Dynamo Action in a Quasi-Keplerian Taylor-Couette Flow

We numerically compute the flow of an electrically conducting fluid in a Taylor-Couette geometry where the rotation rates of the inner and outer cylinders satisfy ${\mathrm{\Omega }}_o/{\mathrm{\Omega }}_i=(r_o/r_i{)}^{-3/2}$. In this quasi-Keplerian regime, a nonmagnetic system would be Rayleigh stable for all Reynolds numbers Re, and the resulting purely azimuthal flow incapable of kinematic dynamo action for all magnetic Reynolds numbers $\mathrm{R}\mathrm{m}$. For $\mathrm{Re}={10}^4$ and $\mathrm{R}\mathrm{m}={10}^5$, we demonstrate the existence of a finite-amplitude dynamo, whereby a suitable initial condition yields mutually sustaining turbulence and magnetic fields, even though neither could exist without the other. This dynamo solution results in significantly increased outward angular momentum transport, with the bulk of the transport being by Maxwell rather than Reynolds stresses.

Figure 1

Magnetic (left) and kinetic (right) energies of the runs at $\mathrm{R}\mathrm{m}={10}^4$ (dashed blue lines) and $\mathrm{R}\mathrm{m}={10}^5$ (solid red lines), as functions of time. The short line segments for $t\le 50$ correspond to the initial condition before the external field is switched off [39]. For the kinetic energies, the laminar profile has been subtracted from $\mathbf{U}$ to better illustrate the relaminarization of the $\mathrm{P}\mathrm{m}=1$ run.

Figure 2

The top row shows different views of $\mathbf{U}$; from left to right, the four panels show isosurfaces of $U_z=\pm 0.01$, contours of $U_z$ and $U_r$ on slices in the ($r$, $z$) meridional plane, and contours of $U_{\varphi }$ on the unrolled cylinder at midgap ($r=1.5$). The bottom row shows equivalent snapshots of $\mathbf{B}$.

Figure 3

Time-averaged kinetic (black) and magnetic (green) energy spectra versus axial (solid) and azimuthal (dashed) wave number for the $\mathrm{R}\mathrm{m}={10}^5$ dynamo case. These spectra are computed at midgap ($r=1.5$) only, but spectra averaged over the entire volume are qualitatively similar.

Figure 4

Normalized torque as a function of time for $\mathrm{R}\mathrm{m}={10}^4$ (dashed) and $\mathrm{R}\mathrm{m}={10}^5$ (solid). The short black segment for $t\le 50$ again corresponds to the initial condition.

Figure 5

Both panels show the time-averaged stresses as functions of radius, with the Maxwell stress green dashed, Reynolds black solid, and viscous red dash-dotted lines. The left panel is for the true Taylor-Couette system, where the $U_{\varphi }$ profile is allowed to adjust as in Fig. 6. The right panel is for the system where the $U_{\varphi }$ profile is forced to remain quasi-Keplerian throughout the interior.

Figure 6

The left panel shows the time-averaged $U_{\varphi }$ radial profile for the dynamo at $\mathrm{R}\mathrm{m}={10}^5$ (red solid), in comparison with the laminar quasi-Keplerian profile (black dashed). The right panel shows the corresponding angular velocities $\mathrm{\Omega }=U_{\varphi }/r$.