Numerical investigation of toroidal plasma response for ELM control via magnetic perturbations in the DTT Tokamak

Linear plasma response modeling is exploited in this work to assess the effect of different coil configurations on edge localized mode (ELM) stability in full power operational scenarios for the Divertor Tokamak Test (DTT) facility, presently under construction at the ENEA site of Frascati (Italy). The MARS-F code is used to compute, in toroidal geometry and including flow, the resistive plasma response to different vacuum fields with toroidal mode numbers n=1,2,3 . Peeling-like response in particular, correlated with ELM control, is found to be significant for n=2,3 perturbations while n=1 induces a large core response in the investigated scenarios. Two metrics are used to link plasma response to ELM control. Namely the local normal plasma displacement in the x-point region and the Chirikov parameter in PEST-like straight-field-line coordinates. These criteria are used to predict optimal phasing of the active coil arrays and current thresholds based on empirical evidence. Depending on the number of active coils and on the scenario, coil currents between 20 and 40 kAt are predicted to be effective for ELM mitigation in DTT.


Introduction
H-mode plasma scenarios, with enhanced energy and particle confinement at plasma edge, are a viable option for fusion energy production and represent the core operational regimes for present day and next generation fusion experiments such as ITER.Good confinement however comes at the price of potentially large pressure and current density gradients in the edge region, leading to the so-called edge localized modes (ELMs).These can be a major challenge for the safe and efficient operation of fusion reactors.ELMs are instabilities that can cause rapid heat loss from the plasma edge and can damage the plasma-facing components.Applying 3D resonant magnetic perturbations (RMPs) with non-axisymmetric saddle coils is a promising method to mitigate or suppress type-I ELMs.Experimentally, the capability of mitigating or suppressing ELMs with small 3D perturbations has been observed on all the present generation devices.Starting with DIII-D [1] different configurations of coils have been successfully used on MAST, ASDEX-Upgrade, KSTAR and EAST [2][3][4][5].
The underlying physics of ELM control with RMPs is different for mitigation or suppression.Magnetic perturbations cause a three-dimensional distortion of flux surfaces and can locally enhance the pressure gradient, thus driving ballooning modes (more) unstable [6].This leads to more frequent but less powerful ELMs.Suppression instead requires that the pedestal pressure is kept below a given threshold for peelingballooning modes.This is achieved with transport and many channels are possible, such as magnetic islands in the pedestal region [7,8] or MHD perturbations with non-linear mode coupling [9].Edge-peeling plasma response is a criterion that can be applied for both applications, since it correlates with plasma displacement in the pedestal region (thus with ballooning instabilities and mitigation) but also with the size of magnetic islands near the edge [10].Experimental results such as those mentioned above led to the design of the ITER ELM control coils [11], which can generate up to n = 3 rotating perturbations.
The Divertor Tokamak Test (DTT) experiment [12,13], presently under construction in Frascati (Italy) with the main mission of developing reactor-relevant power exhaust technologies.Non-axisymmetric in-vessel coils are being designed for the purpose of ELM mitigation and suppression on DTT.The design of these coils is principally driven by the following requirements: (i) physical constrains for in-vessel mounting, (ii) integration with other in-vessel sub-systems, (iii) double purpose of ELM and Error Field (EF) control [14].An advantage of ELM control with RMPs is that it can be used to reduce the frequency and severity of ELMs without affecting the overall performance of the tokamak plasma.This is because RMPs only affect the plasma edge, where ELMs occur, and have little impact on the core if properly tuned.One of the main challenges is to find the optimal RMP configuration that will effectively control ELMs without causing other adverse effects on the plasma.This requires a careful evaluation of the plasma response to different RMP configurations.Linear plasma response modeling is exploited in this work to assess the effect of different coil geometries on ELM stability in a full power DTT scenario.Peeling-like plasma response in particular is found to be correlated with ELM control and is computed by solving single fluid MHD equations in toroidal geometry with the MARS-F code [15], which is a linear single fluid resistive MHD code that combines vacuum perturbations with plasma response including screening effects due to toroidal rotation.These mostly lead to an amplification of the non-resonant (m ⩽ 0) components (i.e. the modes that have opposite helicity to the field lines) but a significant reduction in the resonant components.In the MARS-F modeling the RMP field also causes a 3D distortion of the plasma surface, which potentially leads to the formation of a 3D steady-state equilibrium.The plasma displacement varies with toroidal angle (ϕ) as ξ e inϕ where n is the toroidal mode number (n = 3) and ξ is the amplitude of the normal displacement of the plasma surface which varies as a function of poloidal angle (θ).In the linear response approximation the displacement amplitude is proportional to the current flowing in the coils.Although it is not possible to calculate threshold values, empirical evidence can be used as reference.
In order to evaluate the RMP effect on ELM stability, local plasma displacement in the X-point region is used as the main metric.This criterion has been correlated with ELM mitigation thresholds in MAST and ASDEX-Upgrade.Both devices have shown similar critical values for X-point displacement to achieve ELM mitigation [16], even with different coils geometries and applied fields, a critical value of ∼1.5 mm has been found.By taking as reference a displacement value of 3 mm at the X-point, the optimal operational space for nonaxisymmetric coils in DTT is sketched in terms of coil current and phasing between independent toroidal arrays.A similar modeling approach, also using the plasma displacement among other metrics, has been adopted in studies for EU-DEMO [10].Similarly, since no robust cross-machine scaling law can be proposed for the critical value of X-point displacement, we assume 3 mm as a conservative choice to estimate the required coil current for ELM mitigation in DTT.These thresholds, supported by empirical evidence, are used here to provide physics-based input for coil design.In particular for establishing a first indication of the total coil current required to achieve ELM control in the operational scenarios foreseen for DTT.As a second metric the Chirikov parameter (σ Chir ) is implemented in the present study.This parameter is a measure of the magnetic island overlap, used to define a stochastic layer induced by RMPs where σ Chir > 1.As a general comment on the adopted single fluid model, we highlight that possible screening effects from electron flow [17] are not accounted for.As discussed, for example, in [18], this can be important in the pedestal region and shall be investigated with codes including two-fluid effects.However, effects of parallel (toroidal) and diamagnetic flow on resonant field screening can be put into similar mathematical forms [19], possibly explaining the aforementioned good comparison with experimental results.An MHD model can be thus considered well suited for the present first device-oriented study.
The paper is organized as follows.In section 2 the investigated equilibria are described, and the MHD linearized model implemented in MARS-F.The coil geometry for DTT is presented in section 3, while numerical results are grouped in section 4. The analysis starts with the initial target full power scenario, which is reported for comparison with the present reference case and its variations.These latter cases will be used to provide an estimate of the total coil current required for ELM control.It should be noted that the final requirement shall be set by a combination of all the tasks that this magnetic system will performed, for example the sum of requirements for both ELM control and EF correction.This allows the design of a system with maximum flexibility and capable to simultaneously address more than one task.

