2.1. A von Kármán gallium flow

The experimental set-up is sketched in figure 1 and described in detail in Verhille et al. (Reference Verhille, Plihon, Fanjat, Volk, Bourgoin and Pinton2010). The liquid gallium flow is created by two coaxial counter-rotating impellers (stainless steel disks with radius $R_{d}=82.5~\text{mm}$ fitted with eight straight blades of height 10 mm). The stainless-steel cylinder enclosing the flow has a radius $R_{0}=97~\text{mm}$ (which will be used as a characteristic length scale of the dimensionless numbers of the problem $L\sim R_{0}$ ). The impellers are driven by two 10 kW AC-motors with rotation rate $F$ adjustable up to $F^{max}=20~\text{Hz}$ . For each run, the motors are operated at constant rotation speed and the motors’ electric drives deliver an analogue output proportional to the current in the motor, as a measurement of the applied mechanical torque ${\it\Gamma}$ .

Figure 1.

Sketch of the experimental Bullard–von Kármán set-up. The flow, created by the rotation of impellers at frequency $F$ , induces a component $B_{z}$ (in the axial direction) in the presence of a field $B_{x}$ (in the transverse direction). The value of the induced field $B_{z}$ measured by a Hall sensor is amplified and drives the power supply which delivers the current in the transverse coils.

The integral kinetic Reynolds number computed using the velocity of the impeller and the radius of the vessel $R_{0}$ is $\mathit{Re}\sim 10^{6}$ ; therefore the flow is fully turbulent. The time-averaged flow consists of two counter-rotating toroidal cells with recirculating poloidal cells created by the centrifugal ejection of the fluid by the rotation of the impellers. In an exact counter-rotating regime, these cells form a shear layer in the mid-plane, and detailed experimental investigations of time-averaged flow topologies, fluctuation levels and characteristics may be found in Ravelet, Chiffaudel & Daviaud (Reference Ravelet, Chiffaudel and Daviaud2008).

The magnetic Reynolds number $\mathit{Rm}$ is typically of order 1, well below the critical value of 44 obtained for a similar flow in the VKS experiment (Monchaux et al. Reference Monchaux, Berhanu, Aumaître, Chiffaudel, Daviaud, Dubrulle, Ravelet, Fauve, Mordant, Pétrélis, Bourgoin, Odier, Pinton, Plihon and Volk2009). A set of induction coils (of total resistance $R=0.28~{\rm\Omega}$ ) creates a magnetic field transverse to the axis of rotation of the von Kármán flow, homogeneous to within 13 % over the flow volume, as shown in figure 1. The current $I$ in the coils is driven by a 10 kW bipolar power supply, with analogue control. The resulting transverse magnetic field $B_{x}={\it\kappa}I$ (where ${\it\kappa}$ is a constant depending on the characteristics of the coils) has a maximal intensity equal to 1200 G (0.12 T) in the flow vessel. The interaction of a transverse magnetic field $B_{x}$ with the turbulent motions of the flow induces an axial magnetic component $B_{z}$ through a mechanism, described in Bourgoin et al. (Reference Bourgoin, Volk, Frick, Khripchenko, Odier and Pinton2004) and Verhille et al. (Reference Verhille, Plihon, Bourgoin, Odier and Pinton2010), involving the shear layer dynamics in the mid-plane (omega-effect) and insulating boundary conditions forcing the current to loop back in the liquid gallium volume. This induction effect will be referred to as the BC-effect. In the limit of vanishingly small interaction parameter $N$ and moderate $\mathit{Rm}$ , the time-averaged axial magnetic field $\langle B_{z}\rangle$ scales linearly with $\mathit{Rm}$ ,

(2.5) $$\begin{eqnarray}\langle B_{z}\rangle =k\,\mathit{Rm}\langle B_{x}\rangle \quad (N\rightarrow 0,~B<100~\text{G}),\end{eqnarray}$$

$k$ being a geometric constant of order $10^{-2}$ and $\langle \cdot \rangle$ standing for time averaging.

When the interaction parameter $N$ is large, a nonlinear correction of functional form

(2.6) $$\begin{eqnarray}\langle B_{z}\rangle =k\,\mathit{Rm}(1-{\it\alpha}\sqrt{N})\langle B_{x}\rangle \quad (B>100~\text{G})\end{eqnarray}$$

has to be taken into account, ${\it\alpha}$ also being a geometric constant (Verhille et al. Reference Verhille, Khalilov, Plihon, Frick and Pinton2012).

Magnetic fields are measured in the mid-plane using a Bell 9150 gaussmeter and Hall-effect probes (sketched as a vertical black tube in the figure 1). The probe measuring the $z$ component is located at a radius $r=0.9R_{0}$ close to the wall except in § 5 where it is located deeper. The $B_{x}$ component is measured on the coils’ axis at $r=0.55R_{0}$ (not shown). Data are recorded using a National Instrument digitiser NI 4472 at sampling frequency of 1 kHz and 24 bits resolution.

2.2. Semi-synthetic Bullard–von Kármán dynamo

The Bullard–von Kármán (BvK) set-up relies on an external feedback inspired by Bullard’s disk dynamo (Bullard Reference Bullard1955).

