Geometric dependencies of the mean E × B shearing rate in negative triangularity tokamaks

This paper presents a comparative study of the poloidal distribution of the mean E × B shearing rate for positive triangularity (PT) and negative triangularity (NT) tokamaks. The effects of flux surface up–down asymmetry due to asymmetric upper and lower triangularities are also considered. Both direct eddy straining and the effects on Shafranov shift feedback loops are examined. Shafranov shift increases the shearing rate at all poloidal angles for all triangularities, due to flux surface compression. The maximum shearing rate bifurcates at a critical triangularity δ crit ( ≲ 0 ) . Thus, the shearing rate is maximal off the outboard midplane for NT, while it is maximal on the outboard midplane for PT. For up–down asymmetric triangularity, the usual up–down symmetry of the shearing rate is broken. The shearing rate at the outboard midplane is lower for NT than for PT, suggesting that the shearing efficiency in NT is reduced. Implications for turbulence stabilisation and confinement improvement in high-β p NT and internal transport barrier discharges are discussed.


Introduction
Negative triangularity (NT) discharges have demonstrated Hmode-like pressure and energy confinement times for L-modelike edge conditions [1][2][3][4][5].Improved confinement with an Lmode edge is advantageous to a fusion reactor [6,7].This is because NT L-mode is an attractive operation regime that is naturally free of edge localized modes (ELMs).Also, NT discharges manifest a very weak degradation of confinement with 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.
power [1], along with broader scrape-off layer heat flux widths as compared to conventional PT H-mode cases, reduced fluctuation levels [1][2][3][4], and reduced plasma wall interaction [8].However, the understanding of the physics of confinement improvement and L-H transition in NT is still in its infancy.Improved confinement in L-mode should facilitate easy access to H-mode i.e. with less power than the conventional L-H transition threshold.While this is observed at weak NTs, Hmode becomes completely inaccessible at strong NTs (δ u < −0.18), even if high power is applied.This has been linked to the loss of access to the second stability region of the infinite-n ideal ballooning modes [9,10].This model is built upon a previous study predicting reduced pedestal height, clamped by degraded peeling ballooning (PB) and kinetic ballooning mode (KBM) stability, due to closed access to the second stability region for ballooning modes in the case of NT [11].Some of the past experiences with conventional H-mode discharges suggest that second stability access may not be a necessary requirement for H mode.The loss of second stability, triggered by changing the squareness of the plasma shape, only changes low-frequency-high-amplitude ELMs to highfrequency-low-amplitude ELMs, without eliminating the Hmode [12,13].A recent experimental study using electron cycloton emission (ECE) imaging suggests that NT edge pressure is limited by low-n interchange type magnetohydrodynamics (MHD) modes or resistive ballooning modes [14].Gyrokinetic simulations [3,4,[15][16][17][18][19] attribute the linear stabilisation of trapped electron mode or ion temperature gradient mode to the observed reduction of turbulence and transport in the core of NT configuration.One thus naturally wonders about the role of mean E × B shear in NT confinement and L-H transition physics.Clearly, the ideas in the NT landscape are still evolving, and a consensus on confinement and L-H physics is still lacking.This requires a study of the mechanisms of turbulence saturation and transport in NT shapes.
Zonal flow shear [20][21][22] and mean E × B shear are candidate players in the saturation of drift wave turbulence in tokamaks, and also play significant roles in the L-H transition.Both zonal flow shear and mean E × B shear break up turbulent eddies, thus reducing the turbulence coherence length [23].As a result, transport is reduced, and regulated by both zonal flow and mean E × B shear.Our recent work shows that the zonal flows are weaker in NT than in PT due to enhanced neoclassical polarisation, from an increase in the trapped fraction in NT [24].As zonal flow lowers the threshold power for L-H transition [25], the prediction of reduced zonal flows in NT is consistent with the observation of increased power threshold for L-H transition.Note that no validated first principle theory of L-H transition [26,27] exists.However, the transition is almost always linked to transport bifurcation due to mean E × B shearing [25,28,29].Similarly, core transport barriers in high poloidal beta reversed shear discharges-often called internal transport barriers (ITBs)-are sometimes linked to transport bifurcation induced by the local mean E × B shear [30][31][32].
Thus, E × B shear suppression of turbulence and transport is one of the key elements of the physics of transport barriers!Given the significant role of mean E × B shear in transport barrier formation, one wonders what happens to the shearing rate when the flux surface shape changes from PT to NT?This is precisely the aim of this paper, which illustrates magneticgeometry-dependent features of the mean E × B shearing rate contrasting the effects of PT and NT.It is well known that magnetic geometry plays a role in shearing physics [33].Here, we focus on the interplay of NT configuration with mean E × B shearing.Usually, shearing is considered as a flux surface averaged quantity.Experiments usually report the shearing rate at the outboard midplane.Here, we study the poloidal structure of the mean E × B shearing rate as the flux surface shapes vary from PT to NT, using Miller's parametrised equilibrium model [34].This also allows the study of the local parametric dependence of the shearing rate with triangularity gradient, elongation, elongation gradient squareness, squareness gradient and Shafranov shift gradient as the triangularity changes from PT Our analysis shows that the mean E × B shear increases with increasing Shafranov shift gradient, because of enhanced flux surface compression.Thus, there is a direct boost of mean shear by the Shafranov shift, which complements the conventionally invoked Shafranov shift effect.This observation and the related physics analysis are the major results of this paper.The rest of the paper is organised as follows.The dependencies of the shearing rate on different geometric parameters is calculated in section 2. The results are discussed and conclusions are given in section 3.

