# Heuristic predictions of RMP configurations for ELM suppression in ITER
burning plasmas and their impact on divertor performance

H Frerichs1, J van Blarcum1, Y Feng2, L Li3, Y Q Liu4, A Loarte5, J-K Park6, R
A Pitts5, O Schmitz1, S M Yang7 1 Department of Engineering Physics,
University of Wisconsin - Madison, WI, USA  2 Max-Planck-Institut für
Plasmaphysik, Greifswald, Germany  3 College of Science, Donghua University,
Shanghai, China  4 General Atomics, San Diego, CA, USA  5 ITER Organization,
St. Paul Lez Durance Cedex, France  6 Seoul National University, Korea  7
Princeton Plasma Physic Laboratory, Princeton, NJ, USA<EMAIL_ADDRESS>
###### Abstract

A subspace of resonant magnetic perturbation (RMP) configurations for edge
localized mode (ELM) suppression is predicted for H-mode burning plasmas at 15
MA current and 5.3 T magnetic field in ITER. Perturbation of the core plasma
can be reduced by a factor of 2 for equivalent edge stability proxies, while
the perturbed plasma boundary geometry remains mostly resilient. The striation
width of perturbed field lines connecting from the main plasma (normalized
poloidal flux $<1$) to the divertor targets is found to be significantly
larger than the expected heat load width in the absence of RMPs. This
facilitates heat load spreading with peak values at an acceptable level below
$10\,\mathrm{M}\mathrm{W}\,\mathrm{m}^{-2}$ on the outer target already at
moderate gas fueling and low Ne seeding for additional radiative dissipation
of the $100\,\mathrm{M}\mathrm{W}$ of power into the scrape-off layer (SOL).
On the inner target, however, re-attachment is predicted away from the
equilibrium strike point due to increased upstream heat flux, higher
downstream temperature and less efficient impurity radiation.

††:

Keywords: Resonant magnetic perturbations for ELM suppression, divertor
detachment, 3D plasma boundary modeling

##  1 Introduction

Good particle and energy confinement is required for ITER to achieve and
sustain burning plasma operation with a fusion gain of $Q=10$ (i.e. where the
internally generated fusion power exceeds the externally applied heating power
by an order of magnitude), and the discovery of high-confinement (H-mode)
operation [1, 2] has been essential for fusion applications and ITER in
particular. However, the reference regime exhibits so called type-I edge
localized modes (ELMs) [3] (an MHD instability with fast loss of particles and
energy from the plasma edge), and extrapolations to ITER have indicated that
the ELM peak energy fluence to divertor targets can exceed acceptable values
for lifetime considerations (surface melting) [4, 5, 6, 7]. A number of
different ELM mitigating techniques are being investigated (see e.g. overview
paper [8]).

Suppression of ELMs by application of resonant magnetic perturbations (RMPs)
has been pioneered at DIII-D [9], and it has since been successfully
reproduced at KSTAR [10], EAST [11] and ASDEX Upgrade [12]. It is presently
the main strategy for suppression of ELMs in ITER [13, 14]. Application of
RMPs leads to helical corrugations of the magnetic separatrix as shown in
figure 1, and those appear as lobes in camera images of the X-point region
[15] and as non-axisymmetric striations of particle and heat fluxes onto
divertor targets [16, 17, 18]. The ITER divertor design [19, 20], however, is
based on 2-D (axisymmetric) SOLPS-4.3 and SOLPS-ITER plasma boundary modeling
which neglects these effects. This paper addresses implications of RMPs by 3-D
modeling with EMC3-EIRENE [21, 22]. Previous modeling of the pre-fusion power
operation phase at lower power has shown that detachment onset occurs at lower
upstream density, but at the expense of diverting heat flux to far SOL strike
locations which are more difficult to detach [23, 24]. The remaining challenge
is to verify the compatibility of RMPs for ELM suppression with a partially
detached divertor plasma for safe handling of steady state power loads under
burning plasma conditions with seeded impurities.

Figure 1: Perturbed magnetic separatrix from application of RMPs
(configuration $C_{\textnormal{maxX}}$ from figure 2 for
$I_{\textnormal{RMP}}\,=\,60\,\mathrm{k}\mathrm{A}\textnormal{t}$). The
geometry of the in-vessel ELM control coils is highlighted.

The stabilizing impact of RMPs on the plasma edge is associated with field
penetration at the pedestal top [25, 11], and this requires a certain level of
perturbation depending on the edge safety factor $q_{95}$, density and plasma
shape [26, 27]. In the following we focus on the standard operating scenario
for an ITER burning plasma with $15\,\mathrm{M}\mathrm{A}$ plasma current,
$5.3\,\mathrm{T}$ magnetic field, $q_{95}=3.14$ and fusion gain of $Q=10$.
Furthermore, we focus on perturbations with toroidal mode number $n=3$ and
evaluate the dependence on configuration parameters (for a comparison of
divertor performance between $n=3$ and $n=4$ see [28]). As can be seen in
figure 1, the in-vessel ELM control coils in ITER are arranged in three rows,
and this offers the opportunity to align the perturbation from adjacent rows
with the helical pitch of the magnetic field lines at the plasma edge. This
leaves a 5-dimensional configuration space of RMP parameters: the amplitudes
of the current waveform in each of the three rows, as well as the relative
phases between adjacent rows (barring other symmetry breaking effects, the
global phase of the perturbation can be neglected).

