A new type of resonant magnetic perturbation for controlling edge localized modes

A new type of resonant magnetic perturbation (RMP), generated by helical coils, is proposed for controlling the edge localized mode (ELM) in H-mode tokamak plasmas. The helical coil optimization utilizes the MARS-F code (Liu et al 2000 Phys. Plasmas 7 3681) computed linear resistive fluid response of the plasma to the applied RMP field. The optimal helical coils are found to be located near the outboard mid-plane of the torus, with relatively simple shape but tilted towards the equilibrium magnetic field line pitch. Compared to the window-frame ELM control coils, the optimal helical coils require 2–4 times less current, in order to achieve the same ELM control performance specified by various figures of merit adopted in this work. The results from the present study show a promising path forward in achieving ELM control with RMP fields in tokamak plasmas.

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Introduction
In order to maximize the fusion power, the high-confinement mode (H-mode) is an essential requirement for next step tokamak fusion reactors such as ITER.Standard H-mode plasmas often suffer the so-called type-I edge localized mode (ELM) instability [1,2], which is a quasi-periodic magnetohydrodynamic (MHD) instability that expels high heat fluxes to the divertor and first wall components of the device.In ITER, the ELM-induced heat flux is expected to exceed the material damage limit of plasma facing components [3], resulting in significant reduction of their lifetime.As a key solution to this problem, three-dimensional (3D) resonant magnetic perturbation (RMP) has been found to be a robust technique for ELM mitigation or suppression during the H-mode discharges in several present day tokamak devices [4][5][6][7][8][9][10][11].
So far, RMP fields for controlling ELMs are almost exclusively produced by window-frame type of coils in tokamaks [4][5][6][7][8][9][10][11]37] (with one exception of the helical coils in TEXTOR).The poloidal spectrum of the vacuum RMP field, generated by this type of coils, is therefore limited by the choice of the coil geometry.Due to spatial aliasing, the window-frame geometry can also possibly cause unwanted side bands in the poloidal spectrum.Further flexibility in the field spectrum is provided by tuning the relative toroidal phase (coil phasing) of the coil currents between different rows of the window-frame coils (for a given toroidal mode number n).Even with this flexibity, the RMP spectrum may still be sub-optimal for ELM control due to the coil geometry constraint.For this reason, 3D helical coils have recently been exploited for ELM control, utilizing a stellarator coil optimization tool [38].
In this work, we design optimal helical RMP coils on a 2D surface.In other words, we simply treat the coil current as a (generic) surface current located in the vacuum region outside the plasma.We then optimize the poloidal spectrum of the surface current (for a given toroidal mode number) to maximize certain figures of merit (FoMs), which have previously been tested to serve as reliable indicators for achieving ELM control.As it turns out, the optimal solution requires a helical pattern for the surface current flow.We then propose control coils with geometries that follow this optimal helical current flow pattern, ending up with (optimal) helical ELM control coils instead of the (presumed) window-frame coil geometry.Because the helical coils allow more freedom in the optimization, we expect that the optimal performance (for ELM control) must be better, or at least not worse, than that of the window-frame coils.As will be reported in this work, with the same 'total' current (to be defined later on), the optimal helical coils indeed perform much better than the optimal windowframe coils (with optimal coil phasing) in terms of our FoMs.
This study may be viewed as of a pure academic interest.On the other hand, our optimization results show that the best helical coil geometry is not too complicated to be useful for a practical design-in fact a single toroidal row of helical coils, with relatively simple shape, suffices to produce the best FoMs.This also excludes the need for tuning the coil phasing among multiple rows, since the alignment of the external field to the plasma is naturally incorporated into the coil geometry optimization.
The paper is organized as follows.Section 2 describes the plasma equilibria adopted for this investigation.Section 3 introduces the FoMs for the control coil current optimization, as well as an analytic procedure for optimization in multidimensional parameter space.Section 4 reports the modeling results for the helical RMP coils and compares them with results obtained assuming window-frame coils.Robustness, not to an exhaustive extent though, of the helical coil solution against variation of the plasma boundary shape and q 95 , is established.Section 5 summarizes and discusses the results.

Plasma equilibria
Three numerical equilibria in toroidal tokamak geometry, specified in a semi-analytic manner, are adopted for this study.Two equilibria have a lower single null divertor-like plasma shape, but with different safety factor profiles (in particular different q 95 ).The third equilibrium has an upper single null plasma shape.The purpose of choosing these equilibria is to test the robustness of the proposed helical coil solution.The equilibria are obtained by analytically specifying the plasma boundary shape and kinetic profiles for the plasma current density and pressure, followed by numerically solving the (fixed-boundary) Grad-Shafranov equation.Generic equilibria of conventional aspect ratio are considered in this study, without referencing to a specific device.

Plasma boundary shapes
We consider both an upper and a lower single null divertorlike plasma, with the boundary shape specified in the (R, Z) coordinates on the poloidal plane [39], normalized by the plasma major radius R 0 Two plasma boundary shapes considered in this study, representing the lower (red solid line) and upper (blue dashed line) single null plasma configurations.Shown are also the window frame coil location (black dashed lines, to be used for the comparative study) and the approximate location of the optimal helical coils for the n = 2 configuration (green solid line, resulted from the optimization study to be reported later on).
where θ is the poloidal angle, divided into the upper (θ U ) and lower (θ L ) half-planes.The parameters ε, δ U , δ L , κ define the inverse aspect ratio of the plasma, the upper and lower triangularity values, and the elongation of the plasma boundary shape.For a lower single null plasma configuration, we specify δ U = δ L = 0.3, b = 0.2 and c = 0.4.Note that b = 0.2 and c = 0.4 are assumed here to obtain the lower single null plasma boundary.For an upper single null plasma, we choose δ U = 0.8, δ L = 0.2, b = c = 0.The parameters ε and κ are fixed at 1/3 and 1.6, respectively, in all cases.Note also that for simplicity, we assume the same upper and lower triangularity values in this study.The effect of varying (upper and lower) triangularity values on the plasma response has previously been well studied [26,34,40] and is therefore not the focus of the present work.These two plasma boundary shapes are plotted and contrasted in figure 1, assuming a plasma major radius of R 0 = 3 m.

