2.1 The VKS experiment

Three successful fluid dynamo experiments have been performed so far: the Karlsruhe experiment (Stieglitz & Müller Reference Stieglitz and Müller2001), the Riga experiment (Gailitis et al. Reference Gailitis, Lielausis, Platacis, Dement’ev, Cifersons, Gerbeth, Gundrum, Stefani and Will2001) and the VKS experiment (Monchaux et al. Reference Monchaux, Berhanu, Bourgoin, Moulin, Odier, Pinton, Volk, Fauve, Mordant and Pétrélis2007). In contrast to the Karlsruhe and Riga dynamos, the magnetic field generated in the VKS experiment strongly differs from the one generated if the mean flow were acting alone.

The VKS experiment was designed to study the generation of magnetic field by a von Kármán swirling flow of liquid sodium (figure 1 a). Von Kármán swirling flows are generated by two co-axial rotating disks (Zandbergen & Dijkstra Reference Zandbergen and Dijkstra1987). The fluid is expelled radially outward by the centrifugal force along each disk and recirculates because of incompressibility. This drives an inward flow in the mid-plane between the two disks and an axial flow toward each disk along their axis as in the case of centrifugal pumps. When the disks are counter-rotating, the toroidal component of the mean flow vanishes and changes sign between the two disks. This strong shear generates a high level of turbulent fluctuations with an intensity comparable to the one of the mean flow. The initial motivation to use this von Kármán flow configuration for dynamo studies resulted from:

1. (i) the observation of strong vorticity concentrations (Douady, Couder & Brachet Reference Douady, Couder and Brachet1991; Fauve, Laroche & Castaing Reference Fauve, Laroche and Castaing1993);

2. (ii) the existence of differential rotation related to the toroidal mean flow component;

3. (iii) and the lack of mirror symmetry.

These arguments in favour of a von Kármán flow configuration were qualitative, the first one being related to the questionable vorticity–magnetic field analogy (Batchelor Reference Batchelor1950), the two others being known to act in the most efficient dynamo mechanisms, although not necessary for dynamo action. In its final configuration, the VKS experiment contained approximately $160$ l of liquid sodium in a cylinder of inner radius $R_{0}=289~\text{mm}$ and height $604$ mm, driven by two co-axial impellers made of disks of radius $R=154.5~\text{mm}$ , $371~\text{mm}$ apart and fitted with eight curved blades of height $41.2$ mm in order to increase the flow entrainment. An inner cylinder of radius $R_{c}=206~\text{mm}$ and height $524~\text{mm}$ has been used in earlier runs in order to obtain sodium at rest surrounding the flow. An annulus has also been attached in the mid-plane along the inner cylinder in order to reduce turbulent fluctuations of the shear layer. These appendages did not improve the dynamo efficiency and were removed later on. The $300$ kW motors drive the impellers in counter-rotation at a maximum frequency $25$ Hz. Above a rotation frequency $F_{1}=F_{2}=f\simeq 13$ Hz, i.e. a magnetic Reynolds number $\text{R}_{m}=\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70E}2\unicode[STIX]{x03C0}fRR_{0}\simeq 43$ , a magnetic field is generated by the flow. The kinetic Reynolds number, $Re$ , being larger than $10^{6}$ , the flow is strongly fluctuating as well as the generated magnetic field. Its temporal average is roughly a dipole with its axis parallel to the rotation axis of the experiment (figure 1 b). The geometry of the magnetic field was one of the surprising results of the VKS experiment. It was indeed commonly believed that the mean part of the flow could be more efficient in generating a magnetic field than the turbulent fluctuations. The mean flow generated in the von Kármán configuration with counter-rotating disks is axisymmetric and has the topology of the so-called $s_{2}t_{2}$ flow considered within a sphere by Dudley & James (Reference Dudley and James1989). The mean flow being axisymmetric, the generated magnetic field should break axisymmetry according to Cowling theorem (Cowling Reference Cowling1933). Indeed, it takes the form of an equatorial dipole (see figure 2 a). Therefore, the magnetic field in the VKS experiment is not that generated if the mean flow were acting alone and turbulent fluctuations play an important role.

Figure 1. (a) Sketch of the VKS experiment. The flow is generated by two co-axial impellers counter-rotating at frequencies $f_{1}$ and $f_{2}$ . The grey lines show the location of the Hall probes. (b) Sketch of the mean magnetic field generated with counter-rotating impellers at the same frequency. Both poloidal (blue) and toroidal (red) field lines are displayed.

It should be emphasized that the available motor power of the VKS experiment does not allow us to reach the dynamo threshold using stainless steel impellers. The critical magnetic Reynolds number given above corresponds to impellers made of iron, i.e. with a high magnetic permeability. The effect of iron will be discussed below but we observe that it does not affect the above picture: even with impellers made of iron, an axisymmetric flow generates an equatorial dipole field (Gissinger et al. Reference Gissinger, Iskakov, Fauve and Dormy2008b ) whereas an axial dipole is observed in the experiment. In order to correct misleading statements about the VKS experiment, it should be stressed that iron impellers are not the primary reason for the generation of a time-averaged magnetic field in the form of an axial dipole. Non-axisymmetric fluctuations of the velocity field are the essential requirement. It could seem afterwards obvious that turbulent fluctuations as large as the mean flow cannot be neglected when predicting the dynamo, but when the VKS experiment was developed, many groups were convinced that kinematic dynamo calculations made using the time-averaged velocity field could be enough to predict the geometry of the magnetic field and its dynamo threshold.

Figure 2. (a) Equatorial dipole generated by an axisymmetric mean flow in a cylindrical domain. (b) Axial dipole generated when a non-axisymmetric component in the form of vortices along the blades is added to the mean flow.