Modeling of the impact of RMPs on ELMs requires time dependent, non-linear MHD
models such as JOREK [29], but those need a significant amount of
computational resources and are unsuitable to evaluate a large number of
configurations. Therefore, for the purpose of scanning through the RMP
parameter space, we rely on linear MHD models and figures of merit that
correlate with ELM suppression, while referring to full MHD results only for a
selected operating point as reference. The aim of this paper is to evaluate
the relation between these figures of merit for ELM suppression and divertor
performance. Specifically, we take the radial displacement of the plasma
boundary surface near the X-point (referred to as X-point displacement) from
linear, resistive MHD plasma response (MARS-F) as figure of merit for edge
stability [30, 31, 32, 33]. In section 2, we will map out RMP configurations
for ELM suppression in ITER, and evaluate the relation to the magnetic
footprint on the divertor targets as a proxy for particle and heat loads.
Then, in section 3, we revisit results for KSTAR where the ideal perturbed
equilibrium code (IPEC) [34] has been used to reproduce the experimentally
observed operating window of RMP configurations with ELM suppression [35]. We
introduce code independent figures of merit, and demonstrate the robustness
and compatibility of the two approaches. Finally, we take particular RMP
configurations of interest and follow up with 3-D plasma boundary simulations
in section 4 in order to explore compatibility with divertor operating limits.

##  2 RMP configurations for ELM suppression

With five free parameters, the RMP configuration space in ITER is vast. The
advantage of linear MHD models is that they are fast enough to map out the
configuration space in order to identify operating points that appear
beneficial for ELM suppression. In this paper, we follow up on earlier results
[33, 36] for the $Q=10$ ITER scenario with $15\,\mathrm{M}\mathrm{A}$ current
and $5.3\,\mathrm{T}$ toroidal field, and further analyze the plasma response
obtained from the MARS-F code [37, 38] which is based on a single fluid,
linearized resistive MHD model. Specifically, we pick one case based on a flow
profile predicted by ASTRA transport simulations for
$33\,\mathrm{M}\mathrm{W}$ of NBI power, $20\,\mathrm{M}\mathrm{W}$ ECRH
power, Prandtl number $\Pr\,=\,0.3$ and a ratio of the toroidal momentum to
thermal confinement time of $\tau_{\Phi}/\tau_{E}\,=\,1$.

Figure 2: Phase scan of the RMP waveform in the upper and lower row relative
to the middle row at fixed amplitude $I_{\textnormal{RMP}}$ in all rows: (a)
Normalized X-point displacement $\xi_{X}^{\ast}$, (b) normalized midplane
displacement $\xi_{M}^{\ast}$, (c-d) magnetic footprint width
$s_{\textnormal{max}}$ on the inner and outer target for
$I_{\textnormal{RMP}}=60\,\mathrm{k}\mathrm{A}\textnormal{t}$. The black dot
highlights the configuration $C_{\textnormal{maxX}}$ with
$\max|\bm{\xi}_{X}|$. Contour lines are shown in (a) for $\xi_{X}^{\ast}=1/2$
(blue) and 2/3 (purple) consistent with the threshold range for ELM
suppression found in JOREK simulations [39].

###  2.1 Configuration scan with fixed RMP current

Despite initial success of the vacuum island overlap width (VIOW) as a proxy
for ELM suppression [40], it is now accepted that plasma response plays a key
role [41, 42, 43]. A new figure of merit has been emerging: the plasma
boundary displacement $|\bm{\xi}_{X}|$ near the X-point. This “X-point
displacement“ is caused by the edge-peeling response [44, 30], and maximizing
it is found to be best for ELM suppression. Note that $\bm{\xi}_{X}$ from
MARS-F is a single complex number associated with toroidal mode $n$. The
plasma response is computed once for each row at
$1\,\mathrm{k}\mathrm{A}\textnormal{t}$, and the resulting $\bm{\xi}_{X}$ from
any current ($I_{U},I_{M},I_{L}$) and phase
($\Phi_{\mathrm{U}},\Phi_{\mathrm{M}},\Phi_{\mathrm{L}}$) combination can then
be obtained from a linear superposition

$\bm{\xi}_{X}\,=\,I_{U}\,\bm{\xi}_{X\mathrm{U}}^{(1\,\mathrm{k}\mathrm{A}\textnormal{t})}\,e^{i\,\Phi_{\mathrm{U}}}\,+\,I_{M}\,\bm{\xi}_{X\mathrm{M}}^{(1\,\mathrm{k}\mathrm{A}\textnormal{t})}\,e^{i\,\Phi_{\mathrm{M}}}\,+\,I_{L}\,\bm{\xi}_{X\mathrm{L}}^{(1\,\mathrm{k}\mathrm{A}\textnormal{t})}\,e^{i\,\Phi_{\mathrm{L}}}$
(1)

of the upper (U), middle (M) and lower (L) rows. As linear models cannot
explain or predict a threshold value for ELM suppression, we can instead use
the normalized magnitude

$\xi_{X}^{\ast}\,=\,\frac{|\bm{\xi}_{X}|}{\max|\bm{\xi}_{X}|}$ (2)

for picking the optimal operating point. The magnitude only depends on the
relative phases $\Phi_{\mathrm{U}}\,-\,\Phi_{\mathrm{M}}$ and
$\Phi_{\mathrm{L}}\,-\,\Phi_{\mathrm{M}}$, and figure 2 (a) shows this
dependence at fixed amplitude $I_{\textnormal{RMP}}$ in all rows. Maximizing
$\xi_{X}^{\ast}$ is achieved by aligning the contributions to $\bm{\xi}_{X}$
from each row, i.e.

$\Phi_{\mathrm{U}}\,-\,\Phi_{\mathrm{M}}\Big{|}_{\textnormal{opt}}\,=\,\arg\left(\bm{\xi}_{X\mathrm{M}}^{(1\,\mathrm{k}\mathrm{A}\textnormal{t})}\right)\,-\,\arg\left(\bm{\xi}_{X\mathrm{U}}^{(1\,\mathrm{k}\mathrm{A}\textnormal{t})}\right)$
(3)

and the equivalent for the lower row. An optimal RMP configuration is found at
$C_{\textnormal{maxX}}\approx(200,140)\,\deg$ as shown by the black dot in
figure 2 (a). It has been found that this operating point is insensitive to
the assumed plasma toroidal flow [33, 36]. It will be the baseline for the
following analysis.

