An analytical model of how the negative triangularity cuts off the access to the second stable region in tokamak plasmas

We present an analytical model to evaluate the triangularity-shaping effects in accessing the second stable region for the ideal ballooning mode. Our results indicate that if the triangularity is sufficiently negative, the path from the first to the second stable region will be closed. The reason is that negative triangularity can weaken the stabilizing effect of the ‘magnetic well’, and even convert the ‘magnetic well’ into a ‘magnetic hill’, which will destabilize the ballooning mode. We also show that the synergistic effects of elongation, inverse aspect ratio, and safety factor can reopen the path to the second stable region. Through a variational approach, we derive an analytical expression of the critical negative triangularity for closing the access to the second stable region. Furthermore, our analysis reveals that in the second ballooning stable regime, the negative triangularity tends to inhibit the emergence of quasi marginally stable discrete Alfvén eigenmodes. These findings provide a quantitative understanding of how the negative triangularity configuration impacts the confinement of tokamak plasmas.


Introduction
In magnetically confined fusion plasmas, the region at the edge connecting the core and scrape-off layer plays a crucial role in achieving high performance and steady-state operations.The stability of this edge region is strongly influenced by the Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.plasma shape, specifically the triangularity [1][2][3].Positive triangularity has been demonstrated to enhance the edge magnetohydrodynamic (MHD) stability of high toroidal mode number (n) ballooning modes, enabling access to the second stable region.This facilitates the transition from low confinement (L-mode) to high confinement (H-mode) by allowing for increased pressure gradients and the formation of an edge transport barrier, also known as the pedestal [4,5].The H-mode has become a reference scenario for contemporary and future tokamaks due to its superior performance, but further enhancement of the pressure gradient is limited by the presence of MHD instabilities, particularly edgelocalized modes (ELMs) [6], which have been a longstanding challenge in terms of plasma-wall interactions and divertor heat load tolerance [7].Consequently, extensive efforts have been devoted to exploring various potential approaches [8][9][10], such as ELM mitigation/suppression [11][12][13] and the investigation of naturally 'ELM-free' advanced scenarios [14][15][16], to avoid its detrimental effects.
Recently, experimental results from TCV, DIII-D, and later on ASDEX Upgrade (AUG) have demonstrated that the negative triangularity configurations can achieve high plasma confinement (i.e.high normalized beta β N and confinement scaling factor H 98y2 ) with L-mode edge profiles [17][18][19].As ELMs generally are not triggered with L-mode profiles, the negative triangularity configuration is in a robustly ELM free regime.Specifically, recent DIII-D experiments reveal that ELMs are robustly avoided with strong negative triangularity shaping, featuring a significantly enhanced edge pressure gradient compared to the typical L-mode plasma with positive triangularity shaping [20].Furthermore, the negative triangularity configurations also possess favorable characteristics such as large particle flux, increased major radius for improved power handling, and enhanced target wetted area in the divertor region [21,22].Therefore, this scenario provides an alternative possibility and choice for future fusion reactors.
Analysis of ideal high-n ballooning modes instability in the DIII-D tokamak indicates that decreasing the top negative triangularity raises the threshold for accessing the second stable region, resulting in a higher L-H transition power threshold observed in experiments [23].It is suggested that high-n ballooning instability for H-mode inhibition is a direct consequence of the negative triangularity shaping, as strong negative triangularity prevents the magnetic shear from being small or negative over the entire bad curvature region to access the second stable region [24,25].Simulations based on the EPED-CH model [26][27][28] confirm that the negative triangularity configurations in the TCV tokamak exhibit a smaller pedestal height compared to the standard positive triangularity configuration, indicating a degradation of MHD stability due to the limited access to the second stable region.Additionally, the MHD stability boundary in the negative triangularity stage of AUG tokamak discharges is found to be located at approximately half the position observed in the positive triangularity stage [19].As such, the negative triangularity configuration restricts the edge equilibrium parameters by hindering access to the second stable region, thus impeding H-mode operation.It is an interesting and challenging issue of how to quantitatively describe the triangularity effect in the aforementioned process.
In this work, by analyzing a modified ideal ballooning equation with shaping effects included, we investigated how the marginal stability boundary is affected by the triangularity, particularly how the second stable region is either closed or opened.Our results confirm that the negative triangularity leads to the broadening of the unstable region in the marginal stability diagram.Conversely, an increase in elongation, inverse aspect ratio, and safety factor aids in stabilizing the ideal ballooning mode.These findings provide an analytically tractable framework to study how the second stable region becomes inaccessible in the presence of sufficiently negative triangularity.A stable pathway that connects the first and second stable region is proposed through a combination of enhanced elongation, inverse aspect ratio, and safety factor.Utilizing a variational approach, we derived an analytical formula for the critical negative triangularity required to close the access to the second stable region.Furthermore, by incorporating the effect of shear Alfvén wave into the modified ideal ballooning equation with shaping effects, we investigated how the marginally stable discrete Alfvén eigenmodes are affected by the triangularity in the second stable region.Our investigation reveals that in the second ballooning stable toroidal plasmas, compared with zero and positive triangularity shaping, the negative triangularity shaping imposes a mitigation effect on the high-n quasi marginally stable discrete Alfvén eigenmodes, which are independent of the toroidal Alfvén frequency gap.These findings contribute to a more quantitative understanding of the impact of the negative triangularity configuration on the confinement of tokamak plasmas.

