Measurements and modelling of plasma response field to RMP on the COMPASS tokamak

The COMPASS tokamak is a compact-sized (R  =  0.56 m, a  =  0.2 m) experimental device of an ITER-like cross section, operated in a diverted plasma regime [13] (for more information about the discharge parameters used in this work, see section 4.2). Its RMP coil system consists of a series of independent ex-vessel conductors that cover the whole vacuum chamber and can be connected into a variable saddle coil configuration [14]. This offers a unique variability of the poloidal mode number m spectrum of the generated RMP. The two specific RMP configurations investigated in this work are depicted in figures 1(a) and (b) (plots were made using ERGOS code [15, 16]) and are referred to as on+off-midplane configuration and off-midplane configuration, respectively. All coils are single-turned, with the off-midplane coils being of even parity, while the midplane coils are of opposite polarity to them. Toroidally, the windings cover tokamak quadrants, generating an RMP field with toroidal mode number n  =  1 and 2. In this work, the n  =  2 field is used, since n  =  1 field is more prone to causing mode locking of magnetic islands that are typically present in the plasma [14]. The spectrograms of the RMP field for magnetic equilibria studied in this work, generated by the on+off-midplane and off-midplane coil configuration are depicted in figures 7(a) and (b), respectively.

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The RMP power supplies enable a single DC pulse per tokamak discharge of the same current magnitude in all of the RMP coils. The temporal evolution of the current waveform has the form of a trapezoid, with flat-top phase lasting several tens of milliseconds and current ramps from units to tens of milliseconds. The arrangement of the conductors generating the RMP uses two independent IGBT power supplies based on the design described in [17], but is capable of producing higher voltage.

The tokamak chamber is covered by a set of 104 ex-vessel saddle loops, arranged into four quadrants (radially located below the RMP coil quadrants), as depicted in figure 2. Poloidal and toroidal angles (θ and ϕ, respectively) are shown for reference. Additionally, the radial location of the saddle loops with respect to the separatrix and the RMP coils is shown in figure 3.

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Each of the four quadrant sets consists of 22 large saddle loops covering the whole quadrant in toroidal direction (e.g. SE1-22), and of four smaller saddle loops on low-field side (LFS) that cover both octants per quadrant in two poloidal rows (e.g. SSE1-2 and ESE1-2). In order to cover the whole chamber, it is necessary for the loops to often adopt a more complex shape to avoid vacuum vessel ports. Moreover, the simplified scheme in figure 2 does not take into account that the loops cover a different poloidal range, as seen from the in-scale figure 3. Note, however, that the real loop geometry was taken into consideration in the evaluation of the magnetic field signals presented in this work.

The poloidal cut of the COMPASS tokamak in figure 3 depicts the relative positions of the tokamak chamber, diagnostic saddle loops, RMP coils and typical plasma separatrix. This illustrates that on the COMPASS tokamak:

• Plasma separatrix is located close to RMP coils.
• The RMP coils cover large poloidal sections—especially the midplane coil.
• Diagnostic saddle loops are of different size, with the largest area loops located on high field side (HFS) and the smallest area loops located on the top and the bottom part of the chamber.
• There is a sufficient number of saddle loops located underneath the RMP coils to provide good information about the spatial distribution of the RMP field.

By an appropriate combination of the signals of each poloidal row of the saddle loops across all four toroidal quadrants as illustrated by the signs in figure 2, namely:

$$\begin{eqnarray}&&{{B}_{n2}}=\frac{1}{4}\left(B_{n}^{\text{NW}}-B_{n}^{\text{SW}}+B_{n}^{\text{SE}}-B_{n}^{\text{NE}}\right),\end{eqnarray} \tag{ 1 }$$

the quantity B_n2 is obtained, which represents the n  =  2 harmonic part of the normal component of the magnetic field. Since the quantities $B_{n}^{\text{NW}-\text{NE}}$ represent normal components of the magnetic field measured by the loops of the respective quadrant (on the chosen row), the resultant B_n2 is averaged across both the toroidal and the poloidal span of the used loops. Given the unique diagnostic arrangement on the COMPASS tokamak, it is possible to measure B_n2 on up to 22 different poloidal positions. Moreover, with the two rows of small octant-covering saddle loops on LFS, there are four possible combinations for obtaining the B_n2 quantity (as shown in the bottom part of figure 2). Therefore, measurements on four different toroidal positions ϕ (corresponding to the center of the used octant or quadrant) are provided. Note that the combination of the small loops No. 2 is equivalent to combinations of the large quadrant loops and in fact is a linear combination of combinations No. 1 and No. 3 (just like combination No. 4 is). As the loops are located outside of the vessel, the high-frequency part of B_n2 is cut-off by skin effect at the frequency of approximately 40 kHz. However, this is of no concern since only the flat-top part of the DC RMP pulse is analyzed in this paper. The RMP current driven by the two independent power supplies is measured with a set of two Rogowski coils.