Equilibrium profiles
A set of equilibrium profiles representing H-mode plasmas are also analytically specified.In particular, we consider the following radial profile for the equilibrium plasma pressure [41] where the first and second terms from the right-hand side (RHS) represent the pedestal and core pressure profiles, respectively.The pedestal width (∆), measured in the normalized equilibrium poloidal flux Ψ N , is fixed at 0.05 in this study.The parameters ψ mid N and ψ ped N describe radial locations of the middle and top of the pressure pedestal, respectively, is the Heaviside step function.The other parameters are also fixed at p 1 = 0.6, p 2 = 1.5 and p 3 = 1.5.We emphasize that these choices in the equilibrium profiles, as well as those for the plasma boundary shape reported in section 2.1, are somewhat arbitrary.The main purpose is to obtain equilibria that resemble typical plasmas from the H-mode tokamak operations in divertor configurations.
The radial profile of the (surface-averaged) toroidal current density, including the bootstrap current contribution associated with the pressure pedestal, is specified as where the first RHS term describes the bootstrap current contribution in the pedestal region, with j 1 being related to the pressure pedestal height p 1 .Numerical tests show that setting j 1 = p 1 reasonably well follows the Sauter bootstrap current model [42] in our case.The second and third RHS terms from equation ( 2)) represent the overall current density profile, with j 2 − j 6 being free parameters.These parameters are tailored to obtain the desirable safety factor profiles, e.g. the on-axis qvalue and the q 95 value.In fact, we tune the plasma current density profiles to obtain three different q-profiles as shown in figure 2(a).Note that both analytic formulae (2.4) and (2.5)only specify the shape of the radial profiles, not the overall amplitude.The latter will be tuned as we generate equilibria with desired beta and plasma total current.The plasma toroidal flow also plays an important role in the plasma response (via screening of the resonant field components).For simplicity, a parabolic radial profile of the toroidal rotation frequency, normalized by the toroidal Alfven frequency, is assumed in the present study where the on-axis value is fixed at Ω 0 = 3 × 10 −2 .The plasma density profile is also chosen for a typical H-mode plasma, with a pedestal structure near the plasma edge.Radial profiles for the aforementioned equilibrium quantities are plotted in figure 2.
We emphasize again that the equilibrium profiles, such as that for the safety factor, is numerically self-consistently computed by solving the fixed boundary Grad-Shafranov equation [43].Whilst the same profiles are assumed for all three equilibria, for the plasma pressure, density and toroidal rotation, we do consider three different radial profiles of the safety factor, since this is known to play a leading order role in the plasma response [26,28,33].These three q-profiles have the same onaxis value of q 0 = 1.05 but different q 95 values: q 95 = 2.61 and 3.1 for the two equilibria with the lower single null plasma boundary shape, respectively; q 95 = 3.1 for the equilibrium with the upper single null shape.These low q 95 values are consistent with that of the ITER baseline plasma scenario [44].In Radial profiles of the assumed (a) safety factor q, (b) plasma toroidal rotation frequency normalized by the (toroidal) Alfven frequency, (c) the equilibrium pressure, and (d) the surface-averaged toroidal current density (normalized to unity at the magnetic axis).Considered are three equilibria with different safety factors but the same profiles for the other quantities as shown in (b)-(d).The red solid (blue dash-dotted) line corresponds to an equilibrium with the lower single null plasma boundary shape shown by the red solid line in figure 1, with the safety factor value q 95 = 3.1 (q 95 = 2.61) at the 95% of the equilibrium poloidal magnetic flux.The black dashed line corresponds to an equilibrium with the upper single null plasma boundary shape shown by the blue dashed line in figure 1, with q 95 = 3.1 as well.The vertical dashed lines indicate the last q = m rational surface for each equilibrium.what follows, the equilibrium with the lower single null shape and with q 95 = 3.1 will be considered as the base case, for which the helical coil optimization results will be systematically reported.The other two equilibria have also been fully exploited, but only limited portion of results will be reported (for the comparative purpose).The vacuum toroidal magnetic field of B 0 = 2.5 T (at the magnetic axis) and the total plasma current of 4.5 MA are considered in this work.

FoM and helical current optimization procedure
As mentioned earlier, our goal is to optimize the (generally helical) surface current without constraining the current flow patterns (as the window-frame coils would do).Optimization requires specification of objective functions, which we refer to as FoM in this work.These FoMs are evaluated based on the plasma response computed by linear MHD models, in our case based on the toroidal computations by the MARS-F code [17].The MARS-F RMP response model is reported in detail in [31], and hence will not be repeated here.We note that the code is based on the flux coordinate system, which requires truncation of the equilibrium flux surfaces near the plasma separatrix or geometric smoothing of the Xpoint.The effect of the X-point smoothing on the MARS-F computed plasma response to the RMP field (produced by the window-frame coils) has previously been systematically investigated [28].We do mention the two key physics effects that are included into the model for the plasma response: the finite plasma resistivity (responsible for resonant field penetration and formation of magnetic islands) and the toroidal plasma flow (responsible for the resonant field screening).The MARS-F computed plasma response fields are then utilized to construct four FoMs, which we define below.

Specification of FoM
The first set of FoMs are defined based on the amplitude of the resonant components of the radial RMP field near the plasma edge, with or without including the plasma response.The case without plasma response is dubbed 'vacuum FoM' in this work, which essentially characterize the vacuum magnetic island size near the plasma edge.The quantity including the plasma is budded 'total field FoM', which is finite due to finite plasma resistivity, but typically smaller than the vacuum case due to plasma screening.The third FoM is defined as the maximum amplitude among all poloidal Fourier harmonics of the plasma radial displacement in the pedestal region.This quantity measures the so-called edge-peeling response of the plasma which plays an important role in ELM control [12,22].The last FoM, that we use in this work, is the amplitude of the plasma displacement near the X-point.Typically, there are good correlations among all three plasma response based FoMs [12,45].Maximization of these plasma response based FoMs generally provides the best RMP spectrum for ELM control-either mitigation or suppression-in present day experiments, as have been extensively demonstrated in earlier studies [11,22,25,28,35,46,47].In this study, we will mostly consider three FoMs based on plasma response to show the robustness of the optimal solution.The vacuum FoM is reported as well, largely to contrast results obtained by the other three plasma response based FoMs.

