Equilibrium reconstruction constrained by the consistency of current simulation on EAST

The attainment of a reliable equilibrium is a critical aspect of tokamak experiments and physics analysis. A common method for establishing a reliable equilibrium current involves reconstructing it from indirect measurements, such as those obtained from polarimeter-interferometers (POINT) and motional Stark effect (MSE) systems. However, uncertainties still exist in the reconstruction results. For the equilibrium reconstruction on the EAST tokamak, which is based on the POINT system, the primary sources of uncertainty are the limited scope of measurements and the sensitivity of the reconstruction process. This paper proposes an enhanced approach that utilizes current simulation as a constraint to maintain consistency between the initial equilibrium and the simulated results. The radio frequency waves driven current is identified as a particularly influential component due to its interaction with the q profiles of the equilibrium and the deposition region of the waves. Two specific discharges are presented to illustrate how a new equilibrium can be achieved, which enhances consistency between the equilibrium and the simulated current, taking into account the dependencies of various components.


Introduction
Magnetohydrodynamic (MHD) equilibrium reconstruction is a crucial and fundamental aspect of tokamak experiments and data analysis [1].It plays a pivotal role in determining plasma geometry, global parameters, and profiles derived from diagnostic data, thereby enhancing our understanding of instabilities, transport phenomena, heating mechanisms, and related studies [2][3][4][5][6][7][8][9][10][11][12][13].In toroidal symmetric devices, such as tokamaks, the ideal MHD equations can be expressed in magnetic flux surface coordinates using the Grad-Shafranov equation: where, ψ is the poloidal magnetic flux, R is the radial coordinate, P is the pressure, F = 2π RB ϕ /µ 0 is the poloidal current stream function, j ϕ is the current density in the toroidal direction, B ϕ is the toroidal magnetic field, and the operator The EFIT code [2,14], which has been widely employed in various tokamaks, including DIII-D and JET [14,15], is utilized to solve the Grad-Shafranov equation and obtain the reconstructed equilibrium for the EAST tokamak [16].The EFIT code represents J ϕ as a linear combination of N P and N F terms of basic functions y n : The value of the unknown parameters α n and β n are fitted from the constraints of the measurements by minimizing the error function: Here, M i , C i , and σ i are the measured value, calculated value, and corresponding uncertainty of ith available measurement.The reliability of the reconstruction results hinges on the appropriate selection and combination of constraints.
The diagnostic constraints used in EFIT reconstruction can broadly be categorized into three groups, as illustrated in table 1 [17].Magnetic diagnostics provide essential information about the plasma shape and certain global plasma properties [3,18,19].Kinetic diagnostics, such as Thomson scattering (TS), electron cyclotron emission (ECE), and xray crystal spectrometer (XCS), are valuable for providing the temperature and density profiles and constraining the pressure profile by P = n e T e + (n i + n z ) T i + P f (4) where n e , n i and n z represent the density of electrons, ions and impurities, T e and T i represent the temperature of electrons and ions, and P f represents the pressure component caused by fast ions.The constraints are employed to enhance the reliability of the reconstruction results for MHD and transport physics, particularly in the pedestal region [17].Direct measurement of the current density presents practical challenges.Consequently, the reconstruction method relies on indirect diagnostics such as the motional Stark effect (MSE) and the polarimeter-interferometer (POINT) system to provide a usable current density profile [20][21][22][23].However, these reconstruction results still carry certain uncertainties, which will be discussed in detail in section 2. On the other hand, numerical simulation and physical analysis are conducted based on the obtained equilibrium and place stringent demands on its reliability.
In the case of EAST, radio frequency (RF) waves are often utilized to achieve steady-state operation.These non-inductive components of plasma current are strongly dependent on the plasma and wave parameters and significantly impact the plasma confinement.A variety of physical models, simulation codes, and even integrated modeling frameworks have been developed to calculate the plasma current [13,[24][25][26][27][28][29].However, the consistency between the equilibrium current and simulated current, which is the sum of each calculated component, has received insufficient attention.In this paper, we assume this discrepancy originates from the imprecision of the equilibrium itself, and we propose utilizing this consistency as a current constraint to improve the equilibrium reconstruction results.
In section 2, we provide a summary of the current constraint method and its uncertainties in the EAST equilibrium reconstruction.Section 3 offers a comprehensive explanation of the current simulation and the equilibrium reconstruction method, with assistance from simulations.Section 4 demonstrate the specific steps and details of the reconstruction using two fully non-inductive discharges on EAST.Finally, section 5 concludes the paper with a discussion of the method.

