# A critical transition of two-dimensional flow in toroidal geometry

Wesley Agoua, Benjamin Favier, Jorge Morales, Wouter J.T. Bos

► To cite this version:

Wesley Agoua, Benjamin Favier, Jorge Morales, Wouter J.T. Bos. A critical transition of two-dimensional flow in toroidal geometry. Journal of Fluid Mechanics, 2024, 988, pp.A33. 10.1017/jfm.2024.425. hal-04620096

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# A critical transition of two-dimensional flow in toroidal geometry

Wesley Agoua, Benjamin Favier, Jorge Morales, Wouter J.T. Bos

► To cite this version:

Wesley Agoua, Benjamin Favier, Jorge Morales, Wouter J.T. Bos. A critical transition of two-dimensional flow in toroidal geometry. Journal of Fluid Mechanics, 2024, 988, pp.A33. 10.1017/jfm.2024.425. hal-04620096

HAL Id: hal-04620096

https://hal.science/hal-04620096v1

Submitted on 21 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Banner appropriate to article type will appear here in typeset article

1 A critical transition of two-dimensional flow
2 in toroidal geometry

3 Wesley Agoua¹ Benjamin Favier² Jorge Morales³ and Wouter J.T. Bos¹†

5 ¹CNRS, Univ Lyon, Ecole Centrale de Lyon, Univ Lyon 1 Claude Bernard, INSA Lyon,

6 Laboratoire de Mécanique des Fluides et Acoustique, UMR5509, 69134 Ecully, France

7 ²IRPHE, Aix-Marseille Université, CNRS, Centrale Marseille, 13453 Marseille, France

8 ³IRCF - CEA, Cadarache, France

9 (Received xx; revised xx; accepted xx)

10 **Abstract:** We investigate two-dimensional flow in toroidal geometry, with a
11 focus on a transition between two-dimensional three-component flow and two-
12 dimensional two-component flow. This latter flow-state allows a self-organization
13 of the system to a quiescent dynamics, characterized by long-living coherent
14 structures. When these large-scale structures orient in the azimuthal direction,
15 the radial transport is reduced. Such a transition, if it can be triggered in
16 toroidally confined fusion plasmas, is beneficial for the generation of zonal flows
17 and should consequently result in a flow field beneficial for confinement.

18 **Key words:** toroidal flow, turbulence, transition, plasma

19 1. Two-dimensional turbulence and toroidal fusion plasmas

20 Thermonuclear fusion is a sustainable and carbon-free energy source. It can
21 thereby constitute a game-changer in the context of energy regulation and climate
22 change. Currently the most advanced geometry to achieve the ultimate goal of a
23 sustained large-scale fusion reaction is the tokamak: in a torus-shaped reactor
24 chamber, magnetic fields are used to confine a plasma at a temperature of
25 hundreds of millions degrees, in which energy is produced by fusion of hydrogen
26 isotopes. A schematic of toroidal geometry, indicating the definitions of toroidal
27 and poloidal directions, is shown in Fig. 1.

28 The largest obstacle for fusion is the confinement of a plasma. Indeed, if any
29 reactor is to produce energy by a fusion reaction, the ionized gas of hydrogen
30 isotopes should be kept at a sufficient temperature, with a sufficient density for a
31 long enough period of time. This triple criterion (time, density and temperature)
32 has been known since the 1950s (Lawson 1957) and the goal of almost any
33 magnetically controlled fusion experiment is to enhance this triple product.
34 Currently, tokamaks cannot work without continuous injection of energy in the
35 plasma, and they produce less energy than they need to sustain the reaction. The

† Email address for correspondence: wouter.bos@ec-lyon.fr
Banner appropriate to article type will appear here in typeset article

1

2

# A critical transition of two-dimensional flow in toroidal geometry

3

4 Bos¹†

Wesley Agoua¹ Benjamin Favier² Jorge Morales³ and Wouter J.T. Bos¹†

5

¹CNRS, Univ Lyon, Ecole Centrale de Lyon, Univ Lyon 1 Claude Bernard, INSA Lyon,

6

Laboratoire de Mécanique des Fluides et Acoustique, UMR5509, 69134 Ecully, France

7

²IRPHE, Aix-Marseille Université, CNRS, Centrale Marseille, 13453 Marseille, France

8

³IRCF - CEA, Cadarache, France

9

(Received xx; revised xx; accepted xx)

10

11

12

13

14

15

16

17

**Abstract:** We investigate two-dimensional flow in toroidal geometry, with a focus on a transition between two-dimensional three-component flow and two-dimensional two-component flow. This latter flow-state allows a self-organization of the system to a quiescent dynamics, characterized by long-living coherent structures. When these large-scale structures orient in the azimuthal direction, the radial transport is reduced. Such a transition, if it can be triggered in toroidally confined fusion plasmas, is beneficial for the generation of zonal flows and should consequently result in a flow field beneficial for confinement.

18

**Key words:** toroidal flow, turbulence, transition, plasma

19

## 1. Two-dimensional turbulence and toroidal fusion plasmas

20

21

22

23

24

25

26

27

Thermonuclear fusion is a sustainable and carbon-free energy source. It can thereby constitute a game-changer in the context of energy regulation and climate change. Currently the most advanced geometry to achieve the ultimate goal of a sustained large-scale fusion reaction is the tokamak: in a torus-shaped reactor chamber, magnetic fields are used to confine a plasma at a temperature of hundreds of millions degrees, in which energy is produced by fusion of hydrogen isotopes. A schematic of toroidal geometry, indicating the definitions of toroidal and poloidal directions, is shown in Fig. 1.

28

29

30

31

32

33

34

35

The largest obstacle for fusion is the confinement of a plasma. Indeed, if any reactor is to produce energy by a fusion reaction, the ionized gas of hydrogen isotopes should be kept at a sufficient temperature, with a sufficient density for a long enough period of time. This triple criterion (time, density and temperature) has been known since the 1950s (Lawson 1957) and the goal of almost any magnetically controlled fusion experiment is to enhance this triple product. Currently, tokamaks cannot work without continuous injection of energy in the plasma, and they produce less energy than they need to sustain the reaction. The

<footnote>† Email address for correspondence: wouter.bos@ec-lyon.fr</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1: Tokamaks are torus-shaped fusion reactors where the plasma is confined by a magnetic field. The toroidal component of the magnetic field is dominant in realistic reactors. In the simplified description considered here, we only consider this toroidal field and assume it strong enough to render the plasma-dynamics invariant along the toroidal direction. This reduces the dynamics to a two-dimensional system, with three velocity components: two components in the poloidal plane, $u_p$ and one toroidal component $u_t$. In the present schematic we indicate the major and minor axes $R$ and $a$, respectively. The color plot indicate the (toroidal) vorticity associated with the poloidal velocity field.</figcaption></figure>

ITER experiment aims at showing that tokamaks can reach, and go beyond, the break-even point.

The main actor limiting the confinement in tokamaks is turbulence. The transport of heat and matter by turbulent fluctuations degrades the confinement quality in all existing tokamaks (Liewer 1985; Garbet et al. 2004). The presence of some turbulence does seem inevitable given the enormous gradients of temperature and magnetic fields in the plasma edge, but limiting the intensity of this turbulence as much as possible in the reaction chamber is paramount. This explains the tremendous importance given to a transition between two turbulent states, observed first experimentally in the ASDEX experiment (Wagner et al. 1982). The transition from $a$, highly turbulent, low-confinement mode (or L-mode) to $a$, more quiescent, high confinement mode (H-mode), is observed to increase the confinement time considerably. Knowing how to trigger such a LH-transition, and keep a plasma in H-mode, can thus be essential for the design of a successful fusion-reactor.

The understanding of the LH-transition is still incomplete. Different propositions of theoretical frameworks can be found in reviews on the subject (Wagner 2007; Connor & Wilson 2000). It is now well accepted that in H-mode, confinement is improved by the presence of shearing motion at the edge of the plasma (Groebner et al. 1990; Shaing & Crume 1989; Shats et al. 2007) and that interaction with the walls of the plasma vessel might play a role in this dy-
<figure><img src="image_2.png" /><figcaption>Figure 1: Tokamaks are torus-shaped fusion reactors where the plasma is confined by a magnetic field. The toroidal component of the magnetic field is dominant in realistic reactors. In the simplified description considered here, we only consider this toroidal field and assume it strong enough to render the plasma-dynamics invariant along the toroidal direction. This reduces the dynamics to a two-dimensional system, with three velocity components: two components in the poloidal plane, $u_p$ and one toroidal component $u_t$. In the present schematic we indicate the major and minor axes $R$ and $a$, respectively. The color plot indicate the (toroidal) vorticity associated with the poloidal velocity field.</figcaption></figure>

36

ITER experiment aims at showing that tokamaks can reach, and go beyond, the break-even point.

37

38

39

40

41

42

43

44

45

46

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48

49

50

The main actor limiting the confinement in tokamaks is turbulence. The transport of heat and matter by turbulent fluctuations degrades the confinement quality in all existing tokamaks (Liewer 1985; Garbet et al. 2004). The presence of some turbulence does seem inevitable given the enormous gradients of temperature and magnetic fields in the plasma edge, but limiting the intensity of this turbulence as much as possible in the reaction chamber is paramount. This explains the tremendous importance given to a transition between two turbulent states, observed first experimentally in the ASDEX experiment (Wagner et al. 1982). The transition from $a$, highly turbulent, low-confinement mode (or L-mode) to $a$, more quiescent, high confinement mode (H-mode), is observed to increase the confinement time considerably. Knowing how to trigger such a LH-transition, and keep a plasma in H-mode, can thus be essential for the design of a successful fusion-reactor.

51

52

53

54

55

56

The understanding of the LH-transition is still incomplete. Different propositions of theoretical frameworks can be found in reviews on the subject (Wagner 2007; Connor & Wilson 2000). It is now well accepted that in H-mode, confinement is improved by the presence of shearing motion at the edge of the plasma (Groebner et al. 1990; Shaing & Crume 1989; Shats et al. 2007) and that interaction with the walls of the plasma vessel might play a role in this dy-
57 dynamics (Dif-Pradalier et al. 2022). Such shearing motion allows to decorrelate
58 radially propagating structures by a mechanism called shear-sheltering in the
59 fluid mechanics literature (Hunt & Carruthers 1990; Hunt & Durbin 1999). In
60 the tokamak community this insight has had a major impact (Terry 2000), in
61 particular since magnetized plasmas show the formation of zonal flows, radially-
62 sheared poloidal flow structures, which contribute importantly to this shear-
63 sheltering (Biglari et al. 1990; Diamond et al. 2005; Gürcan & Diamond 2015).

64 Indeed, in addition to zonal flows, another established ingredient of the H-
65 mode is its link with two-dimensional turbulence. It has been known since the
66 works of Kraichnan (Kraichnan (1967); Kraichnan & Montgomery (1980)) that
67 a fluid flow in two space dimensions has the tendency to self-organize into
68 large-scale structures. Examples of such self-organization are cyclonic structures
69 in the atmosphere, and controlled numerical and physical experiments have
70 verified this tendency to self-organization (McWilliams 1984; Sommeria 1986;
71 Paret & Tabeling 1997). The turbulence in plasmas seems to behave in a similar
72 manner (Fyfe & Montgomery 1979), i.e., the turbulence also tends to form large
73 scale structures. This link between the formation of space-filling structures in
74 two-dimensional flows and the H-mode was stressed in experiments (Shats et al.
75 2005). However, the magnetic field is also present in the L-mode, so that the
76 dynamics should not be far from two-dimensional. Therefore it is not only the
77 two-dimensional character of the flow which allows to explain the LH-transition.
78

79 In the present study we use a representation of a toroidal plasma which is delib-
80 erately simplified as much as possible to study the dynamics of two-dimensional
81 flow in bounded toroidal geometry. Thereby we certainly miss a number of
82 features important to describe the dynamics of fusion plasmas, but this approach
83 allows to understand a critical transition between nearly two-dimensional flow
84 with three dominant flow components, to a purely two-dimensional flow and
85 the influence of this transition on confinement properties. The present study is
86 thereby complementary to investigations of simple models of plasma turbulence in
87 periodic slab-geometry, such as the Hasegawa-Mima (Hasegawa & Mima (1978)),
88 Hasegawa-Wakatani (Hasegawa & Wakatani (1987)), which ignore the toroidal
89 boundaries, while it remains simpler to interpret than toroidal simulations us-
90 ing the full three-dimensional MHD system or even more complex gyrokinetic
descriptions Goerler et al. (2011); Grandgirard et al. (2006).