Flux surface geometry dependence of mean E × B shearing rate
The Hahm-Burrell formula for the mean E × B shearing rate [33], ignoring mean parallel flow shear, is obtained from a twopoint correlation calculation for an axisymmetric toroidal system and reads as where ∆ψ 0 is the turbulence correlation width in poloidal magnetic flux ψ and ∆ζ is the toroidal correlation angle of the ambient fluctuations.In fact, ψ is the stream function for poloidal magnetic field and is poloidal magnetic flux (divided by 2π).The mean electrostatic potential is assumed to be a flux function i.e.Φ 0 = Φ 0 (ψ).Since, fluctuation diagnostics (such as beam emission spectroscopy, Doppler back scattering etc) measure correlation length ∆r in the radial co-ordinate r, it is useful to express ∆ψ in terms of ∆r as ∆ψ = ∆r ∂ψ ∂r .
Here, R is the major radius and B θ is the poloidal magnetic field.Similarly, the toroidal correlation angle ∆ζ can be expressed in terms of poloidal correlation angle ∆θ as ∆ζ = ν∆θ, where ν = 2π I d pol is an effective measure of the total poloidal current I d pol (both plasma and toroidal field coil currents) outside the flux surface ψ = const.J is the Jacobian of transformation from toroidal coordinates to orthonormal Euclidean coordinates, ⃗ r(r, θ, ζ) with ⃗ r = R sin ζx + R cos ζŷ + Zẑ being the position vector in the Euclidean space spanned by the unit basis vectors (x,ŷ,ẑ).The coordinate conventions used in the calculations are explained in figure 1.Notice that J /ψ ′ is the Jacobian of the flux coordinates (ψ, θ, ζ).Therefore, where which has been obtained from the definition of the global safety factor q. Thus, the magnetic geometry/topology dependence of the mean E × B shearing rate enters through the Jacobian J , the major radius R and the radial gradient of poloidal flux i.e. ψ ′ for fixed ∂ 2 ∂ψ 2 Φ 0 (ψ).Clearly, ∆ψ0 ∆ζ is not a flux function, and varies with θ on a given flux surface.Therefore, the mean shearing rate varies with θ on a flux surface, such that the shearing rates are not symmetric at the inboard and outboard midplanes.The in-out asymmetry of the mean E × B shearing rate and fluctuations has been observed in DIII-D PT experiments [38].The factor R 2 ψ ′2 J captures the poloidal variation of the mean E × B shearing rate on a flux surface.To specify the flux surface shape, we use the local parametrised model for D shaped plasmas developed by Miller et al [34] and generalised for up-down asymmetric flux surfaces with finite squareness Here, δ u is upper triangularity, δ l is lower triangularity, κ is ellipticity or elongation and σ is the squareness of the flux surface.For up-down symmetric flux surfaces, the shape is parametrised by a single triangularity parameter δ = δ u = δ l .Note that r is the minor radius at θ = 0.The primary advantage of this model as compared to a full numerical equilibrium is that the parameters can be individually varied.This allows for systematic studies of the effects of each parameter upon stability and transport for shaped flux surfaces.Here, we study the effects of each of these shaping parameters on the mean E × B shearing rate.The effects of triangularity receive special focus.The magnetic field is defined in flux coordinates (ψ, θ, ζ) as: The Jacobian J of transformation ⃗ r(r, θ, ζ) is defined as and Z given by equations ( 4) and ( 5), the Jacobian J becomes: where is the triangularity gradient, and S σ = r σ ∂σ ∂r is the squareness gradient.Thus, the mean E × B shearing rate depends not only on local triangularity δ, ellipticity κ and squareness σ but also on their local radial gradients S δ , S κ , and S σ through the Jacobian J .Here, we assume R ′ 0 , S κ and S δ and S σ are independent parameters.For vanishing squareness σ = 0, the Jacobian takes the familiar form shown in [24] i.e.
where x = sin −1 δ(r).In the following, we examine how the poloidal structure of the mean shearing rate varies with the flux surface shaping parameters for fixed ∂ 2 ∂ψ 2 Φ 0 (ψ) and fixed ratio of radial correlation length to poloidal correlation angle, i.e. for fixed ∆r ∆θ .

