Kinetic equilibrium reconstruction withinternal safety factor profile constraints on EAST tokamak

Haochen FAN (樊皓尘),Guoqiang LI (李国强),Jinping QIAN (钱金平),Xuexi ZHANG (张学习),Xiaohe WU (邬潇河),Yuqi CHU (储宇奇),Xiang ZHU (朱翔),Hui LIAN (连辉),Haiqing LIU (刘海庆),Bo LYU (吕波),Yifei JIN (金仡飞),Qing ZANG (臧庆) and Jia HUANG (黄佳)

1 Institute of Plasma Physics,Hefei Institutes of Physical Science,Chinese Academy of Sciences,Hefei 230031,People’s Republic of China

2 University of Science and Technology of China,Hefei 230031,People’s Republic of China

3 Shenzhen University,Shenzhen 518060,People’s Republic of China

4 University of California Los Angeles,Los Angeles 90095,United States of America

Abstract Reconstruction of plasma equilibrium plays an important role in the analysis and simulation of plasma experiments.The kinetic equilibrium reconstruction with pressure and edge current constraints has been employed on EAST tokamak.However,the internal safety factor (q) profile is not accurate.This paper proposes a new way of incorporating q profile constraints into kinetic equilibrium reconstruction.The q profile is yielded from the Polarimeter Interferometer (POINT)reconstruction.Virtual probes containing information on q profile constraints are added to inputs of the kinetic equilibrium reconstruction program to obtain the final equilibrium.The new equilibrium produces a more accurate internal q profile.This improved method would help analyze EAST experiments.

Keywords: equilibrium reconstruction,tokamak data analysis,kinetic equilibrium,q profile,polarimeter-interferometer

1.Introduction and overview

Reconstruction of plasma equilibrium is important for tokamak research.On EAST tokamak,the study of magnetohydrodynamics (MHD),transport process,heating,and the current drive requires an understanding of the basic geometry of pressure,current,safety factor (q),etc.To yield the experimental equilibria,the equilibrium reconstruction using the diagnostic data has been extensively studied [1–6].Equilibrium reconstruction has been successfully developed on EAST,and the kinetic equilibrium reconstruction is an important method for obtaining the pressure and edge current profiles [7].

The internal current profile also plays a major role in tokamak research.The profile ofqdetermined by current is important for research on transport and stability.It is necessary to obtain theqprofile.To fully reconstruct the internalqprofile self-consistently,kinetic measurement constraints must be used in conjunction with the internalqprofile constraints [5].In many tokamaks,the Motion Stark Effect(MSE) measurement is used to constrain the pitch angle of the magnetic field to reconstruct theqprofile [8,9].MSE is now being developed on EAST tokamak [10].Another way to obtain internal current andqprofile information is to measure the Faraday rotation angle [11].EAST tokamak installed the Polarimeter Interferometer (POINT) system with 11 measurement chords in the 2015 experiment,which could provide the Faraday rotation angles [12].Equilibrium reconstruction with POINT diagnostic constraint is also successfully applied to EAST,resulting in a more accurate internal current andqprofile.

The essential equation of the steady-state tokamak plasma is widely known as the Grad-Shafranov (G-S) equation:

The G-S equation includesJφandF,the information on current density,andP,the pressure profile relative to ψ,the poloidal magnetic flux enclosed by a magnetic surface,and Δ*=R2∇·(∇/R2).In this work,the EFIT code is used to solve the G-S equation [13].EFIT is a well-known program that is applied in almost all the major tokamaks worldwide,including EAST tokamak [14].In EFIT,the two stream functionsP′(ψ) andFF′(ψ) are represented by a set of basic stream functionsynin terms of several linear parameters αnand βn,as is seen below:

In EFIT,polynomial or variable tension spline can be used to represent stream functions [15].The polynomial basis functions are:

and the variable tension spline basis functions are:

Here,σTis the tension of the spline. Δxn+1=xn+1-x,Δxn=x-xn,wn+1=xn+1-xn.Thexrepresents the normalized poloidal flux coordinate.

The coefficients αnand βnare calculated with the Picard iteration scheme to iteratively find the solution to the G-S equation and minimize the error quality function:

Here,Mi,σi,andCiare respectively theith measurement value obtained from diagnostics,the uncertainty associated with theith magnetic measurement,and the computed value.

