2.1. General description

The KEPLER experiment (see figure 1) consists of an annular cylindrical channel with an internal diameter $2R_i=12$ cm and an external cylinder diameter $2R_o=38$ cm (mean radius $r_m=12.5$ cm, gap $\Delta r=13$ cm). It is filled with Galinstan, a eutectic alloy of gallium, indium and tin, that is liquid at room temperature and whose physical characteristics are summarized in table 1. A strong electrical current (up to $3000$ A) is injected radially by a direct current generator Power TEN P66 Series 53000 from the inner cylinder to the outer rim. The temperature of the inner cylinder is controlled by a water cooling system. Both electrodes are made of brass and protected from the Galinstan by an electrochemical deposit of nickel, in order to ensure a good electric contact with the liquid metal. The height of the cell is $h=1.5$ cm, corresponding to a relatively large aspect ratio $\varGamma =\Delta r/h= 8.7$. The top and bottom walls are electrically insulating Plexiglas plates. The cylinder is placed between two large Helmholtz coils generating a homogeneous vertical magnetic field along the vertical $z$-axis. The coils are powered by two direct current generators, Ametek RS 20V-250A, producing magnetic fields of up to $110$ mT. The combination of the radial current and the vertical magnetic field generates a strong Lorentz force in the azimuthal direction.

Figure 1. Sketch of the experimental set-up. A thin disc of diameter $D=38$ cm and thickness $h=1.5$ cm is filled with a liquid metal (Galinstan) and placed between two Helmholtz coils generating a vertical homogeneous magnetic field. The cell is confined by top and bottom walls (1) made of electrically insulating Plexiglas and by electrically conducting cylindrical walls acting as an anode (4) and cathode (6). The cross-section of the duct is rectangular (2), and characterized by a large aspect ratio $\Delta r/h\simeq 9$. The flow is entirely driven by the Lorentz force generated by the combined action of the vertical magnetic field ($B_0$) and the strong radial current (I) passing from the anode to the cathode. Velocity measurements are made by potential probes inserted in the top wall up to the middle of the gap (5) thanks to Doppler probes inserted in a rail inclined at an angle of $45^{\circ }$ analogous to (7) not represented here for clarity. Induced magnetic field measurements were done thanks to Hall probes inserted in a rail (7) and the pressure sensor was inserted in the top wall (8).

Table 1. Physical properties of the liquid metal Galinstan used in the experiment.

2.2. Measurement methods

The velocity field is recorded using two different measurement techniques. First, velocity profiles are measured with ultrasonic Doppler velocimetry, by using the commercial system DOP3010 from Signal Processing SA, in which probes send ultrasonic pulses at regular intervals. The phase shift between the sent pulse and its echo on particles in the bulk (mostly oxides in liquid metals) is computed in order to obtain the velocity profile along the direction pointed by the Doppler probe. The Doppler probes are installed into the Plexiglas plates at an angle of $45^{\circ }$ with the flow. Plexiglas and Galinstan have a close acoustic impedance, ensuring a good transmission of the ultrasonic pulses. Note that due to the angle, the measured velocity is a linear combination of the local azimuthal component of the velocity $u_{\theta }$ and the vertical component $u_z$.

In addition, both azimuthal and radial components of the velocity are also measured using potential probes. For a conductive fluid in motion, Ohm's law specifies that the current density $\boldsymbol {j} = \boldsymbol {\nabla }\times \boldsymbol {B}/\mu _0$ is given by the relation $\boldsymbol {j} = -\sigma \boldsymbol {\nabla }\phi + \sigma (\boldsymbol {u}\times \boldsymbol {B})$, where $\mu_0$ is the vacuum magnetic permeability, $\sigma$ is the electrical conductivity and $\phi$ is the electric potential. After having removed the injected current, the local velocity is deduced from the potential difference $(\delta\phi)$ between two probes separated by a small distance $d$ using the relation $U_{\theta } = \delta \phi /(B_0 d)$ where $B_0$ is the magnitude of the vertical magnetic field. Nine small electrodes ($1$ mm diameter, separated from each other by $d=8$ mm) are inserted equidistantly into the top plate through holes located between inner and outer cylinders. These probes are nickel wires electrically insulated everywhere but at the tip. The mean component of the velocity is obtained through measurement of the voltage by a nanovoltmeter, Keithley Model 182, and filtered via a filter amplifier, Stanford Research Systems Model SR560 in DC mode. The turbulent fluctuations are obtained from amplification of the signal through a filter amplifier, Princeton Applied Research Low Noise Amplifier Model 1900, using smaller potential probes ($d=4$ mm).

Pressure fluctuations are measured by a pressure sensor Kistler Type701A (sensitivity of 82.24 pC bar^-1, acquisition frequency $10$ kHz), placed in contact with the liquid metal through a hole in the top Plexiglas plate and coupled to a charge amplifier Kistler Type5018. Finally, induced magnetic field were measured by a Teslameter F.W. Bell Model7030 connected to Hall probes (precision up to $\sim$10 $\mathrm {\mu }$T) inserted in vertical holes in the Plexiglas plate around two millimetres away from the liquid metal.

