Gyrokinetic analysis of turbulent transport by electromagnetic turbulence in finite β plasmas with weak magnetic shear on HL-2A

Turbulent transport and zonal flow (ZF) dynamics in the mixed kinetic ballooning mode (KBM) and ion temperature gradient (ITG) mode dominated turbulence states are analyzed by gyrokinetic simulations based on the experimental observations of KBMs in the HL-2A Internal Transport Barrier (ITB) plasmas with weak magnetic shear configuration. Results have shown that KBMs and ITGs are the main candidates for turbulent transport and the inclusion of fast ions (FIs) can destabilize both instabilities, which is resulted by the decrease of ballooning parameter αMHD hence reduced Shafranov shift stabilization due to the negative FI density gradient, making them more unstable according to the ŝ-αMHD diagram. In addition, the ITGs are suppressed by both dilution and finite-β stabilization effects caused by FIs. Nonlinear simulations excluding the effects of FIs indicate that the transport is minimum at βe=βeexp as the turbulence predominated by ITGs is strongly suppressed by the nonlinear electromagnetic (EM) stabilization and enhancement of ZFs. However, the transport is further increased with β e although the ZFs become stronger due to the transition from ITG to KBM dominated turbulence state. The presence of FIs can modify the relation between ZF shearing rate and β e. The transport level is insensitive to β e when βe⩽βeexp but increases significantly because of the KBM destabilization. Meanwhile, the ZF amplitude reaches maximum at βe=βeexp whereas it suffers an erosion at higher values, implying that the plasma might be a self-organized critical state owning to the interactions among ZFs, KBMs and FI effects hence setting the plasma β around β exp. The results have also provided a possible explanation that the destabilization of EM turbulence is responsible for the ion heat transport stiffness under weak magnetic shear configurations in the off-axis neutral beam injection heated ITB plasmas on HL-2A.

as the turbulence predominated by ITGs is strongly suppressed by the nonlinear electromagnetic (EM) stabilization and enhancement of ZFs.However, the transport is further increased with β e although the ZFs become stronger due to the transition from ITG to KBM dominated turbulence state.The presence of FIs can modify the relation between ZF shearing rate and β e .The transport level is insensitive to β e when β e ⩽ β exp e but increases significantly because of the KBM destabilization.Meanwhile, the ZF amplitude reaches maximum at β e = β exp e whereas it suffers an erosion at higher values, implying that the plasma might be a self-organized critical state owning to the interactions among ZFs, KBMs and FI effects hence setting the plasma β around β exp .The results have also provided a possible explanation that the destabilization of EM turbulence is responsible for the ion heat transport stiffness under weak magnetic shear configurations in the off-axis neutral beam injection heated ITB plasmas on HL-2A.

