3.1. Large-scale structures

We use our numerical data to analyse the flow structures in the convection cell for different fringe widths. Figures 4(a–e) exhibit the contour plots of temperature field on the horizontal midplane for $\delta =0.01$–9. The corresponding contour plots of the vertical velocity field are shown in figures 4(f–j). The data for these plots are averaged over 20–21 time frames (see table 1). The figures show that bulk convection is suppressed completely in the regions of strong magnetic fields. The region of suppressed bulk convection becomes smaller as $\delta$ is increased. This is expected because for large $\delta$, the gradient of the magnetic field is small and hence the region with ${{Ha}}_z(y)$ above the critical Hartmann number is also small.

Figure 4.

Contour plots of time-averaged fields on the horizontal midplane for different fringe widths of the imposed magnetic field. Plots of the temperature field for (a) $\delta =0.01$, (b) $\delta =0.3$, (c) $\delta =1$, (d) $\delta =3$, and (e) $\delta =9$. Plots of the vertical velocity field for (f) $\delta =0.01$, (g) $\delta =0.3$, (h) $\delta =1$, (i) $\delta =3$, and (j) $\delta =9$. The flows are organized into large-scale patterns that extend over scales larger than the domain height. The positions corresponding to ${{Ha}}_z(y)={{Ha}}_{z,c}$, ${{Ha}}_z(y)=0.8 {{Ha}}_{z,c}$, and the edge of the magnetic poles are represented by purple, blue and black horizontal lines, respectively, in decreasing order of thickness. The magnetic poles extend from $y=16$ to $y=\infty$.

Figures 4(a–j) also show that in the regions ${{Ha}}_z(y)<{{Ha}}_{z,max}$, the flow gets organized into superstructures. The superstructures observed in these figures in the weak magnetic flux regions are qualitatively similar to those observed by Pandey et al. (Reference Pandey, Scheel and Schumacher2018) for ${{Pr}}=0.021$. Further, the superstructures are relatively isotropic in the regions of weak magnetic flux, that is, they do not show any preference towards specific orientations. However, as the strength of the vertical magnetic field increases, the superstructures become more elongated and align themselves perpendicular to the sidewalls. The change in spatial structure of the flow is more clearly visible for $\delta =9$, in which the vertical magnetic field changes gradually. The transition in the orientation of the superstructures begins to occur for ${{Ha}}_z(y) \approx 0.8 {{Ha}}_{z,c}$, where the flow is dominated by ascending and descending planar jets originating from the sidewalls. The flow structures in this regime are very similar to those observed by Akhmedagaev et al. (Reference Akhmedagaev, Zikanov, Krasnov and Schumacher2020b) in their simulation data of RBC in a cylindrical cell under a strong uniform vertical magnetic field. They can be attributed to quasi-two-dimensional vortex sheets, which are often found in magnetohydrodynamic flows with strong magnetic fields (Zikanov & Thess Reference Zikanov and Thess1998). As one moves further towards the stronger magnetic flux region, the structures originating from the sidewalls extend less into the bulk, until for ${{Ha}}_z(y) \gtrsim {{Ha}}_{z,c}$, the flow is confined only near the sidewalls. These wall-attached flows will be discussed in detail in § 3.3.

It can be observed visually that the size of the convection patterns decreases along the direction of increasing magnetic field. We focus specifically on the results for $\delta =9$ because the increase of magnetic field along $y$ is gradual in this case, thus enabling us to obtain better statistics for analysing the variations of the structures with the magnetic field strength. In order to analyse quantitatively the evolution of the size of the structures along $y$, we divide the horizontal midplane into 10 subslices along the $y$-direction, and estimate the characteristic length scale of these structures in each subslice using the vertical velocity data. Each subslice spans the entire width of the convection cell. Since the structures are large in the low magnetic flux region, the first two subslices (corresponding to $0< y<5.34$ and $5.34< y<10.67$) are larger than the rest of the subslices; the larger subslices have width $5.34$ units, and the smaller subslices are $2.67$ units wide. We employ the procedure outlined in Pandey et al. (Reference Pandey, Scheel and Schumacher2018) to estimate the length scale, as elaborated below. For each subslice, we compute the Fourier transform of the vertical velocity field at each snapshot to obtain $\hat {u}_z(k,\phi _k,t)$, where $k$ is the magnitude of the wavevector, and $\phi _k$ is the corresponding azimuthal angle in the wavenumber space. We calculate the azimuthally averaged kinetic energy spectra, which are obtained as