2.2 The $\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D714}$ mechanism

Shortly after the observation of the dynamo effect in the VKS experiment (Monchaux et al. Reference Monchaux, Berhanu, Bourgoin, Moulin, Odier, Pinton, Volk, Fauve, Mordant and Pétrélis2007), we proposed an $\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D714}$ mechanism for the generation of the magnetic field (Petrelis, Mordant & Fauve Reference Petrelis, Mordant and Fauve2007). The $\unicode[STIX]{x1D714}$ effect is straightforward to understand in the VKS experiment. It is related to the counter-rotating impellers that generate a strong differential rotation along the axis of the set-up. An applied axial magnetic field therefore generates an azimuthal field that can be easily as large as the axial one (Bourgoin et al. Reference Bourgoin, Marie, Petrelis, Gasquet, Guigon, Luciani, Moulin, Namer, Burguete and Chiffaudel2002). As usual, the difficult step is to convert this azimuthal field to an axial one in order to close the loop of the dynamo mechanism. The idea is to take into account some important non-axisymmetric velocity fluctuations related to the blades of the impellers used in the experiment. A strong radial outflow is generated along the disk between two successive blades because of the centrifugal force. In addition, the shear related to the counter-rotation of the impellers generates radial vorticity such that the radially outward flow that exists between two successive blades is a swirling flow with helicity. These 8 helical vortices provide a way to generate an axial magnetic field from an azimuthal one through the $\unicode[STIX]{x1D6FC}$ effect. It can be easily checked that the respective signs of $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D714}$ are appropriate for a dynamo and that the helicity of the generated magnetic field by this $\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D714}$ mechanism has the opposite sign to the kinetic helicity of the radial vortices along the blades. These properties were observed in the VKS experiment.

This $\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D714}$ mechanism has been confirmed by kinematic dynamo simulations: it has been shown that when strong enough vortices along the blades are added to an axisymmetric mean flow, the generated dynamo is no longer an equatorial dipole but is dominated by an axial dipole (see figure 2 b Gissinger Reference Gissinger2009). It should be noted that, despite its simplicity, this model is the only numerical model of the VKS experiment that has also been able to reproduce the main transitions observed in the parameter space of the VKS experiment, i.e. an alternation of stationary and oscillatory dynamo regimes as the difference in the speeds of the impellers is increased from the counter-rotating case (Berhanu et al. Reference Berhanu, Verhille, Boisson, Gallet, Gissinger, Fauve, Mordant, Petrelis, Bourgoin and Odier2010). To the best of our knowledge, other models, including much more sophisticated ones, have not yet been able to capture these important features.

Following the $\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D714}$ mechanism described in Petrelis et al. (Reference Petrelis, Mordant and Fauve2007), some attempts have been made to model helical fluctuations using mean-field magnetohydrodynamics (MHD) equations including the $\unicode[STIX]{x1D6FC}$ tensor (Laguerre et al. Reference Laguerre, Nore, Ribeiro, Leorat, Guermond and Plunian2008). Although some initial success has been claimed, the same authors and others have later concluded that unrealistically large values of $\unicode[STIX]{x1D6FC}$ should be considered in order to reproduce the experimentally observed threshold value (Giesecke et al. Reference Giesecke, Nore, Plunian, Laguerre, Ribeiro, Stefani, Gerbeth, Leorat and Guermond2010a ). We comment that it could be difficult to get quantitative results assuming a rough distribution of the $\unicode[STIX]{x1D6FC}$ effect as well as a mean flow only in the bulk of the cylinder or no mean flow at all. In addition, it is clear that the $\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D714}$ mechanism does not take into account the rotating iron impellers that are known to decrease the dynamo threshold observed in experiments. It would be more realistic to evaluate the value of $\unicode[STIX]{x1D6FC}$ needed to reproduce the threshold of order $\text{R}_{m}\sim 100$ , experimentally predicted with stainless steel impellers inside an iron cylinder using the decay time of a transient magnetic field.

2.3 The effect of iron impellers

The VKS dynamo has been observed so far only when impellers made of soft iron have been used. More precisely, one iron impeller is enough provided it rotates fast enough. Both the disk and the blades should be made of iron. This of course does not mean that iron impellers are necessary to generate a dynamo. They are required in our experiment in order to reach the dynamo threshold $R_{m}^{c}$ with the available motor power.

Impellers made of iron first modify the boundary conditions for the magnetic field. It is therefore not surprising that this changes the dynamo threshold and this was the motivation to use them. Numerical simulations have shown that magnetic boundary conditions corresponding to the high permeability limit significantly decrease the dynamo threshold in the VKS geometry both for an equatorial or axial dipolar mode (Gissinger et al. Reference Gissinger, Iskakov, Fauve and Dormy2008b ; Gissinger Reference Gissinger2009).

This shift in threshold does not fully explain the experimental results since an experiment performed with two impellers made of stainless steel in an iron inner cylinder of radius $R_{c}=206~\text{mm}$ did not reach the dynamo threshold. However, measurements of the decay rate of a transient magnetic field extrapolate to zero for a frequency of approximately $45$ Hz, therefore predicting a critical magnetic Reynolds number $R_{m}^{c}=\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70E}2\unicode[STIX]{x03C0}fRR_{c}\sim 100$ . This can be compared to the dynamo threshold observed with impellers made of iron in a copper inner cylinder of radius $R_{c}=206~\text{mm}$ , $R_{m}^{c}\sim 36$ . The VKS experiment has therefore shown that, for a given flow of liquid sodium, iron impellers strongly decrease the dynamo threshold.

