Laurent Prat

Pascal Guiraud

Luc Rigal

Christophe Gourdon

A one

Keywords: Twin-screw extruder, Solid/liquid extraction, Two-phase residence time distribution, Transport phenomena, Raw plant substrate

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The twin-screw extruders (TSE) ability to carry out simultaneous combination of several unit operations in a single continuous reactor makes it very valuable for the valorization of agroressources wastes [START_REF] Gourdon | Technologie d'extrusion bi-vis et fractionnement de la matie `re ve ´ge ´tale, 40 ans d'extrusion bi-vis chez CLEXTRAL[END_REF]. For instance, N'Diaye et al. [START_REF] N'diaye | Fractionnement de la matie `re ve ´ge ´tale: mise au point d'un proce ´de ´thermo-me ´cano-chimique et mode ´lisation du re ´ac[END_REF][START_REF] N'diaye | Extraction of hemicelluloses from poplar, populus tremuloides, using an extruder-type twin screw reactor: influence of the main factors[END_REF], but also Bouzid et al. [START_REF] Bouzid | Extraction des huiles essentielles par des techniques non conventionnelles[END_REF] or Isobe et al. [START_REF] Isobe | A new twin-screw press design for oil extraction of dehulled sunflower seeds[END_REF] have demonstrated the feasibility of extraction processes from various raw plant matters.

Part of Fig. 1 shows the process operating principle for this type of two-phase application. This type of machine consists of a seven eight-shaped elements barrel enclosing two corotating intermeshing screws. Raw plant matter and extraction solvent are fed into the barrel by the first transport screws and are mixed together in the transport and kneading elements. As the liquid is absorbed by the matter, solubilization occurs. Further down, reversed transport screw elements (RSE) cause severe compression of the vegetable matter and allow the separation of the two phases. Liquid phase is then retrieved through a barrel filter located just upstream from the RSE.

As it is clearly demonstrated on Fig. 2 for the case under consideration in this paper (extraction of hemicelluloses from hardwood populus via alkaline solubilization), TSE allows to obtain higher extraction yields for reduced contact times in comparison with a batch reactor. This comparison points out the main interest of performing vegetable extraction processes with a TSE.

In a previous work, Prat et al. have proposed a description of the functioning of the TSE for such twophase processes, focusing on Residence Time Distributions (RTD) of the phases and on the relative contribution of the screw elements to the RTD under the influence of the operating parameters (feed rates, screw rotational speed, NaOH concentration in the alkaline solvent).

If the ability of TSE to carry out high performance solutes extraction from vegetable matters does not have to be more demonstrated, no tool allows, up to now, to predict the extraction yields of this process as a function of the operating parameters. The aim of this work is to propose a one dimensional model for this prediction, on the basis of the example of hemicelluloses extraction from hardwood populus by NaOH aqueous solution. This model couples a one dimensional approach for the solid transport along the TSE and a kinetic description of the solubilization process which has been established via batch experiments. These kind of approach has been successfully applied to a lot of other polyphasic processes like liquid Á/liquid extraction columns for instance.

The first paragraph proposes a system analysis, leading to the modeling methodology. In the second paragraph, the models describing transport, reaction and mass transfer are developed. The third paragraph presents the matter, the material and the experimental methods. Then, in the last paragraph, the model results are presented and compared with the experimental data.

Scope and modeling principle

2.1. Two-phase transport by the TSE Prat et al. [START_REF] Prat | Two phase residence time distribution in a modified twin-screw extruder[END_REF] have proposed a qualitative description of the functioning of the TSE and studied the two phase RTD in this particular application. The reactor has been separated into two functional parts: before the RSE, the two phases (liquid and impregnated solid) are present, meanwhile from the RSE to the die, only impregnated solid can be found. In this second part, the underpressured elements (i.e. the RSE, the half transport screw just before the RSE and the die) are totally filled with the solid matter, which is not the case in the first part of the reactor. The TSE is then not totally full of matter under classical working conditions. Visual observation and RTD study prove that the solid accumulates in the kneading elements and in the RSE (as can be seen in Fig. 3 for the central functioning point of the experimental domain). This confirms the known restrictive effects [START_REF] Altamore | An analysis of residence time distribution pattern in a twin-screw cooking extruder[END_REF] of these elements in term of transport. As far as solvent extraction is concerned, the first part is the more important because it is where the extractive reaction occurs. The second part, from the RSE to the die, is necessary for the solid/liquid final separation and, as a consequence, for the correct functioning of the reactor.

Reactive extraction from raw plant substrates

A description and a modeling for the reactive extraction of hemicelluloses from poplar wood in batch reactor have been proposed by Prat et al. [START_REF] Prat | Solid Á/liquid reactive extraction with raw plant substrate[END_REF]. Poplar cells are assumed to be cylinders hollowed by a central canal. The hemicelluloses are situated in the cells walls [START_REF] Sjo | Wood Chemistry, Fundamentals and Applications[END_REF].

The kinetic and transport phenomena taking place during the solid Á/liquid extraction of hemicelluloses from popular wood are separated in five steps [START_REF] Prat | Solid Á/liquid reactive extraction with raw plant substrate[END_REF], as described in Fig. 4: Á/ solvent (NaOH solution) axial diffusion from the bulk phase to the central canal in the cells; Á/ solvent radial diffusion from the central canal to the reactional sites into the walls (primary and secondary walls); Á/ NaOH reaction, liberation and solubilization of the hemicelluloses; Á/ hemicelluloses radial diffusion into the walls; Á/ hemicelluloses diffusion from the central canal to the bulk phase.

The conducted experiments gave data for hemicelluloses concentrations in the central canal and in the bulk phase as a function of time in different operating conditions in term of temperature and NaOH concentration.

From these experimental data, a parameter identification allowed to quantify the diffusion coefficients and the reaction kinetic law which is of the form:

( dC L (t) dt 0kC L (t) 00:047s 3:4 e (18900=RT C L (t) (1)
where C L (t ) is the concentration of the non-solubilized hemicelluloses, s , the NaOH concentration in the solvent and T , the temperature.

Modeling principle

The present work separates the reactor into local functional elements. Two different kinds of elements will be distinguished, the elements where the reaction occurs (the transport screw elements) and the elements where compression and then solid/liquid separation is combined with the reaction (the kneading elements, the reversed screw elements and the half transport element before the RSE).

In a first step, the solid residence time is calculated for each screw element. The equations are described in paragraph Section 3.1.

Then, the resulted local residence time is used with the kinetic law (1) to calculate the liberated mass of hemicelluloses from the wood structure.

Finally, in the restrictive elements, a local mass balance is done in order to quantify the mass transfer and obtain the hemicelluloses mass in the liquid phase. Some simplifications will be done, considering the mechanical effect of the TSE. This model processing is described in Fig. 1.

Global process model

Transport model

Some hypothesis are made in the transport model in order to simplify the coupled phenomena, the physicochemical properties and changes of these properties.

All the elements of the screw are considered as independent from the others (there is no retroaction of the behavior). Consequently, the transport phenomena properties linked to the element only depend on their geometry. This hypothesis has been justified by a first local approach in previous work [START_REF] Prat | Two phase residence time distribution in a modified twin-screw extruder[END_REF] and give a good representation of the matter transport.

The transport will be described in terms of mean residence times. It will then be possible to obtain the two phase masses (the mass is the product of the mean residence time by the mass flow), and, with the values of the densities, the occupied volumes. As the model consider each screw element as an independent module, the physico-chemical properties of the phases keep a constant value in each of the screw elements.