An analytic procedure for multi-dimensional optimization
Given a FoM as defined above, we optimize the poloidal spectrum of the surface current for a given toroidal mode number n, in order to maximize the FoM value.The surface current is specified at a poloidally closed 'control' surface in the vacuum region outside the plasma.For fair comparison, we choose the same radial location for the control surface as that of the window-frame RMP coils.We then assume a unit current on the control surface with the m/n helicity as the source term, and compute each FoM separately for each single-helicity current source.Linear superposition (implied by the linear response model) of the FoM values contributed by all poloidal harmonics, weighted by coefficients C m , then yield the total FoM due to the (generally helical) surface current.The task of optimization then consists of the best choice for C m , in order to maximize the amplitude of a given FoM.
We use the total resonant response field near the plasma edge (referred to as b 1 res(pls) ) as an example to illustrate the optimization procedure.The total b 1 res(pls) , due to multiple poloidal harmonics of the surface current, is calculated as where b 1 res(pls)m is the resonant radial field (at the last rational surface near the plasma edge for a given n) produced by a unit helical current m/n at the control surface, and C m is the coefficient for optimization.Note that both b 1 res(pls)m and C m are generally complex numbers due to our representation of the field perturbations.Our goal is to maximize |b 1 res(pls) | with a constraint on the total current on the control surface, or equivalently with a fixed value of The optimal values for C m are analytically calculated following the Cauchy-Bunyakovsky-Schwarz inequality which is also a constant according to our construction.The values of the Fourier harmonics (that contribute to B 2 ) are obtained assuming a unit current for each single poloidal harmonic.Therefore, B 2 does not depend on the choice of the optimization coefficients C m , thus being a constant.Similar analytic optimization solutions are obtained with the other three FoMs.

Equivalent constraints between helical and window-frame coils
In order to compare the performance (in terms of our four FoMs) between the optimal helical coils and the optimal window-frame coils for a given equilibrium, we need to assume the same constraint on the total coil current.For the window-frame coils, we assume I C = 5 kAt coil current for the single middle row.Fourier decomposition of corresponding toroidal or poloidal current density, as a function of the poloidal angle, results in the poloidal harmonics J ϕ m and J θ m .Note that J ϕ m and J θ m are effectively the coefficient C m that we defined above.Again, the Parseval's theorem implies that fixing the same coil current is equivalent to fixing the same value for m J ϕ In what follows, we mostly report the optimization results using the constraint for the toroidal surface current density, although the same studies were also repeated assuming the m J θ m 2 = 3.5 × 10 −4 constraint.Note that the toroidal and poloidal components of the surface current are related by the current divergence-free condition.As we will show, the choice of the constraint m J ϕ m 2 = 8.5 × 10 −4 or m J θ m 2 = 3.5 × 10 −4 in fact produces similar results.More importantly, with either constraint, the optimal helical coil solution can significantly out-perform the window-frame coils in terms of the maximum achievable FoMs at the same coil current, as will be demonstrated in section 4. Another interpretation of the constraint on I 2 C is to fix the power supply U C I C that drives the coil current, assuming that the power voltage U C is largely proportional to the coil current (for resistive ELM control coils).