Introduction
Turbulent transport in magnetic confined devices is one of the most crucial issues in fusion plasma physics for a long time since it will seriously degrade the confinement performance [1].It has been well-known that the particle and heat transport resulted primarily from drift-wave fluctuations, referred to as microturbulence driven by various kinds of temperature and density gradients [2] in magnetically confined fusion devices usually have orders of magnitude larger than those predicted by neoclassical transport theory [3], which considers purely collisional processes.Plenty of theories and simulations have indicated that the heat and particle transports induced by microinstabilities in the typical tokamak plasma parameter range are mainly the long wavelength toroidal ion temperature gradient (ITG) mode [4] and medium scale trapped electron mode (TEM) [5], where the former is excited once the ITG exceeds a critical value while the latter is driven unstable by the trapped electrons and the electron pressure gradient.In addition to the above two main instabilities, the electron temperature gradient (ETG) mode [6] driven by ETG, which is the counterpart of ITG at electron scales, is suggested to be responsible for the electron thermal transport.Much progress has been made in clarifying the impact of these electrostatic (ES) microinstabilities on transport in the relatively low β plasmas (β is the ratio of plasma pressure to magnetic pressure) over the past several decades [7], however, the situations will be quite different for the next-generation fusion reactors which will operate in the high β scenarios where the electromagnetic (EM) turbulence may dominate, such as ITER and DEMO [8].
High β plasma is an important topic in the field of tokamak physics as it offers the potential for highly confined fusion plasmas thus more efficient and cost-effective fusion power, which could revolutionize our energy supply [9].However, achieving high β plasma is not an easy task as it requires a high pressure, which can lead to the excitation of EM instabilities known as the kinetic ballooning modes (KBMs) [10].KBMs are a type of instability when the plasma pressure gradient exceeds a certain threshold and can lead to fast particle and energy loss as they are strongly destabilized by plasma β, which in turn reduce the confinement and limit the achievement of high performance plasma.The KBMs are also referred to as shear-Alfvén ITGs (AITGs) modes which are excited due to wave-particle interactions with thermal ions [11].It has been commonly accepted that the KBMs exist in the H mode pedestal region where the pressure gradients are large and they strongly restrict the pedestal pressure [12][13][14][15].However, it is proven by both theories and experiments that the KBMs/AITGs can also be driven in the core region of plasmas with finite pressure gradient where the magnetic shear ŝ (≡r/q(dq/dr) with r and q being the radial coordinate and safety factor, respectively) is close to zero [16].These situations generally occur in an off-axis neutral beam heated Internal Transport Barriers (ITBs) or even in the high density Ohmic discharges having weak magnetic shear configurations, such as discovered on the HL-2A tokamak [17,18].The underlying physics can be simply explained by the reason that a low ŝ would reduce the KBM threshold α crit in the ŝ-α MHD diagram [19], where is called the ballooning parameter in the fluid description.β ′ is the radial derivative of β and is related to the local pressure gradient.Here the subscript j donates the jth species, L nj and L Tj are the density and temperature gradient scale lengths of the corresponding species and R 0 is the major radius of the device.The excitation of KBM/AITG at low ŝ and finite pressure gradient regions might be particularly important as the modes will induce rather high ion heat transport hence increase the ion temperature profile stiffness level.It is then quite possible that the profiles are set around a marginally stable state close to a certain threshold due to complex interactions among the transport by KBM turbulence, self-generated sheared flows known as zonal flows (ZFs) [20], rotational shear [21] and so on.The behavior of KBMs/AITGs in tokamak plasmas is complicated depending on a number of factors and these modes have significant impact on the transport properties.Understanding the transport features in KBM turbulence is essential for designing and optimizing tokamak scenarios that are stable and capable of producing fusion energy through the achievement of high performance of plasma confinement, especially for devices like ITER which will operate under weak or negative magnetic configurations.
It is widely known that the self-generated meso-scale ZFs, including the low/zero frequency ZFs (L/ZFZFs) [22][23][24] in the core plasma and the high frequency branch geodesic acoustic modes (GAMs) [25] generally existing at the edge will be excited in fully developed drift-wave turbulence through nonlinear three-wave interactions who are able to regulate or suppress the turbulent transport [26].Although the turbulence is also suppressed by the mean flows, including poloidal sheared flow and toroidal rotational shear [27], the role of ZFs in the core is particularly important as the former is strongly damped in these regions while the shearing effect on turbulence eddies by the latter is usually weaker than that of the former.More importantly, in addition to the rotational shear stabilization, the parallel velocity gradient (PVG) [28] generally has destabilization effect on microinstabilities, such as ITGs [29].From this prospective, it is suggested that the rotation and its velocity gradient cannot be designed to be large in order to avoid the PVG modes.Hence, the shearing effect by ZFs may become the most important ingredient in regulating or suppression turbulence and the turbulent transport properties are largely controlled by the dynamics of ZFs in the core.Another factor that affects the behavior of transport is the presence of energetic particles or also called as fast ions (FIs) created by external heating systems such as neutral beam injection (NBI) or ion cyclotron resonance heating, which can interact with plasma waves and create additional pressure gradients within the plasma.Over the past decades, much important progress has been made in clarifying the role of FIs on ITG dominated plasmas, indicating that the FIs have significant stabilization effect on turbulence [30][31][32] and can even lead to new high confinement regimes such as F-ATB on ASDEX-U [33] and 'FIRE' mode on KSTAR [34].It is identified that the FIs can influence the plasma turbulence through a number of mechanisms, including the dilution effect which always exists once another type of ion species differs with the bulk ions is present due to the quasineutrality condition [35], the ES resonance mechanism that FIs can interact with the main ion microinstability through wave-particle resonance when the FI magnetic drift frequency is close to the linear frequency of the wave [36] and the complex nonlinear interactions among microinstabilities, ZFs and FI driven modes [37].However, it is noted that most of the present simulations focus on the effect of FIs on ITGs or TEMs, which are ES in nature.The presence of FIs can directly drive KBMs [38] or modify the plasma profiles, making the plasma more susceptible to KBMs in the weak magnetic shear region.The unique interplay among transport, ZFs and FIs have potential impact on plasma performance and the development of fusion energy makes them a critical area of research, which is the main scope of the present work.
The reset of the paper is organized as follows: section 2 gives the typical experimental observations of kinetic EM microinstabilities in the HL-2A ITB plasmas with weak magnetic shear configuration.In section 3, the GENE gyrokinetic simulation setup and analysis of the dynamics of transport and ZFs in the mixed KBM-ITG turbulence are presented.Finally, conclusions are drawn in section 4.

