Rayleigh-Bénard convection with uniform vertical magnetic field

We present the results of direct numerical simulations of Rayleigh-Bénard convection in the presence of a uniform vertical magnetic field near instability onset. We have done simulations in boxes with square as well as rectangular cross sections in the horizontal plane. We have considered the horizontal aspect ratio $\eta =L_y/L_x=1$ and 2. The onset of the primary and secondary instabilities are strongly suppressed in the presence of the vertical magnetic field for $\eta =1$. The Nusselt number $\mathrm{Nu}$ scales with the Rayleigh number $\mathrm{Ra}$ close to the primary instability as ${\left[\{\text{Ra}-{\text{Ra}}_c\left(Q\right)\}/{\text{Ra}}_c\left(Q\right)\right]}^{0.91}$, where ${\text{Ra}}_c\left(Q\right)$ is the threshold for onset of stationary convection at a given value of the Chandrasekhar number $Q$. $\mathrm{Nu}$ also scales with $\text{Ra}/Q$ as ${\left(\text{Ra}/Q\right)}^{\mu }$. The exponent $\mu$ varies in the range $0.39\le \mu \le 0.57$ for $\text{Ra}/Q\ge 25$. The primary instability is stationary as predicted by Chandrasekhar. The secondary instability is temporally periodic for $\mathrm{Pr}=0.1$ but quasiperiodic for $\mathrm{Pr}=0.025$ for moderate values of $Q$. Convective patterns for higher values of $\mathrm{Ra}$ consist of periodic, quasiperiodic, and chaotic wavy rolls above the onset of the secondary instability for $\eta =1$. In addition, stationary as well as time-dependent cross rolls are observed, as $\mathrm{Ra}$ is further raised. The ratio $r_o/\mathrm{Pr}$ is independent of $Q$ for smaller values of $Q$. The delay in the onset of the oscillatory instability is significantly reduced in a simulation box with $\eta =2$. We also observe inclined stationary rolls for smaller values of $Q$ for $\eta =2$.

Figure 1

Variation of (a) kinetic energy $K$ and (b) convective entropy $\Phi$ with $\epsilon =\left[\text{Ra}-{\text{Ra}}_c\left(Q\right)\right]/{\text{Ra}}_c\left(Q\right)$ for $\mathrm{Pr}=0.1$ in a square box [$k_x=k_y=k_c\left(Q\right)(\eta =1$)] shown in different colors for different values of $Q$, as computed from DNS. Colored (gray) dots are for different values of $Q$. The dashed lines show the variation of (a) $K$ and (b) $\Phi$ with $\epsilon$ for the stationary convection.

Figure 2

Plot of convective heat flux $(\text{Nu}-1)$ as a function of $\epsilon =\left[\text{Ra}-{\text{Ra}}_c\left(Q\right)\right]/{\text{Ra}}_c\left(Q\right)$ for $\mathrm{Pr}=0.1$ and $k_x=k_y=k_c\left(Q\right)(\eta =1)$ shown for four different $Q$ values, as computed from DNS, for free slip boundaries (dotted lines with points). They are compared with the numerical results (Clever and Busse [13]) for no slip boundaries (solid lines). The dashed line is parallel to the linear region for stationary rolls at onset for all four $Q$ values. $(\mathrm{Nu}-1)$ scales as ${\epsilon }^{0.91}$ for stationary rolls near onset.

Figure 3

Variation of Nusselt number $\mathrm{Nu}$ with $\text{Ra}/Q$ for $\mathrm{Pr}=0.1$ (square box) for three different values of $Q$ (for $\text{Ra}/Q>25$), as computed from DNS. The dashed lines through blue, green and red data points are the best linear fits for $Q=20,50$, and 670, respectively. $\mathrm{Nu}$ scales as ${\left(\text{Ra}/Q\right)}^{\mu }$, where $\mu$ values are $0.39$, $0.48$, and $0.57$ for $Q=20$, 50, and 670, respectively.

