Systematic Route to Subcritical Dynamo Branches

We demonstrate that the nonlinear optimization of a finite-amplitude disturbance over a freely evolving and possibly even turbulent flow, can successfully identify subcritical dynamo branches as well as the structure and amplitude of their critical perturbations. As this approach does not require prior knowledge of the magnetic field amplification mechanisms, it opens a new avenue for systematically probing subcritical dynamo flows.

Figure 1

Time-series of the magnetic energy (thicker line) and kinetic energy (thinner line) densities, from DNS of the quasi-Keplerian Couette flow at $\mathrm{Re}=25$ and $\mathrm{Pm}=75$ (resolution: $N_x,N_y,N_z=\{96,192,96\}$). A blowup on the optimization window (right) shows the same quantities, along with the energies of the transverse fields. Letters mark the times of the various snapshots. The structure of the optimal seed ($A$) is highlighted by magnetic streamlines in transverse cut (left). The snapshots in inset show the amplitude of the streamwise component $b_x$, illustrating the $\mathrm{\Omega }$ effect ($B$), destabilization of the large-scale field to MRI ($C$) and fully 3D saturated dynamo ($D$). (Note that the MRI has already saturated by the optimization time $T$).

Figure 2

Left: bifurcation diagram of the Taylor-Green dynamo, for $\mathrm{Gr}=1875$. Magnetic energy of (shaded square) the saturated dynamo states and (shaded triangle) the smallest magnetic disturbances that trigger a subcritical dynamo, as found by nonlinear optimization with $T=10\sqrt{l/f}$. $▿$: failed dynamo seeds. Right: Typical time series of the kinetic energy (thinner line) and magnetic energy (thicker line) densities, as the minimal seed ($A$) spontaneously evolves toward the saturated dynamo state ($D$). Letters mark the times of the snapshots shown in Fig. 3.

Figure 3

Left to right: snapshots of (A) the minimal seed for the Taylor-Green dynamo at $\{\mathrm{Gr}=1875;\mathrm{Pm}=0.44\}$ (magnetic lines in blue, current lines in red); (B) magnetic field during the algebraic growth (rescaled and time-averaged over the growth period); (C) growing magnetic field during the exponential growth (rescaled and time-averaged); (D) saturated state (recovers [37]). The volume rendering shows the density of magnetic energy. The magnetic energy at saturation displays cigar-shaped structures aligned in two parallel planes.