12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

TURBULENT TRANSITIONS IN AXISYMMETRIC FLOW

Wesley Agoua
Wouter J.T. Bos
LMFALMFA
Ecole Centrale de Lyon, CNRS, Univ Lyon Ecole Centrale de Lyon, CNRS, Univ Lyon
Ecully, France wesley.agoua@ec-lyon.fr
Ecully, France wouter.bos@ec-lyon.fr

Benjamin Favier
Jorge Morales
IRPHE
IRFM
Aix Marseille Univ, CNRS, Centrale Marseille
CEAMarseille, France
Saint Paul-Lez-Durance, France
favier@irphe.univ-mrs.fr
jorge.morales2@cea.fr

ABSTRACT

the ratio of the toroidal and poloidal components of the forcing.

We investigate a transition between two turbulent states inan axisymmetric flow. These are two-dimensional states with
two and three velocity components and will be indicated by
2D2C and 2D3C, respectively. This transition is studied using
direct numerical simulations in toroidal geometry. We showthat in our set-up the 2D2C flow-state is caracterized by reduced fluctuations and reduced radial transport. We suggestthat the 2D3C-2D2C transition is relevant to explain a similartransition in tokamak fusion plasmas.

Even though few flows in nature are close to strictly axisymmetric, there exists a direct and very important, application of
the concept of axisymmetric turbulence. This application ismagnetically confined fusion. A tokamak is a torus-shaped
metallic chamber in which a plasma of hydrogen is confinedby a strong magnetic field in order to produce energy by nuclear fusion. Indeed a strong magnetic field renders a velocity
field invariant in the direction in which it is applied (Moffatt(1967); Favier et al. (2010); Agoua et al. (2021)), here in theazimuthal direction. To maintain a sufficiently intense temperature, turbulence which induces a loss of confinement of
the plasma must be reduced as far as possible. We propose todemonstrate that the 2D3C-2D2C transition can be linked to
the LH transition between a low confined turbulent state (Lmode) and a higher quality confined state (H-mode) observedin tokamaks (Wagner et al. (1982)). For that we show that thisphenomenon observed in a cylindric system (Qin et al. (2020))
can be observed in a torus, and that it can be linked to a loss ofconfinement.

BACKGROUND
Statistically axisymmetric turbulent flow is observed whena strong body force, such as the Coriolis force, renders a system invariant along an axis (Batchelor (1946); Chandrasekhar(1950)). This is different from strictly axisymmetric flow, considered here, where not only the averages but also the instantaneous fields are invariant in the azimuthal direction. This casehas been the subject of recent investigations (Leprovost et al.
(2006); Naso et al. (2010); Thalabard et al. (2014)) in order toextend statistical mechanics, succesfully applied to 2D flows,
to a case between 3D and 2D. Numerical simulations wereperformed (Qu et al. (2017, 2018)) in order to corroborate
these theorical investigations. Axisymmetric flows seem indeed to be an intermediate state between two dimensional andthree dimensional flows. For instance 2D turbulence is caracterised by a self-organization of large structures caused by anenergy transfert from smaller turbulent structures to large vortices (Kraichnan (1967); Kraichnan & Montgomery (1980);
Paret & Tabeling (1997)), which is also observed in axisymmetric turbulence.

MODEL AND NUMERICAL METHOD
The domain of the simulations is a disk, corresponding
to a poloidal cross-section of the torus. We consider a neutral
fluid described by the axisymmetric Navier-Stokes equation(1) in the coordinate system of the disk (er , eθ , eφ ) with a forcing of the velocity F.

In the case of a cylindrical domain (Qin et al. (2020, 2021),
it has been demonstrated that it is possible to induce with alinear forcing a transition between two dimensional states, onewith a three-component velocity (2D3C), and an other withtwo velocity components (2D2C). The second state is caracterised by the large structures of a two dimensional flow (Quet al. (2017, 2018) The first one is an intermediate state between two and three dimensional flow, with large structures
breaking up in smaller vortices. The transition is piloted by1∂u+ (u · ∇)u = − ∇P + ν∆u + F∂tρ
∂T+ (u · ∇)T = κ∆T + S(r)∂tFpol = c p T (r)er − upol × β eφutor y
Ftor = ct utor −eφy1(1)(2)(3)(4)

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
eφ corresponds to the toroidal direction and velocity can be
decomposed in u = upol + utor . The toroidal magnetic field is
not included in the equations, but is at the origin of the axisymmetric nature of the system. In order to introduce a realistic
forcing, we propose that poloidal fluctuations upol are induced(3) by a scalar field T modelled by the heat equation (2). An
anisotropic friction upol × β eφ is added to allow the development of zonal flows observed in the plasma (Lin et al. (1999);
Diamond et al. (2005)). Toroidal perturbations are generated
by a simple linear forcing (4). A correction is added to avoid
the generation of angular momentum uy y with y the distance tothe vertical axis of the tokamak. The source of the scalar field
and the toroidal forcing is chosen such that the generation ofvelocity perturbations increases at the edge of the plasma, asobserved in experiments (Holland et al. (2011)).
Numerical simulations are performed using the open-sourcecode Nek5000. It uses the spectral element method (Patera(1984)) to solve the Navier-Stokes and scalar equations. Velocity fluctuations are initialy chosen randomly around a nullaverage. A Dirichlet boundary condition is imposed at theedge of the domain.