3.1. Parameter space and outline of the results

Figure 2 summarizes the various results obtained with the KEPLER experiment. The colour plot shows interpolated values of $\log (u_\theta )$ in the ($I_0,B_0$) parameter space. Four different domains have been identified. These regimes will be described in more details in the next sections, but it is insightful to give a short overview of the parameter space now: at large magnetic field ($N_t>10$, indicated by the red dashed line) and small Reynolds number ($Re<9000$, indicated by the black solid line), the viscous–ideal regime corresponds to a balance between the Lorentz force and the viscous dissipation, and is characterized by a strong value of the external magnetic field such that the induction term $\boldsymbol {u}\times \boldsymbol {B}_0$ dominates the magnetic diffusion term $\eta \Delta \boldsymbol {b}$ where $\boldsymbol {b}$ is the induced field. The flow is laminar, and the velocity field scales as $u_\theta \propto I_0/r$, independently of $B$.

Figure 2. Map of the logarithm of the mean azimuthal flow, $\log (\overline {u_\theta })$ as a function of $(I_0,B_0)$, measured with the potential probes. The red dashed line ($N_c=10$) and the black solid line ($Re_B=300$) delimitate the transition between the different regimes generated in the experiment. The points indicate the four typical runs discussed in § 6.

At smaller magnetic field but moderate Reynolds number, the inertial–resistive regime is obtained. The Lorentz force is now balanced by inertia, but the boundary layers remain laminar. In addition, a significant amount of current passes through the bulk flow. The corresponding velocity field scales as $u_\theta \propto (IB)^{2/3}r^{-1/3}f(\Delta r/r)$ ($f$ being an unknown function) and remains in a quasi-Keplerian state. Note that the first regime has been identified in previous MHD experiments and well described theoretically (Baylis Reference Baylis1971; Messadek & Moreau Reference Messadek and Moreau2002). Although not discussed in detail by previous experimental studies, analysis of published data suggests that the second regime has also been observed in previous experiments.

However, when the Reynolds number is increased, the flow then bifurcates to a quasi-bidimensional turbulent flow regime, in which both the bulk and the boundary layers are turbulent. In this so-called ultimate regime, we observe a new scaling $u_\theta \propto \sqrt {IB/r}$, corresponding to a turbulent Keplerian velocity profile sharing some similarities with the velocity profile expected in accretion discs. This turbulent regime is observed only if the magnetic field is not too large, for $N_t<10$, and therefore requires an extremely large applied current.

Finally, at large magnetic field ($N_t>10$) and large Reynolds number ($Re>9000$), we observe a bi-dimensional turbulent regime which exhibits a slow dynamics and condensation of the energy at large scale.

4.1. Viscous–ideal flows

At large magnetic field ($Ha\ge 30$), figure 3(a) shows that the velocity scales linearly with the applied current at low $I_0$. This regime has been first described theoretically by Baylis & Hunt (Reference Baylis and Hunt1971) for a laminar axisymmetric flow in a toroidal channel of square cross-section, and later confirmed by most experimental studies mentioned in the introduction.

The theoretical prediction for this laminar flow can be retrieved by simple arguments. When the Elsasser number $\varLambda$ is sufficiently large, the Hartmann layers are much thinner than Bödewadt layers, meaning that most of the velocity gradient occurs in layers of size $\delta _{Ha}$. In addition, if $\varUpsilon$ is small, most of the applied current $I_0$ passes through the two Hartmann layers with a current density $\boldsymbol {j}\simeq ({I_0}/{(2{\rm \pi} r (2\delta _{Ha}))})\boldsymbol {e}_{\boldsymbol {r}}$ in each layer. The azimuthal Lorentz force $F_\theta =j_rB_0$ is then balanced in the Hartmann layer by the viscous dissipation $\rho \nu \Delta \boldsymbol {u}\sim \rho \nu u_\theta /\delta _{\textit {Ha}}^{2}\boldsymbol {e}_{\boldsymbol {\theta }}$, leading to

(4.1)\begin{equation} U_\theta = \frac{I_0}{4{\rm \pi} r\sqrt{\rho\nu\sigma}}. \end{equation}

In this expression, $B_0$ does not appear explicitly, although it controls the bi-dimensionalization of the flow. Note that this expression stands only if the Shercliff boundary layers generated at the surface of the cylinders parallel to the magnetic field do not interfere with the bulk flow. This means $r_2-r_1\gg {h}/{\sqrt {\textit {Ha}}}$, which is well satisfied in the KEPLER experiment. Note that the ideal term used here refers to the fact that magnetic diffusion term $\eta \Delta \boldsymbol {b}$ can be neglected compared with the induction $uB_0$. It is, however, important to mention that this comes from the fact that the field $B_0$ is strong, and is quite different from the classical ideal MHD generally obtained when $Rm\gg 1$.

The velocity field obtained at large $B_0$ and low current agrees very well with this prediction, indicated by the dashed line in figure 3.

4.2. Inertial–resistive flows

Figure 3(b) shows that, in fact, two different scaling laws can be obtained at low forcing: for $Ha\ge 30$, the linear scaling $u_\theta \propto I_0$ of the viscous–ideal regime described above, and a scaling of the form $u_\theta \propto (I_0B_0)^{2/3}$ at smaller $Ha$. A similar regime seems to have been observed in at least one previous experiments (Baylis Reference Baylis1971) and one recent numerical study (Poye et al. Reference Poye, Agullo, Plihon, Bos, Desangles and Bousselin2020). As shown by figure 3, this regime is observed for $I_0B_0<10$ and $B_0<40$ mT. We give below simple arguments to explain this regime, following closely Poye et al. (Reference Poye, Agullo, Plihon, Bos, Desangles and Bousselin2020).