The dynamo mechanism may be split into two induction processes. The first one is the turbulent induction process described above, inducing an axial component $B_{z}$ from a transverse $B_{x}$ component through a turbulent BC-effect. The second one stems from the external amplification of the Hall-probe signal sampling the $B_{z}$ component of the magnetic field, which is used to control the power supply feeding the current into the transverse coils. This external feedback produces a transverse component $B_{x}$ of the magnetic field proportional to the turbulent-induced component $B_{z}$ :

(2.7) $$\begin{eqnarray}B_{x}=AB_{z},\end{eqnarray}$$

$A$ being proportional to the gain of the amplifier. Thus, the complete mechanism of the present dynamo is referred to as an $A{-}BC$ type dynamo. The onset of the dynamo instability may be understood in the following way: combining the induction equation (2.5) with the external feedback (2.7) the magnetic energy grows from a vanishingly small magnetic field perturbation if $\mathit{Rm}>\mathit{Rm}_{c}=(kA)^{-1}$ . Since the amplification coefficient $A$ can be tuned, the threshold of the instability can be set arbitrarily (typically $\mathit{Rm}_{c}=(kA)^{-1}\sim 1$ ) to values much lower than the expected one for homogeneous dynamos. In the remainder of this article, several values of the amplification $A$ will be presented; however, for the sake of simplicity, only the associated critical $\mathit{Rm}$ value, $\mathit{Rm}_{c}=(kA)^{-1}$ , will be specified. A first set of experiments using this arrangement has been reported in Verhille et al. (Reference Verhille, Plihon, Fanjat, Volk, Bourgoin and Pinton2010). Then, one was restricted to kinematic regimes because the maximum current delivered by the power supply limited the magnetic field amplitude to 60 G, and the interaction parameter $N$ to typically $10^{-4}$ ; Lorentz forces were not strong enough to significantly modify the flow. The magnetic field at saturation was then set by the maximum current drawn from the power supply. However, several features were observed, such as dynamo states connected to ‘on–off’ intermittency, and field reversals. In the present set-up, not only is self-generation reached at $\mathit{Rm}\sim 1$ , but it is achieved in a fully MHD regime, so far restricted to large sodium flows (at low magnetic Prandtl number $\mathit{Pm}={\it\nu}/{\it\lambda}\sim 10^{-6}$ in liquid metals) or to computer simulations (at moderate to high  $\mathit{Pm}$ ).

The dynamo magnetic mode chosen in the present configuration (a transverse dipole relative to the axis of rotation of the impellers) should also be discussed further. Kinematic simulations (i.e. solving the induction equation as an eigenvalue problem from a prescribed velocity field) predict that the most unstable magnetic eigenmode is a transverse dipole when using the time-averaged von Kármán flow (Ravelet et al. Reference Ravelet, Chiffaudel, Daviaud and Léorat2005). These simulations predict an onset of dynamo for $\mathit{Rm}$ of order 50. The configuration of the Bullard–von Kármán dynamo investigated in this article allows regimes to be studied where saturation is provided by Lorentz forces, at $\mathit{Rm}$ of order 1 (thanks to the external amplification) in the presence of a transverse dipole. The present study gives information about the full nonlinear coupled problem in the presence of a dynamo mode similar to the one predicted by kinematic simulations, at leading order. This is in sharp contrast with the experimental dynamo magnetic field observed in the VKS experiment (with soft iron impellers), which is an axial dipole (Boisson et al. Reference Boisson, Aumaître, Bonnefoy, Bourgoin, Daviaud, Dubrulle, Odier, Pinton, Plihon and Verhille2012). The exact mechanism is still under investigation, but recent numerical simulations have highlighted that a small- ${\it\alpha}$ effect between the blades of the impellers, together with a high magnetic permeability of the impeller, could explain the observed magnetic dynamo mode (Giesecke et al. Reference Giesecke, Nore, Stefani, Gerbeth, Léorat, Herreman, Luddens and Guermond2012; Nore et al. Reference Nore, Léorat, Guermond and Giesecke2015).

2.3. Interaction parameter

The focus of the current study is on the dynamo characteristics in regimes where the bifurcated flow differs significantly from the original base flow. This occurs when Lorentz forces are sufficient to compete with other terms in the Navier–Stokes momentum equation (2.1). Some care is needed when expressing the interaction parameter $N$ as a function of the observables of the system. Depending on the system and on the values of the control parameters, several estimates of the Lorentz force can be derived in a system with characteristic length $L$ and characteristic velocity $U$ . We will assume that the velocity and magnetic field characteristic length scales have the same order of magnitude since in our gallium flow the magnetic field diffusion length ( $L_{d}=\sqrt{{\it\lambda}/2{\rm\pi}F}\sim 5.9~\text{cm}$ for $F=10~\text{Hz}$ ) is of the order of the shear layer thickness (note that since $L_{d}$ is comparable to the integral scale and to the characteristic length of the system $R_{0}$ , we choose to estimate the space variations of $\boldsymbol{B}$ using $L=R_{0}$ ).

1. (a) In dynamo regimes (high $\mathit{Rm}$ values and absence of external magnetic fields), the current density is expressed as $\boldsymbol{j}=\boldsymbol{{\rm\nabla}}\times \boldsymbol{B}/{\it\mu}_{0}$ . The interaction parameter is then estimated as

   (2.8) $$\begin{eqnarray}N^{dyn}=\frac{|(\boldsymbol{{\rm\nabla}}\times \boldsymbol{B})\times \boldsymbol{B}|}{|{\it\mu}_{0}{\it\rho}(\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}})\boldsymbol{u}|}\sim \frac{B^{2}}{{\it\rho}{\it\mu}_{0}U^{2}}\end{eqnarray}$$
   where $B$ is the amplitude of the total magnetic field created by the flow motions. This is the typical expression used in astrophysics, for which $\boldsymbol{B}$ depends on $\boldsymbol{u}$ through dynamo scaling laws.

