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Interchange destabilization of collisionless tearing modes
by temperature gradient
S. Nasr, A Smolyakov, P. Migliano, D. Zarzoso, X. Garbet, S. Benkadda
To cite this version:
S. Nasr, A Smolyakov, P. Migliano, D. Zarzoso, X. Garbet, et al.. Interchange destabilization of
col-lisionless tearing modes by temperature gradient. Physics of Plasmas, American Institute of Physics,
2018, 25 (7), pp.074503. �10.1063/1.5030799�. �hal-01893532v2�
gradient
S. Nasr,1A. I. Smolyakov,2P. Migliano,1 D. Zarzoso,1 X. Garbet,3and S. Benkadda1
1)Aix-Marseille Université, CNRS PIIM, UMR 7345 Marseille, France
2)Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon,
Canada
3)CEA, IRFM, F-13108 St. Paul-lez-Durance Cedex, France
Using a uid theory, the stability of collisionless tearing modes in plasmas is analyzed in the presence of an inhomo-geneous magnetic eld, electron temperature and density gradients. It is shown that small scale modes, characterized by a negative stability parameter (∆0 < 0), can be driven unstable due to a combination of the magnetic eld and
electron temperature gradients. The destabilization mechanism is identi ed as of the interchange type similar to that for toroidal Electron Temperature Gradient modes.
Tearing modes13 are instabilities that can occur in
fu-sion plasmas in the presence of non-ideal eects (such as resistivity or inertia). They are responsible for change of the topology of magnetic elds4 and lead to the
for-mation of magnetic islands through magnetic reconnection processes5. Particles can then follow the perturbed eld lines
inside the magnetic islands connecting the inner and outer re-gions. This increases the radial transport from the core and causes a degradation in con nement4and can possibly lead to
disruption6. The tearing mode instability has been observed
in a wide variety of astrophysical and laboratory plasmas7
and is thought to be responsible for the reconnection pro-cesses in the Earth magnetotail8. It has been extensively
studied analytically within the framework of magnetohydro-dynamics (MHD) since the seminal work by Furth et al.2
The tearing mode is linearly excited by the radial gradi-ent of the equilibrium parallel currgradi-ent. The radial domain is separated into an ideal region fully described by ideal MHD equations and a narrow resonant region inside which non-ideal eects take place and the perturbed parallel current in highly localised. The tearing mode stability is commonly parametrized by ∆0, a parameter calculated from the solution
of the tearing mode equations in the ideal outer region. It is de ned as the jump in the logarithmic derivative of the par-allel vector potential Ak across the non-ideal region, inside
which Ak is assumed constant. This assumption is known
as the constant-ψ approximation2, where ψ is the parallel
scalar potential of the magnetic eld. Generally, large scale modes with low poloidal number m, may have ∆0> 0. They
are driven by the free energy in the outer region. The sta-bility of such modes has been calculated in the collisionless limit in the framework of uid theory911 as well as kinetic
theory1214. On the other hand, small scale tearing modes
with high m are not much aected by the large scale current density gradient in the outer region. Such modes are charac-terized by a negative ∆0 ≈ −2k
y = −2m/r < 0 where ky is
the perpendicular wave vector and r the minor radius. There-fore the high m modes with ∆0 > 0would be stable. It was
shown previously that high m tearing modes can be driven linearly unstable1517 by the thermal force eects related to
collisions18,19. Such modes were called microtearing modes.
It is important to note that the thermal force destabilization (due to the energy dependence of the Coulomb collision fre-quency) is related to the current contribution in the inner tearing mode layer but not to the outer ideal region (param-eterized by the value of ∆0).
Current and future tokamaks are characterized by weakly collisional scenarios which makes it important to under-stand the stability of tearing modes in such a limit. Re-cently a large body of gyrokinetic simulations have indicated the presence of an additional, collisionless, destabilization mechanism for small scale micro-tearing modes, likely re-lated to magnetic gradients2022. Alternative or additional
mechanisms may be related to nonlinear excitation of mag-netic islands via non-linear coupling from pressure gradient driven microturbulence23or electromagnetic magnetic utter
uctuations2428.
In this paper we investigate the linear collisionless destabi-lization mechanism due to magnetic eld inhomogeneity and plasma gradients with a uid theory.
