Determination of current and rotational transform profiles in a current-carrying stellarator using soft x-ray emissivity measurements

Knowledge of internal plasma current and rotational transform profiles is important for an understanding of magnetohydrodynamic (MHD) stability of toroidal plasma and reliable operation of toroidal magnetic confinement devices. However, it is well known that external magnetic diagnostics have limited sensitivity to the variation of the internal current distribution and typically yield only global current and pressure profile parameters such as poloidal plasma beta $βp$ and internal plasma inductance, $li$⁠,^1–3 although reconstructions of highly shaped tokamak plasma allow distinguishing between some global parameters.⁴ The internal inductance l_i is defined as the volume averaged square of the poloidal field, normalized to the square of the poloidal field at the edge of the plasma. It characterizes the breadth of the current profile in axisymmetric plasmas. Even in stellarator plasmas with flux surface geometries characterized by high elongation and non-axisymmetric 3D shaping, magnetic diagnostics located outside the plasma are not predicted to yield details of the internal current profile.⁵ Previous experiments on the Compact Toroidal Hybrid (CTH) current-carrying torsatron⁶ have demonstrated that external magnetic diagnostics by themselves do not provide well-constrained values of the current and safety factor (q) profiles.⁷ 

As in tokamak discharges, the plasma current profile and thus the q-profile in a stellarator discharge can be inferred from measurements of the internal poloidal magnetic field from motional Stark effect^8,9 (MSE) or Faraday rotation diagnostics,^10,11 for example. However, these diagnostics may require substantial investment and work best in strong magnetic fields and/or dense plasmas, and thus, it is of interest to develop and test alternative diagnostics suitable for smaller, lower-field experiments.

Christiansen and Taylor analytically demonstrated that the current distribution in an axisymmetric discharge with non-circular cross section may be determined completely from the geometric shape and centroids of the magnetic flux surfaces.¹² One way to measure the shapes of the magnetic flux surfaces is to measure the soft x-ray (SXR) emission from the plasma along multiple chords. With the assumption that the plasma temperature, density, and impurity concentrations are constant on a flux surface, the SXR volume emissivity should also be constant on that flux surface. Iso-emissivity contours extracted from SXR imaging can thus be used to reconstruct the current distribution. Based on this idea, plasma current and safety factor profiles were obtained using SXR imaging in JET,¹³ PBX,^14,15 and PEGASUS^16,17 experiments. These applications made use of a two-step algorithm, which combines the extraction of SXR emissivity contours and equilibrium reconstructions. The emissivity contours were first determined using direct tomographic inversion of an SXR image or an Abel inversion of the SXR intensity, and then the iso-emissivity constraints were incorporated in an MHD equilibrium code to reconstruct the current and q-profiles. A comprehensive method to self-consistently reconstruct the equilibrium current profiles with magnetic and SXR data using the equilibrium fitting code EFIT was developed on DIII-D.¹⁸ 

In this paper, we describe the use of SXR data in conjunction with external magnetic diagnostics to determine the plasma current profiles in hybrid discharges in CTH, in which the plasma equilibrium is generated by both the external helical torsatron coil current and an induced ohmic plasma current. The fitting of the plasma parameters to the data is performed using the fully 3D equilibrium reconstruction code V3FIT.¹⁹ The paper is organized as follows: the CTH experiment and SXR camera system are described in Sec. II. The details of the implementation of SXR emissivity measurements in the V3FIT code are described in Sec. III. In Sec. IV, reconstructions using experimental SXR data and external magnetic data are presented and compared to ones using magnetic measurements and prior information of the position of the q = 1 surface from sawtooth inversion. The results are summarized in Sec. V.

The Compact Toroidal Hybrid (CTH) is a low aspect ratio (⁠$R0/aplasma∼3.5$⁠), five-field period torsatron. It can operate as a traditional stellarator with ECRH heating or as a stellarator/tokamak hybrid with inductively generated plasma currents. The magnetic equilibria of CTH plasmas are highly non-axisymmetric, both with and without plasma current. Therefore, fully three-dimensional equilibrium reconstruction is required for their diagnosis.