Current constraint of EAST equilibrium reconstruction
Given the absence of direct current density measurements, indirect tools such as POINT and MSE are selected as constraints to enhance the accuracy of the reconstruction results.These tools provide magnetic quantities such as Faraday rotation or local q value, which can be compared with the equilibrium and integrated into the error function.By including these measurements, the reconstruction results yield a more reliable estimation of the current density.
A POINT system was installed on EAST in 2014, providing electron density and current constraints through the reconstruction method [30].This multi-channel laser system, illustrated in figure 1, enables the measurement of both laser polarization and interference information.The Faraday rotation, denoted by ψ F , represents the polarization change resulting from the different rotation speeds of left and right circularly polarized waves.The phase change relative to the reference light in vacuum is denoted as ϕ n .These values correspond to  path integrals of the electron density n e and magnetic field B along the light path: while λ is the wavelength of laser.To effectively utilize these integral data for equilibrium reconstruction, a modified version of the EFIT code has been developed [31].This enhanced method incorporates both external magnetic data and diagnostic data from the POINT system.The resulting reconstructed equilibrium exhibits improved reliability in terms of the central safety factor and better conformity to MHD instability phenomena [5].
While this method has demonstrated improved reliability in achieving current-reliable equilibrium compared to standard magnetic or pressure-constrained methods in many studies, several concerns need to be mentioned about the uncertainties introduced in the reconstructed equilibrium.First, the results of the reconstruction process, constrained by the POINT system, exhibit lower confidence levels in the edge regions.This is particularly evident in the integral formula of Faraday rotation, as illustrated in 5, where only the component of the magnetic field parallel to the optical path contributes to the integral.Consequently, the Faraday rotation of each channel is primarily influenced by the current at the flux surface tangent to the optical path.For instance, as shown in figure 1, the channel highlighted by the green line encompasses the entire magnetic field along the integration path.But the current within the green shaded region, where the flux surface is nearly tangential to the line, significantly contributes to the Faraday rotation.Due to the limited coverage of the POINT system, the current in the edge region contributes less significantly than the core current.Furthermore, even with an increased number of channels, the values from edge channels may be insufficient to yield a satisfactory signal-to-noise ratio.Second, the sensitivity of the inversion process is a matter of concern.The Faraday rotation value is dependent on the magnetic field, plasma shape, and electron density, with the first two factors being influenced by the current density.This intricate relationship introduces sensitivity and uncertainty into the reconstruction results.As illustrated in figure 2, we observe similar Faraday rotations in (d) calculated from three equilibria (artificially obtained, represented by blue, green, and red curves) with distinct current densities in (a).Notably, the variations of Faraday rotations fall within the tolerance range indicated by the shaded region.Conversely, uncertainty in the Faraday rotations may result in significant discrepancies in the equilibrium current.Consequently, it is necessary to introduce an additional constraint to enhance the reliability of the POINT reconstruction method.
In recent years, a ten-channel MSE system has been installed on EAST and is scheduled to become operational [32,33].The MSE system provides a spatially resolved measurement of the magnetic pitch and imposes constraints on the internal current profile.However, this diagnostic approach has its own limitations.For instance, the optical system is complex and sensitive, requiring precise calibration.Additionally, its reliance on neutral beam injection (NBI) and restrictions on the observation space are factors that cannot be overlooked [34][35][36][37].Due to the potential sensitivities and uncertainties associated with diagnostic methods, our approach of refining equilibrium calculations, as will be demonstrated in section 4, is applicable regardless of the availability of the diagnostics from MSE or POINT.

Reconstruction method with assistance from simulations
Numerous simulation studies have been conducted on the evolution and components of current [7,[38][39][40], necessitating calculations based on a specific equilibrium.However, there are instances where the simulated currents do not align with the equilibrium currents used as inputs, an issue that has been overlooked in certain studies.This paper proposes that these inconsistencies may result from inaccuracies in the equilibrium, which could be attributed to the uncertainties inherent in the constraint diagnostics mentioned previously.A reliable equilibrium should be consistent with the currents simulated from it.Therefore, it is imperative to adjust the equilibrium, with consistency as a guiding principle and objective.To ensure the efficacy and convergence, we propose a manual adjustment of the equilibrium based on a physical analysis.In this section, we will elucidate our methodology for calculating the components of the current, the dependencies of the different components, and the computational procedure we employ to adjust the equilibrium.

