Instabilities and turbulence in stellarators from the perspective of global codes E. S anchez1 A. Ba n on Navarro2 F. Wilms2 M. Borchardt3 R.

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Instabilities and turbulence in stellarators from the

perspective of global codes

E. S´anchez1, A. Ba˜n´on Navarro2, F. Wilms2, M. Borchardt3, R.

Kleiber3, F. Jenko2

1Laboratorio Nacional de Fusi´on - CIEMAT. Avda. Complutense 40, 28040, Madrid,

Spain.

2Max-Planck Insitut f¨ur Plasmaphysik, Garching, Germany.

3Max-Planck Insitut f¨ur Plasmaphysik, Greifswald, Germany.

E-mail: edi.sanchez@ciemat.es

Abstract. In this work, a comparison of the global gyrokinetic codes EUTERPE

and GENE-3D in stellarator conﬁgurations of LHD and W7-X is carried out. In linear

simulations with adiabatic electrons, excellent agreement is found in the mode numbers,

growth rate and frequency, mode structure, and spatial localization of the most

unstable mode in LHD. In W7-X, the dependence of the growth rate and frequency

with the mode number is well reproduced by both codes. The codes are also compared

in linear simulations with kinetic ions and electrons in W7-X using model proﬁles,

and reasonable agreement is found in the wavenumber of the most unstable modes.

A stabilization of small-scale modes in kinetic-electron simulations with respect to

the adiabatic-electron case is consistently found in both codes. Nonlinear simulations

using adiabatic electrons and model proﬁles are also studied and the heat ﬂuxes are

compared. Very good agreement is found in the turbulent ion heat ﬂuxes in both LHD

and W7-X. Two problems that cannot be properly accounted for in local ﬂux tube codes

are studied: the localization of instabilities and turbulence over the ﬂux surface and

the inﬂuence of a background long-wavelength electric ﬁeld. Good agreement between

codes is found with respect to the spatial localization of instabilities and turbulence

over the ﬂux surface. The localization of saturated turbulence is found in both codes

to be much smaller than that of the linear instabilities and smaller than previously

reported in full-surface radially-local simulations. The inﬂuence of the electric ﬁeld on

the localization is also found to be smaller in the developed turbulent state that in

the linear phase, and smaller than in previous works. A stabilizing eﬀect of a constant

electric ﬁeld on the linearly unstable modes is found in both codes. A moderate

reduction of turbulent transport by the radial electric ﬁeld, with small dependence on

the sign of the electric ﬁeld, is also found.

1. Introduction

Many codes have been developed based on the gyrokinetic formalism [1, 2, 4, 5] and have

been used for the simulation of plasma turbulence in toroidal devices. Simulation codes

can result extremely useful for the understanding of plasma turbulence, as it is a complex

arXiv:2210.05467v1 [physics.plasm-ph] 11 Oct 2022

Instabilities and turbulence in stellarators from the perspective of global codes 2

problem with no possible fully analytic treatment, except in simpliﬁed or particular

cases. The physical problem can be treated numerically by using diﬀerent numerical

implementations, each of them having their own weakness and strengths. Besides, these

kinds of numerical models imply some approximations and simpliﬁcations, which makes

it very important to verify the numerical codes by means of comparisons to analytical

models when possible, or against other codes with diﬀerent numerical implementations.

The axial symmetry in tokamaks makes it possible to use of the local, so-called ﬂux

tube approximation [6], consisting of the simulation of a physical domain surrounding

a magnetic ﬁeld line and following the line one poloidal turn, which allows reducing

signiﬁcantly the computational resources required for turbulence simulations with these

codes with respect to the simulation of either the full domain or the full ﬂux surface.

In stellarators, the situation is quite diﬀerent and diﬀerent ﬂux tubes lying over the

same ﬂux surface cannot be considered as equivalent. Although ﬁrst applications of

gyrokinetic codes to stellarators were based on the direct adaptation of the ﬂux tube

paradigm for stellarators [7, 8, 9] and the use of ﬂux tubes with just one poloidal turn

in length, this is not satisfactory in stellarators. The minimum computational domain

required for stellarators was addressed in recent works studying two linear problems

[10, 11] and it was demonstrated that, at ﬁxed radial wavenumber, short ﬂux tubes on the

same ﬂux surface provide diﬀerent results, in general, which only converge to each other

when the ﬂux tube length is suﬃciently increased. In [12] it was shown that the heat ﬂux

computed in full-surface simulations can be signiﬁcantly larger than that obtained in

full global simulations, suggesting that global simulations are required in stellarators, in

general. Furthermore, global codes are, in principle, required in stellarator turbulence

simulations to properly account for the inﬂuence of the long-wavelength electric ﬁeld

and the global density and temperature proﬁles.