Theoretical model of the ideal ballooning mode including shaping effects
In this study, we employ a non-circular cross section for the plasma boundary of the equilibrium, described by the following relations [29,30]: (1) where R 0 is the major radius at the magnetic axis, r is the minor radius (half-width of the surface at the elevation of the centroid) and θ is the poloidal angle.Additionally, κ and δ are the shaping coefficients that control the elongation and triangularity, respectively.In the s − α equilibrium diagram, the normalized magnetic shear s is defined as s = rq ′ /q, where q is the safety factor, and q ′ represents the derivative of q with respect to r.The normalized pressure gradient α is given by α = −2µ 0 q 2 R 0 P ′ 0 /B 2 , where µ 0 is the vacuum magnetic permeability, P 0 represents the equilibrium pressure, and B denotes the magnetic field.The high-n ideal ballooning mode equation, derived from the ballooning representation in the ballooning angle space (−∞ < η < ∞) is [31]: where ζ, γ are the local eigen function and linear growth rate (normalized to the Alfvén frequency ω A ), respectively.In addition, d m takes the form of where ϵ = r/R 0 is the inverse aspect ratio.The second term on the left-hand side of equation ( 3) characterizes the driving pressure force caused by magnetic field curvature, i.e. the ballooning effect.The parameter d m incorporates the effects of elongation and triangularity, and can be related to the ideal Mercier criterion [32], determining whether a 'magnetic well' (d m < 0, stabilizing effect) or a 'magnetic hill' (d m > 0, destabilizing effect) is present.Equation ( 3) serves as the fundamental equation for analyzing the ideal ballooning mode in plasmas with non-circular cross sections.When taking the limit of (κ, δ) → (1, 0), the well-known ideal ballooning equation with a circular cross section can be recovered [33].3) is solved for every point on the s − α space to get the eigen function and linear growth rate γ.The marginal stability boundaries in figure 1 represent the boundaries where γ = 0.The plot displays two distinct stable regions (where γ ⩽ 0), termed as first and second stable region, along with one unstable region (where γ > 0).The first stable region is characterized by low α and high s (which normally corresponds to L-mode), and the second stable region is characterized by high α and low s (which normally corresponds to H-mode).When compared with the circular case (i.e.κ = 1.0, δ = 0.0, black open circles in figure 1), increasing the elongation to κ = 1.4 results in a smaller unstable region.In addition, for a fixed elongation, transitioning from zero to negative triangularity enlarges the unstable region.When the access to the second stable region is not fully closed, the safe path from the first stable region to the second stable region requires sufficiently low magnetic shear.This feature is qualitatively consistent with that in [24].Notably, when the triangularity decrease to δ = −0.4,the stable path connecting the first and second stable regions is closed.This implies that the first and second stable regions are separated by the unstable region, which is the primary reason for the inhibition of H-mode in tokamak plasmas with negative triangularity [21,23,24,28].In contrast, for the positive triangularity (δ = 0.4), the unstable region is much smaller, facilitating access to the second stable region and making it easier to achieve H-mode.Figure 2(a) shows that for fixed values of s = 1, α = 2 (blue star in figure 1) and κ = 1.4,ϵ = 0.1, q = 1.5, positive (negative) triangularity has a stabilizing (destabilizing) effect on the ideal ballooning mode, i.e. increasing (decreasing)  3) with different elongations and triangularities in the s − α space.Here the inverse aspect ratio and safety factor are fixed as ϵ = 0.1, q = 1.5 (whose radial location is between the middle and the edge).The blue star represents the reference location studied in figure 2.