Recently, identification of this optimal RMP configuration has motivated non-
linear MHD simulations with the JOREK core in order to verify ELM suppression
in ITER [39]. These simulations have predicted that the threshold for ELM
suppression is in the range of $I_{\textnormal{RMP}}\,=\,45--
60\,\mathrm{k}\mathrm{A}\textnormal{t}$. This is somewhat larger than the
value of $25--30\,\mathrm{k}\mathrm{A}\textnormal{t}$ predicted by TM1 based
on a $15\,\%$ pedestal pressure reduction [45] (TM1 is another non-linear
time-dependent two-fluid MHD code with cylindrical geometry, albeit with
circular cross-section, and coupled to GPEC [46] for boundary conditions). The
TM1 results are, however, very sensitive to $q_{95}$ (see figure 9 in [45]).
In the following, we will leverage the threshold range identified by JOREK and
refer to the lower bound ($45\,\mathrm{k}\mathrm{A}\textnormal{t}$) as
optimistic scenario and to the upper bound
($60\,\mathrm{k}\mathrm{A}\textnormal{t}$) as conservative scenario. With
$\max|\bm{\xi}_{X}|$ evaluated for the hardware limit of
$90\,\mathrm{k}\mathrm{A}\textnormal{t}$ in ITER, the threshold
$\xi_{X}^{\ast}$ for ELM suppression is 1/2 (optimistic) or 2/3
(conservative). The blue and purple contours in figure 2 (a) show the
corresponding boundary of the configuration space that supports ELM
suppression.

Qualitatively, ELM suppression by RMPs requires enough perturbation at the
plasma edge, but at the same time avoiding too much perturbation in the core
in order to avoid mode locking. The latter imposes an upper threshold for the
applied RMP strength, and this is found to be a particularly strict condition
for low $n$ RMPs in KSTAR [47, 35, 48]. A combined approach for maximizing the
edge impact and minimizing the core impact allows for optimal ELM control [49,
50]. It has been found that the core-kink response is linked to a large
displacement $|\bm{\xi}_{M}|$ of the plasma boundary at the outboard midplane
[51, 30, 31], and in the following we will use this as a figure of merit for
core stability. For the optimization approach below, we will normalize
$\bm{\xi}_{M}$ to its value at $C_{\textnormal{maxX}}$ and
$I_{\textnormal{RMP}}=90\,\mathrm{k}\mathrm{A}\textnormal{t}$:

$\xi_{M}^{\ast}\,=\,\frac{|\bm{\xi}_{M}|}{|\bm{\xi}_{M}|_{\max|\bm{\xi}_{X}|}}$
(4)

where $\bm{\xi}_{M}$ is obtained from a linear superposition similar to (1).
Note that values of $\xi_{M}^{\ast}$ can exceed 1 in this definition, unlike
$\xi_{X}^{\ast}$. The phase dependence of $\xi_{M}^{\ast}$ at fixed
$I_{\textnormal{RMP}}$ is shown in figure 2 (b). It can be seen in comparison
to figure 2 (a) that large $\xi_{X}^{\ast}$ tend to imply large
$\xi_{M}^{\ast}$. This shows that some trade-off between $\xi_{X}^{\ast}$ and
$\xi_{M}^{\ast}$ may need to be considered (as already suggested in [33]). At
KSTAR, an empirical limit for $\xi_{M}^{\ast}$ is obtained from the standard
$90\,\deg$ configuration for $n=1$ RMPs, and this implies that large parts of
the RMP configuration space are excluded [47, 35]. For $n=3$ RMPs in ITER,
this may not be as restrictive. Without a known limit value for ITER, we can
at least map out which regions of the RMP configuration space are less
susceptible for core-kink response than others. We will follow up on this
below in section 2.2.

Another important effect to consider is the impact of RMPs on particle and
heat loads on divertor targets. Application of RMPs lead to helical
corrugations of the magnetic separatrix, and those guide field lines from the
interior to the divertor targets. Transport along those perturbed field lines
results in non-axisymmetric striations of particle and heat loads on divertor
targets which follow the geometry of the perturbed separatrix. Therefore, the
magnetic footprint can be used as a proxy for characterization of particle and
heat load striations. The FLARE code [52] is used to trace field lines from
the divertor targets and to track their deepest connection into the plasma
$\min\psi_{N}|_{\textnormal{fieldline}}$ along the way ($\psi_{N}$ is the
normalized poloidal magnetic flux). The footprint width $s_{\textnormal{max}}$
is then defined as the maximum distance from the equilibrium strike point with
$\min\psi_{N}|_{\textnormal{fieldline}}<1$. Despite the linear superposition
of the MARS-F plasma response from each row, field line tracing and the
computation of $s_{\textnormal{max}}$ is of non-linear nature. Figure 2 (c)
and (d) shows the resulting $s_{\textnormal{max}}$ on the inner and outer
target, respectively, for
$I_{\textnormal{RMP}}\,=\,60\,\mathrm{k}\mathrm{A}\textnormal{t}$. Each
magnetic footprint is sampled with a resolution of $1\,\deg$ toroidally and
$1\,\mathrm{c}\mathrm{m}$ along the target, and the 2-D phase scan is
conducted with a resolution of $20\,\deg$. It can be seen that large
$\xi_{X}^{\ast}$ tends to imply large $s_{\textnormal{max}}$ both on the inner
and outer target. This is consistent with computational observations at DIII-D
[53] based on the high field side plasma response as proxy for ELM
suppression. Furthermore, it can be seen that the magnetic footprints on the
outer target are about a factor of 2 larger than on the inner target. This is
a consequence of the target geometry in relation to the shape of the
equilibrium: flux surfaces have a shallower (poloidal) incident angle on the
outer target which implies a larger distance from the equilibrium strike point
for the same $\psi_{N}$.