2. (b) In induction regimes (moderate $\mathit{Rm}$ values and presence of an external magnetic field $\boldsymbol{B}^{E}$ ), a magnetic field $\boldsymbol{b}^{I}$ is induced by the interaction of the flow with the external and homogeneous magnetic field $\boldsymbol{B}^{E}$ . The current density in the fluid is expressed as $\boldsymbol{j}=\boldsymbol{{\rm\nabla}}\times \boldsymbol{b}^{I}/{\it\mu}_{0}$ , and the interaction parameter is estimated as

   (2.9) $$\begin{eqnarray}N^{ind}\sim \frac{b^{I}(B^{E}+b^{I})}{{\it\mu}_{0}{\it\rho}U^{2}}.\end{eqnarray}$$
   In the linear regime $b^{I}\sim \mathit{Rm}\,B^{E}$ and usually $b^{I}\ll B^{E}$ in liquid metal flows at $\mathit{Rm}\sim 1$ . This leads to
   (2.10) $$\begin{eqnarray}N^{ind~linear}\sim \frac{B^{2}{\it\sigma}L}{{\it\rho}U}\end{eqnarray}$$
   where $B$ is the amplitude of the external magnetic field (or equivalently the amplitude of the applied magnetic field). This is the typical expression used in induction experiments, for which $\boldsymbol{B}$ and $\boldsymbol{u}$ are independent.

3. (c) In the context of the Bullard–von Kármán dynamo considered here, the induced magnetic field ( $b^{I}=B_{z}$ ) actually drives the external magnetic field ( $B^{E}=B_{x}$ ), through the amplification feedback $B_{x}=AB_{z}$ , so that the interaction parameter can be estimated as

   (2.11) $$\begin{eqnarray}N^{BvK}=A\frac{B_{z}^{2}}{{\it\mu}_{0}{\it\rho}U^{2}}\end{eqnarray}$$
   where $B_{z}$ is the magnetic field created (induced) by the flow motions. One recovers the dynamo expression, in which the amplification factor $A$ shows clearly that the external amplification is used to produce a large impact of the magnetic field on the flow. There is an equivalent expression, valid when the magnetic induction remains linear in $\mathit{Rm}$ ( $\langle B_{z}\rangle \sim \mathit{Rm}\langle B_{x}\rangle$ , i.e. for moderate values of the interaction parameter $N$ ), and thus valid close to dynamo onset $N^{BvK\,onset}\sim (B_{x}^{2}{\it\sigma}L/{\it\rho}U)$ . One recovers the induction expression, but in a dynamo configuration, for which $\boldsymbol{B}$ is not independent of $\boldsymbol{u}$ .

These equivalent formulations of $N$ result from the nature of the Bullard–von Kármán set-up: a dynamo set-up from a semi-synthetic feedback relying on a turbulent induction process from an externally applied magnetic field.

3.1. Bifurcation

In this section, we focus on the onset of the instability and on the description of dynamo regimes. The control parameter of the dynamo bifurcation is the magnetic Reynolds number $\mathit{Rm}=2{\rm\pi}{\it\mu}_{0}{\it\sigma}R_{0}^{2}F$ , and is varied by modification of the rotation rate of the impellers $F$ from 0 to $F^{max}$ . In the remainder of this article, we will study the behaviour of the dynamo magnetic field through the evolution of moments of the $B_{x}$ component of the magnetic field, which for the sake of simplicity will be denoted $B$ . Figure 2(a) shows a typical bifurcation diagram, i.e. the evolution of the magnetic field while increasing the magnetic Reynolds number $\mathit{Rm}$ . Following the kinematic approach presented in § 2.2, the critical value of self-excitation is expected to be $\mathit{Rm}_{c}=(kA)^{-1}=1.94$ for the chosen amplification. For reasons that will soon be clear, the behaviour of the dynamo field is tracked using two indicators: the time-averaged value and the most probable value of the norm of the dynamo magnetic field, respectively labelled $\langle |B|\rangle$ and $B_{mp}$ .

Figure 2.

(a) Bifurcation diagram of the Bullard–von Kármán dynamo as a function of $\mathit{Rm}/\mathit{Rm}_{c}$ . Circles stand for the time average of the dynamo magnetic field $|B|$ and diamonds for the most probable value of $|B|$ . Squares mark the values for which time series are plotted in (b) and (c). The threshold for this run is $\mathit{Rm}_{c}=1.94$ . (b) Time series for $\mathit{Rm}/\mathit{Rm}_{c}=0.9$ , in the intermittent regime. (c) Time series for $\mathit{Rm}/\mathit{Rm}_{c}=1.6$ , in the Lorentz force saturation regime. Dashed lines mark the maximum value of the magnetic field $B^{max}$ which could be drawn if the current reached the maximum value allowed by the external power supply.

Below $\mathit{Rm}/\mathit{Rm}_{c}\sim 0.61$ , the magnetic field is vanishingly small. Slightly above, the dynamics of the magnetic field displays irregular bursts of activity, as shown in figure 2(b) at $\mathit{Rm}/\mathit{Rm}_{c}=0.9$ . Briefs periods of non-zero magnetic field are recorded, with both polarities being accessible due to the presence of additive noise in the loop, as noted in Verhille et al. (Reference Verhille, Plihon, Fanjat, Volk, Bourgoin and Pinton2010). In this regime the time-average value of the magnetic field grows steadily with $\mathit{Rm}/\mathit{Rm}_{c}$ (circles in figure 2 a), but its most probable value remains equal to zero (diamonds in figure 2 a). This regime extends for a range of $\mathit{Rm}/\mathit{Rm}_{c}$ values between 0.61 and 1.5 and is referred to as the ‘Intermittent regime’ in figure 2(a).

