# Critical gradient turbulence optimization toward a compact stellarator
reactor concept

G. T. Roberg-Clark<EMAIL_ADDRESS>G. G. Plunk  P. Xanthopoulos  C. Nührenberg
S. A. Henneberg  H. M. Smith  Max-Planck-Institut für Plasmaphysik, D-17491,
Greifswald, Germany

###### Abstract

Integrating turbulence into stellarator optimization is shown by targeting the
onset for the ion-temperature-gradient mode, highlighting effects of parallel
connection length, local magnetic shear, and flux surface expansion. The
result is a compact quasihelically symmetric stellarator configuration,
admitting a set of uncomplicated coils, with significantly reduced turbulent
heat fluxes compared to a known stellarator. The new configuration combines
low values of neoclassical transport, good alpha particle confinement, and
Mercier stability at a plasma beta of almost 2$\%$.

Introduction.– A primary obstacle for the success of magnetic confinement
fusion is the transport caused by instabilities such as the ion temperature
gradient (ITG) mode, which is thought to significantly reduce plasma
confinement in experiments such as the Wendelstein 7-X stellarator (Beurskens
_et al._ , 2021; Bähner  _et al._ , 2021; Carralero  _et al._ , 2021). To
overcome the losses from such turbulence, a given configuration can be scaled
up in size and heating power. A less costly alternative, currently explored,
is to shape the magnetic field to alleviate the turbulence. This option could
be particularly appealing for reactor scenarios, in which it will likely be
difficult to control density profiles via pellet injections (Bozhenkov  _et
al._ , 2020; Xanthopoulos  _et al._ , 2020), since the penetration distance of
pellets may be limited in comparison to the minor radius of a reactor.

To achieve turbulence optimization via shaping, several strategies have been
developed to reduce the rate that turbulent transport increases (“stiffness”)
as a function of the ion temperature gradient (Mynick  _et al._ , 2010;
Xanthopoulos  _et al._ , 2014; Hegna  _et al._ , 2018; Nunami  _et al._ ,
2013; Hegna  _et al._ , 2022). Another approach is to target the onset
(“critical”) gradient of significant turbulent transport (Roberg-Clark  _et
al._ , 2021, 2022a; Zocco  _et al._ , 2022), which relies primarily on linear
physics of ITG modes themselves (Jenko  _et al._ , 2001; Biglari  _et al._ ,
1989; Baumgaertel  _et al._ , 2013; Zocco  _et al._ , 2018) and avoids the
hard problem of solving turbulence in the full range of toroidal geometries.
Successful optimization has been recently demonstrated using a critical
gradient (CG) approach, even leading to reduced stiffness of ITG turbulence in
the nonlinear regime (Roberg-Clark  _et al._ , 2022a, b), but at the apparent
cost of MHD stability via the loss of a vacuum magnetic well.

In this Letter, we show that CG optimization can be achieved without
compromising MHD stability, or other key properties needed by a stellarator
design. Here we target the critical gradient of only the toroidal branch of
the ITG mode, based on the assumption that turbulence intensity will be small
below this threshold. Our model highlights the familiar stabilizing effects of
local shear, but gives greater emphasis to short connection lengths between
regions of ‘good’ and ‘bad’ magnetic curvature. The resulting objective
function is used via optimization to produce a quasi-helically symmetric (QH)
configuration with strongly reduced ITG turbulence, with acceptable levels of
neoclassical confinement, alpha particle confinement, coil complexity, and MHD
stability, completing the picture for an initial stellarator concept with
improved ion confinement.

Definitions.– Following (Plunk  _et al._ , 2014), we use the standard
gyrokinetic system of equations (Brizard and Hahm, 2007) to describe
electrostatic fluctuations destabilized along a thin flux tube tracing a
magnetic field line. The ballooning transform (Dewar and Glasser, 1983; Connor
_et al._ , 1978) is used to separate out the fast perpendicular (to the
magnetic field) scale from the slow parallel scale. The magnetic field
representation in field following (Clebsch) representation reads,
$\mathbf{B}=\bm{\nabla}\psi\times\bm{\nabla}\mbox{$\alpha$}$, where $\psi$ is
a flux surface label and $\alpha$ labels the magnetic field line on the
surface. The perpendicular wave vector is then expressed as
$\mathbf{k_{\perp}}=k_{\alpha}\bm{\nabla}\alpha+k_{\psi}\bm{\nabla}\psi$,
where $k_{\alpha}$ and $k_{\psi}$ are constants, so the variation of
$\mathbf{k_{\perp}}(l)$ stems from that of the geometric quantities
$\bm{\nabla}\alpha$ and $\bm{\nabla}\psi$, with $\ell$ the field-line-
following (arc length) coordinate.