For attached plasmas, RMPs can spread the heat loads over a wider area,
depending on the magnetic footprint width $s_{\textnormal{max}}$ in relation
to the reference heat load width without RMPs. With an upstream heat flux
width of $\lambda_{q}=3.4\,\mathrm{m}\mathrm{m}$ [20] (or below [54]), and a
flux expansion of $f_{x}\approx\,6$, it can be expected that the perturbed
configuration will significantly impact the divertor plasma in ITER. The ITER
divertor, however, is required to operate in a partially detached state in
order to remain below an acceptable level of about
$10\,\mathrm{M}\mathrm{W}\,\mathrm{m}^{-2}$. Previous EMC3-EIRENE simulations
for the pre-fusion power operation phase at lower heating power have shown
that while detachment onset occurs at lower upstream density, heat flux is
shifted along the helical corrugations of the separatrix into the far SOL
strike locations which remain attached towards higher upstream density [23,
24]. Thus, maximizing $s_{\textnormal{max}}$ may not be the best approach for
ITER, if power dissipation from impurity seeding turns out to be insufficient
for reducing heat loads at far SOL strike points. We will follow up on this in
section 4 for the configurations of interest identified below.

###  2.2 Configuration scan with fixed $\xi_{X}^{\ast}$

Figure 3: Phase scan of the RMP waveform in the upper and lower row with
relative to the middle row at fixed $\xi_{X}^{\ast}$ for the optimistic
threshold at $I_{\textnormal{th}}=45\,\mathrm{k}\mathrm{A}\textnormal{t}$: (a)
Required current $I_{\textnormal{RMP}}$, (b) normalized midplane displacement
$\xi_{M}^{\ast}$, (c-d) magnetic footprint width $s_{\textnormal{max}}$ on the
inner and outer target. The black dot marks the reference configuration
$C_{\textnormal{maxX}}$ from figure 2, and the blue dot highlights the
configuration $C_{\textnormal{minM}}^{(45)}$ with minimal $\xi_{M}^{\ast}$
that is compatible with the maximum available $I_{\textnormal{RMP}}$. The
black line marks the iso-$\xi_{M}^{\ast}$ contour through
$C_{\textnormal{maxX}}$, and the green dot highlights the configuration
$C_{\textnormal{isoM}}^{(45)}$ with minimal $s_{\textnormal{max}}$ on the
outer target along that contour.

Based on the threshold(s) identified by JOREK simulations, we will now look at
configurations with the same $\xi_{X}^{\ast}$ in the anticipation that they
imply equivalent ELM suppression. Knowing the phase dependence of
$\xi_{X}^{\ast}$ as shown in figure 2 (a), we can define the RMP current

$I_{\textnormal{RMP}}(I_{\textnormal{th}},\Delta\Phi_{\mathrm{UM}},\Delta\Phi_{\mathrm{LM}})\,=\,\frac{I_{\textnormal{th}}}{\xi_{X}^{\ast}(90\,\mathrm{k}\mathrm{A}\textnormal{t},\Delta\Phi_{\mathrm{UM}},\Delta\Phi_{\mathrm{LM}})}$
(5)

that is required for each phase combination in order to match the
$\xi_{X}^{\ast}$ of the ELM suppression threshold $I_{\textnormal{th}}$ at
$C_{\textnormal{maxX}}$. Figure 3 (a) shows the optimistic scenario
$I_{\textnormal{RMP}}(I_{\textnormal{th}}=45\,\mathrm{k}\mathrm{A}\textnormal{t})$
which reproduces the value of $I_{\textnormal{th}}$ at $C_{\textnormal{maxX}}$
(black dot). Configurations which exceed the
$90\,\mathrm{k}\mathrm{A}\textnormal{t}$ limit are excluded (white), but about
half ($52.2\,\%$) of the phase combinations support ELM suppression.

Figure 3 (b) shows the corresponding $\xi_{M}^{\ast}$ as figure of merit for
the core response. As the normalization of $\xi_{M}^{\ast}$ in (4) is based on
the $C_{\textnormal{maxX}}$ configuration at full current, a value of 0.5 is
found here for this configuration in the optimistic $I_{\textnormal{th}}$
scenario. It can be seen that the impact on the core plasma can be reduced by
a factor of 2 for the configuration
$C_{\textnormal{minM}}^{(45)}=(151,261)\,\deg$ (blue dot) with respect to the
reference configuration $C_{\textnormal{maxX}}$. The maximum of
$\xi_{M}^{\ast}$ is below 0.74, but at this point we do not have an upper
limit for core stability (which in KSTAR significantly reduced the set of
viable configurations). Overall, $29.5\,\%$ of the phase combinations support
ELM suppression at $\xi_{M}^{\ast}\,\leq\,1/2$ (left of the black line).

The corresponding magnetic footprint widths on the inner and outer divertor
targets are shown in figure 3 (c-d). A much smaller variation of
$s_{\textnormal{max}}$ is found now that all configurations have equal
$\xi_{X}^{\ast}$. Thus, decoupling of the footprint width from ELM control may
be challenging, as already pointed out by Munaretto et al. [53]. However, the
2-D phase scan here shows that the three row setup of ITER (and KSTAR)
supports fine tuning that would otherwise be severely limited with a two row
setup such as DIII-D. Here, a higher resolution of $0.1\,\deg$ and
$0.1\,\mathrm{c}\mathrm{m}$ is used to sample field lines for the 3
configurations of particular interest (highlighted by the black, blue and
green dots). The magnetic footprints at $C_{\textnormal{maxX}}$ are smaller
here ($17.6\,\mathrm{c}\mathrm{m}$ on the inner target and
$30.6\,\mathrm{c}\mathrm{m}$ on the outer target) than in figure 2 (c-d)
because of the lower
$I_{\textnormal{RMP}}\,=\,45\,\mathrm{k}\mathrm{A}\textnormal{t}$ applied
here. A minor increase to $18.4\,\mathrm{c}\mathrm{m}$ (inner target) and
$33.2\,\mathrm{c}\mathrm{m}$ (outer target) is found for
$C_{\textnormal{minM}}^{(45)}$ with improved core stability. On the other
hand, a somewhat larger variation of $s_{\textnormal{max}}$ is found along the
black iso-$\xi_{M}^{\ast}$ line where configurations are equivalent in both
ELM suppression and core stability. In particular, for the configuration
$C_{\textnormal{isoM}}^{(45)}=(225,-2)\,\deg$ (green dot),
$s_{\textnormal{max}}$ is increased to $21.6\,\mathrm{c}\mathrm{m}$
($+23\,\%$) on the inner target while $s_{\textnormal{max}}$ is decreased to
$25.4\,\mathrm{c}\mathrm{m}$ ($-17\,\%$) on the outer target compared to
$C_{\textnormal{maxX}}$.