Near the tokamak rational surfaces one can introduce a slab-like geometry with a Cartesian coordinates system (ˆx, ˆy, ˆz). The magnetic eld can then be represented in the form B = B0(x)bz + ∇ψ ×bz (1) ψ =1 2B0 x2 Ls + eAk(x, y, t) (2)
where B0 is the equilibrium magnetic eld along the
z-direction, Aek is the z-component of the perturbed magnetic
vector potential, ψ is the auxiliary vector potential intro-duced to describe the magnetic shear eect for the helical perturbations, kyy = m (θ − ζ/q) , Ls= qR/s, q is the safety
factor, s = (r/q)dq/dr is the magnetic shear, R is the toka-mak major radius and x = r −rsthe distance to the resonant
surface position rs. With Eq. 2, the total parallel gradient
operator along the magnetic eld ∇k= (B0+ ˜B) · ∇/B can
2 parts as follows: ∇k0= B0· ∇ B0 = ikk(x) = i kyx Ls (3) e ∇k= e B · ∇ B0 = ikyAek B0 ∂ ∂x (4)
The electron dynamics is described by standard uid equa-tions in the absence of collisions and neglecting the electro-static potential. Similarly to Ref.13, we neglect the contribu-tion to the parallel current due to the electrostatic potential which is valid for small scale magnetic islands. The electron continuity equation takes the form
∂ne
∂t + ∇⊥· (nev⊥e) + ∇k(neVke) = 0 (5) The parallel electron velocity Vke is found from the electron
parallel momentum balance equation, given by mene
∂Vke
∂t + b · (V · ∇V )
= eneEk− ∇kpe (6)
where me is the electron mass, ne the electron density,
b = B/B0 is the unit vector in the equilibrium magnetic
eld direction, Ek = −∂Ak/c∂t is the parallel electric eld
and peis the electron pressure. In the low frequency regime,
i.e. ω ωce where ωce = mec/eB > 0is the electron
cy-clotron frequency and neglecting the electrostatic potential, the electron perpendicular velocity is the diamagnetic drift. To the lowest order, it is given by
v?e= −
c en0e
b × ∇pe
B0 (7)
The second term in the continuity equation (5) can be written
∇⊥(nev?e) =
1 T0e
vDe· ∇pe (8)
where vDeis the electron magnetic drift velocity given by
vDe= −
2cT0e
eB0
b × ∇ ln B0 (9)
Linearizing Eqs. (5) and (6) gives29
∂ ˜ne ∂t +n0evDe· ∇pee p0e + ∇k0(n0eVek) = 0 (10) men0e " ∂ eVke ∂t − 4cp0e n0eeB0 b × ∇ ln B · ∇ eVke # (11) = −en0eEek− ∇k0pee− e∇kp0e
where p0e= n0eT0eand pee=neeT0e+ n0eTeeare respectively the equilibrium and perturbed pressure. Unless stated oth-erwise, all quantities in this text refer to electrons. The sub-script e to designate electrons will be dropped in the follow-ing.
In the simplest case of neglecting magnetic eld gradient and temperature perturbations these equations read
− ωen + n0kkVek = 0, (12)
men0ω eVk =
en0
c ω eAk+ kkTn,e (13) giving the following parallel electron response in terms of the perturbed magnetic potential
e Vk= − e cme ω2 k2 kv 2 th− ω2 e Ak. (14)
The limit in Eq.(14) reproduces the collisionless tearing mode instability studied by the kinetic theory in Refs.12and13and in the uid theory in Refs.9,10, and 30. The reconnection here is driven by the parallel electron current due to inertia balanced by the inductive electric eld and electron pressure perturbation. The parallel electron current is accompanied by electron density perturbation. When the gradient of the magnetic eld is included, there is an additional contribution to the parallel current which comes from the compressibility of the perpendicular electron diamagnetic current in equation (10).