Variation of mean shearing rate with triangularity δ
Notice that the geometric modulation to the mean E × B shearing rate is manifested through the term R 2 ψ ′2 J , where the explicit formulas for R, ψ ′ , and J are given by equations ( 4), ( 3) and ( 7) respectively.Hence, the variation of mean E × B shearing rate with shaping parameters is inferred based on the dependence of the factor R 2 ψ ′2 J on shaping parameters.Variations of the poloidal distribution of the shearing rate and the flux surface averaged shearing rate with triangularity δ = δ u = δ l for the up-down symmetric equilibria are shown in figure 2. This figure clearly shows that: • The shearing rate is maximal at the outboard midplane for PT δ + .But the shearing rate is maximal off the outboard midplane for NT δ − .So for δ − , the shearing effect is stronger for finite k x modes than for k x = 0 modes.Recall that k x = 0 modes are the most dangerous modes, which balloon at θ = 0 and cause maximum transport by shortcircuiting the plasma radially.• The peak shearing rate bifurcates at a critical triangularity δ crit .This is because the Jacobian (or equivalently the local safety factor ν) is a nonlinear function of δ, which exhibits spontaneous symmetry breaking (i.e. the minimum of the Jacobian bifurcates) for δ < δ crit .For δ < δ crit , the minimum splits into two and locates symmetrically above and below the outboard midplane for up-down symmetric triangularities.The δ crit can be obtained from the solution of equation Clearly, the critical triangularity δ crit for the onset of bifurcation is a function of triangularity gradient S δ , ellipticity gradient S κ and the Shafranov shift gradient R ′ 0 , squareness σ and squareness gradient S σ .The critical triangularity is δ crit ≲ 0 for typical experimental parameters.
• Peak shears move towards the good curvature region and the shearing rate at the outboard midplane θ = 0 decreases with increasing δ − .• The poloidal width of the shear distribution increases, so that the flux surface averaged shear for NT is slightly higher than that for PT.The shearing is weaker at the poloidal midplane for NT, at equal values of radial force balance ∂ 2 ∂ψ 2 Φ 0 (ψ), while the fluctuation intensity balloons at θ = 0.The mismatch in ballooning angle and maximum shear location may reduce the shearing efficiency for NT.This may contribute to the observed increase in the L-H power threshold.Also, the transition might be initiated off the midplane for NT, in contrast to PT shapes.

