3.1 Visualisation of the Rayleigh–Bénard flow in the presence of a vertical magnetic field

Figure 2.

Three-dimensional snapshots of the temperature field under different Hartmann numbers at $Ra=10^{9}$ ; (a–d) refers to the cases $Ha=10,50,200$ and 500 respectively. Temperature scale is shown by the colour bar. Note how the plume morphology changes as $Ha$ increases.

Figure 3.

Two-dimensional cuts of the instantaneous velocity (arrows) and temperature (colour) fields at vertical mid-plane $(y=0.5)$ of the flow field at $Ra=10^{9}$ for different Hartmann numbers; (a–d) refers to the cases $Ha=10,50,200$ and 500 respectively. The arrows indicate the flow velocity, with their length representing the velocity magnitude. The colour bar indicates the temperature scale.

Figure 2 shows three-dimensional snapshots of the temperature field, with the value given in the scale bar. For the cases of $Ha=10$ and $Ha=50$ , we can clearly see the large-scale circulation (LSC) flowing in the diagonal direction (Kadanoff Reference Kadanoff2001; Xi, Lam & Xia Reference Xi, Lam and Xia2004). Regarding figures 2(a) and 2(b) it is difficult to distinguish them from the cases when the magnetic field is absent ( $Ha=0$ ). This is easy to understand since magnetic field suppresses mainly the small-scale flows; the behaviour of the large-scale convective flow does not change much. In these cases, the plumes are flowing in the direction of the LSC. When the magnetic field becomes sufficiently large, e.g. the case with $Ha=200$ , the plume morphology and behaviour start to change. We can see from figure 2(c) that the depth of plume penetration from both the top and bottom plates increases dramatically, which means that the plumes become much larger and more coherent. It is also seen that some of the plumes extend the entire height of the cell. The plumes become more organised, although they do not follow the LSC entirely. When the magnetic field increases further, the picture becomes different again as seen in figure 2(d). First of all, it is clear that the LSC is completely suppressed. There exist a large number of columnar plumes that extend vertically between the top and the bottom plates. The morphology of cold and hot pillars looks very different from the classical mushroom-like plumes. They are narrow and have a clear separation. To understand the columnar plumes, we look at the effect of the Lorentz term in (2.1). The vertical magnetic field in the Lorentz term suppresses the horizontal motion of the fluid only. Therefore, in the case of extremely large $Ha$ , the fluid would move only in the vertical direction, which gives rise to the formation of columnar plumes. This phenomenon was observed in RRB convection (Stevens, Clercx & Lohse Reference Stevens, Clercx and Lohse2013) and in severely confined RB convection (Chong & Xia Reference Chong and Xia2016) as well.

To examine the flow behaviour in more detail, we show in figure 3 vertical cuts of the instantaneous velocity and temperature fields. Although not much difference is observed between the $Ha=10$ and $Ha=50$ cases, a close examination of the length of the arrows indicates that the magnitude of the flow is decreased with the increasing of the Lorentz force. Also, the large-scale circulation becomes less robust. As Hartmann number increases further to $Ha=200$ , it becomes difficult to ascertain whether the LSC still exists, as the plumes are seen to move mostly in the vertical direction. One may conclude that the LSC has already broken down. In the extreme case of $Ha=500$ , only columnar plumes that extend between the top and bottom plates are observed, and it is clear that the LSC does not exist any more at this very large $Ha$ value, as already seen from the 3-D plot. From the above examples of the instantaneous temperature and velocity fields, we obtain a sense of the role of the Lorentz force, which acts to suppress the flow field. To quantify the effect of the Lorentz force in RB convection, in the next section we look at the effect of the magnetic field on the time-averaged global quantities, i.e. the Reynolds number and the Nusselt number.

3.2 Reynolds number and Nusselt number under different $Ha$

In this section we examine the properties of the two global response parameters, Nusselt number and Reynolds number. The Nusselt number is calculated using heat flux across the horizontal plane, which is given by $Nu_{h}=\langle (Ra\,Pr)^{1/2}u_{z}T-\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}z\rangle _{x,y,t}$ , where $\langle \cdot \rangle _{x,y,t}$ is the average over a horizontal plane and over time. $Nu$ is then obtained by averaging $Nu_{h}$ over all horizontal planes. In addition, $Nu$ can also be caculated from the exact relation based on global thermal dissipation, which is given by $Nu_{T}=\langle \unicode[STIX]{x1D716}_{T}\rangle (Ra\,Pr)^{1/2}$ . Because of the additional Lorentz term in (2.1), the additional Joule heating effect will contribute to dissipation. $Nu$ calculated from global viscous dissipation and Joule heating is given by $Nu_{v+b}=1+\langle \unicode[STIX]{x1D716}_{v}\rangle (Ra\,Pr)^{1/2}+\langle \unicode[STIX]{x1D716}_{b}\rangle Ha^{2}(Ra\,Pr)^{1/2}$ . Here, $\langle \cdot \rangle$ denotes time- and volume-average, while the thermal dissipation, viscous dissipation and dissipation from Joule heating are $\unicode[STIX]{x1D716}_{T}=(Ra\,Pr)^{-1/2}\sum _{i}(\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}x_{i})^{2}$ , $\unicode[STIX]{x1D716}_{v}=(Ra/Pr)^{-1/2}\sum _{i}\sum _{j}(1/2)(\unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x_{j}+\unicode[STIX]{x2202}u_{j}/\unicode[STIX]{x2202}x_{i})^{2}$ and $\unicode[STIX]{x1D716}_{b}=(Ra/Pr)^{-1/2}\sum _{i}J_{i}^{2}$ respectively. The error for the Nusselt number is estimated by the standard deviation between $Nu$ , $Nu_{T}$ and $Nu_{v+b}$ . The Reynolds number is defined by $Re=\sqrt{\langle \boldsymbol{u}^{2}\rangle (Ra/Pr)}$ . $Re$ so-defined provides a measure of the overall strength of the flow field.

Figure 4.

Normalised Reynolds number $Re/Re_{0}$ versus $Ha$ , where $Re_{0}$ is the value obtained at $Ha=0$ .

Figure 4 plots $Re$ (normalised by its zero-field value) versus $Ha$ for four cases of $Ra$ . In all the cases, the value of $Re$ remains almost constant for $Ha\leqslant 10$ and after that $Re$ decreases monotonically. This is expected since under small $Ha$ , the effect of Lorentz force is too small to alter the flow dynamics of the system. As $Ha$ increases, the Lorentz effect becomes sufficiently strong to suppress the fluid flow. We also see that for the same value of $Ha$ , the suppression is more severe for smaller values of $Ra$ .

Figure 5.

Nusselt number behaviour and the optimal heat transport enhancement. (a) Normalized $Nu$ versus $Ha$ , where $Nu_{0}$ is the value obtained at $Ha=0$ . (b) A magnified view of (a). (c) Normalised $Nu$ versus $Ha/Ha_{opt}$ , where $Ha_{opt}$ is where the maximum heat transport occurs. The green dashed line indicates $Ha=0.2Ha_{opt}$ for the onset of heat transfer enhancement and the black dashed line indicates $Ha=Ha_{opt}$ . (d) $Ha_{opt}$ versus $Ra$ , where a power law fit produces $Ha_{opt}=0.018Ha^{0.45\pm 0.06}$ .

Although the flow strength is suppressed as $Ha$ increases, the heat transport behaves very differently. We plot the normalised $Nu$ against $Ha$ in figure 5(a) and an enlarged portion in figure 5(b), for the same four $Ra$ cases as in the $Re$ plot. It is seen that the detailed behaviour for each $Ra$ may be slightly different, but for all values of $Ra$ , the overall trend is the same, i.e. $Nu$ remains unchanged at the beginning and starts to increase as $Ha$ is further increased, and it reaches a maximum value at a certain $Ha$ (denoted as $Ha_{opt}$ ). After attaining maximum enhancement, $Nu$ starts to drop sharply. The peak heights are different as well, suggesting that the heat transport enhancement is more prominent for larger $Ra$ . The onset $Ha$ value for enhancement is also seen to be different for different $Ra$ . However, if we normalise $Ha$ by the $Ra$ -dependent $Ha_{opt}$ , then all the curves for different $Ra$ collapse quite well, which is shown in figure 5(c). It is also seen from the figure that the onset Hartmann number for heat transfer enhancement is the same ( $Ha=0.2Ha_{opt}$ ) for all values of $Ra$ , suggesting that $Ha_{opt}$ is a characteristic quantity for transport enhancement. Moreover, we find that the $Ra$ -dependence of $Ha_{opt}$ may be described by a power law, $Ha_{opt}=0.018Ra^{0.45\pm 0.06}$ , as shown in figure 5(d).

Although both are global quantities, the Nusselt and the Reynolds numbers behave very differently. For example, $Re$ changes monotonically and it starts to decrease at a single transition point of $Ha\approx 10$ for all $Ra$ cases. For $Nu$ , on the other hand, the $Ha$ values for the onset of enhancement and for the optimal enhancement are all $Ra$ -dependent. This suggests a decoupling between momentum and heat transport in the system, similar to that seen in the other three systems, i.e. CRB, RRB and DDC. For example, in the $Ra=10^{10}$ and $Ha=500$ case, $Re$ is already suppressed by ${\sim}50\,\%$ compared to its $Ha=0$ value, while the maximum enhancement for $Nu$ occurs for this case. To understand the heat transport behaviour, such as the existence of the optimal $Nu$ enhancement, we next examine the properties of the thermal and momentum boundary layers and the interplay between the stabilising and destabilising forces in the quasistatic magnetoconvection system.

Figure 6.

(a) Mean temperature profile. The dashed lines define the thickness of the thermal boundary $\unicode[STIX]{x1D6FF}_{T}$ . (b) Stress profile; the main figure is for height $0\leqslant z\leqslant 0.5$ , and the inset shows data for both $0\leqslant z\leqslant 0.5$ (red circles) and $0.5\leqslant z\leqslant 1$ (blue triangles). The dashed vertical line indicates the location of stress boundary layer $\unicode[STIX]{x1D6FF}_{P}$ . Both (a) and (b) are for the case of $Ra=10^{10}$ and $Ha=10$ .

Figure 7.

Thermal $\unicode[STIX]{x1D6FF}_{T}$ and momentum (stress) $\unicode[STIX]{x1D6FF}_{P}$ boundary layer thicknesses as a function of the Hartmann number $Ha$ ; (a–d) refers to the cases $Ra=10^{7},10^{8},10^{9},10^{10}$ respectively. The vertical dashed lines denote the positions of $Ha_{opt}$ where the estimated maximum enhancement occurs.

3.3 The role of the thermal and stress boundary layers in optimal heat transport enhancement

Following the idea proposed by Chong et al. (Reference Chong, Yang, Huang, Zhong, Stevens, Verzicco, Lohse and Xia2017), we define the thickness of the stress (momentum) boundary layer as the peak position of the profile of the quantity $(\unicode[STIX]{x2202}_{x}u)^{2}+(\unicode[STIX]{x2202}_{y}v)^{2}+(\unicode[STIX]{x2202}_{z}w)^{2}$ . This quantity is the square of the normal gradient of velocity summing over all components, which measures the overall magnitude of the normal stress (hereafter simply referred to as the stress); in Chong et al. (Reference Chong, Yang, Huang, Zhong, Stevens, Verzicco, Lohse and Xia2017) it is called the momentum boundary layer. Note that this definition is different from the traditionally defined velocity boundary layer, which will be discussed in § 3.5 and which will be called the viscous boundary layer. For the temperature boundary layer, we use the so-called slope method based on the mean temperature profile (Tilgner, Belmonte & Libchaber Reference Tilgner, Belmonte and Libchaber1993; Lui & Xia Reference Lui and Xia1998). For the region very close to the wall, the temperature profiles exhibit linear behaviour. By fitting the data points lying within the linear region $0<z<z_{T,linear}$ and finding the intersection point between this linear line and the bulk temperature, one can evaluate the boundary layer thickness. Here $z_{T,linear}$ is given values of $0.01,0.005,0.002$ and 0.001 for $Ra=10^{7},10^{8},10^{9}$ and $10^{10}$ respectively. The error of $\unicode[STIX]{x1D6FF}_{T}$ is determined by the difference between this result and that when one more data point is included in the fiting. Note that we have also tried using the maximum location of the temperature root-mean-square profile to define the thickness, which gives the same conclusion.

Figure 8.

(a) The thermal boundary layer $\unicode[STIX]{x1D6FF}_{T}$ normalised by the value at $Ha=0$ versus $Ha/Ha_{opt}$ , for different values of $Ra$ . (b) A magnified view of (a) to show more clearly the decrease of $\unicode[STIX]{x1D6FF}_{T}$ , with the position of the minimum of thermal boundary layer corresponding to the maximum $Nu$ shown in figure 5(c). The green dashed line indicates $Ha=0.2Ha_{opt}$ for the onset of heat transfer enhancement and the black dashed line indicates $Ha=Ha_{opt}$ .

Figure 6 plots the profiles of the temperature and of the stress for the case of $Ra=10^{10}$ and $Ha=10$ (the inset shows the peak of stress profile more clearly in log scale for the horizontal axis). From the profile we obtain respectively the thickness $\unicode[STIX]{x1D6FF}_{T}$ of the thermal boundary layer and the thickness $\unicode[STIX]{x1D6FF}_{P}$ of the momentum boundary layer, as indicated by the blue dashed vertical lines. Note that in the inset figure plotted in semi-log scale, data points with $0\leqslant z\leqslant 0.5$ are shown as red circles and those with $0.5\leqslant z\leqslant 1$ are projected under transformation $1-z$ and shown as blue triangles. That the positions of the peaks from near the top and from near the bottom plates coincide with each other indicates a nearly perfect top–bottom symmetry about the middle height of the system. For cases at optimal $Ha$ , the profiles looks qualitatively similar, such that these definitions of boundary layer thickness can still be applied.

Figure 7 plots $\unicode[STIX]{x1D6FF}_{T}$ and $\unicode[STIX]{x1D6FF}_{P}$ as functions of $Ha$ for the four $Ra$ values respectively. An overall feature seen from these plots is that for $Ha$ smaller than a certain value, both $\unicode[STIX]{x1D6FF}_{T}$ and $\unicode[STIX]{x1D6FF}_{P}$ do not change significantly with $Ha$ and that the thermal boundary layer remains nested inside the stress (momentum) layer. When $Ha$ becomes larger than ${\sim}O(10)$ , the momentum layer starts to decrease rapidly, while the thermal boundary experiences an increase. The onset $Ha$ value roughly corresponds to that for the onset of $Nu$ enhancement and $Re$ decrease. With increasing magnetic field, the two boundary layers eventually cross over at certain value of $Ha$ , which itself increases with $Ra$ . This crossover point is very close to the Hartmann number $Ha_{opt}$ for the optimal heat transfer enhancement, which is indicated as dashed vertical lines in the respective figures. A similar feature is also observed for the optimal heat/salt transport enhancement in the systems of CRB, RRB and DDC (Chong et al. Reference Chong, Yang, Huang, Zhong, Stevens, Verzicco, Lohse and Xia2017).

Figure 9.

Correlation of the boundary layer thickness ratio $\unicode[STIX]{x1D6FF}_{T}/\unicode[STIX]{x1D6FF}_{P}$ with various quantities. (a) $\unicode[STIX]{x1D6FF}_{T}/\unicode[STIX]{x1D6FF}_{P}$ versus $Ha/Ha_{opt}$ . (b) Normalised temperature standard deviation versus BL thickness ratio. (c) Normalised $Nu$ versus $\unicode[STIX]{x1D6FF}_{T}/\unicode[STIX]{x1D6FF}_{P}$ . (d) A magnified view of (c).

A well-known property of RB convection is that the thermal BL thickness should decrease when $Nu$ is increased. This feature is not obvious in figure 7, because of the scale of the plots. To show that $Nu$ enhancement is indeed accompanied by a decrease in the thermal boundary layer (BL), we show in figure 8(a) the thermal BL thickness normalised by its value under zero magnetic field versus $Ha/Ha_{opt}$ , and an enlarged portion in figure 8(b). It is evident from the figure that $\unicode[STIX]{x1D6FF}_{T}$ indeed starts to decrease at about the same $Ha$ value ( ${\sim}0.2Ha_{opt}$ ) as $Nu$ starts to increase, reaching a minimum at the optimal Hartmann number and then starts to increase sharply, which corresponds to the sharp $Nu$ decrease. We remark that, as the Reynolds number has decreased sharply under the stabilising Lorentz force, the thermal boundary layer is not thinned by a stronger shear in this case. As shown by Chong et al. (Reference Chong, Huang, Kaczorowski and Xia2015), the thinning of $\unicode[STIX]{x1D6FF}_{T}$ is a result of the increased plume coherence, so that the hot (cold) plumes are able to more efficiently cool (heat) the thermal boundary layer when reaching the opposite plate after traversing the bulk of the fluid within which the stabilising effect takes place. This is another example of what is termed the plume-controlled regime in which the boundary layers are being controlled, via the modification of the bulk flow, rather than controlling (Chong et al. Reference Chong, Huang, Kaczorowski and Xia2015).

To further examine the role played by boundary layer crossing in determining the optimal heat transfer enhancement, we plot in figure 9 the boundary layer ratio $\unicode[STIX]{x1D6FF}_{T}/\unicode[STIX]{x1D6FF}_{P}$ against various quantities. Figure 9(a) plots $\unicode[STIX]{x1D6FF}_{T}/\unicode[STIX]{x1D6FF}_{P}$ versus $Ha/Ha_{opt}$ , which shows that the optimal Hartmann corresponds to the situation when the momentum BL becomes thinner than the thermal BL; and this property holds for all the $Ra$ values explored. Figure 9(b) plots $\unicode[STIX]{x1D70E}_{T}/\unicode[STIX]{x1D70E}_{T,0}$ versus $\unicode[STIX]{x1D6FF}_{T}/\unicode[STIX]{x1D6FF}_{P}$ , where $\unicode[STIX]{x1D70E}_{T}$ is the temperature standard deviation at the edge of the thermal BL and $\unicode[STIX]{x1D70E}_{T,0}$ is its value at $Ha=0$ . This figure shows that when the thicknesses of the two BLs become comparable to each other, temperature fluctuations are enhanced and become maximum at the point of BL crossing. This enables the optimal coupling of the strongest suction with the maximal temperature fluctuations. It is known that thermal plumes are generated by thermal BL instability and detachment. Therefore, the above optimal coupling would enhance thermal plume emission, which in turn enhances heat transfer. Figures 9(c) and 9(d) (an enlarged part of c) plot $\unicode[STIX]{x1D6FF}_{T}/\unicode[STIX]{x1D6FF}_{P}$ versus the normalized $Nu/Nu_{0}$ . Again, the figures show that optimal heat transfer enhancement corresponds to when the two BLs cross each other, or shortly thereafter. These features are exactly the same as those exhibited by the other three systems of stabilising–destabilising turbulent flows, i.e. CRB, RRB and DDC as shown by Chong et al. (Reference Chong, Yang, Huang, Zhong, Stevens, Verzicco, Lohse and Xia2017). This provides very convincing evidence that the present system of quasistatic magnetoconvection belongs to the same universality class as the other three systems.

3.4 Scaling behaviour of the global Nusselt number

In this section we will examine the scaling behaviour of the global heat transport in quasistatic magnetoconvection under a vertical external magnetic field.

Figure 10.

$Nu$ – $Ra$ plot for various values of $Ha$ . (a) $Nu$ versus $Ra$ for the cases of $Ha=0,10,100,200,500$ . The $Ha=0$ ( $Pr=4.38$ ) case is from Kaczorowski & Xia (Reference Kaczorowski and Xia2013) and the data are used here as a baseline. (b) Compensated plot of $Nu$ with $Ra^{-0.3}$ .

Figure 10(a) plots $Nu$ versus $Ra$ for the various values of $Ha$ . For the case of $Ha=0$ , we take the data set from Kaczorowski & Xia (Reference Kaczorowski and Xia2013) and Kaczorowski, Chong & Xia (Reference Kaczorowski, Chong and Xia2014) and use it as the baseline, which gives a scaling $Nu=0.14Ra^{0.30}$ . The range of parameter explored here spans from $Ra=10^{6}$ to $10^{9}$ . Within this range, it is seen that for $Ha=10$ the scaling is essentially the same as the baseline case. For $Ha\geqslant 100$ , below certain value of $Ra$ , the scaling starts to deviate from the baseline. Figure 9(b) plots the $Nu$ data compensated by the scaling exponent from the baseline data. Here, one sees that for $Ha=100$ , the $Nu$ – $Ra$ scaling follows the classical Rayleigh–Bénard scaling for $Ra\geqslant 2\times 10^{7}$ . For $Ra\leqslant 2\times 10^{7}$ , a much steeper scaling of $Nu\sim Ra^{0.42}$ is observed. The $Ha=200$ and $Ha=500$ cases also exhibit a similar transition at $Ra=2\times 10^{8}$ and $2\times 10^{9}$ , respectively. For $Ha=200$ , the steeper scaling is $Nu\sim Ra^{0.50}$ , while for $Ha=500$ , the scaling becomes $Nu\sim Ra^{0.58}$ for lower values of $Ra$ . To better understand the observed scaling transition, we introduce a new parameter called the transition Rayleigh number, $Ra_{T}$ , which, following Chong & Xia (Reference Chong and Xia2016) for the case of severely confined RB convection, can be defined by generalising the relationship between $Ha_{opt}$ and $Ra$ (see figure 5 d) as

(3.1) $$\begin{eqnarray}Ra_{T}=\left(\frac{Ha_{opt}}{0.018}\right)^{1/0.45}.\end{eqnarray}$$

In figure 11 we replot the data for the compensated $Nu$ , but with $Ra$ normalised by $Ra_{T}$ . The figure shows that the transitions for the different values of $Ha$ all occur approximately at $Ra=Ra_{T}$ , suggesting that the $Ha$ -dependent $Ra_{T}$ defined above may be used as a characteristic Rayleigh number for identifying the regime transition. As $Nu$ decreases sharply after reaching the optimal enhancement, this suggests that the steeper scaling for $Ra<Ra_{T}$ is related to the reduction of heat transport. Again, this feature is similar to those found in RRB and CRB (King et al. Reference King, Stellmach, Noir, Hansen and Aurnou2009; Chong & Xia Reference Chong and Xia2016).

With the much steeper $Nu$ – $Ra$ scaling and the suppressed heat transport for $Ra<Ra_{T}$ , one would expect the onset value $Ra_{c}$ for convection will be dependent on $Ha$ . To find out this, we perform an extrapolation taking the lowest three $Nu$ values for each $Ha$ to determine $Ra_{c}$ . The estimated $Ra_{c}$ values are plotted in figure 12. (For $Ha=10$ , the scaling change is so small that we cannot confidently determine an $Ra_{c}$ for this case.) A power-law fit to the data gives $Ra_{c}=1.16Ha^{2.08\pm 0.09}$ . This value is in close agreement with the theoretical value of $Ra_{c}\sim Ha^{2}$ predicted by Chandrasekhar for magnetoconvection (Aurnou & Olson Reference Aurnou and Olson2001; Chandrasekhar Reference Chandrasekhar2013; de Vaux et al. Reference de Vaux, Zamansky, Bergez, Tordjeman and Haquet2017).

Figure 11.

Compensated plot of $Nu\,Ra^{-0.3}$ with $Ra$ normalized by $Ra_{T}$ . Dashed line indicates $Ra=Ra_{T}$ .

Figure 12.

The onset Rayleigh number for convection, $Ra_{c}$ versus $Ha$ . A power law fit represented by the solid line gives $Ra_{c}=1.16Ha^{2.08\pm 0.09}$ .

3.5 The viscous boundary layer in magnetoconvection: Prandtl–Blasius type versus Hartmann type

In this subsection, we examine the properties of the traditionally defined viscous boundary layer thickness $\unicode[STIX]{x1D6FF}_{v}$ , in particular for the boundary layer transition from Prandtl–Blasius type to Hartmann type. Figure 13(a) plots examples of the horizontal velocity profile, which shows clearly the existence of well-defined peaks near both the top and bottom plates, with the maximum velocity decreasing with increasing $Ha$ . Also note that, as a closed system, the mean flow velocity decays to zero at the cell centre. Figure 13(b) shows a magnified view of the profile near the bottom plate ( $Ra=10^{9}$ and $Ha=200$ ), where it is seen that the peak is rather broad under this resolution. We therefore adopt the so-called slope method to determine the BL thickness $\unicode[STIX]{x1D6FF}_{v}$ , which is illustrated in the figure. Again, we fit the data points between $0<z<z_{v,linear}=0.005$ for evaluating the BL thickness. The error of $\unicode[STIX]{x1D6FF}_{v}$ is determined by the difference between this result and that when one more data point is included in the fitting. For cases with large $Ra$ and large $Ha$ , indicated by $^{\ast }$ in table 1, $z_{v,linear}=0.002$ is used and the error is determined similarly.

With the BL thickness determined, we now examine its properties for various values of $Ra$ and $Ha$ . Figure 14(a) plots in log–log scale $\unicode[STIX]{x1D6FF}_{v}$ versus $Ra$ for various values of $Ha$ . If one starts with large $Ha$ , then it is seen that $\unicode[STIX]{x1D6FF}_{v}$ appears to be insensitive to changes in $Ra$ . As $Ha$ decreases, $\unicode[STIX]{x1D6FF}_{v}$ converges gradually to its $Ha=0$ value. For example, for $Ha=0$ , $2$ , and $10$ , the values of $\unicode[STIX]{x1D6FF}_{v}$ are almost the same, indicating that the viscous boundary layer is not appreciably perturbed under the corresponding magnetic field. Furthermore, a simple power law relationship between $\unicode[STIX]{x1D6FF}_{v}$ and $Ra$ is evident for these low $Ha$ data. In figure 14(b) we plot $\unicode[STIX]{x1D6FF}_{v}$ versus $Ha$ for various values of $Ra$ . It is seen that for weak magnetic field, $\unicode[STIX]{x1D6FF}_{v}$ is independent of $Ha$ and its magnitude depends on $Ra$ only. For large values of $Ha$ , on the other hand, $\unicode[STIX]{x1D6FF}_{v}$ for different $Ra$ appears to converge to a value independent of $Ra$ and depends only on $Ha$ . Therefore, it is clear that $\unicode[STIX]{x1D6FF}_{v}$ exhibits two types of asymptotic behaviour, depending on the relative magnitude of $Ra$ (or $Re$ ) and $Ha$ , or the relative strength of driving over stabilising forces.

Figure 13.

(a) Mean horizontal velocity profile for different values of $Ha$ at $Ra=10^{9}$ . (b) An example of mean horizontal velocity profile close to the boundary layer ( $Ra=10^{9}$ and $Ha=200$ ). The lines define the thickness of the viscous boundary layer $\unicode[STIX]{x1D6FF}_{v}$ .

Figure 14.

Viscous boundary layer $\unicode[STIX]{x1D6FF}_{v}$ dependence on $Ra$ and $Ha$ .(a)  $\unicode[STIX]{x1D6FF}_{v}$ versus $Ra$ for various values of $Ha$ . (b) $\unicode[STIX]{x1D6FF}_{v}$ versus $Ha$ for various values of $Ra$ ; the dashed line indicates an $Ha^{-1}$ scaling.

Figure 15.

Scaling properties of the viscous boundary layer and the Reynolds number under zero magnetic field. (a) The dependence of $\unicode[STIX]{x1D6FF}_{v}$ on $Ra$ at $Ha=0$ ; a power law fit represented by the solid line yields $\unicode[STIX]{x1D6FF}_{v}=0.845Ra^{-0.20\pm 0.01}$ . (b) The Reynolds number $Re_{0}$ at $Ha=0$ versus $Ra$ , where a power law fit yields $Re_{0}=0.014Ra^{0.53\pm 0.01}$ .

To gain insight into the dependence of $\unicode[STIX]{x1D6FF}_{v}$ on $Re$ and $Ha$ , we consider two asymptotic limits, i.e. $Re\gg Ha$ and $Re\ll Ha$ . In the classical regime of turbulent thermal convection, i.e. the BL remains laminar, and in the absence of magnetic field (corresponding to $Re\gg Ha$ ), one can obtain the Prandtl–Blasius-type BL by balancing the inertial term and the viscous term in the momentum equation, which gives

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{v}=\frac{H}{\sqrt{Re}}.\end{eqnarray}$$

In the limit of $Re\ll Ha$ , the Lorentz force becomes dominant and the viscous term balances the Lorentz term and one obtains the so-called Hartmann boundary layer Zürner et al. (Reference Zürner, Liu, Krasnov and Schumacher2016):

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{v}=\frac{H}{Ha}.\end{eqnarray}$$

For the intermediate cases, we assume that both the inertial and Lorentz terms are important in the momentum equation and it is the inertial term that balances both the viscous and the Lorentz terms, i.e.

(3.4) $$\begin{eqnarray}(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}\sim \unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}+\frac{1}{\unicode[STIX]{x1D70C}}(\boldsymbol{J}\times \boldsymbol{B}).\end{eqnarray}$$

Using dimensional analysis, we can write (3.4) as

(3.5) $$\begin{eqnarray}\frac{U^{2}}{H}\sim \frac{\unicode[STIX]{x1D708}U}{\unicode[STIX]{x1D6FF}_{v}^{2}}-\frac{\unicode[STIX]{x1D708}UHa^{2}}{H},\end{eqnarray}$$

which gives

(3.6) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{v}\sim \frac{H}{\sqrt{Re+Ha^{2}}}.\end{eqnarray}$$

As the obtained $\unicode[STIX]{x1D6FF}_{v}$ are in terms of the two control parameters $Ra$ and $Ha$ of the study, we need to first determine its $Re$ dependence. To obtain the asymptotic behaviour of $\unicode[STIX]{x1D6FF}_{v}$ in the limit of $Re\gg Ha$ , we fit a power law to its values at $Ha=0$ , which is shown in figure 15(a). The obtained power law exponent of $-0.20$ agrees excellently with previous measured values (see, for example, Xin, Xia & Tong (Reference Xin, Xia and Tong1996), Wei & Xia (Reference Wei and Xia2013)). For the $Re$ – $Ra$ relationship, we show in figure 15(b) a plot of the zero- $Ha$ Reynolds number $Re_{0}$ versus $Ra$ ; a power law fit gives $Re_{0}=0.014Ra^{0.53\pm 0.01}$ . Although the viscous BL in RB convection remains largely laminar in the so-called classical regime, it is rare that the exact Prandtl–Blasius (PB) scaling $\unicode[STIX]{x1D6FF}_{v}\sim Re^{\unicode[STIX]{x1D6FD}}$ with $\unicode[STIX]{x1D6FD}=1/2$ is observed (see, for example, Xin et al. (Reference Xin, Xia and Tong1996), Sun, Cheung & Xia (Reference Sun, Cheung and Xia2008), Wei & Xia (Reference Wei and Xia2013)). Part of the reason for the discrepancy is that the PB boundary layer theory is two-dimensional and most of the experimental and numerical studies are inherently three-dimensional. In general, the exponent $\unicode[STIX]{x1D6FD}$ appears to depend on the geometry and also the location (i.e. sidewall or horizontal plate) of the measurement and its value varies from $-0.32$ to $-0.5$ (Wei & Xia Reference Wei and Xia2013). In the present work we find $\unicode[STIX]{x1D6FF}_{v}\sim Ra^{-0.20}$ and $Re\sim Ra^{0.53}$ for the asymptotic case of $Ha\rightarrow 0$ , which implies $\unicode[STIX]{x1D6FF}_{v}\sim Re^{-0.38}$ in the $Ha\rightarrow 0$ limit. Based on the fact that the power law exponent is not exactly $1/2$ between $\unicode[STIX]{x1D6FF}_{v}$ and $Re$ , we therefore modify the equation (3.6) as follows for the forthcoming fitting:

(3.7) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{v}=\frac{H}{\sqrt{c_{1}Re^{0.76}+c_{2}Ha^{2}}},\end{eqnarray}$$

which would yield the numerically determined Prandtl–Blasius-like scaling and the Hartmann-like scaling in the respective limits of weak and strong magnetic fields, but also gives the $Re$ - and $Ha$ -dependence of $\unicode[STIX]{x1D6FF}_{v}$ for the intermediate cases. In the above the coefficients $c_{1}$ and $c_{2}$ account for the relative contributions of the inertial and Lorentz forces. By fitting the above equation to $\unicode[STIX]{x1D6FF}_{v}$ for the various values of $Re$ and $Ha$ , we obtain $c_{1}=24$ and $c_{2}=0.67$ . The fitting result is shown in figure 16, where the surface represents (3.7) with the fitted parameters; and the circles are the numerically obtained $\unicode[STIX]{x1D6FF}_{v}$ . It is seen that almost all data points fall onto the curved surface. To see the $Re$ -dominant and $Ha$ -dominant regions more clearly, we plot in figure 17 a phase diagram in which the cream-coloured area denotes the region where the boundary layer is controlled by the viscous force and the purple-coloured area denotes the Lorentz force controlled region. The two regions are separated by the dashed line, which is determined by setting $24Re^{0.76}=0.67Ha^{2}$ , i.e. the contributions by the two forces to the boundary layer equal to each other. Thus, the behaviour of the viscous boundary layer in quasistatic magnetoconvection may be understood as follows: when $Ha$ is very small, the viscous boundary layer is dominated by the inertial term and it behaves as a Prandtl–Blasius-type BL which depends only on $Re$ ; as $Ha$ increases, the Lorentz force becomes dominant and the BL becomes a Hartmann-type layer and becomes independent of $Re$ . In the intermediate range, the BL thickness may be represented by (3.7).

Figure 16.

Three-dimensional surface plot of the dependence of the viscous boundary layer on both $Re$ and $Ha$ . Circles represent the numerically obtained $\unicode[STIX]{x1D6FF}_{v}$ and the surface represents equation $\unicode[STIX]{x1D6FF}_{v}=H/\sqrt{24Re^{0.76}+0.67Ha^{2}}$ .

Figure 17.

Boundary layer phase diagram. The cream-coloured area denotes the region where the boundary layer is dominated by the viscous force, and the purple-coloured area denotes the region dominated by the Lorentz force. The dashed line indicates where the contributions by the two forces to the boundary layer equal each other, i.e. $24Re^{0.76}=0.67Ha^{2}$ . The stars represent the parameters for which simulations were made.