Numerical simulations using mean-field MHD with boundary conditions that mimic the iron impellers of the VKS experiment have shown that the required magnitude of the $\unicode[STIX]{x1D6FC}$ -effect for the dynamo threshold decreases when the magnetic permeability of the impellers increases (Giesecke, Stefani & Gerbeth Reference Giesecke, Stefani and Gerbeth2010b ). Below dynamo onset, ferromagnetic impellers lead to an increased decay time of the axisymmetric mode (Giesecke et al. Reference Giesecke, Nore, Stefani, Gerbeth, Leorat, Herreman, Luddens and Guermond2012). It has been claimed that impellers of high magnetic permeability are important ‘to promote axisymmetric modes’. This is true only at low kinetic Reynolds number $Re$ . When $Re$ increases and the flow becomes turbulent, an axial dipolar time-averaged dynamo is favoured compared to an equatorial dipole even without ferromagnetic boundary conditions (Gissinger, Dormy & Fauve Reference Gissinger, Dormy and Fauve2008a ). It has also been argued (Giesecke et al. Reference Giesecke, Stefani and Gerbeth2010b ) that the periodic modulation of the magnetic permeability in the azimuthal direction resulting from the presence of the blades could generate a different dynamo mechanism in which the poloidal and toroidal field components are coupled through the boundary conditions. This is indeed a possible mechanism but it leads to a dynamo threshold orders of magnitude larger than the one observed in the VKS experiment (Gallet, Petrelis & Fauve Reference Gallet, Petrelis and Fauve2012, Reference Gallet, Petrelis and Fauve2013).

An analytical model using mean-field MHD allows us to understand the physical mechanism explaining the decrease of dynamo threshold that results from the presence of an iron disk (Herault & Petrelis Reference Herault and Petrelis2014). It should be first noticed that the differential rotation generating the $\unicode[STIX]{x1D714}$ -effect in the VKS experiment has opposite signs in the bulk and behind the disks. Assuming that the poloidal field does not change sign across the disk, the azimuthal field generated by the $\unicode[STIX]{x1D714}$ -effect should change sign, and therefore should vanish on the disk. The other alternative is that the poloidal field vanishes on the disk, the azimuthal field keeping the same sign on both sides of the disk. In both cases, if the component of the field that vanishes remains small close to the disk, the dynamo efficiency that requires the presence of both components, will decrease. It has been shown in Herault & Petrelis (Reference Herault and Petrelis2014) that increasing the magnetic permeability of a disk results in a more abrupt change of sign of the axial field. Therefore the axial field is large on both sides of the disk, thus leading to a configuration with a good dynamo efficiency. It has also been shown that there is an optimum value of the magnetic permeability for maximum dynamo efficiency, i.e. minimum dynamo threshold. The low dynamo threshold observed when both the disks and the blades are ferromagnetic can be understood along the same line of thought: the easy magnetization direction is azimuthal in the disk and along the blades in the blades. The ferromagnetic disk (respectively blades) leads to a large toroidal (respectively poloidal) component of the magnetic field in the vicinity of the impeller. Therefore both the $\unicode[STIX]{x1D714}$ and $\unicode[STIX]{x1D6FC}$ effects are large in the same region close to the impellers, thus providing a high dynamo efficiency.

Direct numerical simulations taking into account the boundary conditions related to iron impellers have been recently performed (Nore et al. Reference Nore, Quiroz, Cappanera and Guermond2016, Reference Nore, Quiroz, Cappanera and Guermond2017). It has been checked that increasing the magnetic permeability of the impellers (up to $\unicode[STIX]{x1D707}_{r}=100$ ) decreases the dynamo threshold. More precisely, it has been found that for $Re=500$ , $\unicode[STIX]{x1D707}_{r}$ does not affect much the threshold of the equatorial dipole ( $m=1$ mode) whereas increasing $\unicode[STIX]{x1D707}_{r}$ decreases the threshold of the axial dipole ( $m=0$ mode).

More interestingly, it has been found that, even for $\unicode[STIX]{x1D707}_{r}=1$ , the axisymmetric component of the magnetic field is dominant when $Re$ is large and well above the threshold. This is in agreement with Gissinger et al. (Reference Gissinger, Dormy and Fauve2008a ) and shows that a high magnetic permeability is not necessary to generate an axisymmetric time-averaged magnetic field provided $Re$ is large enough. These results give confidence that the VKS dynamo with $Re\sim 5\times 10^{6}$ would involve a dominant time-averaged axial dipole with non-ferromagnetic impellers, the effect of $\unicode[STIX]{x1D707}_{r}$ being just to shift the dynamo threshold without changing the geometry of the saturated magnetic field. In addition, it has been checked (Nore et al. Reference Nore, Quiroz, Cappanera and Guermond2017) that the magnetic field generated by the time-averaged flow alone is an equatorial dipole and therefore strongly differs from the one which is observed in direct simulations. This confirms that the VKS dynamo strongly depends on turbulent fluctuations, as already emphasized (Petrelis et al. Reference Petrelis, Mordant and Fauve2007). In addition, for $\unicode[STIX]{x1D707}_{r}=50$ , a continuous decrease of the dynamo threshold is observed when $Re$ is increased from $500$ to $1500$ and even up to $10^{5}$ using large eddy simulations (LES). Much less data are available for $\unicode[STIX]{x1D707}_{r}=5$ but it is observed that the threshold of the axial dipole is decreased when $Re$ is increased from $500$ to $1500$ whereas the threshold of the equatorial dipole is increased (Nore et al. Reference Nore, Quiroz, Cappanera and Guermond2017). This can confirm that turbulent fluctuations provide an efficient dynamo mechanism for the axial dipole observed in the VKS experiment.