Figure 4

Contour plots of the convective temperature field at $z=0.5$ close to the onset of convection ($r=1.05$) in a simulation box with a square cross section ($\eta =1$) for $\mathrm{Pr}=0.025$ and $Q=50$ [$k_c(Q=50)=3.270$] showing temporally quasiperiodic wavy rolls (QWR). Straight rolls are observed whenever the Fourier mode $W_{111}$ becomes zero.

Figure 5

Properties of QWR corresponding to parameters given in Fig. 4. Temporal variations of the two largest Fourier modes: (a) $W_{011}$ and (b) $W_{111}$, (c) power spectral density (PSD) of the Fourier mode $W_{111}$, and (d) phase portrait in the $W_{111}\text{−}{W}_{011}$ plane show the quasiperiodic nature of the wavy patterns.

Figure 6

Contour plots of the temperature field at midplane ($z=0.5$) in a simulation box with square cross section [$L_x=L_y=2\pi /k_c\left(Q\right)$] for $\mathrm{Pr}=0.1$, $r=1.05$. Periodic wavy rolls (WR) for $Q=10$ [$k_c(Q=10)=2.589$] with time period $\tau =0.262$ (in viscous time units) at four instants: (a) $t=0$, (b) $t=\tau /4$, (b) $t=\tau /2$, and (b) $t=3\tau /4$.

Figure 7

Properties of periodic wavy rolls (WR) in a simulation box with square cross section corresponding to the parameters given in Fig. 6. The temporal variations of the two largest Fourier modes (a) $W_{011}$ and (b) $W_{111}$, (c) the power spectral density (PSD) of the mode $W_{111}$, and (d) the phase portrait in the $W_{111}\text{−}{W}_{011}$ plane show a set of periodic wavy rolls.

Figure 8

Regions of Ra-$Q$ plane showing various patterns for $\mathrm{Pr}=0.1$ and $k_x=k_y=k_c\left(Q\right)$. The onsets of primary as well as secondary instabilities are delayed by magnetic field. Convection at onset is stationary (Rolls) and becomes oscillatory (WR) after $\mathrm{Ra}$ is increased above ${\text{Ra}}_o(Q,\text{Pr})$, the threshold for secondary (oscillatory) instability. Above the region of wavy rolls, we observe time-dependent cross rolls. The dashed lines show polynomial fits for the DNS data (points).

Figure 9

Scaling of oscillatory threshold with $Q$. (a) The combination $r_o(Q,\text{Pr})/\text{Pr}$ is independent of $Q$ for smaller values of $Q$. Circles, squares, and triangles are data points for $\mathrm{Pr}=0.025$, $0.1$, and $0.7$, respectively. (b) The quantity $[{\text{Ra}}_o\left(Q\right)-{\text{Ra}}_o(Q=0)]/\text{Pr}$ varies linearly with $Q$ for different values of $\mathrm{Pr}$.

Figure 10

Midplane ($z=1/2$) contour plots of the convective temperature field for $\mathrm{Pr}=0.025$ and $r=1.1$ in a simulation box of rectangular cross section ($\eta =2$). A quasiperiodic competition between straight and inclined wavy rolls for $Q=60$ [$k_c(Q=60)=3.376$] shown for three different instants. (d) Temporal variations of the Fourier modes $W_{101}$ [blue (black) curve] and $W_{111}$ [red (gray) curve]. The time instants marked by $t_a$, $t_b$, and $t_c$ correspond to the patterns (a), (b), and (c), respectively.

Figure 11

Contour plots of the convective temperature field (at $z=0.5$) in a rectangular simulation box ($\eta =2$) for $\mathrm{Pr}=0.1$ showing (a) stationary straight rolls for $r=1.05$ and $Q=100$ [$k_c(Q=100)=3.702$], and (b) stationary inclined rolls (IR) for $r=1.2$ and $Q=10$ [$k_c(Q=10)=2.589$].