Transition to Turbulent Dynamo Saturation

While the saturated magnetic energy is independent of viscosity in dynamo experiments, it remains viscosity dependent in state-of-the-art 3D direct numerical simulations (DNS). Extrapolating such viscous scaling laws to realistic parameter values leads to an underestimation of the magnetic energy by several orders of magnitude. The origin of this discrepancy is that fully 3D DNS cannot reach low enough values of the magnetic Prandtl number Pm. To bypass this limitation and investigate dynamo saturation at very low Pm, we focus on the vicinity of the dynamo threshold in a rapidly rotating flow: the velocity field then depends on two spatial coordinates only, while the magnetic field consists of a single Fourier mode in the third direction. We perform numerical simulations of the resulting set of reduced equations for Pm down to $2\times {10}^{-5}$. This parameter regime is currently out of reach to fully 3D DNS. We show that the magnetic energy transitions from a high-Pm viscous scaling regime to a low-Pm turbulent scaling regime, the latter being independent of viscosity. The transition to the turbulent saturation regime occurs at a low value of the magnetic Prandtl number, $\mathrm{Pm}\simeq {10}^{-3}$, which explains why it has been overlooked by numerical studies so far.

Figure 1

Threshold magnetic Reynolds number ${\mathrm{Rm}}_c$ for dynamo action as a function of Pm, for $\gamma {\ell }^2/\eta =5.1\times {10}^{-3}$.

Figure 2

Snapshots of the saturated state for $\mathrm{Pm}=4.25\times {10}^{-5}$, $\gamma {\ell }^2/\eta =5.1\times {10}^{-3}$, and $\mathrm{Rm}=0.44$. Top: vertical vorticity in units of $U/\ell$. Middle: vertical velocity in units of $U$. Bottom: dimensionless magnetic field $\mathbf{B}\ell /\sqrt{\rho {\mu }_0{\eta }^2}$ in the plane $z=0$. The arrows indicate the horizontal components while color codes for the vertical one. These arrows correspond to a typical magnitude $4\times {10}^{-2}$ of the dimensionless horizontal magnetic field. Their average direction rotates with $z$.

Figure 3

Magnetic energy as a function of the departure from onset, for several values of the magnetic Prandtl number. The friction is $\gamma {\ell }^2/\eta =5.1\times {10}^{-3}$ and symbols are circle, $\mathrm{Pm}=1.4\times {10}^{-3}$; inverted triangle, $\mathrm{Pm}=7.0\times {10}^{-3}$; star, $\mathrm{Pm}=4.0\times {10}^{-2}$; open diamond, $\mathrm{Pm}=8.9\times {10}^{-2}$. The dashed linear fits allow to extract the slope $S(\mathrm{Pm})$ of each bifurcation curve.

Figure 4

Slope $S$ of the magnetic energy above onset. High-Pm solutions are time independent (blue circles) and obey the quantitative prediction (9) from viscous alpha quenching (thick solid line). For low Pm, the flow is time dependent (red triangles). For $\mathrm{Pm}\lesssim {10}^{-3}$, the dynamo saturation follows the turbulent scaling-law (2), represented as a dashed eye guide. The main figure corresponds to a friction coefficient $\gamma {\ell }^2/\eta =5.1\times {10}^{-3}$. The inset highlights the independence of $S$ on friction in the turbulent saturation regime.