91 The motivation for the present work finds its origin in recent insights from
92 theoretical studies on axisymmetric turbulence, which we will now briefly re-
93 view. We consider purely axisymmetric flows, where not only the average flow
94 quantities, but also every fluctuation is exactly axisymmetric. In the absence of
95 magnetic fields or other body-forces, in neutral fluids such a flow in the turbulent
96 regime is difficult to establish. Therefore, in the fluid mechanics community,
97 this type of flow has received interest only recently, mostly in order to extend
98 ideas from statistical mechanics of two-dimensional flows to a case closer to
99 three dimensions (Leprovost et al. 2006; Naso et al. 2010; Thalabard et al. 2014).
100 Since such a turbulence is hard to reproduce experimentally, the assessment of
101 theoretical ideas has been mainly achieved through direct numerical simulations
102 of the axisymmetric Navier-Stokes equations (Qu et al. 2017, 2018). In a recent
103 investigation (Qin et al. 2020) it was observed that a critical transition between
104 two types of axisymmetric turbulence can be observed, where one of the flow
105 states is characterized by typical two-dimensional behavior, i.e., self-organization
106 of large velocity structures, whereas the other flow is two-dimensional, but in-
57 dynamics (Dif-Pradalier et al. 2022). Such shearing motion allows to decorrelate
58 radially propagating structures by a mechanism called shear-sheltering in the
59 fluid mechanics literature (Hunt & Carruthers 1990; Hunt & Durbin 1999). In
60 the tokamak community this insight has had a major impact (Terry 2000), in
61 particular since magnetized plasmas show the formation of zonal flows, radially-
62 sheared poloidal flow structures, which contribute importantly to this shear-
63 sheltering (Biglari et al. 1990; Diamond et al. 2005; Gürcan & Diamond 2015).

64 Indeed, in addition to zonal flows, another established ingredient of the H-
65 mode is its link with two-dimensional turbulence. It has been known since the
66 works of Kraichnan (Kraichnan (1967); Kraichnan & Montgomery (1980)) that
67 a fluid flow in two space dimensions has the tendency to self-organize into
68 large-scale structures. Examples of such self-organization are cyclonic structures
69 in the atmosphere, and controlled numerical and physical experiments have
70 verified this tendency to self-organization (McWilliams 1984; Sommeria 1986;
71 Paret & Tabeling 1997). The turbulence in plasmas seems to behave in a similar
72 manner (Fyfe & Montgomery 1979), i.e., the turbulence also tends to form large
73 scale structures. This link between the formation of space-filling structures in
74 two-dimensional flows and the H-mode was stressed in experiments (Shats et al.
75 2005). However, the magnetic field is also present in the L-mode, so that the
76 dynamics should not be far from two-dimensional. Therefore it is not only the
77 two-dimensional character of the flow which allows to explain the LH-transition.
78

79 In the present study we use a representation of a toroidal plasma which is delib-
80 erately simplified as much as possible to study the dynamics of two-dimensional
81 flow in bounded toroidal geometry. Thereby we certainly miss a number of
82 features important to describe the dynamics of fusion plasmas, but this approach
83 allows to understand a critical transition between nearly two-dimensional flow
84 with three dominant flow components, to a purely two-dimensional flow and
85 the influence of this transition on confinement properties. The present study is
86 thereby complementary to investigations of simple models of plasma turbulence in
87 periodic slab-geometry, such as the Hasegawa-Mima (Hasegawa & Mima (1978)),
88 Hasegawa-Wakatani (Hasegawa & Wakatani (1987)), which ignore the toroidal
89 boundaries, while it remains simpler to interpret than toroidal simulations us-
90 ing the full three-dimensional MHD system or even more complex gyrokinetic
descriptions Goerler et al. (2011); Grandgirard et al. (2006).

91 The motivation for the present work finds its origin in recent insights from
92 theoretical studies on axisymmetric turbulence, which we will now briefly re-
93 view. We consider purely axisymmetric flows, where not only the average flow
94 quantities, but also every fluctuation is exactly axisymmetric. In the absence of
95 magnetic fields or other body-forces, in neutral fluids such a flow in the turbulent
96 regime is difficult to establish. Therefore, in the fluid mechanics community,
97 this type of flow has received interest only recently, mostly in order to extend
98 ideas from statistical mechanics of two-dimensional flows to a case closer to
99 three dimensions (Leprovost et al. 2006; Naso et al. 2010; Thalabard et al. 2014).
100 Since such a turbulence is hard to reproduce experimentally, the assessment of
101 theoretical ideas has been mainly achieved through direct numerical simulations
102 of the axisymmetric Navier-Stokes equations (Qu et al. 2017, 2018). In a recent
103 investigation (Qin et al. 2020) it was observed that a critical transition between
104 two types of axisymmetric turbulence can be observed, where one of the flow
105 states is characterized by typical two-dimensional behavior, i.e., self-organization
106 of large velocity structures, whereas the other flow is two-dimensional, but in-
<figure><img src="image_3.png" /><figcaption>Figure 2: Spectral element mesh on the poloidal plane. The mesh consists of a central part and a boundary-adapted circular part. The major radius of the torus is $R$ and the minor radius is denoted $a$.</figcaption></figure>

107 involves three velocity components (see Fig. 3(a,b) for an illustration). Even though 108 this latter state is essentially 2D, the large-scale flow structures are inherently 109 unstable and tend to lose their energy to smaller scales, a feature reminiscent 110 of 3D turbulence: this change in cascade direction is a major difference between 111 2D and axisymmetric 2D3C turbulence.

112 Whereas neutral fluid turbulence is rarely in a close to axisymmetric state, 113 this changes for the case of electrically conducting fluids, or plasmas. Indeed, 114 the presence of a strong azimuthal magnetic field limits the variations in the 115 direction of the field (Moffatt 1967; Favier et al. 2010; Gallet & Doering 2015a). 116 In a tokamak, a strong toroidal magnetic field is present, which renders the flow 117 close to axisymmetric. In general the strength of this field is an order of magnitude 118 larger than the poloidal field (associated with a toroidal current) which we will 119 neglect in our approach. The plasma in a tokamak is thereby close to a strictly 120 axisymmetric state and can be described, at first order, by an axisymmetric fluid 121 flow. We suggest that the transition which was discovered between two different 122 axisymmetric turbulent states (Qin et al. 2020) should carry over to the dynamics 123 of tokamaks. To illustrate this possibility, we have set-up a numerical experiment 124 in toroidal geometry and we will show the confinement properties of the two 125 axisymmetric flow states from a pure fluid mechanics perspective.

126 We are not aware that such an elementary fluid set-up has been investigated 127 previously. Most fluid mechanics studies in toroidal geometry have focused on 128 liquid metal flows (Baylis & Hunt 1971), and very few studies consider the 129 axisymmetric limit (Poyé et al. 2020). We think that, even if the transition that 130 we will assess is shown in a too simple set-up to claim a one-to-one correspondence 131 with the LH transition, it does illustrate the robustness of the 2D3C-2D2C 132 transition in toroidal geometry, and its ability to dramatically change the flow- 133 physics and self-organization properties of the flow.

134 In the next section, we describe in detail the model we use to describe a 135 turbulent plasma in toroidal geometry. In Sec. 3 we discuss the numerical details. 136 In Sec. 4 we present the results of our numerical experiments. Finally in Sec. 5 137 we conclude.
<figure><img src="image_3.png" /><figcaption>Figure 2: Spectral element mesh on the poloidal plane. The mesh consists of a central part and a boundary-adapted circular part. The major radius of the torus is $R$ and the minor radius is denoted $a$.</figcaption></figure>

107 involves three velocity components (see Fig. 3(a,b) for an illustration). Even though 108 this latter state is essentially 2D, the large-scale flow structures are inherently 109 unstable and tend to lose their energy to smaller scales, a feature reminiscent 110 of 3D turbulence: this change in cascade direction is a major difference between 111 2D and axisymmetric 2D3C turbulence.

112 Whereas neutral fluid turbulence is rarely in a close to axisymmetric state, 113 this changes for the case of electrically conducting fluids, or plasmas. Indeed, 114 the presence of a strong azimuthal magnetic field limits the variations in the 115 direction of the field (Moffatt 1967; Favier et al. 2010; Gallet & Doering 2015a). 116 In a tokamak, a strong toroidal magnetic field is present, which renders the flow 117 close to axisymmetric. In general the strength of this field is an order of magnitude 118 larger than the poloidal field (associated with a toroidal current) which we will 119 neglect in our approach. The plasma in a tokamak is thereby close to a strictly 120 axisymmetric state and can be described, at first order, by an axisymmetric fluid 121 flow. We suggest that the transition which was discovered between two different 122 axisymmetric turbulent states (Qin et al. 2020) should carry over to the dynamics 123 of tokamaks. To illustrate this possibility, we have set-up a numerical experiment 124 in toroidal geometry and we will show the confinement properties of the two 125 axisymmetric flow states from a pure fluid mechanics perspective.

126 We are not aware that such an elementary fluid set-up has been investigated 127 previously. Most fluid mechanics studies in toroidal geometry have focused on 128 liquid metal flows (Baylis & Hunt 1971), and very few studies consider the 129 axisymmetric limit (Poyé et al. 2020). We think that, even if the transition that 130 we will assess is shown in a too simple set-up to claim a one-to-one correspondence 131 with the LH transition, it does illustrate the robustness of the 2D3C-2D2C 132 transition in toroidal geometry, and its ability to dramatically change the flow- 133 physics and self-organization properties of the flow.

134 In the next section, we describe in detail the model we use to describe a 135 turbulent plasma in toroidal geometry. In Sec. 3 we discuss the numerical details. 136 In Sec. 4 we present the results of our numerical experiments. Finally in Sec. 5 137 we conclude.
## 2. Modeling and governing equations

The system we consider is a model representation of a magnetically confined plasma in toroidal geometry. Since we assume axisymmetry, the dynamics can be described by a three-component dynamics in the poloidal plane. This simplifies the numerical experiments considerably. The poloidal domain on which we focus is shown in Fig. 2, where we indicate the coordinate system. The major radius is $R$ and the minor radius $a$. The cylindrical coordinate system is centered around the major (z-)axis of the torus. In this coordinate system the radial and vertical direction in the poloidal plane are defined $r, z$. We also define a local coordinate system, centered in the circular cross-section, with polar coordinates $\rho, \theta$.

In the present section we will first introduce the fluid-description. Then we will focus on the forcing protocol, representing the plasma instabilities, and we will explain how we measure the confinement quality of the plasma.

### 2.1. A fluid mechanics modeling of toroidal plasmas

Plasmas can be described by a hierarchy of physical models (Boyd et al. 2003). The most precise, but thereby also least tractable, description is a kinetic approach involving all charged particles of the plasma and their nonlocal interactions (Diamond et al. 2010). The coarsest approach is probably a fluid approach, where the plasma is described using continuum mechanics (Biskamp 1997). In the present investigation it is this latter description which is adopted. We will omit all kinetic effects from our system. Furthermore we will assume the dynamics isothermal, solenoidal and we do not model the detailed interaction of the plasma with electrical currents and magnetic fields. The only influence of magnetic fields which is retained in the present system is the influence of a toroidal magnetic field, assumed to be strong enough to render the dynamics perfectly axisymmetric. Physically this corresponds to the fact that charged particles can freely move along magnetic field lines, whereas perpendicular motion is constrained by Coulomb-forces. This quasi-bidimensionalisation of the flow is well documented in magnetohydrodynamical turbulence (Bigot & Galtier 2011; Alexakis 2011) and can even be exact when the magnetic Reynolds number is low enough (Gallet & Doering 2015b).