We assume Boltzmann-distributed (adiabatic) electrons, thus solving for the
perturbed ion distribution $g_{i}(v_{\parallel},v_{\perp},\mbox{$\ell$},t)$,
defined to be the non-adiabatic part of $\delta f_{i}$ ($\delta
f_{i}=f_{i}-f_{i0})$ with $f_{i}$ the ion distribution function and $f_{i0}$ a
Maxwellian. The electrostatic potential is $\phi(\mathbf{\mbox{$\ell$}})$, and
$v_{\parallel}$ and $v_{\perp}$ are the particle velocities parallel and
perpendicular to the magnetic field, respectively. The gyrokinetic equation
reads

$iv_{\parallel}\frac{\partial
g}{\partial\ell}+(\omega-\mbox{$\widetilde{\omega}_{d}$})g=\varphi
J_{0}(\omega-\mbox{$\widetilde{\omega}_{*}$})f_{0}$ (1)

where $\omega$ is the mode frequency,
$\omega_{*}^{T}=(Tk_{\alpha}/q)\mathrm{d}\ln
T/\mathrm{d}\psi\left(v^{2}/\mbox{$v_{\mathrm{T}}$}^{2}-3/2\right)$ is the
diamagnetic frequency, and
$J_{0}=J_{0}(k_{\perp}(\mbox{$\ell$})v_{\perp}/\Omega(\mbox{$\ell$}))$ is the
Bessel function of zeroth order. The thermal velocity is
$\mbox{$v_{\mathrm{T}}$}=\sqrt{2T/m}$, the thermal ion Larmor radius is
$\rho=\mbox{$v_{\mathrm{T}}$}/(\Omega\sqrt{2})$, $n$ and $T$ are the
background ion density and temperature, $q$ is the ion charge,
$\varphi=q\phi/T$ is the normalized electrostatic potential, $\Omega=qB/m$ is
the cyclotron frequency, with $B=|\mathbf{B}|$. The magnetic drift frequency
in the low $\beta$ approximation becomes
$\mbox{$\omega_{d}$}=(1/\Omega)(\mathbf{k_{\perp}}\cdot\mathbf{b}\times\bm{\kappa})(v_{\parallel}^{2}+v_{\perp}^{2}/2)=\omega_{d0}(v_{\parallel}^{2}+v_{\perp}^{2}/2)$,
with $\bm{\kappa}=\mathbf{b}\cdot\bm{\nabla}\ \mathbf{b}$ and
$\mathbf{b}=\mathbf{B}/B$. For simplicity in this analysis we set
$k_{\psi}=0$. We rewrite the drift frequency as
$\mbox{$\omega_{d}$}(\ell)\propto K_{d}(\ell)\equiv
a^{2}{\bm{\nabla}}\alpha\cdot\mathbf{b}\times\bm{\kappa}$, referring to
$K_{d}(\ell)$ as the “drift curvature” and to individual regions of bad
curvature along the field line (where $K_{d}>0$) as “drift wells”. We define a
radial coordinate $r=a\sqrt{\psi/\psi_{edge}}$, with $a$ the minor radius
corresponding to the flux surface at the edge, and $\psi_{edge}$ the toroidal
flux at that location. As usual, the temperature gradient scale length is
measured relative to the minor radius,
$a/L_{T}=-(a/T)\mathrm{d}T/\mathrm{d}r$. To study the most unstable ITG mode
conditions, we neglect certain stabilizing factors such as the density
gradient (Proll  _et al._ , 2022; Jorge and Landreman, 2021) and plasma beta
(electromagnetic effects) (Pueschel  _et al._ , 2008).

Finally, the gyrokinetic system is completed by quasineutrality,

$\int d^{3}{\bf v}J_{0}g=n(1+\tau)\varphi,$ (2)

with $\tau=|q_{e}|T_{i}/(q_{i}T_{e})$, $T_{e}$ the electron temperature, and
$q_{e}$ the electron charge.