Note however that in another direct simulation of the VKS experiment (Kreuzahler et al. Reference Kreuzahler, Ponty, Plihon, Homann and Grauer2017) where it has also been found that the magnetic permeability of the impellers decreases the dynamo threshold, some different trends have been observed: the spectrum of the instantaneous magnetic field for $\unicode[STIX]{x1D707}_{r}=1$ and $Re=1500$ involves a dominant $m=1$ mode and an increase of the dynamo threshold is observed when $Re$ is increased from $500$ to $1500$ for $\unicode[STIX]{x1D707}_{r}=1$ . It should be noted that the time-averaged magnetic field is not computed in these simulations such that the comparison with the VKS experiment is difficult. The instantaneous magnetic field obviously involves many modes since the flow is strongly turbulent. In contrast, the geometry of the time-averaged magnetic field is well defined and one expects an axial dipole to leading order. There is indeed no preferred direction in the equatorial plane for the equatorial dipole and, in the presence of fluctuations, it is likely to drift and therefore to average to zero. It should be noted that the simulations performed by Kreuzahler et al. (Reference Kreuzahler, Ponty, Plihon, Homann and Grauer2017) involve a cubic box with periodic boundary conditions in which the cylindrical flow is embedded. Axisymmetry is therefore broken and this can quench the direction of the equatorial dipole. The different behaviours of the dynamo threshold versus $Re$ observed in the simulations made by the two groups can be understood as follows: turbulent fluctuations increase the dynamo threshold of the equatorial dipole whereas they decrease that of the axial dipole. It would be interesting to check that independently of the value of $\unicode[STIX]{x1D707}_{r}$ .

The simulations made by the two groups strongly disagree on the prediction of the dynamo thresholds. This disagreement seems too large to be explained only by the different boundary conditions that are used. It would also be interesting to study the saturation of the magnetic field. Depending on the magnetic Prandtl number, it should be possible to check the transition between the two scaling laws for the saturated magnetic field predicted in Petrelis & Fauve (Reference Petrelis and Fauve2001). Another problem that has not been addressed is the dependence of the saturated magnetic field on the magnetic permeability. Finally, these simulations have not been able to reproduce so far the different dynamo regimes observed in the VKS experiment or to check simple experimental facts such as the dependence of the dynamo threshold on the direction of rotation of the impellers with curved blades. Checking these experimental observations would provide a useful and comforting benchmark of the numerical simulations.

3.1 Effect of weak fluctuations on the dynamo threshold

A way to study the effect of fluctuations on the dynamo threshold is to calculate it perturbatively in the amplitude of the perturbations. This approach is not restricted to mean-field dynamos and is presented here for an unspecified flow. Using the Reynolds decomposition, we write

(3.1) $$\begin{eqnarray}\boldsymbol{V}(\boldsymbol{r},\boldsymbol{t})=\overline{\boldsymbol{V}}(\boldsymbol{r})+\tilde{\boldsymbol{v}}(\boldsymbol{r},\boldsymbol{t}),\end{eqnarray}$$

where $\overline{\boldsymbol{V}}(\boldsymbol{r})$ is the mean flow and $\tilde{\boldsymbol{v}}(\boldsymbol{r},\boldsymbol{t})$ are the turbulent fluctuations. The overbar stands for a temporal average. The induction equation then becomes

(3.2) $$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}}=\unicode[STIX]{x1D735}\times (\overline{\boldsymbol{V}}\times \boldsymbol{B})+\unicode[STIX]{x1D735}\times (\tilde{\boldsymbol{v}}\times \boldsymbol{B})+\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{B}.\end{eqnarray}$$

We observe that turbulent fluctuations $\tilde{\boldsymbol{v}}(\boldsymbol{r},\boldsymbol{t})$ act as a random multiplicative forcing in the induction equation (3.2). It is well known, both from simple theoretical models (Stratonovich Reference Stratonovich1963; Graham & Schenzle Reference Graham and Schenzle1982; Lucke & Schank Reference Lucke and Schank1985) and from experiments on different instability problems (Kabashima et al. Reference Kabashima, Kogure, Kawakubo and Okada1979; Residori, Berthet, Roman & Fauve Reference Residori, Berthet, Roman and Fauve2002; Berthet et al. Reference Berthet, Petrossian, Residori, Roman and Fauve2003; Petrelis & Aumaitre Reference Petrelis and Aumaitre2003; Petrelis, Aumaitre & Fauve Reference Petrelis, Aumaitre and Fauve2005), that multiplicative noise generally shifts the bifurcation threshold.

The calculation of the shift in threshold has been given by Pétrélis (Reference Pétrélis2002) and Fauve & Pétrélis (Reference Fauve, Pétrélis, Sepulchre and Beaumont2003) using a perturbation expansion in the limit of small fluctuations. Let $\overline{\boldsymbol{V}}$ be the average velocity field at onset, and $\boldsymbol{B}$ the neutral mode of the instability. Let $\boldsymbol{V}^{(0)}$ be the flow leading to the neutral mode $\boldsymbol{B}^{(0)}$ when there are no velocity fluctuations. Our aim is to find how the dynamo threshold of the velocity field $\boldsymbol{V}^{(0)}$ is modified in the presence of small turbulent fluctuations. We write $\tilde{\boldsymbol{v}}=\unicode[STIX]{x1D6FF}\boldsymbol{v}$ where $\unicode[STIX]{x1D6FF}$ is a small parameter that measures the amplitude of the turbulent fluctuations so that $\boldsymbol{v}$ is of order one. The neutral mode is likely to be slightly modified by the fluctuations as well as the dynamo threshold. We therefore expand $\boldsymbol{B}$ and $\overline{\boldsymbol{V}}$ in powers of $\unicode[STIX]{x1D6FF}$

(3.3) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \boldsymbol{B}=\boldsymbol{B}^{(0)}+\unicode[STIX]{x1D6FF}\,\boldsymbol{B}^{(1)}+\unicode[STIX]{x1D6FF}^{2}\,\boldsymbol{B}^{(2)}+\cdots \,,\\ \displaystyle \overline{\boldsymbol{V}}=\boldsymbol{V}^{(0)}(1+c_{1}\unicode[STIX]{x1D6FF}\,+c_{2}\unicode[STIX]{x1D6FF}^{2}\,+\cdots \,),\end{array}\right\}\end{eqnarray}$$