A transition is observed above $\mathit{Rm}/\mathit{Rm}_{c}=1.6$ : the most probable value of the magnetic field is no longer zero. It must be noted that this transition is associated with a steeper increase of $\langle |B|\rangle$ with $\mathit{Rm}/\mathit{Rm}_{c}$ , though without any discontinuity at $\mathit{Rm}/\mathit{Rm}_{c}=1.6$ . A typical time series of the magnetic field is shown in figure 2(c) over $700~\text{s}$ at $\mathit{Rm}/\mathit{Rm}_{c}=1.6$ . In either polarity state, the magnetic field saturates to a value which is less than the maximum value of the magnetic field $B^{max}$ (corresponding to the maximum current, $I^{max}$ , fed by the power supply, i.e. $B^{max}(I^{max})=1200~\text{G}$ ). In this regime, and in contrast with the previous studies, the saturation of the dynamo magnetic field occurs from the action of the Lorentz forces and not from the material limitation imposed by the power supply driving the coils. Note also that the fluctuations in the magnetic field are of the same amplitude as its time-averaged value, a feature often observed in turbulent dynamos (Monchaux et al. Reference Monchaux, Berhanu, Aumaître, Chiffaudel, Daviaud, Dubrulle, Ravelet, Fauve, Mordant, Pétrélis, Bourgoin, Odier, Pinton, Plihon and Volk2009). This regime, for which the dynamo magnetic field is saturated via the action of Lorentz forces acting in the bulk of the flow, extends for $1.6<\mathit{Rm}/\mathit{Rm}_{c}<2.16$ . It is denoted as ‘Lorentz force saturation’ in figure 2(a).

As $\mathit{Rm}/\mathit{Rm}_{c}$ is increased above 2.16, instantaneous values of the dynamo magnetic field reach the amplitude of the material limitation $B^{max}$ . At this point the dynamo field is saturated by material limitations rather than by the action of the Lorentz forces: the most probable value shows a concave evolution instead of the linear evolution in the Lorentz force saturation regime and saturates at $B^{max}$ (since excursions of the magnetic field are limited to $\pm B^{max}$ ). This regime is denoted ‘Power supply limitation’ and its range is shown hatched in figure 2(a).

Since the most probable value $B_{mp}$ acts as an indicator of a transition between the intermittent and Lorentz force saturation regimes, further information is obtained from the probability distribution functions (p.d.f.) of $B$ as displayed in figure 3(a) (curves have been shifted for clarity) and $B^{2}$ in figure 3(b). In the intermittent regime ( $\mathit{Rm}/\mathit{Rm}_{c}=0.89$ , solid curve and $\mathit{Rm}/\mathit{Rm}_{c}=1.5$ , dashed curve) the p.d.f. is maximum at $B=0$ , and displays extended lateral wings. In the saturated regime ( $\mathit{Rm}/\mathit{Rm}_{c}=1.6$ , circles and $\mathit{Rm}/\mathit{Rm}_{c}=1.66$ , squares) the p.d.f. is bimodal with lobes centred at $B=\pm B_{mp}$ . The probability of having $B=0$ is still significant; this might be due to the fact that the magnetic field stays around zero for several turnover times during polarity inversions (see § 4). The transition between the intermittent and the saturated regime is statistically continuous as seen in the evolution of the p.d.f. between $\mathit{Rm}/\mathit{Rm}_{c}=1.5$ and 1.66. The value of the p.d.f. at $B=0$ slowly decreases, while the bimodal lateral wing bumps slowly increase. $B_{mp}$ bifurcates from zero to non-zero values at $\mathit{Rm}/\mathit{Rm}_{c}=1.6$ . In all cases, the p.d.f. displays local peaks at the highest amplitudes of $B\sim B^{max}$ , as a result of the power supply limitations.

Figure 3.

(a) Probability distribution function of the dynamo magnetic field $B$ for various $\mathit{Rm}/\mathit{Rm}_{c}$ values (shifted for clarity). (b) Probability density function of the magnetic energy $B^{2}$ .

The following subsections discuss in detail the different regimes introduced here.

3.2. Intermittent regime

Previous studies on the kinematic regime of a BvK dynamo have characterized the observed intermittent regime close to dynamo onset as ‘on–off’ intermittency (Bourgoin et al. Reference Bourgoin, Volk, Plihon, Augier, Odier and Pinton2006; Verhille et al. Reference Verhille, Plihon, Fanjat, Volk, Bourgoin and Pinton2010). On–off intermittency close to an instability onset is expected for fluctuating dynamical systems presenting an unstable invariant manifold in the phase space (Ott & Sommerer Reference Ott and Sommerer1994). The presence of a multiplicative noise (turbulent fluctuations) in a dynamical system creates fluctuations in such a way that the instantaneous growth rate of the magnetic field can be non-zero. Since, in our system, there is always an additive noise contribution (from the electronic noise and from turbulent induction arising from the ambient field), the intermittent signal close to dynamo onset displays both polarities. This intermittent regime is characterized by the statistics of the magnetic energy; typical time series of $B^{2}$ are plotted in figure 4(a,b). Bursts of dynamo activity occur with durations, amplitude and intervals covering a wide range of scales. Their occurrence increases as $\mathit{Rm}/\mathit{Rm}_{c}$ increases. ‘On–off’ intermittency is usually characterized by the statistics of the moments of the intermittent field and of the duration of the laminar phases (Heagy, Platt & Hammel Reference Heagy, Platt and Hammel1994). The p.d.f. of the magnetic energy is displayed in figure 3(b) for two values of $\mathit{Rm}/\mathit{Rm}_{c}$ . In the intermittent regime, it exhibits a power law scaling of exponent $-0.8\pm 0.05$ with an exponential cut-off (dark solid curve for $\mathit{Rm}/\mathit{Rm}_{c}=0.89$ ), a feature characteristic of on–off intermittency. As $\mathit{Rm}/\mathit{Rm}_{c}$ increases, the exponent of the power law increases, as expected from low-dimensional models of on–off intermittency (Aumaître, Pétrélis & Mallick Reference Aumaître, Pétrélis and Mallick2005). In the Lorentz force saturation regime, the most probable value of the magnetic energy is non-zero and the p.d.f. of the magnetic energy does not display a power law scaling anymore (light curve with squares for $\mathit{Rm}/\mathit{Rm}_{c}=1.66$ ).