Observations of KBMs in the weak magnetic shear ITB plasmas on HL-2A
The typical ITB experiments are performed in deuterium plasmas with plasma current I p ∼ 150 kA, toroidal magnetic field B t ∼ 1.37 T and central line-averaged density ne ≈1.2 × 10 19 m −3 on the HL-2A tokamak [39] which has a major radius of R 0 = 1.65 m and plasma minor radius of a ≈ 0.33 m, respectively.The NBI heating is crucial for the formation of ITB on HL-2A which also provides a significant population of FIs.The HL-2A plasma exhibits nearly a circular cross section equilibrium although a divertor configuration is applied during the discharges.The acceleration voltage and port-through power P NBI of the tangential neutral beam are typically 45 kV and 1 MW, respectively.Figure 1 shows a typical experimental result of an ITB discharge of shot #25803, which is achieved with a NBI power of P NBI ≈ 750 kW and the electron cyclotron resonance heating (ECRH) power P ECRH ≈ 1.2 MW, as shown in figure 1(a).It should be note that the ECRH is not important for the ITB formation or the excitation of kinetic Alfvén instabilities.The KBMs/AITGs are identified on the density fluctuation measured by four-channel microwave interferometry [40], which shows a set of high frequency coherent fluctuations around f = 80-200 kHz in the spectrogram of the core microwave interferometer signal at the chordal distance r d = 5 cm corresponding to ρ = r/a ≈ 0.15 at t = 610-800 ms during the ITB phase, as discovered in figure 1(b).However, they are almost invisible in spectrograms at r d = 11 cm (ρ ≈ 0.33), indicating that the microinstabilities are highly localized in the inner plasma core.These instabilities can also be observed by core soft x-ray (SXR) but occasionally measured by Mirnov coils, as shown by the spectra of SXR in figure 1(c), and this phenomenon is perfectly reproducible [17].The fluctuation are also observable at t = 540-620 ms, during which the fishbone activities driven by FIs are also present who are suggested to be important for the formation and sustainment of ITB [41].Nevertheless, we have limited our simulations based on the profiles at t = 662 ms in order to avoid the strong magnetohydrodynamic instabilities which are not resolved by the simulations in the present work, as shown by the dashed lines in figures 1(b)-(c).The inclusion of FI driven instabilities and their impact on turbulence will be left for another important work in the near future.
Shown in figure 2 are the profiles of shot #25803 at t = 662 ms, which is a typical ITB discharge [42].The barrier starts to form at ∼30 ms and is well developed at ∼50 ms after NBI is turned on.The ion temperature T i together with toroidal angular rotation frequency Ω t are measured by a 32-channel Charge Exchange Recombination Spectroscopy (CXRS) diagnostic system with spatial and temporal resolutions about ∼1.5 cm and 12.5 ms [43].A 32-channel fast Electron Cyclotron Emission (ECE) system provides the electron temperature T e with temporal and spatial resolutions up to 0.8 µs and 1 cm [44].The electron density n e profile is reconstructed from the formic acid (HCOOH) laser interferometer [45] measurements through Abel inversion method.All of these profiles are mapped on to the flux surface coordinates.The safety factor (q) profile shown in figure 2(a) is calculated by the kinetic equilibrium and reconstruction fitting code (kinetic EFIT) in the framework of OMFIT integrated modeling [46].It is discovered that the plasma shape generally shows a weak magnetic shear configuration in the core region due to the off-axis NBI heating.The E × B shearing rate due to toroidal rotation calculated as γ E = (r/q)dΩ t /dr in units of c s /R 0 is shown in figure 2(e), where the maximum value locates at the same position of the largest ITG because the gradient of the rotation also reaches maximum at this time.
Here c s = √ T e /m i is the ion sound velocity and R 0 is the major radius of the device.The profiles of FIs are shown in figures 2(c) and (d) which are calculated by the NUBEAM module incorporated in the ONETWO transport solver in the OMFIT framework [47].The position of maximum FI density (n f ) is near ŝ ∼ 0 as can be concluded from figures 2(a) and (c).The FI temperature (T f ) is almost constant across a wide region corresponding to the inner ITB region, indicating a strong relation between energy deposition of FIs and the ITB, as shown in figure 2(d).It is worth to note that the plasma is characterized by different microinstabilities depending on the position, i.e., the weak magnetic shear region of ρ ≲ 0.23 are suspected to be dominated by KBMs/AITGs while the transport in the core ITB region of 0.3 ≲ ρ ≲ 0.42 are generally considered to be controlled by ITG modes.Moreover, it is shown that the rotational shear is maximum around ŝ ∼ 0 but is about an order magnitude smaller at ρ ∼ 0.15 where the KBMs are unstable, indicating that shearing and suppression effect of turbulence transport by rotation is weak, hence we have neglected the effects induced by rotations, including the stabilization by rotational shear as well as the destabilization effect by PVG.
The above conclusion could also be inferred from the profiles of the main parameters that influence the stability of KBM and ITG, namely, the plasma β, α MHD and ŝ profiles shown in figure 3. The profiles are calculated by the experimental measurements and do not consider the contributions from the FIs, which serves as the reference case for comparing the effect of FIs.The KBMs are suggested to exist at radial locations of 0.1 ≲ ρ ≲ 0.23 due to the combined effects of relative high β shown in figure 3(a) hence α MHD and low ŝ, whereas at ρ ≳ 0.28 the ITGs would be the major microinstabilities because of small α MHD and relatively large positive magnetic shear, as depicted in figure 3(b), which is in quantitative agreement with the observation by microwave interferometry.The simulations are carried out based on the parameters corresponding to the location of minimum ŝ at ρ = 0.15 as indicated by the vertical dashed line in figure 3(b), at where the signatures of the KBMs/AITGs are the clearest on microwave interferometry as previously demonstrated in figure 1(b).It is also suggested that at the simulation location where ŝ is negative and q is small, the ITGs are more stable compared to other positions with positive ŝ as previously demonstrated [48], hence the transport driven by KBMs becomes more significant.The bulk ion profiles are modified when considering the FI contributions due to the requirements of quasineutrality condition.The physical parameters used in the simulations are listed in 12 -15.01.60 18.0 table 1, where the cases in the absence and presence of FIs are shown.Our simulations have focused on the effect of plasma β e in order to clarifying the dependence of transport by KBM turbulence and ZFs on β e .We note that the β j of all ionic species are scaled with β e according to the pressure ratios so as that for the self-consistent α MHD and the Shafranov shift effect is also included in our simulations.Other parameters such as gradients and collision rates are kept unchanged unless otherwise stated.