The plasma response to both RMP configurations was modelled using the linear resistive MHD code MARS-F [12]. This code solves the linearized single-fluid MHD equations, assuming that the resulting perturbation of plasma equilibrium remains small [20]. Essentially, the non-axisymmetric RMP perturbation is imposed on the axisymmetric plasma equilibrium and a forced eigenvalue problem of stability is solved [10].

The unperturbed magnetic equilibrium of both discharges is provided by the numerical code EFIT+ + [21, 22], with the local magnetic measurements, the total plasma current I_plasma and the toroidal magnetic field ${{B}_{\phi}}$ used as inputs. In both cases, the chosen equilibrium corresponds to the temporal moment of 1164 ms, i.e. to the flat-top phase of the RMP current waveform. In addition, the cross-talk of the RMP field on the magnetic measurements used as the input for the equilibrium reconstruction was eliminated in the same manner as shown in equation (3). The magnetic equilibria were remapped to a straight field line coordinate system, using the equilibrium solver CHEASE [23], prior to being used as the input for the MARS-F code. However, since the code requires a finite, well-defined q(a), the plasma X-point was slightly smoothed in the process of re-mapping.

In the model the RMP coils are represented as toroidally aligned straight lines of finite poloidal width that carry a toroidal harmonic current  ∼$\exp \left(\text{i}n\phi \right)$ (with n  =  2 in this paper) [24]. This representation naturally differs from the real coil geometry, and thus the RMP field calculated by MARS-F was compared to the RMP field calculation by the Biot–Savart law-based ERGOS code [15, 16], which takes into account the real coil geometry. Specifically, the comparison between the RMP field components aligned with the pitch angle of the magnetic equilibrium of discharge #7454, generated by the on+off-midplane RMP configuration can be seen in figure 6. Since the difference stays below 8%, the MARS-F coil representation is considered satisfactory.

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The quantity describing the spectrum of the RMP field in this paper (and consequently also plotted in figure 6), is the normal field component [15]:

$$\begin{eqnarray}&&|b_{mn}^{1}|=\left|\frac{2}{R_{0}^{2}{{B}_{0}}}\frac{1}{\text{d}{{\psi}_{p}}/\text{d}s}\frac{\mathbf{b}\cdot \nabla \psi}{{{\mathbf{B}}_{\text{eq}}}\cdot \nabla \phi}\right|.\end{eqnarray} \tag{ 4 }$$

R₀ represents the radial position of the magnetic axis with B₀ being the toroidal field on this position. ${{\psi}_{p}}=\frac{\psi -{{\psi}_{0}}}{{{\psi}_{a}}-{{\psi}_{0}}}$, with the poloidal magnetic flux ψ normalized with respect to the magnetic axis flux ${{\psi}_{0}}$ and to the magnetic flux on the plasma separatrix ${{\psi}_{a}}$. $s=\sqrt{{{\psi}_{p}}}$ (note that $s\approx r/a$). The quantity $\mathbf{b}$ represents the perturbation due to the RMP coils and ${{\mathbf{B}}_{\text{eq}}}$ the magnetic equilibrium field. Note that $|b_{mn}^{1}|$ corresponds to the definition used in the ERGOS code. While n  =  2 is fixed for the applied perturbation field, numerous poloidal mode number m harmonics of variable radial distribution are present. The spectra of the two studied RMP configurations (for the given plasma equilibria) are shown in figure 7.

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Figure 7 shows the vacuum RMP field, i.e. without the effect of plasma. Comparison of figures 7(a) and (b) implies that the large midplane row RMP coils have a significant effect on the amplitude of the field, providing approximately half of the field magnitude. Distribution-wise, absence of the midplane coil row leads to the shift of the pronounced $m\pm 1$ harmonics from the plasma center to the edge regions, leading to much weaker magnitude of the generated RMP field on the midplane, with respect to the poloidal positions of the bottom/top coil row (see vacuum field distribution in figure 4 and discussion in section 3). The white diamond symbols in figure 7 show the location of the resonant surfaces satisfying the condition $q=\frac{m}{n=2}$. Having them located close to the ridge of the spectrum, rather than its valleys implies good resonance between the plasma equilibrium and the vacuum RMP field at the COMPASS tokamak.

In order to model the effect of plasma screening on the $|b_{mn}^{1}|$ spectrum, MARS-F needs to be provided with radial profiles of electron density n_e, electron and ion temperatures T_e and T_i and of toroidal plasma flow. On the COMPASS tokamak, the n_e and T_e profiles are measured by high-resolution Thomson scattering (HRTS, or TS in short) system [25–27], with spatial resolution up to 3 mm at the edge plasma, at the frequency of 60 Hz—see figure 8 for both studied discharges. The direct measurement of the T_i profile is not available presently. However, it is possible to obtain the profiles of T_e and T_i from the METIS code simulations [28]. By overplotting all of the obtained temperature profiles in figures 8(b) and (d), one can see that there is a good agreement between the results by METIS and TS. Therefore, the T_e and T_i profiles by METIS are used as the input for MARS-F in this paper.

