Quasistatic magnetoconvection with a tilted magnetic field

A numerical study of convection with stress-free boundary conditions in the presence of an imposed magnetic field that is tilted with respect to the direction of gravity is carried out in the limit of small magnetic Reynolds number. The dynamics are investigated over a range of Rayleigh number Ra and Chandrasekhar numbers up to $Q=2\times {10}^6$, with the tilt angle of the imposed magnetic field vector fixed at ${45}^{\circ }$ relative to vertical. For a fixed value of $Q$ and increasing Ra, the convection dynamics can be broadly characterized by three primary flow regimes: (1) quasi-two-dimensional convection rolls near the onset of convection, (2) isolated convection columns aligned with the imposed magnetic field, and (3) unconstrained convection reminiscent of nonmagnetic convection. The influence of varying $Q$ and Ra on the various fields is analyzed. Heat and momentum transport, as characterized by the Nusselt and Reynolds numbers, are quantified and compared with the vertical field case. Ohmic dissipation dominates over viscous dissipation in all cases investigated. Various mean fields are investigated and their scaling behavior is analyzed. Provided Ra is sufficiently large, all investigated values of $Q$ exhibit an inverse kinetic energy cascade that yields strong “zonal” flows with an amplitude that scales as $Q^{1/3}$. Relaxation oscillations, as characterized by a quasiperiodic shift in the predominance of either the zonal or nonzonal component of the mean flow, occur when Ra and $Q$ are sufficiently large.

Figure 1

Geometry used in the present study. The fluid layer has depth $H$ and horizontal extent $L$.

Figure 2

Parametric overview of the simulations, as characterized by the ratio ${\text{Ha/Re}}_z$, where $\text{Ha}=\sqrt{Q}$ is the Hartmann number and ${\text{Re}}_z$ is the Reynolds number based on the rms of the vertical component of the velocity field. The critical Rayleigh number is denoted by ${\text{Ra}}_c$. The ratio ${\text{Ha/Re}}_z$ represents the relative size of the Lorentz force and inertia: Cases with ${\text{Ha/Re}}_z\gtrsim O\left(1\right)$ are considered magnetically constrained.

Figure 3

Volumetric renderings of the fluctuating temperature for various cases. Top row ($Q=2\times {10}^3$): (a) ${\text{Ra/Ra}}_c=1.3$, (b) ${\text{Ra/Ra}}_c=13.2$, (c) ${\text{Ra/Ra}}_c=132$. Middle row ($Q=2\times {10}^5$): (d) ${\text{Ra/Ra}}_c=1.4$, (e) ${\text{Ra/Ra}}_c=7.4$, (f) ${\text{Ra/Ra}}_c=27.8$. Bottom row ($Q=2\times {10}^6$): (g) ${\text{Ra/Ra}}_c=1.5$, (h) ${\text{Ra/Ra}}_c=3.9$, (i) ${\text{Ra/Ra}}_c=6.8$. The orientation for all visualizations is shown in panel (a).

Figure 4

Nusselt number data: (a) Nu vs Ra; (b) compensated Nusselt number, $\text{Nu}({\text{Ra}}_c/\text{Ra})$, vs ${\text{Ra/Ra}}_c$. Tilted magnetoconvection cases are denoted by TMC, and the vertical field cases from Ref. [5] are denoted by VMC.

Figure 5

Vertical profiles of the time-averaged temperature for all cases. Top row [(a)–(c)]: horizontally averaged temperature. Bottom row [(d)–(f)]: rms temperature fluctuation. [(a), (d)] $Q=2\times {10}^3$; [(b), (e)] $Q=2\times {10}^5$; [(c), (f)] $Q=2\times {10}^6$. Darker grayscale lines correspond to larger Rayleigh numbers.

Figure 6

Dissipation ratios: (a) $Q=2\times {10}^3$, (b) $Q=2\times {10}^5$, (c) $Q=2\times {10}^6$. Note that the scale of the horizontal axis is different for each figure.