Toroidal mode threshold.– It is expected (Jenko and Dorland, 2002; Plunk  _et
al._ , 2017; Zocco  _et al._ , 2022) that the onset of toroidal ITG modes, as
can be observed in the linear spectrum (Zocco  _et al._ , 2018), should lead
to noticeable increases in nonlinear heat fluxes. That is to say, the
turbulence found below this onset (in the Floquet-like or slab-like regime) is
thought to be more benign, transitioning to significant heat losses only for
larger gradients (in the toroidal regime).

We therefore focus our attention on the toroidal ITG mode (Jenko  _et al._ ,
2001; Plunk  _et al._ , 2014; Zocco  _et al._ , 2018; Sugama, 1999) with
strongly peaked eigenmode structure that decays within a single drift well. In
the standard local (in $\ell$) theory of toroidal ITG modes (Biglari  _et al._
, 1989; Jenko  _et al._ , 2001; Plunk  _et al._ , 2014), the critical gradient
is set by the drive parameter $\kappa_{d}=\mbox{$R_{\mathrm{eff}}$}/L_{T}$,
where $R_{\mathrm{eff}}$ is the local effective radius of curvature determined
by the profile of $K_{d}(\ell)$. This threshold can be computed for general
parameter $b$ by solving the local dispersion relation

$0=2-\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}dx_{\perp}\>x_{\perp}\int_{-\infty}^{\infty}dx_{\parallel}\left[\frac{\omega-\mbox{$\widetilde{\omega}_{*}$}}{\omega-\mbox{$\widetilde{\omega}_{d}$}}\right]J^{2}_{0}\exp(-x^{2}),$
(3)

which upon substitution of $K_{d}=a/\mbox{$R_{\mathrm{eff}}$}$ yields the
critical gradient $\mbox{$R_{\mathrm{eff}}$}/L_{T,crit}=F(b)$, where $F(b)$
can be obtained numerically and is fairly well approximated by

$\displaystyle F(b)=2.84+4.926\>b,b<0.755$ $\displaystyle
0.0371+7.51\sqrt{b},b\geq 0.755$ (4)

In realistic geometry, the threshold is controlled by the extent of drift
wells, i.e. the parallel connection length $L_{\parallel}$, but this can be
related to finite Larmor radius (FLR) stabilization as follows. Note that a
toroidal mode must have a drift frequency $\mbox{$\omega_{d}$}\propto
k_{\alpha}$ that exceeds the parallel transit rate
$\mbox{$k_{\parallel}$}v_{T}\sim\pi v_{T}/\mbox{$L_{\parallel}$}$. Although
this can always be satisfied by choice of $k_{\alpha}$, the increase of
$k_{\alpha}$ comes at the cost of increasing
$b=\mbox{$k_{\perp}$}^{2}\rho^{2}$ as $L_{\parallel}$ is reduced. Thus, to
estimate the critical gradient, we simply determine the minimum value of $b$
for which the resonance condition is satisfied, i.e. that for which
$\mbox{$\omega_{d}$}\sim\pi v_{T}/\mbox{$L_{\parallel}$}$, yielding
$b_{min}=(\pi
a|\bm{\nabla}\alpha|\mbox{$R_{\mathrm{eff}}$}/\mbox{$L_{\parallel}$})^{2}$,
and

$\frac{a}{L_{T,crit}}=\frac{a}{\mbox{$R_{\mathrm{eff}}$}}F((\pi
a|\bm{\nabla}\alpha|\mbox{$R_{\mathrm{eff}}$}/\mbox{$L_{\parallel}$})^{2})$
(5)

