1-\frac{3(\kappa -1)}{4}) \biggr ], \end{array} \end{aligned}$$

(9)

which yields, including Eq. 4:

$$\begin{aligned} \begin{array}{r@{}l} D_R = - \,\frac{2r\,p'}{s^2\,B_0^2}\;s\,q^2\,\,\biggl \{ 1\,-\,q^2\,+\,\frac{3\,q^2\,(\kappa -1)}{4}\,\bigl [1\,-\,\frac{2\,\delta }{\epsilon }+\frac{s}{6}(1+\frac{\delta }{\epsilon })-s\Lambda \,(\frac{4}{3(\kappa -1)}-1) \bigr ]) \bigr \}. \end{array} \end{aligned}$$

(10)

We plot the results using CHEASE of $D_R(\rho )$ in Fig. 7, in the same way as for $d_M$ in Fig. 4.

Fig. 7

Resistive interchange parameter $D_R$ and its analytical approximation (Eq. 6 from Eq. 5.24 of Lütjens (1993))

As expected, $D_R(PT)$ is stable while NT is not (in the core and near the edge). It also shows that the analytic expression, Eq. 8, reproduces quite well the exact evaluation (Eq. 20 of Lütjens et al. (1996)). We also show (dotted red line) the result of using the NT equilibrium with Eq. 8 but where the sign in front of the $\delta$ only terms have been changed. As in the case of $D_M$ and $d_M$, the sign of $D_R$ is changed from positive to negative (unstable in NT to stable in PT). In particular the “bump” near the edge, where there is a large pressure gradient and a low shear, changes sign. Thus the absence of a magnetic well also results in an unstable resistive interchange situation. Note that the dotted red line in Fig. 7 over-estimates the equivalent blue dashed line $D_R^{anal}(PT)$, contrary to Fig. 4, bottom panel. This is because the term proportional to $\Lambda$ (independent of $\delta$) has the $(\kappa -1)/2$ term being cancelled for $D_M$, Eq. 4, while it adds up for $D_R$ leading to a much larger contribution of the Shafranov shift term: $\Lambda \approx \beta _p+l_i/2$. This also means that NT can become more unstable with increasing $\beta _p$, contrary to our standard expectations from PT.

Experimentally, it is not clear yet if there is a fundamental difference between NT and PT for tearing mode stability, or if the latter is indirectly related to the electron temperature profile and whether sawtooth activity can be sustained or not. Early NT experiments already pointed to the appearance of tearing modes when sawteeth disappeared Weisen et al. (1997). The only study computing the dependence of the classical tearing mode index $\Delta '$ on negative triangularity that we are aware of is in Ref. Atanasiu et al. (2001), which shows essentially a quadratic dependence $\Delta '\sim -\delta ^2$ (see Atanasiu et al. (2004) for more details on the model and the triangularity dependence). Clearly more systematic experimental and theoretical studies on the tearing mode stability in NT are needed.

2.2.4 Global MHD limit

Several theoretical and experimental studies, including triangularity dependence, look at the various aspects of plasma stability, from $n=0$ to high n, with the aim of determining the global stability boundaries of a given tokamak shape. We shall review them in this Section with the focus on the effective “$\beta$”-limit, that is the lowest $\beta$-limit determined by ideal MHD $n\ge 1$ modes, and ideal modes which can limit fusion performance. As mentioned earlier this usually means the $\beta$-limit defined by the external $n=1$ kink, where “internal” modes mean modes still unstable with an ideal wall on the plasma surface and “external” modes, those fully stabilized by a wall on the LCFS. Note that internal modes can still have a finite edge displacement for the most unstable eigenfunction, like the $n=1$ internal kink, but the growth rate is only decreased (typically a factor of two for the internal kink) when a wall is added on the LCFS. Related to these definitions, the “no-wall” stability limit, is the $\beta$-limit obtained when the ideal wall is placed very far, practically at more than 10 times the plasma minor radius (in all directions).

Recent experimental results from DIII-D show that $\beta _N\sim 2.7$ can be sustained in NT plasmas Austin et al. (2019); Marinoni et al. (2019) without tearing activity. Within the power available to the DIII-D experiment, the ideal limit has not been reached. Numerical calculations, carried out with the GATO Bernard et al. (1981) and DCONGlasser (2016) codes, based on a DIII-D discharge estimate $\beta _N\sim 3.1$ as the ideal limit without profile optimization Austin et al. (2018); Marinoni et al. (2019). As discussed earlier, wall stabilization was predicted to increase the $\beta _N$ limit by less than 10%, likely due to the L-mode edge yielding small edge currents. DEMO-size NTT also predict ideal limits near or above 3 Medvedev et al. (2015) for inductive type profiles. With reverse shear and bootstrap-aligned current density profile, it is shown in Medvedev et al. (2015) that the magnetic well can be partially recovered, however low shear leads to a lower global low n limit of about $\beta _N\sim 2.1$. Note that PT steady-state plasmas also have a lower $\beta _N$-limit than inductive-type monotonic profiles, which is why they rely on wall stabilization. This has not yet been studied in detail for NT shapes. We mentioned the pioneering Asian DEMO-NTT studies described in Kikuchi et al. (2019) and refs therein. The EU-DEMO team now also studies NT plasmas, following the conclusion that only a no-ELM regime can be compatible with DEMO requirement Siccinio et al. (2020). In this respect an L-mode edge NT plasma with a confinement time similar to an equivalent PT H-mode plasma would be ideal. According to predictions by Merle et al. (2017) and preliminary experiments on TCV Pochelon et al. (2012), a NT H-mode is characterized by a shallow edge pedestal, resulting in small and frequent ELMs which, in case of transient H-mode transitions, might be tolerated long enough for the control system to safely ramp-down the discharge or return the plasma to the nominal L-mode operation. In addition, as discussed in Sec. 2.2.1, if $min(\delta _{top},\delta _{bottom})\lesssim -0.25$ one would not expect H-modes to be triggered. We note that the L$\rightarrow$H power threshold may have a triangularity dependence due to fast ions orbit losses, as explained in Sec. 4. The systems-code BLUEPRINT has now been extended to include NT plasmas and will be used for DEMO studies Coleman and McIntosh (2020). A preliminary target equilibrium has been generated by the EU-DEMO team, although with a very low $q_0$ assuming significant fast particle stabilization of the internal kink. This was used to study the internal kink limit, including kinetic effects, in comparison to a PT DEMO plasma Zhou et al. (2021). No significant difference in the internal kink stability was found between these NT and PT EU-DEMO plasmas, except a larger growth rate in NT consistent with the earlier results Martynov et al. (2005); Martynov (2001). Sawteeth can lead to significant fast ion transport and reduce fusion performance, therefore these studies are important. Other modes which can reduce fusion power, although not reducing the overall $\beta$-limit, are fast ion modes, which will be discussed in Sec. 4.