The transport screw elements

Except for the half transport screw just before the RSE, it will be considered that the matter is pushed by the screws and moves forward at a maximum mean velocity equals to Np (with N , the screw rotational speed, and p the pitch). To represent the actual velocity of the matter, a slip velocity of the matter with the screw is introduced. It will permit to consider the friction of the matter with the screw and with the barrel. The residence time in the transport screw elements t screw will then be modeled by:

t screw 0 L Np(1 ( a) (2)
where, L is the length of the element; N , the screw rotational speed; p , the pitch of the element and Npa the slip velocity of the matter relative to the screw. As this expression does not take the volume into account, a test will be made in order to verify that the matter volume implied by t screw does not exceed the maximum authorized by the geometry of the system. As this never happen, no other equation will be propose in this work.

The restrictive elements in the first part of the reactor

The restrictive elements in the first part of the reactor (i.e. the two kind of kneading elements) are totally filled. As the radial and axial mixing in these elements are important, the general expression of the mean residence time t can be expressed as follows:

t0 V V (3)
where, V is the volume of the element; and V the volumic flow of matter in the element. Since there are two phases in these elements, the volume is shared between them. The volumic flows (/ V sol / for the solid and V liq for the liquid) are of the form:

V sol 0 Q S Q P (4)
and

V liq 0 Q L ( Q S ((1 ( v)=v) r liq (5)
where, Q L is the mass flow of liquid in the reactor; Q S , the mass flow of solid; v the solid weight percentage in the solid phase in the element; r P , the apparent density of the solid particles and r liq , the density of the liquid phase. Furthermore, the total volume V of the element is:

V 0V sol 'V liq (6)
where, V sol is the volume occupied by the impregnated solid; and V liq , the volume occupied by the liquid phase.

As the phases hold up is unknown, b , the ratio between the solid phase and the liquid phase velocities can be expressed by:

b 0 V sol V L V liq V S 0 t liq t sol ( 7
)
where t sol , is the solid phase residence time in the kneading element; and t liq, the liquid phase residence time in the kneading element.

Combining and rearranging Eqs. (3) Á/(7), gives:

t sol 0 V r P Q S ' (r P =r liq )(Q L ( Q S ((1 ( v)=v))b t liq 0bt sol 8 > < > : (8)

The restrictive elements in the second part of the reactor

The restrictive elements in the second part of the reactor (i.e. the RSE, the half transport screw just before the RSE and the die) are supposed to be totally filled. Equation Eq. ( 3) is still available, even if in this part of the reactor, there is only impregnated solid.

From Eq. ( 2), and with the corresponding volumes and density, the residence time in the RSE t rse and in the die t die are:

t rse 0 V rse Q S r rse (9)
and

t die 0 V die Q S r die ( 10
)
where r rse is the apparent solid density in the RSE and r die is the apparent solid density in the die.

Reaction and mass transfer model

In [START_REF] Prat | Solid Á/liquid reactive extraction with raw plant substrate[END_REF], Prat et al. proposed a first order kinetic law for the hemicelluloses liberation [START_REF] Gourdon | Technologie d'extrusion bi-vis et fractionnement de la matie `re ve ´ge ´tale, 40 ans d'extrusion bi-vis chez CLEXTRAL[END_REF].

Concerning the mass transfer, as the solid matter is under pressure in the restrictive elements, the hemicelluloses diffusion is neglected compared with the convection due to the pressure. In these elements, the solid weight percentage goes from v 1 to v 2 (v 1 B/v 2 ). Then, the liquid mass m liq which transfers from a solid mass m sol to the liquid phase is expressed by the following equation:

m liq 0m sol 1 ( v 1 v 1 ( 1 ( v 2 v 2 (11)

Matter, material and experimental methods

The local models previously described need different kind of parameters: Á/ elements geometrical parameters (pitches, length, volumes); Á/ operating parameters (N , Q L , Q S ); Á/ indirect responses of the system (v , the solid weight percentages along the reactor); Á/ physicochemical properties (densities, kinetic laws); Á/ parameters describing the behavior of the matter (the b parameters, and the slip velocity).

Furthermore, experimental data for solid residence time and extraction yields are to be obtained in order to validate the model.

Model parameters acquisition

Geometrical parameters are directly measurable on the elements and operating parameters are given by the experiments conditions.

Solid weight percentage in the impregnated solid

Solid weight percentage v in the impregnated solid along the reactor depends on the mechanic behavior of the screw elements. Therefore, the results obtained during the stopping and opening of the reactor allow to use mean values for all experiments. v will only vary with z , the axis of the extruder. In all experiments, profiles of the solid weight percentage in the solid phase have quite the same values. Consequently, these values (Fig. 3) will be used in the model, considering one value for each of the screw elements.

Densities of the two phases

The liquid density will be considered as a constant, even if its composition changes with the progress of the extractive reaction (at the worst conditions for this hypothesis */i.e. the minimum liquid flow, the maximum solid flow and the maximum hemicelluloses extraction */, the density of the liquid will only varies of 5% from its initial value).

The physicochemical properties of the vegetable matter used in this work are r W the real density (the density of the wood, without considering the empty space due to the porosity), r P the apparent density of a wood particle, and o the porosity. The considered porosity is the useful porosity for the process, i.e. the porosity accessible to the solvent. r P , r W and o are linked by the following equation (without considering the contribution of air):

r P 0r W (1(o) (12)
o and r W are intrinsic properties of the matter and can be obtained with a single experiment: a known mass of vegetable matter m w is mixed with a known mass of solvent m liq . After the transfer equilibrium is reached (i.e. at the maximum of the solid impregnation), the volume of the two mixed phases V and the solid weight percentage v in the solid phase are measured. The porosity is then filled with the solvent and the following equations can be written:

r w 0 m w V ( (m liq =r liq ) (13)
and

1(v 0 r liq r liq ' (1 ( o=o)r w ( 14
)
where r liq is the density of the solvent.

TSE experiments 4.2.1. Feedstock

Debarked chips of populus are ground to an average size of 15 )/2 )/2 (mm) using an Electra type VS1 industrial grinder. The composition of the wood chips is determined using American Standard Test & Measurement Methods (ASTM D-1103, D-1106, D-1787). The wood contains 52% of cellulose, 21Á/24% of lignin, 16 Á/19% of hemicelluloses, and some pectins, proteins, aromatic compounds and aliphatic acid. The moisture content of the initial chips is about 4%. Wood chips are fed in from a bin via a feeding screw and a chute (see Fig. 1 for reactor design).

The extractive solutions are injected using a volumetric pump at a single point (180 mm from the beginning of the screw).

Twin-screw extruder (TSE)

Experiments are performed using a modified CLEX-TRAL BC45 TSE (see Fig. 1). It is a fully intermeshing corotating TSE. Seven sections form the 1.4 m long barrel. Four sections are heated by induction belts and cooled by water circulation. Conical holes (1 mm entry per 2 mm exit) in a plate form the filter element added at the end of the barrel in order to extract the liquid from the slurry. The die consists of a 25 mm long cylindrical hole, 12 mm in diameter. Five different kinds of elements are used in the axis profile: T2F (trapezoidal two-flight screw), C2F (two-flight screw), MAL2 (10 mm bi-lobe kneading element) always positioned with a p /2 shift (i.e. in a neutral position for transport compared with conveying screw elements), MAL0 (10 mm right-handed mono-lobe kneading element), positioned with a p/13 shift (i.e. in a positive position for transport) and C2FC (25 mm pitch reversed two-flight screw element). 100 mm in length elements were principally used but 50 mm in length elements were also available. The pitch of the transport screw elements varies from 66 to 25 mm. All experiments are carried out with a reverse screw element with peripheral slots grooved in the screw flight for leakage flow.