Effect of triangularity on the second stability access
positive (negative) triangularity results in a decrease (increase) in the growth rate.The distinct roles of positive and negative triangularity can be attributed to the d m term, as demonstrated in figure 2(b).For positive triangularity, d m is negative, corresponding to a stabilizing effect of the 'magnetic well'.Conversely, for negative triangularity, d m tends to be less negative and can even become positive if the triangularity is sufficient negative, resulting in a further destabilizing effect of the 'magnetic hill'.According to equation (4), it is important to note that the influence of triangularity is significant only when the elongation is not equal to unity, as the triangularity parameter δ does not affect d m when the elongation is absent (i.e.κ = 1.0).
In order to restore the stabilizing effect of the 'magnetic well' and reopen access to the second stable region in the case of a negative triangularity configuration (δ = −0.4),several strategies are possible.First, we can increase the elongation κ while keeping the other parameters fixed.However, this approach yields mild changes in the stability boundary (figure 3(a)).Second, we can increase the inverse aspect ratio ϵ.Although a narrow stable window with low magnetic shear linking the first and second stable regions emerges, it is too narrow to provide sufficient access (figure 3(b)).Third, we can increase the safety factor q. Similar to the case of increasing inverse aspect ratio, the existence of an access to the second stable region requires low magnetic shear and is limited in parameter space (figure 3(c)).Note that for the situations similar to figure 3(c) with larger q, they indeed open a narrow window for the second stable region access.However, this kind of window is so narrow that it is difficult to safely move from the first stable region to the second stable region.Therefore, we also treat these kind of situations as closed second stability access.Finally, we can simultaneously increase the elongation,  inverse aspect ratio, and safety factor.The results demonstrate that the unstable region shrinks, effectively reopening access to the second stable region from the first stable region, with a significantly expanded stable parameter space (figure 3(d)).Taken together, we employ a spider plot (figure 4) to visually illustrate the possibility to access the second stable region.In the spider plot presented in figure 4, with fixed triangularity δ = −0.4,the dashed orange, yellow and purple lines (with their enclosed space) represent scenarios of increasing elongation κ, inverse aspect ratio ϵ and safety factor q separately, where the access to the second stable region is closed.The solid green line represents the case of increasing κ, ϵ, q simultaneously, which effectively reopens access to the second stable region.Therefore, it can be concluded that the synergistic effect of increasing these three parameters (κ, ϵ, q) is capable of reopening access to the second stable region in the presence of a negative triangularity configuration, in agreement with the simulation results reported in [24].

Critical negative triangularity of closing the second stability access
To calculate the critical negative triangularity for the inhibition of access to the second stable region, we employ a variational approach based on the work of [34][35][36].This approach allows us to solve equation (3) analytically.By defining G = 1 + (sη − α sin η) 2 , A = 2/(κ 2 + 1) and transformation of V = ζG 1/2 , it is convenient to rewrite equation (3) in the following 'Schrödinger-type' form: or equivalently, where γ2 = γ 2 − d m .To make progress, we expand V as V = V 0 + V 1 to the first order, where V 0 is the leading order term and satisfies dV 0 /dη = 0 since it is slowly varying and tends to be constant.For simplicity, we set V 0 = 1.Additionally, in the limit of η ≫ 1, sη dominates over α sin η.Thus α sin η can be ignored and With the expansion and approximation, we choose the trial function for V in the following form: Then equation ( 6) can be rewritten as a variational form: Substituting the trial function of V from equation ( 7) into equation ( 8), and setting H = 0 and γ = 0, we obtain: where only the lowest-order terms have been retained.It needs to be emphasized that the terms proportional to e − 1 |s| account for the coupling of perturbations developed at neighboring resonant surfaces.The eigenfunctions get broadened when the magnetic shear s increases.These terms are important and have been retained in the analysis.The positive (negative) terms in equation ( 9) corresponds to the stabilizing (destabilizing) effect on the ideal ballooning mode.In the case where the path to the second stable region is closed, the marginal stability boundary intersects the horizontal axis in the s − α diagram.By taking the limit s → 0, equation ( 9) is reduced into: This equation establishes a direct linkage among δ, α, κ, ϵ and q.For fixed κ, ϵ and q, there is a functional relationship between δ and α, i.e. δ = F(α).Figure 5(a) illustrates that when the second stable region is closed (i.e. in the limit of s → 0, black solid curve), for a given α, there is a corresponding triangularity.Specifically, the minimum absolute triangularity (here it is δ c = −0.06 with κ = 1.4,ϵ = 0.1, q = 1.5) indicates that below this critical value, accessing the second stable region from the first stable region become inaccessible.
In addition, scenarios akin to figure 3(c) with larger values of q can be model as ds/dα = 0 and s → 0 based on equation ( 9), yielding: As a result, the minimum absolute triangularity in this case (orange dotted curve in figure 5(a)) is close to that of s → 0 (black solid curve in figure 5(a)).It is noteworthy that the minimum absolute triangularity for s → 0 also intersects with the curve of ds/dα = 0, s → 0. This further demonstrates that situations akin to figure 3(c) with larger q can also be considered as closed second stability access with s → 0. Figure 5(b) provides a schematic diagram of different situations (represented by full circle markers) in figure 5(a) with the corresponding colors, aiding in illustrating how the second stability access is closed as the negative triangularity gradually approaches the critical negative triangularity.
In general, the increase of κ, q and ϵ helps to stabilize the ideal ballooning mode, corresponding to an increase of the critical minimum absolute triangularity for closing second stability access.Therefore, on the basis of equation ( 10), we can estimate the critical negative triangularity below which the second stable region is closed.This estimation could serve as a quantitative formula to help identify possible parameter regimes for accessing or cutting off the second stable region.