Target equilibria and model formulation
The main scenarios for the full power operation of DTT are investigated in the following sections.For the purpose of designing the active coil system, two full power and full current cases have been considered.This scenario shall represent the baseline of DTT operations and thus for RMP application.The first case addressed is a DTT single-null target (R 0 = 2.14 m, a = 0.65 m), H-mode scenario with plasma current I p = 5.5 MA, toroidal field B t = 6 T, q 95 ∼ 2.7 and an additional heating power of 45 MW.A second variant of the fullpower baseline is then considered, this is also obtained from integrated modeling but the equilibrium magnetic axis is shifted to R = 2.19 m.This mainly serves a stability improvement purpose, resulting in q 95 ⩾ 3.These two variants of the full power scenario will be named R214 and R219 respectively.The development of these scenarios is detailed in [20,21].The main profiles characterizing these equilibria are reported in figure 1.The two plasma boundaries are instead given in figure 2(a) along with the coil geometry, discussed in the next section.Finally, the robustness of results for vertical plasma shifts is considered by applying rigid displacement of the magnetic axis, for the R = 2.19 m equilibrium, an assessing how this affects plasma response.The rationale behind this lies in ongoing studies and optimization of the DTT baseline scenario.For the purpose of providing useful information for coil design, the present full power baseline has been chosen.This is in fact the reference for H-mode operation and presently the most studied case.In future work, we shall numerically investigate plasma response in scenarios for early machine operation with reduced auxiliary power.
The MARS-F [15,22] code is used to calculate plasma response to RMPs in the aforementioned plasma equilibria.The code solves linearized resistive MHD equations in toroidal geometry including toroidal flow V 0 = RΩ Φ and a viscous term in the momentum equation that models Landau damping due to parallel sound waves and can compare well with experimental results when the target equilibrium is not too close to the Troyon limit [23]; this will be discussed further in section 4. The model can be described by: where Ω RMP is the frequency of the external perturbation (in the present work Ω RMP = 0), n is the toroidal mode number, R the plasma major radius, Φ and Ẑ the unit vectors respectively along the geometrical toroidal angle and the vertical direction.The viscous damping term is included in equation ( 2), where k ∥ = (n − m/q)/R is the parallel wave number, v th,i = 2T i /M i the thermal ion velocity for given ion temperature (T i ) and mass (M i ).The parameter κ ∥ is used to tune the strength of the damping effect, it will be set to κ ∥ = 1.5 in the following simulations in order to model strong sound wave damping.This value has been found suitable for modeling plasma response at low beta [24] and has been used in investigations for ITER and DEMO [10,25].The validity of the low beta limit, with respect to the no-wall limit, will be discussed later in section 4. Plasma resistivity is included in equation ( 3) and indicated as η.The unknown variables ξ, v, b, j, p indicate plasma displacement, perturbed velocity, perturbed magnetic field, perturbed current and perturbed pressure.The equilibrium quantities ρ, B, J, P are the mass density, magnetic field, current density and pressure.Of these, ρ is included as a profile over the minor radius and calculated consistently with the scenario [20], while B, J and P are also processed with the equilibrium code CHEASE [26] which provides the n = 0 equilibrium as input for MARS-F on a 2D grid (R, Z) as a function of s = ψ p and χ, where ψ p is the normalized poloidal flux and χ a generalized poloidal angle.For the following results an equal-arc-length poloidal angle is chosen for computation, for better convergence with fewer poloidal harmonics, and mapping to PEST-like straight-field-line angle is carried out in post-processing for physical meaning of the perturbed magnetic field Fourier harmonics.
Since flux coordinates are used, flux surfaces must be defined in the computational domain.The X-point geometry thus cannot be rigorously treated and the MHD equations are solved within the last closed flux surface (LCFS), or a plasma boundary with slightly smoothed X-point.In the vacuum region conformal surfaces (to the plasma boundary) are defined, where the perturbed magnetic field satisfies: ∇ × b = 0, ∇ • b = 0.An ideal wall boundary condition is placed at infinity, while no resistive boundary is assumed in the present work.MARS-F solves the above equations ( 1)-( 5) as a driven problem with the external coil current (flowing in RMP coils) being defined by: ∇ × b = j RMP , ∇ • j RMP =  0.Where j RMP ∼ e −inΦ along the toroidal angle.The active coils are represented as currents sheets of finite poloidal width and effectively lie on a surface conformal to the plasma boundary, thus the radial component of j RMP vanishes.The divergence-free condition defines the poloidal component of j RMP , while the toroidal one is defined as in [27]: Where r c is the radial position of the coil center and χ c the poloidal extension.Here the aforementioned surface is chosen, and the radial mesh consequently optimized, as to fit at best the radial positions of coil legs extracted from 3D models.For input to MARS-F the coil current waveforms are modeled as sine signals discretized with the number of saddle coils in each toroidal array (i.e. 9 points).If we label the peak current as I c , the value in each coil can be written as: where n is the perturbation toroidal mode number, Φ i the ith coil center angle and ∆Φ the current toroidal phasing.As mentioned above, to allow DTT the maximum flexibility the ELM coils will be all independently fed, thus each can be driven with a specific current.This makes not only the toroidal rows independent bu also fine tuning single coils possible and faults easily observable.The aforementioned Chirikov parameter (σ Chir ) can be calculated as: where w = 2δ are magnetic island widths and ∆ m,m+1 is the distance between the two neighboring rational surfaces located at s 1 and s 2 in terms of (square root of) normalized poloidal flux.The island width can be evaluated in MARS-F straight field line (PEST-like) coordinates, following [25] we can use: where are the Fourier harmonics of perturbed radial field and ψ ′ ≡ d ψ ds = 2 ψ0 s.Thus the Chirikov parameter can be expressed assuming fixed toroidal mode number and in the limit of single magnetic island: As mentioned in the introduction, in this work we first use the maximum plasma displacement in the X-point region to optimize the RMPs in terms of coil phasing.This will be further discussed in section 4.1.Secondly, the vacuum perturbed field is used to calculate the Chirikov parameter and check the stochasticity induced in the magnetic configuration by fully penetrated perturbations, in the aftermath of the external field screening by plasma flow and resistive response [18,22].