Effect of up-down asymmetric triangularity.
The effect of flux surface up-down asymmetry due to up-down asymmetric triangularity is analysed here.The plots in figure 3 show how the poloidal structure of the mean shear and the flux surface averaged mean shear vary with the varying degree of asymmetry in the upper and lower triangularities.These plots clearly show that: • The flux surface averaged shearing rate is higher for negative upper triangularity δ − u than for positive upper triangularity δ + u , for fixed lower triangularity δ l .Also, for fixed upper triangularity δ u , the shearing rate is higher for negative lower triangularity δ − l than for positive lower triangularity δ + l .Flux surface averaged shearing decreases with |δ l | because the flux gradient ψ ′ decreases with |δ l |.
• The poloidal distribution of the shearing rate becomes asymmetric in θ when the upper and lower triangularities are different i.e. δ u ̸ = δ l .This is because the poloidal structure of the Jacobian above the midplane (0 < θ < π) depends on δ u , whereas the poloidal structure of the Jacobian below the midplane (π < θ < 2π) depends on δ l .The structure of the shearing rate in (θ − δ u ) space varies strongly with δ l , as shown in figure 3. Notice the contrast with the updown symmetric flux surface case shown in figure 2, where the max shearing rate (at θ = 0) bifurcates into two equal strength peaks located symmetrically above and below the outer midplane.Comparison of the poloidal distribution of the shearing rate shows that the poloidal width is bigger for δ − u (δ − l ) than for δ + u (δ + l ), for fixed δ l (δ u ).Interestingly, for strong δ − l (>0.2), another peak of the shearing rate appears below the outboard midplane for all δ u .The shearing peak below the outer midplane gets stronger and moves further away for the outboard midplane (θ = 0) on increasing δ − l .For δ l > δ l,crit and δ u > δ u,crit the shearing rate is maximal at the outboard midplane (θ = 0).For δ l > δ l,crit and δ u < δ u,crit the shearing rate is maximal above the outboard midplane (θ > 0).For δ l < δ l,crit and δ u > δ u,crit the shearing rate is maximal below the outboard midplane (θ < 0).For δ l < δ l,crit and δ u < δ u,crit the shearing rate peaks both at θ < 0 and θ > 0. The height and locations of the shearing peaks depend on the values of δ l , δ u and how far they are from their respective critical values δ l,crit and δ u,crit .Here, δ l,crit = δ u,crit = δ crit since all other shaping parameters are up-down symmetric.

Variation of mean shearing rate with triangularity gradient S δ
Variations of poloidal distribution of the shearing rate with triangularity gradient S δ are shown in figure 4. The figure clearly shows that • Increasing S δ moves the critical triangularity δ crit for the onset of bifurcation towards increasing δ − .That is, for higher S δ the geometric bifurcation occurs at higher NT.• The shearing rate at the outboard midplane decreases as S δ increases.
This shows that the radial profile of the triangularity matters, not only its local value.The shearing rate increases significantly upon decreasing the triangularity gradient S δ .• The shearing rate increases with increasing −R ′ 0 for all δ.This is due to an increase in the compression of flux surfaces, which occurs for increasing Shafranov shift.This result is consistent with an earlier observation by Hahm et al [39] for circular flux surfaces.

Variation of mean shearing with
• The critical triangularity δ crit for bifurcation of the max shearing moves towards increasing δ − for increasing −R ′ 0 .That is, the geometric bifurcation occurs at stronger NTs on increasing the Shafranov shift gradient.[40], this effect is significant for high-β p (poloidal beta) ITB regimes.Negative triangularity experiments suggest that β p for δ − is higher than that for δ + [1].
Hence, the Shafranov shift induced boost of mean E × B shear is also important for NT discharges.
A realistic MHD equilibrium study shows that R ′ 0 is not a free parameter and varies with δ, even for fixed β p [41].In fact, −R ′ 0 is higher for δ − than for δ + for fixed β p .As a result, mean shearing should increase when PT→NT, even at fixed β p , owing to the enhanced Shafranov shift gradient.This in turn can improve confinement and increase β p .The increase in β p then drives stronger shearing and Shafranov shift, further increasing confinement and β p .Thus, enhanced mean E × B shearing by Shafranov shift produces positive feedback in the development of Shafranov-shift-induced transport bifurcation [35][36][37].Conversely, the Shafranov shift also has a positive effect on the feedback loop of mean E × B shear-induced transport bifurcation, not only through a reduction of linear growth rate [42], but also through the enhanced E × B shearing rate.Thus, the Shafranov shift gradient affects turbulence in two distinct ways: (i) The Shafranov shift stabilises turbulence by the reduction/reversal of magnetic drifts.(ii) The Shafranov shift directly enhances the mean E × B shear, which causes additional turbulence suppression.
While (i) is well known [42,43], (ii) is a novel finding in this paper.Both (i) and (ii) can cause bifurcation independently to enhance confinement, through their positive feedback loops.They can also work in tandem.This is shown in figure 6.However, (i) is often invoked as a mechanism of turbulence suppression and confinement improvement in high-β p discharges [35][36][37], ignoring the role of mean E × B shear.But given the significant boost of mean E × B shear by Shafranov shift, the two mechanisms (i) and (ii) can reinforce each other     to reduce the critical ∇P for the onset of bifurcation to an ITB state in PT reversed shear plasmas.

Variation of mean shearing with elongation κ and elongation gradient Sκ
Variations of poloidal distribution of shearing rate with elongation κ and elongation gradient S k are shown in figure 7. The figure clearly shows that: • The shearing rate increases with increasing κ and S κ .
• The critical triangularity δ crit for bifurcation of the maximum shearing is independent of κ, while the δ crit moves towards higher δ − upon an increase in S κ .