Effect of triangularity on the discrete Alfvén eigenmodes
It is important to note that in the regime of large-α second ballooning stable toroidal plasmas, a distinct type of eigenmode known as α-induced toroidal Alfvén eigenmodes (αTAEs) emerges [37][38][39].These αTAEs are trapped within the α-induced potential well due to the combined effects of significant pressure gradient and ballooning curvature.Unlike conventional TAEs, αTAEs exhibit quasi marginally stable behavior (referred to as a 'bound state') and possess characteristic Alfvén frequencies determined by the MHD parameters.Remarkably, αTAEs can exist even in the absence of the toroidal frequency gap, as the α-induced potential well effectively decouples these trapped eigenmodes from the Alfvén continuum, allowing them to persist within the continuous spectrum [37].Consequently, αTAEs display qualitative distinctions from conventional TAEs, which typically vanish into the shear Alfvén continuum and experience substantial continuum damping [40].In contrast, αTAEs feature negligible continuum damping (with a damping rate on the order of 10 −4 ω A ) due to wave energy tunneling through finite α-induced potential barriers.As a consequence, αTAEs can be readily destabilized by energetic particles through waveparticle resonances, given that the characteristic Alfvén frequencies align with the characteristic frequencies of energetic or alpha particles generated during heating/ignition experiments involving neutral beam injection, radio-frequency waves, or deuterium-tritium fusion reactions [41].Moving forward, we will investigate the impact of triangularity on the behavior of αTAEs.
In the ideal MHD description of the αTAEs, the effect of shear Alfvén wave [42] can be incorporated into equation (6), resulting in a modified governing equation: where Ω = ω/ω A , ϵ 0 = 2(ϵ + d∆/dr) with ∆ being the Shafranov shift.The effective Schrödinger potential is: V By setting q = 4.5, ϵ = 0.3, ϵ 0 = 0.2, s = 1, κ = 1.7 and a large α = 3 (green star in figure 3(d)), we examine the effective Schrödinger potential barriers (solid lines) along the magnetic field line (ballooning angle) with different triangularities, as shown in figure 6.It can be observed that negative triangularity with δ = −0.4(positive triangularity with δ = 0.4) corresponds to lower (higher) peaked potential barriers compared to the case of zero triangularity (δ = 0.0).This indicates that the shaping factor of triangularity could impact the αTAEs by altering the height of the potential barriers.The dashed lines in figure 6 depicts the corresponding eigenmodes of the αTAEs, which exhibit similar structures characterized by peaks at η = 0 and being bounded by the two sharply peaked potential barriers as shown by the solid lines in figure 6.However, the negative triangularity leads to larger damping due to energy leakage through tunneling across the potential barriers.This can be observed from the smaller mode amplitude within the potential barriers as shown in figure 6. Quantitatively, we find that the αTAEs associated with negative triangularity (δ = −0.4)exhibit lower real frequencies, i.e.Ω r,δ=−0.4< Ω r,δ=0.0< Ω r,δ=0.4 , and higher damping rates, i.e. |Ω i,δ=0.4| < |Ω i,δ=0.0| < |Ω i,δ=−0.4|.These results could provide insights into the impact of triangularity on the characteristics of the αTAEs.
Furthermore, figure 7 displays the α dependence of the real frequency and damping rate of the αTAEs with fixed q = 4.5, ϵ = 0.3, ϵ 0 = 0.2, s = 1, κ = 1.7 and different triangularities.The damping rate is found to be negligibly small, i.e. |Ω i /Ω r | < 10 −3 .The real frequency of the αTAEs exhibits a similar increasing trend with the increase of α.For a fixed α, negative (positive) triangularity corresponds to a smaller (larger) real frequency compared to the case of zero triangularity.Specifically, if the real frequency of αTAEs falls into the toroidal Alfvén frequency gap, i.e.Ω L ⩽ Ω r ⩽ Ω U , where Ω R L = [4(1 ∓ ϵ 0 )] −1/2 , the eigenmode experiences backscattering from the periodic potential structures generated by the finite-ϵ 0 term, resulting in no continuum damping [37].Figure 7 demonstrates that the values of α corresponding to the real frequency falling into the toroidal frequency gap are larger for negative triangularity, while smaller for positive triangularity compared to the case of zero triangularity.
Taken together, in the large α second ballooning stable domain, negative triangularity tends to inhibit the emergence of αTAEs, while positive triangularity has the opposite effect.Based on these findings, it can be speculated that in the presence of energetic particles [43,44], the destabilization of αTAEs will be more difficult with negative triangularity compared to zero and positive triangularity.This is beneficial for sustaining a larger pressure gradient in tokamaks subjected to auxiliary heating or fusion reactions.