Regarding ideal stability limit simulations, there is of course too much interesting work to review fairly here. We shall mention a subset with the main bias of looking at triangularity effects. It is interesting to note the blossoming of papers at the 6th international conference on plasma physics and controlled nuclear fusion research in Berchtesgaden in 1976 (published in 1977) studying ideal MHD stability for non-circular plasmas, most of them considering the triangularity dependence down to zero or negative values Becker and Lackner (1977); Berger et al. (1977); Callen et al. (1976); Chu et al. (1976); Johnson et al. (1977). This was followed by an interesting workshop in Varenna in 1977 on “finite Beta Theory” Coppi and Sadowski (1977) and a review paper in Wesson (1978). These papers show that equilibria with a wide variety of shapes and $\beta$ values are possible, but mainly tested their stability against local modes as a first application of the ideal MHD stability codes being developed. Several of these early findings remain valid, such as PT being stabilizing, squareness less favourable and the peakedness of the profiles having significant effects. In the following 2-3 years, the first numerical results on the limit set by the $n=1$ external kink, including the effects of a conducting wall, were obtained confirming the favourable properties of PT Bernard et al. (1978); Todd et al. (1979); Akkermans (1982) and leading to the studies for JET-shape plasmas Kerner et al. (1981) (doublets were also studied Chu and Miller (1978) but are outside the scope of this review). The next major result confirming the advantage of elongation combined with PT to obtain stable high $\beta$ equilibria was the increasing $\beta$-limit with increasing plasma current (up to the $q=2$ limit) (now known as the Troyon limit Troyon et al. (1984); Troyon and Gruber (1985)).

Apart from the bean shape Manickam et al. (1983), it is only around the time of the design of TCV that new thorough series of standard PT or racetrack shapes up to very high elongation and high current were studied Turnbull et al. (1988); Schultz et al. (1990). This further confirmed the advantage of PT over racetrack and similar studies focusing on high current and high $\beta$ Phillips et al. (1988). Note the new findings during the TCV design phase showing that the stability boundary shrinks and differs from the Troyon limit at very high elongation ($\kappa \ge 2.5$) Turnbull et al. (1988) (the boundary was later confirmed experimentally in TCV with cases up to $\kappa \sim 2.8$Hofmann et al. (1998)). These new numerical studies were aiming at using the codes to find the optimum shape, instead of the operating range of a given shape/experiment, albeit solely in regard to the ideal MHD limit. Integrated performance started to be increasingly important with studies for reactor-size tokamaks Kikuchi et al. (1991) and the start of the ITER project in March 1987 ITER (1988). As such, the required confinement properties were used as a driver for finding the optimal shape Ohkawa et al. (1989). This leads to a geometry reducing the effects of trapped particle modes thanks to drift reversal and thus to an NT plasma, where most of the trapped particles have their bounce tips in the good curvature region Rewoldt et al. (1982); Ohkawa et al. (1989), see Sec. 3. This led to a comet shape yielding drift reversal even at zero $\beta$ Ohkawa et al. (1989), when maximizing $J=\int {v_\parallel dl}$ with plasma shaping Miller et al. (1989). This shape was found stable up to $\beta _N\ge 2$ in Kesner et al. (1995) which also confirmed the favourable role of NT on trapped particle modes (extended to low aspect ratio as well Roberto and Galvão (1994)). Note that this was also associated with lower ITG growth rates in NT plasmas Waltz and Miller (1999), so the 1990s started to have MHD and transport studies integrated to provide a combined physical picture of an “optimum” performance. The transport aspects are detailed in Sec. 3, but we just note here that MHD and transport started to be integrated experimentally in NT plasmas at the same time with the first TCV results showing important differences between NT and PT for both transport and MHD Moret et al. (1995); Weisen et al. (1997).

Regarding ideal global stability boundaries, the main recent results from the last 20 years come from the series of studies conducted by Medvedev et al, first focusing on TCV experimental discharges Medvedev et al. (2008, 2009, 2014) and then to DEMO-NTT size plasmas Medvedev et al. (2014, 2015); Kikuchi et al. (2019). First H-mode like profiles were also considered, but with the goal of an L-mode DEMO-NTT, studies concentrated on L-mode like profiles. The main results are presented in Medvedev et al. (2015) and show stable DEMO-NTT with $\beta _N\ge 3$.

Another important result is the inefficient wall stabilization because of the coupling between internal and external modes, itself related to the lack of magnetic well and to $D_M$ being more unstable in NTs as discussed in Sec. 2.2. This could have an impact for steady-state profiles and resistive wall modes (RWM), since Medvedev et al. (2015) also showed that reverse shear q profiles have a lower limit, $\beta _N\ge 2$, due to the coupling with infernal modes (similar to but possibly more significant than for PT reverse shear q profiles). RWM with advanced scenario q-profiles are of particular interest for steady-state DEMO-NTT studies. At this stage a detailed comparison between NT and PT plasmas with a “standard” monotonic q-profile was performed in Ren et al. (2016). It was also found that the wall-stabilization is less effective for NT than PT, leading to a narrow window for RWMs. Otherwise assuming a wall radius of 1.10a for NT and 1.5a for a PT DEMO-size case and using MARS-F and MARS-K, Ren et al. (2016) found similar behaviour in terms of ideal/resistive wall, plasma flow and drift kinetic effects due to thermal trapped particles. The lack of wall stabilization depends on the effective eigenfunction. In these cases they also find the $n=1$ ideal kink mode amplitude to be quite localized near the “X-points”, similar to the $n=0$ eigenfunction mentioned earlier and in a similar location as the large positive local shear discussed earlier as well, important for ballooning modes. These effects might be responsible for a stronger coupling observed between local and global modes in NT. Non-linear analyses should provide more insights.