In the bulk flow, inertia terms in the azimuthal direction are balanced by the applied Lorentz force ${I_0B_0}/{(2{\rm \pi} rh)}$. In contrast to the previous case, experimental values of $\varUpsilon$ suggest that the fraction of current passing through the bulk should be relatively large. It follows that

(4.2)\begin{equation} u_\theta \simeq \frac{I_0B_0}{2{\rm \pi}\rho h u_r}f\left(\frac{\Delta r}{r}\right), \end{equation}

where the inertial term $\rho u_ru_\theta /r$ is larger than $\rho u_r\partial _ru_\theta$ because $r$ is smaller than the gap $\Delta r$ in the inner part of the disc. At large radii, $r>\Delta r$ and the second term might be dominant, which is taken into account by the function $f$. Because $r\sim \Delta r$ in any cases, the function $f$ is expected to be of order one and the choice of the inertial term will not change the scaling law.

For moderate magnetic fields, when $\varLambda \leq 1$, the radial inflow near the boundaries is mostly dominated by the Bödewadt layers. The corresponding imbalance between the centrifugal force in the layer and the pressure gradient $\partial _rp\sim \rho u_\theta ^{2}$ leads to

(4.3)\begin{equation} u_r^{BL}\sim \frac{\delta_B^{2}u_\theta^{2}}{\nu r}, \end{equation}

where $u_r^{BL}$ is the typical radial velocity in the Bödewadt layer of thickness $\delta _B$. Because $u_z$ is presumably small in the bulk flow, incompressibility implies $\int _0^{h}u_r\,\textrm {d}r\approx 0$ at lowest order and so $u_rh\sim 2\delta _Bu_r^{BL}$. By combining this last result, with expressions (4.2) and (4.3), one finally obtains

(4.4)\begin{equation} u_\theta = \left(\frac{I_0B_0}{4{\rm \pi}\rho\sqrt{\nu}} \right)^{{2}/{3}}r^{-{1}/{3}}f\left(\frac{\Delta r}{r}\right). \end{equation}

Note that the unknown function $f$ may influence the radial dependence, but not the scaling law for the magnitude of the flow because $f\sim 1$. Although the scaling law (4.4) shares some similarities with the one found by Poye et al. (Reference Poye, Agullo, Plihon, Bos, Desangles and Bousselin2020), our full expression is quite different, and leads to values one order of magnitude smaller. This is due to some differences between our experiment and the set-up studied by these authors, in particular the relatively large aspect ratio used here ($\Delta r/h\sim 9$) and the fact that $r_m\sim \Delta r$ in the KEPLER experiment. When plotting the results of Baylis (Reference Baylis1971) in this form, it appears that their data follow a similar dependence $u_\theta \propto (I_0B_0)^{2/3}$. A dependence $u_\theta \propto (I_0)^{2/3}$ based on a different argument was also observed by Potherat & Klein (Reference Potherat and Klein2014), although their set-up is quite different. At the exception of Messadek & Moreau (Reference Messadek and Moreau2002) where a very turbulent flow is reached, we argue that this regime is probably the one mostly observed in previous experiments operating at large Reynolds number.

Prediction (4.4) (with $f=1$) is shown by the dash-dotted line in figure 3. It fits our experimental data with a very good agreement.

4.3. Keplerian turbulence

Finally, when the forcing due to the Lorentz force becomes sufficiently strong, the flow undergoes a new transition from the previous laminar flows toward a presumably turbulent state. Figure 4(a) shows that for $I_0B_0>10$, all the data collapse on a single curve. The best power law to fit the data is $u_\theta \propto (I_0B_0)^{0.54\pm 0.01}$.

Figure 4. (a) Mean azimuthal flow $u_{\theta }$ vs the forcing $I_0B_0$ in the fully turbulent regime. All our data collapse onto the theoretical prediction given by (4.8) (solid line). (b) Turbulent velocity $u^{*}$ estimated from the standard deviation of $u_\theta$ for $B=20$ mT, and compared with (4.5). Here, $Ha\sim 0.57B_0$ with $B_0$ in $\textrm {mT}$.

This result can be understood by assuming that the bulk flow is fully turbulent, such that the Reynolds stress gradient $\rho \partial _i\overline {u_iu_j}$ overcomes the inertial terms involving the mean velocity $\rho (\overline {u_i}\partial _i){\bar u_j}$. In addition, because of the large aspect ratio of the experiment $h\ll r_m,\Delta r$, the Reynolds stress is mostly dominated by the components $\tau _{iz}=\rho \partial _z\overline {u_iu_z}$. The azimuthal component of the Navier–Stokes equation therefore reduces to a balance between the Reynolds stress $\rho \partial _z\overline {u_\theta u_z}$ and the applied Lorentz force

(4.5)\begin{equation} u^{*}=\sqrt{\frac{I_0B_0}{4{\rm \pi} \rho r}}, \end{equation}

where the additional factor $2$ comes from the estimate $\partial _z\overline {u_\theta u_z}\sim 2(u^{*})^{2}/h$, with $u^{*}$ the typical velocity of the turbulent structures. This prediction for the intensity of the turbulent fluctuations matches very well our experimental data, as shown by figure 4(b).