Figure 4.

(a) Time series of $B^{2}$ in the intermittent regime, for $\mathit{Rm}/\mathit{Rm}_{c}=0.9$ and over 100 s; (b) zoom over a 30 s window. The dashed line represents the threshold considered to compute $T_{off}$ . (c) Probability distribution function of the duration of off phases. Red solid line: fit of the form $T_{off}^{-3/2}\text{e}^{-T_{off}/{\it\tau}_{i}}$ with best fit for ${\it\tau}_{i}=5~\text{s}$ .

‘On–off’ intermittent signals also display a universal behaviour of inter-burst time intervals, which will be denoted as $T_{off}$ in the remainder of this paragraph, standing for ‘off’ phases or laminar phases. These phases are defined as follows: one computes the time $T_{off}$ for which the magnetic energy is below an arbitrary threshold that is small compared to the amplitude of the bursts ( $49~\text{G}^{2}$ for the example given in figure 4(a,b)). The p.d.f. of the duration of ‘off’ phases is shown in figure 4(c). Over almost two decades its functional form is algebraic, with an exponent equal to $-3/2$ and an exponential cut-off. A functional distribution of the form $P(T_{off})\propto T_{off}^{-3/2}\exp -(T_{off}/{\it\tau}_{i})$ is plotted in red in figure 4(c), with a least-square fitting procedure giving ${\it\tau}_{i}=5~\text{s}$ . The $-3/2$ power-law evolution is in agreement with modelling (Heagy et al. Reference Heagy, Platt and Hammel1994) for perfect systems with infinite response time. While similar on–off intermittent behaviour has been observed in numerical simulations of dynamo models close to onset (Peffley et al. Reference Peffley, Cawthorne and Lathrop2000; Sweet et al. Reference Sweet, Ott, Antonsen, Lathrop and Finn2001; Alexakis & Ponty Reference Alexakis and Ponty2008; Raynaud & Dormy Reference Raynaud and Dormy2013), it has never been observed in other dynamo experiments. It should also be emphasized, that the ‘on–off’ intermittent regime is observed over an extended range of control parameter, i.e. from $\mathit{Rm}/\mathit{Rm}_{c}=0.61$ to 1.5. This is in sharp contrast with numerical models, for which on–off intermittent regimes have only been observed close to onset. Another interesting feature of the high value of the magnetic field of this arrangement is that the range of control parameter over which intermittent regimes are observed is narrowed as the interaction parameter at saturation increases. For instance, in the previous kinematic studies (Verhille et al. Reference Verhille, Plihon, Fanjat, Volk, Bourgoin and Pinton2010), it was observed up to $\mathit{Rm}/\mathit{Rm}_{c}=5$ . This corroborates similar observations in numerical models of the ABC dynamo (Alexakis & Ponty Reference Alexakis and Ponty2008).

3.3. Lorentz force saturation regime

An interesting feature of this regime is that the bulk von Kármán flow has been modified by Lorentz forces: Verhille et al. (Reference Verhille, Khalilov, Plihon, Frick and Pinton2012) have shown that the nonlinearity of the induction law was linked to a transition from an hydrodynamic regime (where the magnetic field has no influence on the force balance) to an MHD regime (where the Lorentz force becomes a dominant player). The present configuration of the Bullard–von Kármán dynamo is associated with a transition from an hydrodynamic flow below dynamo onset to an MHD flow above onset.

Figure 5.

(a) Bifurcation diagram for three values of amplification corresponding to $\mathit{Rm}_{c}=1.94$ , 1.11 and 0.65. The saturated regime is marked with filled symbols and fitted using a linear scaling, crossing $B=0$ at $\mathit{Rm}=\mathit{Rm}_{c}$ . (b) Normalized bifurcation diagram as a function of $(\mathit{Rm}-\mathit{Rm}_{c})/\sqrt{\mathit{Rm }_{c}}$ ; the linear fit gives a slope of $6.6\times 10^{-2}~\text{G}$ in agreement with (3.2).