In this paper, equilibrium reconstructions of CTH plasmas are performed with SXR emission data and external magnetic data. The set of 50 magnetic diagnostics used in these reconstructions has previously been described.⁷ While these external magnetic diagnostics can be used to determine the boundary and some global properties of the plasma equilibrium, x-ray emission from the core of the plasma will be used to reconstruct the current profile in the plasma interior.

Ten pinhole-type SXR/bolometer cameras are installed on CTH at different toroidal and poloidal locations. All of them utilize photo-diode arrays with twenty detection elements each, giving a total of 200 channels. They are located at either of the two planes of vertical symmetry within a field period of the five-period CTH geometry. Filters with different materials and thicknesses are placed in front of the detection diodes to block low energy photons. The SXR camera suite on CTH consists of three two-color cameras²⁰ and two bolometer/SXR systems.²¹ Figure 1 is a rendering of the last closed flux surface of a CTH plasma showing the location of the poloidally viewing SXR cameras.

A two-color SXR camera system is mounted on a toroidal symmetry plane at $ϕ=252°$⁠. It consists of three sets of two-color SXR cameras, with one set viewing from the outer mid-plane and another two sets at $60°$ above and below the mid-plane. The lines-of-sight of the three cameras are shown in Fig. 2(a). At each poloidal location, two parallel 20-channel diode arrays view the plasma through collimators and compound filters with a $0.5 μm$ carbon layer and either a $1.0 μm$ or $3.0 μm$ aluminum layer. They both observe photons with energy from 0.6 keV to 20 keV, while the cameras with thinner filters have a larger transmission window.

The bolometer-SXR camera shares a similar design as the two-color cameras, but one of the photo-diode arrays has no light filter thus acting as a bolometer, while the other photo-diode array does view the plasma through a filter. One bolometer-SXR diode array is installed at $ϕ=0°, θ=270°$ to have a vertical view of the plasma at its most vertically elongated cross section and uses the same thin aluminum/carbon filters as in the two-color system. An additional camera is mounted at $ϕ=36°, θ=60°$⁠, using $1.8 μm$ thick beryllium filters, observing photons from 0.5 keV to 20 keV Their chordal sight lines are shown in Figs. 2(b) and 2(c).

Sawtooth oscillations characteristic of ohmically driven plasmas are frequently observed with SXR cameras in CTH discharges that are sufficiently dense (normally above $1.5×1019 m−3$⁠).²¹ The presence of a q = 1 surface within the plasmas may be identified under the assumption that the inversion radius of the sawtooth signals corresponds to the magnetic surface at which q = 1, as is the case with tokamaks.²² Previously, we have shown that prior knowledge (subsequently referred to as a “prior”) of the position of q = 1 surfaces provides a strong constraint that can be used to fit plasma current profiles with far greater accuracy than using magnetics alone.⁷ However, the application of this particular constraint on the location of the q = 1 surface is not straightforward in the V3FIT fitting procedure. The inversion surface must first be identified before its location can be used as a prior as discussed in Ref. 6. The prior technique also introduces inaccuracies due to the limitation of physical resolution of the SXR camera system and computational resolution of equilibrium code. The flux surfaces in the VMEC code are labeled by s, the normalized toroidal flux, where s = 0 corresponds to the magnetic axis, and s = 1 the outermost closed magnetic surface. The flux grid evenly spreads in the s coordinate and will be naturally denser at the edge of the plasma than in the center, inducing inaccuracy of identifying the location of q = 1 surface near the center of the plasma. Also, increasing the resolution of the flux grid will significantly increase the time for convergence. Furthermore, the prior method only uses SXR emissivity measurements from one diode array out of the eight available on the CTH system. Given the availability of the other internal SXR measurements, it is desirable to develop a more straightforward way to directly incorporate all SXR measurements in V3FIT to help specify the q = 1 surface. This has been accomplished by reconstructing the shape of magnetic flux surfaces and thus inferring the plasma current distribution using emissivity measurements from all SXR diode arrays in V3FIT.