In the KEPLER experiment, the relevant boundary layers at the vertical wall are either the Hartmann layer at high $\varLambda$ or the Bödewadt layer at low $\varLambda$. In the former, the stability of the layer is controlled by the Reynolds number $Re_H$ based on $\delta _H$, while in the latter, the relevant critical number is the Reynolds number $Re_B$ based on the thickness of the Bödewadt layer. Moresco & Alboussière (Reference Moresco and Alboussière2004b) have predicted that pure Hartmann layers should become turbulent for $Re_H=Ha/Re>400$. Although it is well known that this criterion strongly depends on the Elsasser number when rotation effects are present (Davidson & Potherat Reference Davidson and Potherat2002; Moresco & Alboussière Reference Moresco and Alboussière2004b), we expect a similar value here, since $\varLambda$ is never much larger than one in our experiment (except in the viscous–ideal regime where $\varLambda \sim 10$). In the regime studied here, as long as the forcing $I_0B_0$ remains larger than $10$, $Re_H$ is between $400$ and $2500$ and $R_B$ is larger than $200$. It is therefore reasonable to expect the Hartmann–Bödewadt boundary layers to become turbulent at large forcing. Let assume a classical logarithmic profile for the turbulent boundary layer

(4.6)\begin{equation} \frac{\overline{u_\theta}(z)}{u^{*}}=\frac{1}{\kappa}\ln\left(\frac{u^{*}z}{\nu}\right) +C, \end{equation}

where $\kappa$ is the Kármán constant and $C\approx 5$. Figure 4 shows that the turbulence intensity is relatively small, $u^{*}/\overline {u_\theta }\sim 3.10^{-2}\ll 1$, so (4.6) evaluated in the bulk ($z=h/2$) can be approximated by

(4.7)\begin{equation} \frac{\overline{u_\theta}}{u^{*}}=\frac{\ln (Re_{h})}{\kappa}, \end{equation}

where the Reynolds number is supposed large enough to overcome contributions from neglected terms. By combining relations (4.5) and (4.7), we finally obtain the following prediction for the turbulent regime:

(4.8)\begin{equation} \overline{u_\theta}=\alpha\frac{\ln Re_h}{\kappa}\sqrt{\frac{I_0B_0}{4{\rm \pi}\rho r}}. \end{equation}

Note that because of the assumptions made during the derivation of this law, agreement between this prediction and experiments is only expected up to some pre-factor $\alpha$ of order one. In particular, MHD effects may very well modify the properties of the turbulent layer through the value of the parameters $\kappa$ and $C$.

Despite these approximations, solution (4.8) with $\alpha =1.5$, represented by the solid blue line in figure 4, matches remarkably well the experimental data. In particular, figure 4 shows that the logarithmic correction is necessary to fit the data, as it clearly increases the exponent of the power law.

Several remarks can be made on this prediction. First, except for the logarithmic correction, it is independent of $h$. This is reminiscent of the turbulent nature of the boundary layers. This expression is only valid in the limit of a thin-disc geometry, which leads to take $h$ rather than $r$ in the nonlinear term. When $h$ is increased (experiments not reported here), we observed that this Keplerian profile (the radial scaling in $r^{-1/2}$) disappears. However, the square-root dependency in $I_0B_0$ stands because it comes from a natural balance between the advection and the Lorentz force (Boisson et al. Reference Boisson, Monchaux and Aumaitre2017). Note also that because this expression requires a destabilization of the boundary layers, it relies on a 3-D cascade of energy. This forward transfer of energy towards small scales is, however, only required for scales smaller than the typical thickness $h$, and no constrains exists on the larger scales. This explains why the scaling law (4.8) can be associated with quasi-bidimensional turbulence. On the other hand, this regime is only observed for interaction parameter not too large, namely $N_t<10$. It means that although quasi-bidimensional, it cannot be described by purely 2-D Navier–Stokes equations.

Because the angular velocity profile is $\varOmega \propto r^{-3/2}$, we refer to this solution as Keplerian turbulence. Although generated through a different mechanism, such profiles are expected in many astrophysical objects in which the centrifugal force balances gravity forces, as discussed in more details in § 5.

4.4. Transitions between the different regimes

Figure 5 is very insightful to understand what controls the transition between the different regimes discussed above. To obtain this figure, we calculate the characteristic exponent of the scaling law

(4.9)\begin{equation} \xi=\frac{\partial \ln (\overline{u_\theta})}{\partial \ln I_0} \end{equation}

and plot it in the parameter space $(I_0,B_0)$. The red area corresponds to the viscous–ideal regime ($\xi \sim 1$), the light blue to the inertial–resistive case ($\xi \sim 2/3$) and the dark blue to the Keplerian turbulence ($\xi \sim 1/2$). The red dash-dotted line corresponds to $N_t=10$ and delimitates the region between strong and moderate magnetic field. Indeed, for large interaction parameter, most of the current passes through the thin Hartmann layers where the corresponding Lorentz force is balanced by viscosity (rather than inertia). As $N_t$ is reduced below $10$, the Bödewadt boundary layer generates a strong radial inflow near the wall. The interaction between this wind and the inertia in the bulk leads to the $2/3$ scaling law, as explained in the previous section. Note that in this regime, the Reynolds number $Re$ is sufficiently small so the flow remains laminar. Alternatively, this transition can be predicted by balancing the two expressions for the mean azimuthal velocity given by (4.1) and (4.4), leading to the (velocity-free) dimensionless ratio $4{\rm \pi} r^{2}B_0^{2}\sqrt {\nu \sigma ^{3}}/I_0\sqrt {\rho }$. The transition between the viscous–ideal and the inertial–resistive regime then occurs when this ratio is equal to one. Note that this ratio is an Elsasser number $\varLambda$ in which the velocity is evaluated thanks to (4.1).