In such an axisymmetric set-up, the dynamics are entirely described by the two velocity components in the poloidal plane $u_P = (u_r, u_z)$, and one component $u_T$ perpendicular to it (see Fig. 1). Such a system does not represent the instabilities associated with temperature, density and magnetic field gradients. It is in fusion plasmas these gradients which are at the origin of the turbulent fluctuations and these sources of instabilities are here modeled explicitly by appropriate external force terms.

We start by writing the axisymmetric Navier-Stokes equations.

$$ \frac{\partial u_P}{\partial t} + u_P \cdot \nabla u_P + \nabla P - \nu \Delta u_P = \mathcal{N}_P + F_P \quad (2.1) $$

$$ \frac{\partial u_T}{\partial t} + u_P \cdot \nabla u_T - \nu \Delta u_T = \mathcal{N}_T + F_T. \quad (2.2) $$

The pressure $P$, in which we absorbed the constant density, ensures incompressibility through the condition

$$ \frac{1}{r} \frac{\partial r u_r}{\partial r} + \frac{\partial u_z}{\partial z} = 0. \quad (2.3) $$
## 2. Modeling and governing equations

The system we consider is a model representation of a magnetically confined plasma in toroidal geometry. Since we assume axisymmetry, the dynamics can be described by a three-component dynamics in the poloidal plane. This simplifies the numerical experiments considerably. The poloidal domain on which we focus is shown in Fig. 2, where we indicate the coordinate system. The major radius is $R$ and the minor radius $a$. The cylindrical coordinate system is centered around the major (z-)axis of the torus. In this coordinate system the radial and vertical direction in the poloidal plane are defined $r, z$. We also define a local coordinate system, centered in the circular cross-section, with polar coordinates $\rho, \theta$.

In the present section we will first introduce the fluid-description. Then we will focus on the forcing protocol, representing the plasma instabilities, and we will explain how we measure the confinement quality of the plasma.

### 2.1. A fluid mechanics modeling of toroidal plasmas

Plasmas can be described by a hierarchy of physical models (Boyd et al. 2003). The most precise, but thereby also least tractable, description is a kinetic approach involving all charged particles of the plasma and their nonlocal interactions (Diamond et al. 2010). The coarsest approach is probably a fluid approach, where the plasma is described using continuum mechanics (Biskamp 1997). In the present investigation it is this latter description which is adopted. We will omit all kinetic effects from our system. Furthermore we will assume the dynamics isothermal, solenoidal and we do not model the detailed interaction of the plasma with electrical currents and magnetic fields. The only influence of magnetic fields which is retained in the present system is the influence of a toroidal magnetic field, assumed to be strong enough to render the dynamics perfectly axisymmetric. Physically this corresponds to the fact that charged particles can freely move along magnetic field lines, whereas perpendicular motion is constrained by Coulomb-forces. This quasi-bidimensionalisation of the flow is well documented in magnetohydrodynamical turbulence (Bigot & Galtier 2011; Alexakis 2011) and can even be exact when the magnetic Reynolds number is low enough (Gallet & Doering 2015b).

In such an axisymmetric set-up, the dynamics are entirely described by the two velocity components in the poloidal plane $u_P = (u_r, u_z)$, and one component $u_T$ perpendicular to it (see Fig. 1). Such a system does not represent the instabilities associated with temperature, density and magnetic field gradients. It is in fusion plasmas these gradients which are at the origin of the turbulent fluctuations and these sources of instabilities are here modeled explicitly by appropriate external force terms.

We start by writing the axisymmetric Navier-Stokes equations.

$$ \frac{\partial u_P}{\partial t} + u_P \cdot \nabla u_P + \nabla P - \nu \Delta u_P = \mathcal{N}_P + F_P \quad (2.1) $$

$$ \frac{\partial u_T}{\partial t} + u_P \cdot \nabla u_T - \nu \Delta u_T = \mathcal{N}_T + F_T. \quad (2.2) $$

The pressure $P$, in which we absorbed the constant density, ensures incompressibility through the condition

$$ \frac{1}{r} \frac{\partial r u_r}{\partial r} + \frac{\partial u_z}{\partial z} = 0. \quad (2.3) $$
183 The last terms on the left hand side of Eq. (2.1) and Eq. (2.2) represent the viscous stresses, with $\nu$ the kinematic viscosity. In these terms the $\Delta$ indicates the axisymmetric vector-Laplacian in polar coordinates.

186 The left-hand-sides (LHSs) of Eq. (2.1) and Eq. (2.2), respectively, describe purely two-dimensional fluid motion, represented by the velocity vector-field $\mathbf{u}_P(\mathbf{x}, t)$ advecting the toroidal (out of plane) component of the velocity $u_T(\mathbf{x}, t)$. In a toroidal geometry the curvature introduces the $\mathcal{N}$ terms, which couple the two fields. The curvature terms are

$$ \mathcal{N}_P = u_T^2 / r e_r \qquad (2.4) $$

$$ \mathcal{N}_T = -u_T u_r / r. \qquad (2.5) $$

191 These terms are reminiscent of the vortex-stretching terms, essential in three-dimensional energy transfer, but absent in purely two-dimensional systems. All toroidal derivatives, $\partial/\partial\phi$ are zero since we consider the axisymmetric case. Physically this assumption is justified by the presence of a strong toroidal magnetic field.

199 Before discussing the forcing terms $F_P$ and $F_T$, we will focus on the invariants of the system. In the absence of forcing and dissipation the nonlinear interaction conserves a certain number of integral quantities. these quantities are not the same in the 2D2C or the 2D3C state. In the 2D2C case, the inviscid poloidal dynamics conserve the poloidal energy defined as

$$ E_P = \frac{1}{2} \langle \| \mathbf{u}_P \|^2 \rangle, \qquad (2.6) $$

205 where the brackets denote a volume averaging. Furthermore, in this limit the enstrophy $Z = \langle \| \nabla \times \mathbf{u}_P \|^2 \rangle$ is conserved. Furthermore, an infinite number of Casimir's (functions of the vorticity $\nabla \times \mathbf{u}_P$) can be defined (Kraichnan & Montgomery 1980). These latter quantities play an important role in statistical mechanics, but are less important in determining the cascade directions of the system.

211 In the (curved) 2D3C case, the nonlinear dynamics conserve the total energy $E_T + E_P$, where the toroidal energy is defined as

$$ E_T = \frac{1}{2} \langle u_T^2 \rangle. \qquad (2.7) $$

214 The enstrophy is no longer a conserved quantity, but the helicity,

$$ H = \langle (\nabla \times \mathbf{u}) \cdot \mathbf{u} \rangle \qquad (2.8) $$

216 becomes an invariant of the system. In cylindrical geometry (Qin et al. 2020), it was shown using energy and transfer spectra that the 2D3C to 2D2C transition changes the cascade properties associated with the above invariants. The inverse transfer of poloidal energy $E_P$ in the 2D2C case changed towards a direct transfer of the total energy $E_P + E_T$ to small scales in the 2D3C configuration. The assessment of energy cascades and fluxes is less convenient in the present set-up since we will consider solid boundaries and spatially localized forcing-terms. We will therefore focus on a physical space characterization of the system.

224 An interesting feature is that, when the major radius of the torus tends to infinity (more precisely the ratio $R/a$, see Fig. 1), the curvature and the associated $\mathcal{N}$ terms tend to zero. The axisymmetric 2D3C system tends to a cartesian 2D3C system where it is no longer the total energy which is conserved, but $E_T$ and $E_P$
183 The last terms on the left hand side of Eq. (2.1) and Eq. (2.2) represent the viscous stresses, with $\nu$ the kinematic viscosity. In these terms the $\Delta$ indicates the axisymmetric vector-Laplacian in polar coordinates.

186 The left-hand-sides (LHSs) of Eq. (2.1) and Eq. (2.2), respectively, describe purely two-dimensional fluid motion, represented by the velocity vector-field $\mathbf{u}_P(\mathbf{x}, t)$ advecting the toroidal (out of plane) component of the velocity $u_T(\mathbf{x}, t)$. In a toroidal geometry the curvature introduces the $\mathcal{N}$ terms, which couple the two fields. The curvature terms are

$$ \mathcal{N}_P = u_T^2 / r e_r \qquad (2.4) $$

$$ \mathcal{N}_T = -u_T u_r / r. \qquad (2.5) $$

191 These terms are reminiscent of the vortex-stretching terms, essential in three-dimensional energy transfer, but absent in purely two-dimensional systems. All toroidal derivatives, $\partial/\partial\phi$ are zero since we consider the axisymmetric case. Physically this assumption is justified by the presence of a strong toroidal magnetic field.

199 Before discussing the forcing terms $F_P$ and $F_T$, we will focus on the invariants of the system. In the absence of forcing and dissipation the nonlinear interaction conserves a certain number of integral quantities. these quantities are not the same in the 2D2C or the 2D3C state. In the 2D2C case, the inviscid poloidal dynamics conserve the poloidal energy defined as

$$ E_P = \frac{1}{2} \langle \| \mathbf{u}_P \|^2 \rangle, \qquad (2.6) $$

205 where the brackets denote a volume averaging. Furthermore, in this limit the entrophy $Z = \langle \| \nabla \times \mathbf{u}_P \|^2 \rangle$ is conserved. Furthermore, an infinite number of Casimir (functions of the vorticity $\nabla \times \mathbf{u}_P$) can be defined (Kraichnan & Montgomery 1980). These latter quantities play an important role in statistical mechanics, but are less important in determining the cascade directions of the system.

211 In the (curved) 2D3C case, the nonlinear dynamics conserve the total energy $E_T + E_P$, where the toroidal energy is defined as

$$ E_T = \frac{1}{2} \langle u_T^2 \rangle. \qquad (2.7) $$

214 The entrophy is no longer a conserved quantity, but the helicity,

$$ H = \langle (\nabla \times \mathbf{u}) \cdot \mathbf{u} \rangle \qquad (2.8) $$

216 becomes an invariant of the system. In cylindrical geometry (Qin et al. 2020), it was shown using energy and transfer spectra that the 2D3C to 2D2C transition changes the cascade properties associated with the above invariants. The inverse transfer of poloidal energy $E_P$ in the 2D2C case changed towards a direct transfer of the total energy $E_P + E_T$ to small scales in the 2D3C configuration. The assessment of energy cascades and fluxes is less convenient in the present set-up since we will consider solid boundaries and spatially localized forcing-terms. We will therefore focus on a physical space characterization of the system.

224 An interesting feature is that, when the major radius of the torus tends to infinity (more precisely the ratio $R/a$, see Fig. 1), the curvature and the associated $\mathcal{N}$ terms tend to zero. The axisymmetric 2D3C system tends to a cartesian 2D3C system where it is no longer the total energy which is conserved, but $E_T$ and $E_P$
are conserved independently, and the poloidal dynamics behave as in the 2D2C case. In such a 2D3C geometry, the helicity degenerates to a correlation between the toroidal vorticity and the toroidal velocity, a quantity recently investigated in Linkmann et al. (2018) and Yin et al. (2024). It is not known at present how large $R/a$ must be to neglect the transfer. However, in the present investigation we will not vary the geometry, and the $N$ terms cannot be neglected. The transfer between the components is thus possible and the total energy is an inviscid invariant of the system.