Design of non-axisymmetric coils
Preliminary to the present modeling work, an initial conceptual design phase has been carried out.Many coil geometries have been considered, varying the poloidal and toroidal widths of window frames, and locations, from ex-vessel to invessel.In-vessel positioning has been preferred due to mounting obstacles posed by the alternative outer option.This intuitively should lead to better performance per unit current but to more demanding cooling requirements at the same time.
In the following, only the latest available coil dimensions are considered, while the number of actuators is varied.We first present a configuration counting 18 coils organized in two arrays of 9 coils each, placed above and below the equatorial plane.We shall refer to this as the 9 × 2 configuration (toroidal direction × poloidal direction).This design evolved then towards a 9 × 3 coil system, following both the results presented in this work and the concurrent requirements of EF control.These called for increased flexibility and power which led to the implementation of the third coil set, located on the equatorial mid-plane.Furthermore, 9 coils per toroidal set is a common feature with ITER, allowing the device to test ITER relevant control or correction schemes.The same tools applied in the present work have been used for extensive analyses of ELM control with RMPs in ITER [28].
For the function of ELM control, on which the present work is focused, external fields produced by the coils should interact mainly with plasma in the pedestal region and avoid resonant and non-resonant amplification effects with core plasma.This calls for toroidal mode number (n ⩾ 2) field distributions in order to maximize the coupling to outer regions, while tailoring capabilities for poloidal mode number m spectrum are required to adapt to different plasma scenarios.A set of 27 independent power supplies will allow the required flexibility.Slow (f < 10 Hz) rotation of external fields is suggested to avoid localized plasma-wall phenomena during operations.The present coil design takes into account geometrical and technical constraints, assembly procedures, but also reflects the main physics driven functional specifications.Integration with other in-vessel components has been partially addressed and a detailed study is presently ongoing.This could lead to a final design which slightly differs from the configuration studied here.Nevertheless these possible changes are foreseen to be small (∼cm) with respect to the overall coil size and not invalidating the results.As we can grasp from the sketch in figure 2(a), the coil system is not symmetrical with respect to the equatorial plane.The upper coils are designed to run around two upper ports while the lower coils lie between the equatorial and the lower ports.Given that the lower vacuum vessel ports are devoted to divertor handling, it is not desirable to have other systems possibly interfering.Both the upper and lower coils have a toroidal width of ∼40 • .As for the mid-plane coils, these are narrower (∼20 • ) and wound around equatorial ports.In this case, the driving constraint is the possibility of installing the coils through the ports themselves.
As we mentioned above, compatibility of this coil geometry with other subsystems, such as heating and current drive actuators, is presently under investigation.This study has been useful for the design of the power supply of non-axisymmetric coils, which has already been completed.However, since the coil geometry in DTT is mostly set by the aforementioned constrains, the present study will not address the optimal geometry in terms of size and radial position, but rather provide physics-based input for the choice of conductors and assess the applicability of the subsystem to full-power scenarios.In the following, configurations with either two or three sets of 9 coils each will be compared.Each set of 9 coils will be indicated as an array: the upper array winding around upper ports, the mid-plane array around equatorial ports and the lower one located between equatorial and lower ports.The present design of coils, adopted for this work, is reported in figure 2(b) for clarity.Here we note that two of the mid-plane coils are slightly larger (>20 • ) than the others, this feature is due to the coils being wound around ICRH antennas.This detail will be neglected in the following modeling, since equally spaced conductors with the same size are assumed to calculate coil currents.

Numerical results
Vacuum modeling for n = 1, 2, 3 static perturbations has been carried out for the initial characterization of the coil spectra.Sinusoidal current waveforms have been used in CARIDDI for preliminary 3D design, comparing window-frame shapes with different poloidal and toroidal widths.The poloidal spectra of vacuum DC perturbations have been compared with the same results from MARS-F, resulting in a good agreement between the two codes, with differences understood as toroidal mode number coupling not captured by the 2D vacuum solution.A fluid model is adopted for plasma response in the present work, extended with a viscous term in the momentum equation representing Landau damping from thermal ions.In principle a more complete drift-kinetic model would be more accurate in determining the plasma response to RMPs.The major issue with a purely MHD response model is the prediction of a strong amplification on external fields close to the Troyon no-wall limit, which disagrees with experimental observations [23,29].All the target scenarios considered here however operate safely far from the aforementioned limit, this has been assessed numerically as reported in figure 3(a) for one of the equilibria under investigation.
Since the target equilibria all include a q = 1 surface, a procedure has been established to modify the safety factor and isolate the essential physics for external kink stability.This procedure starts with tailoring a safety factor profile safely above the q = 1 value, using a parametric expression that reproduces the features of the initial profile.A self-consistent equilibrium is calculated by imposing the new safety factor with CHEASE, while keeping constant the total pressure and aiming at approximately the same output 0D parameters as the original equilibrium (e.g.plasma current, internal inductance).This approach allows the estimation, although approximated by the modified equilibria, of stability threshold for pressure-driven instabilities.For ideal stability of mode numbers n = 1, 2, 3 the MARS eigenvalue code has been used.The achieved results show reasonable agreement with typical internal inductance empirical scaling laws (e.g.β no−wall ∼ 3 ÷ 4l i ) and are useful in the scope of this work for determining the validity region of the adopted response model.The plasma β N = β(%)a(m)B 0 (T)/I p (MA) is scaled from an arbitrarily high value to the target number (β N ≃ 1.5 for the case in figure 3) while keeping the other equilibrium quantities reasonably constant (e.g.fixed total plasma current and safety factor variations within 10%).The boundaries of this scan in β N are chosen so that the equilibrium safety factor is not altered in a way that would change the stability properties (i.e. the current-driven drive).For n = 3 this implies that only one unstable point could be recovered, for the maximum chosen pressure (β N ≃ 3.7 in figure 3(a), and corresponding to 3 × p 0 with p 0 being the target equilibrium initial pressure).This is nevertheless enough to roughly locate the threshold for n = 3 ideal kink, which is larger than n = 1, 2. An ideally conducting boundary is found to be stabilizing these modes up to the maximum pressure, confirming the external kink nature, although mixed internal-external eigenfunctions are found.The β N ≃ 3.7 point, as reported in figure 3(b) for n = 1, is still found to be unstable with an ideal boundary conformal to the LCFS.This indicates that the value of β N ≃ 3.7 is idealkink-unstable for any position of the ideal wall.A close fitting ideal wall (r/a ∼ 1.2) would instead stabilize the n = 1 kink up to β N ≃ 3.3.
With the ratio of target β N to the limit β no-wall N being βN β no-wall N ≈ 0.5, we can apply the fluid MHD model with Landau damping of the plasma response to provide a first estimation of the external fields required for ELM control in the main DTT scenarios and its variations.Drift-kinetic response shall be investigated in future work.Detailed linear MHD analyses of the reference scenarios is also ongoing as reported in [30].Results for two different scenarios are presented in the following, with the main differences being magnetic axis position (R = 2.14-2.19m) and safety factor profiles varying both onaxis and at the edge.The first equilibrium presented in the following section has been slightly modified to avoid the q = 1 rational surface, this allows avoidance of the internal kink instability while not affecting stability at the boundary.All investigated equilibria and variations have been solved with the CHEASE code [26].The safety factor modification effectively leads to a pre-sawtooth crash profile and only changes the core plasma response, while the edge components are not affected.The full power single-null equilibrium with R = 2.19 m instead shows an unstable ideal mode with m/n = 1/1 pattern.This mode has an internal kink nature, confirmed by its eigenfunction, and is associated with the q = 1 surface being present in the plasma (i.e.q(r = 0) < 1). Figure 4 depicts plasma response characteristics for the three toroidal mode numbers in the R219 scenario, the mid-plane coil array is used for exemplification.Screening of the resonant perturbation components is observed in the plasma response spectra, as we can see by comparing the vacuum spectra in figures 4(a)-(c) with those including plasma response in (d)-(f ).In particular the white dashed line shows the safety factor profile, with rational surfaces marked with circles.To highlight this better, we report the radial profiles of resonant magnetic field components in figure 5. Conversely, amplification of the perturbed field is induced for the negative spectrum (m < 0 harmonics) and for the non-resonant m > 0 components.This can be noted in particular by comparing figures 4(b), (e) and (c), (f ), corresponding to n = 2 and n = 3 respectively.As for n = 1 perturbations, these mainly induce a strong m = 1 and m = 2 response, linked to the unstable or marginally stable ideal internal kink.Resistive plasma response has been compared with the ideal case, in which a perfectly conducting plasma is assumed.In this limit the field-aligned components of the total field perturbation vanish at rational surfaces.Ideal results are found to be similar to the resistive case, in which the Spitzer model for resistivity is used, in particular for inner rational surfaces.This indicates that in the investigated scenarios plasma resistivity plays a minor role at least for core response.The role of plasma flow [18] shall be further investigated in terms of both amplitude and shear, following the modeling of momentum sources in DTT scenarios.

Plasma response with 9 × 2 coils
As mentioned in section 3, in the following we discuss results for the 9 × 2 coil configuration.The vacuum field and plasma response has have been calculated with MARS-F for DC unit current perturbations with toroidal mode numbers n = 1, 2, 3.This has been done separately for upper and lower coils, so that the perturbed fields could be linearly combined with any given toroidal phasing.Given the linearity of the model, results can be re-scaled to any value of coil current.Two rows of in-vessel coils can give ξ X n ∼ 3 mm with coil currents ranging from 10 kAt to 30 kAt depending on the toroidal mode number.The ξ X n parameter used to evaluate the effectiveness of RMPs is defined here as the maximum plasma displacement in the X-point region [24].Such region is indicated by vertical lines in figures 6(b)-8(b), where the plasma displacement as a function of poloidal angle is considered, for each toroidal mode number, at the LCFS.For consistency, the choice of the angles defining the X-point region is the same for all toroidal mode numbers and equilibria.Since the Xpoint is not rigorously represented in the model, we define this region for a lower single null configuration, as that within the range [χ X − δχ, χ X + δχ].Where χ X is the poloidal angle corresponding to the lowest LCFS point along the vertical axis and δχ is chosen here to be 30 • .In this region we calculate the maximum displacement as max(|ξ n |).Results are found to be robust with respect to changes of the X-point region width, since the same local maximum is captured with δχ ∈ [10 • , 30 • ].It is worth noting that n = 1 perturbations trigger important core response as well as edge peeling response.This is because the target equilibrium, with q a < 1, is unstable to the (n = 1, m = 1) internal kink.This is clear when considering the displacement over the whole poloidal cross section, as we can appreciate in figure 6(a).Perturbations with n = 1 toroidal mode number induce large global plasma response, with core components often referred to as kink response.This is due to the fact that the underlying equilibrium is unstable to the (m = 1, n = 1) internal kink mode, driven by the plasma current profile and thus safety factor which has q 0 < 1 (i.e.on-axis safety factor).
In terms of plasma displacement and comparing with n > 1 a higher ξ /I C ratio is found for n = 1.The boundary displacement in n = 1 case is non negligible on the torus outboard, although being dominated by upper and lower regions.For n = 2, 3 perturbations the poloidal maps and profiles at optimal angle are reported in figures 7 and 8.Here we can note that by increasing the perturbation's toroidal mode number (n > 1) we increase the ξ X n /ξ mid n ratio, where ξ mid n is the boundary displacement at the equatorial plane, transitioning from a kink dominated response to a peeling-like response.Exploiting the linearity of the applied metric it is possible to estimate the achieved plasma displacement while modulating coil current.ξ X n is given as a function of toroidal coil phasing and coil current in figure 9 for the R214 and n = 1, 2, 3 respectively.Using R219, although the plasma scenario is still the same H-mode reference for DTT, the target equilibrium has a higher q 95 (and in this limiter-like model higher q a ) compared to the previous case.This implies a different response of the plasma to RMPs, with increased stability to current-driven modes.Therefore higher coil currents are required to have a reasonable effect of the RMPs, as can be appreciated from the summary plots in figure 10.The minimum coil current and optimal phasing needed to achieve a 3 mm X-point displacement with the three considered mode numbers, with two arrays of in-vessel coils in an upper-lower configuration are summarized in tables 1 and 2.

