Numerical simulations of current generation and dynamo excitation in a mechanically forced turbulent flow

The role of turbulence in current generation and self-excitation of magnetic fields has been studied in the geometry of a mechanically driven, spherical dynamo experiment, using a three-dimensional numerical computation. A simple impeller model drives a flow that can generate a growing magnetic field, depending on the magnetic Reynolds number $\mathrm{Rm}={\mu }_0\sigma Va$ and the fluid Reynolds number $\mathrm{Re}=Va∕\nu$ of the flow. For $\mathrm{Re}<420$, the flow is laminar and the dynamo transition is governed by a threshold of ${\mathrm{Rm}}_{\mathrm{crit}}=100$, above which a growing magnetic eigenmode is observed that is primarily a dipole field transverse to the axis of symmetry of the flow. In saturation, the Lorentz force slows the flow such that the magnetic eigenmode becomes marginally stable. For $\mathrm{Re}>420$ and $\mathrm{Rm}\sim 100$ the flow becomes turbulent and the dynamo eigenmode is suppressed. The mechanism of suppression is a combination of a time varying large-scale field and the presence of fluctuation driven currents (such as those predicted by the mean-field theory), which effectively enhance the magnetic diffusivity. For higher Rm, a dynamo reappears; however, the structure of the magnetic field is often different from the laminar dynamo. It is dominated by a dipolar magnetic field aligned with the axis of symmetry of the mean-flow, which is apparently generated by fluctuation-driven currents. The magnitude and structure of the fluctuation-driven currents have been studied by applying a weak, axisymmetric seed magnetic field to laminar and turbulent flows. An Ohm’s law analysis of the axisymmetric currents allows the fluctuation-driven currents to be identified. The magnetic fields generated by the fluctuations are significant: a dipole moment aligned with the symmetry axis of the mean-flow is generated similar to those observed in the experiment, and both toroidal and poloidal flux expulsion are observed.

Figure 1

(Color online) (a) A schematic of the Madison Dynamo Experiment. The sphere is $1\mathrm{m}\mathrm{diam}$ and filled with $105–110°\mathrm{C}$ liquid sodium. High-speed flows are created by two counterrotating impellers. Two sets of coils, one coaxial and one transverse to the drive shafts, are used to apply various magnetic fields for probing the experiment. (b) Contours of the toroidal velocity $v_{\varphi }$ and contours of the poloidal flow stream function $\Phi$ where ${\mathbf{v}}_{\mathrm{pol}}=\nabla \Phi \times \nabla \varphi$, of the axisymmetric double vortex flow generated by the impeller model. The region of forcing is shown schematically along the symmetry axis.

Figure 2

(Color online) (a) The kinetic and magnetic energy densities shown versus time with $\mathrm{Rm}=159$ and $\mathrm{Pm}=1$. The time is in units of the resistive diffusion time ${\tau }_{\sigma }={\mu }_0\sigma a^2$. (b) The contributions to the total magnetic energy density from the $m=1$ transverse dipole and the axisymmetric $m=0$ modes. (c) Magnetic field lines of a saturated dynamo state for a laminar flow with $\mathrm{Rm}=150$.

Figure 3

The dependence of the linear growth rate of the least-damped magnetic eigenmode on impeller pitch $ϵ$. The transition from damped to growing ($\lambda =0$ point) defines the critical magnetic Reynolds number; $\mathrm{Rm}>{\mathrm{Rm}}_{\mathrm{crit}}$ for a growing magnetic eigenmode.

Figure 4

$\mathrm{Rm}$ and ${\mathrm{Rm}}_{\mathrm{crit}}$ evolution during the saturation phase of a laminar dynamo. ${\mathrm{Rm}}_{\mathrm{crit}}$ is calculated from linear stability for each instantaneous velocity field during the simulation. In saturation, $\mathrm{Rm}={\mathrm{Rm}}_{\mathrm{crit}}$.

Figure 5

(Color online) The magnetic and kinetic energy densities for runs with fixed Rm $(\mathrm{Rm}=165\pm 3\%)$ but different $\mathrm{Pm}$ versus time in ${\tau }_{\sigma }$. Note that $\mathrm{Pm}=0.33$ shows a relaminarization of turbulent flow, whereas $\mathrm{Pm}=0.22$ is barely amplifying the initial noise and is shown multiplied by $50000$.