## 2.2. Forcing protocol

An important feature associated with a heated magnetized plasma is the presence of a number of instabilities leading to the generation of turbulent fluctuations. The forcing terms $F_P$ and $F_T$ are added to our system to reproduce the main features of sources of turbulent fluctuations in realistic plasmas (such as the interchange instability (Boyd et al. 2003)), which are located at the tokamak edge, where the pressure, density and temperature gradients are large. Modeling the ensemble of these instabilities together by artificial force terms might not represent certain important features associated with the feedback between the flow and the forcing, and this is necessarily left for future study.

To generate the poloidal velocity fluctuations, we add a Rayleigh-Bénard type instability as follows. We compute the advection of a scalar-field by the poloidal velocity field,

$$ \frac{\partial b}{\partial t} + \mathbf{u}_P \cdot \nabla b = \kappa \Delta b + S'. \qquad (2.9) $$

The term $S'$ represents a source of scalar in the edge region of the plasma. More precisely, the value $S'$ is constant and is non-zero only in a shell near the boundary. The boundary condition at $r = a$ is the Dirichlet condition $b = 0$. Thereby, on average a negative radial gradient builds up between the radial location of the source term and the boundary. The poloidal forcing term is

$$ F_P = C_P b e_\rho + F_\beta. \qquad (2.10) $$

The first term in this expression leads then to the Rayleigh-Bénard-type (linear) instability, through the coupling of the poloidal velocity (Eq. (2.1)) and the scalar (Eq. (2.9)).

The second term, $F_\beta$ is a symmetry-breaking term, reminiscent of the anisotropic nature of a magnetized plasma. Indeed, in magnetized plasmas, a natural tendency to organize into concentric, toroidally invariant structures is observed related to the radial density gradients. The resulting zonal-flows are equivalent to the zonal-flows in rapidly rotating flows, such as the bands on Jupiter or the earth. To mimic this effect in the present set-up a body-force is added to the poloidal forcing in the spirit of the Hasegawa-Mima equation (Hasegawa & Mima 1978; Gürcan & Diamond 2015),

$$ F_\beta = \beta' \rho e_T \times \mathbf{u}_P. \qquad (2.11) $$

This contribution to the poloidal force does therefore not inject energy in the system. As we will see below, the term $F_\beta$ term is not essential to trigger the transition from 2D3C to 2D2C, but it leads to enhanced confinement if the transition to a 2D2C state is obtained. Simulations with and without this force will be presented.
are conserved independently, and the poloidal dynamics behave as in the 2D2C case. In such a 2D3C geometry, the helicity degenerates to a correlation between the toroidal vorticity and the toroidal velocity, a quantity recently investigated in Linkmann et al. (2018) and Yin et al. (2024). It is not known at present how large $R/a$ must be to neglect the transfer. However, in the present investigation we will not vary the geometry, and the $N$ terms cannot be neglected. The transfer between the components is thus possible and the total energy is an inviscid invariant of the system.

## 2.2. Forcing protocol

An important feature associated with a heated magnetized plasma is the presence of a number of instabilities leading to the generation of turbulent fluctuations. The forcing terms $F_P$ and $F_T$ are added to our system to reproduce the main features of sources of turbulent fluctuations in realistic plasmas (such as the interchange instability (Boyd et al. 2003)), which are located at the tokamak edge, where the pressure, density and temperature gradients are large. Modeling the ensemble of these instabilities together by artificial force terms might not represent certain important features associated with the feedback between the flow and the forcing, and this is necessarily left for future study.

To generate the poloidal velocity fluctuations, we add a Rayleigh-Bénard type instability as follows. We compute the advection of a scalar-field by the poloidal velocity field,

$$ \frac{\partial b}{\partial t} + \mathbf{u}_P \cdot \nabla b = \kappa \Delta b + S'. \qquad (2.9) $$

The term $S'$ represents a source of scalar in the edge region of the plasma. More precisely, the value $S'$ is constant and is non-zero only in a shell near the boundary. The boundary condition at $r = a$ is the Dirichlet condition $b = 0$. Thereby, on average a negative radial gradient builds up between the radial location of the source term and the boundary. The poloidal forcing term is

$$ F_P = C_P b e_\rho + F_\beta. \qquad (2.10) $$

The first term in this expression leads then to the Rayleigh-Bénard-type (linear) instability, through the coupling of the poloidal velocity (Eq. (2.1)) and the scalar (Eq. (2.9)).

The second term, $F_\beta$ is a symmetry-breaking term, reminiscent of the anisotropic nature of a magnetized plasma. Indeed, in magnetized plasmas, a natural tendency to organize into concentric, toroidally invariant structures is observed related to the radial density gradients. The resulting zonal-flows are equivalent to the zonal-flows in rapidly rotating flows, such as the bands on Jupiter or the earth. To mimic this effect in the present set-up a body-force is added to the poloidal forcing in the spirit of the Hasegawa-Mima equation (Hasegawa & Mima 1978; Gürcan & Diamond 2015),

$$ F_\beta = \beta' \rho e_T \times \mathbf{u}_P. \qquad (2.11) $$

This contribution to the poloidal force does therefore not inject energy in the system. As we will see below, the term $F_\beta$ term is not essential to trigger the transition from 2D3C to 2D2C, but it leads to enhanced confinement if the transition to a 2D2C state is obtained. Simulations with and without this force will be presented.
273

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275

The toroidal fluctuations are also assumed to originate from a linear mechanism and are simulated by a linear forcing term. More precisely, for the toroidal forcing we use

276

$$F_T = C_T \left[ u_T - \tau_{\sigma}^{-1} \frac{\langle u_T r \rangle}{R} \right] \qquad (2.12)$$

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and is applied on the same shell as the source term $S'$. The notation $\langle \cdot \rangle$ indicates a spatial average over the poloidal domain. The second term allows to avoid the build-up of toroidal angular momentum. Indeed, the spontaneous generation of angular momentum $\langle u_T r \rangle$ (or spin-up) in the system is of major interest (Rice et al. 2007), but we want to disentangle this effect from the investigation of the transition between 2D2C and 2C3C flows. The present form of the toroidal forcing term does therefore dominantly excite the toroidal velocity fluctuations avoiding the build-up of mean angular momentum. In plasma experiments, a transition to a 2D2C flow, like the one we discuss here, might be accompanied by a global rotation associated with this angular momentum, which makes the identification of the transition less trivial.

288

## 2.3. Passive tracer to measure confinement

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Eventually, we are interested in the confinement quality of the plasma. In practice, a good confinement in our system is associated with a small value of the radial turbulent diffusion. To measure turbulent diffusion, a passive tracer is injected continuously in the center of the domain, while homogeneous Dirichlet conditions are imposed on the wall. The quantity $\xi$, which follows the flow as a small amount of ink in a water-flow, does not affect the flow, but allows to measure the diffusion associated with the turbulent fluctuations. The governing equation is, as Eq. (3.3) an advection-diffusion equation,

297

$$\frac{\partial \xi}{\partial t} + \mathbf{u}_P \cdot \nabla \xi = \kappa \Delta \xi + f_{\xi} \qquad (2.13)$$

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where $f_{\xi} = C_{\xi}X(\rho_{\xi} - \rho)$, where X is the heaviside function, $\rho_{\xi}$ the radius of the source and $C_{\xi}$ a constant. When the turbulent fluctuations are strong, the diffusion allows efficient transport of the scalar, thus bad confinement, and the temperature in the center of the domain drops. Thereby the center-temperature ($\xi(\rho = 0)$) directly measures the confinement quality of the flow. In the remainder of this article, we will call the scalar $\xi$ temperature, since we introduce it to measure the confinement of heat by the system. It is important to distinguish it from the other scalar b, associated with the poloidal forcing, since in our numerical set-up we decouple the dynamics of both scalars. This decoupling is voluntary here since we want to independently measure the confinement and adjust the forcing terms. In a fusion plasma the improved confinement of heat will necessarily lead to modified temperature gradients in the plasma, so that confinement and forcing are there coupled.

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## 3. Normalization and Numerical set-up

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314

Before performing the numerical simulations, it is convenient to introduce an appropriate normalization of the governing equations. This allows to identify the key parameters that will be varied. In this section we discuss the non-
The toroidal fluctuations are also assumed to originate from a linear mechanism and are simulated by a linear forcing term. More precisely, for the toroidal forcing we use

$$F_T = C_T \left[ u_T - \tau_{\sigma}^{-1} \frac{\langle u_T r \rangle}{R} \right] \qquad (2.12)$$

and is applied on the same shell as the source term $S'$. The notation $\langle \cdot \rangle$ indicates a spatial average over the poloidal domain. The second term allows to avoid the build-up of toroidal angular momentum. Indeed, the spontaneous generation of angular momentum $\langle u_T r \rangle$ (or spin-up) in the system is of major interest (Rice et al. 2007), but we want to disentangle this effect from the investigation of the transition between 2D2C and 2C3C flows. The present form of the toroidal forcing term does therefore dominantly excite the toroidal velocity fluctuations avoiding the build-up of mean angular momentum. In plasma experiments, a transition to a 2D2C flow, like the one we discuss here, might be accompanied by a global rotation associated with this angular momentum, which makes the identification of the transition less trivial.

## 2.3. Passive tracer to measure confinement

Eventually, we are interested in the confinement quality of the plasma. In practice, a good confinement in our system is associated with a small value of the radial turbulent diffusion. To measure turbulent diffusion, a passive tracer is injected continuously in the center of the domain, while homogeneous Dirichlet conditions are imposed on the wall. The quantity $\xi$, which follows the flow as a small amount of ink in a water-flow, does not affect the flow, but allows to measure the diffusion associated with the turbulent fluctuations. The governing equation is, as Eq. (3.3) an advection-diffusion equation,

$$\frac{\partial \xi}{\partial t} + \mathbf{u}_P \cdot \nabla \xi = \kappa \Delta \xi + f_{\xi} \qquad (2.13)$$

where $f_{\xi} = C_{\xi}X(\rho_{\xi} - \rho)$, where $X$ is the heaviside function, $\rho_{\xi}$ the radius of the source and $C_{\xi}$ a constant. When the turbulent fluctuations are strong, the diffusion allows efficient transport of the scalar, thus bad confinement, and the temperature in the center of the domain drops. Thereby the center-temperature ($\xi(\rho = 0)$) directly measures the confinement quality of the flow. In the remainder of this article, we will call the scalar $\xi$ temperature, since we introduce it to measure the confinement of heat by the system. It is important to distinguish it from the other scalar $b$, associated with the poloidal forcing, since in our numerical set-up we decouple the dynamics of both scalars. This decoupling is voluntary here since we want to independently measure the confinement and adjust the forcing terms. In a fusion plasma the improved confinement of heat will necessarily lead to modified temperature gradients in the plasma, so that confinement and forcing are there coupled.

# 3. Normalization and Numerical set-up

Before performing the numerical simulations, it is convenient to introduce an appropriate normalization of the governing equations. This allows to identify the key parameters that will be varied. In this section we discuss the non-
dimensionalization, the parameters used in our simulations and the numerical method.