Figure 4: Phase scan of the RMP waveform in the upper and lower row with
relative to the middle row at fixed $\xi_{X}^{\ast}$ for the optimistic
threshold at $I_{\textnormal{th}}=60\,\mathrm{k}\mathrm{A}\textnormal{t}$: (a)
Required current $I_{\textnormal{RMP}}$, (b) normalized midplane displacement
$\xi_{M}^{\ast}$, (c-d) magnetic footprint width $s_{\textnormal{max}}$ on the
inner and outer target. The black dot marks the reference configuration
$C_{\textnormal{maxX}}$ from figure 2, and the blue dot highlights the
configuration $C_{\textnormal{minM}}^{(60)}$ with minimal $\xi_{M}^{\ast}$
that is compatible with the maximum available $I_{\textnormal{RMP}}$. The
black line marks the iso-$\xi_{M}^{\ast}$ contour through
$C_{\textnormal{maxX}}$, and the green dot highlights the configuration
$C_{\textnormal{isoM}}^{(60)}$ with minimal $s_{\textnormal{max}}$ on the
outer target along that contour.

The conservative scenario for $I_{\textnormal{th}}=\,60\,kAt$ is shown in
figure 4. A smaller set of configurations ($31.6\,\%$) supports ELM
suppression compared to the optimistic scenario due to the higher
$I_{\textnormal{th}}$ at $C_{\textnormal{maxX}}$ here. Nevertheless, figure 4
(b) shows that a significant reduction of core perturbation is still possible
with an optimum of $\xi_{M}^{\ast}=0.45$ at
$C_{\textnormal{minM}}^{(60)}=(159,229)\,\deg$ ($-32\,\%$ relative to
$C_{\textnormal{maxX}}$). Overall, $16.8\,\%$ of the phase combinations
support ELM suppression with improved core stability ($\xi_{M}^{\ast}<2/3$).
Magnetic footprints in figure 4 (c-d) are larger than in figure 3 (c-d)
consistent with the larger $\xi_{X}^{\ast}$ of the conservative scenario. For
the reference point $C_{\textnormal{maxX}}$,
$s_{\textnormal{max}}\,=\,24.9\,\mathrm{c}\mathrm{m}$ on the inner target and
$s_{\textnormal{max}}\,=\,52.7\,\mathrm{c}\mathrm{m}$ on the outer target. The
same trend of shifting the footprint size from the outer to the inner target
is found when moving along the black iso-$\bm{\xi}_{M}$ line towards
$C_{\textnormal{isoM}}^{(60)}=(162,16)\,\deg$ (green dot). For the improved
core configuration $C_{\textnormal{minM}}^{(60)}$ (blue dot), the footprint
increases to $s_{\textnormal{max}}\,=\,64.6\,\mathrm{c}\mathrm{m}$ ($+23\,\%$)
on the outer target. It should be noted that the extent of the beveled
tungsten monoblock region (the dedicated high heat flux region) on the outer
vertical target is only $57.9\,\mathrm{c}\mathrm{m}$ (for this equilibrium).
This implies that too much optimization for core perturbation may not be
compatible with tolerable divertor heat loads.

##  3 Figures of merit

The optimization of RMP configurations for ELM suppression relies on robust
figures of merit. At KSTAR, the experimentally observed operating window of
RMP configurations with ELM suppression has been reproduced by linear, ideal
MHD plasma response [35]. Specifically, figures of merit for the edge
($\mathcal{E}$) and for the core ($\mathcal{C}$) have been constructed from
the perturbed flux $\Phi_{mn}$:

$\mathcal{E},\mathcal{C}\,=\,\sqrt{\frac{1}{N}\,\sum\,|\Phi_{mn}|^{2}}$ (6)

where the sum is taken over $N$ rational surfaces that have to be properly
chosen for each of the two figures. For $n=1$ RMPs in KSTAR, the $m=5,6$
resonances are located in the edge and the $m=2,3$ resonances are located in
the core. Viable configurations require
$\mathcal{E}\,\geq\,\mathcal{E}_{\textnormal{th}}$ for ELM suppression and
$\mathcal{C}\,<\,\mathcal{C}_{\textnormal{crit}}$ to avoid disruptive core
field penetration. The critical values $\mathcal{E}_{\textnormal{th}}$ and
$\mathcal{C}_{\textnormal{crit}}$ are determined from the empirical ELM
suppression window in the $90\,\deg$ standard configuration from
$I_{\textnormal{RMP}}\,=\,1.8\,\mathrm{k}\mathrm{A}$ to
$2.1\,\mathrm{k}\mathrm{A}$.

Figure 5: (a) Edge figure of merit with threshold for ELM suppression (green),
(b) core figure of merit with stability limit (red), and (c) operation window
for ELM suppression (blue). Contour lines are based on $R_{mn}$ (dashed) and
$\varepsilon_{m+1\,n}$ (dotted) in comparison to the $\Phi_{mn}$ reference
(solid) from Park et. al [35]. Here,
$\Delta\Phi=\Phi_{M}\,-\,\Phi_{L}\,=\,\Phi_{U}\,-\,\Phi_{M}$ while
$I_{U}\,=\,I_{L}\,=5/\sqrt{2}\,\mathrm{k}\mathrm{A}$ is kept fixed and $I_{M}$
is scanned.