Experimental processing

The domain of work is limited by several conditions [START_REF] N'diaye | Fractionnement de la matie `re ve ´ge ´tale: mise au point d'un proce ´de ´thermo-me ´cano-chimique et mode ´lisation du re ´ac[END_REF]. First, clogging of the reactor has to be avoided, but the more important is the correct functioning of the RSE. One function of this element is to allow the matter to form a 'plug'. This plug is necessary to press the solid in order to separate the two phases, but unfortunately, it only appears under specific conditions. Furthermore, the RSE can totally stop the transport of solid matter and, as a consequence, the TSE. Consequently, the 'authorized' range of parameters is organized around a central functioning point in terms of screw rotational speed N , solid feed rate Q S , liquid feed rate Q L and NaOH concentration s . The ranges of these parameters are reported on Table 1. All the experiments are conducted at a fixed barrel temperature of 55 8C (the best temperature to obtain good extraction yields and to limit degradation of hemicelluloses). Furthermore, for the study of the profile influence, only the kneading elements and RSE are replaced in half or totality by 33 mm pitch transport elements.

For all the experiments, the following parameters are measured after the permanent operating conditions have been reached: Á/ liquid and solid rates at the exit of the reactor in order to perform mass balances; Á/ v , the solid weight percentage in the exiting solid phase characterizing the solid/liquid separation and also needed in order to perform the mass balance; Á/ R , the extraction yield (in term of hemicelluloses mass recovered in the liquid phase compared with the mass contained in the initial wood) [START_REF] N'diaye | Fractionnement de la matie `re ve ´ge ´tale: mise au point d'un proce ´de ´thermo-me ´cano-chimique et mode ´lisation du re ´ac[END_REF];

Á/ t sol , the mean residence times of the solid phase (the mean residence times determination is described in [START_REF] N'diaye | Extraction of hemicelluloses from poplar, populus tremuloides, using an extruder-type twin screw reactor: influence of the main factors[END_REF]).

After these measurements, the extruder is brutally stopped and opened in order to observe the solid situation.

Modeling results

Transport model results

Model analysis

The global residence times for the solid and the liquid phases are of the form:

t S 0 X transport screws L Np ( v S ' X kneading elements  V r P Q S ' (r P =r liq )(Q L ( Q S (1 ( v=v))b ' V rse r rse Q S ' V die r die Q S (15)
and

t L 0 X transport screws L Np ( v S ' X kneading elements  V r P b Q S ' (r P =r liq )(Q L ( Q S (1 ( v=v))b (16)
In [START_REF] Prat | Two phase residence time distribution in a modified twin-screw extruder[END_REF], the important decrease of t sol with Q S and N , the decrease of t liq with Q L and N and the neutrality of s towards t sol and t liq (as already said in paragraph Section 2) have been found. The mathematical expressions (Eqs. ( 15) and ( 16)) are in agreement with the main experimental observations upon the influences of the operating parameters on the mean residence times.

Adjustable parameters

The proposed model has three non-intrinsic parameters which will be considered as adjustable: Á/ a , the slip velocity between the matter and the transport elements; Á/ b 1 , the ratio between the solid and the liquid phase velocities in the kneading elements MAL2; Á/ b 2 , the ratio between the solid and the liquid phase velocities in the kneading elements MAL0.

N pa the slip velocity, arises from the friction between the matter and the screw and between the matter and the barrel. It can also help to consider the transport limitation due to the phenomena in the two screw intermeshing area. It is, of course, a simplified way of representing these phenomena. The introduction of the b parameter into the modeling of the kneading elements allows to represent the hold up of the two phases. Due to its expression, the volumes ratio can be written in a form which implies the volume flows in a proportional way:

V L V S 0b V liq V sol (17)

Model adjustment

Due to the great numbers of the operating parameters, in order to analyze the results obtained, the calculated times versus the experimental mean residence times will be represented for the two phases.

The difference between experimental and calculated values will be characterized by a criterion of the form:

h 0 1 n X n j01 2jt j exp ( t j mod j (t j exp ' t j mod ) (18)
where, n is the number of experiments; t j exp , the solid experimental residence time of the experiment j; and t j mod is the solid modeled residence time of the experiment j .

Model results

The criterion for the solid residence time a is minimized by the Gauss Á/Newton based method and the results are plotted in Fig. 5. The three adjustable parameters become a 0/0.3, b 1 0/1.42 and b 2 0/2.05. The criteria is h 0/0.17.

The obtained value of a shows that the velocity in the transport screw element is a compromise between the flight velocity and the rotational speed due to the shearing. In the MAL elements, the matter velocity is due to the upstream pressure and the transport contribution of the flight. The values of b1 and b2 are higher than one, showing that the solid phase runs out quicker than the liquid phase. This is due to the fact that the friction is more important for the solid than for the liquid phase. Furthermore, as the MAL2 is positioned with a p /2 shift, it has a low transport effect and has a value of b lower than the MAL0 element.

The results are good concerning the solid residence times, except for low values of the screws rotational speed (experiments with N 0/100, the lower limit). Total liquid residence times are overvalued by the model in these conditions. Furthermore, when tracing the mass of solid phase along the reactor, (the mass is the product of the residence time in each element by the solid feed rate), the global tendencies of the experiments can be found, as shown in Fig. 3 for the central functioning point.

In the model, N only influences the global residence times by the transport screws local residence times (Eqs. ( 15) and ( 16)). But the contributions of these elements represent only 4Á/10% of the modeled solid residence time which can not explain the experimental variations with N . As it has be shown in [START_REF] N'diaye | Extraction of hemicelluloses from poplar, populus tremuloides, using an extruder-type twin screw reactor: influence of the main factors[END_REF], the screw rotational speed has a major influence upon v , the solid weight percentage in the solid exiting phase, and then upon the behavior of the RSE which conditions the solid/liquid separation and consequently, v. This means that N should intervene in the RSE element. But, for the purpose of solid/liquid extraction modeling in the TSE, only the first part of the reactor residence time is important as the extractive reaction occurs in this part.

The transport model presents some default for the prediction of solid residence time in the RSE at low values of the axis speed rotation (N B/100 rpm). Even if there is a lot of simplifying hypothesis, it gives good results in the main part of the experimental domain. The model needs a single adjustable parameter for each screw element type. These three parameters have the same value for all the experimental conditions. This model will then be used for the global prediction of extraction yields.

Extraction yields results

The extraction yields prediction is obtained by coupling the transport model, the reaction and the transfer model, as described in Fig. 1. The temperature used for the kinetic law is the experimental temperature measured in the TSE at the interface barrel-matter. The results are plotted in Fig. 6 for all the experiments. The global model gives a good prediction of the experimental data in the whole domain investigated.

These results lead to several observations on the adopted method and on the studied process. First of all, the good prediction justifies the utilization of coupling between a one dimensional model for transport by screw element and a first order kinetic. Besides, the tentative method in batch reactor for the acquirement of the kinetic gives results exploitable in continuous reactors, presumably because of the simplified reactional way adopted. Finally, the obtained results suggest that the contribution of TSE to solidÁ/liquid extraction processes is firstly located in its ability to separate the two phases and then suppress the limitative diffusion step.