Simulation set up
In this section, the linear and nonlinear simulations of the effects of FIs on the dynamics of ZFs and transport are investigated using gyrokinetic code GENE [49,50] based on the parameters listed in table 1, mainly focusing on the effect of β for the cases with and without FI contribution through scanning β e with a self-consistent α MHD depending on β.GENE is a δf formulated gyrokinetic code that solves the gyrokinetic Vlasov equation coupled self-consistently to Maxwell's equations [51].The local field-aligned coordinate system (x, y, z) were applied where x is the radial coordinate, z is the coordinate along the field line and y is the binormal coordinate [52].Collisions are modeled by a linearized Landau-Boltzmann operator.All the simulations performed here were local and with collisions.An analytical ŝ-α equilibrium [53] is used and it is believed that other choices such as Miller geometry [54] would not affect the nature of transport and ZFs as the configuration of the HL-2A is well characterized by a circular flux surface shape and the geometrical effects are generally small.Such treatment is important especially for the case where the magnetic shear is low and the pressure gradient is relatively large, which will lead to the onset of KBM/AITG even though β is not very high.Besides, although previous simulations have indicated that the parallel magnetic fluctuation (δB || ) can destabilize KBMs, this effect is neglected in the present simulations since it has been demonstrated that the δB || has only weak effect on both ITGs and KBMs at low β e values.At rather high β we can reasonably suspect that the transport is higher and the ZF amplitude is lower than that in the present simulation.Nevertheless, similar β-dependence of transport features can be obtained when δB || is taken into account, despite that the levels of turbulent transport and ZFs might be different.For such reason, the ∇B drift frequency is thus set to be equal to the curvature drift frequency.Convergence tests were carried out for typical linear simulations with radial grid points n x = 64, 64 point discretization in the parallel direction, 48 points in the parallel velocity direction and 16 magnetic moments in order to resolve the mode structure.In addition, typical grid parameters for nonlinear runs were as follows: perpendicular box sizes [L y , L x ] = [210, 136] in units of ρ s = c s /Ω ci with Ω ci = eB/m i being the ion Larmor radius, perpendicular grid discretization [n x , n ky ] = [128, 24], 32 × 32 × 8 point discretization in the parallel, parallel velocity and magnetic moments direction.A reduced electron-to-deuterium mass ratio of m e /m i = 1/800 is used in order to reduce the computational effort.The minimum poloidal and radial wavenumbers are set to k y,min = 0.03 and k x,min = 0.046 with maximum value of k x,max = 2.91, respectively.The heat fluxes are time averaged values over the quasisteady state of the GENE simulations normalized by gyroBohm units as Q GB = T 2.5  e n e m 2 i /(eBR) 2 .The wavenumbers k y and k x are in units of 1/ρ s while the eigenvalues γ and ω are normalized by c s /R 0 .The pure rotational plasma is assumed and the effects of impurities are neglected unless otherwise stated.

