The Unsteady Taylor--Vortex Dynamo is Fast

Abstract

Astrophysical and geophysical fluids commonly generate organized magnetic fields, despite having enormous magnetic Reynolds numbers $\rm{Rm}$ and abundant small-scale turbulence. Flow-induced dynamo action produces these fields, with the ``kinematic dynamo problem'' devoted to determining the rate at which a flow exponentially amplifies weak magnetic fields. However, previous studies on high-Rm kinematic dynamos have generated flows via imposed volumetric forcing or oscillatory boundary conditions. In this letter, we investigate a system with three important attributes: realistic flow conditions, fast dynamo action (operational for $\rm{Rm}\to\infty$), and a subharmonic spatio-temporal structure. We show that unsteady Taylor--vortex flow, a regime observed in laboratory experiments, gives rise to fast dynamos with time and length scales twice those of the flow at high $\rm{Rm}$. By numerically integrating a Floquet system driven by periodic oscillations of Taylor vortices, we solve the kinematic dynamo problem up to $\rm{Rm} = 3.2 \cdot 10^6$, calculating the dynamo's growth rate as a function of Rm and streamwise wavenumber. We find the onset of instability and compute Finite-Time Lyapunov Exponents, which identify the regions of Lagrangian chaos required for fast dynamo action. To our knowledge, unsteady Taylor--vortex flow produces the most physically motivated fast dynamo to date.

Figures (5)

• Figure 1: Snapshots of the dynamo cycle are shown at the beginning ($t=0$, panes A, B, and C) and after one quarter of a period ($t=T/4$, panes D, E, and F). At both times, we plot the streamwise velocity (panes A and D) and streamwise magnetic field at $\rm{Rm} = 1.5\cdot 10^3$ (panes B and E) and $\rm{Rm} = 1.5\cdot 10^5$ (panes C and F). Both dynamos were generated at streamwise wavenumber $k_y = 0.18$. We observe Taylor vortices in the streamwise velocity field, given by two pairs of counter-rotating cells which transport momentum between thin boundary layers. As the plumes of these cells oscillate in $z$, the magnetic field is amplified in ring-like structures in alternating regions between plumes, such that the rings co-rotate. At $t=0$, we observe two rings rotating clockwise. At $t=T/4$, there is a transition toward counterclockwise rotating rings which form in regions where the magnetic field was initially weak. The $\rm{Rm}=1.5\cdot 10^3$ dynamo (panes B and E) has a more diffuse structure than $\rm{Rm}=1.5\cdot 10^5$ (panes C and F). Both dynamos are dominated by the $k_z = 1$ mode, whereas the velocity field is primarily $k_z = 2$.
• Figure 2: Growth rates vs. streamwise wavenumber ($k_y$) for various $\rm{Rm}$. The onset of instability occurs at $\rm{Rm}=36.9$ and $k_y=0.29$ while the dominant $k_y=1.0$ mode becomes marginally stable when $\rm{Rm}=861$. As $\rm{Rm}$ increases, the maximizing $k_y$ decreases for individual extrema while higher $k_y$ modes become dominant.
• Figure 3: The magnetic growth rate is plotted as a function of Rm for streamwise wavenumbers $k_y=0.18, \, 0.29, \, 1.0$. For $k_y=1.0$, we observe cusp points where the growth rate's derivative with respect to $\rm{Rm}$ appears discontinuous. Evidence for a fast dynamo is observed at high $\rm{Rm}$, where the growth rate exhibits no appreciable dependence on $\rm{Rm}$. Triangle markers denote cases where the Floquet modes' period is equal to the flow's oscillation period $T$. Circular points denote Floquet modes with period $2T$ and the cases marked by stars have even longer periods or exhibit quasiperiodicity.
• Figure 4: We plot the FTLE as a function of position for $t=0$ and $t=T/4$. These results were obtained using particle tracking initialized with the flow states shown in Figure \ref{['big']}. Positive FTLEs with sensitive spatial dependence indicate large regions of Lagrangian chaos enveloping the Taylor vortices.
• Figure 5: $k_{\rm{eff}}^2 / \rm{Rm}$ as a function of $\rm{Rm}$ for $k_y = 0.18, \, 0.29,$ and $1.0$. This quantity becomes constant for large $\rm{Rm}$, implying $k_{\rm{eff}} \sim \rm{Rm}^{1/2}$. Marker and color conventions match those described in the caption of Figure \ref{['gamma_rm']}.