Calculated current components
In tokamaks, the total plasma current is typically categorized into three components: Ohmic current (j Ohmic ), auxiliary driven current (j CD ), and bootstrap current (j BS ).Each of these components can be simulated independently for a specific discharge, with the total simulated current being the sum of these contributions: This paper employs neoclassical transport models to calculate the bootstrap and Ohmic currents.The bootstrap current is determined using the Sauter model [24,41], as represented by: Here, j ∥ , E ∥ , B represent the parallel current, parallel electric field and the magnetic field.The angle brackets ⟨. ..⟩ denote flux surface averaging, and R pe = p e /p, I(ψ) = RB ϕ .The specific expressions for conductivity σ neo and the coefficients L can be found in [24].The terms in the square brackets correspond to the bootstrap currents driven by the gradients of electron density (n e ), electron temperature (T e ), and ion temperature (T i ) [42,43].The first term, which is the flux surface averaged Ohmic current, is described by a simple model [24,44]: where V loop represents the loop voltage and R 0 represents the the major radius at the magnetic axis.Both the bootstrap and Ohmic currents are calculated using the ONETWO transport code [45].It is noteworthy that in future reactors, fully noninductive discharges are required due to limitations in voltseconds, and therefore the Ohmic current can be disregarded.
The EAST tokamak is equipped with four auxiliary heating systems [46]: NBI, ion cyclotron (IC) resonance heating, lower hybrid (LH) current drive and heating system, and electron cyclotron (EC) resonance heating/current drive system.In previous experiments, the LH and EC systems have been found to contribute significantly to the driven current [25].These two auxiliary heating systems have a wide range of possible deposition regions, making them effective for controlling the current profile in experiments.The propagation of these waves can be simulated using a combination of ray-tracing and Fokker-Planck solvers.
For LH and EC waves, the propagating electromagnetic field can be approximated as a ray based on the WKB (Wentzel-Kramers-Brillouin) approximation [47,48].And the wave propagation can be described by the eikonal equation: Here, r, k, ω represent the position vector, wave vector and the angular frequency of wave, and the D(ω, k) = 0 is the real part of the dispersion relation.While the driven current can be calculated from the damping rate using the ray-tracing method, a more reliable driven current profile is obtained by solving the Fokker-Planck equation.This paper utilizes the GENRAY code [49] to solve the refractive index and trace of waves, while the CQL3D code [50,51] is employed to solve the Fokker-Planck equation and present the plasma response to wave-particle interaction.For brevity, this paper ignores the contribution of the ICRH and NBI systems to the current.However, if necessary, the effects of these two systems can be simulated using the fullwave code TORIC [52] and the Monte-Carlo particle code NUBEAM [53].All these codes are integrated in the OMFIT Framework [27,54].

Dependencies of current components on plasma parameters
When adjusting the equilibrium, it is essential to consider the relationship between the equilibrium and the current calculation, taking into account the dependencies and uncertainties associated with each component.These components must be discussed separately due to their distinct characteristics.
The bootstrap current is primarily influenced by the gradients of temperature and density profiles [24].In this study, the profiles of temperature and density are not considered as quantities that require adjustment; therefore, the primary determinant of the bootstrap current is the equilibrium, with particular emphasis on the changes in magnetic flux surfaces.Consequently, with each iteration of equilibrium adjustment, the bootstrap current should be recalculated.As the equilibrium converges through iterative refinement, the calculations of the bootstrap current will also approach stability.In principle, due to changes in the magnetic flux surfaces, a re-fitting of the diagnostic data should be performed with each iteration.However, in practice, we have arrived at a conclusion analogous to that presented in [17]: iterations subsequent to the first do not significantly impact the fit to the diagnostic data.Thus, the step of re-fitting can be omitted.
The Ohmic current is neglected in the non-inductive discharges discussed in this paper.For long-pulse steady-state discharges, even if there is some Ohmic current, it is typically broad and flat, which provides evidence for its relatively low sensitivity to equilibrium adjustments.The EC resonant position, determined by the equation ω − ω ce = k ∥ v ∥ , is also not highly sensitive to equilibrium adjustments.
In contrast to the aforementioned components, the LH driven current exhibits greater sensitivity to the initial equilibrium [8].Therefore, when conducting equilibrium adjustments, special attention should be given to the LH driven current component.
The propagation of the LH wave in plasma is a complex process.Instead of simulating each individual case, a parametric analysis can be performed to study the potential power deposition (PPD) region, which represents the deposition of the LH wave [55].The dispersion relation provides a potential domain in the (ρ, N ∥ ) space, where ρ is the normalized minor radius and N ∥ is the parallel refractive index.The upper and lower boundaries of N ∥ are given by:

|B|
) .(10) Here, the '−' and '+' signs represent the upper and lower boundaries respectively.N ϕ,axis is toroidal refractive index on axis, n e represents the plasma density, and B θ is the poloidal , where ω pe is the plasma frequency, ω is the LH wave frequency, and S is the Stix cold plasma dielectric tensor.If the wave deposits through Landau damping, the refractive index should match the electron temperature [56]: Figure 3 illustrates that only a portion of the Landau damping condition lies between the upper and lower boundaries of N ∥ , and this region, known as the PPD region, represents where the wave can deposit.
For typical parameters of the EAST tokamak, the approximate solution for the upper boundary is given in [8] by: Here, q represents the local safety factor, r/a is normalized plasma radius, a is the minor radius, f is the wave frequency, N e ≡ n e /10 20 , B 0 is the axis magnetic field, and N ϕ,0 is toroidal refractive index at launcher.It is evident that the profiles of n e , T e , and q play crucial roles in LH current drive.Apparently, the driven current profile depends on the q profile which is mainly determined by the total equilibrium current where the driven current is the dominant component.This interplay provides guidance on how to adjust the equilibrium accordingly.