3 Transport

As described in Sec. 2, plasmas with a negative triangularity shape have not received extensive theoretical or experimental attention, at least compared to their standard-dee shaped counterparts, on the grounds that they were expected to have poor MHD stability properties. However, as pointed out by T. Okhawa, the transport characteristics of elongated dee-shaped plasmas are, in general, not significantly better than those of circular configurations once the indirect benefit of having a larger toroidal current is factored out. Instead, Okhawa proposed an oblate configuration at negative triangularity in an effort to improve confinement by reducing the fraction of trapped particles that orbit in the bad curvature region Okhawa (1988). Further analysis work, particularly of MHD stability properties, on such configurations, also termed comet shapes, ensued Kesner et al. (1995). It is interesting to note that, even before trapped particle instability theories were widely accepted, based only on stability considerations outlined in Laval et al. (1971), an oblate shape at negative triangularity was adopted for a detailed design of a power plant edited by R.G. Mills in 1974 Mills (1974). Such design work, however, appeared to have been abandoned in favor of dee-shaped plasmas, which became the prominent configuration in the late 1970s.

3.1 Thermal transport and turbulence

The first work in the realm of transport investigating the effect of $\delta <0$ can be found in the analysis performed by Rewoldt Rewoldt et al. (1982) who, among other classes of instabilities, considered the kinetic linear stability of the trapped-electron drift-wave regime using an electromagnetic code interfaced with a general magneto-hydro-dynamic equilibrium solver. Although the effect of $\delta <0$ was found to be stabilizing, as shown in Fig. 8, the actual decrease in the growth rate was a factor $3-4$ smaller than that due to an increase in elongation, which was therefore deemed to be a more important parameter to focus on in future optimizations.

Fig. 8

Mode frequency ($\omega$) and growth-rate ($\gamma$), normalized to the electron diamagnetic frequency ($\omega *$), for trapped electron drift modes as a function of triangularity (Reproduced from Rewoldt et al. (1982), with the permission of AIP Publishing)

Negative triangularity configurations were experimentally investigated in the late 1970s on the Poloidal Divertor eXperiment (PDX) tokamak, which was designed to create discharges with a poloidal cross section having standard or elongated dee shapes, as well as with square and reversed-dee configurations. The control system was such that, by progressively varying the radius of the magnetic axis by modifying the vertical magnetic field, plasmas could be moved from one configuration to another continuously. The main design parameters of the PDX tokamak were major radius $R = 1.3-1.5$ m, minor radius $a=$ 0.4 m, confining magnetic field $B_T=$ 2.5 T and plasma current $I_P=$ 0.5 MA for a typical pulse length of one second duration. Plasmas were characterized by line averaged density in the range $\langle n_e\rangle =2-4\cdot 10^{19}$ cm$^{-3}$ and on-axis electron temperature $T_e\le$ 1 keV. All surfaces in contact with the plasma were made from 99% pure titanium. The PDX team reported that the confinement properties of $\delta <0$ plasmas were quantitatively similar to those measured in more standard configurations, both in ohmic and neutral beam heated discharges Meade (1980).

Reversed-dee configurations were also realized in the late 1970s on the Tokapole II tokamak, although research work investigating the impact of triangularity appears to have focused on vertical stability, as discussed in Sec. 2.

3.2 Experiments on the TCV tokamak

After early work in the 1970–1980s, $\delta <0$ configurations essentially disappeared from the radar of the fusion community until dedicated experiments on the TCV tokamak began in the mid 1990s. The electron energy confinement time of Ohmically heated L-mode plasmas in a wide range of shapes, featuring $1.06<\kappa <1.86$, $-0.41<\delta <0.72$, was measured to improve with increasing elongation and degrading rather strongly with positive triangularity Moret et al. (1997); Weisen et al. (1997). However, the fact that the inferred thermal conductivity was found to be rather independent of plasma shaping led to interpreting the observed confinement improvement as mainly due to a variation of the real space electron temperature gradient due to flux surface compression. More specifically, as displayed in Fig. 9 a large fraction of the shape dependence was removed from the confinement time upon normalization to a shape enhancement factor (SEF), defined as the ratio of the confinement time of a shaped plasma to that of the reference cylindrical plasma (see Eq.1 in Moret et al. (1997)). This result appears to be consistent with experiments on the PDX tokamak Meade (1980) for which confinement was not observed to strongly depend on the sign of triangularity. It has to be noted, however, that the experiments on PDX appear to have been run at moderate values of triangularity and in a somewhat collisional regime, i.e. in conditions for which the stabilizing effect of negative triangularity is not expected to be apparent.

Fig. 9

(Left) Electron energy confinement time as a function of triangularity, sorted by electron density and edge safety factor. (Right) Same data-set with normalization using the shape enhancement factor (Reprinted figure with permission from Moret et al. (1997), Copyright by the American Physical Society)

Shortly thereafter, however, Ohmic confinement studies were extended to strong negative triangularity (to $\delta \sim -0.5$) at lower density in MHD-quiescent conditions, and a strong increase in confinement with negative triangularity was observed Pochelon et al. (1998). This improvement could not be explained by the SEF. At the same time, a refurbishment of the TCV wall allowed the safe utilization of auxiliary heating in $\delta <0$ discharges, although only in inner-wall limited (IWL) configurations. Subsequent experiments explored the dependence of the electron energy confinement time on plasma shaping in L-mode plasmas with applied centrally deposited electron cyclotron heating (ECH). The effect of plasma shaping was studied by varying the elongation and triangularity of the last closed flux surface (LCFS) in the range $1.1< \kappa < 2.15$ and $-0.65< \delta < 0.55$, for values of the engineering safety factor corresponding to 1.7 and 3. The central electron density was maintained below $2.5 \times 10^{19}\,\hbox {m}^{-3}$, in contrast to the initial ohmically heated experiments previously described, for which the central density was $\mathrm 2-3$ times larger. A clear improvement of confinement with negative triangularity was documented also in these conditions. The overall scaling of the energy confinement time on triangularity was cast in the form Pochelon et al. (1999)

$$\begin{aligned} \tau _E \propto (1+\delta )^{-0.35\pm 0.3}, \end{aligned}$$

(11)

which illustrates the beneficial impact of $\delta <0$ on the overall confinement.