$\boldsymbol{B}^{(i)}$ are the corrections at order $i$ to the neutral mode due to the presence of the turbulent fluctuations. $c_{i}$ are constants that express the shift in the dynamo threshold caused by turbulence. We emphasize that we study the modification of the dynamo threshold of a mean flow with prescribed geometry due to the presence of fluctuations. When one inputs these expressions in (3.2), the zeroth-order part can be written

(3.4) $$\begin{eqnarray}L\,\boldsymbol{B}^{(0)}={\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{B}^{(0)}}{\unicode[STIX]{x2202}t}}-\,\unicode[STIX]{x1D735}\times (V^{(0)}\times \boldsymbol{B}^{(0)})-\unicode[STIX]{x1D702}\,\unicode[STIX]{x1D6FB}^{2}\boldsymbol{B}^{(0)}=0,\end{eqnarray}$$

where $L$ is a linear operator. This is the laminar dynamo problem. By hypothesis, the instability onset is the one without turbulent perturbation. At next order in $\unicode[STIX]{x1D6FF}$ we get

(3.5) $$\begin{eqnarray}L\,\boldsymbol{B}^{(1)}=c_{1}\,\unicode[STIX]{x1D735}\times (V^{(0)}\times \boldsymbol{B}^{(0)})+\unicode[STIX]{x1D735}\times (\boldsymbol{v}\times \boldsymbol{B}^{(0)}).\end{eqnarray}$$

We now introduce a scalar product $\langle f|g\rangle$ and calculate $L^{+}$ the adjoint of $L$ . As $LB^{(0)}=0$ , $L^{+}$ also has a non-empty kernel. Let $\boldsymbol{C}$ be in this kernel. Then

(3.6) $$\begin{eqnarray}\langle \boldsymbol{C}|L\boldsymbol{B}^{(1)}\rangle =\langle L^{+}\boldsymbol{C}|\boldsymbol{B}^{(1)}\rangle =0\end{eqnarray}$$

and this solvability condition gives the first-order correction in the threshold

(3.7) $$\begin{eqnarray}c_{1}=-{\displaystyle \frac{\langle \boldsymbol{C}|\unicode[STIX]{x1D735}\times (\boldsymbol{v}\times \boldsymbol{B}^{(0)})\rangle }{\langle \boldsymbol{C}\unicode[STIX]{x1D735}\times (V^{(0)}\times \boldsymbol{B}^{(0)})\rangle }}.\end{eqnarray}$$

We use a scalar product in which the average over the realizations of the perturbation is made. In that case, the average over the realizations of $\langle \boldsymbol{C}|\unicode[STIX]{x1D735}\times (\boldsymbol{v}\times \boldsymbol{B}^{(0)})\rangle$ is proportional to the average of $\boldsymbol{v}$ , the value of which is zero by hypothesis. Thus, the dynamo threshold is unchanged up to first order in $\unicode[STIX]{x1D6FF}$ , $c_{1}=0$ . This result is obvious in many simple cases. For instance if $\tilde{\boldsymbol{v}}$ is sinusoidal in time, the threshold shift cannot depend on the phase which implies that it is invariant if $\tilde{\boldsymbol{v}}\rightarrow -\tilde{\boldsymbol{v}}$ . This is also true if $\tilde{\boldsymbol{v}}$ is a random noise with equal probabilities for the realizations $\tilde{\boldsymbol{v}}$ and $-\tilde{\boldsymbol{v}}$ . Note however that simple symmetry arguments do not apply for asymmetric fluctuations about the origin although the threshold shift vanishes to leading order if the fluctuations have zero mean.

To calculate the next-order correction, we write (3.2) at order two in $\unicode[STIX]{x1D6FF}$ and get

(3.8) $$\begin{eqnarray}L\,\boldsymbol{B}^{(2)}=c_{2}\,\unicode[STIX]{x1D735}\times (V^{(0)}\times \boldsymbol{B}^{(0)})+\unicode[STIX]{x1D735}\times (\boldsymbol{v}\times \boldsymbol{B}^{(1)}).\end{eqnarray}$$

We then get the second-order correction

(3.9) $$\begin{eqnarray}c_{2}=-{\displaystyle \frac{\langle \boldsymbol{C}|\unicode[STIX]{x1D735}\times (\boldsymbol{v}\times \boldsymbol{B}^{(1)})\rangle }{\langle \boldsymbol{C}\unicode[STIX]{x1D735}\times (V^{(0)}\times \boldsymbol{B}^{(0)})\rangle }},\end{eqnarray}$$

where $\boldsymbol{B}^{(1)}$ is solution of

(3.10) $$\begin{eqnarray}L\boldsymbol{B}^{(1)}=\unicode[STIX]{x1D735}\times (\boldsymbol{v}\times \boldsymbol{B}^{(0)}).\end{eqnarray}$$

Here, there is no simple reason for the correction to be zero. Its computation requires the resolution of (3.10). In some simple cases, an analytical expression for $c_{2}$ can be calculated and both signs can be found, thus showing that fluctuations can in general increase or decrease the dynamo threshold (see below).

The shift in threshold occurring at second order, we understand why the dynamo thresholds observed in the Karlsruhe and Riga experiments were in agreement with the predictions made from the mean flow alone. The turbulent fluctuations related to the mean flow are indeed small in these experiments (not higher than $10\,\%$ ).