It should be noted that the $\Phi_{mn}$ from IPEC do not represent the
resulting resonant fields (which are screened in ideal MHD), but rather their
values when the singular currents at the resonances were to dissipate. As
such, $\Phi_{mn}$ is specific output of IPEC. In order to link those results
to ours - and to demonstrate the robustness of these figures of merit - we
introduce a code independent approach below. The Fourier transformed perturbed
flux

$\varepsilon_{mn}(\psi)\,=\,\frac{1}{\left(2\pi\right)^{2}}\,\oint\\!d\theta
d\varphi\,\mathcal{J}\,\frac{\delta{\bf
B}\cdot\nabla\psi}{||\nabla\psi||}\,e^{-i\left(m\theta\,-\,n\varphi\right)}$
(7)

is computed in a post-processing step from the perturbation field $\delta{\bf
B}$ ($\mathcal{J}$ is the flux surface Jacobian in straight field line
coordinates $\theta$ and $\varphi$ for the equilibrium poloidal flux $\psi$).
The FLARE code [52] has interfaces for $\delta{\bf B}$ from IPEC, MARS-F and
other plasma response codes, and can evaluate (7) in support of field line
tracing. The magnitude of the resonant field $|\varepsilon_{mn}(\psi_{mn})|$
for $q(\psi_{mn})\,=\,m/n$ is typically small due to plasma response (or zero
in ideal MHD). For the new figures of merit, we replace $\Phi_{mn}$ in (6)
with either the kink amplitude $|\varepsilon_{m+1\,n}(\psi_{mn})|$, or with
the resonant response

$R_{mn}\,=\,\frac{|\varepsilon_{mn}^{\prime}|}{m}\,\frac{q}{q^{\prime}}\Bigg{|}_{\psi_{mn}}$
(8)

as a measure for the kinking of the resonant surface that can be attributed to
the plasma response. Derivatives (′) in (8) are taken with respect to the
radial direction $\psi$. Figure 5 shows an RMP configuration scan for the new
edge and core figures of merit based on $R_{mn}$. Contour lines for critical
values are also shown for figures of merit based on $\varepsilon_{m+1\,n}$
(dotted) and for the $\Phi_{mn}$ reference (solid). Note that the RMP
configuration scan here is different from figures 2 \- 4, but it is chosen in
order to match the one in [35] as baseline for this benchmark. It can be seen
that the simultaneous constraints on $\mathcal{E}$ (left of green lines) and
$\mathcal{C}$ (right of red lines) result in a narrow window of suitable RMP
configurations for ELM suppression (blue). In particular, it can be seen that
the new figures of merit based on $R_{mn}$ and $\varepsilon_{m+1\,n}$ are
consistent with the IPEC ones based on $\Phi_{mn}$.

Figure 6: Benchmark of figures of merit for ELM suppression. Blue and purple
lines represent the boundary of viable configurations for an optimistic and
conservative threshold, respectively. Black dots represent the configuration
with maximum edge figure of merit at fixed RMP current, and black lines
represent configurations with equivalent core perturbation at fixed edge
figure of merit. Colored dots represent configurations with minimal core
perturbation.

In order to improve confidence in the previously identified RMP configurations
for ELM suppression in ITER, figure 6 shows the results from section 2 in
comparison to the new figure of merit. For $n=3$ RMPs in ITER, $\mathcal{E}$
is determined from the edge resonances $m=9,10$ and 11, while $\mathcal{C}$ is
determined from the core resonances $m=4,5$ and 6. It can be seen that our
earlier results based on the plasma displacement at the X-point and at the
midplane are reproduced by the new figures of merit based on the kink
amplitude $\varepsilon_{m+1\,n}$ and the resonant response $R_{mn}$. In
particular, the black dots in figure 6 mark the optimal configuration obtained
from maximizing each of the edge figures of merit at fixed RMP current
$I_{\textnormal{RMP}}$. By rescaling $I_{\textnormal{RMP}}$ to match the edge
perturbation of the optimal configurations, the blue and purple lines
highlight the boundary of viable configurations for ELM suppression for an
optimistic threshold of $45\,\mathrm{k}\mathrm{A}\textnormal{t}$ and a
conservative threshold of $60\,\mathrm{k}\mathrm{A}\textnormal{t}$,
respectively. At this fixed level of edge perturbation, the blue and purple
dots mark the configurations with minimal core perturbation. Furthermore, it
is shown that similar contour lines (black) are found for configurations with
both equivalent core and edge perturbation. Even though some deviation may be
noted for the optimistic window, this has little impact on the corresponding
magnetic footprints in figure 3 (c,d).

##  4 Impact on divertor heat loads

Figure 7: Cross section of the plasma density (a) and electron temperature (b)
at $\varphi\,=\,0\,\deg$ for the $C_{\textnormal{maxX}}$ RMP configuration at
$I_{\textnormal{RMP}}\,=\,45\,\mathrm{k}\mathrm{A}\textnormal{t}$.