(3.2)\begin{equation} E_u(k,t) = \frac{1}{2{\rm \pi}} \int_0^{2{\rm \pi}} |\hat{u}_z(k,\phi_k,t)|^2\,{\rm d}\phi_k. \end{equation}

These computed kinetic energy spectra are averaged over $20$ snapshots. The wavenumber $k_{max}$ corresponding to where the spectrum is maximum is the characteristic wavenumber. (In case the kinetic energy spectrum has multiple maxima, the characteristic wavenumber is given as the average of the wavenumbers corresponding to the maxima weighted by the kinetic energy contained in these wavenumber shells.) The characteristic length scale $\lambda$ is computed as

(3.3)\begin{equation} \lambda = \frac{2{\rm \pi}}{k_{max}}. \end{equation}

The contour plot of $u_z$ is redrawn in figure 5(a), and the computed characteristic length scale for each subslice is plotted versus $y$ in figure 5(b). Figure 5(b) shows that $\lambda$ decreases with the increasing magnetic field strength along $y$, which is consistent with the visual observation that the size of the patterns decreases along $y$. Note that the characteristic length scale in the weak magnetic flux region is $\lambda =5.5$, which is of the same order as the value $\lambda \approx 4$ computed by Pandey et al. (Reference Pandey, Scheel and Schumacher2018) using their numerical data for ${{Ra}}=10^5$ and ${{Pr}}=0.021$. Figure 5(b) also shows that the characteristic length scales computed using our data are larger than the critical wavelength $\lambda _c$ (Chandrasekhar Reference Chandrasekhar1981) at the onset of convection for the corresponding Hartmann number. In the wall-attached convection region where ${{Ha}}_z(y)>{{Ha}}_{z,max}$, the computed length scale ranges from $\lambda \approx 1.5$ to $\lambda \approx 2$ and denotes the characteristic size of the wall modes. These values are fairly close to the analytically derived value of $\lambda _{c,w} = \sqrt {2}$ for wall modes in magnetoconvection with one sidewall (Busse Reference Busse2008).

Figure 5.

Pattern analysis for $\delta =9$. (a) Contour plot of the time-averaged vertical velocity field on the horizontal midplane. Solid lines are replotted from figure 4. (b) The evolution of the characteristic horizontal length scale of the superstructures (red circles) and the critical wavelength $\lambda _c$ (black curve) at the onset of convection along the horizontal $y$-direction. The length scale of the convection patterns becomes smaller along the direction of the increasing magnetic field, and is larger than the critical wavelength.

We now examine the convection structures on different vertical planes along $y$ for $\delta =9$. In figures 6(a–c), we show the density plots of vertical velocity on the planes at $y=3.4$, $y=11.9$ and $y=19.7$. The local Hartmann numbers at these locations are ${{Ha}}_z(y=3.4)=25$, ${{Ha}}_z(y=11.9)=45$ and ${{Ha}}_z(y=19.7)=75$. The velocity fluctuations are strong at $y=3.4$ where the flow is highly turbulent. At $y=11.9$, the flow structures consist of quasi-two-dimensional upward and downward streams occupying the entire plane. The velocity fluctuations become weak at $y=19.7$, and the flow structures become elongated along the $x$-direction, consistent with the observation from figures 4(e,j). We plot the mean horizontal velocity $\left \langle \sqrt {u_x^2 + u_y^2} \right \rangle _x$, the mean absolute vertical velocity $\langle |u_z| \rangle _x$, and the mean temperature $\langle T \rangle _x$ versus $z$ on the above three planes in figures 7(a–c), respectively. These figures further reinstate that the velocity fluctuations decrease along $y$. Moreover, the temperature approaches the linear conduction profile as the local magnetic field strength increases.

Figure 6.

Contour plots of the time-averaged vertical velocity field for $\delta =9$ in vertical planes at (a) $y=3.4$ (${{Ha}}_z(y)=25$), (b) $y=11.9$ (${{Ha}}_z(y)=45$), and (c) $y=19.7$ (${{Ha}}_z(y)=75$). The velocity fluctuations decrease with the increasing magnetic field strength along $y$.