Based on these early results, the TCV program embarked on a large set of experiments aimed at studying the dependence of confinement on triangularity while disentangling other factors that are predicted to affect the underlying turbulence, such as safety factor, collisionality, electron temperature, density as well as their spatial scale lengths. Experiments were limited to L-mode plasmas because that would make it easier to extract the dependence of core transport on triangularity. Indeed, as in PT H-mode the pedestal height was expected to depend on triangularity, as was later experimentally confirmed Pochelon et al. (2012) and quantified by modeling Merle et al. (2017), the direct triangularity dependence of the overall confinement would be entangled with the confinement dependence on the edge pedestal height. More specifically, the dependence of the electron heat diffusivity was probed with respect to the electron temperature and its radial gradient, as well as to density within values compatible with near full EC first pass absorption, by varying the amount of ECH coupled power and its deposition location. The electron heat diffusivity was observed to significantly decrease at negative triangularity, even when normalizing values to a $T^{3/2}$ gyro-Bohm dependence to account for local variations of the electron temperature. Notably, in $\delta >0$ and $\delta <0$ plasmas with matched values of elongation and absolute value of triangularity at the LCFS, the same electron density and temperature profiles were obtained when half the EC power was coupled into the latter case, thus demonstrating that $\delta <0$ L-mode plasmas sustain an H-mode level of confinement Camenen et al. (2005); Camenen (2006). The observed dependence of the mid-radius electron heat diffusivity on the electron temperature, density and effective charge was cast in a unique dependence on the plasma effective collisionality, as displayed in Fig. 10. This was found to be consistent with linear simulations performed with the GLF23 model Waltz et al. (1997) which indicated trapped electron modes (TEM) as the dominant instability at play in those discharges Camenen et al. (2007).

Fig. 10

Mid-radius electron heat diffusivities vs inverse collisionality for a number of EC heated IWL discharges at varying values of triangularity of the LCFS (Reproduced from Camenen et al. (2007), with the permission of IAEA Publishing)

Later experiments compared $\delta <0$ to $\delta >0$ configurations in terms of the characteristics of turbulent fluctuations measured by the correlation electron cyclotron emission (CECE) and tangential phase contrast imaging (TPCI) Marinoni et al. (2006) diagnostics. In ohmic discharges, the intensity of electron temperature fluctuations is strongly reduced at $\delta <0$ in the outer half of the plasma volume, while at mid-radius the $\delta <0$ broadband fluctuation level falls below the instrumental noise floor, precluding a quantitative assessment of the difference between the two cases there. Additionally, a threshold behavior was observed in the outer half of the minor radius only at $\delta >0$ , suggesting that, at least in the ohmic cases considered, the $\delta <0$ case was characterized by a higher threshold for the onset of the instability Fontana et al. (2017); Fontana (2018).

Similar results were obtained when monitoring electron density fluctuations using the TPCI diagnostic in EC heated discharges. More specifically, the absolute intensity of fluctuations was observed to halve at $\delta <0$ as compared to $\delta >0$ , along with a reduction of the spectral index. As opposed to the ECE measurements in ohmic discharges, the TPCI diagnostic could resolve $\delta <0$ fluctuations well into the core, and showed that the reduction at $\delta <0$ extends to mid radius, despite the fact that the local value of $\delta$ therein essentially vanishes Huang and S.C. (2018); Huang (2017). The stabilizing effect of the effective collisionality was also observed with TPCI measurements, as displayed in Fig. 11, consistent with results reported in Camenen et al. (2007); Marinoni et al. (2009).

Fig. 11

Relative density fluctuation amplitude as a function of the reciprocal of effective collisionality for plasmas with triangularity values at the LCFS equal to $\delta = 0.5$ (red) and $\delta = -0.4$ (blue), Ohmic (open symbols) and ECH L-mode (full symbols) TCV discharges ($\copyright$ IOP Publishing. Reproduced with permission from Huang and S.C. (2018). All rights reserved)

These investigations of core turbulence were recently complemented by measurements of turbulence in the scrape-off layer (SOL) with both wall-mounted Langmuir probes and a gas puff imaging system. There too, a strong reduction in fluctuation amplitudes is observed for $\delta <-0.25$, accompanied by nearly full suppression of plasma interactions with the first wall Han et al. (2021). An analysis of different possible correlations and causations points at the shorter connection length in $\delta <0$ plasmas as possibly the most crucial element behind this phenomenon. It should be noted that a negative-triangularity TCV plasma features among a set of shapes that were used for validation of SOL turbulence simulations by the GBS code by means of reciprocating probe measurements Riva et al. (2020). More recently, thanks to the installation of a neutral beam heating system, the TCV team started exploring the impact of triangularity on confinement in regimes where the ion temperature is close to or higher than that of electrons, thereby complementing earlier results for which $T_i\ll T_e$. Comparison discharges with edge triangularity $\delta =\pm 0.4$ were executed at elongation $\kappa =1.4$ and line averaged density $\langle n_e\rangle \simeq ~2 \times 10^{19}\,\hbox {m}^{-3}$. It was chosen to approximately match the safety factor between the two plasma shapes, with $q_{95}\simeq 3.4$, thereby increasing the plasma current in the $\delta >0$ discharge by about 18%; this is a conservative choice when it comes to comparing confinement improvements at $\delta <0$ because of the positive correlation between stored energy and plasma current which is generally observed in tokamaks. Improved confinement was demonstrated both at fixed auxiliary power, which yielded higher stored energy at $\delta <0$ , and with matched profiles, which required lower coupled power at $\delta <0$ . Consistent with the improved confinement, the intensity of electron temperature fluctuations measured by the CECE system was always observed to decrease at $\delta <0$ at comparable values of collisionality, electron to ion temperature ratio, and density and temperature inverse scale lengths Fontana et al. (2019); Fontana (2018). With NBI, the highest performances have been achieved on TCV in terms of both $\beta _N$ and confinement enhancement $H_{98,y2}$, reaching, respectively, 2 and 1.3 with 0.5 MW power, albeit in a non-stationary manner Coda et al. (2019). Among the latest reported developments on TCV is also the establishment of a fully diverted shape in the upper part of the chamber, which, with Ohmic heating, confirms the favorable properties of negative triangularity Coda et al. (2019) (Fig. 12). Studies of H-mode $\delta <0$ plasmas were also conducted in TCV using hybrid diverted shapes with positive lower triangularity and upper triangularity varying from positive to negative Pochelon et al. (2012). Discharges where heated using EC system with $500-700$ kW at the third harmonic (X3) for distributed core heating, and $300-900$ kW of second harmonic (X2) for heating localized near the $\rho =0.9$ surface close to the X-point. The resulting mid-radius collisionality was close to unity which, based on results from Camenen et al. (2007), is expected to result in a modest stabilizing effect of $\delta <0$ . Correspondingly, global confinement was found to be reduced in such scenarios, compared with standard $\delta >0$ cases, and this could be imputed to the 20% lowered edge pedestal pressure. The lower pedestal height resulted in a concomitant passive mitigation effect on ELMs, as discussed in Sec. 2.