Since αnand βnappear linearly,we can explicitly relate the unknown parameter vectordirectly to the measurement vectorthrough the response matrixThe EFIT program can construct the response matrix.The equilibrium is then calculated using Picard iteration.The detailed explanation is given in reference [13].

For equilibrium constraints,three kinds of diagnostic data are used: magnetic,kinetic,and current constraints.By using magnetic diagnostics,we could determine the plasma shape and other global information such as total plasma currentIp,internal inductanceli,and poloidal beta βp[3,16].The pressure profile reconstruction is constrained by kinetic diagnostics.The kinetic equilibrium reconstruction based on the EFIT code available on EAST can provide a more accurate pressure profile and edge current profile with kinetic constraint [7],which we refer to as the kinetic EFIT.The kinetic EFIT is described in detail in section 3.

In order to incorporate more internal current information into the reconstructed equilibrium,it is necessary to add theqprofile constraints to the kinetic EFIT reconstruction.In EFIT code,the MSE constraint can be deployed in kinetic EFIT reconstruction [13].The MSE diagnostics requires the Neutral Beam Injection (NBI) for spectrum measurement[9].However,the experiments performed without the use of the NBI system lack MSE diagnostic data.An alternative is to use the POINT constraint [11,17].Combining the POINT constraint with kinetic information would be beneficial for equilibrium reconstruction on EAST tokamak.

In the previous work,a method for combining POINT reconstruction and kinetic EFIT was developed [18].The current profile reconstructed with POINT constraint is first calculated.The equilibrium is obtained with constraints from both the current profile and the pressure profile.However,it uses the polynomial to represent the stream function,which cannot effectively match the edge pedestal structure effectively in the high confinement mode (H-mode) experiment.To improve the profile for H-mode experiments,this paper proposes a new approach based on the use of virtual magnetic probes [19].

2.Safety factor profile reconstruction

This section discusses the equilibrium reconstruction independently using POINT and virtual probes on EAST.These two methods have been shown to be effective in reconstructing safety factor profile.However,the accurate pressure profile cannot be obtained by the use of POINT or virtual probes alone.

2.1.Equilibrium reconstruction with POINT constraint

The POINT system operating on EAST tokamak can simultaneously provide 11 chords of phase shift and Faraday rotation for density and magnetic field information [12].The internal poloidal magnetic field profile of the plasma can be determined by measuring the rotation of a linearly polarized probe beam.According to the Faraday effect,the phase shift and the Faraday rotation angle of insight beam in relation to the plasma density and the parallel magnetic field are determined by the following formulae:

Here,φnerepresents the phase shift,θnerepresents the Faraday rotation angle in radians,λrepresents the wavelength of the source laser beam in meters,nerepresents the electron density in m-3,B‖represents the magnetic field parallel to the laser beam direction in Tesla,and the quantities are in MKS units.Figure 1 shows the installation position of the POINT system at the poloidal cross-section.The green dashed lines represent the 11 chords laser beam,which are located from 0.425 m to -0.425 m in the vertical position.Note that the Cotton–Mouton effect on EAST could be negligible [17].The Cotton-Mouton effect should be considered in when the current and density are high.

Figure 1.Location of the virtual probes and POINT chords on the poloidal cross-section.Red arrowheads represent the location of the probes and Z component of the magnetic field (BZ).Green dashed line represents the installation position of the POINT system and the laser beam line,which are located from 0.425 m to -0.425 m in the vertical position.The blue contour represents the magnetic flux surface,and the black line represents the first wall on EAST tokamak.

Based on the Faraday effect,JET tokamak has employed the NICE code and the Stokes model for equilibrium reconstruction [20,21].On EAST,the EFIT code has been developed to self-consistently incorporate density and Faraday rotation data from POINT with magnetic data into the equilibrium reconstruction [11].The phase shift and Faraday rotation angle are treated as parts of the measured value and added to the response matrix for Picard iteration as described in the introduction section.This EFIT version has been verified to agree well with the POINT diagnostic and to produce a reasonable density profile and safety factorqprofile in the EAST experiments [17].

2.2.Virtual probe

The virtual probe method is a suitable way of incorporatingqprofile constraints into the kinetic EFIT reconstruction.It has been shown that equilibrium can be reconstructed with safety factor constraints [19].We know that the safety factor is given by the formula:

Here,BφandBpare the toroidal and poloidal magnetic fields.So that theBpvalue affects theqprofile.In addition,the magnetic probes installed on most tokamaks measure the poloidal magnetic field.If we have a series of poloidal virtual magnetic probes inside the plasma separatrix,the EFIT code could take into account the internal magnetic information.With these virtual probes,we can also control theqprofile of EFIT output.When the exact geographical location ofqis known,as analyzed through diagnostics indicating MHD instability,the virtual probes provide a set of internal poloidal magnetic field constraints for EFIT reconstruction iteration.In each iteration,the virtual probes are automatically adjusted to bring theqprofile closer to the target profile until an optimal convergence is achieved.

For example,electron cyclotron emission (ECE) signals could exhibit periodic sawtooth behavior in the center of the plasma,along with the well-known inverted behavior in the outer region [22].This periodic variation is due to the presence of aq=1 surface near the location of the sawtooth inversion.This allowed us to obtain the position of theq=1 surface.EFIT reconstruction employs external magnetic diagnostics in conjunction with the virtual probes positioned at theq=1 location on the middle plane.The iteration formula is:

Here,themis the number of iterations,qmis the value of the safety factor calculated in themth iteration,is the value of virtual probe set in each iteration,andis the target safety factor value.Relax is a relaxation parameter in the range of 0.0–1.0,which can increase the computation time but reduce the risk of oscillation or failure of program.After several iterations,an equilibrium withq=1 surface coinciding with the sawtooth inverse location is determined [19].

3.Kinetic equilibrium reconstruction with internal safety factor profile constraints

3.1.Kinetic equilibrium reconstruction on EAST

The kinetic equilibrium reconstruction,which we call the kinetic EFIT,employed in EAST tokamak experiments plays an important role in research [7].Both the pressure profile and the edge current profile constraints are considered in this reconstruction.

The pressure profiles are determined using the following formula:

Thexrepresents the normalized poloidal flux coordinate.In this formula,the impurity temperature is assumed to be identical to the main ion temperature.Currently,on EAST tokamak,electron densityneand temperatureTeprofiles are obtained from Thomson scattering measurement [23,24].The electron density edge pedestal in the H-mode experiment is also obtained from microwave reflectometry [25,26].Theneprofile is checked with line average density measurement from POINT.Ion densityniand impurity densitynZare determined by quasi-neutrality condition withneprofile and effective ion chargeZeffmeasurement.The ion temperatureTiprofiles are obtained from both the X-ray imaging crystal spectrometer (XCS) [27,28] and the charge exchange recombination spectroscopy (CXRS) [29].In addition,the pressurePfcontributed by the fast ions from the neutral beam injection is calculated by the NUBEAM code [30].For EAST tokamak,the fast ion slowing down time is much less than the fast ion confinement time,so the fast ions have a slowing-down distribution and it is almost isotropic.The anisotropic pressure is not considered.

The edge current profile is another important constraint of the kinetic EFIT.Even with the MSE diagnostics,it is frequently difficult to capture the details of the current profile in the narrow pedestal region of H-mode plasma.However,we can use the bootstrap current to constrain the edge current.In general,the current consists of three components:

The components represent Ohmic current,auxiliary driven current,and bootstrap current,respectively.At the pedestal region,the current is dominated by the bootstrap current,and a local current peak is generated.This was confirmed by the experimental analysis [31–33].The Sauter model [34,35] is applied to calculate the bootstrap current.The Ohmic current is calculated with a model [31,34]:

Vlooprepresents the loop voltage measured by the flux loop,σneorepresents the neoclassical resistivity as is described in reference [34].

The kinetic EFIT reconstruction with magnetic data,pressure and edge current constraints has been verified to be effective for H-mode experiments on EAST [7].

3.2.Kinetic equilibrium reconstruction with virtual probe constraint

The combination of the kinetic EFIT reconstruction with the internalqprofile constraint is important for plasma research.This work uses the virtual probes yielded from the reconstruction with POINT constraint for internal current constraint,which we consider a good way to obtain the experimentalqprofile.The equilibrium reconstruction with all constraints is also developed.