Figure 7.

Profiles for $\delta =9$ of (a) the mean horizontal velocity $\left \langle \sqrt {u_x^2+u_y^2} \,\right \rangle _x$, (b) the mean absolute vertical velocity $\langle |u_z| \rangle _x$, and (c) the mean temperature along the vertical $z$-direction, at $y=3.4$ (black dotted curves), $y=11.9$ (red dashed curves), and $y=19.7$ (blue solid curves). As the magnetic field strength increases along $y$, the velocity fluctuations decrease and the temperature approaches the linear conduction profile.

We now analyse quantitatively the isotropy of the flow using our numerical data. Towards this objective, we compute the following ratios: the local vertical anisotropy parameter $2E_z(y)/E_h(y)$, the local horizontal anisotropy parameter $E_y(y)/E_x(y)$, and the ratio of the kinetic energies along the $z$- and $y$-directions. In these, $E_z(y) = 0.5 \langle u_z^2 \rangle _{x,z}$, is the vertical kinetic energy, $E_h(y) = 0.5 \langle u_x^2 + u_y^2 \rangle _{x,z}$ is the horizontal kinetic energy, $E_x(y) = 0.5 \langle u_x^2 \rangle _{x,z}$ is the kinetic energy along the $x$-direction, and $E_y(y) = 0.5 \langle u_y^2 \rangle _{x,z}$ is the kinetic energy along the $y$-direction, all averaged over the $x$–$z$ plane. To understand the variation of flow anisotropy with magnetic field strength, the aforementioned anisotropy factors are plotted versus ${{Ha}}_z(y)/{{Ha}}_{z,c}$ for different $\delta$ in figures 8(a–e). In these plots, we focus on the regions where $0.25{{Ha}}_{z,c} < {{Ha}}_z(y) < {{Ha}}_{z,c}$ because, as discussed later, in § 3.2, the flow undergoes significant changes in this parameter interval. Further, we add horizontal tick labels with the coordinates of $y$ on the top of each plot to better understand the evolution of anisotropy along the horizontal direction.

Figure 8.

Mean profiles of the local vertical anisotropy parameter $2E_z(y)/E_h(y)$ (solid green curves), the local horizontal anisotropy parameter $E_y(y)/E_x(y)$ (dashed red curves), and the ratio of the kinetic energy along the $z$- and $y$-directions $E_z(y)/E_y(y)$ (dotted blue curves), versus the normalized Hartmann number ${{Ha}}_z(y)/{{Ha}}_{z,c}$ based on the mean vertical magnetic field $B_z(y)$, for (a) $\delta =0.01$, (b) $\delta =0.3$, (c) $\delta =1$, (d) $\delta =3$, and (e) $\delta =9$. Both the vertical and horizontal anisotropy parameters increase with the local vertical magnetic field strength.

The above plots show that the flow is roughly isotropic in the regions of weak vertical magnetic field corresponding to ${{Ha}}_z(y)<0.3{{Ha}}_{z,c}$. However, with increasing ${{Ha}}_z(y)$, the vertical velocity fluctuations become more dominant compared to the horizontal ones. This property is consistent with the anisotropy observed by Yan et al. (Reference Yan, Calkins, Maffei, Julien, Tobias and Marti2019) and Akhmedagaev et al. (Reference Akhmedagaev, Zikanov, Krasnov and Schumacher2020b) for strong Hartmann number convection with uniform magnetic fields, and is reminiscent of the stable cellular regime described by Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020). The increase of anisotropy with ${{Ha}}_z(y)$ becomes more pronounced as the fringe width, governed by $\delta$, increases. This is because as the fringe width increases, the span of $y$ falling in the regime $0.25{{Ha}}_{z,c} < {{Ha}}_z(y) < {{Ha}}_{z,c}$ also increases, as exhibited in figures 8(a–e). Thus we get better statistics, and hence clearer trends in this regime for large $\delta$.