We focus in this subsection on the Lorentz force saturated regime extending from $\mathit{Rm}/\mathit{Rm}_{c}=1.6$ to $\mathit{Rm}/\mathit{Rm}_{c}\sim 2.16$ in figure 2(a). As already described in § 3.1, the most probable value of the magnetic field $B_{mp}$ is non-zero and the p.d.f. of the magnetic field amplitude and of the magnetic energy display a wide distribution around their mode. This regime is investigated by varying both the rotation rate of the impeller $F$ and the amplification factor $A$ of the electronic loop (or equivalently the critical magnetic Reynolds number $\mathit{Rm}_{c}=(kA)^{-1}$ ). Figure 5(a) shows bifurcation diagrams for three critical values, namely $\mathit{Rm}_{c}=1.94$ , 1.11 and 0.65. In each case there is a significant range of $\mathit{Rm}$ values over which the time-averaged magnetic field grows linearly with the distance to threshold:

(3.1) $$\begin{eqnarray}\langle B\rangle \propto (\mathit{Rm}-\mathit{Rm}_{c}),\end{eqnarray}$$

for which the prefactor depends on the amplification factor $A$ . We will now show that this scaling can be recovered from the induction equation (2.6) and the feedback constitutive equation (2.7). Let us first discuss the induction equation (2.6) which was expressed in Verhille et al. (Reference Verhille, Khalilov, Plihon, Frick and Pinton2012) as $\langle B_{z}\rangle =k\,\mathit{Rm}(1-{\it\alpha}\sqrt{N})\langle B_{x}\rangle$ . In the original expression, the authors used $N^{ind}=(B_{x}^{2}{\it\sigma}R_{0})/({\it\rho}U)$ ; in our work, we will use the more accurate expression $N^{BvK\,onset}=(kB_{x}^{2}{\it\sigma}R_{0})/({\it\rho}U)$ , which takes into account the $k$ geometrical factor for a more accurate estimate of the induced current; it is then necessary to replace ${\it\alpha}$ by ${\it\alpha}/\sqrt{k}$ in the induction law. We showed in § 2 that, close to dynamo onset, $N^{BvK\,onset}$ is equivalent to $N^{BvK}=B_{x}^{2}/(A{\it\mu}_{0}{\it\rho}U^{2})$ . The strength of the Lorentz force is equivalently taken into account when using both expressions, but more accurately estimated using the relation $B_{x}=AB_{z}$ ascribed by the amplification feedback loop than assuming any functional form for the induction law. Close to dynamo onset, we combine the induction law (2.6) and the feedback law (2.7), using the expression of $N^{BvK}$ for the interaction parameter, which gives

(3.2) $$\begin{eqnarray}\langle B\rangle =\sqrt{\frac{{\it\rho}}{{\it\mu}_{0}}}\frac{1}{{\it\alpha}{\it\sigma}R_{0}}\left(\frac{\mathit{Rm}-\mathit{Rm}_{c}}{\sqrt{\mathit{Rm }_{c}}}\right).\end{eqnarray}$$

This formula brings out that this scaling is independent of the amplification factor when choosing the control parameter $(\mathit{Rm}-\mathit{Rm}_{c})/\sqrt{\mathit{Rm }_{c}}$ as the relevant control parameter of the saturated regime. The prefactor only depends on the fluid physical characteristics and on geometric parameters determined empirically; this gives a value of $6.2\times 10^{-2}~\text{G}\pm 10\,\%$ with ${\it\alpha}=2.73\pm 0.23$ .

A good agreement between the experimental data and the proposed scaling is observed in figure 5(b), when the dynamo field is plotted as a function of $(\mathit{Rm}-\mathit{Rm}_{c})/\sqrt{\mathit{Rm }_{c}}$ : in the saturated regime, curves collapse on a single master curve. The black line in figure 5(b) is a least-square fit of the data in the saturated regime for three values of the amplification, with slope $6.6\times 10^{-2}~\text{G}$ , within the error given above.

3.4. Power budget

3.4.1. Channels for power dissipation in homogeneous and semi-synthetic dynamos

Precise estimation of the power budget in dynamo regimes is a long-standing issue in dynamo theory. In homogeneous dynamo regimes, dissipation occurs via two channels: viscous dissipation in the fluid flow (denoted as $P_{F}$ in the remainder) and Joule dissipation from electrical currents generating the dynamo magnetic field (denoted as $P_{J}$ ). For natural and experimental dynamos, these two dissipation terms should be estimated from sources of kinetic energy (convection, mechanical stirring, etc.) which are coupled to the magnetic energy.

For instance, for an experimental homogeneous dynamo driven by mechanical stirring, the total dissipated power $P_{T}$ is equal to the power delivered by the stirring system, denoted $P_{M}$ , and the power balance reads

(3.3) $$\begin{eqnarray}P_{T}=P_{M}=P_{F}+P_{J}\quad (\text{homogeneous dynamo}).\end{eqnarray}$$

In these homogeneous systems (which can be seen as isolated systems), the only directly measurable power is $P_{M}$ .

On the other hand, semi-synthetic dynamos, such as the Bullard–von Kármán set-up are more complex: they can be analysed as open or isolated systems depending on which energy sources are considered. They can be seen as open systems when taking into account only the mechanical power $P_{M}$ or as isolated systems when considering both the mechanical power $P_{M}$ and the contribution of the power supply feeding the coils $P_{PS}=RI^{2}$ . A global power budget of the isolated system is then

(3.4) $$\begin{eqnarray}P_{T}=P_{M}+P_{PS}=P_{F}+P_{J}+P_{PS}\quad (\text{semi-synthetic dynamo}).\end{eqnarray}$$

One asset of the Bullard–von Kármán set-up, when considering only the mechanical power (and thus an open system), is that the dynamo may be switched off (on) at will by opening (closing) the feedback loop, giving access to the mechanical power for two distinct flow regimes driven by the same rotation rate $F$ of the impellers (one being an hydrodynamic regime, the other being an MHD regime). In the following subsection, we will first investigate the modification of mechanical power between hydrodynamic regimes (in the open-loop, dynamo-free, configuration) and MHD (or dynamo) regimes, then focus on the global power budget.