Fig. 12

(Left) Positive- and negative-triangularity diverted TCV plasmas and (right) comparison of plasma current (top), normalized $\beta$ (middle), and $H_{98,y2}$ confinement-enhancement factor (bottom)

3.3 Theoretical studies

A non-linear gyro-kinetic analysis carried out using the flux-tube GS2 Kotschenreuther et al. (1995) code confirmed that the TCV ECH L-mode discharges were TEM dominated, and quantitatively reproduced the dependence of the core electron heat diffusivity both on triangularity and collisionality Marinoni et al. (2009); Marinoni (2009), as displayed in Fig. 13. However, agreement with experiments within error bars was found only in the outer part of the poloidal cross section, with the predicted stabilizing effect of $\delta <0$ progressively vanishing towards the magnetic axis. Such result was interpreted to be due to the modest radial penetration length of triangularity (at $\rho =0.7$ it is one-half the value at the boundary), which causes the flux tube simulations to fail to discern significant differences between the two scenarios as the flux surfaces progressively approach the shape of an oval at inner radii. This result suggested that profile shearing effects might be important in determining the overall stability of these discharges Marinoni et al. (2009).

Fig. 13

Predicted electron heat diffusivity as a function of the inverse effective collisionality for $\delta >0$ TCV discharge #28014 (- -) and $\delta <0$ counterpart #28008 (-), computed for identical density and temperature profiles of all kinetic species ($\copyright$ IOP Publishing. Reproduced with permission from Marinoni et al. (2009). All rights reserved)

Subsequent studies carried out with the flux tube version of the GENE non-linear gyro-kinetic code Jenko et al. (2000) confirmed the GS2 results Merlo et al. (2015); Merlo (2016). In the GENE simulations, the impact of triangularity on micro-stability was evaluated also near the LCFS, where the largest stabilizing effect of $\delta <0$ was found, consistent with the largest absolute variation of triangularity. Additionally, by varying the density and temperature scale lengths and taking linear extrapolations with respect to the driving gradient, it was estimated that the critical temperature gradient increases at $\delta <0$ , while no significant impact of the density inverse scale length was found in either equilibrium. The fact that the largest impact of $\delta <0$ was found, from a gyro-kinetic stand-point, near the LCFS is qualitatively consistent with dedicated experiments, reported in Sauter et al. (2014), designed to obtain high resolution electron and density profiles by executing repeat discharges at slightly different vertical positions in the vacuum chamber so as to effectively enhance the spatial resolution of the electron pressure profiles measured by the Thomson Scattering diagnostic. While the plasma region within 60-70% of the total volume featured a rather constant logarithmic temperature gradient for both triangularities, indicating a high degree of profile resiliency, the outer region was characterized by a constant gradient, with the $\delta <0$ case exhibiting steeper radial profiles for very similar heat flux values.

Further insights into the GS2 modeling revealed that, at fixed density and temperature profiles, the stabilization observed at negative triangularity is due to a combined effect of the perpendicular drifts and of the ballooning eikonal, which reflects the effect of magnetic shear and can be interpreted as the perpendicular wave-vector $k_\perp$. More specifically, the overall stabilization is a result of the modification exerted by triangularity on the toroidal precession drift of trapped electrons. This is intuitively in agreement with the basic destabilization mechanism of the collisionless TEM which, as first derived in Kadomtsev and Pogutse (1967), is given by the resonance between a given wave (e.g. a pressure driven drift wave) and the toroidal precession drift of trapped electrons, with the resulting dispersion relation in a pure plasma being

$$\begin{aligned} \left( \frac{1}{T_e}+\frac{1}{T_i}\right) \tilde{\phi }= \sqrt{\epsilon }\left( \frac{1}{T_i}\frac{\omega -\omega _{*,i}}{\omega -\omega _{D,i}} +\frac{1}{T_e}\frac{\omega +\omega _{*,e}}{\omega +\omega _{D,e}}\right) \langle \tilde{\phi }\rangle . \end{aligned}$$

(12)

In Eq. 12, $T_{x}$ is the temperature of the generic plasma species x, $\omega _{*,x}$ and $\omega _{D,x}$ are, respectively, the diamagnetic drift and the toroidal precession drift angular frequencies, $\epsilon$ the inverse aspect ratio of the machine, $\tilde{\phi }$ the perturbed electrostatic potential, and the angular bracket refers to averaging along the bounce trajectory of the trapped particle. In the particular case of perfect equipartition, i.e. $T_e=T_i$, so that $\omega _{*,i}=-\omega _{*,e}\equiv -\omega _*$ and $\omega _{D,i}=-\omega _{D,e}\equiv \omega _D$, Eq. 12 reduces to

$$\begin{aligned} \omega ^2 = \omega _D^2+\sqrt{\epsilon }\omega _D\omega _*, \end{aligned}$$

(13)

from which it is apparent that a necessary condition for the instability to arise is that the toroidal precession drift and the diamagnetic drift have to be oriented in opposite directions. In the limit of a circular plasma at large aspect ratio and low $\beta$, the toroidal precession drift of a trapped electron of mass m and velocity v orbiting on a given flux surface $\psi$ can be written as Kadomtsev and Pogutse (1967)

$$\begin{aligned} \omega _{D,e} = \frac{m}{2e}\frac{\partial J}{\partial \psi }/\frac{\partial J}{\partial v^2}\propto G(s,\varkappa )v^2 \end{aligned}$$

(14)

where J is the second adiabatic invariant, s the magnetic shear, $\varkappa$ a modified pitch angle variable and G is defined as

$$\begin{aligned} G(s,\varkappa )\equiv \left[ 2\frac{E(\varkappa )}{K(\varkappa )}-1\right] +4s\left[ \frac{E(\varkappa )}{K(\varkappa )}-1+\varkappa \right] , \end{aligned}$$