Figures 8(a–e) also show that for strong magnetic fields, there is anisotropy among the horizontal components of velocity as well. For ${{Ha}}_z(y)> 0.65 {{Ha}}_{z,c}$, $E_y(y)$ begins to dominate $E_x(y)$, implying that the velocity fluctuations in the $y$-direction are stronger compared to those in the $x$-direction. This is also consistent with figures 4(a–j), which show that for moderately strong Hartmann numbers, the superstructure rolls orient themselves perpendicular to the longitudinal sidewalls. In the next subsection, we discuss the effects of the fringing magnetic fields on the local as well as global momentum and heat transport.

3.2. Heat and momentum transport

We analyse the spatial variation of the large-scale heat and momentum transport in our numerical set-up. Towards this objective, we first compute the local planar Reynolds number ${{Re}}(y)$ and Nusselt number ${{Nu}}(y)$ for every $\delta$. These are given by

(3.4a,b)\begin{equation} {{Re}}(y) = \sqrt{\frac{{{Ra}}}{{{Pr}}}}\langle u_x^2 + u_y^2 + u_z^2 \rangle_{x,z,t_N}^{1/2}, \quad {{Nu}}(y) = 1 + \sqrt{{{Ra}}\,{{Pr}}}\,\langle u_z T \rangle_{x,z,t_N}. \end{equation}

It must be noted that in order to avoid clutter, ${{Nu}}(y)$ and ${{Re}}(y)$ are computed for every 120th $x$–$z$ plane. Moreover, ${{Nu}}(y)$ exhibits strong spatial fluctuations even after averaging over all the time frames; these fluctuations are smoothed by computing the moving average of ${{Nu}}(y)$ using a window size of 5 $x$–$z$ planes.

We plot ${{Nu}}(y)$ and ${{Re}}(y)$ versus $y$ in figures 9(a) and 9(b), respectively. Figure 9(a) shows that ${{Nu}}(y)$ fluctuates between 2.0 and 4.0 in the region of weak magnetic flux, drops steeply in the fringe zone, and becomes close to unity in the strong magnetic flux region. Figure 9(b) shows that ${{Re}}(y)$ follows a similar trend; it is large in the weak magnetic flux region, drops steeply in the fringe zone, and becomes negligibly small in the strong magnetic flux region. The gradients of both ${{Re}}(y)$ and ${{Nu}}(y)$ curves in the fringe zone decrease with an increase of $\delta$. This is expected because $B_z$, which suppresses kinetic energy and heat transport in the fluid, has steeper gradients for small $\delta$.

Figure 9.

Variations of the local planar Nusselt and Reynolds numbers with the longitudinal direction $y$ . For $\delta =0.01$–$9$: (a) local Nusselt number ${{{Nu}}}(y)$, and (b) local Reynolds number ${{{Re}}}(y)$, versus $y$.

We now look for a universal dependence of the local heat and momentum transport on the local magnetic field strength. Taking inspiration from the recent experimental work of Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020) on thermal convection under uniform vertical magnetic fields, we plot the normalized local Nusselt number $\widetilde {{{Nu}}}(y)$ versus the normalized local vertical Hartmann number ${{Ha}}_z(y)/{{Ha}}_{z,c}$. The normalized local Nusselt number is given by

(3.5)\begin{equation} \widetilde{{{Nu}}}(y) = \frac{{{Nu}}(y)-1}{{{Nu}}_{max}-1}, \end{equation}

where ${{Nu}}_{max}=2.66$ (see § 2.2). Figure 10(a) shows that for all $\delta$, the points collapse well into a single curve, thus showing a universal dependence of ${{Nu}}(y)$ on ${{Ha}}_z(y)$. The plot shows that for ${{Ha}}_z(y)\lesssim 0.3{{Ha}}_{z,c}$, the Nusselt number does not change significantly and remains close to its maximum value. This regime corresponds to the turbulent and isotropic regime. For ${{Ha}}_z(y)\gtrsim 0.3{{Ha}}_{z,c}$, the flow transitions into the cellular regime (as per the nomenclature of Zürner et al. Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020) and the local Nusselt number starts to drop sharply as ${{Ha}}_z(y)$ increases. For ${{Ha}}_z(y)<0.85 {{Ha}}_{z,c}$, the best-fit curve for our data is