The pressure perturbation (the second term in Eq. (10)) needs a closure for the temperature evolution. In general, one needs a closure which is uniformly valid in the whole range of the kkvT e/ω parameter, which changes from zero
at the rational surface to some nite value at the inertial layer width. Similarly to earlier work3133 we adopt here a
constant temperature model along the perturbed magnetic surface giving
∇k0T + ee ∇kT0= 0 (15)
This closure is appropriate for small scale magnetic islands as in Ref. 16for which the condition ω kk(w)vT e is
sat-is ed, where w is the magnetic island width or the inertial layer width in the linear case determined by the electron skin depth parameter9,34. In a more accurate model, the closure
of the Hammett-Perkins type35valid for ω ≈ k
kvT e, could be
used to account for the Landau damping eects. The latter however is outside of the scope of our work here. We note also that the condition of the electron temperature atten-ing across the magnetic islands seems well satis ed for small scale islands as shown in gyrokinetic simulations36.
The electron continuity and parallel momentum balance, Eqs. (10) and (11) together with the closure on the temper-ature (15) read −(ω − ωD)en + ωDω?T en0 ckk e Ak+ n0kkVek = 0 (16) men0(ω − 2ωD) eVk = en0 c (ω − ω?n) eAk+ kkT0en (17) kkT +e ky B0 e Ak∂T0 ∂x = 0 (18)
where plasma and magnetic eld gradients enter the system of equations through ω?n,T and ωD, respectively given by
ω?T ,n= − cky eB0 T0 ∂ ln T0, n0 ∂x = −kyρe vth LT ,n (19) ωD= − cky eB0 T0 ∂ ln B0 ∂x = −2kyρe vth LB (20)
Here ρe= mcvth/eB0 is the thermal electron Larmor radius,
vth = pT0/me is the electron thermal velocity, LB,n,T is
the gradient length scale of the magnetic eld, density and temperature respectively, de ned by LG = −(1/G)∂G/∂x,
where G = (n, T, B).
The above system of uid equations is closed with Am-père's law. Projected onto the direction parallel to the equi-librium magnetic eld, it takes the form
− ∇2⊥Aek= 4π
c Jek. (21) Here Jek = −en0Vek is the perturbed electron parallel
cur-rent which contains the destabilizing eect of the collision-less tearing mode. The perturbed electron parallel velocity is calculated by coupling the system of Eqs. (16), (17) and (18)
e Vk= − e cme (ω − ω?n)(ω − ωD) + ωDω∗T k2 kvth2 − (ω − 2ωD)(ω − ωD) e Ak (22) For small scale islands ∂/∂x ky, Eq. (21) is written
∂2Aek ∂x2 = − ω2 pe c2 (ω − ω?n)(ω − ωD) + ωDω∗T k2 kv 2 th− (ω − ωD) (ω − 2ωD) e Ak (23)
where ωpe=p4πe2n0/meis the electron plasma frequency.
The tearing mode dispersion relation is obtained by in-tegrating Ampère's law across the resonant layer. We em-ploy the constant-ψ approximation which consists in assum-ing that in the non-ideal region the perturbed parallel vector potential does not vary signi cantly. The collisionless tearing mode dispersion relation then reads
∆0+ω 2 pe c2 Z ∞ −∞ dx (ω − ω?n)(ω − ωD) + ωDω?T k02 kv 2 thx2− (ω − ωD) (ω − 2ωD) = 0 (24) Here k0
k = kys/qR and ∆0 is the tearing mode stability
parameter2 that matches the solutions in the ideal outer
re-gion and non-ideal inner rere-gion. It is de ned as the jump in the logarithmic derivative ofAek, solution in the outer region,
across the resonant layer, i.e. ∆0= lim ε→0 ∂ ln eAk ∂x rs+ε rs−ε (25) The x-integral in Eq. (24) is converging for complex ω. Us-ing the properties of the complex logarithm37one can obtain
0 1 2 3 4 5 6 1/L T (R) 0 1 2 3 4 5 6 7 8 ( te ) 10-3 0 0.01 0.02 0.03 -5 0 5 10 15 20 10-5 m=20, <0 m=10, <0 m=2, >0
(a) Growth rate(γ)
0 1 2 3 4 5 6 1/L T (R) -6 -5 -4 -3 -2 -1 r ( te ) 10-3 m=20 m=10 m=2 (b) Frequency(ωr)
Figure 1: Normalized (a) growth rate γ, and (b) real frequency ωras a function of the temperature gradient scale
length 1/LT; m = 10, ∆0= −1.1360, kyρe= 0.0011;
m = 20, ∆0 = −2.2718, kyρe= 0.0023; m = 2, ∆0 = 0.5,
kyρe= 2.2891 × 10−4. The density and magnetic eld
gradient scale lengths are xed to resp. 1/Ln= 2and
1/LB= 1. the expression Z ∞ −∞ dx k02 k x2vth2 − Ω2 = iπ k 0 kvth Ω for =(Ω) > 0 − iπ k 0 kvth Ω for =(Ω) < 0 (26) where we have introduced the notation Ω2 ≡
(ω − ωD) (ω − 2ωD).