Equilibrium calculation method
The equilibrium calculation method in this paper is illustrated in figure 4. The calculation necessitates an initial equilibrium, which can be provided by magnetic EFIT or POINTconstrained EFIT.Based on the initial equilibrium and other information such as the auxiliary heating setup and kinetic profiles, the total simulated current can be determined (indicated by the red dashed box in figure 4), which is considered the sum of individual current components mentioned in section 3.1.This total current is then compared with the initial equilibrium current.If a significant discrepancy is observed, a targeted current profile is manually derived through physical analysis.This target current, in conjunction with the target pressure obtained from the kinetic profiles, is then input into EFIT for recalculation to yield a new equilibrium, serving as the input for the next iteration step.An error function is defined to quantify the difference between these two current profiles: Here, j Sim r,z and j Equi r,z represent the simulated and initial equilibrium current densities respectively, and (r, z) denote the sampling grids on the small cross-section.The summation is performed over all mesh grids within the last closed surface (LCS).If the value of χ 2 j is small enough, it indicates that the input equilibrium is consistent.However, if the value is large, it implies that the equilibrium needs to be corrected.In practical applications, the use of a 33 × 65 grid in the EFIT code is sufficient to capture the differences between two similar equilibria.Approximately one fourth of the grids lie within the LCS.Moreover, with this number of sampling grids, a threshold of χ 2 j < 10 is stringent enough to differentiate the consistency of the two current densities.
In the methodology employed, target currents are set based on physical analysis, and equilibrium is generated using the EFIT code to achieve the objective of adjusting the equilibrium currents.Specifically, the target current and pressure are utilized to calculate P ′ and FF ′ terms in (1) on the basis of the existing magnetic flux.These distributions are incorporated into the equilibrium file and used in conjunction with the initial equilibrium to recalculate a new self-consistent solution.Throughout this process, global parameters such as I p (plasma Flow chart of current component calculation and equilibrium correction method.The red dashed box highlights the conventional simulation of current components respectively.Subsequently, the simulated current and equilibrium current are compared for consistency.In cases of inconsistency, a target current will be given for equilibrium correction to the next iteration.current) and l i (internal inductance) that have proven to be accurate even in magnetic reconstruction [3] are monitored.
The adjustment of current necessitates a multi-step process, with subsequent steps potentially influencing the analysis and rationale derived from prior steps.Therefore, in order to mitigate the mutual impact between different steps, it is prudent to segment these multi-step adjustments into distinct regions.
Typically, adjustments commence in the edge region as the first step, where the current component is singular.As discussed in section 3.2, the edge region warrants special consideration due to the dramatic profile changes within the narrow pedestal region, particularly in H-mode discharges.The calculation on EAST showed generally the driven current is inside ρ = 0.8 [17,57].This conclusion can also be corroborated by estimating using the PPD theory.By substituting the typical parameters of EAST into (12), assuming f = 4.6 GHz, N ϕ,0 ∼ 2.0, B 0 ∼ 2.5 T, N e ∼ 3.0 and q ∼ 8 at ρ ∼ 0.8, we can obtain an upper limit of N ∥ as 3.9.According to (11), this implies that the T e at this location needs to be greater than 2 keV, which is uncommon for EAST.The EC system is mainly used for very localized current drive, typically in the inner region [58], which can be considered separately for different cases.Therefore, it is also possible to disregard it in the edge region.Consequently, we focus on calculating the edge current solely based on the bootstrap current and the Ohmic current, which relies on the kinetic profiles.In the case of this fully non-inductive discharge, the Ohmic current can be omitted.
Subsequently, adjustments are made to the inner region as the second step, where the current components are more complex.However, as per the analysis in section 3.2, the LH driven current should be given primary consideration.One factor to consider is that the alterations in flux surfaces resulting from the second-step adjustment may influence the edge bootstrap current.This, in turn, could potentially impact the foundation for the first-step adjustment.However, in practice, this impact is typically restricted, and the bootstrap current is recalculated at each iteration.Consequently, there is no cause for concern that the second-step adjustment will disrupt the foundation for the first-step adjustment.
The following section presents two examples to illustrate how to achieve a new equilibrium with the assistance of simulations.