Field line tracing is a relatively cheap method to determine where particle
and heat loads would be expected from RMP application, but ultimately, 3D
plasma edge and boundary modeling is required. The EMC3-EIRENE code [55]
combines EMC3 [56] for the fluid edge plasma with EIRENE [57] for a kinetic
neutral particle transport. Reactions include ionization, charge exchange,
recombination, molecular processes such as dissociation, ion conversion and
elastic $D_{2}\,-\,D^{+}$ collisions, and neutral-neutral collisions (in BGK
approximation). The edge plasma model is a time-independent, simplified
version of Braginskii’s fluid equations for particle, momentum and energy
balance. Quasi neutrality $n_{e}=n_{i}$ is assumed, along with no plasma
current (which implies $u_{e}=u_{i}$). Classical heat conductivity
$\kappa_{e}\sim T_{e}^{5/2}$ and $\kappa_{i}\sim T_{i}^{5/2}$ is assumed for
electrons and ions, respectively, but cross-field particle and energy
transport is assumed to be anomalous with model parameters $D$, $\chi_{e}$ and
$\chi_{i}$. Impurity transport with $T_{Z}\,=\,T_{i}$ is included in an
approximation for small concentrations $Z^{2}\,n_{Z}\,\ll\,n_{e}$, but
corresponding radiation losses are coupled to the energy balance for
electrons. The balance equations for $n_{i},u_{i},T_{i},T_{e}$ and $n_{Z}$ are
cast into a Fokker-Planck form which is suitable for a Monte Carlo method
[58]. A reversible field line mapping algorithm is applied for fast
reconstruction of field lines from the 3D mesh [59], which is block-structured
for applications in poloidal divertor geometry [60].

Simulations have been conducted for each of the three configurations of
interest ($C_{\textnormal{maxX}}$, $C_{\textnormal{minM}}$ and
$C_{\textnormal{isoM}}$) from figure 3 and 4 for both $I_{\textnormal{th}}$
scenarios. All simulations have been conducted with the same model parameters:
$100\,\mathrm{M}\mathrm{W}$ input power into the edge, $\textnormal{D}_{2}$
gas puffing of $5\,\cdot 10^{22}\,\mathrm{s}^{-1}$ from the top, core fueling
of $9.1\,\cdot 10^{21}\,\mathrm{s}^{-1}$, $100\,\%$ recycling of $D^{+}$ on
material surfaces, $0.72\,\%$ absorption (pumping) of D neutrals below the
divertor dome, anomalous cross-field transport of
$D\,=\,0.3\,\mathrm{m}^{2}\,\mathrm{s}^{-1}$ and
$\chi_{e}\,=\,\chi_{i}\,=\,1\,\mathrm{m}^{2}\,\mathrm{s}^{-1}$, and a Ne
concentration of $c_{\textnormal{sepx}}\,=\,0.8\,\%$ at the separatrix. This
is consistent with the range of parameters used in SOLPS-4.3 and SOLPS-ITER
simulations for the physics basis of the ITER divertor [20]. The position of
the gas puff and the pumping location is marked in figure 7. Self consistent
impurity throughput is not implemented in EMC3-EIRENE at this point. Rather,
impurity recycling of $99\,\%$ is applied on material surfaces, and the source
strength from Ne gas puffing is adjusted in order to maintain a given
concentration $c_{\textnormal{sepx}}$ evaluated in the outermost cell layer of
the core zone of the computational mesh.

Figure 8: Magnetic footprint and plasma conditions on the inner divertor
target (left column) and outer divertor target (right column) for the
$C_{\textnormal{maxX}}$ configuration at
$I_{\textnormal{RMP}}\,=\,45\,\mathrm{k}\mathrm{A}\textnormal{t}$: (a) radial
field line connection, (b) plasma density, (c) electron temperature, and (d)
heat loads. Color bars are chosen to accommodate the value range in figure 9.

The plasma density and electron temperature for the $C_{\textnormal{maxX}}$
RMP configuration at
$I_{\textnormal{RMP}}\,=\,45\mathrm{k}\mathrm{A}\textnormal{t}$ are shown in
figure 7. The lobe structure of the perturbed separatrix is clearly visible in
the SOL plasma. It can be seen that the density below the X-point is of the
order $10^{15}\,\mathrm{c}\mathrm{m}^{-3}$. This is about 25 times higher than
the value of $4.2\cdot 10^{13}\,\mathrm{c}\mathrm{m}^{-3}$ at the last closed
flux surface. The high density facilitates low temperatures in front of the
divertor targets, but more so near the equilibrium strike point. Plasma
conditions at the divertor targets are shown in figure 8 (b-d) in comparison
to the magnetic footprint in (a) where red colors indicate field lines
connecting from the plasma interior ($\psi_{N}<1$). The density peak is
located within a few $\mathrm{c}\mathrm{m}$ from the equilibrium strike point
as a result of recycling in the private flux region (favored by the field line
incident on the vertical targets), and the temperature drops below
$2\,\mathrm{eV}$ there with heat loads of about $2--
3\,\mathrm{M}\mathrm{W}\,\mathrm{m}^{-2}$. Along the lobe strike locations
further out in the SOL ($s\gtrsim 10\,\mathrm{c}\mathrm{m}$), on the other
hand, the density is lower - but still in the range $0.5--1\,\cdot
10^{15}\,\mathrm{c}\mathrm{m}^{-3}$ \- and the temperature remains above
$10\,\mathrm{eV}$. Heat loads of up to
$6\,\mathrm{M}\mathrm{W}\,\mathrm{m}^{-2}$ can be found, which is still at an
acceptable level. This is significantly different from earlier results at
lower upstream density [61] where the plasma was still attached with peak
loads of over $20\,\mathrm{M}\mathrm{W}\,\mathrm{m}^{-2}$ \- but only because
at that time EMC3-EIRENE was not able to handle partially detached conditions
with strong neutral gas interactions. Meanwhile, recombination has been
implemented into EMC3-EIRENE, and the algorithm has been improved which now
permits convergence at low divertor temperatures. Comparisons to SOLPS-ITER
show good agreement for the pre-fusion power operation phase [62], and we will
follow up with a benchmark of impurity seeding at higher power at a later
point.

Figure 9: Magnetic footprint and plasma conditions on the inner divertor
target (left column) and outer divertor target (right column) for the
$C_{\textnormal{maxX}}$ configuration at
$I_{\textnormal{RMP}}\,=\,60\,\mathrm{k}\mathrm{A}\textnormal{t}$: (a) radial
field line connection, (b) plasma density, (c) electron temperature, and (d)
heat loads.