The expression (26) can be better understood by integrat-ing in the complex plane and usintegrat-ing the residue theorem. For this purpose, we extend the integration domain over a given real interval [x1, x2]to the complex plane
Z x2 x1 dx x2− Ω2 + Z Γ dz z2− Ω2 = Z C dz z2− Ω2 (27)
4 where we have de ned Γ as a half-circle that lies in the upper
(resp. lower) complex half-plane for = (Ω) > 0 (resp. < 0) and C is the closed contour C = [x1, x2] ∪ Γ. The integral
over the closed contour in both the upper and lower halves of the complex plane can be calculated by the residue theorem. Therefore, we decompose the integrand as follows
Z C dz z2− Ω2 = 1 2 1 Ω Z C dz 1 z − Ω− 1 z + Ω (28) Depending on the sign of the imaginary part of Ω the rst or the second term on the right hand side de nes the value of the integral, 1 2 1 Ω Z C dz 1 z − Ω− 1 z + Ω = 1 2Ω2πi, for =(Ω) > 0 (29) 1 2 1 Ω Z C dz 1 z − Ω− 1 z + Ω = − 1
2Ω2πi, for for =(Ω) < 0. (30) Finally, taking the limit x1→ −∞and x2→ ∞and noticing
that within this limit the integral over Γ vanishes, we obtain the expression (26). The dispersion relation in Eq. (24) is therefore expressed as ∆0±ω 2 pe c2 iπ k 0 kvT e (ω − ω∗n) (ω − ωD) + ωDω∗T p(ω − ωD) (ω − 2ωD) = 0. (31) The ± sign has to be chosen according to the condition for =(Ω)given in Eq. (26). When curvature eects are neglected, i.e. ωD= 0, an unstable solution is found only for ∆0> 0
∆0= ω 2 pe c2 π k 0 kvth γ ωr= ω?n (32)
Such a solution has been found for the collisionless tearing mode using a uid model in Refs. 10, 9, and 30. The dis-persion relation in Eq. (32) obtained from uid theory is not fully identical to the kinetic result12,13 which is
∆0= ω 2 pe c2 2√π k 0 kvth γ ωr= ω?n(1 + ηe/2) (33)
where ηe= Ln/LT. The kinetic 2
√
πand uid π coe cients are quite close. The dierence in the real part of the frequency, ωr= ω?n(1 + ηe/2)in kinetic theory and ω = ω?n
in uid theory, is due to our approximation in Eq. (15) on the closure which is not uniformly valid for all kk(x).
It is interesting to note that the solution in Eq. (31) for ωD= 0is in fact ∆0= ω 2 pe c2 π k 0 kvth |γ| . (34)
Thus it means that there are two solutions; with positive and negative gamma, for ∆0 > 0, and no solutions exist for
∆0 < 0.
In the general case with ωD 6= 0 and ω∗T 6= 0, there are
two solutions which may have the additional destabilization mechanism due to the magnetic drift gradient and the tem-perature gradient, provided ω?TωD > 0. The modes with
∆0 > 0can be driven by the free energy from the outer re-gion. However, the high m modes have negative ∆0, but they
can be driven by temperature and magnetic eld gradient ef-fects. This destabilizing mechanism can be easily illustrated for the marginal stability case ∆0 = 0.In this case the
solu-tion of Eq. (31) is found as
ω = ω∗n+ ωD 2 ± s (ω∗n− ωD) 2 4 − ω∗TωD, (35) which has the instability for su ciently large values of ω?TωD> 0. Thus, the additional destabilization mechanism
can be identi ed of the toroidal ETG (interchange) type33
rather than the tearing type due to ∆0 > 0. Furthermore,
when ∆0 is negative and small, the approximate solution of
Eq. (31) has the same tendency as Eq. (35), as follows from numerical solutions.