(3.6)\begin{equation} \widetilde{{{Nu}}}(y) = \left[1 + \chi_1 \left\{\frac{{{Ha}}_z(y)}{{{Ha}}_{z,c}} \right\}^{\gamma_1} \right]^{{-}1}, \end{equation}

where $\gamma _1 = 2.1 \pm 0.3$ and $\chi _1 = 4.0 \pm 0.7$. Interestingly, the values of $\chi _1$ and especially $\gamma _1$ are close to those observed by Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020), who reported $\gamma _1 = 2.03 \pm 0.06$ and $\chi _1 = 5.9 \pm 0.3$ for uniform magnetic fields. For ${{Ha}}_z(y) > 0.85 {{Ha}}_{z,c}$, $\widetilde {{{Nu}}}(y)$ decreases more steeply than (3.6) and is described by the power law

(3.7)\begin{equation} \widetilde{{{Nu}}}(y) = 0.028 \left\{\frac{{{Ha}}_z(y)}{{{Ha}}_{z,c}} \right\}^{{-}10.5\pm 0.9}. \end{equation}

This regime corresponds to the wall-mode regime in which the flow gets further suppressed and begins to confine itself near the sidewalls. It must be noted that Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020) could not obtain a best-fit expression for ${{Nu}}$ in the wall-mode regime due to lack of data close to the wall.

Figure 10.

Variations of the normalized local Nusselt and Reynolds numbers with the normalized local vertical magnetic field strengths. For $\delta =0.01$–$9$: (a) normalized local Nusselt number $\widetilde {{{Nu}}}(y)$ versus the normalized local Hartmann number ${{Ha}}(y)/Ha_c$, based on vertical magnetic field strength; and (b) normalized local Reynolds number $\widetilde {{{Re}}}(y)$ versus the local free-fall interaction parameter $N_{f}(y)$. The best-fit curves for the above data are also shown.

We conduct a similar analysis for ${{Re}}(y)$. We compute the normalized Reynolds number given by

(3.8)\begin{equation} \widetilde{{{Re}}}(y) = \frac{{{Re}}(y)}{{{Re}}_{max}}, \end{equation}

where ${{Re}}_{max}=1115$ (see § 2.2). The results of Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020) suggest that $\widetilde {{{Re}}}(y)$ should scale as ${{Ha}}_z(y)/\sqrt {{{Ha}}_{z,c}}$ instead of ${{Ha}}_z(y)/{{Ha}}_{z,c}$. Let us now consider the local free-fall interaction parameter $N_{f}(y)$, which is given by

(3.9)\begin{equation} N_{f}(y) = {{Ha}}^2_z(y)\,\sqrt{\frac{{{Pr}}}{{{Ra}}}}. \end{equation}

For large Rayleigh numbers, the critical Hartmann number can be approximated as ${{Ha}}_{z,c} \approx \sqrt {{{Ra}}}/{\rm \pi}$ (Chandrasekhar Reference Chandrasekhar1981). Using this relation, we get

(3.10)\begin{equation} N_{f}(y) = {{Ha}}^2_z(y)\,\frac{\rm \pi}{\sqrt{{{Ra}}}}\,\frac{\sqrt{{{Pr}}}}{\rm \pi} \approx \frac{{{Ha}}^2_z(y)}{{{Ha}}_{z,c}}\,\frac{\sqrt{{{Pr}}}}{\rm \pi} \sim \frac{{{Ha}}^2_z(y)}{{{Ha}}_{z,c}}. \end{equation}

This relation shows that $N_f(y)$ is proportional to ${{{Ha}}^2_z(y)}/{{{Ha}}_{z,c}}$, implying that $\widetilde {{{Re}}}(y)$ scales with $N_f(y)$. Thus we plot $\widetilde {{{Re}}}(y)$ versus $N_f(y)$. The plot in figure 10(b) shows that, similar to the Nusselt number, the data points for all $\delta$ collapse into a single curve. For $N_{f}(y) < 0.17$, which corresponds to ${{Ha}}_z(y) < 0.22 {{Ha}}_{z,c}$, the Reynolds number retains its maximum value (${{Re}}_{max}$). At higher $N_{f}(y)$, the Reynolds number starts to fall sharply with the interaction parameter. For $N_{f}(y) < 3.2$, which corresponds to ${{Ha}}_z(y) < 0.95 Ha_{z,c}$, the best-fit curve for our data is given by