### 3.1. Dimensionless equations

In order to non-dimensionalize the equations, we choose as a typical timescale the inverse of the poloidal forcing rate $T^* = C_P^{-1}$. As lengthscale we use the minor radius $L^* = a$. This allows to normalize the equations using $\tilde{u} = uT^*/L^*$, $\tilde{\nabla} = L^*\nabla$, and analogous for $\Delta, P, b, \rho, \partial_t$. Removing after normalization all tildes, for notational ease, we obtain the non-dimensional set of equations,

$$ \frac{\partial \mathbf{u}_P}{\partial t} + \mathbf{u}_P \cdot \nabla \mathbf{u}_P + \nabla P - \frac{1}{Re} \Delta \mathbf{u}_P = \qquad (3.1) $$

$$ u_T^2 / r e_r + b e_r + \beta \rho e_T \times \mathbf{u}_P, \qquad (3.2) $$

and

$$ \frac{\partial b}{\partial t} + \mathbf{u}_P \cdot \nabla b = \frac{1}{Pe} \Delta b + S. \qquad (3.3) $$

These two equations define the poloidal dynamics. For the toroidal velocity component we have in normalized form,

$$ \frac{\partial u_T}{\partial t} + \mathbf{u}_P \cdot \nabla u_T - \frac{1}{Re} \Delta u_T = -u_T u_r / r + \gamma \left[ u_T - \tau_{\sigma}^{-1} \frac{\langle u_T r \rangle}{R} \right] \qquad (3.4) $$

and for the passive scalar

$$ \frac{\partial \xi}{\partial t} + \mathbf{u}_P \cdot \nabla \xi = \frac{1}{Pe} \Delta \xi + f_{\xi}. \qquad (3.5) $$

In these equations we define,

$$ Re = \frac{C_P a^2}{\nu}, \quad Pe = \frac{C_P a^2}{\kappa} \qquad (3.6) $$

$$ \gamma = \frac{C_T}{C_P}, \quad \beta = \frac{\beta' a}{C_P}, \quad S = \frac{S'}{a C_P^2}. \qquad (3.7) $$

The parameter $\gamma = C_T/C_P$ measures the toroidal forcing strength compared to the poloidal forcing. This means that if $\gamma$ is small, the instability mechanisms mainly drive the fluid flow in the poloidal plane. This ratio $\gamma$ is the main control-parameter of our system.

### 3.2. Parameters

The major radius of the torus is $R = 2$ and the minor radius $a = 1$. The ratio $R/a = 2$ is of the order of magnitude of typical tokamaks. For instance the JET tokamak is characterized by an aspect ratio $R/a = 2.4$, ITER by a value close to three, while spheromaks have $R/a \approx 1$.

The Reynolds number is $Re = 5000$, which is a high enough value to ensure turbulent motion in our system. A change in its value does not qualitatively change the main results of the present investigation, as long as the flow remains turbulent. The Péclet number is chosen equal to the Reynolds number $Pe = Re$. The value of $C_P = 10$ is fixed and the forcing ratio $\gamma$ is varied in the range
dimensionalization, the parameters used in our simulations and the numerical method.

### 3.1. Dimensionless equations

In order to non-dimensionalize the equations, we choose as a typical timescale the inverse of the poloidal forcing rate $T^* = C_P^{-1}$. As lengthscale we use the minor radius $L^* = a$. This allows to normalize the equations using $\tilde{u} = uT^*/L^*$, $\tilde{\nabla} = L^*\nabla$, and analogous for $\Delta, P, b, \rho, \partial_t$. Removing after normalization all tildes, for notational ease, we obtain the non-dimensional set of equations,

$$ \frac{\partial \mathbf{u}_P}{\partial t} + \mathbf{u}_P \cdot \nabla \mathbf{u}_P + \nabla P - \frac{1}{Re} \Delta \mathbf{u}_P = \qquad (3.1) $$

$$ u_T^2 / r e_r + b e_r + \beta \rho e_T \times \mathbf{u}_P, \qquad (3.2) $$

and

$$ \frac{\partial b}{\partial t} + \mathbf{u}_P \cdot \nabla b = \frac{1}{Pe} \Delta b + S. \qquad (3.3) $$

These two equations define the poloidal dynamics. For the toroidal velocity component we have in normalized form,

$$ \frac{\partial u_T}{\partial t} + \mathbf{u}_P \cdot \nabla u_T - \frac{1}{Re} \Delta u_T = -u_T u_r / r + \gamma \left[ u_T - \tau_{\sigma}^{-1} \frac{\langle u_T r \rangle}{R} \right] \qquad (3.4) $$

and for the passive scalar

$$ \frac{\partial \xi}{\partial t} + \mathbf{u}_P \cdot \nabla \xi = \frac{1}{Pe} \Delta \xi + f_{\xi}. \qquad (3.5) $$

In these equations we define,

$$ Re = \frac{C_P a^2}{\nu}, \quad Pe = \frac{C_P a^2}{\kappa} \qquad (3.6) $$

$$ \gamma = \frac{C_T}{C_P}, \quad \beta = \frac{\beta' a}{C_P}, \quad S = \frac{S'}{a C_P^2}. \qquad (3.7) $$

The parameter $\gamma = C_T/C_P$ measures the toroidal forcing strength compared to the poloidal forcing. This means that if $\gamma$ is small, the instability mechanisms mainly drive the fluid flow in the poloidal plane. This ratio $\gamma$ is the main control-parameter of our system.

### 3.2. Parameters

The major radius of the torus is $R = 2$ and the minor radius $a = 1$. The ratio $R/a = 2$ is of the order of magnitude of typical tokamaks. For instance the JET tokamak is characterized by an aspect ratio $R/a = 2.4$, ITER by a value close to three, while spheromaks have $R/a \approx 1$.

The Reynolds number is $Re = 5000$, which is a high enough value to ensure turbulent motion in our system. A change in its value does not qualitatively change the main results of the present investigation, as long as the flow remains turbulent. The Péclet number is chosen equal to the Reynolds number $Pe = Re$. The value of $C_P = 10$ is fixed and the forcing ratio $\gamma$ is varied in the range
$\gamma \in [0, 1.8]$. The value of $\beta = 0; 2; 8$. The relaxation time for the suppression of angular momentum is $\tau_\sigma = 0.25$.

For the active scalar $b$ the injection shell near the boundary is defined by inner and outer radii [$\rho_1 : \rho_2$] = [0.87a : 0.90a] and the value of the source term is $S = 0.8$. For the passive scalar $\xi$ the source term is confined to a circular surface of radius $\rho_\xi = 0.1$ in the center of the poloidal plane with injection rate $C_\xi = 0.1$.

### 3.3. Numerical set-up

Direct numerical simulations are performed using the Nek5000 code (Fischer et al. 2008), a robust and well-tested open source code, based on the spectral element method (Patera 1984). We solve the discretized Navier-Stokes equations and two scalar advection-diffusion equations. We impose simple non-slip boundary conditions on the circular walls of the numerical domain for the velocity, and trivial Neumann-conditions for the scalars.

All simulations of the axisymmetric system are performed on a 2D grid which allows fast computations compared to a full three-dimensional description. The computational domain is a disk representing a poloidal cross section of the tokamak (see Fig. 2). The numerical grid consists of 640 spectral elements, with $n = 12$ the order of Lagrangian interpolant polynomials. The time-step is adaptive with a Courant-Friedrichs-Lewy condition CFL= 0.3. All results are reported during statistically steady states.

## 4. Numerical experiments of the 2D3C-2D2C transition

Now that the system is modeled and the numerical set-up is specified, we will here discuss the results of our simulations.

### 4.1. Characterization of the 2D3C-2D2C transition

As we will show, enhanced confinement needs, from the fluid mechanics viewpoint, two ingredients. We will first focus on the first part, the transition from a 2D3C to a 2D2C flow. We illustrate this by changing the anisotropy of the forcing, $\gamma = C_T/C_P$. In Fig. 3(a) we show a time-series of the toroidal and poloidal kinetic energy for a representative case ($Re = 5000$, $\beta = 0$, $C_P = 10$).

For $t < 2850$ the flow is in the 2D3C regime, with a value $\gamma = 1.7$. During this time interval the order of magnitude of the two components of the kinetic energy is comparable with a somewhat more bursty behavior of the toroidal kinetic energy. At $t = 2850$ the strength of the toroidal forcing is instantaneously lowered resulting in $\gamma = 1.35$. This value of $\gamma$ is apparently below the critical value for the transition and the flow becomes purely poloidal as is illustrated by the purely poloidal dynamics in Fig. 3(d). Indeed, the value of the toroidal energy drops to zero. The poloidal energy is not significantly affected. Decreasing the value of the toroidal force-coefficient can therefore trigger the 2D2C state.

In Fig. 3(b) we report the results of a parameter-sweep for the parameter $\gamma = C_T/C_P$ for a fixed value of $C_P$. All the data for the energy corresponds to temporal averages in a statistically steady state. We observe, when increasing $\gamma$, a critical transition from the 2D2C state (characterized by $E_T/E_P = 0$) to a 2D3C state, where the toroidal energy is non-zero. The influence of the parameter $\beta$ will be discussed below.

Visualizations of the flow-field in the two regimes are shown in Fig. 3(c,d). The
$\gamma \in [0, 1.8]$. The value of $\beta = 0; 2; 8$. The relaxation time for the suppression of angular momentum is $\tau_\sigma = 0.25$.

For the active scalar $b$ the injection shell near the boundary is defined by inner and outer radii [$\rho_1 : \rho_2$] = [0.87a : 0.90a] and the value of the source term is $S = 0.8$. For the passive scalar $\xi$ the source term is confined to a circular surface of radius $\rho_\xi = 0.1$ in the center of the poloidal plane with injection rate $C_\xi = 0.1$.

### 3.3. Numerical set-up

Direct numerical simulations are performed using the Nek5000 code (Fischer et al. 2008), a robust and well-tested open source code, based on the spectral element method (Patera 1984). We solve the discretized Navier-Stokes equations and two scalar advection-diffusion equations. We impose simple non-slip boundary conditions on the circular walls of the numerical domain for the velocity, and trivial Neumann-conditions for the scalars.

All simulations of the axisymmetric system are performed on a 2D grid which allows fast computations compared to a full three-dimensional description. The computational domain is a disk representing a poloidal cross section of the tokamak (see Fig. 2). The numerical grid consists of 640 spectral elements, with $n = 12$ the order of Lagrangian interpolant polynomials. The time-step is adaptive with a Courant-Friedrichs-Lewy condition CFL= 0.3. All results are reported during statistically steady states.

## 4. Numerical experiments of the 2D3C-2D2C transition

Now that the system is modeled and the numerical set-up is specified, we will here discuss the results of our simulations.

### 4.1. Characterization of the 2D3C-2D2C transition

As we will show, enhanced confinement needs, from the fluid mechanics viewpoint, two ingredients. We will first focus on the first part, the transition from a 2D3C to a 2D2C flow. We illustrate this by changing the anisotropy of the forcing, $\gamma = C_T/C_P$. In Fig. 3(a) we show a time-series of the toroidal and poloidal kinetic energy for a representative case ($Re = 5000$, $\beta = 0$, $C_P = 10$).

For $t < 2850$ the flow is in the 2D3C regime, with a value $\gamma = 1.7$. During this time interval the order of magnitude of the two components of the kinetic energy is comparable with a somewhat more bursty behavior of the toroidal kinetic energy. At $t = 2850$ the strength of the toroidal forcing is instantaneously lowered resulting in $\gamma = 1.35$. This value of $\gamma$ is apparently below the critical value for the transition and the flow becomes purely poloidal as is illustrated by the purely poloidal dynamics in Fig. 3(d). Indeed, the value of the toroidal energy drops to zero. The poloidal energy is not significantly affected. Decreasing the value of the toroidal force-coefficient can therefore trigger the 2D2C state.

In Fig. 3(b) we report the results of a parameter-sweep for the parameter $\gamma = C_T/C_P$ for a fixed value of $C_P$. All the data for the energy corresponds to temporal averages in a statistically steady state. We observe, when increasing $\gamma$, a critical transition from the 2D2C state (characterized by $E_T/E_P = 0$) to a 2D3C state, where the toroidal energy is non-zero. The influence of the parameter $\beta$ will be discussed below.

Visualizations of the flow-field in the two regimes are shown in Fig. 3(c,d). The
<figure><img src="image_4.png" /><figcaption>Figure 3: (a) Time-evolution of the volume-averaged poloidal energy $E_P$ and toroidal energy $E_T$. For time $t < 2850$ the value $\gamma = 1.7$. For time $t \ge 2850$ the value the ratio of the forcing strength is lowered to $\gamma = 1.35$. The volume averaged energies illustrate a transition from a 2D3C (c) to a 2D2C state (d), respectively. The movement is in these visualizations plotted in the poloidal plane by colors indicating the strength of the stream-function. The toroidal velocity is illustrated by the out-of-plane morphology. (b) Influence of the forcing anisotropy on the ratio $E_T/E_P$ for $\beta = 0$. The two values of the forcing anisotropy $\gamma = 1.35; 1.7$ associated with the timeseries in Fig. 3 are indicated by red and green symbols, respectively.</figcaption></figure>

main feature is the non-zero value of the toroidal velocity fluctuations in Fig. 3(c). However, another outstanding feature is the tendency to self-organization. Indeed, as observed by inspecting the stream-function associated with the velocity pattern in the poloidal plane, in the 2D2C regime [Fig. 3(d)] a large scale self-organization is observed consisting of two counter-rotating toroidal vortex rings.