Conclusion and discussion
In summary, we have investigated how negative triangularity cuts off the access to the second stable region for the ideal ballooning mode.We found that this behavior arises from the weakening of the stabilizing effect from the 'magnetic well' or the emergence of a further destabilizing effect from the 'magnetic hill'.Our study has also demonstrated that increasing the elongation, inverse aspect ratio, and safety factor simultaneously could establish a substantial and stable path for accessing the second stable region.Additionally, we have employed a variational approach to simplify the evaluation of second stable region access through an analytical formula.Moreover, our analysis revealed that in the second ballooning stable toroidal plasmas, the emergence of quasi marginally stable αTAEs tends to be more difficult in the case of negative triangularity compared to zero and positive triangularity.These findings could provide valuable insights for a quantitative understanding of how negative triangularity configuration impacts the confinement both in present and future fusion plasmas.
It should be noted that the ideal ballooning equation presented in equation ( 3) can be extended to study the non-ideal effects if the ion viscosity, resistivity, current diffusivity and thermal conductivity are included [31].In particular, the resistivity and current diffusivity can destabilize the ideal ballooning mode, resulting in resistive ballooning mode and current diffusive ballooning mode [31,45].These non-ideal effects may interact with the shaping effects and affect the stability boundary of the ballooning mode.In addition, it is important to emphasize that incorporating shaping effects in the ideal ballooning mode equation also pertains to modifications of the fluid-like potential energy (as discussed for the αTAE case).This modification can have implications for the vorticity equation governing Alfvén eigenmodes in the presence of energetic particles [18,41,43,44].Thus, the stability properties of Alfvén eigenmodes resonant with energetic particles can be modified by shaping effects, presenting an intriguing avenue for future research.
Furthermore, it is essential to acknowledge that this work primarily focuses on a local analysis of the ideal ballooning mode to provide the physical understanding of how the key elements affect the second stability access.However, the impact of global effects from the equilibrium profiles on the stability of the ballooning mode in the negative triangularity configuration requires further investigation.In general, the self-consistently generated bootstrap current will modify the magnetic shear as well as the current drive of the kink/peeling mode, which may impact the stability of the edge region and the access criterion of the second stable region [24].Our model may be expanded to include the bootstrap current by using Sauter model [46] and investigate the edge peeling-ballooning mode [47] with shaping effects incorporated.The coupling between the finite toroidal mode number peeling mode and the high toroidal mode number ballooning mode plays a significant role in constraining the stability boundary [47,48] and pedestal pressure gradient [23,28].Therefore, exploring how shaping effects affect the linear stability boundary and nonlinear dynamic behaviors of the peeling-ballooning mode in realistic tokamak devices is of great importance for future studies.In addition, more quantitative comparison with experiments will also be pursued for future work.Such investigations would provide a more comprehensive understanding of the overall stability of the plasma system in the presence of shaping effects and shed light on their impact on the nonlinear dynamic process of MHD modes.