As a non-axisymmetric, torsatron/tokamak hybrid device, CTH is a useful platform to benchmark and validate the fully 3D equilibrium reconstruction code, V3FIT. Using the VMEC code^23,24 as the equilibrium solver, V3FIT solves the classical inverse problem: using known experimental measurements or data, $d$⁠, to determine plasma equilibrium parameters, $p$⁠. V3FIT optimizes the parameter set, $p$⁠, to achieve the best agreement between modeled signals and experimental measurements. In this optimization process, the value of $χ2$⁠, defined as

is minimized. Here, $SiO(d)$ are the observed signals derived from the experimental data, and the uncertainties in the measurements are expressed as $σiS$⁠, the variance of the signals under the assumption of uncorrelated Gaussian noise. Modeled signals, $SiM(p)$⁠, are determined from the VMEC equilibrium, or from an auxiliary model imposed on top of that equilibrium. Auxiliary models are usually implemented assuming profiles where the values are constant on a flux surface. Profiles in VMEC and V3FIT are parameterized as either functions of the flux surface or as interpolated segments such as splines or line segments.

The SXR chordal model used in V3FIT is a one-dimensional line integration through the center of the solid angle observed by a particular photo-diode detector. To utilize SXR emissivity measurements, both geometric information (geometric input) and experimental measurements (data input) of all SXR channels are required in V3FIT. The position and orientation of each SXR camera diode are measured with a coordinate-measuring machine to a nominal accuracy of 0.25 mm with respect to the CTH magnet coil system. Based on those measurements, coordinates of all the chordal sight lines with respect to the magnetic geometry are calculated and formatted as geometric inputs to V3FIT. Due to the flat geometry of the diode array and the placement of the viewing slit in the center of the diode array, the angular extent of the poloidal view is different for each diode within each SXR camera. Because the edge channels are further away from the slit than the central channels, there results an effective decrease in the slit width. This so-called geometric factor is accounted for when comparing integrated emissivity measurements in V3FIT²¹ and is included in the geometric input file. The data input contains processed SXR measurements from all chords of the cameras illustrated in Fig. 2. This post-processing includes deconvolution to correct the phase shifts and account for gains from the amplifiers and low-pass filters at 10 kHz. The data are further smoothed by averaging over a time window of 1 ms before being used in V3FIT.

With the known geometry of SXR chordal sight lines, parametrized emissivity profiles are specified in flux geometry to calculate the simulated SXR emission from a given VMEC equilibrium. Because the SXR cameras record emission above a certain energy threshold that is filter dependent, each set of SXR diodes with filters of the same thickness and material is used to create a filter-specific piecewise-linear emissivity profile. The large number of chords available expands the number of parameterizations that can be used. Each piecewise-linear emissivity profile consists of ten line segments, the height of each segment representing the emission intensity at specific flux surface. Line segments of length 0.1 (in normalized flux units) are reconstructed from the axis at s = 0 to the plasma boundary at s = 1, where the emission intensity is set to be zero. For each emissivity profile, there are ten reconstructed parameters.

Using specified emissivity profiles and plasma equilibrium provided by VMEC, modeled SXR signals are calculated using line integration. With processed experimental inputs and modeled responses calculated by V3FIT, the corresponding value of $χ2$ is calculated. The emissivity profiles along with other plasma parameters are optimized/reconstructed iteratively within V3FIT until a minimum value of $χ2$ is obtained.