(3.11)\begin{equation} \widetilde{{{Re}}}(y) = \left[1 + \chi_2 \left\{N_{f}(y) \right\}^{\gamma_2} \right]^{{-}1}, \end{equation}

with $\chi _2 = 1.40 \pm 0.03$ and $\gamma _2 = 1.01 \pm 0.03$. The value of $\gamma _2$ is again close to that observed by Zürner et al. (Reference Zürner, Schindler, Vogt, Eckert and Schumacher2020), who reported $\gamma _2 = 0.87 \pm 0.03$ for uniform magnetic fields. For ${{Ha}}_z(y)>0.95 {{Ha}}_{z,c}$, which corresponds to the wall-attached convection regime, $\widetilde {{{Re}}}(y)$ falls even more steeply than (3.11) and is described by the power law

(3.12)\begin{equation} \widetilde{{{Re}}}(y) = 18\{N_{f}(y)\}^{{-}4.0\pm 0.2}. \end{equation}

We now analyse the impact of the local heat and momentum transport on the evolution of the viscous and thermal boundary layers near the top and bottom walls of our convection cell. Again, we concentrate on $\delta =9$ to obtain better statistics and a clearer picture of the evolution. The thermal boundary layer thickness $\delta _T$ is the depth where a linear fit of the mean temperature profile near the wall intersects with the mean bulk temperature ($T=0.5$). The viscous boundary layer thickness is the depth where a linear fit of the horizontal velocity ($u_h = \langle (u_x^2+u_y^2)^{1/2} \rangle _x$) profile near the wall intersects with its local maximum. See Breuer et al. (Reference Breuer, Wessling, Schmalzl and Hansen2004), Ahlers et al. (Reference Ahlers, Grossmann and Lohse2009) and Scheel et al. (Reference Scheel, Kim and White2012) for a detailed procedure to compute the boundary layer thicknesses. We compute the thermal and viscous boundary layer thicknesses at different $y$-coordinates for $\delta =9$, and plot them versus the corresponding local Hartmann number ${{Ha}}_z(y)$ in figures 11(a,b), respectively.

Figure 11.

Boundary analysis for $\delta =9$: (a) local thermal boundary layer thickness $\delta _T(y)$, and (b) local viscous boundary layer thickness $\delta _u(y)$, near the top and bottom walls, versus the local Hartmann number ${{Ha}}_z(y)$. The thermal boundary layer thickness increases with the increase of ${{Ha}}_z(y)$. The viscous boundary layer thickness decreases with increasing ${{Ha}}_z(y)$ and approaches the Hartmann-type scaling $\delta _u \sim {{Ha}}_z^{-1}$ at large Hartmann numbers.

Figure 11(a) shows that the thermal boundary layer thickness increases with ${{Ha}}_z(y)$. This is expected because the thermal boundary layer thickness is inversely proportional to the local Nusselt number ${{Nu}}(y)$, which decreases with increasing ${{Ha}}_z(y)$. The viscous boundary layer, on the other hand, exhibits an interesting behaviour. As is evident in figure 11(b), the boundary layer thickness $\delta _u$ decreases with ${{Ha}}_z(y)$, with the decrease becoming steeper at large ${{Ha}}_z(y)$. To gain insight into this behaviour, we take inspiration from the work of Lim et al. (Reference Lim, Chong, Ding and Xia2019), and using dimensional analysis on the momentum equation, we obtain the following scaling for $\delta _u$:

(3.13)\begin{equation} \delta_u \sim \frac{1}{\sqrt{{{Re}}(y) + {{Ha}}_z^2(y)}}. \end{equation}

This scaling suggests that for ${{Ha}}_z(y) \ll \sqrt {{{Re}}(y)}$, $\delta _u$ follows the Prandtl–Blasius-type scaling $\delta _u \sim {{Re}}^{-1/2}$, in which the boundary layer thickness should increase with the decrease of ${{Re}}(y)$, and hence with the increase of ${{Ha}}_z(y)$. However, we do not observe such a regime in our convection cell. When ${{Ha}}_z(y) \sim \sqrt {{{Re}}(y)}$, the boundary layer thickness varies marginally with $y$, as observed for ${{Ha}}_z(y) < 50$ from our data. For ${{Ha}}_z(y) \gg \sqrt {{{Re}}(y)}$, the boundary follows the Hartmann type scaling $\delta _u \sim {{Ha}}_z^{-1}$, hence $\delta _u$ decreases sharply with ${{Ha}}_z(y)$. From figure 11(b), it can be seen that $\delta _u$ approaches the Hartmann type scaling at ${{Ha}}_z(y) \approx 70$.

Having studied the local variations of heat and momentum transport along with their associated boundary layers, we now analyse the global Reynolds and Nusselt numbers (${{Re}}_{global}$ and ${{Nu}}_{global}$, respectively) and their dependence on $\delta$. These global quantities are computed numerically using (2.12) and (2.13) from our simulation data, and are plotted versus $\delta$ in figures 12(a,b) as black squares. The figures show that both ${{Re}}_{global}$ and ${{Nu}}_{global}$ decrease as $\delta$ increases. This result is counter-intuitive because for large $\delta$, bulk convection is completely ceased only in a small region (as seen in figure 4), hence one would expect the overall heat and momentum transport to be stronger. However, a careful look at figure 2(a) reveals that the vertical magnetic field is stronger in the weak magnetic flux region for large fringe widths, which, in turn, weakens convection in that region. The increase in convection in the strong magnetic flux region is unable to compensate the suppression of convection in the weak magnetic flux region, resulting in a decrease of overall heat and mass transport for large fringe widths.

Figure 12.

Plots versus $\delta$ of (a) the global Reynolds number ${{Re}}_{global}$ (black squares for ${{Ha}}_{z,max}=120$ and purple circles for ${{Ha}}_{z,max}=30$), computed from our numerical data using (2.12), and (b) the global Nusselt number ${{Nu}}_{global}$ (black squares for ${{Ha}}_{z,max}=120$ and purple circles for ${{Ha}}_{z,max}=30$), computed from our numerical data using (2.13). Also shown in the plots are the estimates (solid curves) of ${{Re}}_{global}$ and ${{Nu}}_{global}$ computed using (3.14) and (3.15) for different values of ${{Ha}}_{z,max}$.

The above results, however, do not indicate whether such a variation of global Reynolds and Nusselt numbers with $\delta$ holds for all ${{Ha}}_{z,max}$. To explore this aspect, we need to estimate ${{Re}}_{global}$ and ${{Nu}}_{global}$ for given values of ${{Ha}}_{z,max}$ and $\delta$. Towards this objective, we make use of the best-fit relations for ${{Nu}}(y)$ and ${{Re}}(y)$ given by (3.6), (3.7), (3.11) and (3.12) obtained earlier, and integrate them numerically over the entire domain to estimate global Reynolds and Nusselt numbers as

(3.14)\begin{gather} {{Re}}_{global} = \frac{1}{l_y}\int_0^{l_y} {{Re}}(y)\,{{\rm d} y} = \frac{1}{l_y}\int_0^{l_y} {{Re}}({{Ha}}_z(y))\,{{\rm d} y}, \end{gather}

(3.15)\begin{gather}{{Nu}}_{global} = \frac{1}{l_y}\int_0^{l_y} {{Nu}}(y)\,{{\rm d} y} = \frac{1}{l_y}\int_0^{l_y} {{Nu}}({{Ha}}_z(y))\,{{\rm d} y}, \end{gather}

where we recall that

(3.16)\begin{equation} {{Ha}}_z(y) = {{Ha}}_{z,max}\,\frac{\langle B_z \rangle_{x,z}}{B_{z,max}}, \end{equation}