Conclusions

Problems generated by the complexity of the vegetable matter and the important coupling of phenomena in a reactor where the matter can not be directly observed are numerous. This work proposes a first approach model for the prediction of solid Á/liquid extraction yields in a modified TSE. The model has been constructed from visual observations and chemical engineering analysis by coupling transport, reaction and transfer descriptions.

This work proposes a one dimensional model for solid transport phenomena. The model needs a single adjustable parameter for each screw element type. These parameters characterize the mechanic phenomena between matter and TSE and depend on the couple (chosen matter-screw element). A theoretical study of these parameters could be the next step in modeling the two-phase flow transport in the TSE. As far as solid Á/ liquid extraction is concerned, the solid residence time prediction gives usable values in all studied domain. Reaction is based on a previous work upon hemicelluloses extraction in a batch reactor from poplar wood. The hemicelluloses diffusion is neglected and mass transfer is due to the under pressure elements which cause solid Á/liquid separation all along the TSE.

This study allows to point out that the main interest of the TSE for solid Á/liquid extraction lies in the suppression of the diffusion step during the operation. Furthermore, it validates the experimental method for the obtaining of kinetic parameters in batch reactor.

For other TSE applications with solidÁ/liquid flows, a more precise study of the RSE will be needed, in order to understand the behavior of such a screw element. This understanding will be of first interest for raw material for instance. As there is only solid in the RSE, developed models for polymer industries or agro-industry [START_REF] Tayeb | A basic model for a twinscrew extruder[END_REF] will certainly be usable with some modifications due to the compressibility of the wood particles.

Fig. 1 .

 1 Fig. 1. Global process in a modified CLEXTRAL BC45 TSE and model processing.

Fig. 2 .

 2 Fig. 2. Hemicelluloses extraction yields in TSE and in batch reactor.

Fig. 3 .

 3 Fig. 3. TSE opening experiments results ( + , solid weight percentage, *, solid mass, */calculated solid mass from experimental residence time).

Fig. 4 .

 4 Fig. 4. Hemicelluloses extraction steps.

Fig. 5 .

 5 Fig. 5. Experimental vs. calculated solid residence times. Fig. 6. Experimental vs. calculated extraction yields.

Table 1

 1

	Experimental domain
		Symbol Central	Minimum Maximum
			point
	Speed screw (rpm)	N	175	110	250
	Solid feed (kg h (1 )	Q S	4.8	3.2	6.7
	Liquid feed(kg h (1 )	Q L	36.5	21.5	51.5
	NaOH massic % in	s	5	3	7
	liquid


Robert Chahine

Wouter J T Bos

Nicolas Plihon

Dynamo-properties of RFP velocity field The dynamo properties of the RFP velocity field

The reversed field pinch (RFP) is a toroidal device aiming at the magnetic confinement of a plasma in order to reach conditions of thermonuclear reactions. In RFPs the magnetic and velocity field self-organize to a saturated state determined by their nonlinear interplay and the values of the transport-coefficients. The question addressed in this article is whether this saturated velocity-field is capable of amplifying a seed magnetic field, the so-called dynamo-effect for the astrophysical community. It is shown, using numerical simulations in periodic cylinders, that the RFP velocity field can amplify a passively advected seed-field, but that this is only observed for values of the magnetic Prandtl number above unity. These observations are reported for both laminar and turbulent RFP flows. We further assess the difference in behaviour between a passively advected vector field and the true magnetic field and show that their difference is associated with the detailed alignment properties of the fields.

I. INTRODUCTION

In the context of geophysics and astrophysics, the dynamo instability is the process, now widely accepted, to be at the origin of magnetic fields of astrophysical bodies [START_REF] Larmor | How could a rotating body such as the sun become a magnet[END_REF] . The dynamo instability converts kinetic energy into magnetic energy for certain types of flows of electrically conducting fluids [START_REF] Moffatt | Magnetic field generation in electrically conducting fluids[END_REF] . There exist a number of anti-dynamo theorems which restrict the occurrence of the dynamo instability to specific classes of flows [START_REF] Moffatt | Magnetic field generation in electrically conducting fluids[END_REF][START_REF] Rincon | Dynamo theories[END_REF] and the observation of the dynamo instability in experiments is currently restricted to a few realisations in liquid metals [START_REF] Verhille | Laboratory dynamo experiments[END_REF] or plasma flows [START_REF] Tzeferacos | Laboratory evidence of dynamo amplification of magnetic fields in a turbulent plasma[END_REF] which were carefully optimized to obtain critical values for the magnetic Reynolds number of the order of a few dozens. While there has been a great deal of interest for dynamo features in spherical geometries in the context of the geodynamo [START_REF] Roberts | On the genesis of the earth's magnetism[END_REF] or the solar dynamo [START_REF] Charbonneau | Solar dynamo theory[END_REF] , dynamos were also investigated in toroidal geometries. Torus dynamos were in particular proposed as a paradigm for the accretion disk dynamo [START_REF] Deinzer | Torus dynamos for galaxies and accretion disks. I. The axisymmetric alpha omegadynamo embedded into vacuum[END_REF][START_REF] Mikhailov | Torus dynamo in the outer rings of galaxies[END_REF] . At the experimental level, several studies considered the screw dynamo in a torus [START_REF] Chupin | The screw dynamo in a thick torus[END_REF] . In this configuration, a liquid metal helical flow is generated from the sudden deceleration of the torus, and the critical magnetic Reynolds was determined to be above 18 after careful optimization of the liquid flow [START_REF] Dobler | Screw dynamo in a timedependent pipe flow[END_REF][START_REF] Sokoloff | Dynamos: from an astrophysical model to laboratory experiments[END_REF] .

The dynamical growth of the dynamo magnetic energy is usually separated as a two-step evolution. During a first step, also called the kinematic dynamo, a small magnetic field is amplified by a velocity field, which is negligibly influenced by the Lorentz force. During this first step we can therefore individually evolve the velocity field and the magnetic field, since the latter is too weak to influence the former. The second step consists of the saturation phase, where the magnetic field has grown strong enough to back-react on the velocity field by means of a) Corresponding author: nicolas.plihon@ens-lyon.fr the Lorentz-force. In this second phase the simultaneous evolution of both fields should therefore be considered in detail to correctly describe the coupled dynamics of the fields.

In the plasma fusion community, dynamo regimes usually refer to regimes where an electromotive force is induced by flow and magnetic field fluctuations due to turbulence, and has been reported for most of the magnetically-confined fusion plasma configurations, such as Reversed Field Pinches (RFPs), spheromaks or tokamaks. The current profile observed in RFPs was proposed to be sustained by a dynamo regime due to continuous chaotic or turbulent motion [START_REF] Gimblett | Mhd turbulence theory and its implications for the reversed field pinch[END_REF][START_REF] Strauss | The dynamo effect in fusion plasmas[END_REF][START_REF] Yoshizawa | A turbulent dynamo model for the reversed field pinches of plasma[END_REF] . Investigations in various regimes were reported over the last two decades [START_REF] Bonfiglio | Dominant electrostatic nature of the reversed field pinch dynamo[END_REF][START_REF] Escande | What is a reversed field pinch?[END_REF][18][19][START_REF] Sarff | Overview of results from the MST reversed field pinch experiment[END_REF] . More recently, a dynamo electromotive force was proposed to explain stationary non-sawtoothing regimes in tokamaks, often referred to as "flux pumping" [START_REF] Jardin | Self-organized stationary states of tokamaks[END_REF][START_REF] Krebs | Magnetic flux pumping in 3d nonlinear magnetohydrodynamic simulations[END_REF][START_REF] Piovesan | Role of a continuous MHD dynamo in the formation of 3d equilibria in fusion plasmas[END_REF] . However, in essence, in the context of fusion plasma, the nature of the dynamo is fairly different from the astrophysical dynamo, in particular in the kinematic phase, since the initial magnetic field is not a weak small-scale background, but a strong large-scale imposed field.