3. 1 .Figure 1
Figure 1 illustrates the direct numerical solutions of equation (3) with the boundary condition that ζ → 0 as |η| → ∞.The ideal ballooning equation (3) is solved for every point on the s − α space to get the eigen function and linear growth rate γ.The marginal stability boundaries in figure1represent the boundaries where γ = 0.The plot displays two distinct stable regions (where γ ⩽ 0), termed as first and second stable region, along with one unstable region (where γ > 0).The first stable region is characterized by low α and high s (which normally corresponds to L-mode), and the second stable region is characterized by high α and low s (which normally corresponds to H-mode).When compared with the circular case (i.e.κ = 1.0, δ = 0.0, black open circles in figure1), increasing the elongation to κ = 1.4 results in a smaller unstable region.In addition, for a fixed elongation, transitioning from zero to negative triangularity enlarges the unstable region.When the access to the second stable region is not fully closed, the safe path from the first stable region to the second stable region requires sufficiently low magnetic shear.This feature is qualitatively consistent with that in[24].Notably, when the triangularity decrease to δ = −0.4,the stable path connecting the first and second stable regions is closed.This implies that the first and second stable regions are separated by the unstable region, which is the primary reason for the inhibition of H-mode in tokamak plasmas with negative triangularity[21,23,24,28].In contrast, for the positive triangularity (δ = 0.4), the unstable region is much smaller, facilitating access to the second stable region and making it easier to achieve H-mode.Figure2(a) shows that for fixed values of s = 1, α = 2 (blue star in figure1) and κ = 1.4,ϵ = 0.1, q = 1.5, positive (negative) triangularity has a stabilizing (destabilizing) effect on the ideal ballooning mode, i.e. increasing (decreasing)

Figure 1 .
Figure 1.The marginal stability boundaries calculated from the ideal ballooning equation (3) with different elongations and triangularities in the s − α space.Here the inverse aspect ratio and safety factor are fixed as ϵ = 0.1, q = 1.5 (whose radial location is between the middle and the edge).The blue star represents the reference location studied in figure2.

Figure 2 .
Figure 2. (a) The growth rate of ideal ballooning mode respective to the triangularity with the other parameters fixed as s = 1, α = 2 (i.e.blue star in figure 1), κ = 1.4,ϵ = 0.1, q = 1.5 .(b) The corresponding dm term respective to different δ, where positive dm corresponds to 'magnetic hill', and negative dm corresponds to 'magnetic well'.The vertical dashed line indicates the separation between positive triangulariry and negative triangularity.

Figure 3 .
Figure 3.The modification of ideal ballooning mode stability boundary when only increasing (a) elogation κ, (b) inverse aspect ratio ϵ and (c) safety factor q with the negative triangularity configuration δ = −0.4.(d) The second stability access is reopened when increasing these three parameters (κ, ϵ and q) simultaneously with δ = −0.4.The green star in figure 3(d) represents the reference location studied in figure 6.

Figure 4 .
Figure 4. Spider plot displaying four axes (in clockwise direction):the triangularity δ, elongation κ, inverse aspect ratio ϵ and safety factor q. The dashed lines (with their enclosed space) indicate that the access to the second stability region is closed, and the solid green line represents that the access to the second stability region is reopened.

Figure 5 .
Figure 5. (a)The triangularity δ respective to normalized pressure gradient α when the access to the second stable region is closed (i.e.s → 0 in equation(10) and ds/dα = 0, s → 0 in equation (11)).The minimum absolute triangularity (black dash-dotted line) gives the critical negative triangularity for closing the second stability access.Note that the minimum absolute triangularity for s → 0 also intersects with the curve of ds/dα = 0, s → 0. (b) A schematic diagram of different situations (represented by full circle markers) in figure5(a) with the corresponding colors, aiding in illustrating how the second stability access is closed as the negative triangularity gradually approaches the critical negative triangularity.