Plasma response to RMP field with helical coil currents
We consider helical surface coil currents with the n = 1-4 toroidal wave forms.The current optimization is performed for each individual n.The choice of these low-n perturbations is motivated by typical toroidal spectra utilized for ELM control in present day experiments as well as in ITER.Following the procedure outlined in section 3, the plasma response to the basis and single-helicity m/n surface current source are computed by MARS-F, followed by the coil current optimization.The optimal solutions are summarized in figure 3, including m = 1-10 poloidal harmonics and applying four different FoMs.
A key observation is that all three plasma response based FoMs result in similar optimal solutions for the helical current, in terms of both amplitude and phase of each individual poloidal Fourier harmonic.This shows the robustness of the optimum found by our procedure.
Another interesting observation is that the amplitude of the optimal poloidal harmonics quickly decreases with increasing m.In fact, the amplitude of the poloidal harmonics with m > 6 is already close to zero, thus making relatively small contributions to the optimal solution.The fact that only a small number of low-m poloidal Fourier harmonics of the surface current play an important role in the optimization also excludes the possibility of having helical coils with complicated shapes (e.g. with small-scale structures), which is a positive feature and is indeed the case as will be shown later on.We mention that only m > 0 harmonics are included here.Examination of the m = 0 and m < 0 harmonics revealed that their contributions are small.This is partly related to the fact that a positive equilibrium field helicity (with q > 0) is assumed in this study.
As a further confirmation of the convergence of the optimal solution, figure 4 plots the optimal (maximal) FoMs achieved with the helical surface current optimization while including m = 1,…, M poloidal harmonics.While varying M, we achieve a good convergence at M ∼ 6, showing that our default choice of M = 10 is sufficient to obtain reliable optimal solutions.We also note that the m < 0 Fourier harmonics do not provide much contribution to the FoMs (figure 4) and therefore have largely been neglected in our helical coil optimization.
As an important remark, we note that the number of poloidal harmonics assumed in MARS-F plasma response computations is much larger than the aforementioned number M for specifying the helical current source, in order to cover the response of all the resonant harmonics (and beyond).For instance, we use 163 poloidal harmonics for computing the plasma response to the n = 4 RMP.The above number M only refers to the total number of helical current sources that we use (as the basis) to represent the eventual helical coil current.In fact, we include the same (large) number of poloidal harmonics when MARS-F computes the plasma response to each given, single-m basis current source.The toroidal coupling effect in a torus means that a single-m helical current can excite plasma response field that contains multiple poloidal harmonics.Our numerical procedure thus properly captures this coupling effect.
Next, we report in figure 5 the optimal surface current flow patterns obtained with various FoMs.Shown here is an example for the n = 1 configuration.Not surprisingly, the optimal patterns, predicted by the three FoMs based on the plasma response (figures 5(b)-(d)), are all similar.This holds for both the current flow directions (shown by arrow plots) and the current density amplitude (shown by contour plots).Based on the current flow patterns and the contour plots, we can propose the shape for helical coils as indicated by the black closed-loop curves in figure 5. Note that the optimal surface current is spatially distributed, and therefore cannot be fully represented by closed-loop curves.But the far-field (with respect to the surface current location), i.e. that generated in the plasma region, should be similar.This will be further quantified later on.Choosing a large (outer) loop, that encloses the dominant portion of the helical current flow, also ensures sufficiently large RMP field to be generated by these helical coils.We emphasize the relatively simple shape of the proposed helical coils, based on our optimization procedure.We envisage that such coils are realizable for practical applications.
Another important observation is that the resulting optimal helical coils are generally located near the outboard midplane.Because the optimally distributed helical coil current is always located at the low-field-side of the torus, this automatically excludes the possibility of achieving good ELM control (according to our FoMs) with coils placed at the highfield-side.We emphasize that this poloidal location, as well as the one-row solution instead of multiple rows that are often designed for the window-frame ELM control coils, comes out naturally as the result of our optimization procedure.The single-row solution also eliminates the need for coil phasing optimization between different rows-the optimal solution already takes that into account.
The last interesting observation is that the optimal coil shape does not exactly follow the helicity of the equilibrium field lines near the plasma edge (shown by red lines in figure 5 in a non-straight-field-line coordinate system where the geometric toroidal angle is assumed and the poloidal angle is measured in the arc-length along the poloidal circumference)-the overall shape of the helical coil is simpler than the equilibrium field line variation.Certain alignment is however evident, showing that the optimal helical coils tend to 'tilt' towards the direction of the equilibrium field line pitch.This (approximate) alignment, however, becomes less obvious with higher-n RMP configurations, as will be shown in figure 6.
Figure 6 shows the optimal helical current solutions for the n = 2-4 configurations.For simplicity, we only compare the results obtained with the total response field FoM.Besides the mis-alignment between the helical coils and the equilibrium magnetic field line pitch as mentioned earlier, we make two more observations.(i) The poloidal extension of the optimal helical coils is about 1/3 of the total poloidal circumference (as measured in the geometric poloidal angle defined here), for all n = 2-4 configurations.The coils roughly cover the poloidal    angle from the −60 • to +60 • near the outboard mid-plane.For the n = 1 configuration, the poloidal coverage is slightly wider as shown in figure 5. Note also that more coils are needed along the toroidal angle with increasing n, for a better resolution of the toroidal wave form as expected.(ii) The optimal helical coils tilt more towards the vertical direction with increasing n.For the n = 4 case as shown in figure 6(c), the shape (but not the orientation) of the helical coils already starts to resemble that of the window-frame coils, implying that the gap of the performance improvement, between the optimal helical coils and the optimal window-frame coils, may shrink with increasing n.This is indeed the case as will be shown later on.Before that, we briefly illustrate the optimal coil current flow patterns for the window-frame coils in the next subsection.
For the purpose of ELM control, it is often important to maximize the edge-peeling response while maintaining a low level of the plasma core-kink response (to minimize the influence of RMPs on the plasma core transport) [22,25].Following [25], we also consider to maximize a metric A peel /A kink that measures the ratio of the edge to core plasma response amplitude.The quantity A peel here is defined as the maximum amplitude of the plasma radial displacement (among all poloidal Fourier harmonics) in the edge region, specified by the normalized equilibrium poloidal flux Ψ N = [0.8,1].The quantity A kink is calculated as the maximum amplitude of the radial displacement in the plasma core region of Ψ N = [0, 0.5].
We point out a key difference, in terms of the optimization procedure, between the FoMs that we defined in section 3.1 and the new one considered here: all the previous FoMs are 'linear' quantities in the sense that they are linearly proportional to the computed plasma response.This 'linearity' property implies that the linear superposition rule applies to those FoMs and thus allowing analytic optimization based on the Cauchy-Bunyakovsky-Schwarz inequality, as shown earlier.
On the other hand, the edge-to-core response ratio is a 'nonlinear' quantity with respect to the plasma response, which does not allow linear superposition.Consequently, the analytic optimization procedure, which we outlined in section 3.2 to obtain the optimal helical coil geometry, does not apply anymore.Instead, we carry out numerical multi-dimensional (dimension = 20 in our case) optimization (with constraint) for maximizing A peel /A kink .A set of results are reported in figure 7, assuming the n = 1 RMP and with the same constraint of m J ϕ m 2 = 8.5 × 10 −4 as before.
The key finding from figures 7(a) and (b) is that, by maximizing A peel /A kink , the resulted optimal helical current distribution shifts to the high-field side (HFS) of the torus (figure 7(b)).Upon careful examination, we find however that the plasma response, produced by such an optimal helical current (with the same total current as imposed by the aforementioned constraint) is very small (figure 7(a)).For instance, the individual values of A peel and A kink are ∼33 and 5054 times smaller than the nominal values obtained by other FoMs, respectively.A too small A peel value certainly indicates poor ELM control, despite the nearly vanishing core response (figure 7(a)) which is good for core transport-the ratio A peel /A kink in fact reaches a very large number of 127 for the optimal solution and for the considered case.
In order to find a more balanced solution where we maximize the edge-to-core ratio but at the same time ensure sufficiently large edge-peeling response (to facilitate ELM control), we consider a compound FoM (A peel /A kink + cA peel ) with c being a positive constant, and again carried out numerical optimization.The results show that even with a small value of c (=0.02), the optimal helical current is again largely located at the low-field side (figure 7(d)), similar to that found by other FoMs.More importantly, the optimal A peel value (figure 7(c)) now also becomes comparable to that found earlier, ensuring a good ELM control.The ratio A peel /A kink for the optimal solution is somewhat larger than that predicted by other FoMs but not significantly larger.
Results shown in figure 7 thus suggest that, in order to maintain the capability of ELM control, one has to allow certain degree of the core response by placing the helical coils at the LFS.Placing the helical coils at the HFS allows significant increase of A peel /A kink but at the substantial sacrifice of the ELM control performance.