The astrophysical and fusion plasma communities do therefore not address exactly the same subject when they discuss the dynamo effect. It is this semantic difference which motivated the present study, but we think that the findings might be important beyond this semantic motivation. The precise question we address is whether the velocity field, spontaneously generated through MHD instabilities in RFP geometries can give rise to a dynamo effect in the astrophysical sense. To do so, we carry out MHD simulations in a periodic cylinder in the RFP regime and characterize its capacity to amplify a seed field. For this we will consider an auxiliary magnetic field which is passively advected by the plasma flow.

Our approach is similar to the one adopted in the investigation by Cattaneo and Tobias 24 who investigated a convectively driven saturated dynamo-velocity field. Indeed, this approach allows to explore how the capacity of Dynamo-properties of RFP velocity field 2 a velocity field to amplify a seed-field changes when the induced magnetic field is large enough to saturate the velocity field. In the present investigation we transpose these ideas to the flow generated in an RFP.

The remainder of this article is constructed as follows. The governing equations and normalization are given in the next section, together with a description of the numerical method, set-up and boundary conditions. The results are presented in Secs. III and IV, where we consider the kinematic dynamo in a frozen laminar velocity field, and for a turbulent velocity field, respectively. Sec. V concludes this investigation.

II. RFP SIMULATIONS AND ADVECTION OF PASSIVE VECTOR FIELDS

The goal of this article is to investigate whether the velocity field spontaneously generated through MHD instabilities in RFPs may sustain a dynamo effect. We thus adopted the following methodology:

• A first set of MHD equations (Eqs. ( 1)-( 4) below) are solved for the magnetic field B and the velocity field u in a cylindrical geometry, with initial and boundary conditions corresponding to that of an RFP (applied axial magnetic field and imposed axial electric current). The initial condition for the velocity field is small random noise and, as time evolves, a kink instability develops and generates a flow. The resulting RFP flow back reacts on the magnetic field and finally reaches a statistically stationary state, referred to as the RFP velocity field.

• We then use this statistically stationary RFP velocity field as an externally prescribed flow in an induction equation for an auxiliary passive magnetic field D (Eqs ( 5)-( 6) below). The questions addressed in this article are thus similar to a kinematic dynamo study: (i) does the RFP velocity field lead to exponential growth of passive magnetic energy starting from an initially infinitesimal seed passive magnetic field ?, (ii) what are the growing eigenmodes of the passive induction equation ?

Let us now describe in detail the set of equations, numerical methods, and the boundary and initial conditions.

A. MHD simulations of the RFP magnetic and velocity fields

The simplest geometry in which the RFP can be investigated theoretically and numerically is the periodic cylinder. By solving the MHD equations in such a domain, combined with imposed axial magnetic and electric fields, the resulting dynamics bear a resemblance with actual RFP dynamics. Indeed, for certain values of the imposed fields, MHD instabilities lead to the generation of a velocity field, which in turn back-reacts on the magnetic field by means of the Lorentz force. The periodic cylinder constitutes therefore a paradigm in RFP research, which originates from pioneering works [START_REF] Aydemir | Three-dimensional nonlinear incompressible mhd calculations[END_REF][START_REF] Schnack | Numerical simulation of reversed-field pinch dynam-ics[END_REF][START_REF] Dahlburg | Driven, steady-state rfp computations[END_REF][START_REF] Cappello | Nonlinear plasma evolution and sustainment in the reversed field pinch[END_REF] , but is still actively considered [START_REF] Piovesan | Role of a continuous mhd dynamo in the formation of 3d equilibria in fusion plasmas[END_REF][START_REF] Veranda | Magnetohydrodynamics modelling successfully predicts new helical states in reversed-field pinch fusion plasmas[END_REF] , since it allows to disentangle the effects of curvature or toroidicity from the already complex dynamics [START_REF] Morales | The effect of toroidicity on reversed field pinch dynamics[END_REF] .

In the following we will thus consider MHD simulations in a periodic cylinder. The RFP velocity field u is generated by a kink-instability resulting from its interaction with the RFP magnetic field B and current density j. These fields are governed by the equations:

∂u ∂t + (u • ∇)u = -∇P + j × B + ν∇ 2 u, (1)
∂B ∂t = ∇ × (u × B) + λ∇ 2 B, (2)
∇ • u = 0, (3)
∇ • B = 0 ( 4
)
where ν is the kinematic viscosity, λ is the magnetic diffusivity, and P is the pressure. This form of the equations corresponds to a normalization of the velocity by the Alfvén velocity C A = B (0) / √ µ 0 ρ, with B (0) the imposed magnetic field, ρ the fluid density and µ 0 the vacuum magnetic permeability. Analogously, an Alfvén timescale is introduced as t A = L/C A . The magnetic field consists of a freely evolving component plus an imposed mean field, described below. The characteristic lengthscale L is equal to the diameter of the cylinder 2R. The control parameter of these equations is the Lundquist number S = C A L/λ, which controls the amplitude of the velocity field, and thus of the kinetic Reynolds number Re = U L/ν, where the characteristic velocity U is chosen equal to the RMS velocity. For simplicity, and following a long-standing practice in dynamo theory [START_REF] Rincon | Dynamo theories[END_REF][START_REF] Cattaneo | Dynamo properties of the turbulent velocity field of a saturated dynamo[END_REF] we restrict ourselves to the case where ν = λ, or in other words, the magnetic Prandtl number is taken to be equal to unity.

This assumption allows to avoid scale separation between the magnetic and velocity fields, thus limiting the already large computational cost and allowing to explore several regimes. Note that since the magnetic Prandtl number is unity, the magnetic Reynolds number Rm associated with the B-field is equal to the kinetic Reynolds number Re.

Eqs. ( 1)-( 4) are solved in a periodic cylinder of length 8π, and diameter 2R = 2. The MHD-domain is solved in a larger rectangular box of π × π × 8π, where the solid boundaries are imposed using a penalization method. The evolution equations are computed using a pseudospectral solver with a 3rd order Adams-Bashforth timeintegration scheme. Details on the numerical methods and its convergence properties can be found in a previous publication [START_REF] Morales | Simulation of confined magnetohydrodynamic flows with dirichlet boundary conditions using a pseudo-spectral method with volume penalization[END_REF] . In the present simulations the resolution is 64 × 64 × 512 grid-points in the x, y, z-directions, respectively. The pressure is computed from the resolution of the Poisson equation in the spectral domain [START_REF] Chahine | On the role and value of β in incompressible mhd simulations[END_REF] The initial magnetic field is a combination of an axial field This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. p (R) = 1.4 at the wall, associated with a uniform imposed axial current density j (0) z (r) = 2.8. The axial component of the magnetic field at the boundaries is left unconstrained. The velocity-field is satisfying no-slip boundaries at the wall.

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For all simulations the initial conditions are small amplitude random noise for the velocity field and zero value for the non-imposed contribution of the magnetic field. After a short transient, the non-linear interactions between the velocity field u and the magnetic field B lead to the generation of flows u RFP in the RFP, whose features are discussed in Secs. III A and IV for diffferent values of the Lundquist number S, which is varied through the value of the diffusivities λ = ν. The resulting statistically stationary velocity field is then used as a prescribed velocity field in the auxiliary passive induction equation (see next subsection) in kinematic dynamo simulations.