Figure 2.

Stream function field in L mode

γ. When γ increases, toroidal fluctuations are more and more
predominant and the flow is in a 2D3C state. Figure 3 shows acritical transition for α above a critical value αcrit ≈ 1.

RESULTS
8.5

1.8

81.6Temperature

7.5

1.4

7

1.2

6.5
1ET/EP
Temperature

60.8

5.5
0.6

5

0.4

4.5

0.2

4
3.5

00

0.2

0.4

0.6

0.8

1FT/FP

1.2

1.4

1.6

1.8

Figure 3. Relation of γ and the temperature of the tracer at
the center of the disk with the control parameter α.

Figure 1.
friction

The transition between 2D3C and 2D2C states, which we
assimilate with the transition between L-mode and H-mode isillustrated in figure 4. From a first simulation with cT /cP =3.52 which induces a 2D3C state with important toroidal perturbations, α is set to cT /cP = 0.24 at t = 90s. This modification induces a dramatic and fast drop of the toroidal fluctuating energy to zero and then a transition to a 2D2C state forthe flow. A remarkable feature is that the poloidal forcing iskept constant, but nevertheless the poloidal fluctuations drop
dramatically. This shows that the nature of the poloidal flow ischanged and not only its amplitude.

Stream function field in H mode without and with

The parameter γ = ET /EP , with ET and EP energy fluctuations in the toroidal direction and poloidal plane respectively,
is used to determinate quantitatively the behaviour of the flow
between 2D3C and 2D2C states. The relation between α andγ is illustrated on figure 3. When γ is closed to zero, the flow
is purely poloidal without toroidal perturbations, which corresponds to a 2D2C state. Its nature is similar to the H-mode intokamas. In figure 3 this state coincides with a low parameter

In order to investigate the link between this transition and aloss of confinement in the tokamak, a passive scalar field modelled by equation (2) is introduced in the disk. Its source islocated at the center. Figure 5 shows the behaviour of thistracer in the flow for the two states. The radial density profile
represents the diffusion of the tracer in the flow from the center

2ET/EP

We study the evolution of the behaviour of the flow with theparameter α = cT /cP which determines the relative intensityof toroidal and poloidal forcing strengths. The evolution ofthe behaviour of the flow is illustrated by the stream function
φ defined by ∆φ = ωφ with ωφ = (∇ × u) · eφ the azimuthalcomponent of the vorticity. The role of the friction is observedin 1. Without friction in the figure at left two counter-rotating
vortices are obtained as it is usually observed in 2D flow. The
presence of the friction occurs the formation of two large andconcentric structures in the 2D2C regime as observed in thefigure at right. In contrary when α increases the behaviour ofthe flow changes as shown in figure 2. In this case the structures are smaller and more chaotic as observed in 3D turbulence, which should weaken the confinement.

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
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2014 Statistical mechanics of the 3d axisymmetric euler
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Wagner, F., Becker, G., Behringer, K., Campbell, D., Eberhagen, A., Engelhardt, W., Fussmann, G., Gehre, O., Gernhardt, J., Gierke, G. v., Haas, G., Huang, M., Karger, F.,Keilhacker, M., Klüber, O., Kornherr, M., Lackner, K., Lisitano, G., Lister, G. G., Mayer, H. M., Meisel, D., Müller,E. R., Murmann, H., Niedermeyer, H., Poschenrieder, W.,

L

H

Figure 4.

Transition between the L-mode and the H-mode.

98Temperature76
5H-modeL-mode

4
32
1
0
0

0.2

0.4

0.6

0.8

1

RadiusFigure 5.

Temperature of the tracer in H-mode and L-mode

to the edge caused by the turbulent transport. For the H-mode,
the turbulent transport is weaker and the tracer remains moreconcentrated around the center than for the L-mode. The relation between temperature at the center and α is represented onfigure 3, which shows the coı̈ncidence between H-mode/2D2Cstate and high temperature.

CONCLUSIONThe link between the LH transition observed in tokamaks
and a 2D3C-2D2C transition is illustrated by simulations of anaxisymmetric flow in a torus. This transition is associated witha better confinement of the temperature in the 2D2C state similar to the H-mode behaviour. In the full paper we will further
investigate the turbulent dynamics underlying the transition byassessing the energy cascade of both toroidal and poloidal flow
components.

REFERENCES
Agoua, Wesley, Favier, Benjamin, Delache, Alexandre, Briard,
Antoine & Bos, Wouter J. T. 2021 Spontaneous generation
and reversal of helicity in anisotropic turbulence. Phys. Rev.
E 103, L061101.
Batchelor, GK 1946 The theory of axisymmetric turbulence.

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
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