Effect of electromagnetic boundary conditions on the onset of small-scale dynamos driven by convection

We present a high-order numerical study of the dependence of the dynamo onset on electromagnetic boundary conditions in convecting Boussinesq flows forced by temperature gradients. Perfectly conducting boundaries, vacuum, and mixed boundary conditions are considered, treating fields and boundary conditions with close to spectral accuracy. Having one or two conducting boundaries greatly facilitates dynamo action. For the mixed case it is shown that the critical magnetic Reynolds number becomes independent of the Rayleigh number Ra for sufficiently large Ra.

Figure 1

Rendering using the software vapor [46, 47] of the temperature fluctuations $\theta$ inside the domain, together with the magnetic field lines (in green) for simulation M22 [48]. Note that the magnetic field permeates into the vacuum above $z=1$.

Figure 2

Magnetic energy $E^b$ as a function of time for all simulations at $\mathrm{Ra}=2.25\times {10}^6$. Line colors and markers denote the magnetic Reynolds number of each simulation. (a) The case of conducting boundaries, (b) mixed boundaries, and (c) vacuum surroundings. Simulations with decaying magnetic energy are drawn with discontinuous lines.

Figure 3

Growth rate $\sigma$ as a function of Rm for simulations with $\mathrm{Ra}=2.25\times {10}^6$. Markers denote electromagnetic boundary conditions: crosses for conducting boundaries, circles for mixed conditions, and squares for vacuum.

Figure 4

Vertically averaged magnetic energy spectra ${\langle E^b(k_{\parallel },z)\rangle }_z$ for wave numbers parallel to the wall, $k_{\parallel }$. (a) Magnetic spectra for conducting boundaries. (b) Same for mixed and for (c) vacuum boundaries. Decaying solutions are marked with dashes, and colors and symbols reference different values of Rm ($\mathrm{Ra}=2.25\times {10}^6$ in all cases). The inset in (c) shows normalized spectra for three simulations at the same Rm differing on boundary conditions (runs C2, M1, and V5). Wave numbers use $L_x=L_y=2\pi L_0$ as the reference length. The correlation length of the velocity is similar in $\widehat{\mathbf{x}}$, $\widehat{\mathbf{y}}$, and $\widehat{\mathbf{z}}$, and is $\sim 0.6L_0$ for $u_z$, and ${\ell }_{\parallel }\approx 0.75L_0$ (in $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$) and ${\ell }_z\approx 0.5L_0$ for $u_x$ and $u_y$.

Figure 5

(a) Growth rate of the magnetic energy $\sigma$ as a function of the Rayleigh number Ra, for mixed boundaries. Increasing values of Rm are denoted with a darker shade and bigger markers. (b) Critical magnetic Reynolds number ${\mathrm{Rm}}^{\text{crit}}$ as a function of Ra. Error bars denote the Rm values corresponding to the pair of simulations whose growth rates are closer to zero, and filled symbols correspond to ${\mathrm{Rm}}^{\text{crit}}$ obtained from a linear interpolation of $\sigma \left(\mathrm{Rm}\right)$ between those values. Labels for filled symbols are as in Fig. 3, and open error bars denote lower bounds for ${\mathrm{Rm}}^{\text{crit}}$.

Figure 6

Vertically averaged energy spectra ${\langle E(k_{\parallel },z)\rangle }_z$ as a function of $k_{\parallel }$ for simulations M4 (blue with crosses), M12 (orange), and M22 (red with empty circles), which operate at approximately the same Rm. (a) Kinetic energy spectrum $E^u$, and (b) magnetic energy spectrum $E^b$. Spectra are normalized by their total energy.