Several global gyrokinetic codes have been speciﬁcally designed for stellarators or

adapted from tokamak codes for thee dimensional geometries, and there is presently a

number of them available for stellarators: XGC-S [13], GTC [14], GT5D [15], EUTERPE

[16, 17], GENE-3D [18] and GKNET [19]. While in tokamaks gyrokinetic simulations

have a reasonable degree of maturity and there is a set of codes cross-benchmarked

and validated, the number of available codes able to target the stellarator geometry

has been limited until recently, and they still lack of veriﬁcation and validation, in

general. In this work, we present the results of a eﬀort carried out during the last

months for the cross-veriﬁcation of the codes EUTERPE [16, 17] and GENE-3D [18].

Both are global codes designed speciﬁcally for stellarator geometry. They are based on

diﬀerent numerical models but they also share common features that permit an in-detail

comparison from which both codes can beneﬁt. In addition to the cross veriﬁcation

of these codes, we address two problems that are considered relevant for turbulent

transport in stellarators and for which the global codes are particularly suited, namely

the localization of instabilities and turbulence over the ﬂux surface and the inﬂuence of

the electric ﬁeld on the instabilities and turbulence.

The structure of the work is as follows. In Section 2, we brieﬂy describe the

Instabilities and turbulence in stellarators from the perspective of global codes 3

codes under comparison, their numerical models and capabilities and also the stellarator

magnetic conﬁgurations that we use in this work. In Section 3, we compare both codes

in linear simulations of Ion-Temperature-Gradient-driven modes (ITGs). The codes are

compared in a nonlinear setting in Section 4. In Section 5, we study the localization of

instabilities and turbulence over the ﬂux surface with both codes. Section 6 is devoted

to study the inﬂuence of a radial electric ﬁeld on the localization of instabilities and

turbulence, the stabilization of linear modes and its eﬀect on the turbulent transport.

Finally, in Section 7, we draw some conclusions.

2. The magnetic conﬁgurations and codes used

2.1. Magnetic conﬁgurations

In this work, we compare simulations carried out in two diﬀerent stellarator magnetic

conﬁgurations: a standard conﬁguration of LHD, with major radius, R= 3.7m

and minor radius, a= 0.6m, and a standard conﬁguration (Ref 168) of W7-X with

R= 5.5m,a= 0.52 m. The magnetic ﬁeld strength over the last closed ﬂux surface of

both conﬁgurations is shown in Figure 1. Figure 2 shows the radial proﬁles of rotational

transform in these conﬁgurations.

Figure 1. Magnetic ﬁeld strength on the last closed ﬂux surface for the two magnetic

conﬁgurations used in this work: a standard conﬁguration of LHD (left) and a standard

conﬁguration of W7-X (right).

LHD has a small rotational transform ι= 0.3 at the center that increases toward

the edge up to ι= 1.25, thus having a signiﬁcant magnetic shear ˆs= 0.08 −0.17,

while for W7-X the rotational transform is in the range 0.86 < ι < 0.97 with very

low magnetic shear, ˆs= 0.0015, in all radii (see Figure 2). In both conﬁgurations we

consider a vacuum equilibrium which is calculated with the code VMEC.

2.2. The codes EUTERPE and GENE-3D

In this section we brieﬂy describe the codes EUTERPE [16, 17] and GENE-3D [18, 26],

their numerical models and their capabilities, and highlight the diﬀerences that are

Instabilities and turbulence in stellarators from the perspective of global codes 4

Figure 2. Radial proﬁles of rotational transform (ι) for the standard magnetic

conﬁgurations of LHD and W7-X used in this work.

relevant for this comparison. Both are global and δf codes allowing electrostatic or

electromagnetic global simulations in stellarators.

In this work, the gyrokinetic equation in the collsionless limit,

∂Fσ

∂t +∂~

∂t ∇Fσ+∂vk

∂t

∂Fσ

∂vk

= 0,(1)

for the kinetic species σis solved together with the ﬁeld equations (quasineutrality and

Amp`ere’s Law) in both codes.

Aδf splitting of the distribution function, F=FMσ +δfσ(t), is used, with FMσ

the Maxwellian, which is assumed as the equilibrium distribution function. With this

splitting, an equation for the δfσcan be obtained:

∂δfσ

∂t =−∂~

∂t ∇δfσ−∂vk

∂t

∂δfσ

∂vk−∂~

∂t ∇Fσ−∂vk

∂t

∂Fσ

∂vk

.(2)

EUTERPE uses a particle-in-cell scheme and the equation for δf is solved by

discretizing the distribution function using quasi-particles or markers. The equation is

solved along the characteristic curves (deﬁned by ∂~

∂t and ∂vk

∂t ), which are the equations

of motion of the markers. PEST (s, θ, φ) magnetic coordinates [20] are used for the

description of the ﬁelds, with s= Ψ/Ψ0, Ψ the toroidal ﬂux, Ψ0the toroidal ﬂux at

the last closed ﬂux surface, φthe toroidal angle, and θthe poloidal angle. Cylindrical

coordinates are used for the markers. Several diﬀerent formulations (pk,vko mixed

variable scheme) are presently implemented in the code [21, 22]. In this work, the pk

formulation is used and the equations solved are those used in references [23, 24], to

which we refer the reader for details.