Figure 6

(a) $\mathrm{Rm}$ as a function of time. Taking a series of flows over the range shown, the mean flow is calculated giving $\mathrm{Rm}=193$ and $\mathrm{Re}=863$. (b) The kinematic analysis of the average flow, where only Eq. (6) is evolved, and $\mathrm{Rm}$ is varied, to determine ${\mathrm{Rm}}_{\mathrm{crit}}=91.5$.

Figure 7

(Color online) $\mathrm{Re}–\mathrm{Rm}$ phase diagram. A number of simulations whose hydrodynamic and final saturated states are documented in Fig. 5. ${\mathrm{Rm}}_{\mathrm{crit}}$ for the mean flow $⟨\mathbf{V}⟩$ is essentially independent of Re, whereas the effective dynamo threshold grows with Re. The dashed line shows the qualitative behavior of the dynamo threshold in turbulent flows $\left(V_T\right)$.

Figure 8

Growth rate of the dominant eigenmode calculated for a time-series of flow profiles.

Figure 9

(Color online) (a) The energy density of a turbulent dynamo with $\mathrm{Rm}=337$ and $\mathrm{Re}=674$. (b) The energy density of the axisymmetric magnetic field $(m=0)$ and the nonaxisymmetric dynamo $(m=1)$. (c) Magnetic field lines of the turbulent saturated dynamo.

Figure 10

(Color online) Simulations with an externally applied, axisymmetric magnetic field. (a) Kinetic and magnetic energy densities for a $\mathrm{Re}=116$ (laminar), $\mathrm{Rm}=70$ (subcritical) simulation. (b) The resulting flow. (c) Kinematic and magnetic energy densities for an $\mathrm{Re}=1803$ (turbulent), $\mathrm{Rm}=108$ (subcritical) simulation. (d) The axisymmetric, time-averaged velocity field for the turbulent simulation. The time average is over the time interval $0.3–2.4$ ${\tau }_{\sigma }$, which is roughly $30$ decorrelation times.

Figure 11

(Color online) (a) The magnetic field for a laminar flow described in Fig. 10. The resulting total magnetic field (the sum of the externally applied field and those generated by the currents in the liquid metal) is shown as a multiple of the applied magnetic field. (b) The time-averaged magnetic fields, scaled to the applied field, for a turbulent flow [$\mathrm{Rm}=107$ (subcritical), $\mathrm{Re}=1803$]. The peak internal poloidal magnetic field is $9.3$ times larger than the applied field.

Figure 12

(Color online) The wave-number spectrum computed from frequency spectrum of fluctuations from $6{\tau }_{\sigma }$ of flow (fitted with the red $k^{-5∕3}$ curve) output at a position ($r\sim 0.75$ $a$, $\theta \sim \pi ∕2$, $\varphi =0$) with a weak applied magnetic field of $B_0\sim 51.3\mathrm{G}$, fitted with the blue $k^{-5∕3}$ at low $k$ and $k^{-11∕3}$ at large $k$. For this simulation, $\mathrm{Rm}=130$, and $\mathrm{Re}=1450$ and fluctuations are assumed to be due to convection of spatial variations in the field. The dispersion relation is $\omega =k⟨V⟩$.

Figure 13

(Color online) The magnetic field (a) generated by the mean-flow EMF $⟨\mathbf{V}⟩\times ⟨\mathbf{B}⟩$. The peak internal poloidal magnetic field is $11.2$ times larger than the applied field. The magnetic field (b) generated by the turbulent EMF $⟨\tilde{\mathbf{v}}\times \tilde{\mathbf{b}}⟩$. (c) The currents associated with (a), and (d) the currents associated with (b). The turbulent flow has [$\mathrm{Rm}=107$ (subcritical), $\mathrm{Re}=1803$] and an externally generated magnetic field applied along the symmetry axis. The time average is taken over $2.1{\tau }_{\sigma }$. Magnetic fields are scaled to the strength of the applied field.

Figure 14

(Color online) The turbulent and helical fluctuations for the simulation described in Fig. 12. (a) The average of the squared turbulent fluctuations as a multiple of the peak squared mean flow. (b) The time average of the kinetic helicity fluctuations $⟨\tilde{\mathbf{v}}\bullet \nabla \times \tilde{\mathbf{v}}⟩$ as a multiple of the volume-averaged helicity of the mean flow.