In order to deconvolve the flux surface geometry from the SXR emission, we make the reasonable assumption that X-ray emissivity is taken to be constant on a magnetic flux surface in non-pathological discharges. However, experimental indications of X-ray emission not being constant in a flux surface have been reported in larger experiments.^25,26 For example, a discrepancy between the X-ray contours and magnetic flux surfaces has been observed in Alcator C-Mod tokamak plasmas with MARFEs, leading to the conclusion that X-ray emissivity is not always constant on a flux surface, particularly in plasmas evidencing local thermal instability. Nonetheless, we will make the assumption that at least within the plasma core, the iso-emissivity contours may be aligned with the magnetic flux surfaces. SXR emission is a function of electron temperature, density, and impurity concentrations. Since electron thermal transport parallel to the magnetic field is strong, considering electron temperature to be a good flux quantity is a reasonable approximation. Also, because there is no symmetry or quasi-symmetry in the magnetic field other than the usual stellarator field-period symmetry, we expect flows to be significantly damped, and hence centrifugal effects that might lead to density not being a good flux surface function should be small. The assumption of impurity concentration as a flux function is the one most difficult to justify. Without further measurements to independently quantify this assumption, we simply assume it and investigate the ability to fit the profiles.

Examples of the final fitting between experimental measurements and modeled signals derived from reconstructed equilibrium are shown in Fig. 3. This particular discharge has an averaged plasma density around $1.8×1019 m−3$ and plasma current, $Ip∼43 kA$⁠. SXR cameras that use the same filters and the same emissivity profile are identified with the similar markings in the upper right corner of the figures (circles, triangles, and a square). Here, three emissivity profiles are introduced. The two sets of signals match well for cameras in different poloidal and toroidal positions. The reasonable profile fit results support the assumption that SXR emissivity may be taken to be a constant on a flux surface for CTH plasmas.

To demonstrate the effect of including SXR emissivity measurements in V3FIT, reconstructions of sawtoothing plasma are presented with and without the use of SXR data in the V3FIT. The reconstruction process returns the toroidal magnetic flux enclosed by the plasma, the safety factor profile, and the plasma current profile, all subject to the parameterizations allowed by the model. A two-power model has been employed to model the current density profile, as follows:

Here, I(s) is the net toroidal current enclosed by the flux surface labeled by s, the normalized toroidal flux. Integration of $I′(s)$ over s gives the total plasma current, the amplitude of which is measured by full Rogowski coils. The current profile parameterization is based on a single fitting parameter α. Throughout this work, β is chosen to have a value of 6. Lower values of the current profile parameter α lead to a more peaked current profile while larger values of α model broader current profiles.

Figure 4 shows an example of a sawtoothing plasma with time traces of the plasma current, electron density, and two SXR signals from the SXR camera at $ϕ=252°,θ=0°$⁠. The higher signal is from a channel viewing the plasma core, while the lower signal is from a channel viewing the plasma near the edge. The sawtooth signal in the core and inverted sawtooth signal near the edge are clearly seen in Fig. 4(d), which shows the SXR signals expanded at the time highlighted by the gray box in Figs. 4(a)–4(c).

An equilibrium reconstruction was first performed using external magnetic diagnostics alone. The reconstruction was performed at the time t = 1.653 s in Fig. 4(c) when the plasma is sawtoothing. The resulting current and safety factor profiles along with their error bars are shown colored in gray in Figs. 5(a) and 5(b), respectively.

In the simple case of uncorrelated error in the data, the signal covariance matrix is diagonal,²⁷ 

By assuming a Gaussian distribution with a small variance, the error in signal space maybe mapped into parameter space

where J is the Jacobian of the modeled signals with respect to the model parameters

The uncertainty of reconstructed parameters is taken from the diagonal elements of the parameter covariance matrix, $Cparam$⁠.

In this reconstruction, the central safety factor is found to be $q≈2$⁠, which is inconsistent with the well accepted premise that the central safety factor of sawtoothing discharges should be $q≈1$⁠. Also, the large fitting uncertainties present in both the current profile and central q values highlight the limitation of using external magnetics to determine the current distribution in the plasma core. The unsymmetrical limits in the fitted current profile are an artifact of the two power model, in which the physical breadth of the profile increases nonlinearly as the value of α increases. At a lower value of α, the same increase or decrease in the α parameter results in a larger modification of the breadth of the profile.