Linear stability features of KBM and ITG.
The dependence of the linear stability of the microinstabilities on β e is depicted in figure 4 for parameters given in table 1, where both the cases without and with FIs are shown.The KBMs are not present in the ES simulations, namely, β e = 0.As β e is increased even at β e = 0.25%, the KBMs with small poloidal wavenumbers are destabilized while the modes with large k y ρ s are identified as the ITG modes, which can be inferred by the large and small real frequency branches illustrated in The positive frequency is defined as the ion diamagnetic direction in GENE.Further increasing β e would destabilized the KBMs at large k y ρ s whereas the ITGs are stabilized because of the finite-β stabilization [55], which is the general phenomena widely observed in numerous simulations, as indicated by figure 4(a).The results have shown that the dominate microinstabilities are long wavelength KBMs and ITGs under the experimental β exp e = 0.32%, thus it is suggested that the plasma is a mixed KBM-ITG turbulence state which is also found in other simulations at low ŝ [56].The simulated mode frequency is in the range f mode = ω/2π ∼ 70-120 kHz peaking around 85 kHz (k y ρ s = 0.1), which is in quite good agreement with the experimental values calculated as f exp = f lab -nf v ∼ 85-135 kHz, where n is the toroidal mode number who is estimated in the range of n ∼ 2-5.The f lab and f v are the mode frequency in lab frame and the toroidal rotation frequency, respectively.The situation is similar when considering the FI effects, expect for that the KBMs are further destabilized while the ITGs are stabilized for EM simulations when β e ̸ = 0.For ES case at β e = 0, the ITGs are stabilized since the presence of FIs reduces the thermal ion drive, known as the dilution effect, as shown in figures 4(c) and (d).A rather high β e (≳0.7%) will lead to the excitation of electron modes having frequencies around ω ≈ −10 c s /R 0 at k y ρ s ≳ 0.3 who are identified as the microtearing modes (MTMs) [57].For this reason, we have limited our simulations at β e ⩽ 0.55% thus the turbulence is predominated by the mixture of KBMs and ITGs within the wavenumber range of k y ρ s < 0.75.
The physical mechanism that the FIs can destabilize the KBMs is mainly due to the negative FI density gradient which reduces α MHD , thus making the KBMs more unstable.This can be examined by scanning the linear stability properties under both situations in the two dimensional ŝ-α MHD phase space, as shown in figure 5 for k y ρ s = 0.1.The KBMs exist at small negative ŝ region and are strongly unstable when ŝ is close to zero regardless of the FIs effects are included or not.The types of the modes can be distinguished by their real frequencies (not shown here).The main underlying mechanism is that although the α MHD is decreased hence a reduced KBM drive through β ′ , the Shafranov shift stabilization due to smaller α MHD is weakened at the same time.In ŝ-α geometry the selfconsistent modification of the magnetic equilibrium is directly linked to the α MHD dependence of magnetic drift frequency ω d .For the case with FIs, the reduced Shafranov shift at lower α MHD due to negative FI density gradient modifies the ω d and decreases the ITG drive known as the geometric Shafranov shift stabilization.A smaller local α MHD also corresponds to a reduced electromagnetic α-stabilization of ITG modes which is distinct and unrelated to the former.The values here are α MHD ≈ 0.187 and 0.07 in the absence and presence of FIs as donated by the red and blue symbols in figures 5(a) and (b), respectively, suggesting that the weakening of Shafranov shift stabilization is responsible for the larger growth rate when considering the FI contribution.This is the key mechanism for the destabilization of both ITGs and KBMs, since the negative FI pressure can reduce the total pressure gradient thus β ′ drive for KBMs, and decrease the stabilization factor for both instabilities simultaneously.As a result, the KBMs have largest growth rate at α MHD = 0 as the Shafranov shift stabilization effect vanishes when scanning the α MHD at fixed β and gradients, which can be found in both situations.There is a small region that both the KBMs and ITGs are stable in figure 5(a) which is the so-called second stable region of KBMs.The transport properties of the turbulence in this parameter space are especially important for high confinement performance scenarios in further devices and will be discussed elsewhere.
It has been known that the KBM/AITG can be excited below the ideal ballooning mode (IBM) beta limit [58].The character is also found in the β e scans for two typical wavenumbers k y ρ s = 0.1 and 0.25 who are KBM and ITG branches under experimental β exp e , respectively, as depicted in figure 6 employing parameters without the inclusion of FIs.The ITG is continuously stabilized by β e due to the finite-β stabilization whereas the KBMs are strongly destabilized once the β e exceeds a critical value around β crit ≈ 0.2%, which is well below the IBM threshold around β IBM ≈ 0.42%, as can be found in figure 6.For β e = 0 both the two modes belong to the ITG branches and the KBMs have much higher frequencies than that of ITGs when β e ̸ = 0, as shown in figure 6(b).Although such feature is observed in our β e scans, it should be pointed out that the KBM can also be excited after IBM [59,60].Whether the KBM threshold is below or above that of the IBM is still an open question at present.The corresponding mode structures under experimental β exp e = 0.32% are shown in figure 7.Both the KBM and ITG exhibit the ballooning and tearing structures in terms of ES potential φ and vector potential Ã|| along the field line as predicted by theory [61].The mode structure is more localized around θ = 0 for ITG partially due to the difference in the wavenumbers.Moreover, the structures characterized by the oscillating sidebands may reduce the efficiency of the nonlinear coupling between ZFs and turbulence and such phenomena are also found for the TEM turbulence whose linear mode structure is more extended along the ballooning angle, as discussed in [62].