with $F(b)$ defined as above. The quantities $R_{\mathrm{eff}}$ and
$L_{\parallel}$ are in practice computed by identifying and searching the
drift wells along a particular field line as presented in Roberg-Clark  _et
al._ (2022a), while $a|\bm{\nabla}\alpha|$ is evaluated at the center of the
drift well, effectively approximating it as a constant. In the small-$b$
limit, $F(b)$ is dominated by the constant term $2.84$, close to the value of
$2.66$ determined or assumed in other works (Romanelli, 1989; Jenko  _et al._
, 2001; Roberg-Clark  _et al._ , 2022a), at least for the case $\tau=1$. In
the large $b$ limit, which applies to the flux tubes of both W7-X and HSX, we
find, ignoring the small constant $\simeq 0.04$, that the formula effectively
predicts $a/L_{T,crit}\sim a^{2}|\bm{\nabla}\alpha|/\mbox{$L_{\parallel}$}$.
Perhaps unsurprisingly, then, the threshold for toroidal modes in this regime
is dominated by both the gradient of the bi-normal coordinate (linked to
expansion of surfaces as well as local magnetic shear (Roberg-Clark  _et al._
, 2022b)) and the parallel connection length, which can be reduced by simply
increasing the number of field periods in a configuration. Indeed, an
experimental realization of this strategy can already be seen in the $10$
field-period LHD heliotron, whose favorable ITG turbulence properties relative
to W7-X have been demonstrated (Warmer  _et al._ , 2021). More generally,
equation (5) can now be used to rapidly estimate the toroidal ITG threshold on
a given magnetic field line.

Optimization results.– We use the SIMSOPT software framework (Landreman  _et
al._ , 2021) to generate a vacuum stellarator configuration with quasi-helical
symmetry (QHS) and target a toroidal critical gradient of $a/L_{T,crit}=2.90$
at the surface $r/a=0.5$. The stellarator magnetic field is described by a
boundary surface given in the Fourier representation
$R(\vartheta,\phi)=\sum_{m,n}R_{m,n}\cos(m\vartheta-6n\phi),Z(\vartheta,\phi)=\sum_{m,n}Z_{m,n}\sin(m\vartheta-6n\phi)$,
where we have assumed stellarator symmetry and have set the number of field
periods to $n_{fp}=6$. This number was chosen to minimize the parallel
connection length without significantly compromising the QH symmetry quality
($n_{fp}=4$ cases were also studied). Global zero-$\beta$ equilibria are
constructed at each iteration by running the VMEC (Hirshman and Whitson, 1983)
code, which solves the MHD equations using an energy-minimizing principle,
setting the angular resolution to be $m_{pol}=n_{tor}=7$.

Optimization proceeds by treating the above-mentioned Fourier coefficients as
parameters and varying them in order to find a least-squares minimization of
the specified objective function, which reads

$f=f_{QS}+f_{A}+f_{\text{crit}}+f_{\text{well}}$ (6)

where $f_{A}=[\Theta(A-7.50)]^{2}$ is the aspect ratio penalty with $A=R/a$
the aspect ratio output by VMEC, $\Theta(x)$ is defined to be $xH(x)$ with
$H(x)$ the Heaviside step function, $f_{QS}$ is the quasisymmetry residual
defined in (Landreman and Paul, 2022) for QHS with $n_{fp}=6$, and
$f_{crit}=\sum_{\alpha_{j}}[\Theta\left(2.90-a/L_{T,crit}(\alpha_{j})\right)]^{2}$
[eq. (5)] is taken at the surface $r/a=0.5$ and summed over the field lines
$\alpha=[0,\pi/8,\pi/4]$, with each field line extending for $8$ poloidal
turns, in an attempt to cover the surface. The vacuum magnetic well penalty is
$f_{well}=\sum_{r_{k}}[\Theta\left(1+V^{\prime\prime}(r_{k})/0.001\right)]^{2}$
with the surfaces $(r/a)^{2}=[0,0.1,...,0.90]$ targeted and
$V^{\prime\prime}(r)$ the second derivative of the flux surface volume with
radius. We calculate the residual $f_{QS}$ on the surfaces
$(r/a)^{2}=[0,0.1,0.2,0.3,0.4,0.5]$ as in Roberg-Clark  _et al._ (2022b). The
input equilibrium for the optimization was produced by taking a four field
period, aspect ratio 6 QH configuration (the “warm start” file from SIMSOPT),
optimized for increased $|\bm{\nabla}\alpha|$ on the outboard side (see e.g.
Roberg-Clark  _et al._ (2022b); Stroteich  _et al._ (2022)), and then
increasing the number of field periods to 6 in the VMEC input file. The final
optimization of the six field period configuration proceeded in two steps,
with the boundary Fourier coefficients varied up to $m_{pol}=n_{tor}=(6,7)$.