Plasma response to RMP field with window-frame coils
In order to better quantify the performance of helical coils, we have also computed the plasma response to the n = 1-4 RMP fields produced by representative window-frame coils.In particular, the two rows of coils, located above and below the low-field-side mid-plane, resemble those installed in present days devices such as DIII-D (I-coils) [4], ASDEX Upgrade (Bcoils) [7], MAST [6] and EAST [11].In our case, we assume that the center of the upper (lower) row of window-frame coils is located at the geometric poloidal angle of 40 • (−40 • ), and each coil spans the poloidal angle of 30 • .This coil geometry is not far from the optimum as found in earlier studies [45,48] where the window-frame coil geometry optimization has been performed for ITER and EU DEMO, respectively.It is important to note that, to fairly compare the performance between the window-frame coils and helical coils, we place all coils at the same radial location, i.e. r/a = 1.25 in this study.The total coil current, specified in terms of either m J ϕ m 2 or m J θ m 2 , is also kept the same.
The optimal current flow patterns (following the coil phasing optimization) for the two-row window-frame coil configuration are shown in figure 8, for the n = 1-4 RMPs.The constraint of m J ϕ m 2 = 8.5 × 10 −4 is adopted for the coil current.For simplicity, we only report results with FoM derived from the total response field here.The optimal results are similar among three FoMs based on the plasma response.It is evident that the coil flow pattern shown here is different from that in figures 5 and 6 for the optimal helical coils, due to the coil geometry constraint in the former.What is essential is that these optimal current flows produce much lower FoM values as compared to those by the optimal helical coils, as reported in the next subsection.We now back to the question of how well the thinwire helical coils, which we propose in figures 5 and 6, can represent the distributed helical current for the plasma response.Since MARS-F cannot directly include the thin-wire helical coil current as the source term, we have to perform the plasma response computations using an equivalent surface current procedure [49], where the vacuum field is evaluated via the Biot-Savart law.Figures 9 and 10 compare the plasma response between the distributed helical current, the thin-wire helical coils, as well as the window-frame coils, for the n = 2 configuration (cf figure 6(a)) as an example.The poloidal spectra of the perturbed radial field agree reasonably well between the distributed helical currents and the thinwire helical coils (figure 9), in particular for the total response field.This is even more so if we compare with the field produced by the window-frame coils.The plasma displacement produced by the distributed helical currents and the thin-wire helical coils is also similar (figure 10).But the window-frame counterpart is substantially different.These comparisons thus demonstrate the feasibility of replacing the distributed helical currents by the thin-wire helical coils as we propose in figures 5 and 6.Again, the ultimate reason for this feasibility is that only the far field is concerned with respect to the source currents.

Comparison of FoMs between helical and window-frame coils
With the same constraint of m J ϕ m 2 = 8.5 × 10 −4 for both types of coils, figure 11 compares the optimal values for all n = 1-4 RMP configurations.Note that the results for the single middle row and two off-middle rows are also compared.The key result is that, whilst the two-row window-frame coil configuration is somewhat better than the single row in terms of our FoMs, the optimal helical coils perform much better.Since the values of FoMs are proportional to the assumed coil current amplitude, to achieve the same FoMs (i.e.effectively the same level of ELM control), the optimal helical coils require 2-4 times smaller current as compared to the two-row window-frame coils.The gap in the performance enhancement is particularly larger for the low-n (n = 1 or 2) RMP configurations, showing the promising potential of application of helical RMP coils for ELM control.
As discussed before, there are two possibilities in imposing the coil current constraint between the helical and windowframe coils.We have so far been constraining m J ϕ m 2 .
Figure 12 reports the optimization results, if we choose to constrain m J θ m 2 = 3.5 × 10 −4 which again corresponds to 5 kAt for the single-row window-frame coils.It is again evident that the optical helical coils substantially outperform the window-frame coils for ELM control, with all n = 1-4 RMP fields and with all three FoMs.For the low-n cases (n = 1-2), the maximum FoM values with the helical coils are about twice larger than that with window-frame coils.The performance gap between these two types of coils decreases with increasing toroidal mode number, for a reason that the high-n helical coil shape starts to resemble that of window-frame coils, as pointed out earlier in this study.We also point out that for practical applications, the constraint on m J ϕ m 2 is probably more reasonable than that on m J θ m 2 .

Influence of q 95 on helical coil optimization
We have so far considered base case equilibrium with q 95 = 3.1 for the single null plasma boundary shape.Given the importance of q 95 in controlling ELMs [28,[50][51][52], we have repeated the above studies for another equilibrium with   Comparison of the maximum FoM values, obtained by optimizing the helical coil geometry (curves in red) or the coil phasing for the two-row window-frame coils (in blue), with that of the single-row window-frame coils (without optimization, in black).The single-row coils are located near the low-field-side mid-plane while the two-row coils are located above/below the mid-plane.Considered are four FoMs: (a) the edge resonant radial field including the plasma response, (b) the edge-peeling component associated with the plasma radial displacement and (c) the plasma normal displacement near the X-point.The optimization is carried out for the m = 1-10 poloidal harmonics of the toroidal surface current density for the helical coils.In all cases, the same constraint of m|J ϕ m | 2 = 8.5 × 10 −4 (in MARS-F dimensionless quantity) is applied that corresponds to 5 kAt for the single-row window-frame coils, for the n = 1-4 toroidal harmonics, respectively.All coils are located at the same minor radius of r/a = 1.25.Considered is a lower single null plasma equilibrium with q 95 = 3.1.q 95 = 2.61.An example (for n = 3) of the optimal current flow pattens, together with the proposed helical coil geometry, is shown in figure 13, with the eventual maximum FoMs reported in figure 14 for all n = 1-4 configurations.Note that the same single null plasma shape and the same coil current constraint of m J ϕ m 2 = 8.5 × 10 −4 as before are assumed here.
We find similar optimal current flow patterns between this case (figure 13) and the previous case with q 95 = 3.1 (figure 6(b)), suggesting certain robustness in the proposed shape for the helical coils against the q 95 -variation.
The ELM control performance gain by the optimal helical coils, in terms of our three FoMs, is on the other hand less pronounced for the case of q 95 = 2.61 (figure 14), especially for the low-n RMP configurations and for the total response FoM as shown in figure 14(a).There is, however, a good reason for this.For the case of q 95 = 2.61, the last rational surface for the n = 1 resonant perturbation is located relatively far away from the plasma edge (the q a value at the plasma boundary is 2.93 for this case).The plasma screening is therefore much enhanced due to higher electron temperature and thus lower plasma resistivity (the Spitzer resistivity model is adopted in MARS-F modeling).As a result, the total resonant field amplitude is small for the n = 1 configuration, as well as for the n = 2 case following the same reason.Large improvement by the helical coil optimization, in the absolute value of b 1 res(pls) , is therefore not expected.The relative enhancement, however, is still significant.For the n = 1 case for instance, the |b 1 res(pls) | value is 0.0009 G for the windowframe coils and 0.0035 G for the helical coils-a factor of 4 increase.We can therefore conclude that the q 95 = 2.61 case may not be within the ELM suppression window for the lown RMP field, but the optimal helical coils still out-performs the window-frame coils in the relative sense (in terms of our FoMs).