Reconstruction of the EAST equilibrium
Long-pulse fully non-inductive discharges are considered a viable scenario for future tokamak reactors, and EAST has conducted numerous experiments in this regime.This section utilizes two time slices from non-inductive discharges #110488 and #91729 to demonstrate the reconstruction process.In numerous studies, it is desirable to have as many constraints as possible for the equilibrium reconstruction.However, it is often challenging to ensure the availability and stability of all diagnostics in every experiment.In such instances, the method previously described can provide valuable current constraints.In the first example, discharge #110488, a basic magnetic-constrained equilibrium is used as the initial input.The POINT diagnostic is assumed to be unavailable during the calculation process, and it is only utilized to verify the final results.
Even when the POINT constraint is available, there is still some uncertainty, and the equilibrium may not exhibit satisfactory consistency with the simulation results, as discussed in section 2. In the second example, discharge #91729, a POINTconstrained equilibrium is utilized to demonstrate how the equilibrium can be further enhanced with the assistance of simulation.

Reconstruction from the magnetic-only equilibrium of #110488
In the case of discharge #110488, which is a 320 s longpulse experiment, various plasma parameters are depicted in figure 5.The plasma current reaches 0.30 MA, accompanied by a line-averaged electron density of approximately 3.2 × 10 19 m −3 .The heating mechanisms employed in this experiment include 1.6 MW EC, 1.2 MW IC, and 1.5 ∼ 2.0 MW LH waves, with the LH wave power being self-adaptive to maintain the loop voltage zero.To analyze the equilibrium, we focus on the time slice at t = 6.0 s, which provides temperature and density profiles illustrated in figure 6.
Following the computational process outlined by the red dashed square in figure 4, we calculate the simulated current components based on the magnetic equilibrium, as presented in figure 7. It is observed that the simulated current (red (downward-pointing triangle) and TS [60] (circle) measurements, T i (red curve) is measured by the XCS [61] (upward-pointing triangle), (b) electron density, ne (blue curve) is reconstructed from the POINT system [30], while the edge is constrained by the reflectometry [62] (downward-pointing triangle).The first step, as outlined in section 3.3, involves adjusting the edge region where ρ > 0.8.This region, indicated by the shaded area in figure 7, is primarily characterized by the bootstrap current.The current components based on the edge-adjusted equilibrium are displayed in figure 8(a).It is evident that the equilibrium current already exhibits a pedestal structure, thereby reducing the error function of χ 2 j to 74.Alternatively, a pressure-constrained kinetic equilibrium would yield similar results.However, further adjustments are required in the inner region, particularly around ρ ∼0.3 and 0.6, where the RF current component is most likely involved.The power deposition of EC and LH, calculated using GENRAY, TORAY, and CQL3D codes, is depicted in figure 8(b).Special attention should be given to two segments of the 4.6 GHz LH wave, namely ρ < 0.5 and ρ > 0.5.These regions warrant particular focus and analysis.
The observed two regions can be explained by the Landau resonance condition and the PPD region.In figure 9, the traces, N ∥ evolution, and power deposition of two representative rays are illustrated.The PPD region is demarcated by a shadow in (b), with the inner and outer boundaries identified as ρ = 0.05 and 0.65.The two deposition regions mentioned above correspond to phenomena occurring near these boundaries.When the LH wave initially propagates through the plasma, N ∥ gradually increases, yet it remains below the Landau condition.As a result, a portion of the wave's energy is deposited closest to the axis, where the N ∥ is closest to satisfying the Landau condition.This phenomenon corresponds to the cyan region in figure 9(b).After reflection on the high field side, the wave's  N ∥ further increases and deposits energy when it encounters the Landau condition near the outer boundary.This phenomenon corresponds to the green region.The two deposition regions arise from this phenomenon.
While the aforementioned analysis appears plausible, it is essential to evaluate the sensitivity of the calculation due to potential alterations in the result when modifying the equilibrium.Figure 10 presents several sensitivity tests conducted on T e , n e , and q, which were identified as the most influential parameters affecting the LH wave in section 3.2.Each profile is scanned through the application of a multiplicative factor.For the variables T e and q, the factors used in the tests are −20%, −10%, +10%, and +20%.The factor for q is achieved by altering the plasma current.For the variable n e , a more restricted testing range is necessary to ensure the subsequent conclusions.The factors used on n e are −10%, −5%, +5%, and +10%.The density affects both the wave propagation and the quantity of resonant particles, leading to more stringent range.This suggests a more stringent requirement for experimental density determination.The test results indicate that, despite the potential variation in the numerical value of the driven current, there is no significant change in the deposition region, suggesting that the previous analysis is robust.Therefore, we can adjust the equilibrium current in the inner region to align it with the simulated results and achieve better agreement.
After adjusting the equilibrium guided by the LH simulation, the new equilibrium current and the corresponding simulated results are presented in figure 11(a).The error function χ 2 j has been reduced to 14, indicating an increased level of consistency.However, further local adjustments can be made to  optimize the equilibrium further.The final result, as depicted in figure 11(b), yields an even lower error function of χ 2 j = 1.9.The initial and final equilibria are compared in figure 12.In (a), the plasma configurations are compared, revealing some variations in the LCS.In (b), pressure exhibits significant variations, which can be attributed to the initial pressure in the magnetic equilibrium being unreliable, while the pressure in the results is constrained by ( 4).(c) and (d) illustrate the comparisons of q and current profiles.(e) and (f ) depict the comparisons of two types of magnetic diagnostics, flux loops and magnetic probes, indicating differences in magnetic field at specific positions.(g) presents the Faraday rotation calculated from the final equilibrium, which aligns well with the POINT data, despite the fact that the entire calculation process did not rely on the measurements.