Transport and ZFs in KBM-ITG dominated plasmas.
The detailed turbulent heat fluxes and ZF dynamics in the β e scans without FI contributions are shown in figure 8, where the traces of E × B turbulent ion heat flux by ES fluctuations and ZF shearing rate ω are plotted in figures 8(a) and (b), respectively.It is identified that the EM flux is much smaller than that of the ES flux in all cases hence the turbulent transport is almost controlled by the ES component.It is seen that a small finite β e is very effective in reducing the transport in the range β e ⩽ β exp e .This is because although long wavelength KBMs are present, their growth rates and corresponding transport would not be very large thus the transport by ITGs is still significant.A finite β can stabilize the linear ITGs and more importantly, it is more effective in suppressing the nonlinear transport which is the well-known nonlinear EM stabilization of ITG turbulence [63], as can be concluded from figure 8(a).The transport is minimum at β exp e which might be explained as follows: although KBMs are unstable, they are localized at limited wavenumbers and their growth rates are not very high as can be found in figure 4(a).Besides, at this β e value the amplitude of ZF generated by ITG branches is relatively large who has strong shearing effect on both the KBM and ITG turbulence.A further increase in β e will lead to a strong destabilizing of the KBMs, causing a higher saturation level.The ZF strengths become higher due to the larger turbulence amplitudes, however, the relatively stabilizing effect of ZF on these turbulence states is weakened.The most important saturation mechanism in these simulations is the nonlinear turbulence driven ZFs and the dynamics of them are analyzed in the quasisteady state, as depicted in figures 8(b) and (c).It is shown that shearing rate by ZFs in terms of ω ZF E×B continuously increases with β e .However, it is suggested that the dynamics of ZFs and turbulence are different as β e varies.For small β e values the turbulence suppression is mainly induced by the enhancement of ZFs as the transport is still predominated by ITG.Nevertheless, in spite of the fact that the ZF becomes stronger at higher β e since the background turbulence amplitude is larger thus an enhanced ZF drive which is seen from the clear increase of ZF shear in KBM dominated plasmas, i.e. at β e = 0.42% and 0.55%., it is ineffective in suppressing or regulating the KBM turbulence, which can be inferred from figure 8(b).Although the turbulence is a mixture of ITGs and KBMs in all simulations, the transport is dominated by the long wavelength KBMs particularly at high β e where the low k y KBMs are much more unstable than high k y ITGs.Moreover, the EM turbulence is less affected by the ZF shearing whereas the turbulence induced by the essentially ES ITGs is subjected to stronger nonlinear EM stabilization at larger β e .The two combined effects will lead to a more pronounced EM turbulence than that inferred from linear growth rate spectrums hence the transport is determined by the KBMs at high β e .An increase with respect to the lower β e cases is especially noteworthy in the low-k x region, where the KBM/ITG-induced transport is mainly concentrated, as discovered in figure 8(c).Therefore, the increase of the ZF activity is suspected to be related to the dynamics of turbulent transport controlled by the ITG branches.The eddy decorrelation by ZF shear is not the major factor for suppressing the KBM turbulence, and other mechanisms such as mean flow shear and energy transfer from unstable to stable modes might be crucial under such conditions.
Similar to figure 8, analysis of the transport and ZF features has been performed where the FI effects have been considered, using the parameters given in table 1 labeled as with FIs.It is seen in figure 9(a) that the transport is enhanced in EM simulations whereas decreased for the ES case, which are suggested to be resulted from the further destabilization of KBMs when β e ̸ = 0 and the dilution effect on ES-ITG turbulence when FIs are included as previously demonstrated by linear simulations.The ZF amplitude and its shearing rate are much smaller in the ES case compared with EM simulations, indicating that a finite β e can strongly reduce the transport through enhancing the ZFs, as shown in figure 9(b).However, the ZF shearing rates for the cases with FIs have some different features: the ZF shearing rate increases and has significant impact on large scale turbulence when β e is not large, then it reaches to largest value at around β e = 0.32% while the transport is insensitive to β e .However, at high β e > 0.32%, the ZF shearing rate starts to decrease, whereas the ion heat transport increases rapidly because the contribution from KBMs becomes dominating.Moreover, the ZFs are more easily to be excited in ITG turbulence than in the KBM turbulence hence the ZF becomes weaker in the latter situation, as shown in figure 9(c).The detailed analysis of the characteristics of ZF excitation and transport behavior in the completely KBM induced turbulence as well as the effects of FI populations will be carried out in the future.
Although the linear eigenvalue spectrums and the heat fluxes from the nonlinear simulations differ in most cases, the shapes of the flux spectrums in wavenumber spaces are generally similar to that of the linear growth rate spectrum, except that the peak positions move to a smaller wavenumber for nonlinear simulations due to the nonlinear coupling among different modes.The comparisons for which the spectra of the ion heat fluxes showing the role of FI on transport is illustrated in figure 10, where the spectrums are extracted in the quasistates of the nonlinear simulations during a time span typically larger than 30 R 0 /c s .The transport is mainly induced by the long wavelength modes.It is shown that very similar predictions of ion heat flux spectrums in both poloidal and radial wavenumber spaces are found regardless of the inclusion of FIs or not.Moreover, it is clear that the FI dramatically affects the heat transport, especially for relative large values of β e , as can be concluded by the comparison between figure 10(a   the fluxes as well as the spectrum width is found when KBMs become significant, which can also be partially inferred from the linear analysis given previously. The well-known saturation mechanism of the ITG turbulence is the nonlinearly self-generated ZF in fully developed turbulence regime [20], which can in turn suppress the turbulence through the shearing effect.The ZF structures in terms of ES potential fluctuations φ are shown in figure 11 for three typical values of β e (=0%, 0.32% and 0.55%), corresponding to the turbulence states of pure ITG, ITG dominated and KBM dominated situations.It is clearly indicated that the long-range correlation in ES cases is relatively small and do not show a clear uniform structure along the file line so that the ZF shearing rate ω ZF E×B is small whether the FIs are considered or not, which can be concluded from figures 11(a1) and (a2).The ZF intensity is enhanced with β e excluding the contributions from FIs, as can be found in figures 11(b1)-(c1).However, the strongest ZF is discovered at β e = β exp e = 0.32% once the FIs are taken into account, showing the distinct feature of longrange correlation in the binormal direction together with an evident corrugation in the radial direction hence a large shearing, as can be seen in figure 11(b2).In this case, the large scale turbulent eddies are teared apart and scattered into high wavenumber domain, thereby reducing the ion heat transport and being beneficial to the confinement.
Based on the simulations above, the transport and ZF shearing rate versus β e are shown in figure 12, where the cases with and neglecting the FI contributions are compared.When the FI effects are neglected, the heat flux is almost fully suppressed when β e = β exp e (0.32%) primarily owing to the EM stabilization of ITG dominated turbulence at β e ⩽ β exp e , as seen in figure 12(a), while the ZF shearing rate shows a continuously increase with β e since the saturation level become larger which provides a stronger ZF drive, as can be found in figure 12(b).However, the presence of FIs will change the features of transport and ZFs, i.e., the transport flux increases rapidly when the KBMs becomes important for β e > 0.32% whereas the ω ZF E×B reaches maximum at β e = β exp e .Moreover, it is also noted under case with FIs, the change in the transport levels is not obvious when β e ≲ 0.32% together with ZF erosion when β e exceeds further above β exp e .The underlying physical mechanisms are complicated regarding such situation thus it is somewhat difficult to distinguish them.Nevertheless, it is suggested that two important factors are responsible for explaining the observed results.One is that the rapid increase in the heat flux is ascribed to the large turbulent transport by KBM turbulence.The other is that in addition to the ZF erosion due to flux-surface breaking induced by magnetic flutter [64], the self-consistent change of the equilibrium magnetic field induced by the Shafranov shift weakens the suppression effect of β on ITG growth and it also reduces the nonlinear ZF level by reducing the Maxwell stress in the quasisteady turbulent state at finite β, especially for cases with low magnetic shear which was confirmed through gyrokinetic simulations by Ishizawa et al [65].Such feature of transport and ZF is quite important as it is implied that the in the weak magnetic shear regions the transport properties is strongly controlled by the interplay among KBM-ITG turbulence, ZFs and FIs.Although the destabilization of KBMs in the presence of FIs would lead to a larger transport level, the ZFs are also increased which stabilize the turbulence if β e is not high.It is also implied that the plasma is characterized by a self-organized state [66][67][68] because of the complex interactions among ZFs, KBMs and FIs effects, which will set the plasma β e (β total ) around β exp e (β exp total ) as even a slightly larger total β will result in the rapid particle and energy losses, causing a high ion profile stiffness level and in turn decrease the plasma confinement and β.The results have also provided a possible way to control the KBM induced transport in weak magnetic configurations through adjusting the FI profiles for the future devices.Nevertheless, we emphasize again that the other roles such as fast particle driven modes [69] on turbulence are neglected in the present simulations, which have been identified as an important factor for achieving high performance plasmas through enhanced ZFs [70,71].This field will be left for a crucial task in the future.