Figure 7

Normalized dissipation profiles averaged in time for select cases from $Q=2\times {10}^3$. (a) normalized viscous dissipation; (b) normalized ohmic dissipation.

Figure 8

Reynolds number data for both TMC and VMC. ${\text{Re}}_z$ is shown for TMC and Re is shown for VMC. (a) Reynolds number ${\text{Re}}_z$ vs ${\text{Ra/Ra}}_c$; (b) rescaled Reynolds number, ${\text{Re}}_z{Q}^{-1/4}$, vs ${\text{Ra/Ra}}_c$.

Figure 9

Time-averaged velocity ratios: (a) $Q=2\times {10}^3$, (b) $Q=2\times {10}^5$, (c) $Q=2\times {10}^6$. All quantities are volumetric rms values and the error bars show the standard deviation. Note that the scale of the horizontal axis is different for each figure.

Figure 10

Time and horizontally averaged $x$ component of the velocity field and magnetic field: [(a), (d)] $Q=2\times {10}^3$; [(b), (e)] $Q=2\times {10}^5$; [(c), (f)] $Q=2\times {10}^6$. Darker lines correspond to higher Rayleigh numbers.

Figure 11

Scalings of the rms values for the horizontally averaged quantities: (a) ${\overline{u}}_{\text{rms}}$; (b) ${\overline{u}}_{\text{rms}}{Q}^{1/2}$; (c) ${\overline{b}}_{x,\text{rms}}$; (d) ${\overline{b}}_{x,\text{rms}}{Q}^{1/2}$.

Figure 12

Volumetric renderings of the $y$ component of the velocity, $v$, illustrating the development of the zonal flow for $Q=2\times {10}^6$ and increasing Ra: (a) $\text{Ra}=1.04\times {10}^7$; (b) $\text{Ra}=1.1\times {10}^7$; (c) $\text{Ra}=1.3\times {10}^7$. The orientation for all visualizations is shown in panel (a).

Figure 13

Structure of the zonal flow from representative cases. [(a), (d)] $Q=2\times {10}^3, \text{Ra}=1\times {10}^5$; [(b), (e)] $Q=2\times {10}^5, \text{Ra}=8\times {10}^6$; [(c), (f)] $Q=2\times {10}^6, \text{Ra}=7\times {10}^7$. The top row shows instantaneous views of the zonal velocity, ${\overline{v}}^y$, and the bottom row shows instantaneous views of the $y$ component of the velocity, $v$, in the horizontal plane at depth $z\approx 0.5$.

Figure 14

Scaling behavior of the rms value for the zonal flow: (a) raw data and (b) rescaled data.

Figure 15

Behavior of various measurements of flow speed with increasing horizontal dimension of the simulation domain for $Q=2\times {10}^5$ and $\text{Ra}=6\times {10}^6$. The horizontal length of the simulation domain is represented by the number of unstable horizontal wavelengths, $n$; the aspect ratio for the cases shown is $\mathrm{\Gamma }=(3.31,4.30,4.97,5.63,6.62)$. Dotted lines show the corresponding least squares fits to the data.

Figure 16

Times series data illustrating the temporal behavior of relaxation oscillations for $Q=2\times {10}^5$ and $\text{Ra}=1.5\times {10}^7$: (a) Reynolds number, $\text{Re}\left(t\right)$; (b) Nusselt number, $\text{Nu}\left(t\right)$.

Figure 17

Instantaneous plots of the horizontal velocity components illustrating relaxation oscillations for $Q=2\times {10}^5$ and $\text{Ra}=1.5\times {10}^7$. The two plots on the left [(a), (c)] are taken at time $t=5.200$, just prior to the peak in Reynolds number. The two plots on the right [(b), (d)] are taken at time $t=5.228$, just after the Reynolds number reaches a peak. [(a), (b)] $y$-velocity component; [(c), (d)] $x$-velocity component.