## 4.2. Assessment of the confinement quality of the flow

Indeed, the double toroidal vortex rings observed in Fig. 3(d) are a generic feature of fluid simulations in toroidal geometry (Bates & Montgomery 1998; Morales et al. 2012). Such a self-organization of the flow into two toroidal vortex rings does not seem beneficial for confinement in the center of the fusion-device, since the fluid or plasma between the large-scale structures will be rapidly expelled. We have tested this by measuring the turbulent diffusion of a passive scalar, injected in the center of the poloidal cross-section.

We solve the additional advection-diffusion equation (3.5), with a constant source term in the center of the poloidal cross-section. By measuring the average profile of the scalar, the confinement is quantified: a large value of the temperature
<figure><img src="image_4.png" /><figcaption>Figure 3: (a) Time-evolution of the volume-averaged poloidal energy $E_P$ and toroidal energy $E_T$. For time $t < 2850$ the value $\gamma = 1.7$. For time $t \ge 2850$ the value the ratio of the forcing strength is lowered to $\gamma = 1.35$. The volume averaged energies illustrate a transition from a 2D3C (c) to a 2D2C state (d), respectively. The movement is in these visualizations plotted in the poloidal plane by colors indicating the strength of the stream-function. The toroidal velocity is illustrated by the out-of-plane morphology. (b) Influence of the forcing anisotropy on the ratio $E_T/E_P$ for $\beta = 0$. The two values of the forcing anisotropy $\gamma = 1.35; 1.7$ associated with the timeseries in Fig. 3 are indicated by red and green symbols, respectively.</figcaption></figure>

main feature is the non-zero value of the toroidal velocity fluctuations in Fig. 3(c). However, another outstanding feature is the tendency to self-organization. Indeed, as observed by inspecting the stream-function associated with the velocity pattern in the poloidal plane, in the 2D2C regime [Fig. 3(d)] a large scale self-organization is observed consisting of two counter-rotating toroidal vortex rings.

## 4.2. Assessment of the confinement quality of the flow

Indeed, the double toroidal vortex rings observed in Fig. 3(d) are a generic feature of fluid simulations in toroidal geometry (Bates & Montgomery 1998; Morales et al. 2012). Such a self-organization of the flow into two toroidal vortex rings does not seem beneficial for confinement in the center of the fusion-device, since the fluid or plasma between the large-scale structures will be rapidly expelled. We have tested this by measuring the turbulent diffusion of a passive scalar, injected in the center of the poloidal cross-section.

We solve the additional advection-diffusion equation (3.5), with a constant source term in the center of the poloidal cross-section. By measuring the average profile of the scalar, the confinement is quantified: a large value of the temperature
<figure><img src="image_5.png" /><figcaption>Figure 4: (a) Stream-function patterns for 2D2C flows with two different values of the symmetry-breaking force ($\beta = 2$ and $\beta = 8$). (b) Ratio of the scalar profiles associated with a passive scalar injected in the center of the domain. In addition to values associated with (a) and (b) we also show the profile for $\beta = 0$. In this representation $T_H(\rho)$ is the scalar profile in the 2D2C regime and $T_L(\rho)$ the profile in the 2D3C regime. These profiles are obtained by averaging over time and over the poloidal angle $\theta$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: (a) Overview of the dependence of the system on the parameter $C_T/C_P$ for fixed $C_P$ and $\beta = 8$, where we also show how the confinement is enhanced by this transition, as measured by the temperature in the center of the toroidal domain. (b) Influence of the forcing anisotropy on the nature of the flow for three different values $\beta$, associated with the symmetry-breaking term.</figcaption></figure>

### 4.3. *The importance of symmetry breaking*

<footnote n="413">413 in the center corresponds to good confinement and, conversely, a low core-temperature indicates bad confinement. Indeed, as illustrated in Fig. 4, for $\beta = 0$.</footnote>

<footnote n="414">414 the confinement is changed at most 10% between the two regimes.</footnote>

<footnote n="415">415 Switching from a 2D3C state to a 2D2C flow is thus not enough to enhance</footnote>

<footnote n="417">417 the confinement properties of an axisymmetric toroidal fluid flow. The case of</footnote>

<footnote n="418">418 non-zero $\beta$, corresponding to the presence of an anisotropic force in the poloidal</footnote>

<footnote n="419">419 dynamics, will be discussed now.</footnote>

<footnote>420</footnote>

<footnote n="421">421 Indeed, one additional effect is needed to enhance the confinement. This is the</footnote>

<footnote n="422">422 symmetry breaking, allowing to modify the double vortex-pattern, observed in</footnote>

<footnote n="423">423 Fig. 3(d) to a concentric pattern in the poloidal plane. In tokamak plasmas,</footnote>

<footnote n="424">424 this symmetry breaking is associated with the strong radial gradients of density,</footnote>

<footnote n="425">425 pressure and temperature. We show the effect of an anisotropic force-term in</footnote>
<figure><img src="image_5.png" /><figcaption>Figure 4: (a) Stream-function patterns for 2D2C flows with two different values of the symmetry-breaking force ($\beta = 2$ and $\beta = 8$). (b) Ratio of the scalar profiles associated with a passive scalar injected in the center of the domain. In addition to values associated with (a) and (b) we also show the profile for $\beta = 0$. In this representation $T_H(\rho)$ is the scalar profile in the 2D2C regime and $T_L(\rho)$ the profile in the 2D3C regime. These profiles are obtained by averaging over time and over the poloidal angle $\theta$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: (a) Overview of the dependence of the system on the parameter $C_T/C_P$ for fixed $C_P$ and $\beta = 8$, where we also show how the confinement is enhanced by this transition, as measured by the temperature in the center of the toroidal domain. (b) Influence of the forcing anisotropy on the nature of the flow for three different values $\beta$, associated with the symmetry-breaking term.</figcaption></figure>

413 in the center corresponds to good confinement and, conversely, a low core- 414 temperature indicates bad confinement. Indeed, as illustrated in Fig. 4, for $\beta = 0$.

415 the confinement is changed at most 10% between the two regimes.

416 Switching from a 2D3C state to a 2D2C flow is thus not enough to enhance

417 the confinement properties of an axisymmetric toroidal fluid flow. The case of

418 non-zero $\beta$, corresponding to the presence of an anisotropic force in the poloidal

419 dynamics, will be discussed now.

### 4.3. *The importance of symmetry breaking*

420 Indeed, one additional effect is needed to enhance the confinement. This is the

421 symmetry breaking, allowing to modify the double vortex-pattern, observed in

422 Fig. 3(d) to a concentric pattern in the poloidal plane. In tokamak plasmas,

423 this symmetry breaking is associated with the strong radial gradients of density,

425 pressure and temperature. We show the effect of an anisotropic force-term in
426 Fig. 4. The presence of this force allows to re-organize the large-scale structuring in a more concentric pattern, beneficial for confinement. In Fig. 5(b) we show the results of a parameter-scan for $C_T/C_P$ for the values of $\beta = 0, 2, 8$. For all three values, a critical transition is observed as in Fig. 3(b) around a given ratio $\gamma$. The transition is therefore present for simulations with and without the symmetry breaking force, but the value of this critical ratio $\gamma$ decreases as a function of $\beta$.

427
428
429
430
431 Most importantly, the resulting $2D2C$ flow, for the non-zero values of $\beta$ considered here, confines the scalar significantly better. Indeed, in Fig. 5, it is observed that for a same constant scalar injection rate, the temperature in the center of the domain increases by a factor around two.

432
433
434
435
436
437
438
439
440
441 The transition of the flow is therefore triggered by the anisotropy of the forcing. This transition allows to enter a fully poloidal flow regime for small values of $C_T/C_P$. Such a purely poloidal flow has a tendency to self-organize. Indeed, in the absence of toroidal flow, the system can be described by purely two-dimensional hydrodynamics. The shape of the self-organized structures does depend on other factors, such as here the poloidal $\beta$-effect.

## 5. Conclusion

442
443
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447
448
449
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451
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454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473 From the results presented in the previous paragraphs, it can be concluded that a recently discovered critical transition in axisymmetric turbulence (Qin et al. 2020) survives in toroidal geometry, forced by instabilities near the toroidal boundaries. Indeed, this mimics in a crude way the dynamics of a tokamak, where a toroidally confined plasma develops instabilities at the boundaries where the pressure, density and temperature gradients are most important.

The present results share some properties with the LH-transition, but given the complexity of a tokamak plasma, the ideas cannot be carried over directly. In the present simplified fluid system to enhance confinement, one needs two ingredients. First, a dominance of the poloidal forcing over the toroidal forcing is required to be under a threshold for the critical $2D3C$ instability. Secondly, the self-organization resulting from the purely two-dimensional two-component-dynamics needs a symmetry-breaking mechanism, allowing the system to organize into a concentric pattern in the poloidal plane (zonal flows). The main observations can then be summarized by Fig. 5(a), showing simultaneously the dependence of the energy ratio and the confinement quality as a function of $\gamma$.

A feature which is deliberately removed from the dynamics by adding the damping in Eq. (2.12) is intrinsic rotation. Indeed, linear forcing mechanisms lead, in toroidal geometry, quite easily to build-up of toroidal rotation. While this is certainly an important feature in tokamak operation, the addition of a global rotation in the toroidal direction does not necessarily add to the three-dimensional character of the flow. Indeed, in the presence of global rotation, the present transition should be assessed in a frame-of-reference moving with the global rotation. In the presence of helically twisted field-lines this will result in a far less obvious observation of a transition from $2D2D$ to $2D3C$ and an investigation of this is left for future studies.

An important point to be improved if the present set-up is to be compared to more realistic approaches is to replace the artificial forcing terms by a self-consistent flux-driven forcing, with an injection of temperature in the core of the plasma. This would possibly allow to add to the critical transition some of the tokamak features which are absent in the present set-up, such as hysteresis
426 Fig. 4. The presence of this force allows to re-organize the large-scale structuring in a more concentric pattern, beneficial for confinement. In Fig. 5(b) we show the results of a parameter-scan for $C_T/C_P$ for the values of $\beta = 0, 2, 8$. For all three values, a critical transition is observed as in Fig. 3(b) around a given ratio $\gamma$. The transition is therefore present for simulations with and without the symmetry breaking force, but the value of this critical ratio $\gamma$ decreases as a function of $\beta$.

427
428
429
430
431 Most importantly, the resulting $2D2C$ flow, for the non-zero values of $\beta$ considered here, confines the scalar significantly better. Indeed, in Fig. 5, it is observed that for a same constant scalar injection rate, the temperature in the center of the domain increases by a factor around two.

432
433
434
435
436
437
438
439
440
441 The transition of the flow is therefore triggered by the anisotropy of the forcing. This transition allows to enter a fully poloidal flow regime for small values of $C_T/C_P$. Such a purely poloidal flow has a tendency to self-organize. Indeed, in the absence of toroidal flow, the system can be described by purely two-dimensional hydrodynamics. The shape of the self-organized structures does depend on other factors, such as here the poloidal $\beta$-effect.

## 5. Conclusion

442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473 From the results presented in the previous paragraphs, it can be concluded that a recently discovered critical transition in axisymmetric turbulence (Qin et al. 2020) survives in toroidal geometry, forced by instabilities near the toroidal boundaries. Indeed, this mimics in a crude way the dynamics of a tokamak, where a toroidally confined plasma develops instabilities at the boundaries where the pressure, density and temperature gradients are most important.