3.2 Effect of velocity fluctuations on an $\unicode[STIX]{x1D6FC}$ -dynamo

The effect of statistical fluctuations of the number of cyclonic cells on the $\unicode[STIX]{x1D6FC}$ -effect has been considered first by Parker (Reference Parker1969) in order to explain field reversals of the Earth’s magnetic field. Phenomenological mean-field models with a random component of the $\unicode[STIX]{x1D6FC}$ -effect have been studied by Hoyng (Reference Hoyng1993), Hoyng, Schmitt & Teuben (Reference Hoyng, Schmitt and Teuben1994) as models of the variability of the solar cycle or to describe field reversals (Hoyng, Schmitt & Ossendrijver Reference Hoyng, Schmitt and Ossendrijver2002; Hoyng & Duistermaat Reference Hoyng and Duistermaat2004). Dynamos generated by rapid fluctuations in the $\unicode[STIX]{x1D6FC}$ -effect in the presence of shear have been studied by Sokolov (Reference Sokolov1997), Vishniac & Brandenburg (Reference Vishniac and Brandenburg1997), Silant’ev (Reference Silant’ev2000), Proctor (Reference Proctor2007), Kleeorin & Rogachevskii (Reference Kleeorin and Rogachevskii2008), Richardson & Proctor (Reference Richardson and Proctor2010), Sridhar & Singh (Reference Sridhar and Singh2010), Richardson & Proctor (Reference Richardson and Proctor2012).

We will not consider these problems here but study how velocity fluctuations in a Roberts flow affect the dynamo for which some conclusions can be drawn using the perturbation approach presented above. Indeed, in the presence of small-scale fluctuations (3.10) simplifies as the diffusive part is the dominant term of the operator $L$ and the gradient of the velocity is the dominant part of the source term (right-hand side of the equation). We thus obtain for $\boldsymbol{B}^{(1)}$ the expression of the small-scale field in a standard calculation of the $\unicode[STIX]{x1D6FC}$ -effect for a flow in scale separation. The shift in onset at second order, (3.9), is then proportional to the $\unicode[STIX]{x1D6FC}$ -effect of the fluctuations. Therefore, if the fluctuations do not contribute to an $\unicode[STIX]{x1D6FC}$ -effect, for instance if they are parity invariant, the change in onset vanishes at second order and will be cubic or smaller in the amplitude of the fluctuations.

These properties can be checked on the following example (Pétrélis & Fauve Reference Pétrélis and Fauve2006)

(3.11) $$\begin{eqnarray}\boldsymbol{v}=\boldsymbol{v}_{0}(y,z)+\boldsymbol{v}_{f}(y,z,t)=\left(\begin{array}{@{}c@{}}V(\cos (ky)-\cos (kz))(1+\unicode[STIX]{x1D6FF}_{v}\cos (\unicode[STIX]{x1D714}_{v}t+\unicode[STIX]{x1D719}_{v}))\\ U\,\sin (kz)(1+\unicode[STIX]{x1D6FF}_{u}\cos (\unicode[STIX]{x1D714}_{u}t+\unicode[STIX]{x1D719}_{u}))\\ U\,\sin (ky)(1+\unicode[STIX]{x1D6FF}_{u}\cos (\unicode[STIX]{x1D714}_{u}t+\unicode[STIX]{x1D719}_{u}))\end{array}\right).\end{eqnarray}$$

The constant part of this flow, $\boldsymbol{v}_{0}$ ( $\unicode[STIX]{x1D6FF}_{u}=\unicode[STIX]{x1D6FF}_{v}=0$ ), is the Roberts’ flow (Roberts Reference Roberts1970, Reference Roberts1972). It consists of a square periodic array of counter-rotating eddies in the $y$ – $z$ plane, with axial flow in each of them such that all vortex have helicity of the same sign (we take $U>0$ , $V>0$ ). This flow is close to the time-averaged flow of the Karlsruhe experiment (Stieglitz & Müller Reference Stieglitz and Müller2001). Such a flow is a quite efficient dynamo because a large-scale magnetic field can be generated by an $\unicode[STIX]{x1D6FC}$ -effect (Moffatt Reference Moffatt1978). As $R_{m}$ is small at dynamo onset (see discussion in § 1), analytical progresses can be performed. The perturbation $\boldsymbol{v}_{f}$ is here a time-periodic function. The expansion of § 3.1 remains essentially unchanged provided the average over realizations is replaced by a time average. Calculating the $\unicode[STIX]{x1D6FC}$ -effect $\langle \boldsymbol{v}\times \boldsymbol{b}\rangle$ where $\boldsymbol{b}$ is the small-scale field, we obtain $\langle \boldsymbol{v}\times \boldsymbol{b}\rangle =\langle \boldsymbol{v}_{0}\times \boldsymbol{b}_{0}\rangle +\langle \boldsymbol{v}_{f}\times \boldsymbol{b}_{f}\rangle$ .

The first term is related to the $\unicode[STIX]{x1D6FC}$ -effect of the basic flow. Indeed we have

(3.12) $$\begin{eqnarray}\langle \boldsymbol{v}_{0}\times \boldsymbol{b}_{0}\rangle =\unicode[STIX]{x1D6FC}_{0}\left(\begin{array}{@{}c@{}}0\\ \langle B_{y}\rangle \\ \langle B_{z}\rangle \end{array}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}_{0}=-UV/(\unicode[STIX]{x1D702}k)$ relates the average of the electromotive force to the averaged magnetic field. The second term is

(3.13) $$\begin{eqnarray}\langle \boldsymbol{v}_{f}\times \boldsymbol{b}_{f}\rangle =\unicode[STIX]{x1D6FC}_{f}\left(\begin{array}{@{}c@{}}0\\ \langle B_{y}\rangle \\ \langle B_{z}\rangle \end{array}\right),\end{eqnarray}$$

where

(3.14) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{f}=-{\displaystyle \frac{\unicode[STIX]{x1D702}k^{3}UV\unicode[STIX]{x1D6FF}_{u}\unicode[STIX]{x1D6FF}_{v}}{2(\unicode[STIX]{x1D714}_{u}^{2}+(\unicode[STIX]{x1D702}k^{2})^{2})}}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D714}_{u},\unicode[STIX]{x1D714}_{v})\cos (\unicode[STIX]{x1D719}_{u}-\unicode[STIX]{x1D719}_{v})\end{eqnarray}$$

and $\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D714}_{u},\unicode[STIX]{x1D714}_{v})$ is zero if $\unicode[STIX]{x1D714}_{u}\neq \unicode[STIX]{x1D714}_{v}$ and is one if $\unicode[STIX]{x1D714}_{u}=\unicode[STIX]{x1D714}_{v}$ . We can relate $\unicode[STIX]{x1D6FC}_{f}$ to $\unicode[STIX]{x1D6FC}_{0}$ by