Variation of mean shearing with squareness σ and squareness gradient Sσ
Variations in the poloidal distribution of the shearing rate with squareness σ and squareness gradient S σ are shown in figure 8.The calculations are done for up-down symmetric flux surface shapes i.e. δ = δ u = δ l to clearly delineate the effect of squareness σ.The figures clearly show that: • The flux surface averaged shearing rate increases with increasing squareness and decreases with decreasing squareness, such that the shearing rate is higher for positive squareness σ + thanfor negative squareness σ − .• However, the shearing rate at θ = 0 decreases with increasing σ such that the shearing rate is higher for σ − than for σ + .• On increasing σ + the critical triangularity δ crit for the onset of geometric bifurcation increases i.e. moves towards higher δ + .The maximal shearing peaks, located symmetrically about the outboard midplane, get stronger and the poloidal width of the shearing rate increases with increasing σ + .• On decreasing σ below zero, the geometric bifurcation of the maximal shearing rate in δ disappears.The shearing at θ = 0 becomes the maximal shearing and the poloidal width of the shearing distribution becomes narrow on increasing σ − .
Variations in the shearing rate with the squareness gradient S σ are shown in figure 8.The figure clearly shows that the shearing rate increases with increasing S σ , for all δ.However, the rate of increase of shearing with S σ is weak.This is because S σ appears in multiples of σ in the Jacobian (see (7)) and since |σ| < 1, the effect of S σ is weakened.Notice that the maximum shearing rate and the shearing rate at θ = 0 increases with increasing S σ .Also, the critical triangularity δ crit for the onset of geometric bifurcation moves towards increasing δ − .

Variation of mean shearing with inverse aspect ratio ϵ and safety factor q
Variations in the poloidal distribution of the shearing rate with inverse aspect ratio ϵ and safety factor q are shown in figure 9.
The figure clearly shows that: • The shearing rate increases with increasing ϵ, while it decreases with increasing q.This is because the flux gradient ψ ′ increases with increasing ϵ and decreases with increasing q. • The critical triangularity δ crit for bifurcation of the maximum shearing is independent of ϵ and q.

Discussion and conclusions
Mean E × B shear is well known to reduce turbulent transport and improve confinement, even in L-mode discharges [44].
The observation of improved confinement in the L-mode and diverging threshold power for L-H transition in strongly NT (δ < −0.18) discharges has motivated this research to evaluate the mean E × B shear strength in matched PT and NT shapes.
Here, we study the flux surface shape-dependent features of the mean E × B shearing rate, which are relevant to L-H transition physics in different plasma shapes.The Hahm and Burrell formula (1) for the mean E × B shearing rate [33] for an axisymmetric toroidal system is analysed for NT and PT flux surface shapes, including the effects of up-down asymmetry, using the locally parametrised equilibrium model of Miller et al [34].Here, the radial electric field shear ∂ 2 ∂ψ 2 Φ 0 (ψ) is taken as fixed, and set by ion radial force balance.Detailed results are inferred from the dependence of the term R 2 ψ ′2 J on the shaping parameters.The prefactor R 2 ψ ′2 J yields a purely geometrical modification to the mean E × B shearing rate.We study this factor as triangularity varies from δ > 0 to δ < 0. The results are summarised in table 2. Notice that the shearing peaks symmetrically above and below the outboard midplane for up-down symmetric NT flux surfaces.This up-down symmetry of shearing is broken, and the shearing is strongest above the outboard midplane for up-down asymmetric NT shapes.Geometric modifications of the mean shearing can have the following important implications.
• The Shafranov shift gradient directly boosts the shearing rate, for all δ.This is due to an increase in flux compression with increasing Shafranov shift gradient.This effect is significant for high-β p regimes as well as NT discharges.This means that this effect will be even more important for NT high-β p ITB regimes.Mean shear enhancement due to the Shafranov shift gradient provides additional turbulence suppression.This mechanism complements the commonly invoked mechanism for confinement improvement in highβ p regimes, which is based on stabilisation due to curvature drift reduction/reversal by Shafranov shift.• Shearing is weaker at the outboard midplane for NT, at equal values of radial force balance ∂ 2 ∂ψ 2 Φ 0 (ψ), while fluctuations balloon at θ = 0. Thus, shearing efficiency is reduced for NT.This may contribute to the increase of the L-H power threshold for NT.This mechanism complements the one based on the loss of second stability of ideal MHD ballooning modes [9,10].
However, notice that going from the PT to the NT edge produces two competing effects.The direct effect of going from PT to NT is a reduction of the mean E × B shear at the outboard midplane.The indirect effect of going from PT to NT is a global (in θ) boost of the mean E × B shear by self-consistently increased Shafranov shift.The parameters in Miller's model are not all free.Thus, MHD equilibrium codes should be used for a more accurate calculation of the mean E × B shearing rate variations with the shaping parameters.
Finally, we present some suggestions for the experimentalists.
• Since the mean shearing is maximal off the outboard midplane for δ < δ crit (∼NT), the eddy tilting should be maximum off the outboard midplane.For up-down symmetric shapes, eddy tilting should maximise symmetrically above Inverse aspect ratio ϵ (figure 9) Increases with increasing ϵ