The present results share some properties with the LH-transition, but given the complexity of a tokamak plasma, the ideas cannot be carried over directly. In the present simplified fluid system to enhance confinement, one needs two ingredients. First, a dominance of the poloidal forcing over the toroidal forcing is required to be under a threshold for the critical $2D3C$ instability. Secondly, the self-organization resulting from the purely two-dimensional two-component-dynamics needs a symmetry-breaking mechanism, allowing the system to organize into a concentric pattern in the poloidal plane (zonal flows). The main observations can then be summarized by Fig. 5(a), showing simultaneously the dependence of the energy ratio and the confinement quality as a function of $\gamma$.

A feature which is deliberately removed from the dynamics by adding the damping in Eq. (2.12) is intrinsic rotation. Indeed, linear forcing mechanisms lead, in toroidal geometry, quite easily to build-up of toroidal rotation. While this is certainly an important feature in tokamak operation, the addition of a global rotation in the toroidal direction does not necessarily add to the three-dimensional character of the flow. Indeed, in the presence of global rotation, the present transition should be assessed in a frame-of-reference moving with the global rotation. In the presence of helically twisted field-lines this will result in a far less obvious observation of a transition from $2D2D$ to $2D3C$ and an investigation of this is left for future studies.

An important point to be improved if the present set-up is to be compared to more realistic approaches is to replace the artificial forcing terms by a self-consistent flux-driven forcing, with an injection of temperature in the core of the plasma. This would possibly allow to add to the critical transition some of the tokamak features which are absent in the present set-up, such as hysteresis
474 between increasing or decreasing the drive and the fact that zonal-flows can
475 exhibit a predator-prey like dynamics (Gürcan & Diamond 2015).

476 In the present set-up it is not the zonal flows which trigger the transition: 477 they are the consequence of the 2D2C nature of the flow. In the 2D3C mode we 478 observe a forward energy cascade, which will destroy the coherence of large-scale 479 structures, preventing thereby the emergence of zonal flows. Once these zonal 480 flows appear, they enhance the confinement. With respect to these observations, 481 a link can possibly be found between the importance of the poloidal forcing to 482 reach a mode of enhanced confinement and the observation that in gyrokinetic 483 simulations the injection of vorticity can trigger transport-barriers and thereby 484 improve confinement (Strugarek et al. 2013; Lo-Cascio et al. 2022).

485 The observations reported here are simple and robust. We have not added 486 any more physics to the system than an axisymmetric fluid-description with 487 linear force terms. We think that this identification of the essential ingredients 488 of the transition is the most important insight that we have gained. The fact 489 that the observations do not contain any precise plasma-instability, magnetic- 490 field structure or kinetic effects allows to transpose the present observations to 491 more specific plasma configurations.

492 Note finally that while magnetohydrodynamical and kinetic effects are 493 obviously fundamental in understanding the details of the LH transition 494 (Connor & Wilson 2000), the objective here is to show that a transition observed 495 in a minimal fluid dynamics model can reproduce certain of its characteristics. 496 Further understanding can be gained by transposing these ideas to a more 497 realistic setting. For instance, in future experimental campaigns it can be tried 498 to, either enhance poloidal energy injection, or to reduce toroidal fluctuations, 499 leading to possibly unexplored magnetic fusion confinement protocols.

500 Acknowledgements. All simulations were carried out using the facilities of the PMCS2I (École 501 Centrale de Lyon).

502 For the purpose of Open Access, a CC-BY public copyright licence has been applied by the 503 authors to the present document and will be applied to all subsequent versions up to the Author 504 Accepted Manuscript arising from this submission.

505 Declaration of interests. The authors report no conflict of interest.

REFERENCES

506 ALEXAKIS, A. 2011 Two-dimensional behavior of three-dimensional magnetohydrodynamic flow 507 with a strong guiding field. *Phys. Rev. E* **84**, 056330.

508 BATES, JASON W & MONTGOMERY, DAVID C 1998 Toroidal visco-resistive magnetohydrody-509 namic steady states contain vortices. *Physics of Plasmas* **5** (7), 2649–2653.

510 BAYLIS, JA & HUNT, JCR 1971 Mhd flow in an annular channel; theory and experiment. 511 *Journal of Fluid Mechanics* **48** (3), 423–428.

512 BIGLARI, H., DIAMOND, P. H. & TERRY, P. W. 1990 Influence of sheared poloidal rotation on 513 edge turbulence. *Physics of Fluids B: Plasma Physics* **2** (1), 1–4.

514 BIGOT, B. & GALTIER, S. 2011 Two-dimensional state in driven magnetohydrodynamic 515 turbulence. *Phys. Rev. E* **83**, 026405.

516 BISKAMP, DIETER 1997 *Nonlinear magnetohydrodynamics*. Cambridge University Press.

517 BOYD, TJ, BOYD, TJM & SANDERSON, JJ 2003 *The physics of plasmas*. Cambridge University 518 Press.

519 CONNOR, JW & WILSON, HR 2000 A review of theories of the lh transition. *Plasma physics 520 and controlled fusion* **42** (1), R1.

521 DIAMOND, PATRICK H, ITOH, SANAE-I & ITOH, KIMITAKA 2010 *Modern Plasma Physics: 522 Volume 1, Physical Kinetics of Turbulent Plasmas*. Cambridge University Press.
474 between increasing or decreasing the drive and the fact that zonal-flows can
475 exhibit a predator-prey like dynamics (Gürcan & Diamond 2015).

476 In the present set-up it is not the zonal flows which trigger the transition: 477 they are the consequence of the 2D2C nature of the flow. In the 2D3C mode we 478 observe a forward energy cascade, which will destroy the coherence of large-scale 479 structures, preventing thereby the emergence of zonal flows. Once these zonal 480 flows appear, they enhance the confinement. With respect to these observations, 481 a link can possibly be found between the importance of the poloidal forcing to 482 reach a mode of enhanced confinement and the observation that in gyrokinetic 483 simulations the injection of vorticity can trigger transport-barriers and thereby 484 improve confinement (Strugarek et al. 2013; Lo-Cascio et al. 2022).

485 The observations reported here are simple and robust. We have not added 486 any more physics to the system than an axisymmetric fluid-description with 487 linear force terms. We think that this identification of the essential ingredients 488 of the transition is the most important insight that we have gained. The fact 489 that the observations do not contain any precise plasma-instability, magnetic- 490 field structure or kinetic effects allows to transpose the present observations to 491 more specific plasma configurations.

492 Note finally that while magnetohydrodynamical and kinetic effects are 493 obviously fundamental in understanding the details of the LH transition 494 (Connor & Wilson 2000), the objective here is to show that a transition observed 495 in a minimal fluid dynamics model can reproduce certain of its characteristics. 496 Further understanding can be gained by transposing these ideas to a more 497 realistic setting. For instance, in future experimental campaigns it can be tried 498 to, either enhance poloidal energy injection, or to reduce toroidal fluctuations, 499 leading to possibly unexplored magnetic fusion confinement protocols.

500 Acknowledgements. All simulations were carried out using the facilities of the PMCS2I (École 501 Centrale de Lyon).

502 For the purpose of Open Access, a CC-BY public copyright licence has been applied by the 503 authors to the present document and will be applied to all subsequent versions up to the Author 504 Accepted Manuscript arising from this submission.

505 Declaration of interests. The authors report no conflict of interest.

REFERENCES

506 ALEXAKIS, A. 2011 Two-dimensional behavior of three-dimensional magnetohydrodynamic flow 507 with a strong guiding field. *Phys. Rev. E* **84**, 056330.

508 BATES, JASON W & MONTGOMERY, DAVID C 1998 Toroidal visco-resistive magnetohydrody-509 namic steady states contain vortices. *Physics of Plasmas* **5** (7), 2649–2653.

510 BAYLIS, JA & HUNT, JCR 1971 Mhd flow in an annular channel; theory and experiment. 511 *Journal of Fluid Mechanics* **48** (3), 423–428.

512 BIGLARI, H., DIAMOND, P. H. & TERRY, P. W. 1990 Influence of sheared poloidal rotation on 513 edge turbulence. *Physics of Fluids B: Plasma Physics* **2** (1), 1–4.

514 BIGOT, B. & GALTIER, S. 2011 Two-dimensional state in driven magnetohydrodynamic 515 turbulence. *Phys. Rev. E* **83**, 026405.

516 BISKAMP, DIETER 1997 *Nonlinear magnetohydrodynamics*. Cambridge University Press.

517 BOYD, TJ, BOYD, TJM & SANDERSON, JJ 2003 *The physics of plasmas*. Cambridge University 518 Press.

519 CONNOR, JW & WILSON, HR 2000 A review of theories of the lh transition. *Plasma physics 520 and controlled fusion* **42** (1), R1.

521 DIAMOND, PATRICK H, ITOH, SANAE-I & ITOH, KIMITAKA 2010 *Modern Plasma Physics: 522 Volume 1, Physical Kinetics of Turbulent Plasmas*. Cambridge University Press.
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529 FAVIER, BENJAMIN, GODEFERD, FABIEN S, CAMBON, CLAUDE & DELACHE, ALEXANDRE 2010 On the two-dimensionalization of quasistatic magnetohydrodynamic turbulence. *Physics of Fluids* **22** (7), 075104.

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525 DIF-PRADALIER, GUILHEM, GHENDRIH, PHILIPPE, SARAZIN, YANICK, CASCHERA, ELISABETTA, CLAIRET, FRÉDÉRIC, CAMENEN, YANN, DONNEL, PETER, GARBET, XAVIER, GRANDGIRARD, VIRGINIE, MUNSCHY, YANN & OTHERS 2022 Transport barrier onset and edge turbulence shortfall in fusion plasmas. *Communications Physics* **5** (1), 1–12.

529 FAVIER, BENJAMIN, GODEFERD, FABIEN S, CAMBON, CLAUDE & DELACHE, ALEXANDRE 2010 On the two-dimensionalization of quasistatic magnetohydrodynamic turbulence. *Physics of Fluids* **22** (7), 075104.

532 FISCHER, PAUL F, LOTTES, JAMES W & KERKEMEIER, STEFAN G 2008 nek5000 web page.

533 FYFE, DAVID & MONTGOMERY, DAVID 1979 Possible inverse cascade behavior for drift-wave turbulence. *The Physics of Fluids* **22** (2), 246–248.

535 GALLET, BASILE & DOERING, CHARLES R 2015a Exact two-dimensionalization of low-magnetic-reynolds-number flows subject to a strong magnetic field. *Journal of Fluid Mechanics* **773**, 154–177.

538 GALLET, B. & DOERING, C. R. 2015b Exact two-dimensionalization of low-magnetic-reynolds-number flows subject to a strong magnetic field. *Journal of Fluid Mechanics* **773**, 154–177.

540 GARBET, X, MANTICA, P, ANGIONI, C, ASP, E, BARANOV, Y, BOURDELLE, C, BUDNY, R, CRISANTI, F, CORDEY, G, GARZOTTI, L & OTHERS 2004 Physics of transport in tokamaks. *Plasma Physics and Controlled Fusion* **46** (12B), B557.

543 GOERLER, TOBIAS, LAPILLONNE, XAVIER, BRUNNER, STEPHAN, DANNERT, TILMAN, JENKO, FRANK, MERZ, FLORIAN & TOLD, DANIEL 2011 The global version of the gyrokinetic turbulence code gene. *Journal of Computational Physics* **230** (18), 7053–7071.

546 GRANDGIRARD, VIRGINIE, BRUNETTI, MAURA, BERTRAND, PIERRE, BESSE, NICOLAS, GARBET, XAVIER, GHENDRIH, PHILIPPE, MANFREDI, GIOVANNI, SARAZIN, YANICK, SAUTER, OLIVIER, SONNENDRÜCKER, ERIC & OTHERS 2006 A drift-kinetic semi-lagrangian 4d code for ion turbulence simulation. *Journal of Computational Physics* **217** (2), 395–423.