When ωD is nite Eq. (31) is solved numerically by
look-ing for unstable solutions, i.e. solutions with =(ω) = γ > 0. We nd the zeros of the left hand-side of Eq. (31) by plot-ting the inverse of its modulus in the complex (ωr, γ) plane
and looking for its poles as outlines in Refs. 3840. We nd that for nite ωDthe temperature gradient generally
destabi-lizes the mode while the density gradient modi es the mode growth rate and becomes stabilizing for stronger gradients around 1/Ln ≈ 6, as can be seen from the general tendency
of the curves in Fig. 2a. Here, frequencies are normalized to the electron transit frequency ωt= vth/R0, lengths are
nor-malized to the major radius R0, ∆0 is normalized to d2 e/ρe
where de= c/ωpe is the electron skin depth. Figs. 1 and 2
show a scan of the normalized growth rate (a) and frequency (b) with the temperature and density gradient scale lengths respectively, at the low eld side (i.e. LB, Ln,T > 0). The
dierent lines correspond to dierent modes, at the same res-onant surface rs = 0.22such that q(rs) = m/n = 2. Fixing
the values of the shear to s = 1.44 and the major radius to R0 = 3m, one can determine the values of kyρe and kk.
The magnetic drift and the diamagnetic frequencies are de-termined by choosing the values of L−1
B , L −1 n and L
−1 T
respec-tively. The choice of 1/Ln,T ,B> 0implies a negative gradient
for the respective quantities. The destabilizing mechanism occurs when the destabilization condition ωDωT ?> 0holds.
For the chosen numerical values, this corresponds to the low eld side position where pro le gradients and the magnetic eld gradient are of the same sign.
The numerical solution shows that, in agreement with Eq. (35), unstable solutions for high-m modes are present only if the combined eect of magnetic eld inhomogeneity
0 1 2 3 4 5 6 1/L n (R) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
(
te)
10-3 m=20 m=10 m=2(a) Growth rate (γ)
0 1 2 3 4 5 6 1/L n(R) -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 r ( te ) 10-3 m=2 m=10 m=20 (b) Frequency (ωr)
Figure 2: Normalized (a) growth rate γ; and (b) real frequency ωr, as a function of the normalized density
gradient scale length 1/Ln; m = 10, ∆0= −1.1360,
kyρe= 0.0011 ; m = 20, ∆0 = −2.2718, kyρe= 0.0023;
m = 2, ∆0= 0.5, kyρe= 2.2891 × 10−4. The temperature
and magnetic eld gradient scale lengths are xed to resp. 1/LT = 2and 1/LB = 1.
(ωD) and plasma temperature gradient (ω?T) is considered.
This can be seen in Fig. 1awhere at null temperature gra-dient the growth rate of both high-m modes (dashed orange and dotted purple lines) is equal to zero. Figs. 1b and 2b
show that the mode's frequency varies very little against tem-perature gradient variations and is linear in 1/Ln, which is
consistent with Eqs. (32) and (35). Notice that for 1/Ln = 0
the mode's frequency is not null, contrary to the case with-out magnetic drift (Eq. 32). This shows that the frequency eect is indeed a toroidal one, as was observed in Ref. 41. In addition, the instability has an upper threshold in the den-sity gradient at 1/Ln ≈ 6after which the mode is stable with
=(ω) = 0 and oscillates only with a real frequency. Note that the growth rate and real frequency scale linearly with kyρe. This inde nite increase would be terminated by the
-nite electron Larmor radius eects which are neglected here.
Furthermore, there is a threshold for the instability for low values of ω∗TωDsuch that
(ω∗n− ωD) 2
4 > ω∗TωD, (36) when the mode becomes stable. For ω∗TωD → 0, only the
mode with ∆0 > 0remains unstable as is shown on the insert
in Fig. 1a.
It is important to note that our uid model results in the non-analytic dispersion relation in Eq. (34). In fact, the same result is obtained in kinetic derivations of Refs. 12 and13. In the latter case, the non-analytic nature of the dispersion relation is not apparent since the result was formulated in terms of the Plasma Dispersion Function which was de ned for =(ω) > 0 and analytically continued into the =(ω) < 0 plane. In the kinetic theory, the non-analytic nature of the dispersion relation is related to the presence of the singular (continuous spectrum) eigen-functions which stipulates the use of the Landau pole rule (or Laplace transform for the ini-tial value problem). In uid theory, one can introduce some dissipation which would also regularize the singular eigen-functions. We should note however that our main result of the interchange destabilization for ωDω∗T > 0is not aected
as the ∆0
= 0 limit given by Eq. (35) shows.