Conclusions
This work has investigated the properties of turbulence transport and ZF in the weak magnetic shear configurations based on the experimental observations of the KBM/AITG in HL-2A ITB plasmas.GENE linear simulations have demonstrated that the plasmas are dominated by the mixture of KBM and ITG in the weak ŝ regions and the inclusion of FI contributions can further destabilize KBMs.The mechanism is identified to be the decrease of ballooning parameter α MHD thus the weakening of Shafranov shift stabilization due to the negative FI density gradient, which makes the KBM more unstable in terms of the ŝ-α MHD diagram while the ITG is mainly subjected to the dilution effect and finite-β stabilization.As for the situations excluding FIs, a small finite β e is effective in reducing the transport in the range β e ⩽ β exp e as the turbulence predominated by ITGs is strongly suppressed by the nonlinear EM stabilization and enhancement of ZFs.Although a further increase of ZF at higher β e is observed, it is not effective in suppressing the KBM dominated turbulence.For the cases with FIs, the KBMs are shown to be more unstable as a negative FI gradient reduces the Shafranov shift stabilization effect.The ZF shearing rate ω ZF E×B increases when β e is not large, reaching to a largest value at around β e = β exp e and it has significant suppression effect on transport.However, the ω ZF E×B starts to decrease at larger β e values and the heat transport is enhanced due to stronger KBM turbulence.It is suggested that the larger ZF shearing rate at β e ⩽ β exp e with the inclusions of FI effects is due to the decreased α-stabilization of ITG modes, hence the ZF amplitude is larger because of the higher saturation level predominated by the ITG turbulence which leads to a stronger ZF drive.The mechanisms for ZF degradation at higher β e are suggested to be mainly caused by the combined roles of ZF erosion due to flux-surface breaking in strong EM turbulence, and change of the equilibrium magnetic field induced by the Shafranov shift which weakens the effect of β-stabilization on linear ITG modes as well as reduces the nonlinear ZF level by reducing the Maxwell stress at finite β in low magnetic shear plasmas.The nonmonastic relationship between ZF shearing and β e suggests that the plasma is characterized by a self-organized critical state due to the complex interactions among ZFs, KBMs and FIs thus limit the plasma β around β exp .The mechanism might also be important in determining the ion profile stiffness levels thus a carefully designed FI profiles could be an effective way in reducing the KBM induced transport through enhanced ZFs, which is also meaningful for future devices employing high power ion heating as well as operating with weak or reversed magnetic shear configurations.