Figure 5. Colour map of the exponent $\xi$ of the mean azimuthal flow $u_{\theta }\propto I_0^{\xi }$ in the parameter space ($I_0,B_0$). The three regimes $\xi \sim 1$, $\xi \sim 2/3$ and $\xi \sim 1/2$ can be clearly distinguished (see the text).

The transition from the $2/3$ regime to the turbulent one is less trivial. At small magnetic field, the Bödewadt layer is much smaller than the Hartmann one, and its stability is fully controlled by a critical Reynolds number $Re_B=u\delta _B/\nu$ based on the Bödewadt layer thickness (black solid line). In contrast, at large magnetic field, the stability of the thinner Hartmann layer is measured by the Reynolds number $Re_H$ based on the thickness of this layer. We observe that the flow consequently bifurcates to the Keplerian turbulent regime along the line $Re_B\sim 400$ (solid black line) at low field, while the transition rather takes place at a critical $Re_H\sim 300$ (dashed black line) at large field (for $B_0>50$ mT). Figure 5 therefore strongly suggests that the transition to the Keplerian turbulence is therefore ultimately triggered by the instability of Hartmann–Bödewadt layers.

Figures 6(a) and 6(b) show that the transition to turbulence in the experiment is associated with a strong (yet continuous) increase of the fluctuations of both induced magnetic field and pressure, without hysteresis. These results suggest that the Keplerian turbulence always occurs through a smooth supercritical transition.

Figure 6. (a) Fluctuations (standard deviation) of the three components of the magnetic field as a function of the injected current $I$ for an imposed magnetic field $B_0=800G$ (80 mT). (b) Fluctuations (standard deviation) of the pressure as a function of the injected current $I$ for an imposed magnetic field $B_0=700G$ (70 mT).

4.5. Energy budget

Because conducting fluids can either dissipate energy viscously or by ohmic dissipation, the question of the energy budget of such electromagnetically driven flows is highly non-trivial. In electromagnetically driven flows, the energy equation is (Reddy, Fauve & Gissinger Reference Reddy, Fauve and Gissinger2018)

(4.10)$$\begin{gather} \partial_tE_u ={-}D_{\nu} + T, \end{gather}$$

(4.11)$$\begin{gather}\partial_tE_b ={-}D_{\eta} - T + F, \end{gather}$$

where $E_u = \rho \int _V\textrm {d}^{3}x({\boldsymbol {u}^{2}}/{2})$, (respectively $E_b = \int _V\textrm {d}^{3}x({\boldsymbol {B}^{2}}/{2\mu _0})$) is the total kinetic (respectively magnetic) energy, $D_\nu =\rho \nu \int _V\textrm {d}^{3}x(\boldsymbol {\nabla }\times \boldsymbol {u})^{2}$ (respectively $D_\eta =({1}/{\sigma }) \int _V\textrm {d}^{3}x\,\boldsymbol {j}^{2}$) is the viscous (respectively ohmic) dissipation; $T=({1}/{\mu _0})\int _V\textrm {d}^{3}x\boldsymbol {B}\otimes \boldsymbol {B}:\boldsymbol {\nabla }\otimes \boldsymbol {u}$ is the term which couples the two equation and corresponds to the transfer of energy between the two fields and $F=\eta \oint _{\partial V}\textrm {d}\boldsymbol {S}\cdot (\,\boldsymbol {j}\times \boldsymbol {B})$ is the injected power.

In addition, one can also define the efficiency $\gamma$ of the transformation as the ratio between the work of the Lorentz force and the total injected power $\gamma = T/F$. In stationary state, this measure of the amount of energy which is transferred to the kinetic part can also be written $\gamma = D_\nu /(D_\nu + D_\eta )$. This energy budget essentially depends on the flow regime, or more precisely on the degree of turbulence in the flow. First, in the viscous–ideal regime, the velocity gradients are essentially confined to the two top and bottom Hartmann boundary layers and the electrically current almost entirely passes through these layers. At the leading order, the current density is therefore given by $j\sim {I_0}/{(4{\rm \pi} r \delta _{Ha})}$, and the dissipations are given by

(4.12)$$\begin{gather} D_\nu \sim \frac{2\rho\nu S U^{2}}{\delta_{Ha}}, \end{gather}$$

(4.13)$$\begin{gather}D_\eta \sim \frac{2SI_0^{2}}{(4{\rm \pi} r )^{2} \delta_{Ha}}, \end{gather}$$

where $S$ is the surface of the endcap. In the laminar regime, the velocity $U$ is given by expression (4.1), and the dissipation ratio $D_\nu /D_\eta$ is simply equal to one. The efficiency $\gamma$ of the magnetic to kinetic energy conversion is therefore $50\,\%$.