Some restrictions that limit direct application of our theory to the realistic tokamak geometry are essential to note. Our theory assumes the local (constant) value of ωD. In the
toka-mak geometry, the main part of the ωD is oscillating in the
poloidal direction (or equivalently along the magnetic eld line) and the mean (constant part) only appears in the next order of the plasma pressure β parameter. These oscillations have been taken into account in Refs. 4244in the framework of resistive MHD where an average curvature is calculated to the ε2 order after toroidal coupling of the central (low-m,
low-n) mode to its side bands. Its eect was found to be stabilizing for the large scale tearing mode.
Our local ωD is the same approximation as made in the
uid theory of ITG/ETG modes4547 which shows good
agreement45,48 with results from fully nonlocal kinetic
sim-ulations. One can think that the local approximation may be acceptable for high m modes, for which the mode spans a narrow poloidal region and any nonlocal corrections come as as the next order toroidal eects49. In general more accurate
account of the structure of the magnetic eld gradient would be required.
Another two assumptions of our model, namely, the neglect of the electrostatic potential and the simpli ed temperature response model in Eq. 15will also have to be abandoned in a more accurate model. The integral formulation in Ref. 50
which does take into account the eects of the electrostatic potential predicts stabilization with increasing values of den-sity gradient (albeit no magnetic eld gradient was included there). The isothermal approximation for electrons should be improved with a more complete energy equation as in Ref. 51. Including all the terms of the pressure equation
6 of the Hammett-Perkins35 closure would give a more
accu-rate description to the model. Such closure could be added to the gyro uid models of the magnetic reconnection, e.g. those studied in Refs. 52 and 53. It would be interesting to investigate whether collisionless closures might also lead to the regularization of the singular eigen-functions. These questions and improvements of our model are left for future work.
In this paper, a uid model has been used for the descrip-tion of linear collisionless tearing modes taking into account magnetic eld, plasma density and temperature gradients. The linear dispersion relation has been derived using the constant-ψ approximation. For a uniform magnetic eld this dispersion equation reduces to the previous uid and kinetic results that predict the instability of ∆0 > 0 modes only.
When the magnetic eld gradient is included, our dispersion equation predicts an additional destabilization mechanism linearly driven by the combination of the nite temperature and magnetic eld gradients persisting even for small scale tearing modes with ∆0 < 0. Eects of large density
gradi-ents have been found to be stabilizing, which is in agreement with general tendencies of earlier results9,14. The instability
mechanism identi ed in our work requires the condition of the interchange type: ωDω∗T > 0, which is similar to the
instability of ETG modes. The ETG instability was also suggested as a source of small scale magnetic islands due to the nonlinear energy transfer54. A somewhat similar idea of
the "mesoscopic" reconnection was also proposed in Ref.32. Furthermore, it was suggested that the unstable ETG type modes can be responsible for the nonlinear excitation of the linearly stable microtearing modes55. The de ning feature of
stable microtearing modes was their independence of the elec-trostatic potential φ (which can be omitted for such modes) contrary to the ETG modes which involve essential pertur-bations of φ. In our work we show that in neglect of φ, the tearing type perturbations can be eectively destabilized by the interchange type mechanism related to the temperature and magnetic eld gradients. We conjecture here that the in-terchange destabilization identi ed in our paper may be oper-ative in numerical simulations that demonstrate collisionless destabilization of micro-tearing modes. We note that linear drift-kinetic and gyro-kinetic simulations with local magnetic eld gradient in Ref. 22also show the growth rate of nega-tive ∆0 tearing mode increasing with the value of the local
magnetic eld gradient in very-low collisionality regimes. Al-ternative explanations, based on the nonlinear coupling to unstable modes2328,55,56 are also plausible.
Useful discussions with Dr. Y. Camenen and Dr. O. Agullo are gratefully acknowledged.
This work has been carried out thanks to the support of the A*MIDEX project (n◦ ANR-11-IDEX-0001-02) funded by
the Investissements d'Avenir French Government program, managed by the French National Research Agency (ANR).
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