Influence of X-point location (upper single null configuration)
We also investigate whether the enhanced performance by helical coils also applies to upper single null plasma configurations.We consider one such equilibrium shown in figures 1 and 2. Figure 15 reports a representative result (for n = 2) on the optimal helical current flow patterns, obtained by maximizing three FoMs.The optimal helical coil geometry, in terms of both the overall shape and the poloidal extension, is similar to that obtained for the lower single null plasma (figure 6(a)).This robustness in the coil geometry is essential for ensuring practical usefulness of this type of coil design.
Figure 16 compares the maximum FoMs between the optimal helical coils and the two-row window-frame coils for the upper single null plasma configuration, assuming the same current constraint of m J ϕ m 2 = 8.5 × 10 −4 .In general, we again observe a 2-3 times enhancement of FoMs with the helical coils.Note that the maximum total resonant field for the n = 1 case (figure 16(a)) is small, for the same reason that the last rational surface is located more inside the plasma and thus subject to more plasma screening (the edge safety factor is q a = 3.55 for this equilibrium).For the other toroidal mode numbers (n = 2-4), the last rational surfaces are closer to the plasma edge and the screening is therefore weaker.The performance improvement by helical coils is significant.

Helical coil optimization for a DIII-D equilibrium
Finally, we also apply our optimization procedure to a plasma from the present-day experiment.We consider an equilibrium reconstructed from the DIII-D discharge 158 103 at 3796 ms [28].This is a lower-single-null H-mode plasma at 1.34 MA plasma current and 1.9 T toroidal field.The q 0 and q 95 values are 1.1 and 4.2, respectively.The on-axis toroidal rotation frequency normalized by the Alfvén frequency is 6.25 × 10 −2 .The upper and lower triangularity values of the plasma boundary shape are 0.42 and 0.6, respectively.The RMP coil  current (in the so-called I-coils) was 4 kAt in this DIII-D discharge.
We again perform helical coil optimization for the n = 1-4 toroidal waveforms (although the RMP experiments are typically carried out with n = 2 or n = 3 in DIII-D).The results are summarized in figure 17 for three FoMs.Note that the window-frame coils here refer to the DIII-D I-coils.It is evident that, assuming the same coil current constraint, the optimal helical coils always out-perform the window-frame coils (with the optimal coil phasing) in DIII-D as well.Importantly, we find no substantial changes in the geometry and current of the optimal helical coils, as compared to that shown in figures 5 and 6, when applied the same technique to a real device.

Summary and discussion
Numerical experiments are carried out to design the best ELM control coils in a generic tokamak geometry.By eliminating the constraints on the current flow path imposed by the window-frame coils, we allow free choices in the poloidal spectrum of the applied RMP field.Not surprisingly, this freedom leads to helically flowing current pattern when we optimize the RMP spectrum for best ELM control in terms of the four figure of merits adopted in this study.In order to ensure robustness of the results, we have taken a systematic approach and compared (i) three plasma equilibrium configurations (upper versus single null, different safety factor profiles), and (ii) different toroidal spectra (n = 1-4) of the applied field.A plasma from the DIII-D RMP-ELM control experiment is also considered.The study utilizes the MARS-F code for computing the linear resistive MHD response to the RMP field in the presence of toroidal equilibrium flow (and thus the flow screening effect).Two crucial requirements to ensure fair comparison of the ELM control performance, between the helical coils and the window-frame coils, that we adopt are (i) the same radial locations of the coils, and (ii) the same total current which we specify via either m J ϕ m 2 or m J θ m 2 .
Our optimization results show that the helical coils can significantly out-perform the window-frame coils in terms of the maximum achievable FoMs at the same coil current.Consequently, for achieving the same level of ELM control, the current requirement for the optimal helical coils is about several times smaller than that for the windowframe coils (with the optimal coil phasing).This performance enhancement is more pronounced for low-n (n = 1-2) RMPs than for higher-n (e.g.n = 4) RMPs.The reason is that the shape of the optimal helical coils starts to resemble that of the window-frame coils at higher-n.We also find that the helical coils, optimized by maximizing the FoMs considered in this study, produce similar edge-to-core ratio for the plasma response amplitude, as compared to the (optimal) windowframe coils.This implies that the side effect of the helical coils on the plasma core transport should not be worse than the optimal window-frame coils.In fact, the side effect is likely weaker with the helical coils since the latter require less current to achieve the same level of ELM control according to our criteria.
As for key geometrical features of the optimal helical coils, we find that they should be (i) located near the outboard midplane of the torus, (ii) covering about 1/3 of the poloidal circumference with the geometric poloidal angle extending from about −60 • to 60 • , (iii) simply connected to form a single closed current flow loop for each coil, and finally (iv) tilted towards the equilibrium field line pitch at the plasma edge but no good alignment with the field line is required (in particular for higher-n's).
We remark that our primary goal in this study is of academic nature-to identify possible best shapes for the helical RMP coils, which in turn provide the upper bound for the performance of (any kind of) window-frame coils for ELM control.The fact that the optimal helical coils turn out to possess relatively simple shapes, and being relatively robust against variation of the plasma conditions, makes the practical design of such coils encouraging.Detailed design of such coils, however, is beyond the scope of the present work.For practical applications, engineering constraints may imply certain modifications to the theoretically-optimal coil shape, as also sometimes the case even with the window-frame coils.
Finally, we do point out one disadvantage of these helical coils, that the optimal shape depends on the toroidal mode number as we found here.This means that in practice, we have to a-priori decide the toroidal spectrum for the RMP field before designing the helical coil shape.This is less flexible compared to the window-frame coils, although this may not be a significant issue for a fusion reactor where the plasma scenario as well as many engineering components have to be fixed at the design phase.On the other hand, the gain in terms of the ELM control performance (or the required coil current) can be substantial with this new type of coil design.
We also emphasize that the present work only reports an initial attempt to design helical coils for ELM control.Many other aspects, such as the sensitivity of the optimal helical coil geometry to the variation of plasma scenarios (in particular q 95 ), the engineering constraints in implementing such coils, application of such coils for controlling error fields or other MHD modes such as the Alfvén eigenmodes [53], deserve further investigations.

