Gyrofluid Turbulence Modelling of the Linear Device VINETA

Abstract

The two-moment version of the three-dimensional gyrofluid code GEM3 has been adapted for the simulation of drift-wave turbulence in the linear device VINETA [1]. In the modified GEM3 code gyrofluid equations for the density and parallel velocity of electrons and ions are solved for the cylindrical annulus. Computations are done for the electrostatic case using an exponential background density profile observed experimentally. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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Contrib. Plasma Phys. 46, No. 7-9, 739–743 (2006) / DOI 10.1002/ctpp.200610071
Gyrofluid Turbulence Modelling of the Linear Device VINETA
G. N. Kervalishvili∗1, R. Kleiber1, R. Schneider1, B. D. Scott2, O. Grulke1, and T. Windisch1
1Max-Planck-Institut f¨ ur Plasmaphysik, EURATOM Association, Wendelsteinstraße 1, D-17491, Greifswald,
Germany
2Max-Planck-Institut f¨ ur Plasmaphysik, EURATOM Association, Boltzmannstraße 2, D-85748, Garching,
Germany
Received 19 October 2005, accepted 9 February 2006
Published online 22 August 2006
Key words Plasma turbulence, VINETA, cylindrical coordinates, gyrofluid simulations.
PACS 52.35.Ra, 52.65.Tt
The two-moment version of the three-dimensional gyrofluid code GEM3 has been adapted for the simulation
of drift-wave turbulence in the linear device VINETA [1]. In the modified GEM3 code gyrofluid equations for
the density and parallel velocity of electrons and ions are solved for the cylindrical annulus. Computations are
done for the electrostatic case using an exponential background density profile observed experimentally.
c ? 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1Introduction
Active control and detailed studies of turbulence is one of the major research topics in fusion plasmas. Numerical
simulations of the turbulence represent one of the most challenging tasks in this field (e.g. [2-5]). Study and
understanding of turbulence under simplified conditions (e.g. for devices with linear magnetic geometry [6,7]) is
a pre-requisite for understandingturbulence in the complicatedgeometry of nuclear fusion devices like tokamaks
or stellarators. VINETA [1] (Versatile Instrument for studies on Nonlinearity, Electromagnetism,Turbulenceand
Applications) is especially suited to investigate plasma waves and instabilities in a linear magnetic geometry. As
an example, observations of drift wave modes in VINETA have been recently reported [8,9].
In order to obtain a better understanding of the drift wave instability and drift wave turbulence in this linear
device, we used a two-moment version of the three-dimensional gyrofluid flux tube code GEM3 [2]. The proper
metrics for the cylinder geometry is implemented. Computations are done for an exponential density profile as
observed experimentally.
The paper is organized as follows: The model is described in section 2. A first application of the modified
code for VINETA parameters and results are presented in section 3. The paper finishes with the conclusions.
2 Model
For the global simulation of the linear device VINETA the two-moment gyrofluid equations for density and
parallel velocity of electrons and ions are used [2]. The original code has been modified for the geometry of a
cylindrical annulus (r,θ,z).
We consider the electrostatic case. The corresponding particle and momentum conservation equations in
gyrofluid formulation are given as follows:
∂˜ ni
∂t
ˆ ?∂˜ u?
∂t
+ (uE· ?) ˜ ni= − ??˜ u?,
(1)
+ ˆ ?(uE· ?) ˜ u?= − ??
?˜φG+ τi˜ ni
?
− C˜J?,
(2)
∗Corresponding author: e-mail: gmk@ipp.mpg.de
c ? 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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740G.N. Kervalishvili et al.: Gyrofluid turbulence modelling
∂˜ ne
∂t
−ˆ µ∂˜ v?
+ (vE· ?) ˜ ne= − ??˜ v?,
(3)
∂t
− ˆ µ(vE· ?) ˜ v?= − ??
?2and ˆ µ = (me/Mi)ˆ ?, and ˜ ni,e, ˜ u?,, ˜ v?representthe ion andelectrondensity andvelocity
C =0.51νe
cs/L⊥ˆ µ.
Ions and electrons are connected by the polarization equation:
?˜φ − ˜ ne
?
− C˜J?.
(4)
Here, ˆ ? =?L?/L⊥
fluctuations, respectively.˜J?= ˜ u?− ˜ v?is the current and the parameter C represents effective collisionality
(5)
Γ1˜ ni+Γ0− 1
τi
˜φ = ˜ ne,
(6)
where τi = Ti/Teand˜φ is the potential which acts on the electrons (their gyroradius is neglected) and˜φG =
Γ1(˜φ) is the gyroaveragedpotential acting on the ions. The operators ??, Γ0and Γ1are defined as follows:
??=
∂
∂z,
Γ0=?1 − ρ2
s?2
⊥
?−1,
Γ1=
?