3.4.2. Mechanical power dissipation

Let us first focus on the injected mechanical power $P_{M}$ required to sustain a given rotation of the impellers $F=\mathit{Rm}/({\it\mu}{\it\sigma}2{\rm\pi}R_{0}^{2})$ . Due to the two dissipation channels from mechanical power, $P_{M}=P_{F}+P_{J}$ , the natural variables for the power dissipation terms are $\mathit{Rm}$ (accounting for the intensity of turbulence) and $B$ (accounting for the intensity of the Joule dissipation at a given value of $\mathit{Rm}$ ). The mechanical power is computed as $P_{M}=2{\rm\pi}F{\it\Gamma}$ with the mechanical torque ${\it\Gamma}=K_{{\it\Gamma}}I_{mot}$ computed from the measurement of current consumption of the motor drives $I_{mot}$ , with $K_{{\it\Gamma}}$ a motor-specific factor.

Figure 6.

(a) Normalized power difference ${\rm\Delta}P_{M}(\mathit{Rm})/P_{M}(\mathit{Rm},0)$ as a function of $\mathit{Rm}/\mathit{Rm}_{c}$ , see text for details. (b) Evolution of the time-averaged mechanical power ( $\langle P_{M}(\mathit{Rm},B)\rangle$ , circles), electric power injected through the coils ( $R\langle I\rangle ^{2}$ , squares) and the total power ( $\langle P_{T}\rangle$ , stars) as a function of $\mathit{Rm}/\mathit{Rm}_{c}$ for $\mathit{Rm}_{c}=1.94$ . The lines corresponds to the scalings given by (3.7) (solid) and (3.8) (dashed) and a combination of the two (dash-dotted).

In the absence of external feedback, which will be referred to as the ‘open-loop’ configuration, the only channel is viscous dissipation (hydrodynamic flow), and $P_{M}(\mathit{Rm},B=0)=P_{F}(\mathit{Rm},0)$ ; this is also valid below dynamo onset. In dynamo regimes (MHD flow), $P_{M}(\mathit{Rm},B)=P_{F}(\mathit{Rm},B)+P_{J}(\mathit{Rm},B)$ , where $B$ depends on $\mathit{Rm}$ through the dynamo scaling law (note that $P_{F}(\mathit{Rm},B)$ is the power required to sustain the flow modified by the Lorentz forces, not taking into account the Joule dissipation, and that $\mathit{Rm}$ is based on the tip velocity of the impellers and not the actual flow velocity in the MHD regime). As previously pointed out, we have access to the difference between mechanical power in the BvK dynamo configuration and in the ‘open-loop’ configuration, ${\rm\Delta}P_{M}(\mathit{Rm})=P_{M}(\mathit{Rm},B)-P_{M}(\mathit{Rm},0)$ , for a given value of $\mathit{Rm}$ . The normalized mechanical power difference ${\rm\Delta}P_{M}(\mathit{Rm})/P_{M}(\mathit{Rm},0)$ is displayed in figure 6(a). A clear increase of the mechanical power consumption of a few per cent is observed in regimes where saturation of the BvK dynamo occurs through Lorentz force modification of the bulk flow. This increase can be understood following the previous study in Verhille et al. (Reference Verhille, Khalilov, Plihon, Frick and Pinton2012), which showed, in induction regimes, that the mechanical power in the MHD case, $P_{M}(\mathit{Rm},B)$ , is related to the mechanical power in the hydrodynamic regime, $P_{M}(\mathit{Rm},0)$ , as

(3.5) $$\begin{eqnarray}P_{M}(\mathit{Rm},B)=(1+{\it\gamma}N^{ind})P_{M}(\mathit{Rm},0)\end{eqnarray}$$

with ${\it\gamma}=140$ an empirical constant. We previously showed that close to dynamo onset, the scaling laws determined from induction measurements remain valid using $N^{BvK}$ . Accordingly, ${\rm\Delta}P_{M}(\mathit{Rm})/P_{M}(\mathit{Rm},0)={\it\gamma}N$ , and $N$ can be recast using the dynamo scaling law (3.2). The expected evolution

(3.6) $$\begin{eqnarray}{\rm\Delta}P_{M}(\mathit{Rm})/P_{M}(\mathit{Rm},0)=\frac{{\it\gamma}k}{{\it\alpha}^{2}}\frac{(\mathit{Rm}-\mathit{Rm}_{c})^{2}}{\mathit{Rm}^{2}}\end{eqnarray}$$

is displayed in figure 6(a) as a dashed line and shows good agreement with experimental data. The open circles symbols in figure 6(b) display the evolution of the time-average injected mechanical power as a function of $\mathit{Rm}$ , for $\mathit{Rm}_{c}=1.94$ . These points can be compared to the scaling of (3.5), using both the dynamo scaling law (3.2) and the turbulent scaling of the hydrodynamic regime $P_{M}(\mathit{Rm},0)=2K_{p}{\it\rho}/({\it\mu}^{3}{\it\sigma}^{3}R_{0})\mathit{Rm}^{3}$ , following the notation of Ravelet et al. (Reference Ravelet, Chiffaudel, Daviaud and Léorat2005), which gives

