Angular Momentum Transport by Keplerian Turbulence in Liquid Metals

We report a laboratory study of the transport of angular momentum by a turbulent flow of an electrically conducting fluid confined in a thin disk. When the electromagnetic force applied to the liquid metal is large enough, the corresponding volume injection of angular momentum produces a turbulent flow characterized by a time-averaged Keplerian rotation rate $\overline{\mathrm{\Omega }}\sim r^{-3/2}$. Two contributions to the local angular momentum transport are identified: one from the poloidal recirculation induced by the presence of boundaries and the other from turbulent fluctuations in the bulk. The latter produces efficient angular momentum transport independent of the molecular viscosity of the fluid and leads to Kraichnan’s prediction ${\mathrm{Nu}}_{\mathrm{\Omega }}\propto \sqrt{\mathrm{Ta}}$. In this so-called ultimate regime, the experiment, therefore, provides a configuration analogous to accretion disks, allowing the prediction of accretion rates induced by Keplerian turbulence.

Figure 1

Left: the experimental setup is an annular cylindrical channel with inner radius $R_1=6\mathrm{cm}$, outer radius $R_2=19\mathrm{cm}$ and height $h=1.5\mathrm{cm}$, subjected to a radial current ($I_0=[0–3000]\mathrm{A}$) and a vertical magnetic field ($B_0=[0–110]\mathrm{mT}$). Right: a series of potential probes extending from the top plate to midheight provide measurements of both azimuthal and radial velocity field in the midplane. The blue probes measure product $u_r\mathrm{\Omega }$ and derivative ${\partial }_r\mathrm{\Omega }$ involved in $J_{\mathrm{\Omega }}$.

Figure 2

Wind Reynolds number ${\mathrm{Re}}_w$ versus Ta for an applied magnetic field $B_0=60\mathrm{mT}$, and compared to theoretical predictions. Inset: Same, but compensated by $\sqrt{Ta}$.

Figure 3

Nusselt number ${\mathrm{Nu}}_{\mathrm{\Omega }}$ versus Taylor number Ta for typical applied magnetic fields $B_0=[40,80,110\mathrm{mT}]$, compared to theoretical predictions.

Figure 4

Fluctuation-based Nusselt number ${\mathrm{Nu}}_{\mathrm{\Omega }}^{*}$ versus Taylor number Ta for typical applied magnetic fields $B_0=[40,80,110\mathrm{mT}]$. The red dashed line corresponds to the ultimate regime ${\mathrm{Nu}}_{\mathrm{\Omega }}\propto \sqrt{\mathrm{Ta}}$, only observed for large enough $B_0$ and Ta. Open symbols correspond to the oscillatory regime (see text). Inset: compensated scaling law.

Figure 5

Dimensionless energy dissipation $G/{\mathrm{Re}}^2$ and $G^{*}/{\mathrm{Re}}^2$ versus Re. The dashed lines indicate the range of values obtained by Paoletti et al. [13] for both quasi-Keplerian and Rayleigh-unstable flows. The flow tends to an ultimate regime as the applied field is increased. (*) denotes $G^{*}/{\mathrm{Re}}^2$ values. Open symbols correspond to the oscillatory regime (see text).