Figure 2 .
Figure 2.Radial profiles of the assumed (a) safety factor q, (b) plasma toroidal rotation frequency normalized by the (toroidal) Alfven frequency, (c) the equilibrium pressure, and (d) the surface-averaged toroidal current density (normalized to unity at the magnetic axis).Considered are three equilibria with different safety factors but the same profiles for the other quantities as shown in (b)-(d).The red solid (blue dash-dotted) line corresponds to an equilibrium with the lower single null plasma boundary shape shown by the red solid line in figure1, with the safety factor value q 95 = 3.1 (q 95 = 2.61) at the 95% of the equilibrium poloidal magnetic flux.The black dashed line corresponds to an equilibrium with the upper single null plasma boundary shape shown by the blue dashed line in figure1, with q 95 = 3.1 as well.The vertical dashed lines indicate the last q = m rational surface for each equilibrium.

2 ,
where the maximum is achieved if and only if

m 2 and m J θ m 2 ,
that correspond to 5 kAt single row window-frame coil current.It turns out that, in MARS-F dimensionless quanties, we need to assume a constraint of either m J ϕ m 2 = 8.5 × 10 −4 or m J θ m 2 = 3.5 × 10 −4 .

Figure 3 .
Figure 3.The optimal helical coil current, in terms of the amplitude (left panels) and phase (right panels) of the m = 1-10 poloidal Fourier harmonics for J ϕ m , is obtained and plotted for the four different figures of merit (FoMs): (a)-(b) the edge resonant vacuum radial field, (c)-(d) the edge resonant radial field including the plasma response, (e)-(f ) the edge-peeling component associated with the plasma radial displacement, and (g)-(h) the plasma normal displacement near the X-point.With each FoM, the optimal solutions for n = 1-4 RMP fields are also compared.The surface current constraint of m|J ϕ m | 2 = 8.5 × 10 −4 (in MARS-F dimensionless quantity) is applied for optimization in all cases.The helical surface current source is located at a closed control surface in the vacuum region outside the plasma, at the minor radius of r/a = 1.25.Considered is a lower single null plasma equilibrium with q 95 = 3.1.

Figure 4 .
Figure 4.The MARS-F computed amplitude of three different FoMs: (a) the edge resonant radial field including the plasma response, (b) the edge-peeling component associated with the plasma radial displacement, and (c) the plasma normal displacement near the X-point, assuming a helical surface current source (at a closed control surface located in the vacuum region outside the plasma) with a single toroidal mode number n but containing multiple poloidal harmonics m = −10,…,M.The current distribution among M harmonics is optimized to achieve the maximum FoM for each case, while constraining the total (squared) amplitude of the poloidal harmonics to m|J ϕ m | 2 = 8.5 × 10 −4 (in MARS-F dimensionless quantity).Compared are multiple cases with n = 1-4 and with progressive increase of M. The control surface is located at the minor radius of r/a = 1.25.Considered is a lower single null plasma equilibrium with q 95 = 3.1.

Figure 5 .
Figure 5.The helical current flow patterns, plotted on the control surface along the geometric toroidal angle and the geometric poloidal angle, and obtained by maximizing the three FoMs respectively: (a) the edge resonant radial field including the plasma response, (b) the edge-peeling component associated with the plasma radial displacement, and (c) the plasma normal displacement near the X-point.Shown in red are also the equilibrium magnetic field lines located at the last rational surface near the plasma boundary, together with contour plots for the amplitude of the optimal helical current density.The black contour line indicates the proposed optimal helical coil geometry based on the FoM-maximization.The optimization is carried out for the m = 1-10 poloidal harmonics of the toroidal surface current density with the constraint of m|J ϕ m | 2 = 8.5 × 10 −4 (in MARS-F dimensionless quantity), for the n = 1 toroidal harmonic.The control surface is located at the minor radius of r/a = 1.25.Considered is a lower single null plasma equilibrium with q 95 = 3.1.

Figure 6 .
Figure 6.The helical current flow patterns, plotted on the control surface along the geometric toroidal and the geometric poloidal angles, and obtained by maximizing the FoM of the edge resonant radial field including the plasma response.Shown in red are also the equilibrium magnetic field lines located at the last rational surface near the plasma boundary, together with contour plots for the amplitude of the optimal helical current density.The black contour line indicates the proposed optimal helical coil geometry based on the FoM-maximization.The optimization is carried out for the m = 1-10 poloidal harmonics of the toroidal surface current density with the constraint of m|J ϕ m | 2 = 8.5 × 10 −4 (in MARS-F dimensionless quantity), for the (a) n = 2, (b) n = 3, and (c) n = 4 toroidal harmonic, respectively.The control surface is located at the minor radius of r/a = 1.25.Considered is a lower single null plasma equilibrium with q 95 = 3.1.

Figure 7 .
Figure 7. Plotted in (a), (c) are the poloidal Fourier harmonics of the radial displacement of the plasma, and in (b), (d) the helical current flow patterns, for the optimal solutions obtained by maximizing the A peel /A kink + cA peel with (a)-(b) c = 0 (a) and (c)-(d) c = 0.02.The optimization is carried out with m = 1-10 poloidal harmonics representing the helical source current with the constraint of m|J ϕ m | 2 = 8.5 × 10 −4 (in MARS-F dimensionless quantity), for the n = 1 toroidal harmonic.The control surface is located at the minor radius of r/a = 1.25.Considered is a lower single null plasma equilibrium with q 95 = 3.1.