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Similarly to T_i, the measurement of toroidal plasma flow is not available either. Although a well-established inter-machine empirical scaling relation exists for H-mode regimes [29], no such study was carried out for ohmic L-mode plasmas. However, on the Tokamak à Configuration Variable (TCV) an empirical relation for radial profile and magnitude of toroidal rotation velocity for ohmically heated L-mode discharges [30]

$$\begin{eqnarray}&&{{v}_{\phi}}(s)=12.5\frac{{{T}_{\text{i}}}(s)}{{{I}_{\text{plasma}}}}~\left[\text{km}~{{\text{s}}^{-1}},\text{eV},\text{kA}\right]\end{eqnarray} \tag{ 5 }$$

was found. We use this relation to obtain the toroidal plasma flow in this paper, taking into account that the TCV and COMPASS tokamaks are not too different in size (${{R}_{\text{COMPASS}}}=0.56$ m, ${{a}_{\text{COMPASS}}}=0.2$ m, ${{R}_{\text{TCV}}}=0.88$ m, ${{a}_{\text{TCV}}}=0.25$ m), as well as the similarity of the discharge parameters used to derive the relation (5) in [30] to the parameters shown in table 1. Therefore, the radial profile shapes of the toroidal plasma rotation frequency of the studied discharges are the same as those of T_i, with central ${{f}_{\phi}}(0)=7.8$ kHz and ${{f}_{\phi}}(0)=7.2$ kHz for discharges #8078 and #9655, respectively. This also yields central toroidal plasma rotation velocities of ${{v}_{\phi}}(0)=27$ km s⁻¹ for discharge #8078 and ${{v}_{\phi}}(0)=25$ km s⁻¹ for discharge #9655, which also agrees with typical ${{v}_{\phi}}(0)$ magnitudes expected for ohmic L-mode tokamak plasmas (table 1 in [31]). Although the estimates from relation (5) developed for the TCV tokamak might not be entirely valid for COMPASS, in section 4.3 it is shown that the principal modelling results are fairly robust with respect to the possible ${{f}_{\phi}}$ uncertainty.

The resulting $|b_{mn}^{1}|$ spectrograms, with the plasma response included, are shown in figures 9(a) and (b) for the on+off-midplane configuration and the off-midplane configuration, respectively. By comparison with the spectra of the original vacuum perturbations in figure 7, it can be seen that the screening effect of plasma is strong in both studied discharges—the pitch-aligned components of the perturbation field, whose positions are depicted by the white diamonds, are having low magnitudes. Note also the shift of field spectrum distribution from the resonant components of positive m towards negative m values of the non-resonant components. The relation of the described RMP spectrum distortions with the $B_{n2}^{\text{resp}}$ quantity is further discussed in section 5.

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Additional insight into the nature of RMP screening on COMPASS is provided by figure 10. Here, the magnitudes of pitch-aligned components of $|b_{mn}^{1}|$ across the q  =  m/n resonant surfaces are shown—the original perturbation versus the perturbation with plasma response is included. First, comparison of the on+off-midplane configuration in figure 10(a) to the off-midplane configuration in figure 10(b) once again shows that the magnitude of generated resonant field is significantly lower in the absence of the large midplane coil row. Also, both plots show shallow RMP penetration into plasma, which takes place at approximately the same depth, regardless of the RMP configuration. Simulations with the quasi-linear MARS-Q code [20] are planned within the scope of future work, where the modelled penetration of the RMP is expected to reach deeper into the plasma [32].

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While it is safe to assume that the radial profile of toroidal plasma rotation is similar to that of T_i, the magnitude from the relation (5) might not entirely characterize the plasma rotation on the COMPASS tokamak. Therefore, an uncertainty by a factor of 2 was assumed and simulations with ${{v}_{\phi 1}}(0)={{v}_{\phi}}(0)/2$ and ${{v}_{\phi 2}}(0)=2{{v}_{\phi}}(0)$ were carried out. Specifically, central rotations ${{f}_{\phi 1}}(0)=3.9$ kHz and ${{f}_{\phi 2}}(0)=15.6$ kHz were assumed for discharge #8078 and ${{f}_{\phi 1}}(0)=3.6$ kHz and ${{f}_{\phi 2}}(0)=14.4$ kHz for discharge #9655, respectively.

The results of the simulations are shown in figure 11 on the pitch-aligned components of $|b_{mn}^{1}|$, together with the original rotation simulations plotted for reference. It can be seen that within the tested frequency range, the effect on the magnitude of screened pitch-aligned components of $|b_{mn}^{1}|$ is linear and small for both of the RMP configurations. Moreover, the depth of the penetration did not significantly change either. It is therefore concluded that the results are not significantly sensitive to the possible uncertainties in the determination of the ${{f}_{\phi}}$.

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