An exact calculation is not possible in the turbulent regime, but a general trend can be obtained: our scaling laws in this regime suggest that most of the viscous dissipation occurs in the Bodewadt boundary layer of typical size $\delta _\varOmega$, and that the electrical current flows through the entire height of the channel. At leading order one would get $j \sim {I_0}/{(2{\rm \pi} r h)}$ and

(4.14)$$\begin{gather} D_\nu \sim \frac{\rho\nu S U^{2}}{\delta_\varOmega}\sim \frac{\rho\nu S U^{5/2}}{\sqrt{\nu r}}, \end{gather}$$

(4.15)$$\begin{gather}D_\eta \sim \frac{SI_0^{2}}{\sigma 4 {\rm \pi}^{2} r^{2} h}. \end{gather}$$

This leads to the dissipation ratio: ${D_\nu }/{D_\eta } \sim 4{\rm \pi} ^{2}r^{3/2}\nu ^{1/2}h\sigma ({U^{5/2}}/{I_0^{2}})$ where the prefactor is equal to $0.1$ in our set-up. By setting $U\sim 1$ m s$^{-1}$ and $I_0\sim 1000$ A as typical values of the turbulent regime, one gets $D_\nu /D_\eta \sim 10^{-7}$. In other words, the turbulence ‘destroys’ the efficiency of the energy transformation, and the dissipation in the turbulent regime is almost entirely due to ohmic dissipation, thus leading to a strong Joule heating of the liquid metal at large current and large magnetic field. As explained in § 2, the inner cylinder is therefore cooled in order to keep the experiment at variations of $\sigma$ less than one per cent.

These results of an efficiency bounded by $50\,\%$ and decreasing with the level of turbulence are surprisingly similar to the one obtained by Reddy et al. (Reference Reddy, Fauve and Gissinger2018) in a very different MHD set-up, suggesting some universal behaviour of the energy budget of electromagnetically driven flows.

4.6. Low frequency oscillations

At large magnetic field and moderate current, there is a small region of the parameter space in which both $Re_B$ and $N_t$ are beyond their critical values $Re_B^{c}$ and $N_t^{c}$. Interestingly, in this region, the velocity field exhibits a slow dynamics characterized by low frequency oscillations and interpreted as large scale structures being advected by the mean flow.

Indeed, for $B_0>95$ mT, $30\ \textrm {A}< I_0<100\ \textrm {A}$, the frequency spectra generally show a peak of energy at low frequencies ($f\sim 1 - 5$ Hz). Time series of figure 7(a) show that this condensate of energy at large scale corresponds to a periodic oscillation of the flow with its lowest frequency at $f\sim 2$ Hz, quite comparable to the advection time $t_u=\Delta r/u_{\theta }\sim 5\times 10^{-1}$ s. As shown by figure 7(b), these oscillations appear only at a critical current $I_0\sim 30$ A, corresponding to $Re_B>Re_B^{c}=400$. The transition to this regime takes the form of a supercritical bifurcation which occurs only in a given range of current with no sign of hysteresis. As the current is increased at fixed magnetic field, the amplitude of these oscillations suddenly goes to zero (for $N_t\sim 10$) and they are replaced by the turbulent flow described above (see figure 7c).

Figure 7. (a) Time series of the fluctuations of $u_\theta$, measured with a potential probe for $B_0=110$ mT and $I=30$ A. (b) Amplitude of the low frequency oscillation as a function of the magnetic field $B_0$ for different values of $I_0$. (c) Same as a function of the current, together with the standard deviation of the azimuthal component of the velocity.

These results are very close to the ones obtained by Messadek & Moreau (Reference Messadek and Moreau2002), who reported similar low frequency oscillations. In their experiment, the current was injected by an array of electrodes located at an intermediate radius between the inner electrode and the outer cylinder of the experiment. These oscillations were therefore attributed to an instability of the shear layer generated near the electrodes between the active flow and the fluid at rest in the outer part of the experiment. There is, however, no steady region in our experiment, since the current is injected directly between the inner and the outer cylinder. These oscillations are more likely to be the consequence of some boundary layer instability. In this perspective, it is interesting to note that the Reynolds number based on the thickness of Shercliff layers generated at the radial boundaries is also relatively large ($Re_{sh}>10^{3}$), such that an instability of these layers could be responsible for the oscillations. In this regard, these results are much closer to the periodic oscillations found by Tabeling near the onset of turbulence in electromagnetically driven Taylor–Couette flows (Tabeling & Chabrerie Reference Tabeling and Chabrerie1981).

In any case, these oscillations can be attributed to large scale bi-dimensional structures of typical size $l\sim 6$ cm which are advected by the mean azimuthal flow. Because they occur in the region delimited by $(N_t>10, Re>9000)$, they could in principle be described by purely 2-D Navier–Stokes equations.

6.1. Power spectra and energy fluxes

The above results show that, at large forcing $I_0B_0$, the flow is clearly turbulent. The dimensionality of this flow is, however, less trivial. Most previous experiments on similar electrically driven flows reported the generation of quasi-bidimensional turbulent flows at large magnetic field. There are, however, two important differences with the results reported here: our experiment operates at a significantly smaller interaction parameter $N_t$, and takes place in a very thin layer of thickness $h\ll \Delta r$, leading to a different regime.

Figure 10(a) shows typical examples of power spectra obtained in the turbulent regime. These spatial spectra were obtained from time series using a Taylor hypothesis based on the fact that the mean azimuthal flow is much larger than the fluctuations. Note that the four cases studied here are indicated by circles in figure 2.

