Numerical investigation of quasistatic magnetoconvection with an imposed horizontal magnetic field

Quasistatic Rayleigh-Bénard convection with an imposed horizontal magnetic field is investigated numerically for Chandrasekhar numbers up to $Q={10}^6$ with stress-free boundary conditions. Both $Q$ and the Rayleigh number (Ra) are varied to identify the various dynamical regimes that are present in this system. We find three primary regimes: (i) a two-dimensional (2D) regime in which the axes of the convection rolls are oriented parallel to the imposed magnetic field; (ii) an anisotropic three-dimensional (3D) regime; and (iii) a mean flow regime characterized by a large scale horizontal flow directed transverse to the imposed magnetic field. The transition to 3D dynamics is preceded by a series of 2D transitions in which the number of convection rolls decreases as Ra is increased. For sufficiently large $Q$, there is an eventual transition to two rolls just prior to the 2D-3D transition. The 2D-3D transition occurs when inertial forces become comparable to the Lorentz force, i.e., when $\sqrt{Q}/\text{Re}=O\left(1\right)$; 2D, magnetically constrained states persist when $\sqrt{Q}/\text{Re}\gtrsim O\left(1\right)$. Within the 2D regime, we find heat and momentum transport scalings that are consistent with the hydrodynamic asymptotic predictions of Chini and Cox [Phys. Fluids 21, 083603 (2009)]: the Nusselt number (Nu) and Reynolds number (Re) scale as $\text{Nu}\sim {\text{Ra}}^{1/3}$ and $\text{Re}\sim {\text{Ra}}^{2/3}$, respectively. For $Q={10}^6$, we find that the scaling behavior of Nu and Re breaks down at large values of Ra due to a sequence of bifurcations and the eventual manifestation of mean flows.

Figure 1

Convective structure versus Rayleigh number (Ra): (a) number of convection rolls, $n_r$ and (b) convection roll aspect ratio, ${\mathrm{\Gamma }}_r$. Black stars denote the values of Ra for which three-dimensional motion is first observed for each value of $Q$.

Figure 2

Horizontal slices of the fluctuating temperature at the thermal boundary layer depth (top rows) and the midplane (bottom rows) for (a)–(d) $Q={10}^2$ and (e)–(h) $Q={10}^3$. The imposed magnetic field points up the page; gravity points into the page. The Rayleigh number increases from left to right in each row: (a) $\mathrm{Ra}={10}^3$; (b) $\mathrm{Ra}=7\times {10}^3$; (c) $\mathrm{Ra}=3\times {10}^4$; (d) $\mathrm{Ra}={10}^6$; (e) $\mathrm{Ra}={10}^4$; (f) $\mathrm{Ra}=3\times {10}^4$; (g) $\mathrm{Ra}={10}^5$; (h) $\mathrm{Ra}=3\times {10}^5$.

Figure 3

Visualizations of the fluctuating temperature for large $Q$ cases that exhibit three-dimensional motion: (a), (c) $Q={10}^4, \text{Ra}={10}^6$ and (b), (d) $Q={10}^6, \text{Ra}=1.5\times {10}^7$. Horizontal $x\text{−}y$ slices are shown in (a) and (b); vertical $y\text{−}z$ slices are shown in (c) and (d). The top (bottom) panels in (a) and (b) are taken at the depth of the thermal boundary layer (midplane).

Figure 4

Heat transport data versus Rayleigh number (Ra) for all cases: (a) Nusselt number (Nu); (b) compensated Nusselt number, $\text{Nu}/{\text{Ra}}^{1/3}$; (c) fraction of ohmic dissipation, $f_{\mathrm{\Omega }}$; and (d) thermal boundary layer thickness. The starred data points in (a) indicate the first observation of three-dimensional flows for the given value of $Q$. The dashed horizontal line in (b) is the prefactor computed by Chini and Cox [34], corresponding to wave number $k\approx 2.22$.

Figure 5

Momentum transport versus Rayleigh number for all cases: (a) Reynolds number, Re; (b) compensated Reynolds number, $\text{Re}/{\text{Ra}}^{2/3}$; (c) ratio of the vertical Reynolds number to transverse Reynolds number. In (b), the horizontal line at $\text{Re}{\text{Ra}}^{-2/3}=0.11$ is the asymptotic result from Ref. [35]. Black stars in (c) denote the values of Ra for which three-dimensional motion is first observed for each value of $Q$.

Figure 6

Instantaneous one-dimensional (along the direction of the magnetic field) power spectra of the vertical component of the velocity vector showing the onset of three-dimensional motion: (a) $Q={10}^2$; (b) $Q={10}^4$; (c) $Q={10}^6$. The spectra are averaged over the depth of the layer. For each value of $Q$, the smallest value of Ra shown corresponds to the first observed onset of three-dimensional motion.

Figure 7

Characterization of the transition from 2D to 3D convection with the ratio $\text{Ha}/\text{Re}=\sqrt{Q}/\text{Re}$. Starred symbols denote the smallest value of Ra for which 3D motion is observed for the indicated value of $Q$.

Figure 8

Heat transport and flow speeds from the first observed case that shows a substantial mean flow ($Q={10}^6$ and $\text{Ra}=2\times {10}^7$): (a) time series of the Nusselt number; (b) vertical profiles of the rms vertical velocity; (c) time series of various measures of flow speed; and (d) vertical profiles of the $y$ component of the horizontally averaged flow. Black arrows in (a) and colored dots in (a) and (c) delineate the times corresponding to the profiles shown in (b) and (d).

Figure 9

Heat transport and flow speeds from the second observed case that shows a substantial mean flow with relaxation oscillations ($Q={10}^6$ and $\text{Ra}=3\times {10}^7$): (a) time series of the Nusselt number; (b) vertical profiles of the rms vertical velocity; (c) time series of various measures of flow speed; (d) vertical profiles of the $y$-component of the horizontally averaged flow. Black arrows in (a) and colored dots in (a) and (c) delineate the times corresponding to the profiles shown in (b) and (d).