Figure 1: Boundary surface of QSTK with one field period removed, showing
contours of $B$ in color. Figure 2: Properties of QSTK in vacuum. (a)
Rotational transform profile. (b) Neoclassical transport coefficient
$\mbox{$\epsilon$}_{\text{eff}}$ as a function of radius. (c) Toroidal cuts of
flux surfaces within one field period. (d) Alpha particle losses for QSTK,
rescaled to ARIES-CS minor radius and magnetic field, at various radii
[$s=(r/a)^{2}$].

We call the resulting configuration “QSTK” (Quasi-Symmetric Turbulence
Konzept), the boundary surface of which is shown in Fig. 1. QSTK has an aspect
ratio of 7.5, a volume-averaged magnetic well ($0.7\%$), large rotational
transform $>1.60$, a neoclassical transport coefficient
$\mbox{$\epsilon$}_{\text{eff}}<1\%$ (Nemov  _et al._ , 1999) up to roughly
half radius, unusually expanded flux surfaces, and low alpha particle losses
$(<1\%)$ for particles initialized at $(r/a)^{2}=0.1$ when rescaled to an
ARIES-CS-equivalent (Mau  _et al._ , 2008) minor radius and volume averaged
magnetic field strength, using the NEAT code (Jorge, 2022; Albert  _et al._ ,
2020) (Fig. 2). Increased neoclassical transport at the edge [reaching
$\mbox{$\epsilon$}_{\text{eff}}=4\%$, Fig. 2(b)] may in fact be beneficial, as
it can prevent a particle transport barrier from forming, which might
otherwise hinder refueling of plasma in an experimental scenario (Maaßberg
_et al._ , 1999). All flux tubes evaluated for QSTK, using the model equation
(5), are predicted to have $a/L_{T,crit}\geq 2.90$.

Figure 3: Full-surface nonlinear gyrokinetic simulations of ITG turbulence
comparing HSX to QSTK. (a) Log plot of ion heat flux multiplied by the
respective aspect ratio of each configuration. (c) ITG density fluctuations
$\tilde{n}/n_{0}$ in one field period of HSX with $a/L_{T,crit}=3$, plotted in
Boozer toroidal ($\phi$) versus poloidal ($\theta$) angles. (d) Same as in (c)
but for QSTK.

ITG turbulence.– To evaluate the performance of QSTK in vacuum with regard to
ITG turbulence we run full-surface nonlinear electrostatic gyrokinetic
simulations using the GENE code (Jenko  _et al._ , 2000; Xanthopoulos  _et
al._ , 2016) in comparison with the HSX stellarator experiment (Talmadge  _et
al._ , 2008). We assume adiabatic electrons, zero density gradient,
$T_{e}=T_{i}$, and simulate fluctuations on the surface $r/a=0.5$ for the
temperature gradients $a/L_{T}=[1.5,2.0,2.5,3.0]$. In Figs. 3 (a)-(b) we plot
the ion heat fluxes in gyro-Bohm units times the aspect ratio of the
associated configuration, to adjust for the dependence of energy confinement
time on aspect ratio implied by the gyro-Bohm scaling of heat fluxes (a factor
of $4/3$ in favor of QSTK). We find that $Q_{i}A$ is significantly reduced (by
a factor 2-3) in QSTK compared to HSX for the range of gradients studied,
demonstrating the success of the optimization strategy. The fluctuations in
QSTK [fig. 3(d)] are relatively weak, and also less localized on the surface,
compared to HSX [and most optimized stellarators, e.g. W7-X (Xanthopoulos  _et
al._ , 2020)], where the density fluctuations lie within a strip near the
outboard midplane, $\theta\sim 0$ [fig. 3(c)], owing to the more pronounced
“bean-shaped” plane. Evaluation of the toroidal critical gradient (eq. 5) for
HSX indicates a threshold of $a/L_{T,crit}=1.15$ on the outboard field line
$\alpha=0$, while a flux tube selected on the inboard ($\alpha=\pi/4$) has the
much larger predicted threshold of $a/L_{T,crit}=2.50$. This suggests that the
model (eq. 5) still captures the basic physics controlling the intensity of
fluctuations on the surface in HSX, even far above the threshold.

Figure 4: Bootstrap current and MHD stability of QSTK with an applied pressure
profile (see text) at volume-averaged $\beta=1.78\%$. The dashed curves
correspond to the case without boostrap current and the solid curves to the
case with current. (a) Enclosed toroidal bootstrap current. (b) Rotational
transform profile. (c) Mercier stability criterion.