Reconstruction from the POINT-constrained equilibrium of #91729
In contrast to the magnetic equilibrium, a more prevalent equilibrium approach in studies combines POINT measurements with kinetic measurements to provide reliable pressure constraints and some current constraints [63].Despite these constraints, the POINT-constrained equilibrium often fails to yield satisfactory consistency with simulation results.As discussed in section 2, this inconsistency can be attributed to the uncertainties inherent in the reconstruction process and the limited current constraints available.In this section, we demonstrate how the constrained equilibrium can be enhanced by incorporating current simulations.
Shot #91729 is a multi-phase discharge characterized by plasma parameters illustrated in figure 13.The second phase of the discharge, initiated at t = 4.6 s, achieves a fully noninductive state.The plasma current remains constant at 0.30 MA, with a line-averaged electron density of approximately 4.0 × 10 19 m −3 .The plasma is heated using 0.9 MW of EC and 2.2 MW of 4.6 GHz LH waves.The temperature and density profiles at 6.0 s are illustrated in figure 14.The equilibrium for this discharge is obtained using a POINT & pressureconstrained approach, as presented in figure 15.Initially, the equilibrium is calculated using POINT-constrained EFIT [31], and subsequently modified based on pressure constraints (depicted in figure 15(a)) and pedestal current using kinetic-EFIT [17].The comparison between the equilibrium and the POINT diagnostic data is presented in figure 15(d).
Figure 16(a) displays the simulated current, derived from the previously established constrained equilibrium.A significant discrepancy is observed between the equilibrium and simulated currents, potentially posing challenges for future applications.The error function, χ 2 j = 1.2 × 10 2 , indicates the extent of the difference.When comparing the two current profiles, it is evident that the equilibrium current in the ρ = 0.6 ∼ 0.8 region significantly deviates from the simulated result, which is nearly zero.As elucidated in section 3.3, this current region primarily reflects the bootstrap current, which is relatively low.This observation is corroborated by the PPD region of the 4.6 GHz LH wave, as depicted in figure 16(b), which confirms the absence of driven current in that area.Consequently, the equilibrium current in this region must be reduced to align with the bootstrap current, while the inner current must be augmented accordingly to maintain a constant total plasma current.The resulting edge-corrected equilibrium and simulated current are presented in figure 17(a), and the corresponding RF power deposition is depicted in figure 17(b).Despite these modifications, the equilibrium and simulated currents continue to exhibit incongruity, particularly in the RF deposition region.The error function χ 2 j = 13 represents the remaining mismatch.Unlike the previous case, in this scenario, it is feasible to adjust both the equilibrium current and consider  the LH deposition region, based on the relationship with the q profile.
Upon analyzing the form of the simulated current, it is observed that the RF driven current is larger at ρ = 0.5 compared to the equilibrium current, while it is smaller at ρ < 0.4.A minor adjustment, which shifts the deposition region towards the axis, can eliminate both the excess in the outer region and the deficiency in the inner region.Based on experimental experience and analyses in [8,64], it is known that LH power deposition tends to occur closer to the magnetic axis under lower q values in the inner region.To achieve a lower q value, the current profile should be more nearaxis, resulting in a larger poloidal magnetic field according to Ampere's circuital theorem, while the toroidal magnetic field remains constant.
The simulation results of the dependency of power deposition on equilibrium current presented in figures 18(a) and (b) align with this analysis.After the adjustment, the power deposition is more concentrated near the axis, suggesting a better alignment with the desired power deposition region.Figure 18(c) presents the current comparison of the final result.The adjustments have effectively increased the consistency between the equilibrium and the simulated currents, resulting in an improved error function value of χ 2 j = 1.0.This indicates a high level of agreement between the adjusted equilibrium and the simulation results.To maintain the initial constraints throughout the adjustment process, the plasma configuration, pressure, and magnetic diagnostics are compared between the initial and final equilibria in figures 19(a), (b), (e) and (f ).The variations in q and current profile are illustrated in (c) and (d), where q 0 on the axis remains relatively constant.The final Faraday rotation is depicted in (g), and remains within the acceptable error region throughout the adjustment process, confirming that the improved equilibrium maintains compliance with the previous kinetic and POINT constraints.However, differences are observed in the flux loops, which is attributed to the adjustment of equilibrium current, leading to a reduction in the constraints of magnetic diagnostics.When utilizing this result, it is essential to make trade-offs based on the specific research question and region.