(3.7) $$\begin{eqnarray}P_{M}(\mathit{Rm},B)=\frac{2K_{p}{\it\rho}}{{\it\mu}^{3}{\it\sigma}^{3}R_{0}}\left(1+\frac{{\it\gamma}k}{{\it\alpha}^{2}}\frac{(\mathit{Rm}-\mathit{Rm}_{c})^{2}}{\mathit{Rm}^{2}}\right)\mathit{Rm}^{3},\end{eqnarray}$$

where the power is expressed in kW, and ${\it\gamma}=140$ and $K_{p}=5.03\times 10^{-2}$ (a value similar to the value $6.1\times 10^{-2}$ reported by Ravelet et al. Reference Ravelet, Chiffaudel, Daviaud and Léorat2005 in a similar water flow). The good agreement between experimental data and (3.7) shows once again the consistency of our results and that the features of the BvK dynamo can be fully understood from the effect of Lorentz forces in the MHD regime. ${\rm\Delta}P_{M}(\mathit{Rm})$ is linked to modifications of viscous dissipation between the hydrodynamic regime and the MHD regime ( ${\rm\Delta}P_{F}$ ) and to the occurrence of Joule dissipation $P_{J}$ . While Joule dissipation obviously increases the mechanical power, the sign of ${\rm\Delta}P_{F}=P_{F}(\mathit{Rm},0)-P_{F}(\mathit{Rm},B)$ cannot be easily estimated. The modifications of viscous dissipation could thus lead to both an increase or a decrease of the mechanical power (depending on the details of the flow modifications at saturation). A precise estimate of the ratio of the two dissipation channels is thus outside the scope of the present study and would require precise measurements of the velocity field and current density. We emphasize here that, investigating only the modification of the injected mechanical power, one considers the Bullard–von Kármán dynamo as an open system. Extrapolations to homogeneous dynamos, seen as isolated systems, should thus be done with care.

3.4.3. Global power budget

Let us now consider the Bullard–von Kármán dynamo as an isolated system and investigate the total injected power $P_{T}=P_{M}+P_{PS}$ . While for an homogeneous dynamo $P_{J}$ accounts for the power required to sustain the dynamo magnetic field, in semi-synthetic dynamos the power required to sustain the dynamo magnetic field is provided by an external power supply and $P_{PS}=RI^{2}$ (where $R$ is the resistance of the coils). Note that the feedback to the induced magnetic field by the flow in the coils is neglected since $\mathit{Rm}^{eff}=B_{z}/B_{x}$ remains small and the power supply is current-controlled. In these cases, $P_{PS}$ is not universal and is system-dependent through its dependence on $R$ (and thus the way the system is being built). However, since the electrical current in the coils is proportional to the dynamo magnetic field $B$ , $P_{PS}$ is proportional to the Joule dissipated power. A precise estimate of $P_{J}$ would require the measurement of the current at each location in space, which is outside the scope of the present work. However, an estimation of the Joule power dissipation $P_{J}^{e}$ can be made from the measurement of the magnetic field at one location using $|\boldsymbol{j}|=|\boldsymbol{{\rm\nabla}}\times \boldsymbol{B}_{\boldsymbol{z}}/{\it\mu}_{0}|\sim B_{x}/({\it\mu}_{0}AR_{0})$ . Consequently, $P_{J}^{e}\sim (B^{2}R_{0}/{\it\mu}_{0}^{2}{\it\sigma}A^{2})$ (recall that $B=B_{x}$ ).

It thus follows that $P_{PS}={\it\chi}A^{2}P_{J}^{e}$ , with ${\it\chi}$ a non-universal constant (depending on geometrical factors and the resistivity of the external coils). The $A^{2}$ is reminiscent of the external amplification loop: for a given value of induced magnetic field in the fluid volume, the dynamo magnetic field is $A$ -fold larger; the power needed to sustain the dynamo is thus a factor $A^{2}$ larger than the estimated Joule dissipated power in the flow volume. The experimental evolution of the power delivered by the external power supply $P_{PS}$ as a function of $\mathit{Rm}$ , for $\mathit{Rm}_{c}=1.94$ , is displayed as square symbols in figure 6(b). The experimental points are correctly described in the regime of Lorentz force saturation using the dynamo scaling law (3.2) for the estimate of $P_{PS}$ :

(3.8) $$\begin{eqnarray}P_{PS}=C_{I}\frac{(\mathit{Rm}-\mathit{Rm}_{c})^{2}}{\mathit{Rm}_{c}},\end{eqnarray}$$

with formula: $C_{I}=(R{\it\kappa}^{2}{\it\rho})/({\it\mu}_{0}{\it\sigma}^{2}{\it\alpha}^{2}R_{0})=1.52$ (with $R=0.28~{\rm\Omega}$ ) and $P_{PS}$ expressed in kW, which is displayed as a dashed line in figure 6(b). It should be noted that a departure from the expected scaling is observed for $\mathit{Rm}/\mathit{Rm}_{c}$ values above 2.16, due to power limit saturation, as already discussed in § 3.1. Finally, the experimental data for the total injected power $P_{T}=P_{M}+P_{PS}$ are also displayed in figure 6(b) as stars. These points are described to a very good accuracy by the scaling law given by the sum of (3.7) and (3.8) and displayed as a dash-dotted line in figure 6(b). Finally, the $A^{2}$ factor in $P_{PS}$ can be understood as a way to control the ratio $P_{M}/P_{PS}$ in the Lorentz force saturation regime. For $\mathit{Rm}_{c}=(kA)^{-1}=1.94$ , the electrical power required to sustain the dynamo is roughly equal to the injected mechanical power in the saturated regime. As the critical magnetic Reynolds number is reduced, $P_{PS}$ becomes 2.3 times larger than $P_{M}$ for $\mathit{Rm}_{c}=1.11$ and 9 times larger for $\mathit{Rm}_{c}=0.65$ . We emphasize that these numbers should be taken with care since they depend on the value of ${\it\chi}$ and thus on the design of the system (using superconductive coils would for instance strongly decrease $P_{PS}$ for the same value of the dynamo magnetic field).