(15)

where $E(\varkappa )$ and $K(\varkappa )$ are the complete elliptic integrals of the first and second kind, respectively. The shape factor $G(s,\varkappa )$, which is displayed in Fig. 14 as a function of the pitch angle variable $\varkappa$, does not depend on the magnetic shear in the limits of deeply and barely trapped particles, i.e. $\lim _{\varkappa \rightarrow 0,1} G(s,\varkappa )$, and is monotonically decreasing across the entire pitch angle space for decreasing values of the magnetic shear. Based on Eq. 13, TEM modes are expected to be weakened or suppressed at negative values of the magnetic shear, a condition often associated with reduced turbulence characterizing Internal Transport Barriers (ITB) Zucca et al. (2008). The physics of ITBs is not the subject of this paper; we note, however, that turbulence quenching mechanisms in ITBs have also been widely associated to other factors, of which the most common is increased flow shear. A general analytic expression for the toroidal precession drift in general geometry and finite pressure was derived in Graves (2013), in which the impact of elongation and triangularity on the toroidal precession drift were evaluated for the IWL TCV discharges described above Camenen et al. (2007). While triangularity is found to have a negligible effect on passing electrons, the impact on trapped electrons is far greater. More specifically, the drift increases at $\delta <0$ across most of the pitch angle space, whereas deeply trapped electrons drift more slowly when the triangularity is negative. This is consistent with the pitch angle dependence of the non-linear energy flux computed by GS2 and reported in Marinoni et al. (2009) for which, when comparing $\delta <0$ to $\delta >0$ equilibria, deeply trapped electrons are on average closer to the resonance condition in Eq. 12, while they are further away everywhere else in the pitch angle space, resulting in a net overall stabilization. It is interesting to note that, consistent with the drift precession predictions Graves (2013), also the GS2 non-linear gyrokinetic simulations found a negligible stabilization exerted by negative triangularity on passing electrons Marinoni et al. (2009).

Fig. 14

Pitch angle dependence of the $G(s,\varkappa )$ factor from Eq. 15 for a few values of the magnetic shear s

The interpretation based on the modification of the toroidal precession drift is valid locally, meaning that the extent of the induced stabilization depends on the triangularity value of the given flux surface under consideration. Indeed, while the experimental results reported in  Camenen et al. (2007) reconstructed a uniform reduction of the electron heat diffusivity across the plasma cross section, non-linear local gyro-kinetic modeling work Marinoni et al. (2009); Merlo et al. (2015) recovered the experimental results only in the outer part of the cross section, consistent with the local value of triangularity quickly decreasing towards the magnetic axis. While it was immediately realized that such discrepancy hinted at global effects, such impact has only been very recently demonstrated using gradient driven global non-linear gyro-kinetic simulations, which found quantitative agreement across a large fraction of the minor radius between predicted non-linear fluxes and the corresponding experimental values derived from a power balance analysis Merlo et al. (2021). In that work it was also concluded that global effects appeared important for the $\delta <0$ case but not for the $\delta >0$ case, for which flux-tube simulations yielded adequate results.

The stabilizing effect of $\delta <0$ is not restricted to TEM dominated turbulence, as evidenced by non-linear gyrokinetic simulations of the more recent TCV discharges which, being NBI heated, are characterized by mixed TEM/ITG Merlo et al. (2019). This is consistent with NBI-EC heated DIII-D discharges, reported in Sec. 3.4, and the corresponding gyrokinetic modeling Marinoni et al. (2019). It is interesting to point out that, even in the case of ITG dominated turbulence that will likely characterize future reactors, the overall transport level is expected to be reduced by decreasing the strength of the sub dominant TEM, as long as their drive is not negligible. The fact that ITG modes are strengthened in the presence of sub dominant TEM turbulence appears to be a rather general property of plasmas in various configurations and has been obtained by a number of authors with both gyro-fluid and gyro-kinetic simulations Beer (1995); Waltz and Miller (1999). This may have beneficial consequences for a $\delta <0$ reactor, for which the sub dominant TEM are expected to be weakened by the effect exerted by plasma shape, although this effect has yet to be quantified.

3.4 Experiments on the DIII-D tokamak

In 2016 the DIII-D team began a series of experiments aimed at validating the results obtained on TCV. All plasma discharges were carried out, as on TCV, in an IWL configuration because portions of the outer-wall on DIII-D are not armored to withstand the power flux at the strike locations at any power level. Plasmas were designed to be up-down symmetric with the LCFS featuring moderate triangularity $\delta =-0.4$, while coil positioning on DIII-D is such that plasma elongation was fixed at $\kappa =1.33$. Comparison discharges at $\delta >0$ were made by developing a mirrored shape with identical elongation but opposite value of triangularity, resulting in the two shapes having the same cross sectional area. To establish a link with the TCV results, experiments were first carried out with pure electron heating, resulting in $\delta <0$ configurations featuring about 30% improved confinement as compared to $\delta >0$ counterparts at matched values of plasma current, confining magnetic field, line averaged density and auxiliary power. The confinement improvement was significantly lower than that observed in Camenen et al. (2007), consistent with the fact that, due to the much lower power per particle available in these EC-only heated discharges on DIII-D as compared to those on TCV, plasmas were operated in an intermediate collisionality region, corresponding to $\nu _{eff}\simeq 0.5$ near mid-radius, where the beneficial effect of $\delta <0$ is marginal Camenen et al. (2005); Marinoni and H-mode-like confinement with L-mode edge in negative triangularity plasmas on DIII-D, (2017).

Auxiliary power from the Neutral Beam Injection (NBI) system was subsequently gradually increased, contingent on no sign of overheating of the outer-wall. At the maximum power level allowed, NB heating resulted in discharges sustaining H-mode grade confinement and pressure levels equal to $\beta _N\simeq 3$ and $H_{98,y2}\simeq 1$, respectively, for several energy confinement times, despite relaxed edge pressure profiles typical of an L-mode regime Austin et al. (2019); Marinoni et al. (2019). The persistence of the L-mode regime at auxiliary power levels that far exceed the expected H-mode power threshold is attributed, as explained in  Sec. 2, to the stability of ballooning modes Saarelma et al. (2021). An example of the time evolution of a typical plasma discharge is displayed in Fig. 15, where high confinement is sustained with an L-mode edge as evidenced by the ELM–free $D_\alpha$ signal.

Fig. 15

Time evolution of DIII-D discharge #176283. From top to bottom, auxiliary power (NBI-ECH), normalized pressure $\beta _N$ and internal inductance $l_i$, confinement enhancement factor $H_{98,y2}$, $D_\alpha$ signal (Reproduced from Marinoni et al. (2019), with the permission of AIP Publishing)

The fact that the confinement enhancement factor seen in Fig. 15 improves with increasing auxiliary power is understood in terms of a power degradation of thermal confinement that, although finite, is weaker than that expected from conventional power scaling laws. When the fast ion energy is accounted for, these $\delta <0$ discharges displayed a near-zero power degradation of total confinement Austin et al. (2019); Marinoni et al. (2019); this is in contrast to the ITER-89P scaling law that features a total stored energy proportional to the squared root of the auxiliary coupled power.