B. Auxiliary passive magnetic field dynamics

In addition to the set of Eqs.( 1)-( 4), we solve the evolution of an auxiliary passive field D, satisfying the equations,

∂D ∂t = ∇ × (u × D) + λ ′ ∇ 2 D, (5)
∇ • D = 0. (6)
where λ ′ is the magnetic diffusivity of the passive field D (quantities related to the passive magnetic field D are primed in order to avoid any confusion with quantities related to B). Note that the equations governing the dynamics of D are identical to the induction equations (Eqs. ( 2),( 4)). The three differences between D and B are that firstly, D does not back-react on the velocity field by the Lorentz force, secondly that D does not contain an imposed contribution. The third difference is that the magnetic diffusivities λ, related to B, and λ ′ , related to D, are not equal. This is also a difference with the investigation by Cattaneo and Tobias 24 , which focused on flows sustaining a dynamo field and for which λ = λ ′ . The question that we will answer is whether the velocity field resulting from the interplay of u and B is able to amplify a small seed-field D. The control parameter for dynamo action is the magnetic Reynolds number, which compares the induction term to the dissipative term, here for the passive magnetic field D, and defined as Rm ′ = U L/λ ′ . For a given value of the Lundquist number, the amplitude and structure of the RFP velocity field u RFP is prescribed by the values of the imposed axial field B (0) z and axial current j (0) z , thus the ability of u RFP to sustain a dynamo is assessed by changing Rm ′ by a change of λ ′ . The initial passive magnetic field D is a random noise similar to the velocity field, but with initial energy D 2 = V D 2 dV ≈ 10 -10 , where V is the total volume. At the boundary, the poloidal component of D is set to zero.

The set of Eqs. ( 5)-( 6) are solved similarly to Eqs. ( 1)-( 4), in a rectangular box of π × π × 8π using the same pseudo-spectral numerical method, and where the solid boundaries are imposed using a penalization method.

Depending on the value of the Lundquist number S, two types of statistically stationary RFP flows are observed after a transient. At low values of S, the flows are are steady and laminar, as reported in Sec. III. In this regime, the laminar steady-state flow u RFP is prescribed and fixed in time for the auxiliary passive induction equation. Thus, once the laminar flow has been computed, we stopped the computation of the set of MHD equations ( 1)-( 4), and only evolve the auxilary induction equation ( 5)-( 6). At higher values of the Lundquist number S, the RFP flows u RFP are dynamical and the velocity field exhibits strong fluctuations, as reported in Sec. IV. In this regime, the flow u RFP prescribed for the auxiliary passive induction equation evolves with time and the set of MHD equations ( 1)-( 4) and passive induction equations ( 5)-( 6) are solved simultaneously. We recall here, that the velocity field u and magnetic field B are fully coupled and that the velocity and magnetic modes reported here are non-linearly saturated. On the other hand, the passive field D does not back react on the velocity field: the modes presented here for the passive field are thus linearly stable or unstable modes, similarly to usual kinematic dynamo computations.

III. LAMINAR DYNAMO SIMULATION FOR THE PASSIVE FIELD.

In this section we will consider cases at low values of the Lundquist number S, for which a steady laminar RFP flow is generated. The features of the velocity field u and magnetic field B are first described, before discussing the onset of dynamo action and the features of the unstable growing mode for the passive dynamo field D.

A. A steady Quasi-Single-Helicity state At first, the steady state laminar flow is computed from the MHD equations ( 1)-( 4), which are solved over a time interval equal to 1500 τ A . A steady state was reached typically after 200 τ A . At t = 1500 τ A , the RFP magnetic field B RFP and the flow u RFP are extracted from the simulations.

The structures of the magnetic field B RFP and velocity field u RFP are displayed in Fig. 1(a) and (b) for S = 150, where isocontours of the magnetic energy (respectively kinetic energy) are color-coded by the sign of the axial component of the field (which would correspond to the toroidal component in a torus). The magnetic field B has a strong constant axial component (the applied magnetic field), which has been subtracted from the total field in Fig. 1 for the sake of clarity. Both the velocity and the This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. In particular, the flow has a complex structure: it consists of two strongly interlaced helical flows of opposite polarities, as shown by the two red and blue screwed tubes, with a strong shear at the interface between the red and the blue tubes. The steady-state MHD fields generated at S = 150 correspond to an intensity of the velocity field leading to a value of the Reynolds number Re = 6.37. Such helical flow fields are observed in all simulations, but at low values of S these fields are stationary and laminar, whereas at large values they become time-dependent.

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Equations ( 1)-( 4) conserve total energy, magnetic helicity and cross helicity in the limit of vanishing dissipation. The process by which the RFP velocity field u RFP is created is a conversion of magnetic energy to kinetic energy through a kink instability. [START_REF] Escande | What is a reversed field pinch?[END_REF] As expected for a RFP equilibrium, the field reversal parameter F = B z / B z decreases as a function of the pinch ratio, where B z denotes the wall averaged and B z the volume-average axial magnetic field. In the present investigation, in both the laminar and turbulent statistically steady states, the equilibrium profile is characterized by a reduced axial magnetic field at the walls B z as compared to the imposed field. This is characteristic of RFPs. However, no reversal is observed (F remains positive) and details on this can be found in Ref. [31].

The flow intensity is measured through the kinetic Reynolds number, shown as a function of the Lundquist number in Fig. 2. The kinetic Reynolds number increases with the Lundquist number, with an empirical scaling Re ∝ S 0.74 . Since in these simulations the integral length scale and imposed magnetic field are fixed, this scaling is associated with a dependence of the RMS velocity on the magnetic diffusivity, U ∼ λ 0.26 . For the lowest values of S and Re this corresponds to steady laminar flow and for the largest values to fluctuating movement with several dominating modes.

B. The laminar kinematic dynamo.

In this subsection, we investigate the capability of the laminar RFP velocity field u RFP described in the previous subsection, to induce dynamo action.

The steady state flow u RFP extracted from the simulations of Eqs. ( 1)-( 4) at t = 1500τ A , is used as a prescribed field for the passive induction equation ( 5) starting at time t = 1500τ A . Th initial condition for the passive field D at t = 1500τ A is small Gaussian white noise with initial energy D 2 ≈ 10 -10 . The control parameter governing Eq. ( 5) is the passive magnetic Reynolds number Rm ′ , which depends upon the passive magnetic diffusivity λ ′ , or equivalently the passive mag-This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. netic Prandtl number P m ′ . Dynamos are usually observed above a critical magnetic Reynolds number; for a given flow u RFP , we thus expect to excite a passive dynamo field D when increasing P m ′ . As an illustration, for S = 150, i.e. Re = 6.37, the time evolution of the total passive magnetic energy D 2 is displayed in the inset of Fig. 3 for four values of the passive magnetic Prandtl number P m ′ . The passive magnetic energy decreases with time for P m ′ ≤ 3 and increases for P m ′ ≥ 4.

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For P m ′ ≥ 4, after a small transient (around 150 τ A ), i.e. for t > 1650τ A , the energy of the passive magnetic field increases exponentially with time. For each value of the passive magnetic Prandtl P m ′ , the growth rate of the passive magnetic energy is computed from the exponential fit of the time evolution of D 2 . The critical passive magnetic Prandtl number P m ′c number is then defined from the linear interpolation of the growth rates between the last negative value and the first positive value, e.g. P m ′c = 3.2 for Re = 6.37. This leads to a critical passive magnetic Reynolds number Rm ′c = Re P m ′c = 20.4.