Summary and discussion
To improve the reliability of the current density profile, conventional equilibrium reconstruction methods rely on indirect measurements, such as those provided by the POINT systems.However, these measurements often exhibit uncertainties due to the sensitivity of the instruments and the constraints imposed by different regions of the plasma.To address these uncertainties and improve the consistency between the equilibrium reconstruction and current simulation, the proposed method utilizes current components simulation as an additional constraint.This approach not only considers the simulation results themselves, but also the consistency between the initial equilibrium and the calculated current components.By examining the agreement between the equilibrium and simulated current components, the reconstruction process can be guided towards a more reliable and self-consistent solution.
In this approach, EFIT serves as the core program for equilibrium calculations, with ONETWO, GENRAY, and CQL3D utilized to compute various current components based on the input equilibrium.Naturally, other programs providing current profiles can also be integrated into this approach.Upon comparison of the simulated currents with the equilibrium currents, a new target current is artificially proposed and subsequently calculated using EFIT to generate a new input equilibrium for the subsequent iteration.In proposing the target current artificially, the LH driven current is identified as a particularly influential component.Its behavior and impact on the equilibrium are meticulously analyzed and utilized as a guide during the adjustment process.
This study analyzes two distinct discharge scenarios to illustrate the methodology proposed.In instances where reliable data from alternative measurements are not accessible, the adjustment process can commence from a magnetic equilibrium.Furthermore, even a well-constrained equilibrium can be enhanced to yield improved consistency with simulated current components.During the adjustment process, the bootstrap current assumes a pivotal role in the edge region and is typically adjusted in the first step.The correlation between the LH driven current and the inner region adjustments can serve as a guide in the subsequent step.In certain situations, the simulated current may not exhibit sufficient sensitivity, necessitating adjustments to the equilibrium to align it with the simulation.Conversely, in other cases, the simulated current evolves in conjunction with the equilibrium, facilitating the attainment of consistency between the two.
A common concern revolves around ensuring the validity of the calculations.This paper employs extensively tested models and codes that have a proven track record of generating reliable results in previous studies.Additionally, it is important to note that the simulated results are not considered completely reliable, but the focus is on achieving consistency.
Typically, in the ShenMa High Performance Computing Cluster with 8 cores, each iteration takes approximately 3 h, with the majority of the time dedicated to simulating the driven current.In practice, achieving a sufficiently stable equilibrium typically requires 4-6 iterations of adjustment.In future work, we aim to reduce the computational time through the optimization of the algorithm.For instance, the process could be refined so that not every iteration necessitates a comprehensive simulation.Alternatively, the adoption of more efficient adjusting methods could potentially decrease the number of iterations required.In each iteration, it is necessary to manually provide the target current based on physical analysis, which is a significant limitation of this method.Future work could involve further quantification of the existing current error function and its rewriting in a format that can be directly compared with error functions of the diagnostics in EFIT.Additionally, inspiration from the CAKE code [13] could lead to a more parameterized adjustment process, potentially even automated.The inconsistency of the equilibrium current could be converted into an error range, and the uncertainties of temperature and density diagnostics (not considered in this paper) could also be incorporated into the calculation of kinetic EFIT.

Figure 1 .
Figure 1.Lines of sight of the POINT diagnostic in EAST.The black and blue solid lines represent the walls and the flux surfaces.The red dashed lines show the 11 horizontal channel POINT chords.The green solid line highlights the highest chord as an example whose Faraday rotation is significantly affected by the current in green shadow where the flux surface is almost tangent to the sight.

Figure 2 .
Figure 2. Comparison of three different equilibria with similar Faraday rotations.Three different cases are shown in blue, green, and red curves.(a) Current densities; (b) flux surfaces; (c) safety factor profiles; (d) Faraday rotations, where the black points represent the measurement data and the shadow represents the statistic error range.

Figure 3 .
Figure 3.An example of the PPD region for 4.6 GHz LH wave with N ∥,0 = 2.08.The black solid lines denote the upper and lower boundaries for N ∥ while propagating.The blue dashed line represents the Landau damping condition.The black arrows indicate the intersection of the above two conditions, and the range between them where the Landau damping condition falls within the allowed range of N ∥ is delineated by shading, which is the potential power deposition domain.

Figure 4 .
Figure 4. Flow chart of current component calculation and equilibrium correction method.The red dashed box highlights the conventional simulation of current components respectively.Subsequently, the simulated current and equilibrium current are compared for consistency.In cases of inconsistency, a target current will be given for equilibrium correction to the next iteration.