(3.15) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{f}=\unicode[STIX]{x1D6FC}_{0}{\displaystyle \frac{(\unicode[STIX]{x1D702}k^{2})^{2}\unicode[STIX]{x1D6FF}_{u}\unicode[STIX]{x1D6FF}_{v}}{2(\unicode[STIX]{x1D714}_{u}^{2}+(\unicode[STIX]{x1D702}k^{2})^{2})}}\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D714}_{u},\unicode[STIX]{x1D714}_{v})\cos (\unicode[STIX]{x1D719}_{u}-\unicode[STIX]{x1D719}_{v}).\end{eqnarray}$$

We now look for unstable modes of the form $\langle \boldsymbol{B}\rangle ={\mathcal{B}}\text{e}^{\text{i}Kx}$ where $K$ is supposed to be small compared to $k$ such that the averaged magnetic field evolves on a larger scale than the velocity field. The threshold is given by

(3.16) $$\begin{eqnarray}\left|{\displaystyle \frac{\unicode[STIX]{x1D6FC}_{0}+\unicode[STIX]{x1D6FC}_{f}}{\unicode[STIX]{x1D702}K}}\right|=1.\end{eqnarray}$$

If the modulation of the velocity field is assumed to be small, the onset is at lowest order

(3.17) $$\begin{eqnarray}{\displaystyle \frac{UV}{\unicode[STIX]{x1D702}^{2}kK}}=1-\unicode[STIX]{x1D6FF}_{u}\unicode[STIX]{x1D6FF}_{v}{\displaystyle \frac{(\unicode[STIX]{x1D702}k^{2})^{2}}{2(\unicode[STIX]{x1D714}_{u}^{2}+(\unicode[STIX]{x1D702}k^{2})^{2})}}\cos (\unicode[STIX]{x1D719}_{u}-\unicode[STIX]{x1D719}_{v})\unicode[STIX]{x1D6FF}(\unicode[STIX]{x1D714}_{u},\unicode[STIX]{x1D714}_{v}).\end{eqnarray}$$

At this order in the expansion, there is a shift in the onset only if the modulations have the same pulsation. The shift can be positive or negative and its sign is determined by that of $\unicode[STIX]{x1D6FF}_{u}\unicode[STIX]{x1D6FF}_{v}\cos (\unicode[STIX]{x1D719}_{u}-\unicode[STIX]{x1D719}_{v})$ . For in-phase modulations, the onset is lowered, whereas it is increased if the pulsations are out of phase. This effect can be understood by evaluating the helicity of the fluctuating field. If it has the same sign as the basic flow, the $\unicode[STIX]{x1D6FC}$ -effects cooperate and the onset is lowered. In contrast, if the helicities have opposite signs, the onset is increased. The amplitude of the shift decreases with the frequency of the modulation. It varies like $(\unicode[STIX]{x1D714}^{2}/(\unicode[STIX]{x1D702}k^{2})^{2}+1)^{-1}$ .

Another study of the effect of velocity fluctuations on the dynamo generated by a Roberts flow has been performed by Tilgner (Reference Tilgner2007). Superposition of Roberts flows with different wavelengths but with the same sign of helicity has been considered in order to mimic the multiple-scale character of turbulence. It has been found that adding small scales to the Roberts flow at the largest scales can be both helpful and detrimental to dynamo action.

3.3 Phase fluctuations on the Roberts flow

When a turbulent flow is not externally confined and thus can develop in a fully three-dimensional way, it is well known that its root-mean-square (r.m.s.) velocity fluctuations are of integral scale i.e. comparable to the largest velocity scale (Landau & Lifshitz Reference Landau and Lifshitz1987). The above calculations are then of little help to predict a dynamo threshold for the full velocity field $\boldsymbol{v}(\boldsymbol{r},\boldsymbol{t})$ after having computed the one of $\overline{\boldsymbol{v}}(\boldsymbol{r})$ . Indeed, we do not expect that a general relation exists between the threshold for $\boldsymbol{v}(\boldsymbol{r},\boldsymbol{t})$ and the one for $\overline{\boldsymbol{v}}(\boldsymbol{r})$ when the fluctuations are not small compared to the mean flow. Large fluctuations are indeed likely to amplify another dynamo mode that is not related to the one generated by the mean flow. In non-confined flows, some large fluctuations are related to the erratic motion of large eddies. Instead of using Reynolds decomposition, $\boldsymbol{v}(\boldsymbol{r},\boldsymbol{t})=\overline{\boldsymbol{v}}(\boldsymbol{r})+\tilde{\boldsymbol{v}}(\boldsymbol{r},\boldsymbol{t})$ , it is then tempting to model this type of disturbances writing $\boldsymbol{v}(\boldsymbol{r},\boldsymbol{t})=\overline{\boldsymbol{v}}[\boldsymbol{r}+\boldsymbol{s}(\boldsymbol{r},\boldsymbol{t})]+\tilde{\boldsymbol{u}}(\boldsymbol{r},\boldsymbol{t})$ , thus keeping in the mean field the motion of the large eddies. In the language of cellular flows, $\boldsymbol{s}(\boldsymbol{r},\boldsymbol{t})$ represents phase perturbations.