Plasma response with 9 × 3 coils
The set of non-axisymmetric coils being studied here is designed to serve a double purpose: contributing to the control of ELMs in H-mode scenarios and correcting spurious EFs arising from misalignment or deformation of equilibrium field          coils.For this reason it is desirable to design the system with enough flexibility.An additional array of coils, located on the machine equator and winding around mid-plane ports, is a possible answer to such a need for operational space.Not only allowing concurrent EF and ELM control, this solution possibly provides more power for the single tasks when performed separately.When added to the simulations so far presented, the equatorial coils are effectively an additional degree of freedom.The displacements induced by each coil are combined as follows: Thus the numerical results in the following will be reported in the 2D space of {∆Φ E−U , ∆Φ E−L }, indicating phase shifts of the upper and lower coils with respect to the equatorial array (respectively E-U and E-L).Coil current requirements are overall reduced when adding the equatorial array.This might not always imply a lower minimum coil current but instead a larger optimal region or safety margin.Compared to previous equilibrium however, the increased stability of this case leads to higher coil currents to satisfy the same criteria.Figure 11 reports the edge X-point displacement induced by n = 1, 2, 3 perturbations in the plasma described above, with a (tor × pol) = 9 × 3 coil configuration.In this case the coil current would be a third coordinate which is not shown for the sake of simplicity, the same currents of table 1 are chosen instead.The additional coil array secures rather wide regions in which the RMPs satisfy the ξ X n > 3 mm threshold.In the following, variations of the DTT reference scenario are introduced.At first a shifted magnetic axis to R = 2.19 m is considered.This eventually leads to a larger q 95 and consequently improved stability.Since the 9 × 3 configuration has been eventually selected as the reference design for ELM coils, the following results all refer to this geometry.In the R219 scenario, plasma response results indicate higher coils currents are required for n = 2, 3 perturbations to achieve ξ X n > 3 mm.This results, reported in figure 12, is probably linked to the lower pedestal pressure and higher edge safety factor of the target equilibrium compared to the R219 one.This improves both the pressure and current-driven stability at the same time.The mid-plane coil array helps in this sense to reduce the peak current threshold with respect to a 9 × 2 configuration, as can be seen for n = 2 by recalling the results in table 2. Conversely n = 1 perturbations seem to induce a stronger response in this scenario.This can be explained once again with the safety factor in the core region, which is lower in R219 and thus gives a more global n = 1 response.
Given the specific task DTT has in exploring divertor configurations, special care is taken in designing scenarios that are compatible with different magnetic geometries of the lower region.This is the main motivation behind the modeling of vertically shifted plasmas.In the following we investigate the response to RMPs using the same equilibrium profiles of the R219 case, but applying a rigid 10 cm vertical shift to the plasma boundary.This is meant to be a representative of scenarios that are presently being studied as a new baseline.In figure 13 the 2D phase space for this plasma is shown.Interestingly enough the response to n = 1 perturbations is more significant compared with the original R219 equilibrium; for better comparison the figure is shown with the same scale.The results for n = 2, 3 mode numbers are instead similar to the non-shifted case or even lead to lower current requirements as reported in figure 13(b) for n = 2.