A first important observation is that the RFP flow has favorable properties for dynamo generation on a passive magnetic field, which can be qualitatively understood since RFP flows are characterized by high helicity and high shear, two properties which are not necessary but enhance dynamo action 2,3 .

Similar simulations have been run for several values of S from 150 to 550 in the laminar regime, and the evolution of the critical passive magnetic Reynolds number as a function of Re is shown in Fig. 3 (only the red points correspond to laminar regimes).

Let us now describe the spatial features of the passive dynamo mode. Figure 1(c) shows the isocontours of the passive magnetic energy D 2 , color-coded by the sign of the axial component of the passive field D z (which would correspond to the toroidal component in a torus). The spatial structure of the passive magnetic field is more complex than the spatial structure of the velocity field. This is further investigated in Fig. 4, left column, where the axial spectra of the magnetic, kinetic and passive magnetic energy are displayed, from top to bottom, for Re = 6.37. The discrete modes observed in the spectra for the magnetic and kinetic energies are identical: the dominant RFP mode is the n = 9 mode, and harmonics of this mode are clearly observed. The strong harmonics observed in the spectrum of B 2 are a signature of the complex spatial structure observed in Fig. 1a) where the blue stripes are larger than the red stripes. We stress here that most of the magnetic energy is contained in the applied n = 0 mode (which is not displayed in Fig. 4), less than 11% of the magnetic energy is present at positive mode numbers (precisely 11%, 6.5% and 4.2% for Re = 6.37, 10.2 and 15.3 respectively). The dominant mode for the passive magnetic energy D 2 is the n = 2 mode, as clearly observed in Fig. 1(c). The induction term ∇ × (u × D) in the passive induction equation ( 5) leads to mode mixing between u and D, which, for the case at Re = 6.37, leads to a complex spectrum for D 2 , with strong contributions of modes 7, 11, 16, 20, etc. The central column of Fig. 4 displays the spectra for the case S = 276, e.g. Re = 10.2, while the right column corresponds to the case S = 417, e.g. Re = 15.3. The dominant RFP mode is at n = 8, which couples with the dominant n = 0 or n = 10 modes for D 2 , also leading to a complex spectrum for D 2 .

We have observed in our simulations that the influence of P m ′ on the spectra of D is very weak: the same modes are excited for simulations with values P m ′ = 2, 4, 7 and only small variations are observed in the relative amplitudes of the modes (not shown).

The strong coupling between the velocity field u and the passive magnetic field D is emphasized in Fig. 5, in which isocontours of the kinetic energy and of the passive magnetic energy are displayed simultaneously for S = 150, e.g. Re = 6.37. A first observation is that the structure of the passive magnetic field, which is dominated by a n = 2 mode is strongly affected by the n = 9 dominant velocity mode. This is observed in the zoom of Fig. 5, where the separation between black and white ribbons for the passive magnetic energy (opposite values of D z ) is a zone of strong shear for the velocity field (transition between the red and blue ribbons of the kinetic energy isosurfaces). A second observation is that the kinetic energy is mostly concentrated in the central part of the cylindrical domain, while most of the passive magnetic energy is concentrated in outer radial regions. Another observation is that the structures of the passive magnetic field connect in high velocity shear regions, and the axial component of passive magnetic field reverses sign at the same location as the reversals of the axial This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. component of the velocity field.

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To summarize, the laminar RFP flow is a kinematic dynamo velocity field for high enough P m ′ . The next question is how the passive and magnetic fields act when the flow is not steady, and how the back-reaction of the magnetic field on the velocity field changes the dynamics.

IV. TURBULENT DYNAMO SIMULATIONS

In this section, we address a regime at large Lundquist number, for which the RFP velocity field u RFP is no longer steady and laminar. Thus, similarly to the case of convectively driven turbulence investigated by Cattaneo and Tobias 24 , simulations are carried out with all three fields u, B and D evolving in time. We will first focus on the flow characterization, then we will assess the dynamo threshold and we will finish by evaluating the alignment properties of the system.

A. Flow characterization

Simulations at S = 2000, resulting in an RFP flow with Re = 50, are carried out, and a snapshot of the kinetic energy is shown in Fig. 6(top). The velocity field clearly shows a trace of a periodic structure, but small scales are also observed. This assessment of the flow-behavior from the flow-visualizations is however of qualitative nature. A more quantitative characterization of the turbulent nature of the different fields is given by the energy spectra. The axial-mode spectra are shown in Fig. 7. These spectra illustrate that the dynamics of the fields u and B are not restricted to a small number of excited modes. Indeed, the broad-band character of these spectra suggests that the flow is in a turbulent state. Nevertheless, the power-spectra of the velocity and the magnetic field show some well-defined peaks. These peaks are associated with the close to periodic-structure observed in the visualization. Altogether, these broad-band spectra over two decades in wavenumber show that the dynamics in this section are turbulent in the presence of close-toperiodic velocity and magnetic field structures.

In the next subsection we will determine the dynamothreshold for the present RFP flow.

B. Dynamo growth rate and threshold

The evolution of the passive vector D was solved simultaneously to the dynamical evolution of the u and D fields for P ′ m = 1.43 and P ′ m = 2 The time evolution of the mean passive magnetic energy D 2 for these two runs is plotted in Fig. 8. It is observed that for P ′ m = 2 the RFP field in the dynamic regime is capable of amplifying a seed magnetic field, and the RFP velocity field can therefore be called, as for the laminar case, a dynamo velocity field. In the case of P ′ m = 1.43, the passive magnetic energy fluctuates around a constant value over a long-time interval. The threshold for dynamo action at Re = 50 is therefore close to P ′ m = 1.43, corresponding to a critical magnetic Reynolds number R ′ mc = 71.5. The new result is added to the data previ-This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. ously obtained in the laminar regime (Figure 3). The critical magnetic Reynolds number in this dynamical case is therefore somewhat higher but of comparable magnitude to the critical value of the laminar kinematic dynamo, observed around Re = 15, where R ′ mc ≈ 50. We stress here that the resolutions required to well-resolve the dynamics of the passive field D are higher than that required to resolve the u and B fields, resulting in high computationalresource-requirement for turbulent dynamo simulations. As a consequence, higher Re regimes are not explored in this manuscript.

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A snapshot of the passive magnetic energy at P ′ m = 2 is shown in Fig. 6 at the same time as the kinetic energy shown in the top panel. As for laminar dynamos, the structure of the passive vector-field D is much more complex than that of the velocity field u and shows a less periodic behavior. This is highlighted by the energy spectrum shown in Fig. 7 which does not display energetic peaks.

The fact that we observe the growth of the amplitude of D 2 in Fig. 8 is in agreement with the observations of [START_REF] Cattaneo | Dynamo properties of the turbulent velocity field of a saturated dynamo[END_REF] , who showed that the saturation of the magnetic field B does not modify the velocity field in such a way that dynamo-action is prohibited. Rather it modifies the simultaneous dynamics of B and u, such as the alignment properties described later in this section.

We stress here that even though the velocity field is able to amplify a seed-field for certain parameters, this is not the case for P m ′ = 1. This shows that the diffusive effects acting upon the passive field need to be smaller in our simulations than those acting upon the RFP magnetic field. This seems to indicate that in our simulations the B-field in the statistically steady state is unaffected by the dynamo effect, since its magnetic Reynolds number is below the threshold for dynamo action. This is a major difference compared to the results of [START_REF] Cattaneo | Dynamo properties of the turbulent velocity field of a saturated dynamo[END_REF] , where dynamo amplification of a passive field by the saturated velocity field was observed for P m ′ = 1.