Next, the same discharge is again reconstructed using the 160 SXR chordal measurements along with the same set of magnetic signals used above. In addition to the magnetic equilibrium parameters of the total enclosed toroidal flux and the current density profile parameter α, three emissivity profiles prescribed with a total thirty parameters are also reconstructed. The reconstructed current and safety factor profiles are plotted in red in Figs. 5(a) and 5(b), and are significantly different from the results from the reconstruction using external magnetics alone. Inclusion of SXR emissivity measurements in the reconstruction results in a substantially more peaked current profile with a fitted value of the central q to be near unity, consistent with the sawtoothing nature of the discharge. In Figs. 5(a) and 5(b), the fitted profiles using both magnetics and SXR data exhibit lower uncertainties than in the case with magnetics alone, demonstrating the effectiveness of the SXR profiles in constraining the shape of the current profile, which again is significantly narrower than case where only magnetic data were used.

The signal effectiveness of each measurement channel is defined in V3FIT¹⁹ as

where $σjP$ is taken from the diagonal of the parameter covariance matrix $Cparam$⁠. It is a dimensionless, normalized ratio of the fractional reduction in the variance of the reconstructed parameter (α in this case) to the fractional reduction in the signal variance.

In a test of the sensitivity of different diagnostics to the change of current profile, a set of 140 discharges with different plasma parameters are reconstructed with all magnetic and SXR diagnostics. The averaged values of the signal effectiveness for reconstructing the current profile width, or α parameter, for all of the magnetic and SXR diagnostics are calculated and compared. Not surprisingly, the SXR measurements are generally much more effective in determining the current profile parameter α compared to magnetic measurements. The most effective magnetic diagnostics are saddle coils, the locations of which were optimized to be most sensitive to changes in the current profile using the V3FIT signal effectiveness calculations. However, the signal effectiveness of the saddle coils is two orders of magnitude or smaller than those from the SXR channels.

The values of the signal effectiveness of all SXR channels are shown in Fig. 6. Generally, the effective SXR chords are those where the radial gradients of the signals are largest. It is interesting that the central cameras from the two-color system have the most effective chords, and the thickness of the filters appears to have no effect on the strength of the signal effectiveness. The camera with beryllium filters has a different spatial dependence of the signal effectiveness, which is consistent with the fact that the SXR signals from this camera have a different spatial profile (broader). It is also important to point out that the camera at $ϕ=0°$ is less sensitive compared to other cameras except the very central channels that are very sensitive to the horizontal shift of the axis. This is due to the fact that the geometry of the magnetic flux surfaces at $ϕ=0°$ [see Fig. 2(b)] is vertically elongated and mostly parallel to the sight lines of the SXR camera at this location and because the flux surfaces at this location experience a smaller horizontal Shafranov shift with a change of the plasma current, compared to the shift of the flux surfaces at the half period of CTH (⁠$ϕ=36°$ and $252°$⁠).

Similar reconstructions incorporating fitting of the SXR emissivity profiles are performed for a set of sixty-nine current-driven plasmas that all exhibit sawtoothing behavior. These discharges are very different in terms of plasma current, density, and external rotational transform settings (imposed stellarator transform). Reconstructions of each discharge are performed in three different ways: using external magnetic diagnostics only, using magnetic diagnostics along with a prior constraint of the location of the q = 1 surface from the sawtooth inversion radius, and directly using all SXR and magnetic data. The reconstructed edge safety factor values, $qedge$⁠, are seen to be similar for the three methods of reconstruction, showing good agreement with less than 5% difference between methods. That all three reconstruction methods give similar values for the edge safety factor again highlights the fact that external magnetic diagnostics are effective in determining characteristics on the magnetic equilibrium close to the last closed flux surface.

The resulting central safety factor q₀ for the same set of sawtoothing shots using the three methods are plotted in Fig. 7 against the edge safety factor. It is seen that the central safety factors, q₀, obtained from reconstructions by directly using magnetic and SXR data are consistent with results using the prior constraint of the location of q = 1 surface to a percentage difference within 15%. Also from Fig. 7 it is seen that the q₀ values obtained from reconstructions using only magnetic diagnostics increase with the edge safety factor and are far from unity for these sawtoothing discharges. When one recalls the large variance exhibited in Fig. 5(b) in the fit of the value of the central safety factor, it is a clear indication that external magnetics are incapable of accurately diagnosing the details of the q profile, i.e., the internal current distribution in 3D plasmas, as in axisymmetric plasmas.