A larger magnetic footprint is expected for the higher
$I_{\textnormal{RMP}}\,=\,60\,\mathrm{k}\mathrm{A}\textnormal{t}$ of the
conservative scenario. Figure 9 shows the resulting impact on plasma
conditions at the divertor targets. Conditions near the equilibrium strike
point are comparable to figure 8, but the impact of the lobes reaches further
out into the SOL. Higher temperatures of $15-20\,\mathrm{eV}$ can be found
with peak heat loads of around $10\,\mathrm{M}\mathrm{W}\,\mathrm{m}^{-2}$ on
the inner divertor target - just around the limit of acceptable values. Those
peaks result from heat flux that is routed into the helical lobes upstream,
and very little dissipation from Ne impurities is found along the way to the
target. A combination of higher density from additional $D_{2}$ gas puffing
and higher impurity concentration may mitigate the impact on the inner
divertor target. This will be done in future work.

Figure 10: Heat load profiles along the inner target at $\varphi=10\,\deg$ (a)
and along the outer target at $\varphi=-50\,\deg$ (b) for the
$C_{\textnormal{maxX}}$ (black), $C_{\textnormal{isoM}}$ (green) and
$C_{\textnormal{minM}}$ (blue) RMP configurations for the optimistic ELM
suppression threshold (solid) and the conservative one (dashed) in comparison
to a simulation without RMPs (red).

Figure 10 shows a comparison of heat load profiles for the 3 different RMP
configurations for both the optimistic ELM suppression threshold (solid) and
the conservative one (dashed) in comparison to a simulation without RMPs
(red). Without RMPs, the plasma at the inner target is detached with a peak
heat load of $4\,\mathrm{M}\mathrm{W}\,\mathrm{m}^{-2}$ but remains attached
at the outer target at $10\,\mathrm{M}\mathrm{W}\,\mathrm{m}^{-2}$ for the
model parameters discussed above. The total head loads are summarized in
figure 11, which shows that only $9.3\,\mathrm{M}\mathrm{W}$ are deposited on
the inner target while $39.7\,\mathrm{M}\mathrm{W}$ are deposited on the outer
target without RMPs. With RMPs, dissipation from impurity radiation is less
effective in inner divertor, which leads to higher peak and integral values in
figure 10 (a) and 11, respectively. In the outer divertor, on the other hand,
impurity radiation becomes more effective with RMPs, which leads to a lower
integral heat load. The main heat load peak near the equilibrium strike point
is reduced to about $2\,\mathrm{M}\mathrm{W}\,\mathrm{m}^{-2}$, but at the
expense of secondary peaks of $5--7\,\mathrm{M}\mathrm{W}\,\mathrm{m}^{-2}$
further away at $s=12--15\,\mathrm{c}\mathrm{m}$. Tuning of the RMP spectrum
appears to have marginal impact on the heat loads for the same
$\xi_{X}^{\ast}$, although a small reduction of the peak heat load on the
inner target can be found for the $C_{\textnormal{isoM}}$ configuration
(green). In any case, the simulations show that both optimistic and
conservative ELM suppression scenarios are (marginally) compatible with
divertor limits - at least for the selected set of model parameters.

Figure 11: Total head loads on inner and outer divertor targets for all
simulations.

##  5 Conclusions

A conservative and an optimistic subspace of RMP configurations for ELM
suppression has been predicted for H-mode burning plasmas at 15 MA current and
5.3 T magnetic field in ITER. This is based on the X-point displacement
$\xi_{X}^{\ast}$ from linear MHD plasma response (MARS-F) as figure of merit
for edge stability, which is then compared to a threshold value. Based on
published results from non-linear MHD modeling (JOREK), this threshold value
is taken from a particular RMP configuration that shows onset of ELM
suppression in the range of $45\,\mathrm{k}\mathrm{A}\textnormal{t}$
(optimistic) to $60\,\mathrm{k}\mathrm{A}\textnormal{t}$ (conservative). The
plasma boundary displacement at the outboard midplane has been used as a proxy
for perturbation of the core, and it has been shown that it can be reduced by
a factor of 2 for equivalent edge stability proxy. At the same time,
variations of the magnetic footprint size within the ELM suppression subspace
remain within $20\,\%$. It has been demonstrated that the predictions of the
RMP subspace for ELM suppression are robust by comparing to figures of merit
based on the kink amplitude and on the resonant response. Those are also
consistent with the RMP subspace for ELM suppression in KSTAR.

The magnetic footprint of perturbed field lines connecting from the main
plasma (normalized poloidal flux $<1$) to the divertor targets is found to be
significantly larger than the expected heat load width in the absence of RMPs.
This facilitates heat load spreading with peak values at an acceptable level
below $10\,\mathrm{M}\mathrm{W}\,\mathrm{m}^{-2}$ on the outer target already
at moderate gas fueling and low Ne seeding for additional radiative
dissipation of the $100\,\mathrm{M}\mathrm{W}$ of power into the scrape-off
layer (SOL). On the inner target, however, re-attachment is predicted away
from the equilibrium strike point due to the heat flux that is routed into the
helical lobes upstream, but with less efficient impurity radiation at higher
downstream temperatures. Tuning of the RMP spectrum appears to have marginal
impact on the heat loads for the same $\xi_{X}^{\ast}$. In any case, the
simulations show that both optimistic and conservative ELM suppression
scenarios are (marginally) compatible with divertor limits - at least for the
selected set of model parameters. Future work is planned to explore the impact
of higher density from additional $D_{2}$ gas puffing and higher impurity
concentration on the heat load peaks away from the equilibrium strike point.

## Acknowledgments

This work was supported by the U.S. Department of Energy under award No. DE-
SC0020284 and DE-SC0020357. This work was done under the auspices of the ITER
Science Fellow Network. The views and opinions expressed herein do not
necessarily reflect those of the ITER Organization.

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