1 −1
2ρ2
i?2
⊥
?−1
.
(7)
For simulation in the cylindrical annulus geometry the following modifications have been implemented. The
E × B advection operators (given in terms of Poisson brackets)
?˜φ,
now have the following form
?∂f
and the perpendicular part of the Laplacian is given by
?
The details of the numerical implementation of these equations follows the original GEM3 code (see Ref.
[10]).
vE· ? =
?
uE· ? =
?˜φG,
?
(8)
[f, g] =1
r ∂r
∂g
∂θ−∂f
∂θ
∂g
∂r
?
,
(9)
?2
⊥=1
r
∂
∂r
r∂
∂r
?
+1
r2
∂2
∂θ2.
(10)
3Results
In order to test the modified code we have performed test runs. For the simulation presented here we used
parameters typical for VINETA (Table 1). The background density profile in the simulation is chosen to be
exponential (Fig. 1) corresponding to those observed in experiments.
The simulation domain extends for r ∈ [2.3ρs,6.2ρs], θ ∈ [−π,π], z ∈
451.2 cm is the system length. The number of grid points is 32 × 128 × 16 in {r,θ,z} and the time step
is 10−3L⊥/cs(L⊥/cs = 1.5 · 10−5s). The θ direction is periodic, for the r-direction Dirichlet boundary
conditions are used (i.e. fluctuation variables are set to zero) and periodicity in the z-direction is assumed. The
duration of the run is 1000L⊥/cs. The simulation results are shown in figures 2-6.
In figure 2 the temporal behaviour of the squared amplitude of˜φ is plotted. As it was expected the simulation
first gives linear growth and later a phase of saturated turbulence. The transition between both phases takes place
at ≈ 570L⊥/cs. The fluctuation spectra and radial profiles of fluctuation amplitude for different quantities are
shown in Figures 3 and 4. In the linear phase (Fig. 3) we see a strong peaking of the spectrum at m = 6, which
corresponds to the experimental observations m = 1 − 8 [11]. The fluctuation is localized in the region with the
?−0.5L?,0.5L?
?, where L?=
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Contrib. Plasma Phys. 46, No. 7-9 (2006)741
strongest gradient which is near the inner boundary. In Fig. 4 are shown the profiles of the same quantities during
the saturatedturbulencestage, wherethe spectra are flat andthe radial profileextendsnearlyoverthe wholeradial
simulation profile.
Figures 5 and 6 show 2D poloidal profiles for z = 0.031 of density and potential fluctuations for the linear
and the turbulent phase, respectively. According to figure 5 one can easily verify that the linear instability shows
a phase shift between density and potential perturbation, which is typical for the drift-wave instability [12].
Fig. 1 Background plasma density profile used in simulation.
Table 1 Typical plasma parameters for VINETA.
Plasma parameterTypical value
Peak electron density ne
6.12 ·1012cm−3
0.2 eVIon temperature Ti
Electron temperature Te
3.0 eV
Drift-scale ρs
1.43 cm
Ion gyroradius ρi
0.37 cm
Magnetic field B
78.9 mT
Fig. 2 Time trace of the squared amplitude of (˜φ). In order to see both phases the linear and the nonlinear ones, we show the
same picture using logarithmic (left figure) and linear (right figure) scales.
4Conclusions
We have modified the original GEM3 code for cylindrical annulus geometry in order to use it for the simulation
of the linear device VINETA. The modified code has been tested with a qualitative simulation using VINETA
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c ? 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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742G.N. Kervalishvili et al.: Gyrofluid turbulence modelling
parameters. This code will allow us to make quantitative comparisons of simulations with experimental results
from VINETA.
Fig. 3 Averaged fluctuation spectrum and radial profile
of fluctuation amplitudes in the linear phase. n, u and
p represents ˜ n2
(Online colour: www.cpp-journal.org).
e+ ˜ n2
iTi/Te, ˆ ?˜ u2
?and˜φ2, respectively.
Fig. 4 Same as Figure 3 but for the saturated turbulence
phase. (Online colour: www.cpp-journal.org).
Fig. 5 2D profiles of the electron density (upper fig-
ure) and potential (lower figure) fluctuations in the linear
phase. (Online colour: www.cpp-journal.org).
Fig. 6 Same as Figure 5 but for the saturated turbulence
phase. (Online colour: www.cpp-journal.org).
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Contrib. Plasma Phys. 46, No. 7-9 (2006)743
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www.cpp-journal.org
c ? 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim