3.1 Spatial structure of the flow

The flow structure for several typical cases is illustrated in figure 1. Flows without a magnetic field (see figure 1a,b) are turbulent. The spatial structure in the horizontal cross-section (middle column) and the correlation between the velocity signals measured along a vertical line indicate the presence of a large-scale circulation (LSC) with upward or downward flow zones.

The magnetic field drastically changes the flow. We observe the suppression of small-scale velocity gradients and the formation of large-scale anisotropic structures dominating the flow (see the left and middle columns of figure 1c–f). This is associated with a reduction of amplitude and frequency of velocity fluctuations (see the right column of figure 1c–f) and, as we will discuss in § 3.2, an increase of $Nu$ and $Re$. Similar flow transformations were detected in the recent measurements of Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020) and in the simulations of Liu et al. (Reference Liu, Krasnov and Schumacher2018), Yan et al. (Reference Yan, Calkins, Maffei, Julien, Tobias and Marti2019) for lower Rayleigh numbers.

The strength of the magnetic field effect determining the degree of the flow transformation correlates with the ratio between $Ra$ and $Ra_{c}$. At $Ra\gg Ra_{c}$ (see figure 1d,f), the large-scale velocity structures are anisotropic (elongated in the direction of the magnetic field) but susceptible to three-dimensional (3-D) instabilities and evolve on the background of 3-D small-scale velocity fluctuations. At $Ra$ larger, but not much larger than $Ra_{c}$ (see figure 1c,e), the anisotropy is much stronger and 3-D fluctuations are much weaker.

In the latter case, nearly Q2D upward and downward streams occupy the flow domain. Interestingly, these flows, in particular the flow at $Ra/Ra_{c}\approx 5$ shown in figure 1(e), are principally different from the other recently identified Q2D magneto-convection regimes: the wall modes of Liu et al. (Reference Liu, Krasnov and Schumacher2018) and the cellular and columnar regimes found in an infinite horizontal layer by Yan et al. (Reference Yan, Calkins, Maffei, Julien, Tobias and Marti2019). The former is not surprising, since the simulations of Liu et al. (Reference Liu, Krasnov and Schumacher2018) find the wall-mode regimes at $Ra/Ra_{c}\sim 1$ or ${<}1$, but not at higher $Ra/Ra_{c}$. For example, the wall modes were not found at $Ra/Ra_{c}=4.05$.

The discrepancy with the results of Yan et al. (Reference Yan, Calkins, Maffei, Julien, Tobias and Marti2019) obtained for an infinite horizontal layer appears more significant. Unlike the ‘cells’ reminiscent of the linear instability modes or the ‘columns’ found in their simulations, the dominant feature of the flow in figure 1(e) can be described as a system of ascending and descending planar jets often originating at the sidewall and extending into the bulk of the flow. We should mention that the horizontal velocity is strong in this flow ($\langle u_{r}^{2}+u_{\unicode[STIX]{x1D703}}^{2}\rangle _{V,t}/\langle u_{z}^{2}\rangle _{V,t}\approx 0.44$ in comparison to ${\approx}0.27$ for the case in figure 1f). The regime is reminiscent of Q2D extended vortex sheets often found in magnetohydrodynamic (MHD) turbulent flows with strong magnetic field effect (see e.g. Zikanov & Thess Reference Zikanov and Thess1998). The fact that this regime rather than the cellular or columnar regimes of Yan et al. (Reference Yan, Calkins, Maffei, Julien, Tobias and Marti2019) is realized in our flow can be attributed to smaller $Pr$ ($Pr=1$ was used by Yan et al. (Reference Yan, Calkins, Maffei, Julien, Tobias and Marti2019)) or, much more likely, the presence of sidewalls.

The wall-mode regime is found in our simulations at $Ra/Ra_{c}\rightarrow 1$. As an illustration, figure 2 shows the instantaneous distributions of convective flux $u_{z}T$ in flows with $Ra=10^{7}$ (also used by Liu et al. (Reference Liu, Krasnov and Schumacher2018)) and several values of $Ha$. The structure of the wall modes is qualitatively similar to the structure observed by Liu et al. (Reference Liu, Krasnov and Schumacher2018) (see figure 2c). Significant flow and heat transfer are limited to tongue-like zones attached to the sidewall. There are noticeable differences, though. The entire wall-mode structure rotates clockwise or anticlockwise in our system. The effect of rotation is illustrated by the almost axisymmetric distribution of the time-averaged heat flux shown in figure 2(d). The travelling waves in the azimuthal direction were not observed in the linear stability analysis (Houchens et al. Reference Houchens, Witkowski and Walker2002). Another difference is the transformation of wall modes at increasing $Ra/Ra_{c}$. As in the simulations of Liu et al. (Reference Liu, Krasnov and Schumacher2018), the wall modes extend further from the wall. The new feature that can be plausibly attributed to rotation is that the wall-mode zones curve (see figure 2b) and form Q2D vortices (see figure 2a). We leave a detailed analysis of the properties and physical mechanisms of wall modes to future studies and only mention here that observed behaviour is qualitatively similar to that in RBC with rotation (Zhong et al. Reference Zhong, Ecke and Steinberg1991; Ecke et al. Reference Ecke, Zhong and Knobloch1992; Zhang et al. Reference Zhang, van Gils, Horn, Wedi, Zwirner, Ahlers, Ecke, Weiss, Bodenschatz and Shishkina2019).

Figure 2.

Snapshots of the convective flux $u_{z}T$ in the horizontal mid-plane of the cylinder at $Ra=10^{7}$ with: $Ra/Ra_{c}=5.0$, $Ha=450$ (a); $Ra/Ra_{c}=2.4$, $Ha=650$ (b); and $Ra/Ra_{c}=1.4$, $Ha=850$ (c). Time-averaged convective flux at $Ra=10^{7}$ and $Ha=850$ is shown in (d).

3.2 Global transport properties

All computed values of $Nu$ and $Re$ are listed in the supplementary material. They are also summarized and compared with the available experimental and numerical data in figure 3. The qualitative agreement between the numerical and experimental data is good. In particular, the increase of the slope of $Nu(Ra)$ and $Re(Ra)$ curves in flows with the magnetic field, which we discuss in detail below, is consistent between the simulations and the experiments. At the same time, the quantitative agreement is less convincing, with the computed values being consistently higher than in the experiments, especially in the experiment of Cioni et al. (Reference Cioni, Chaumat and Sommeria2000). This situation is typical for thermal convection in low-$Pr$ fluids and can be attributed to several known difficulties of the experimental procedure: (1) the ideal boundary conditions of constant temperature at the top and bottom plates cannot be accurately maintained; (2) the temperatures at the top and bottom boundaries are not measured directly, which usually leads to a slight overestimation of the Rayleigh number; (3) complete avoidance of heat losses through the sidewall is impossible; and (4) physical properties within a system are not constant, which results in inaccurate estimates of the Prandtl, Hartmann and Rayleigh numbers.

Figure 3.

Nusselt number $Nu$ versus $Ra$ (a) and Reynolds number $Re$ versus $Ra$ (b) with and without a magnetic field. Experimental and numerical data for $\unicode[STIX]{x1D6E4}=1$ are shown for comparison. Lines with slope ${\sim}Ra^{\unicode[STIX]{x1D6FD}}$ are plotted for guidance.

Estimates of the scaling behaviour of $Nu$ and $Re$ based on the data of our simulations are presented in table 1. Some of the respective lines are also shown in figure 3. The common regression method is applied to the computed time-averaged values to determine the exponents $\unicode[STIX]{x1D6FD}_{Nu}$ and $\unicode[STIX]{x1D6FD}_{Re}$ and their standard errors, and the constants $\unicode[STIX]{x1D6FC}_{Nu}$ and $\unicode[STIX]{x1D6FC}_{Re}$ in the approximations $Nu\approx \unicode[STIX]{x1D6FC}_{Nu}Ra^{\unicode[STIX]{x1D6FD}_{Nu}}$ and $Re\approx \unicode[STIX]{x1D6FC}_{Re}Ra^{\unicode[STIX]{x1D6FD}_{Re}}$.

For flows with $Ha=0$, some disagreement with the data of Cioni et al. (Reference Cioni, Chaumat and Sommeria2000) is found (see figure 3a). At the same time, the computed value of $\unicode[STIX]{x1D6FD}_{Nu}$ is in good agreement with the value $0.29\pm 0.01$ for $Pr=0.025$ found in the experiments with the largest range $2\times 10^{5}\leqslant Ra\leqslant 8\times 10^{10}$ by Glazier et al. (Reference Glazier, Segawa, Naert and Sano1999) performed at $\unicode[STIX]{x1D6E4}=0.5,1,2$. The exponent $\unicode[STIX]{x1D6FD}_{Re}$ is consistent with $0.46\pm 0.02$ for $Pr=0.025$ and $0.42\pm 0.03$ for $Pr=0.029$ found in measurements by Takeshita et al. (Reference Takeshita, Segawa, Glazier and Sano1996) and Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2019), respectively, and with $0.45\pm 0.04$ for $Pr=0.021$ in numerical simulations by Scheel & Schumacher (Reference Scheel and Schumacher2017).

An imposed magnetic field reduces the rate of heat transfer and kinetic energy in the entire studied range of $Ra$. At the same time, growth of $Nu$ and $Re$ becomes faster. In particular, the slope $\unicode[STIX]{x1D6FD}_{Nu}$ increases from $0.298$ at $Ha=0$ to approximately $0.574$ at $Ha=1400$. A similar increase is observed for $\unicode[STIX]{x1D6FD}_{Re}$ (see table 1 and figure 3). The results are consistent with the data of Cioni et al. (Reference Cioni, Chaumat and Sommeria2000), the only experiment in our range of interest of $Ra$ and $Ha$.

Figure 4.

Normalized Nusselt number $\widetilde{Nu}$ versus $Ha/Ra^{1/2}$ (a) and Reynolds number $\widetilde{Re}$ versus $Ha/Ra^{1/2}$ (b). Experimental data $\unicode[STIX]{x1D6E4}=1$ are shown for comparison. Lines with slope ${\sim}(Ha/Ra^{1/2})^{\unicode[STIX]{x1D6FD}}$ are plotted for guidance.

Table 1.

Scaling coefficients in the approximations $Nu\approx \unicode[STIX]{x1D6FC}_{Nu}Ra^{\unicode[STIX]{x1D6FD}_{Nu}}$ and $Re\approx \unicode[STIX]{x1D6FC}_{Re}Ra^{\unicode[STIX]{x1D6FD}_{Re}}$ based on the data of the present simulations (see figure 3).

The Nusselt number $\widetilde{Nu}=(Nu-1)/(Nu_{Ha=0}-1)$ and the Reynolds number $\widetilde{Re}=Re/Re_{Ha=0}$ normalized by their reference values at $Ha=0$ are shown in figure 4. The universal scaling behaviour of $\widetilde{Nu}=1/[1+\unicode[STIX]{x1D6FC}_{\widetilde{Nu}}(Ha/Ha_{c})^{\unicode[STIX]{x1D6FD}_{\widetilde{Nu}}}]$ and $\widetilde{Re}=1/[1+\unicode[STIX]{x1D6FC}_{\widetilde{Re}}(Ha/\sqrt{Ha_{c}})^{\unicode[STIX]{x1D6FD}_{\widetilde{Re}}}]$ was proposed by Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020). To verify this hypothesis, we plot $1/\widetilde{Nu}-1$ and $1/\widetilde{Re}-1$ versus $Ha/Ra^{1/2}$ in logarithmic scales. The computed data provide $\unicode[STIX]{x1D6FD}_{\widetilde{Nu}}=1.305\pm 0.049$ and $\unicode[STIX]{x1D6FD}_{\widetilde{Re}}=0.549\pm 0.034$.

The qualitative agreement between the simulations and the experiments, especially with the experiments by Cioni et al. (Reference Cioni, Chaumat and Sommeria2000), is observed for $\widetilde{Nu}$ at $Ra>Ra_{c}$ ($Ha/Ra^{1/2}\sim 0.1$) and $Ra\gg Ra_{c}$ ($Ha/Ra^{1/2}<0.1$) in figure 4(a). The results indicate that a universal power is approached for flows at high Rayleigh numbers. At the same time, the standard error is higher and the coefficient of determination ($R^{2}$) is lower than for the scaling $Nu\sim Ra^{\unicode[STIX]{x1D6FD}_{Nu}}$ presented above ($R^{2}=0.943$ in comparison to $R^{2}=0.998$ for $Nu\sim Ra^{\unicode[STIX]{x1D6FD}_{Nu}}$ at $Ha=1400$). A similar situation is observed for the normalized Reynolds numbers in figure 4(b).

We expect an increase of the exponents $\unicode[STIX]{x1D6FD}_{\widetilde{Nu}}$ and $\unicode[STIX]{x1D6FD}_{\widetilde{Re}}$ in the wall-mode regime at $Ra\sim Ra_{c}$ ($Ha/Ra^{1/2}\sim 0.25$). However, the measurements of Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020) for $1/\widetilde{Re}-1$ show a line with higher slope ${\sim}(Ha/Ha_{c})^{1.73\pm 0.05}$ for a wide range of $Ra$ and $Ha$. The discrepancy with this result and the presence of even stronger deviations in experimental data for moderate $Ra$ can be attributed to the limitations of the ultrasound Doppler velocimetry used to probe the flow field discussed by Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020).

Estimates of the thickness of the thermal boundary layer are presented in the supplementary material. The common slope method used in the experiments (Takeshita et al. Reference Takeshita, Segawa, Glazier and Sano1996) and simulations (Liu et al. Reference Liu, Krasnov and Schumacher2018) is applied, i.e. the intersection point of the tangent of the time-averaged mean temperature profile near the sidewall and the horizontal line drawn through the mean value is taken. We only mention here that the results of the present simulations deviate by no more than $2\,\%$ from the theoretical value $\unicode[STIX]{x1D6FF}_{T}\approx 1/(2Nu)$ of Grossmann & Lohse (Reference Grossmann and Lohse2001), even in the presence of a magnetic field.