Figure 10. (a) Power spectral densities measured with the potential probes at four different points in the parameter space. The lines correspond to scalings $E(k)\propto k^{-5/3}$ and $E(k)\propto k^{-3}$. (b) Coefficient of the slope of the power spectral density as a function of the injected current $I$ in the range: $k=[100 - 300]$. Here, $Ha\sim 0.57B_0$ with $B_0$ in $\textrm {mT}$.

In three of the four cases presented in figure 10, $Re>1$, $5\times 10^{5}$, $R_B>360$ and $\varLambda < 1$, such that they all belong to the turbulent regime described in the previous section, but the interaction parameter $N_t$ remains between $0.25$ and $16$. Such values are much smaller than the ones used previously in similar experiments, but sufficiently large for quasi-2-D turbulence to be generated.

In all cases except for the low frequency region, the velocity field $u_\theta$ exhibits important turbulent fluctuations ($u_\theta '/\overline {u_\theta }\sim 10\,\%$) and all energy spectra computed at $B_0<100$ mT follow closely $E(\,f)\propto f^{-5/3}$ for $2{\rm \pi} /\Delta r< k<2{\rm \pi} /h$ with a much steeper slope for $k>2{\rm \pi} /h$. The latter can probably be attributed to the cutoff of the potential probes, which are limited by the finite distance $l_p$ between the two electrodes measuring the local voltage and generate a $k^{-11/3}$ spectrum for wavenumbers $k>2{\rm \pi} /l_p$ (Berhanu et al. Reference Berhanu, Gallet, Mordant and Fauve2008). Figure 10(b) reports the slope $\alpha$ of the power spectral density (PSD) computed for wavenumbers $k=[100 - 300]$, which correspond to distances midway between the largest (horizontal) scale $\Delta r$ and the smaller vertical scale $h$ of the experiment. It shows that this $k^{-5/3}$ exponent is observed for a large range of the parameters ($I_0,B_0$), and deviations from it occur only at very large field and small current.

This $k^{-5/3}$ spectrum could correspond either to a direct 3-D turbulent cascade of energy as predicted by Kolmogorov, or to the Kraichnan prediction of an inverse cascade of energy towards the large scales.

In order to access the energy flux characterizing the turbulent cascade, one can compute the third order structure functions of the velocity field defined by

(6.1)\begin{equation} S_3(\ell)=\langle (\boldsymbol{u} (\boldsymbol{x}, \ell+\boldsymbol{d}\ell)-\boldsymbol{u}(\boldsymbol{x}, \ell))^{3}\rangle_x, \end{equation}

where $\langle \cdots \rangle _x$ denotes a spatial average. Since we do not have access to well resolved spatial measurements, we rather computed $S_3$ from very long time series of the velocity field acquired from the potential probes. To do so, we introduce the velocity increment $\delta u(t,\tau ) = \boldsymbol {u} (t+\tau )-\boldsymbol {u}(t)$ as the difference between the values of the flow velocity at time $t+\tau$ and $t$. One can then take advantage of the strong mean azimuthal flow to transform the quantity $S_3(\tau )=\langle \delta u(t,\tau )^{3}\rangle _t$ into $S_3(\ell )$ by use of the Taylor hypothesis $\ell =\tau \overline {u_\theta }$. Depending on the direction of the flux of energy (in the $k$-space), two different behaviours are possible: in 3-D turbulence, the famous Kolmogorov $4/5$ law $S_3(\ell )=-\frac {4}{5}\epsilon \ell$ is expected from the injection scale to the viscous dissipation scale. In the case of an inverse cascade, Kraichnan rather predicted an inverse cascade towards large scale with a positive structure function $S_3(\ell )=-\frac {3}{2}\epsilon \ell$, where $\epsilon$ is now negative.

Figure 11 shows typical structure functions $S_3(l)$, obtained for the same set of parameters as for figure 10. They all correspond to different physical situations. For $B_0=20$ mT and $I_0=400$ A, the energy is injected at a scale slightly smaller than $h$. It then essentially cascades towards the large scales, as evidenced by the positive value of $S_3(\ell )$ and its linear dependence $S_3(\ell )\propto \ell$ in the range $\ell =[h-\Delta r]$. The corresponding $k^{-5/3}$ spectrum of the PSD therefore corresponds to an inverse cascade of energy as predicted by Kraichnan.

Figure 11. Structure function $S_3(l)$ at four different points in the parameter space. Inset: log–log plot of the modified structure functions $\langle |\delta V_L|^{3}\rangle$. Here, $Ha\sim 0.57B_0$ with $B_0$ in $\textrm {mT}$.