Theqprofile of the equilibrium reconstructed by EFIT with POINT constraint includes the internal current information.A set of theZcomponents of poloidal magnetic field values (BZ) obtained from this equilibrium are assigned to the virtual probes.They are treated in the same way as other real magnetic probes.Using these virtual probes and external magnetic diagnostics,the EFIT code reconstructs an equilibrium that is quite consistent with the former one,including the flux surface and theqprofile [19].Figure 1 shows the location of the virtual probes.Figure 2 shows the comparison of the equilibrium profiles reconstructed with POINT constraint and virtual probe constraint,both without kinetic constraint.In this case,six virtual probes were used,located from 1.95 m to 2.2 m.Figure 2(c) shows the recalculated Faraday rotation angles from the reconstruction with virtual probe constraint,compared to the POINT reconstruction.It is shown that the reconstruction with virtual probe constraint matches the POINT reconstruction with a good quality.It should be noted that the recalculated Faraday rotation angles from the reconstruction with POINT constraint,ignoring the 4th and 8th chords of bad data,could not fully agree with the POINT diagnostics due to the algorithm of the modified EFIT code with POINT constraint mentioned in section 2.1.So,the virtual probes have the information on the safety factor from the reconstruction with POINT constraint.Here,ρrepresents the square root of the normalized toroidal magnetic flux coordinates.Thus,it is believed that virtual probes could be used in kinetic EFIT reconstruction.In this case,the positions of the POINT chords onρaxis locate from 0 to 0.65,atR=1.85 m on the large radius.

Figure 2.Comparison of the equilibrium reconstructed with POINT constraint (blue line) and virtual probe constraint (red line).Both reconstructions use the external magnetic diagnostics without the kinetic constraint.Figure (a) shows the cross-section of the magnetic flux surface contour,and the flux surface reconstructed with virtual probe constraint is similar to that with POINT constraint.Figure (b) shows the safety factor profile.Figure (c)shows the recalculated Faraday rotation angles with raw POINT data.

Figure 3.Flowchart of kinetic equilibrium reconstruction with POINT and virtual probes.

The procedure of kinetic equilibrium reconstruction is shown in figure 3.First,an equilibrium is reconstructed with POINT constraint.The density,temperature diagnostics,and fast ion components are then mapped onto the flux surface of the previous equilibrium to calculate the pressure profile.With the pressure profile,the edge current profile is also calculated.Values of virtual probes are extracted from reconstruction with POINT constraint to constrain theqprofile.These internal virtual probes combined with external magnetic diagnostics,as well as the pressure profile and the edge current profile,are incorporated into EFIT code inputs and run.Finally,the kinetic EFIT reconstruction withqprofile constraint is completed.To validate this method,we calculate the Faraday rotation angles using this equilibrium and the reconstructed electron density.The results are usually in good agreement with those from the POINT diagnostic.

3.3.Reconstruction results

For example,we calculate the kinetic equilibrium reconstruction with internalqprofile constraint for the long-pulse shot #90949 on EAST at 20 s.The plasma current is 350 kA.The stored energy is 170 kJ.The line-integrated electron density is 3.6×1019m-3.Figure 4 shows diagnostics data of density from microwave reflectometry,electron temperature from Thomson scattering,and ion temperature from XCS.The data points are transformed into flux surface coordinates,and the discrete data points are fitted with smooth functions.The fitted density is also calibrated by POINT.Figure 5 illustrates the line-integrated electron density and Faraday rotation as measured by 11 POINT chords,with the blue circles.Due to the poor quality,the 7th and 11th chords of the line-integrated density are not included in the POINT reconstruction.The 4th and 8th chords of the Faraday rotation angle signal trembled heavily in shot #90949,making them untrustworthy for consideration.

Figure 4.The diagnostics data and the fitted profile.Figure (a)shows the temperature.Te_TS is the electron temperature from Thomson scattering and Te_fit is the fitted electron temperature.Ti_XCS is the ion temperature from XCS and Ti_fit is the fitted ion temperature.Figure (b) shows the electron density.The ne_Refl is electron density from microwave reflectometry and ne_fit is the fitted electron density.The fitted density is also calibrated by POINT.

Figure 5.POINT system raw data of shot #90949 compared to the reconstruction with internal q constraint.Figure (a) is the Faraday rotation angles and figure (b) is the line integrated electron density.The locations of 11 chords are described in figure 1.Raw is the raw data from POINT.Kinetic is the recalculated values from the kinetic reconstruction with virtual probe constraint.Magnetic is the reconstruction with external magnetic diagnostics only.