With improved knowledge of the current distribution within the plasma, the effect of strong 3D shaping magnetic fields on CTH plasmas can be explored. A group of discharges are reconstructed at the times when their plasma currents reach maximum values. These plasmas are selected to have similar values of density (⁠$1.2−1.5×1019m−3$⁠) and current (⁠$50–52 kA$⁠) but varying vacuum rotational transforms (from 0.02 to 0.09). The internal inductance l_i, familiar from the specification of tokamak equilibria, characterizes the breadth of the current profile in tokamaks. However, since much of the confining poloidal magnetic field in a hybrid stellarator is provided by the current in the external coils, l_i is relatively insensitive to the current profile in hybrid stellarator plasmas, and thus is not particularly useful for distinguishing between different current profiles in CTH plasmas. Instead, we use the full width at half maximum, FWHM, of the current profile in the normalized flux surface parameter s as a measure of the breadth of the current profile.

The FWHMs of reconstructed current profiles and internal inductances are shown in Fig. 8. We find that the current profile becomes significantly broader with addition of vacuum rotational transform, i.e., increasing level of stellarator fields. By comparison, the internal inductance, l_i decreases only slightly as the current profile broadens. That the FWHM and l_i approach the similar magnitude at low values of vacuum transform is just a coincidence.

With the increase in vacuum rotational transform, the portion of the central transform from the central plasma current density must necessarily decrease to keep the net central transform near or below unity, thus leading to a broader current profile for a given value of plasma current. The variation of neoclassical resistivity may also play an important role here. The resistivity depends on the trapped electron fraction, which is strongly affected by the level of toroidal field ripple. These toroidal ripples in a stellarator will be the effective sum of several spectral terms of non-zero toroidal periodicity. The profile of the toroidal magnetic ripples may be different when the magnetic field configuration changes.

Determining the internal current distribution in a magnetically confined fusion plasma is useful to infer the MHD properties of the discharge. Because direct measurement of the internal poloidal magnetic field is not available in CTH, we have developed a straightforward way to incorporate SXR emissivity measurements to constrain the parameters that describe the plasma current profile through 3D equilibrium reconstruction using the V3FIT code. In this paper, measurements from eight SXR camera arrays at different toroidal and poloidal locations are used to reconstruct the geometry of flux surfaces and thus infer the current distribution within the plasma. The original approach of deriving the plasma current distribution from the geometric shape of the flux surfaces through consideration of the MHD equilibrium requirement is based on toroidally symmetric, non-circular cross section configurations. We have shown that the concept remains valid and practical in a non-axisymmetric stellarator with significant plasma current. Equilibrium reconstructions of sawtoothing plasmas using SXR and magnetic data predict the central safety factors to be around unity, in agreement with reconstruction results where external magnetic data were used in conjunction with the prior constraint of the location of the q = 1 surfaces extracted from SXR emissivity data. Improved knowledge of the current distribution within the plasma indicates that the current profile of these hybrid plasmas becomes broader with increasing vacuum rotational transform, or level of 3D stellarator fields, at given plasma density and current. Also, there is evidence showing that the internal inductance, l_i, which is a useful measure of the breadth of the current profile in axisymmetric plasmas, is not as useful for hybrid stellarator equilibria. Many small-scale or low-field shaped plasmas do not have the ability to directly measure the internal magnetic field, but are equipped with x-ray diagnostic systems that have good temporal and spatial resolutions. The technique described in this paper provides an alternative method that can be used to measure the current distribution, 3D in our case, with reasonable accuracy as demonstrated.

This work was supported by the U.S. DoE Contract No. DE-FG02–00ER54610. We thank John Dawson of Auburn University for his technical assistance with diagnostics and in operating and maintaining the CTH facility.