In the following we investigate the effect on the dynamo onset of random phase perturbations of the cellular flow. To wit, we successively study two Roberts flows modified by phase fluctuations (Pétrélis & Fauve Reference Pétrélis and Fauve2006). The first case is a time-dependent phase fluctuation and we write the velocity field as

(3.18) $$\begin{eqnarray}\boldsymbol{v}=\left(\begin{array}{@{}c@{}}V(\cos (ky+\unicode[STIX]{x1D719})-\cos (kz+\unicode[STIX]{x1D713}))\\ U\,\sin (kz+\unicode[STIX]{x1D713})\\ U\,\sin (ky+\unicode[STIX]{x1D719})\end{array}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D719}$ are two random functions that depend on time only. This amounts to randomly switching in time the origin of the flow.

Assuming the gradients to be small, i.e. $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D713}/(\unicode[STIX]{x1D702}k^{2})\ll 1$ and $\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D719}/(\unicode[STIX]{x1D702}k^{2})\ll 1$ , we obtain for the $\unicode[STIX]{x1D6FC}$ -effect

(3.19) $$\begin{eqnarray}\langle \boldsymbol{v}\times \boldsymbol{b}\rangle =-{\displaystyle \frac{UV}{\unicode[STIX]{x1D702}k}}\left(\begin{array}{@{}c@{}}0\\ \langle B_{y}\rangle \left(1-{\displaystyle \frac{1}{\unicode[STIX]{x1D702}^{2}k^{4}}}\langle (\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D719})^{2}\rangle \right)\\ \langle B_{z}\rangle \left(1-{\displaystyle \frac{1}{\unicode[STIX]{x1D702}^{2}k^{4}}}\langle (\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D713})^{2}\rangle \right)\end{array}\right).\end{eqnarray}$$

We first remark that this effective $\unicode[STIX]{x1D6FC}$ -effect is smaller than the $\unicode[STIX]{x1D6FC}$ -effect of the unmodulated flow. Therefore, the dynamo onset is postponed to

(3.20) $$\begin{eqnarray}{\displaystyle \frac{UV}{\unicode[STIX]{x1D702}^{2}kK}}=1+{\displaystyle \frac{1}{\unicode[STIX]{x1D702}^{2}k^{4}}}\langle (\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D719})^{2}\rangle +{\displaystyle \frac{1}{\unicode[STIX]{x1D702}^{2}k^{4}}}\langle (\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D713})^{2}\rangle .\end{eqnarray}$$

Equation (3.20) is obtained when the phase fluctuations are random in time but act coherently in space. We now turn to a space-dependent phase that drives a random detuning between the cells of the flow. Indeed we expect that one of the effects of turbulence on a periodic flow will be to reduce the power spectrum density of the velocity field at wavenumber $k$ . Random fluctuations acting on the phase is a possible, although rough, model of this effect. To investigate this situation, we consider a Roberts flow for which the origin of the cellular flow depends randomly on the axial coordinate and write

(3.21) $$\begin{eqnarray}\boldsymbol{v}=\left(\begin{array}{@{}c@{}}V(\cos (ky+\unicode[STIX]{x1D719})-\cos (kz+\unicode[STIX]{x1D713}))\\ U\,\sin (kz+\unicode[STIX]{x1D713})-{\displaystyle \frac{V}{k}}\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D719}\cos (ky+\unicode[STIX]{x1D719})\\ U\,\sin (ky+\unicode[STIX]{x1D719})+{\displaystyle \frac{V}{k}}\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D713}\cos (kz+\unicode[STIX]{x1D713})\end{array}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D713}$ are functions of $x$ only. Derivatives of the phases appear explicitly in the expression for the velocity in order to insure incompressibility of the flow. Assuming the gradients to be small, i.e. $\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D713}/k\ll 1$ and $\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D719}/k\ll 1$ , the calculation of the fluctuating magnetic field can be performed perturbatively. In this limit, the effect of phase fluctuations is to reduce the part of the $\unicode[STIX]{x1D6FC}$ -effect that drives the instability and the onset of dynamo action is postponed to

(3.22) $$\begin{eqnarray}{\displaystyle \frac{UV}{\unicode[STIX]{x1D702}^{2}kK}}=1+{\displaystyle \frac{\langle (\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D713})^{2}\rangle }{k^{2}}}+{\displaystyle \frac{\langle (\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D719})^{2}\rangle }{k^{2}}}.\end{eqnarray}$$

Note that the averaged helicity of the flow is

(3.23) $$\begin{eqnarray}\langle \boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{v}\rangle =UVk\left(2+{\displaystyle \frac{\langle (\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D719})^{2}\rangle }{k^{2}}}+{\displaystyle \frac{\langle (\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D713})^{2}\rangle }{k^{2}}}\right),\end{eqnarray}$$

such that it is increased by phase fluctuations but this does not result in an increase of the part of the $\unicode[STIX]{x1D6FC}$ -effect that drives the instability.

The results (3.20) and (3.22) are valid for both random and deterministic functions, $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D713}$ , provided that their scale of variation is much larger than that of the flow and much smaller than that of the whole system.

Note that the time (respectively $x$ ) average of (3.18) (respectively (3.21)) gives a mean velocity field $\langle \boldsymbol{v}\rangle$ that depends on $\langle \cos \unicode[STIX]{x1D713}\rangle$ , $\langle \sin \unicode[STIX]{x1D713}\rangle$ , $\langle \cos \unicode[STIX]{x1D719}\rangle$ and $\langle \sin \unicode[STIX]{x1D719}\rangle$ . Thus, these terms explicitly appear in the dynamo threshold of $\langle \boldsymbol{v}\rangle$ that differs from the predictions (3.20) and (3.22) which involve phase gradients, $\unicode[STIX]{x1D702}$ and $k$ .

The examples presented here in the context of mean-field magnetohydrodynamics, show that large-scale fluctuations due to random displacement of eddies within a cellular flow (phase fluctuations), always increase the dynamo threshold, whereas fluctuations of the amplitude of the velocity field can shift the threshold in both directions.