Figure 6 illustrates the equilibrium result withqprofile constraint compared to the reconstructions with external magnetic diagnostics and POINT constraint respectively.The reconstructed pressure profile shows an apparent edge pedestal structure,and the reconstructed current profile shows the pedestal bootstrap current.The reconstructions using external magnetic diagnostics or POINT constraint could not provide accurate pressure profiles.The value of the error quality function calculated by EFIT is χ2=45,which indicates the computational quality of the magnetic diagnostics data.Since there are more than seventy magnetic measurements,on average,these errors are smaller than uncertainty.The error quality function could be further minimized to χ2＜10 when using a smaller number of virtual probes,but this would lead to an unreasonable equilibrium result such as negative pressure profile at core region.Figure 7 shows the reconstructedqprofile result.The internalqprofile of kinetic equilibrium withqprofile constraint agrees well with the equilibrium with POINT constraint well in ρ ＜0.7,and is superior to the magnetic reconstruction.Theqprofile in the pedestal region is slightly different due to the pedestal bootstrap current calculated in the previous phase.The recalculation of the Faraday rotation angles from the kinetic reconstruction with virtual probe constraint is shown in figure 5.It can be observed that the rotation angles from kinetic reconstruction are in better agreement with the diagnostics data than the external magnetic reconstruction from the 3rd to the 9th chords.As shown in figure 7,this means that theqprofile is closer to the POINT reconstruction in ρ ＜0.7.This establishes the self-consistency of the kinetic EFIT withqprofile constraint and POINT reconstruction.

Figure 6.Kinetic reconstruction with internal safety factor profile constraints of shot #90949 at 20 s on EAST.Figure (a) shows the cross-section of reconstructed magnetic flux surface contour with red lines.Figure (b) shows the comparison of reconstructed pressure profile with magnetic diagnostics constraint,POINT constraint and kinetic reconstruction with internal safety factor profile constraint.Figure (c) shows the reconstructed current profile.The kinetic reconstruction shows the pressure pedestal structure and the pedestal bootstrap current.

Figure 7.Comparison of safety factors q from magnetic,POINT,and kinetic reconstruction with virtual probes.The q profile reconstruction with virtual probe constraint matches well with the result of reconstruction with POINT constraint,and better than the magnetic reconstruction.

4.Discussion and summary

A viable way to improve the kinetic EFIT is proposed by using the virtual probe method,which has the ability to constrain theqprofile.The equilibrium reconstructed with POINT constraint provides the virtual probes to constrain the internalqprofile.The virtual probes can be considered as a magnetic diagnostic analogous to the outer ones.The EFIT reconstruction is then performed with the magnetic data,the pressure profile,and the edge current profile constraints.The new equilibrium has both current and pressure information from the diagnostics.It outperforms the previous kinetic EFIT reconstruction by showing a more accurateqprofile.It also has a pressure pedestal structure which is important for H-mode experiments by using spline representation for stream function.This method can be used in the experiments when the MSE data is not available.This is because the MSE measurement requires the specific NBI beamline,while on EAST most discharges only have the RF heating.

However,this method could be further improved.The cumulative error introduced by the two steps,POINT reconstruction and kinetic reconstruction with virtual probes,cannot be avoided for the time being.In figure 6(c),the reconstructed current profile has a larger value in ρ ＜0.2 compared to that from the reconstruction with POINT constraint.The reconstructed Faraday angle of the 2nd and 10this slightly smaller than the raw data value.This may be due to the higher localized current density.This phenomenon requires further investigation.While theqprofile has shown some achievements compared with the previous kinetic equilibrium reconstruction.Adjusting the calculation weight of the virtual probes takes a long time to improve the iteration convergence,so making the procedure more convenient and speeding up the convergence will be our next goal,and the value of error quality function χ2should be further minimized.

This method will be further validated in subsequent equilibrium reconstruction research.This project will advance the analysis of EAST experiments.In the future,after the MSE system is actually deployed on the EAST experiment,this method will be further cross-validated with MSE measurements and continuously improved.

Acknowledgment

This work was supported by National Key R&D Program of China (Nos.2019YFE03040004 and 2017YFE0300404).This work was also supported by Comprehensive Research Facility for Fusion Technology Program of China (No.2018-000052-73-01-001228).The authors would like to thank Dr.C.T.Holcomb from LLNL for his valuable suggestion.The numerical calculations in this paper were performed on the ShenMa High Performance Computing Cluster in Institute of Plasma Physics,Chinese Academy of Sciences.