C. Alignment of the different fields

As mentioned above, in ref. [START_REF] Cattaneo | Dynamo properties of the turbulent velocity field of a saturated dynamo[END_REF] the same system of equations was investigated as in the present study, but in a different geometry, using a different forcing. It was observed that the variance of the D-field continues to increase after the saturation of the B-field. One of the This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0078860 key insights obtained thereby is that the saturation is not only due to the modification of the flow-field by the Lorentz-force. Indeed, if this was the case, the strength of D would saturate simultaneously with that of B. Rather is it due to some intricate interplay between the velocity field and the magnetic field. It is therefore not visible from the velocity field alone that the magnetic field saturates, and at which level. One should thus consider the simultaneous dynamics of u and B.

One way to consider the simultaneous dynamics of the velocity and the magnetic field is to introduce the Elsasser variables, defined as, z ± = u±B , and the passive Elsasser variables, defined as, ξ ± = u ± D

The dynamical equations for the Elsasser variables are obtained from the sum and the difference of Eqs ( 1) and (2) yielding

∂ t z ± + z ∓ • ∇z ± = ν∇ 2 z ± -∇P * , (7)
with P * = P + B 2 /2. The dynamical equations for the passive Elsasser variables are obtained from the sum and the difference of Eq (1) and ( 5), and, using the simplifying assumption

P m = P ′ m = 1 write ∂ t ξ ± + ξ ∓ ∇ξ ± = ν∇ 2 ξ ± -∇P * - 1 4 (ξ + -ξ -) • ∇(ξ + -ξ -) -(z + -z -) • ∇(z + -z -) , (8)
It is insightful to look at the structure of Eqs. ( 7) and ( 8) which explicitly expresses the coupling of the different fields. The equations for the Elsasser variables (Eq. ( 7)) is more symmetric than the equations for the passive Elsasser variables (Eq. ( 7)), due to the presence of the Lorentz force (and the absence of its passive counterpart).

What this difference exactly implies for the dynamics is not straightforward to infer from the equations. We can however mention the specific case where strong correlations exist between the magnetic field and the velocity field [START_REF] Grappin | Alfvenic fluctuations as asymptotic states of mhd turbulence[END_REF][START_REF] Dobrowolny | Fully developed anisotropic hydromagnetic turbulence in interplanetary space[END_REF] , such as observed in the solar-wind. In the extreme case of u = ±B, the nonlinear term vanishes for the case of the full system, but does not for the passive advected system. This shows a concrete example of different behavior of the two systems: if the fields are aligned and of equal magnitude, the nonlinearity of the full system is decreased, but not for the passive case.

If the fields align, but are not of equal magnitude, the nonlinearity is weakened, but does not vanish. The tendency for rapid alignment was reported in several studies of incompressible MHD [START_REF] Neffaa | The decay of magnetohydrodynamic turbulence in a confined domain[END_REF][START_REF] Matthaeus | Rapid alignment of velocity and magnetic field in magnetohydrodynamic turbulence[END_REF][START_REF] Servidio | Depression of nonlinearity in decaying isotropic mhd turbulence[END_REF] . We will here investigate whether the alignment shows a qualitatively different behavior. For this we show in Fig. 9 the probability functions of the cosine of the angle between the different fields at t = 1200τ A . At instant t = 0, the PDF is almost constant (not shown), hence no (u, D) alignment is observed. At instant t = 1200τ A , the PDF reaches two peaks at ±1 for all three quantities, reflecting preferential alignment and anti-alignment between all three quantities. This local alignment is strongest between the velocity field and the passive field D. This illustrates that all three vectors locally align with each-other. However, the quantitative difference is not enormous, and the fact that the saturated velocity field is able to amplify the passive seed field is likely to depend on subtle correlations between the magnetic field and the velocity field. We recall that, for the numerical simulations reported here, P ′ m = 1.43 > P m = 1 and that the simplifying assumptions leading to the formulation of Eqs. (8) are not strictly fulfilled and that the study of these correlations would require reaching a dynamo regime at P ′ m = 1.

V. CONCLUSION

The question which motivated the present study was whether the RFP velocity field can amplify a seed-field and can thereby called a dynamo field. The answer to the question is not straightforward. If we do not put any constraints on the diffusivity of the passively advected field, the answer is yes, since both in the laminar, steady case and in the dynamic case we have succeeded to determine a finite value of the critical Reynolds number, as shown in Fig. 3.

In all our simulations we needed however to raise the magnetic Prandtl number of the passive field P m ′ above the unity value of the corresponding value for the RFP magnetic field. This shows that fluctuations of the Bfield are not subject to the dynamo-effect in the present set-up.

We also traced down a difference in the description of the B and the D field, by considering the Elsasser variables. In particular we showed that alignment of the velocity field and the B-field reduces more strongly the This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0078860

nonlinear interactions than alignment of the D-field with the velocity does.

Exploring in more detail the saturation-mechanism of the turbulent dynamo in terms of detailed alignment properties and Elsasser-dynamics seems a promising direction for further research.

FIG. 1 .

 1 FIG. 1. Visualizations of the kinematic dynamo system. The subfigures (a), (b), (c) show iso-surfaces of B 2 , u 2 and D 2 , respectively. These isosurfaces are colored by the sign of the axial component. For instance in (a) Bz > 0 is colored in red, Bz < 0 in blue.

3 FIG. 2 .

 32 FIG. 2. Lundquist dependence of the Reynolds number inthe kinematic dynamo investigation. The Reynolds number is based on the root-mean-square velocity in the domain.

FIG. 3 .

 3 FIG. 3. Evolution of the critical magnetic Reynolds number R ′cm as a function of the (kinematic) Reynolds number Re. The red, circular data points correspond the kinematic dynamo simulations with a fixed velocity field. The blue, star-shaped data-point corresponds to the result for a fully dynamic simulation, discussed in Sec. IV. The inset shows the time evolution of the passive magnetic energy for different values of P ′ m of a given flow with Re = 6.37.

FIG. 4 .

 4 FIG. 4. Normalized axial spectra of B 2 (top), u 2 (center) and D 2 (bottom) for different values of Re and for P ′ m = 4.

FIG. 5 .

 5 FIG.5. Visualization of the normalized kinetic energy isosurface, u 2 / u 2 = 6.6 and the normalized passive magnetic energy isosurface D 2 / D 2 = 9, colored by the axial velocity field uz and axial passive magnetic field Dz, respectively, for Re = 6.37 and P ′ m = 4.

FIG. 6 . 2 FIG. 7 . 2 .

 6272 FIG.6. Visualizations of an instantaneous snapshot of the (a) iso-surfaces of the kinetic energy u 2 and (b) passively advected magnetic energy D 2 , colored by the axial component, in the turbulent RFP simulation. Whereas the velocity field seems to be dominated by a small number of energetic modes, the advected field shows a more turbulent behavior.

2 FIG. 8 .

 28 FIG. 8. Time evolution of the variance of the passive field in the dynamic runs for Re = 50 and P ′ m = 1.43; 2.

1 FIG. 9 .

 19 FIG.9. Probability density functions of the alignment between the three fields during the statistically steady state of the dynamic RFP simulations.

32 2

 2 This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0078860

ACKNOWLEDGMENTS

Simulations were carried out using the facilities of the PMCS2I (École Centrale de Lyon), IDRIS (Project No. 22206) and P2CHPD. We further acknowledge support by the French Federation for Magnetic Fusion Studies (FR-FCM) and of the Eurofusion consortium from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.