Linear gyro-kinetic and gyro-fluid analysis performed with the CGYRO Candy et al. (2016) and TGLF Staebler et al. (2007) codes, respectively, showed that TEM is the dominant ion scale instability both in EC-only and mixed EC-NB heated discharges, with decreased growth-rates at $\delta <0$ consistent with results exposed in Sec. 3.2 Marinoni et al. (2019). Nevertheless, the use of both beam heating and the more collisional regime in which the DIII-D discharges were operated even in the EC-only heated phase, as compared to the collisionless discharges on TCV, lowered the $\mathrm T_e/T_i$ ratio: this caused ETG modes to be linearly unstable even in the core, whereas modeling of the TCV discharges indicated no such feature Marinoni et al. (2009); Merlo et al. (2015).

The impurity confinement time was estimated by ablating suitable Aluminium targets using the Laser Blow-Off system, yielding a particle to energy confinement time ratio, $\tau _P/\tau _E$, of order unity Marinoni et al. (2019). As opposed to H-mode regimes, for which such ratio is typically measured in the range $\mathrm 2-4$, this scenario makes impurity retention less problematic although fueling might be more difficult in large devices depending on how the main ion particle transport scales compared to that of impurities. The low $\tau _P/\tau _E$ value is believed to be due to the absence of a density edge barrier, rather than to the negative value of triangularity, as it is consistent with results in the I-mode regime Whyte et al. (2010). The intensity of fluctuations was monitored by the phase contrast imaging Dorris et al. (2009) (PCI), the beam emission spectroscopy McKee et al. (2010) (BES) and the CECE Sung et al. (2016) diagnostics. All systems consistently detected a decreased intensity of fluctuations in $\delta <0$ plasmas as compared to matched $\delta >0$ discharges at fixed conditions, consistent with lower transport coefficients Marinoni et al. (2019). While the radial dependence of the reduction of the intensity of fluctuations appears to differ between the BES and CECE systems (see Figure 5 of Marinoni et al. (2019)), no dedicated study was executed trying to determine whether the beneficial effect of $\delta <0$ on the intensity of fluctuations is uniform across the minor radius or decreases towards the axis following the finite penetration length of triangularity. As a consequence, due to the paucity of data for which both the BES and the CECE systems were simultaneously measuring the same plasma region with good signal to noise ratio, it is not possible to determine whether the observed difference is real or, rather, is an artifact caused by uncertainties in the measurements.

4 Energetic particles

To assess whether a given cross sectional shape is beneficial for confinement, its impact on the transport coefficients must be evaluated both for thermal and for supra-thermal species. Indeed, if fast ion modes caused enhanced anomalous transport in $\delta <0$ plasmas, this could potentially reduce the impact of the turbulence stabilization discussed in Sec. 3.1.

At present, only one published work Van Zeeland et al. (2019), whose main findings are summarized hereafter, has reported on a DIII-D experiment dedicated to fast ion physics in $\delta <0$ plasmas. Inner-wall limited (IWL) plasmas with opposite values of triangularity were compared while holding actuators such as line averaged density ($\langle n_e\rangle$), plasma current ($I_P$), toroidal field ($B_T$) and auxiliary power the same. Fast ion modes were probed using beam heating during the current ramp-up phase, which is the reference scenario used on DIII-D Van Zeeland et al. (2006) for fast ion studies because it slows down the current penetration, thereby creating a region with reverse magnetic shear, and generates a good fraction of energetic ions. The discharge evolution is such that the LCFS in plasmas at $\delta >0$ converges more slowly to its final form than its $\delta <0$ counterpart, resulting in 12% larger elongation for the former. All experiments reported in the DIII-D work were carried out with $B_T=$ 2.0 T, $\langle n_e\rangle ~ < 5 \times 10^{19}\,\hbox {m}^{-3}$, $I_P~< 1$ MA as well as electron and ion temperatures below 5 keV. All $\delta <0$ plasmas with early beam heating display numerous Alfvén eigenmodes (AE), including Beta induced Alfvén Acoustic Eigenmodes (BAAEs) Gorelenkov et al. (2007), Beta induced Alfvén Eigenmodes (BAEs) Turnbull et al. (1993), Reversed Shear Alfvén Eigenmodes (RSAEs) Kusama et al. (1998) and Toroidicity induced Alfvén Eigenmodes (TAEs) Cheng and Chance (1986). The number of unstable Alfvén modes as well as their amplitude was found to increases in power scans, as expected. Additionally, the amplitude of the AE is quantitatively comparable between the two shapes, consistent with the fact that, although the fraction of thermal particles that are trapped is larger in $\delta <0$ configurations, the fraction of fast ions that are trapped does not significantly depend on triangularity. However, as displayed in Fig. 16, many of the modes in $\delta <0$ plasmas, especially BAEs and RSAEs, feature a quasi-coherent or chirping frequency evolution which, on DIII-D, is more typically found in H-mode plasmas. Some authors argue that such behavior is attributable to a lower intensity of fluctuations in the background plasma which, by decreasing the amount of stochastic scattering energetic particles are subject to, allows coherent structures in phase space to shine through the background noise Duarte et al. (2017). Values of fast ion transport were deduced using a dedicated beam modulation technique Heidbrink et al. (2016) and comparing the corresponding response measured by fast ion diagnostics to that expected by classical calculations from the TRANSP code Breslau et al. (2018). The measured modulated neutron emission signal departs from the TRANSP predictions at increasing auxiliary beam power, consistent with the fact that AEs transport beam ions faster than the classical expectations. However, by comparing the inferred transport levels of $\delta <0$ discharges to those computed for oval plasmas at similar values of auxiliary power, no dependence on triangularity was found despite a large difference in the spectra; this could be a manifestation of the fact that local fast ion gradients relax to a marginally unstable value Van Zeeland et al. (2019). Based on such findings, at least in the cases examined, fast ion transport does not appear to worsen at $\delta <0$ . As a result, the improved thermal confinement displayed by $\delta <0$ plasmas is not expected to be deteriorated by the fast ion population in a greater amount than the $\delta >0$ counterpart.