Figure 7 .
Figure 7.Current profiles comparison calculated from the initial magnetic equilibrium of #110488.The black line represents the equilibrium current as input, while the red line represents the simulated total current.The shadowed indicates the edge area where RF power will not be deposited, thereby defining the bootstrap current domain.

Figure 8 .
Figure 8.Current profiles comparison and power deposition of discharge #110488 calculated from the equilibrium after the adjustment in the first step, which mainly focus on the edge region.(a) Current profiles comparison.The black line represents the input equilibrium current, while the red line corresponds to the simulated total current.(b) Power deposition.The black solid line represents the total RF power, the red dashed line represents the 4.6 GHz LH wave power, and the blue dotted line represents the EC wave power.The green and cyan shadows represent two segments of the LH wave power, which will be discussed.

Figure 9 .
Figure 9. Propagation of two LH wave rays.(a) Ray traces on the small cross, the red and green solid lines represent two typical rays.(b) Phase space analysis, the black lines represent the upper and lower boundary of N ∥ while propagating, the blue dashed line represents the Landau damping condition, the shadow between ρ ∼ 0.05 and 0.65 represents the PPD region.(c) Power in the rays, dashed lines represent the linear results from GENRAY code, and the solid lines represent the quasi-linear results from CQL3D code.

Figure 11 .
Figure 11.Current profiles comparison of adjustment steps of #110488.(a) The second step, which mainly focus on the inner region.(b) The third step, which is the final equilibrium.The black line represents the input equilibrium current, while the red line corresponds to the simulated total current.

Figure 12 .
Figure 12.The comparison between the final and the initial equilibria of #110488.(a) Plasma configuration, blue and red represent the flux surfaces of the initial and final equilibria.(b) Pressure.(c) q.(d) Current density.(e) Magnetic flux at 35 flux loops.Blue triangles and red circles represent the calculated flux from the initial and final equilibria respectively, and the black squares represent the unused outliers.(f ) Magnetic field at probes, marked the same as (e).(g) Faraday rotation calculated from the final equilibrium(red triangles), measured data by POINT(black circles) and the error range(gray shadow).

Figure 14 .
Figure 14.Kinetic profiles of #91729 at 6.0 s.Use the same measurements mentioned in figure 6.

Figure 15 .
Figure 15.Constrained equilibrium of discharge #91729 at 6.0 s.(a) Pressure profile calculated from the temperature and density profiles, used as a constraint in the kinetic-EFIT reconstruction process.(b) Magnetic flux surface.(c) q profile.(d) Faraday rotation, indicated by the blue line, represents the calculated result from the reconstructed equilibrium, while the black points and shaded region represent the measurement data and associated error range, respectively.

Figure 16 .
Figure 16.Current profiles comparison and PPD analysis of discharge #91729 calculated from the initial equilibrium constrained by POINT and pressure.(a) Current profiles comparison.The black line represents the input equilibrium current, while the red line corresponds to the simulated total current.(b) The Phase space analysis.The black lines represent the upper and lower boundary of N ∥ while propagating, the blue dashed line represents the Landau damping condition, the shadow between ρ ∼ 0.02 and 0.65 represents the PPD region.

Figure 17 .
Figure 17.Current profiles comparison and power deposition of discharge #91729 calculated from the equilibrium after the adjustment in the first step, which mainly focus on the edge region.(a) Current profiles comparison.The black line represents the input equilibrium current, while the red line corresponds to the simulated total current.(b) Power deposition.The black solid line represents the total RF power, the red dashed line, green dash-dot line, and the blue dotted line represent the 4.6 GHz LH, 2.45 GHz LH, and the EC wave power, respectively.

Figure 18 .
Figure 18.The process and results of the second step of adjustment.(a) Adjustment of the equilibrium current.The blue/green curve represents the case before/after adjustment.(b) Variation in the power deposition of 4.6 GHz LH wave.(c) Current comparison of the new equilibrium and the corresponding simulated current.

Figure 19 .
Figure 19.The comparison between the final and the initial equilibria of #91729.(a) Plasma configuration, blue and red represent the flux surfaces of the initial and final equilibria.(b) Pressure.(c) q.(d) Current density.(e) Magnetic flux at 35 flux loops.Blue triangles and red circles represent the calculated flux from the initial and final equilibria respectively, and the black squares represent the unused outliers.(f ) Magnetic field at probes, marked the same as (e).(g) Faraday rotation calculated from the final equilibrium(red triangles), measured data by POINT(black circles) and the error range(gray shadow).

Table 1 .
Three kinds of diagnostics and constraints in EFIT reconstruction.
i , βp Current MSE, POINT Internal current profile, q profile Kinetic TS, ECE, POINT, reflectometry, XCS, etc Pressure profiles (from ne, Te, T i , nz, n i profiles)