Analysis of field stochastization
The Chirikov parameter (σ ch ) is a measure of magnetic island overlap, used to define a stochastic layer where σ ch > 1.The phasing of perturbed field resonant components is applied to the upper and lower coils, with respect to the equatorial array, similarly to what is done for the plasma displacement: The vacuum perturbed field b m,n is used to calculate σ ch , linearly scaling with the coil current.Contour plots of σ ch are reported in the following, again varying {∆Φ E−U , ∆Φ E−L } for n = 1, 2, 3 perturbations.The vacuum Chirikov parameter metric can be interpreted as the effect that the penetrated RMP has on the underlying equilibrium.At the end of the field penetration dynamic, i.e. when resistive plasma response can no longer shield the external field, the vacuum perturbation is effectively present.Thus the Chirikov parameter phase space optimization has been done for the same coil currents that satisfy the displacement criterion.This leads to understanding whether these currents also provide stochastization of the magnetic configuration and to which depth into the plasma.
In the following study the Chirikov parameter as defined in section 2 is implemented for vacuum RMPs.For the three cases so far investigated (R214, R219 and up-shifted respectively) the parameter σ ch is calculated on the outermost resonant surface for n = 1, 2, 3 using a coil current I c (vac) ⩽ I c (ξ X n > 3).Assuming that I c (ξ X n > 3) is required for significant plasma response, this new vacuum criterion shall ensure that this current (or even lower) is sufficient for field stochastization.Figure 14 shows the result for the early R214 equilibrium, where only n = 3 RMPs satisfy the σ ch > 1 criterion.In figures 14(a) and (b) the relevant q = 3 rational surface is further away from the coils with respect to all the other cases, leading to an insufficient effect of the RMP in terms of magnetic chaos generation.For this specific case n = 3 perturbations would be more effective.This also indicates though that reducing the plasma-coil gap, for example by increasing the magnetic axis radius to R = 2.19 m, is beneficial for ELM control purposes.The applied coil currents for R214 are up to the threshold indicated by plasma response and shown in figure 11. Results related to the R214 scenario are summarized in table 3.While the current set by the displacement empirical threshold is not enough to satisfy the Chirikov criterion for n = 1, 2, it is more than sufficient for the n = 3 case.This apparent contradiction of the previously reported result can be partially explained by two considerations: the Chirikov parameter is calculated with the vacuum resonant components whose magnitude depends on the coil distance, a sharp transition seems to be take place across q = 3 between the last n = 2 and n = 3 rational surfaces, respectively (m, n) = (5, 2) and (m, n) = (10, 3).The plasma displacement instead is an intrinsically different quantity calculated on the LCFS, for all toroidal mode numbers, with constant coil distance.This aspect can be important, and requires a dedicated numerical investigation focused on the edge safety factor dependence, to further understand the DTT operational space and ELM control capabilities.The improved stability for plasma response in R219 is not relevant for the discussion of results obtained from vacuum modeling.Hence for the Chirikov parameter higher currents leads linearly to higher values of σ ch .Furthermore the plasmacoil gap, as already discussed above, affects the results and makes n = 1 perturbation also capable of satisfying the criterion in this scenario.This conclusion is valid for both versions of the R219 scenario, as we can appreciate from figures 15(a) and 16(a) comparing Z 0 ≃ 0 cm and Z 0 ≃ 10 cm respectively, where Z 0 is the vertical position of the magnetic axis.This rigid displacement of the plasma region overall reduces the plasma-coil distance, in particular for the upper coil.This is reflected in the coil current values being lower than in the R219 scenario and lower than the displacement criterion prediction.This is reported in figures 15(b), (c) and 16(b), (c) and can be compared with figures 14(b) and (c).To clarify the message from the two criteria, these currents are summarized in table 4.
The implemented formula for σ ch is approximated in the limit of a single magnetic island, this can be confusing since the definition implies overlap of magnetic islands.To support the analysis presented so far and verify the achieved stochastization, magnetic field lines need to be traced.The REORBIT module [31], developed for runaway electron orbits, has been applied for the purpose of field line tracing by following the guiding center trajectory of 200 test particles in the vacuum RMP field.These particles are initially distributed uniformly along the poloidal flux coordinate ψ p .Assuming the coil current required for maximizing the peeling response (identified above with the edge plasma displacement)and the optimal phase-space from the Chirikov criterion, we trace the test particles for each toroidal mode number.The result is reported  in the Poincaré plots of figure 17.As already seen in the Chirikov parameter results, the strong n = 1 plasma response is not present here, and these perturbations only provide a certain degree of chaos to the outermost rational surface q = m/n = 4/1.Higher toroidal mode numbers on the other hand show stronger effect on the outer magnetic configuration, with n = 2 affecting the volume between the last two resonant surfaces and n = 3 going deeper to the third (ψ p ∼ 0.96).If field line tracing in the vacuum field shows large induced islands and field stochastization, the picture changes when introducing plasma response.With the Spitzer resistivity model applied in this study, only the very outer part of the plasma is affected by the RMP, which nevertheless induces a stochastic region here.To better resolve this, denser field line tracing simulations have been carried out, with 300 test particles in the range 0.9 ⩽ ψ p ⩽ 1.As we can appreciate in figures 17(d)-(f ) the stochastic region is limited to outer rational surfaces and some island penetration is observed for n > 1 perturbations.We note that n = 1 perturbations give very limited magnetic island penetration and the stochastic layer is confined at the very edge of the plasma.This, along with the results shown in previous sections, indicate that n = 1 is not indicated for ELM control in the scenarios so far investigated.As a final remark, we note that a full picture of RMP-plasma interaction and first-principle prediction of coil current for ELM control can only be achieved with non-linear MHD modeling.An example of this is reported in [9] where the JOREK code is applied to ITER.This is at the moment beyond the scope of the study presented here and can be the subject of future work.