Increases with increasing ϵ
No effect Safety factor q (figure 9) Decreases with increasing q Decreases with increasing q No effect and below the outboard midplane.For up-down asymmetric flux surface shapes, eddy tilting should be maximised above the outboard midplane for δ u < δ crit and δ l > δ u .Eddy tilting should maximise below the outboard midplane for δ l < δ crit and δ u > δ l .This can be directly visualised in experiments using gas puff imaging [45].The poloidal distribution of the tilt angle of the joint PDF of the radial and poloidal velocity fluctuations should also exhibit symmetry/asymmetry about the outboard midplane, depending on the flux surface symmetry.This implies that the poloidal envelope of the Reynolds stress should also exhibit similar symmetry/asymmetry due to flux surface shaping effects.• Reassess the role of mean E × B shear in high-β p reversed shear ITB discharges given that Shafranov shift boosts the mean E × B shear.

0
Shafranov shift gradient R ′ Variations of the poloidal distribution of shearing rate with Shafranov shift gradient R ′ 0 are shown in the plots in figure 5.The figure clearly shows that:

Figure 3 .
Figure 3. (a) Flux surface averaged shearing rate vs upper triangularity (δu) for various lower triangularities (δ l ) as parameters.(b)-(h) Structure of mean E × B shearing rate in (θ − δu) space for different δ l 's.The solid red line tracks the local maxima of the shearing rates at θ > 0 and θ < 0 and the dotted red line tracks the shearing rate at θ = 0.For δu < δ u,crit and δ l > δu shearing rate is maximal above the outer midplane.For δ l < δ l,crit and δu > δ l shearing rate is maximal below the outboard midplane.Other parameters are the same as in figure 2.

Figure 4 .
Figure 4. (a) Variations in (θ − δ) structure of the max shearing rates (solid lines) and the shearing rate at θ = 0 (dashed lines) with triangularity gradient S δ .The max shear bifurcation point δ crit moves along δ − on increasing S δ .Shearing at θ = 0 decreases on increasing S δ .(b) Projection of (a) on δ-axis.Other parameters are the same as in figure 2.

Figure 5 .
Figure 5. (a) Maximum shearing rates structure in (θ − δ) space with Shafranov shift gradient R ′ 0 as parameter.Shearing rate increases with −R ′ 0 .The bifurcation point moves along increasing δ − on increasing R ′ 0 .Solid lines are the loci of maxima of the mean E × B shearing rate in (θ − δ) space.The dashed lines track the shearing rate at θ = 0. (b)-(d) 3d plots of shearing rates at different R ′ 0 .Other parameters are the same as in figure 2.

Figure 6 .
Figure 6.Feedback loops of mutual interactions of Shafranov shift, mean E × B shear and turbulence.Shafranov shift and mean E × B shear reinforce each other.

Figure 7 .
Figure 7. (a) Shearing rate elevates on increasing the elongation κ.(b) Shearing rate elevates and the bifurcation point moves along the δ − direction on increasing the elongation gradient Sκ.Solid lines are the loci of maxima of the mean E × B shearing rate in (θ − δ) space.The dashed lines track the shearing rate at θ = 0.The solid red line tracks the maximum shearing rate and the dotted red line tracks the shearing rate at the mid-pane.Other fixed parameters: R ′ 0 = −0.4,S δ = 0.8, κ = 1 for (b), Sκ = 1 for (a), σ = 0, Sσ = 0, ϵ = 0.18, q = 3.

Table 1 .
Shaping parameters and their meanings.