Figure 1 .
Figure 1.Typical discharge waveforms and observation of KBM/AITG on HL-2A.(a) Time traces of the plasma current (blue), line-averaged density (red), ECRH power (magenta) and NBI power (black).(b) Spectrograms of the core microwave interferometry at chordal distance r d = 5 cm and (c) spectrogram of the core SXR signal.The profiles used in the simulations are measured at t = 662 ms as shown by the vertical dashed line.

Figure 2 .
Figure 2.Profiles of #25803 at t = 662 ms.(a) q (blue) and ne (black), (b) temperatures of T i (red) and Te (blue), (c) FI density n f (black) and its ratio to ne (blue), (d) FI temperature T f (black) and its ratio to Te (blue), (e) rotation frequency Ωt (black) and the corresponding shear γ E (magenta).

Figure 4 .
Figure 4. Poloidal wavenumber spectrums of the growth rate and real frequency in the βe scans based on parameters given in table 1.(a) and (b) Are the results without fast ions while (c) and (d) are that for the cases in the presence of FIs, respectively.

Figure 5 .
Figure 5.The stability of KBM and ITG in two dimensional ŝ-α MHD diagram for the case (a) without and (b) with fast ion contribution.The dashed curves donate the stability boundary.

Figure 7 .
Figure 7.Typical mode structure of the KBM and ITG under experimental βe = 0.32%.(a) and (b) are the structures of KBM at kyρs = 0.1 while (c) and (d) are that of ITG at kyρs = 0.25, respectively.

Figure 8 .
Figure 8.Time evolutions of (a) ES flux, (b) ZF shearing rate ω ZF E×B and (c) ω ZF E×B as a function of the radial wavenumber kxρs for different values of βe.The time span used for the analysis of ZF shearing rates is shown by the vertical dashed lines.

Figure 9 .
Figure 9.The same as in figure 8 except for that the contribution of FIs has been taken into account.(a) ES flux, (b) ZF shearing rate ω ZF E×B and (c) ω ZF E×B as a function of the radial wavenumber kxρs for different values of βe.

Figure 10 .
Figure 10.Comparison of spectrums of ion heat flux for the cases of without and with FIs at different βe values.(a) and (b) are the spectrums in poloidal and radial wavenumber spaces in the absence of FIs while (c) and (d) are that with FI contributions, respectively.

Figure 12 .
Figure 12.(a) Ion heat flux and (b) ZF shearing rate versus βe.The results without and with FI contributions are donated by red and blue curves, respectively.

Table 1 .
Dimensionless parameters used in the simulations for the cases of with and without the contribution from fast ions.