Figure 8 .
Figure 8.The current flow patterns in the two rows of window-frame ELM control coils, plotted along the toroidal and poloidal angles and obtained by maximizing the FoM of the edge resonant radial field including the plasma response.Shown are also contour plots for the current amplitude.The black contour lines indicate the geometry window-frame coils.The optimization is carried out for coil phasing with the constraint of m|J ϕ m | 2 = 8.5 × 10 −4 (in MARS-F dimensionless quantity) that corresponds to 5 kAt current, for the (a) n = 1, (b) n = 2, (c) n = 3, and (d) n = 4 RMP configurations, respectively.The ELM control coils are located at the minor radius of r/a = 1.25.Considered is a lower single null plasma equilibrium with q 95 = 3.1.

Figure 9 .
Figure 9.The poloidal spectrum of the n = 2 perturbed radial field for (a)-(c) the vacuum field and (d)-(f ) the total field including the plasma response, between (a), (d) the distributed helical currents, (b), (e) the thin-wire helical coils, and (c), (f ) the window-frame coils.

Figure 10 .
Figure 10.The MARS-F computed n = 2 plasma displacement assuming (a) the distributed helical currents, (b) the thin-wire helical coils, and (c) the window-frame coils.

Figure 11 .
Figure 11.Comparison of the maximum FoM values, obtained by optimizing the helical coil geometry (curves in red) or the coil phasing for the two-row window-frame coils (in blue), with that of the single-row window-frame coils (without optimization, in black).The single-row coils are located near the low-field-side mid-plane while the two-row coils are located above/below the mid-plane.Considered are four FoMs: (a) the edge resonant radial field including the plasma response, (b) the edge-peeling component associated with the plasma radial displacement and (c) the plasma normal displacement near the X-point.The optimization is carried out for the m = 1-10 poloidal harmonics of the toroidal surface current density for the helical coils.In all cases, the same constraint of m|J

Figure 12 .
Figure 12.Comparison of the maximum FoM values, obtained by optimizing the helical coil geometry (curves in red), or the coil phasing for the two-row window-frame coils (in blue) located above/below the low-field-side mid-plane.Considered are three FoMs: (a) the edge resonant radial field including the plasma response, (b) the edge-peeling component associated with the plasma radial displacement, and (c) the plasma normal displacement near the X-point.The optimization is carried out for the m = 1-10 poloidal harmonics of the toroidal surface current density for the helical coils.In all cases, the same constraint of m|J θ m | 2 = 3.5 × 10 −4 (in MARS-F dimensionless quantity) is applied that corresponds to 5 kAt for the window-frame coils, for the n = 1-4 toroidal harmonics, respectively.All coils are located at the same minor radius of r/a = 1.25.Considered is a lower single null plasma equilibrium with q 95 = 3.1.

Figure 13 . 2 = 8 . 5 ×
Figure 13.The helical current flow patterns, plotted on the control surface along the toroidal and poloidal angles, and obtained by maximizing the three FoMs respectively: (a) the edge resonant radial field including the plasma response, (b) the edge-peeling component associated with the plasma radial displacement, and (c) the plasma normal displacement near the X-point.Shown are also contour plots for the helical current amplitude, with the black contour line indicating the proposed optimal helical coil geometry based on the FoM-maximization.The optimization is carried out for the m = 1-10 poloidal harmonics of the toroidal surface current density with the constraint of m J ϕ m

Figure 14 .
Figure 14.Comparison of the maximum FoM values, obtained by optimizing the helical coil geometry (curves in red), or the coil phasing for the two-row window-frame coils (in blue) located above/below the low-field-side mid-plane.Considered are three FoMs: (a) the edge resonant radial field including the plasma response, (b) the edge-peeling component associated with the plasma radial displacement, and (c) the plasma normal displacement near the X-point.The optimization is carried out for the m = 1-10 poloidal harmonics of the toroidal surface current density for the helical coils.In all cases, the same constraint of m|J ϕ m | 2 = 8.5 × 10 −4 (in MARS-F dimensionless quantity) is applied that corresponds to 5 kAt for the window-frame coils, for the n = 1-4 toroidal harmonics, respectively.All coils are located at the same minor radius of r/a = 1.25.Considered is a lower single null plasma equilibrium with q 95 = 2.61.

Figure 15 . 2 = 8 . 5 ×
Figure 15.The helical current flow patterns, plotted on the control surface along the toroidal and poloidal angles, and obtained by maximizing the three FoMs respectively: (a) the edge resonant radial field including the plasma response, (b) the edge-peeling component associated with the plasma radial displacement, and (c) the plasma normal displacement near the X-point.Shown are also contour plots for the helical current amplitude, with the black contour line indicating the proposed optimal helical coil geometry based on the FoM-maximization.The optimization is carried out for the m = 1-10 poloidal harmonics of the toroidal surface current density with the constraint of m J ϕ m

Figure 16 .
Figure16.Comparison of the maximum FoM values, obtained by optimizing the helical coil geometry (curves in red), or the coil phasing for the two-row window-frame coils (in blue) located above/below the low-field-side mid-plane.Considered are three FoMs: (a) the edge resonant radial field including the plasma response, (b) the edge-peeling component associated with the plasma radial displacement, and (c) the plasma normal displacement near the X-point.The optimization is carried out for the m = 1-10 poloidal harmonics of the toroidal surface current density for the helical coils.In all cases, the same constraint of m|J ϕ m | 2 = 8.5 × 10 −4 (in MARS-F dimensionless quantity) is applied that corresponds to 5 kAt for the window-frame coils, for the n = 1-4 toroidal harmonics, respectively.All coils are located at the same minor radius of r/a = 1.25.Considered is an upper single null plasma equilibrium with q 95 = 3.1.

Figure 17 .
Figure 17.Comparison of the maximum FoM values, obtained by optimizing the helical coil geometry (curves in red) or the coil phasing for the two-row window-frame coils (in blue).Considered are three FoMs: (a) the edge resonant radial field including the plasma response, (b) the edge-peeling component associated with the plasma radial displacement, and (c) the plasma normal displacement near the X-point.Considered is an equilibrium reconstructed from the DIII-D discharge 158103 at 3796 ms.