MHD stability and coils.– QSTK, as a result of the $f_{\text{well}}$ objective
in the optimization [eq. 6)], also possesses a vacuum magnetic well and
satisfies $V^{\prime\prime}<0$ at all radial locations (not shown). An
artificial, nearly linear (in $r^{2}$) pressure profile
$T_{i}=T_{e}=1\>\text{keV}*[1-(r/a)^{2}],n_{i}=n_{e}=4.4*10^{20}\>m^{-3}*[1-(r/a)^{10}]$,
corresponding to a volume-averaged $\beta=1.78\%$ (with volume-averaged
$B\simeq 1\>T$), is applied to the configuration. The resulting bootstrap
current profile calculated with DKES (Hirshman  _et al._ , 1986; Van Rij and
Hirshman, 1989; Beidler  _et al._ , 2021) amounting to an integrated current
of roughly $80\>\text{kA}$ [fig. 4(a)]. Both the vacuum and bootstrap
configurations produce rotational transform profiles that avoid crossing the
resonance $m/n=6/4=3/2$ [Figs. 2(a), 4(b)]. The method of Landreman  _et al._
(2022), which relies on the isomorphism between quasisymmetry and axisymmetry,
was found to produce a similar bootstrap current profile for QSTK, although
the results (not shown) were not in as good agreement as in cases with
“precise” quasisymmetry (Landreman and Paul, 2022). Evaluations
(N$\backslash$”uhrenberg and Zille, 1987) of the Mercier criterion (Mercier,
1962; Landreman and Jorge, 2020) indicate that, for the configuration with
pressure and bootstrap current included, the Mercier criterion is satisfied
[fig. 4(b)] and the bootstrap current produces a stabilizing up-shift in
$D_{M}$ (Ku and Boozer, 2011). This result is in contrast to previous QH
configurations with significant ITG turbulence reduction compared to HSX
(Roberg-Clark  _et al._ , 2022b; Bader  _et al._ , 2020), which, unlike HSX,
violated Mercier stability. We also apply the coil optimization features of
SIMSOPT to produce the magnetic field of QSTK, finding that the relative
maximum field error can be reduced below $5\%$ and a relative mean error below
$1\%$ with four unique coils (48 coils in total). The initial coil and MHD
studies show that QSTK has potential for finite-$\beta$ (reactor-relevant)
operation scenarios.

Figure 5: Boundary surface of QSTK surrounded by electromagnetic coils

Discussion and conclusions.– We have not explored trapped electron mode
stability (Proll  _et al._ , 2012; Mackenbach  _et al._ , 2022) as we expect
flat density profiles in ECRH heated plasmas (Beurskens  _et al._ , 2021),
especially at reactor scales. Electron temperature gradient (ETG) turbulence
in QSTK is likely to benefit from the ITG optimization both linearly (from the
isomorphism with ITG modes (Jenko  _et al._ , 2001)) and nonlinearly (from the
short connection length (Plunk  _et al._ , 2019)) with regard to ETG losses.
At high $\beta$ values, turbulence is expected to transition from ITG to
kinetic ballooning mode turbulence (Aleynikova  _et al._ , 2018; Pueschel  _et
al._ , 2008). This physics is delegated to a separate publication, in the
framework of a QSTK reactor study. More generally, the method developed here
for calculating ITG critical gradients can also be applied to optimization of
quasi-axisymmetric and quasi-isodynamic configurations. The design of
turbulence optimized stellarators with improved ion confinement thus appears
to now be within reach.

The authors thank P. Helander, A. Zocco, M. Landreman, and C. Beidler for
helpful conversations. We thank J.F. Lobsien for help with initial coil
optimization. This work was supported by a grant from the Simons Foundation
(No. 560651, G. T. R.-C.). Computing resources at the Cobra cluster at IPP
Garching and the Marconi Cluster were used to perform the simulations. This
work has been carried out within the framework of the EUROfusion Consortium,
funded by the European Union via the Euratom Research and Training Programme
(Grant Agreement No 101052200 — EUROfusion). Views and opinions expressed are
however those of the author(s) only and do not necessarily reflect those of
the European Union or the European Commission. Neither the European Union nor
the European Commission can be held responsible for them.

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