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558 HUNT, JCR & DURBIN, PA 1999 Perturbed vortical layers and shear sheltering. *Fluid dynamics research* **24** (6), 375.

560 HUNT, JULIAN CR & CARRUTHERS, DAVID J 1990 Rapid distortion theory and the ‘problems’ of turbulence. *Journal of Fluid Mechanics* **212**, 497–532.

562 KRAICHNAN, ROBERT H 1967 Inertial ranges in two-dimensional turbulence. *The Physics of Fluids* **10** (7), 1417–1423.

564 KRAICHNAN, ROBERT H & MONTGOMERY, DAVID 1980 Two-dimensional turbulence. *Reports on Progress in Physics* **43** (5), 547.

566 LAWSON, J D 1957 Some criteria for a power producing thermonuclear reactor. *Proceedings of the Physical Society. Section B* **70** (1), 6–10.

568 LEPROVOST, NICOLAS, DUBRULLE, BÉRENGÈRE & CHAVANIS, P-H 2006 Dynamics and thermodynamics of axisymmetric flows: Theory. *Physical Review E* **73** (4), 046308.

570 LIEWER, PAULETT C 1985 Measurements of microturbulence in tokamaks and comparisons with theories of turbulence and anomalous transport. *Nuclear Fusion* **25** (5), 543.

572 LINKMANN, MORITZ, BUZZICOTTI, MICHELE & BIFERALE, LUCA 2018 Nonuniversal behaviour of helical two-dimensional three-component turbulence. *The European Physical Journal E* **41**, 1–9.

575 LO-CASCIO, G, GRAVIER, ETIENNE, RÉVEILLÉ, T, LESUR, M, SARAZIN, Y, GARBET, X, VERMARE, L, LIM, K, GUILLEVIC, A & GRANDGIRARD, V 2022 Transport barrier in 5d gyrokinetic flux-driven simulations. *Nuclear Fusion* **62** (12), 126026.

578 McWILLIAMS, JAMES C 1984 The emergence of isolated coherent vortices in turbulent flow. *Journal of Fluid Mechanics* **146**, 21–43.
580 MOFFATT, HK 1967 On the suppression of turbulence by a uniform magnetic field. *Journal of fluid Mechanics* **28** (3), 571–592.

582 MORALES, JORGE A, BOS, WOUTER JT, SCHNEIDER, KAI & MONTGOMERY, DAVID C 2012
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585 NASO, AURORE, MONCHAUX, ROMAIN, CHAVANIS, PIERRE-HENRI & DUBRULLE, BÉRENGÈRE 2010 Statistical mechanics of beltrami flows in axisymmetric geometry: Theory reexamined. *Physical Review E* **81** (6), 066318.

588 PARET, JÉRÔME & TABELING, PATRICK 1997 Experimental observation of the two-dimensional inverse energy cascade. *Physical review letters* **79** (21), 4162.

590 PATERA, ANTHONY T 1984 A spectral element method for fluid dynamics: Laminar flow in a channel expansion. *Journal of Computational Physics* **54** (3), 468 – 488.

592 POYÉ, ALEXANDRE, AGULLO, OLIVIER, PLIHON, NICOLAS, BOS, WOUTER JT, DÉSANGLES, VICTOR & BOUSSELIN, GUILLAUME 2020 Scaling laws in axisymmetric magnetohydrodynamical duct flows. *Physical Review Fluids* **5** (4), 043701.

595 QIN, ZECONG, FALLER, HUGUES, DUBRULLE, BÉRENGÈRE, NASO, AURORE & BOS, WOUTER JT 2020 Transition from non-swirling to swirling axisymmetric turbulence. *Physical Review Fluids* **5** (6), 064602.

598 QU, BO, BOS, WOUTER JT & NASO, AURORE 2017 Direct numerical simulation of axisymmetric turbulence. *Physical Review Fluids* **2** (9), 094608.

600 QU, BO, NASO, AURORE & BOS, WOUTER JT 2018 Cascades of energy and helicity in axisymmetric turbulence. *Physical Review Fluids* **3** (1), 014607.

602 RICE, JE, INCE-CUSHMAN, A, DEGRASSIE, JS, ERIKSSON, L-G, SAKAMOTO, Y, SCARABOSIO, A, BORTOLON, A, BURRELL, KH, DUVAL, BP, FENZI-BONIZEC, C & OTHERS 2007 Inter-machine comparison of intrinsic toroidal rotation in tokamaks. *Nuclear Fusion* **47** (11), 1618.

606 SHAING, K. C. & CRUME, E. C. 1989 Bifurcation theory of poloidal rotation in tokamaks: A model for l-h transition. *Phys. Rev. Lett.* **63**, 2369–2372.

608 SHATS, MG, XIA, HUA & PUNZMANN, HORST 2005 Spectral condensation of turbulence in plasmas and fluids and its role in low-to-high phase transitions in toroidal plasma. *Physical Review E* **71** (4), 046409.

611 SHATS, MG, XIA, HUA, PUNZMANN, HORST & FALKOVICH, GREGORY 2007 Suppression of turbulence by self-generated and imposed mean flows. *Physical review letters* **99** (16), 164502.

614 SOMMERIA, JOEL 1986 Experimental study of the two-dimensional inverse energy cascade in a square box. *Journal of fluid mechanics* **170**, 139–168.

616 STRUGAREK, A, SARAZIN, Y, ZARZOSO, D, ABITEBOUL, J, BRUN, AS, CARTIER-MICHAUD, T, DIF-PRADALIER, GUILHEM, GARBET, X, GHENDRIH, PH, GRANDGIRARD, V & OTHERS 2013 Ion transport barriers triggered by plasma polarization in gyrokinetic simulations. *Plasma Physics and Controlled Fusion* **55** (7), 074013.

620 TERRY, P. W. 2000 Suppression of turbulence and transport by sheared flow. *Rev. Mod. Phys.* **72**, 109–165.

622 THALABARD, SIMON, DUBRULLE, BÉRENGÈRE & BOUCHET, FREDDY 2014 Statistical mechanics of the 3d axisymmetric euler equations in a taylor–couette geometry. *Journal of Statistical Mechanics: Theory and Experiment* **2014** (1), P01005.

625 WAGNER, FRITZ 2007 A quarter-century of h-mode studies. *Plasma Physics and Controlled Fusion* **49** (12B), B1.

627 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., RAPP, H., RÖHR, H., SCHNEIDER, F., SILLER, G., SPETH, E., STÄBLER, A., STEUER, K. H., VENUS, G., VOLLMER, O. & YÜ, Z. 1982 Regime of improved confinement and high beta in neutral-beam-heated divertor discharges of the asdex tokamak. *Phys. Rev. Lett.* **49**, 1408–1412.

635 YIN, XI-YUAN, AGOUA, WESLEY, WU, TONG & BOS, WOUTER JT 2024 The influence of the vorticity-scalar correlation on mixing. *arXiv preprint arXiv:2402.16762*.
580 MOFFATT, HK 1967 On the suppression of turbulence by a uniform magnetic field. *Journal of fluid Mechanics* **28** (3), 571–592.

582 MORALES, JORGE A, BOS, WOUTER JT, SCHNEIDER, KAI & MONTGOMERY, DAVID C 2012
Intrinsic rotation of toroidally confined magnetohydrodynamics. *Physical review letters*
**109** (17), 175002.

585 NASO, AURORE, MONCHAUX, ROMAIN, CHAVANIS, PIERRE-HENRI & DUBRULLE, BÉRENGÈRE 2010 Statistical mechanics of beltrami flows in axisymmetric geometry: Theory reexamined. *Physical Review E* **81** (6), 066318.

588 PARET, JÉRÔME & TABELING, PATRICK 1997 Experimental observation of the two-dimensional inverse energy cascade. *Physical review letters* **79** (21), 4162.

590 PATERA, ANTHONY T 1984 A spectral element method for fluid dynamics: Laminar flow in a channel expansion. *Journal of Computational Physics* **54** (3), 468 – 488.

592 POYÉ, ALEXANDRE, AGULLO, OLIVIER, PLIHON, NICOLAS, BOS, WOUTER JT, DÉSANGLES, VICTOR & BOUSSELIN, GUILLAUME 2020 Scaling laws in axisymmetric magnetohydrodynamical duct flows. *Physical Review Fluids* **5** (4), 043701.

595 QIN, ZECONG, FALLER, HUGUES, DUBRULLE, BÉRENGÈRE, NASO, AURORE & BOS, WOUTER JT 2020 Transition from non-swirling to swirling axisymmetric turbulence. *Physical Review Fluids* **5** (6), 064602.

598 QU, BO, BOS, WOUTER JT & NASO, AURORE 2017 Direct numerical simulation of axisymmetric turbulence. *Physical Review Fluids* **2** (9), 094608.

600 QU, BO, NASO, AURORE & BOS, WOUTER JT 2018 Cascades of energy and helicity in axisymmetric turbulence. *Physical Review Fluids* **3** (1), 014607.

602 RICE, JE, INCE-CUSHMAN, A, DEGRASSIE, JS, ERIKSSON, L-G, SAKAMOTO, Y, SCARABOSIO, A, BORTOLON, A, BURRELL, KH, DUVAL, BP, FENZI-BONIZEC, C & OTHERS 2007 Inter-machine comparison of intrinsic toroidal rotation in tokamaks. *Nuclear Fusion* **47** (11), 1618.

606 SHAING, K. C. & CRUME, E. C. 1989 Bifurcation theory of poloidal rotation in tokamaks: A model for l-h transition. *Phys. Rev. Lett.* **63**, 2369–2372.

608 SHATS, MG, XIA, HUA & PUNZMANN, HORST 2005 Spectral condensation of turbulence in plasmas and fluids and its role in low-to-high phase transitions in toroidal plasma. *Physical Review E* **71** (4), 046409.

611 SHATS, MG, XIA, HUA, PUNZMANN, HORST & FALKOVICH, GREGORY 2007 Suppression of turbulence by self-generated and imposed mean flows. *Physical review letters* **99** (16), 164502.

614 SOMMERIA, JOEL 1986 Experimental study of the two-dimensional inverse energy cascade in a square box. *Journal of fluid mechanics* **170**, 139–168.

616 STRUGAREK, A, SARAZIN, Y, ZARZOSO, D, ABITEBOUL, J, BRUN, AS, CARTIER-MICHAUD, T, DIF-PRADALIER, GUILHEM, GARBET, X, GHENDRIH, PH, GRANDGIRARD, V & OTHERS 2013 Ion transport barriers triggered by plasma polarization in gyrokinetic simulations. *Plasma Physics and Controlled Fusion* **55** (7), 074013.

620 TERRY, P. W. 2000 Suppression of turbulence and transport by sheared flow. *Rev. Mod. Phys.* **72**, 109–165.

622 THALABARD, SIMON, DUBRULLE, BÉRENGÈRE & BOUCHET, FREDDY 2014 Statistical mechanics of the 3d axisymmetric euler equations in a taylor–couette geometry. *Journal of Statistical Mechanics: Theory and Experiment* **2014** (1), P01005.

625 WAGNER, FRITZ 2007 A quarter-century of h-mode studies. *Plasma Physics and Controlled Fusion* **49** (12B), B1.

627 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., RAPP, H., RÖHR, H., SCHNEIDER, F., SILLER, G., SPETH, E., STÄBLER, A., STEUER, K. H., VENUS, G., VOLLMER, O. & YÜ, Z. 1982 Regime of improved confinement and high beta in neutral-beam-heated divertor discharges of the asdex tokamak. *Phys. Rev. Lett.* **49**, 1408–1412.

635 YIN, XI-YUAN, AGOUA, WESLEY, WU, TONG & BOS, WOUTER JT 2024 The influence of the vorticity-scalar correlation on mixing. *arXiv preprint arXiv:2402.16762*.