GENE-3D solves the GK Equation (2) in a ﬁxed grid in the ﬁve-dimensional

phase space (plus time), consisting of two velocity coordinates vk(velocity parallel to

the magnetic ﬁeld) and µ(magnetic moment), and the three magnetic ﬁeld-aligned

Instabilities and turbulence in stellarators from the perspective of global codes 5

coordinates (x, y, z), with xthe radial coordinate, deﬁned as x=a√s,ythe coordinate

along the binormal direction and zthe coordinate along the ﬁeld line.

In this work, the electrostatic equations are always solved in EUTERPE, both

in adiabatic-electron and kinetic-electron simulations. Then, the gyrokinetic equation

(2) is solved together with the quasineutrality equation. In the case of GENE-3D,

the electromagnetic version of equation 2 is solved together with the quasineutrality,

Amp`ere’s law and an equation for the inductive electric ﬁeld (Ohm’s law) in the kinetic-

electron simulations with GENE-3D presented in Section 3.4 while the electrostatic

version of the equation 2 is solved in the simulations with adiabatic electrons.

EUTERPE can simulate the entire conﬁned plasma or a reduced volume covering a

radial annulus with inner and outer radial boundaries other than the magnetic axis and

the last closed ﬂux surface [16, 17]. For the ﬁeld solver, natural boundary conditions

are set at the inner boundary for the full volume simulations, while Dirichlet conditions

are used in the simulation considering an annulus. Dirichlet conditions are always used

at the outer boundary. Density and temperature proﬁles, depending only on the radial

coordinate, are used as input for both kinds of simulations. GENE-3D can be used in

either the ﬂux tube, the full ﬂux surface or the radially global domains [18, 12, 26]. In

this work, only simulations in the global domain will be used and Dirichlet conditions

are used at both the inner and the outer boundaries.

Both GENE-3D and EUTERPE use a real space representation of the ﬁelds

(in their internal coordinates). In EUTERPE, the electrostatic potential is Fourier

transformed on each ﬂux surface, and ﬁltering the potential in Fourier space (with a

variety of diﬀerent ﬁlters) is possible. The code allows extracting a phase factor in linear

simulations, thus allowing for a signiﬁcant reduction in the computational resources [16].

Two diﬀerent implementations of the quasi-neutrality equation and the ﬁeld solver are

available, one assuming a long wavelength approximation and another one using a Pad´e

approximant, which allows resolving modes with arbitrarily large wavenumber (see [27]

for details on these approximations). The Pad´e approximation [27] is used in this work,

which allows resolving modes k⊥ρ > 1. Keeping these modes in the simulation is required

because in W7-X the spectrum of linearly unstable ITG modes shows peak growth rates

for k⊥ρ1 [11].

The equilibrium magnetic ﬁeld is obtained from a magneto-hydrodynamic

equilibrium calculation carried out with the code VMEC [28], and selected magnetic

quantities are mapped for its use in the gyrokinetic codes with the intermediate programs

VM2MAG and GVEC for EUTERPE and GENE-3D, respectively. In GENE-3D the

resolution used in the equilibrium magnetic ﬁeld is coupled to the resolution in the

electrostatic potential, while in EUTERPE, both resolutions are independent.

Several tools are implemented in EUTERPE for the control of numerical noise and

stabilization of the proﬁles in gradient-driven nonlinear simulations [24]. In this work, a

heating source term is used to keep the temperature proﬁle stable during the simulation

and the weight smoothing is used for improving the signal to noise ratio. Particle and

heat source terms as described in [29] are used in GENE-3D for sustaining the density

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InstabilitiesandturbulenceinstellaratorsfromtheperspectiveofglobalcodesE.Sanchez1,A.Ba~nonNavarro2,F.Wilms2,M.Borchardt3,R.Kleiber3,F.Jenko21LaboratorioNacionaldeFusion-CIEMAT.Avda.Complutense40,28040,Madrid,Spain.2Max-PlanckInsitutfurPlasmaphysik,Garching,Germany.3Max-PlanckInsitutfurPlasmaphy...

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Instabilities and turbulence in stellarators from the perspective of global codes E. S anchez1 A. Ba n on Navarro2 F. Wilms2 M. Borchardt3 R.

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