Conclusions
In this work we presented a first exploration of the DTT capabilities in terms of ELM control with magnetic actuators.Results indicate that the presented design of nonaxisymmetric coils is adequate for the purpose with reasonable safety margins.The calculations are based on some of the robust early versions of the DTT baseline scenario and its variations, mostly in shape and position.Although integrated modeling studies are continuously evolving and deepening the understanding of DTT scenarios, the techniques and metrics presented here will be applied for quick evaluation of RMP performance.The investigated scenarios are all robustly stable for ideal global MHD modes that typically impact performance, with the n = 1 ideal kink being triggered first at β N > 2.8.This does not take into account possible resistive MHD instabilities, such as Tearing Modes.Dedicated studies shall be carried out both with linear and non-linear models to assess the stability of DTT baseline scenarios with respect to these modes.Alternative scenarios will also be possible during the DTT scientific exploitation, which might vary significantly in terms of stability properties and consequently in the interaction with external fields.With this in mind we included the calculation for n = 1 RMPs in the present work, although in the studied scenarios this leads to strong core response and does not couple well with the plasma edge.In a broader perspective, n = 1 perturbations could be dedicated to the control of EFs in DTT using one of the available coil arrays such as the mid-plane one, while ELM control could be sought simultaneously with the optimized action of the other two arrays.This is possible in principle, given the results presented here for the 9 × 2 configuration.Plasma response calculations are overall consistent in the target equilibria, yielding a coil current requirement in the range 20-40 kAt for ELM control.This result is achieved using the empirical plasma displacement metric with a factor of 2 as safety margin with respect to the results of presently operating mid-size tokamaks.Two rows of coils in upper-lower configuration are found to be capable of satisfying the selected threshold already, although adding the mid-plane row allows significant expansion of the optimal phase space and overall lower coil currents.Field line tracing in the penetrated vacuum fields that satisfy the displacement criterion shows how large magnetic islands are induced in the edge region, leading to full stochastization of the field at the outermost rational surfaces, which could impact transport and this ELM suppression.Physics-based (conceptual) design of the non-axisymmetric coils has been carried out using the numerical study presented here, while also taking into account the requirements coming from vacuum EF correction assessment.A more detailed analysis is being planned involving the investigation of plasma scenarios for the first phases of DTT operation, which will rely on reduced heating capabilities.In terms of plasma response physics, studies of RMP-induced Neoclassical Toroidal Viscous torque are being carried out.Furthermore, the possible effect of toroidal sidebands on ELM control has not been taken into account so far.Sidebands are inevitably generated when making a given single-n perturbation with a discrete set of actuators, for example n = 3 RMPs will have n = 6, 12, 15, . . .sidebands.Moreover, coil failure may induce further spurious components in the applied spectra.These could have an impact on the coil current requirements or cause undesirable and non always predictable effects.An example regarding the n = 1 sideband is reported in [10], where it is suggested to treat this as an EF component, while n > 1 sidebands are found to have little impact on the Xpoint displacement results.On the other hand recent modeling results suggest that n > 1 sidebands could impact fast ion confinement [32].These aspects require further investigation and assessment of possible correction strategies for DTT [33,34], taking advantage of the flexibility of independently fed coils.The DTT non-axisymmetric coils system will significantly broaden the device operational space, allowing investigation of many MHD related issues, starting with ELM and EF control, and implementation of diagnostic techniques such as MHD spectroscopy.The system itself, although specifically tuned for the DTT device, features a 9 × 3 configuration which makes it promising for ITER relevant studies.

Figure 1 .
Figure 1.Main equilibrium and kinetic profiles used in the MARS-F simulations, solid and dashed lines indicate the R219 and R214 scenario respectively.(a) Equilibrium density profiles, normalized to on-axis value, (b) safety factor profiles, (c) normalized electron density profiles, (d) normalized electron (blue) and ion (red) temperatures.Data is consistent with [20, 21].

Figure 2 .
Figure 2. (a) Plasma boundary for R214 and R219 equilibria with coils on poloidal section.(b) Preliminary 3D design of ELM coils.

Figure 4 .
Figure 4. Poloidal spectra over the plasma domain for a 10 kAt DC perturbation using mid-plane coils with (a), (d) n = 1 (b), (e) n = 2 and (c), (f ) n = 3 patterns.The upper row shows vacuum spectra, lower one includes plasma response in the R219 equilibrium.The radial component harmonics |b 1 m | are calculated in PEST-like poloidal angle, units are normalized to the on-axis toroidal field.

Figure 5 .
Figure 5. Profiles of resonant radial magnetic field components in PEST-like coordinates for (a) n = 1, (b) n = 2 and (c) n = 3 perturbations.Rational surfaces are marked by vertical dotted lines.Black squares indicate the vacuum perturbation amplitude at each rational surface, while total field amplitudes including resistive plasma response are marked by black diamonds.Plots (b) and (c) are zoomed from the first rational surface to the LCFS.

Figure 9 .
Figure 9. Normal plasma displacement in X-point region induced by off-midplane coils with varying toroidal phase and linear scaling of total applied current, for (a) n = 1, (b) n = 2, (c) n = 3 DC perturbations in R214 equilibrium.

Figure 10 .
Figure 10.Normal plasma displacement in X-point region induced by off-midplane coils with varying toroidal phase and linear scaling of total applied current, for (a) n = 1, (b) n = 2, (c) n = 3 DC perturbations in R219 equilibrium.

Figure 11 .
Figure 11.Normal plasma displacement in X-point region, for the equilibrium with R = 2.14 m, induced by three sets of coils with varying relative phasing and fixed coil current to (a) 10 kAt for n = 1 (b) 20 kAt for n = 2 and (c) 20 kAt for n = 3.

Figure 12 .
Figure 12.Normal plasma displacement in X-point region, for the equilibrium with R = 2.19 m, induced by three sets of coils with varying relative phasing and fixed coil current to (a) 10 kAt for n = 1 (b) 30 kAt for n = 2 and (c) 30 kAt for n = 3.

Figure 13 .
Figure 13.Normal plasma displacement in X-point region, for the equilibrium with R = 2.19 m with Z = +10 cm shift, induced by three sets of coils with varying relative phasing and fixed coil current to (a) 10 kAt for n = 1 (b) 20 kAt for n = 2 and (c) 20 kAt for n = 3.

Table 1 .
Minimum coil current to achieve 3 mm displacement and corresponding optimal phasing.R214 scenario with upper and lower coils.

Table 2 .
Minimum coil current to achieve 3 mm displacement and corresponding optimal phasing.R219 scenario with upper and lower coils.

Table 3 .
Summary of coil current requirements for R219 scenario with 9 × 3 coil configuration.

Table 4 .
Summary of coil current requirements for R219 scenarios with 9 × 3 coil configuration.