For larger forcings ($I_0B_0=20$ or $I_0B_0=65$), the structure functions now clearly exhibit negative values in the range $\ell =1 - 30$ mm, and positive values for $\ell >3$ cm. The injection energy presumably takes place at an intermediate scale, leading to a (limited) scale separation $\Delta r>\ell _{inj}> h$, which is typical of thin layer turbulence. The negative values of $S_3(\ell )$ are therefore the signature of a direct cascade of energy which takes place from $\ell _{inj}$ to smaller scale. On the other hand, some part of the energy also cascades upscale from $\ell _{inj}$ to $\Delta r$, as shown by the positive values of $S_3$ in this range. This split cascade of energy, in which the energy is injected at an intermediate scale $h<\ell _{inj}<\Delta r$ and cascades in both directions, is very similar to the one predicted for turbulence in very thin layers when the injection scale is much larger than the thickness $h$ but much smaller than the largest scale (here $\Delta r$) (Celani, Musacchio & Vincenzi Reference Celani, Musacchio and Vincenzi2010; Xia et al. Reference Xia, Byrne, Falkovich and Shats2011; Benavides & Alexakis Reference Benavides and Alexakis2017). Here, the natural integral scale for the 3-D structures is the height of the disc, such that we expect $\ell _{inj}$ not too far from $h$. In this regime, the $k^{-5/3}$ spectrum of the PSD for $k>2{\rm \pi} h$ describes a forward cascade of energy, while the rest of the spectrum still corresponds to the inverse cascade. Note that, in thin layer turbulence, the forward cascade of energy at small scale is accompanied by a co-directional cascade of enstrophy in the range $[l_{inj}-h]$. This enstrophy spectrum is not visible in our data, probably because of the very limited scale separation of the experiment.

Finally, the regime observed at $B_0=110$ mT, $I_0=60$ A and reported in figures 10 and 11 obviously corresponds to a different regime than the double cascade described above. These parameters are located deep inside the regime of low frequency oscillations described in section $4.5$. When this large scale instability occurs, it produces bi-dimensional vortices of the size of the gap $\ell \sim \Delta r$ which are advected by the mean flow and periodically passes through the probe. These vortices can be regarded as a type of condensate resulting from the inverse cascade (Sommeria Reference Sommeria1986). The $k^{-3}$ spectrum shown in figure 10 may therefore be a signature of this condensate, which takes the form of localized vortices travelling along the azimuthal direction.

This scenario is very different from the one given by Messadek et al., where the $k^{-3}$ spectrum is interpreted as an inverse cascade of energy modified by the strong Hartmann damping when the dissipation in the Hartmann layer is dominant at each wavenumber within the inertial range. We, however, emphasize that, because the low frequency oscillations strongly dominate the dynamics of the flow at large scale, the values of the structure function in the range $\ell =1 - 10$ cm probably does not accurately describe the energy flux. It is then difficult to determine which of these scenarios is correct. In any case, this regime is associated with a very small energy flux at scales smaller than $h$.

6.2. Transition from direct to inverse cascade

In the recent years, the question of the transition from direct to inverse energy transfer in quasi-2-D systems has been at the centre of many studies, as summarized in the review of Alexakis & Biferale (Reference Alexakis and Biferale2018). To follow the transition between these different regimes, we estimate the energy flux at a given scale $\ell ^{*}$ by computing the quantity

(6.2)\begin{equation} \epsilon={-}\left\langle \frac{S_3(\ell)}{\ell} \right\rangle, \end{equation}

where the average is taken on a given range of $\ell$ centred on $\ell ^{*}$. With this definition, a positive (respectively negative) value of $\epsilon$ corresponds to a flux of energy towards the small (respectively large) scales. Figure 12(a) shows the flux $\epsilon _f$ calculated on the range $\ell =10^{-3}-10^{-2}$ m at small scale, while figure 12(b) shows the flux $\epsilon _i$ calculated on the range $\ell =10^{-2}-10^{-1}$ m, at large scale. We plot this energy flux as a function of the ratio $\overline {U_\theta }/\sqrt {B_0}$ which rescales on a single curve the data obtained for different Hartmann numbers.

Figure 12. (a) Energy flux at large wavenumber ($\ell =[10^{-3} - 10^{-2}]$ m) as a function of $\overline {U_\theta }/\sqrt {B_0}$, showing a direct cascade of energy. (b) Energy flux at small wavenumber ($\ell =[10^{-2} - 10^{-1}]$ m) as a function of $\overline {U_\theta }/\sqrt {B_0}$, showing a transition from a direct to an inverse cascade of energy. The two black dashed lines correspond to the scaling $|\epsilon |\propto \overline {U_\theta }^{3}$. Here, $Ha\sim 0.57B_0$ with $B_0$ in $\textrm {mT}$.

Two different states are clearly identified: the regime generated at $\overline {U_\theta }/\sqrt {B_0}<1$ by the large scale oscillations of the flow is associated with a very weak flux of energy at small scale (figure 12a) and a condensate at large scale ($S3>0$, see figure 12b). For $\overline {U_\theta }/\sqrt {B_0}>1$, this regime is replaced by a state characteristic of thin layer turbulence: scales smaller than $h$ are characterized by a direct cascade of energy ($\epsilon >0$, see figure 12a). The energy flux strongly increases with the velocity, and seems to follow a turbulent scaling $\epsilon \propto \overline {U_\theta }^{3}$. At large scales $h<\ell <\Delta r$, the flux is negative in this regime, as expected for an inverse cascade of energy and follows the same scaling $|\epsilon |\propto \overline {U_\theta }^{3}$ but with a larger amplitude: most of the injected energy cascades upscale.

At $\overline {U_\theta }/\sqrt {B_0}\approx 1$, a sharp bifurcation is observed between these two states. Although it describes a bifurcation between the condensate and some sort of thin layer turbulence, it is interesting to note that it takes the form of a second-order phase transition, similarly to what was observed in numerical shell models (Benavides & Alexakis Reference Benavides and Alexakis2017) for the transition between direct and inverse cascade.