Fig. 16

ECE spectrograms displaying the time evolution of a variety of Alfvén modes in the current ramp-up phase of $\delta >0$ DIII-D discharge #170680 (a-c) and $\delta <0$ discharge #170660 (b-d). (Reproduced from Van Zeeland et al. (2019), with the permission of IAEA Publishing)

Fast ions losses entail a number of potential issues in any tokamak, both under safe operation and physics standpoints, and are briefly discussed below. To establish safe protocols for operation in $\delta <0$ plasmas, prompt losses were predicted in Van Zeeland et al. (2019) to be quantitatively similar between the two IWL shapes at $\delta <0$ and $\delta >0$ to be run on DIII-D, although differences were found in the region of the vessel in which power would be deposited. More recently, ion orbit losses were computed in the case of double-null diverted configurations in Nishimura et al. (2020), where it was found that, in contrast to $\delta >0$ plasmas in which fast ions typically complete their banana orbits, $\delta <0$ plasmas are expected to suffer from enhanced losses primarily due to the X-point diverting trapped fast ions to the inner plate. The subsequent creation of edge electric fields is argued to inhibit the formation of H-mode pedestals in diverted $\delta <0$ configurations. This mechanism, however, would not explain the higher L$\rightarrow$H power threshold observed in IWL $\delta <0$ configurations when compared to $\delta >0$ counterparts Marinoni et al. (2019).

5 Power exhaust

Comparatively little work has been performed to date on characterizing the power exhaust properties of negative-triangularity plasmas. In fact, to our knowledge there is a single published article on the subject, which examined some features of this issue in TCV Faitsch et al. (2018).

This work focused on the SOL power fall-off length, $\lambda _q$, and in particular on its dependence on triangularity in L-mode. The scenario used in this study featured a lower-single-null diverted shape with lower positive triangularity and varying upper triangularity $\delta _\mathrm{up}$, from -0.28 to 0.47. The value of $\lambda _q$ at the outer target was found to decrease monotonically with $\delta _\mathrm{up}$ going from positive to negative, independently of the direction of the magnetic $\nabla B$ drift and of the main ion species being deuterium or helium. By contrast, the inner power fall-off length features a maximum at $\delta _\mathrm{up}\sim 0$ and decreases with finite triangularity of either sign Faitsch et al. (2018).

The value of $\lambda _q$, particularly at the outer target which generally features a larger heat flux, is of great importance in a reactor as it directly affects the heat load on a reactor’s first wall, smaller values being increasingly worrisome. The decrease of the outer fall-off length with negative triangularity can thus be seen as a detrimental effect. However, the values of power fall-off lengths are related to the spatial scale lengths of pressure profiles which, in turn, are set by the turbulence levels for a given flux. As such, $\delta <0$ discharges with improved confinement over matched $\delta >0$ counterparts are characterized by shorter $\lambda _q$. However, the TCV team reported that $\delta <0$ L-mode plasmas with H-mode grade confinement are characterized by wider $\lambda _q$ Faitsch et al. (2018) than $\delta >0$ plasmas with comparable core performance Maurizio et al. (2021). Similarly, it has been recently reported on DIII-D that the SOL power fall-off length in $\delta <0$ L-mode plasmas characterized by H-mode grade confinement and pressure levels ($H_{98,y2}\simeq 1$, $\beta _N\simeq 3$) exceeds by 50% values from the ITPA scaling law Marinoni et al. (2021). This result is understood by considering that, as explained above, $\lambda _q$ widens with stronger turbulence near the edge, as is typically measured in L-mode vs H-mode plasmas. When evaluating the requirements relative to exhaust power in L-mode $\delta <0$ vs H-mode $\delta >0$ plasmas, for similar values of global confinement, such observations add further to the merits of the former scenario vs the latter.

To conclude, we note that $\delta <0$ plasmas are also particularly attractive from a power exhaust stand-point because the strike points naturally impinge on the divertor on the low field side of the machine. As such, for a given value of $\lambda _q$ on at the separatrix, the heat flux foot print is wider than that in $\delta >0$ counterparts by the ratio of the major radii at which strike points impinge on the divertor in these two configurations.

6 Conclusions and future perspectives

The negative-triangularity tokamak is a concept that has existed for over five decades and has been actively explored experimentally for twenty-five years. Yet, in some ways it can be said to be in its infancy because, for most of this time, it has been a fringe configuration studied as little more than a curiosity from a theoretical standpoint and investigated experimentally with perseverance only on the TCV tokamak.

Indeed, based on initial MHD estimates, plasmas at negative triangularity were not expected to achieve reactor relevant pressure levels. Furthermore, although early experiments did succeed in controlling NT plasmas Lipschultz et al. (1979); Meade (1980); Takahashi et al. (1982)), plasmas were most probably too collisional and not sufficiently triangular, or elongated, to observe the confinement improvement that was much later obtained on TCV and DIII-D Camenen et al. (2007); Austin et al. (2019). As such, negative triangularity plasmas, along with other non standard configurations like the bean-shaped Bol et al. (1986), could not compete with promising results obtained in standard-dee plasmas. The discovery of the H-mode regime Wagner et al. (1982), which happened at about the same time, understandably steered the fusion community towards configurations at positive triangularity.

As the realm of energy and particle confinement began to be experimentally explored and theoretically developed, it was realized that tokamak performances are mostly curtailed by confinement properties and density limits, rather than by ideal MHD boundaries, obviously provided the latter are above the target equilibria. Thus, between the counteracting properties of better confinement and decreased stability in NT plasmas, the former began to have a dominant role. High $\beta$ equilibria that benefited from an improved confinement due to stabilization of micro-instabilities from banana tips being in the good curvature region were proposed in the mid-1990s Hsu et al. (1996); Ohkawa et al. (1989); Kesner et al. (1995), but failed to attract vast interest worldwide.

Only with the recent development of integrated scenarios that need to simultaneously optimize all aspects of a reactor, from pressure levels sustained in the core to wall exhaust, has the fusion community started to realize the immense challenges that a reactor based on conventional scenarios will face. Indeed, an optimal design of a reactor should by definition be near all the related limits, taking into account the respective risks, and should not favor a particular parameter, e.g. MHD limit with respect to exhaust. Consequently, the need for alternative options for DEMO has lately caused a decisive paradigm shift in how negative triangularity is perceived by the fusion community. The important additional contributions of DIII-D and the very recent developments on HL-2M Xue et al. (2019) and on ASDEX Upgrade Happel et al. (2020) bear witness to a renewed worldwide interest and the option of a negative-triangularity reactor is now being looked at very seriously indeed Kikuchi et al. (2019).

There is every reason to believe that the coming decade will see rapid progress towards a thorough characterization of the negative-triangularity tokamak concept, and bring us close to a proper reactor feasibility assessment.