and $B_z=B_z(x,y,z,\delta )$ is given by (2.3). Thus for given ${{Ha}}_{z,max}$ and $\delta$, one can estimate ${{Re}}_{global}$ and ${{Nu}}_{global}$ using (3.14) and (3.15). We compute these estimates for ${{Ha}}_{z,max}=120$, 60, 50 and 30, and for $\delta$ ranging from 0.005 to 9.1 in increments of 0.01. The estimated values of ${{Re}}_{global}$ and ${{Nu}}_{global}$ are plotted versus $\delta$ in figures 12(a,b) as solid curves. The curves for ${{Ha}}_{z,max}=120$ fit the data points of directly computed ${{Re}}_{global}$ and ${{Nu}}_{global}$ (black squares) very well, implying that (3.14) and (3.15) provide fairly accurate estimates. In the process of computing the estimates of ${{Re}}_{global}$ and ${{Nu}}_{global}$ for ${{Ha}}_{z,max} < 120$, we assume that the equations and coefficients given in (3.6), (3.7), (3.11) and (3.12) remain unchanged as the maximum Hartmann number is varied. We also assume that the values ${{Ha}}_z(y) = 0.85 {{Ha}}_{z,c}$ and $N_f(y)=3.2$, which were observed to separate the two components of the piecewise functions for ${{Nu}}(y)$ and ${{Re}}(y)$ for ${{Ha}}_{z,max}=120$, remain the same for all values of ${{Ha}}_{z,max}$. These assumptions are based on the fact that the dependence of ${{Nu}}(y)$ and ${{Re}}(y)$ on ${{Ha}}_z(y)$ is suggested to be universal since the normalized local Nusselt and Reynolds number curves for different $\delta$ collapse as shown in figures 10(a,b).

Figure 12(a) also shows that the decrease of ${{Re}}_{global}$ with $\delta$ becomes less apparent with a decrease of ${{Ha}}_{z,max}$. In fact, for ${{Ha}}_{z,max}=30$, ${{Re}}_{global}$ increases marginally with $\delta$. Figure 12(b) indicates that ${{Nu}}_{global}$ increases with $\delta$ for ${{Ha}}_{z,max}=60$, 50 and 30, with this increase becoming more apparent with the decrease of ${{Ha}}_{z,max}$. To test the correctness of the above predictions, we conduct two more direct numerical simulations of our magnetoconvection set-up for ${{Ha}}_{z,max}=30$, and $\delta =0.01$ and 9. The grid size and the numerical schemes are the same as those described in § 2.2. We run the simulations for 10 free-fall time units after attaining a statistically steady state. We compute the time-averaged global Reynolds and Nusselt numbers using the above simulation data, and plot them in figures 12(a) and 12(b), respectively. The figures show that the simulation results are in close agreement with the predictions, thereby reinforcing our theory. It is therefore clear that for smaller maximum magnetic field strengths, the overall heat and momentum transport increases with the increase of fringe width. This implies that for small values of ${{Ha}}_{z,max}$, the suppression of convection in the weak magnetic flux region is small compared to the increase of convection in the strong magnetic flux region. Thus we can conclude that the variation of ${{Re}}_{global}$ and ${{Nu}}_{global}$ with the fringe width depends critically on ${{Ha}}_{z,max}$. Finally, we discuss the dynamics of wall-attached convection in the strong magnetic flux region in the next subsection.

3.3. Wall-attached convection

We plot the time-averaged isosurfaces of $u_z = 0.01$ (red) and $u_z = -0.01$ (blue) for $\delta =1$ and $\delta =9$ in figures 13(a) and 13(b), respectively. The figures show that in the strong magnetic flux region, the flow in the bulk is completely suppressed, with alternating upwelling and downwelling flow regions attached to the sidewalls. The structure of the wall modes is consistent with that observed by Liu et al. (Reference Liu, Krasnov and Schumacher2018) in their simulations of thermal convection in a rectangular domain with uniform vertical magnetic field. However, unlike for rotating convection (Horn & Schmid Reference Horn and Schmid2017; Zhang et al. Reference Zhang, van Gils, Horn, Wedi, Zwirner, Ahlers, Ecke, Weiss, Bodenschatz and Shishkina2020) or convection in cylindrical domains (Akhmedagaev et al. Reference Akhmedagaev, Zikanov, Krasnov and Schumacher2020b), the wall modes do not oscillate or move along the sidewalls (Grannan et al. Reference Grannan, Cheng, Aggarwal, Hawkins, Xu, Horn, Sánchez- Álvarez and Aurnou2022; Schumacher Reference Schumacher2022).

Figure 13.

Isosurfaces of $u_z=0.01$ (red) and $u_z=-0.01$ (blue) for (a) $\delta =1$ and (b) $\delta =9$. In the regions with strong magnetic fields, convective motions are confined near the sidewalls.

The wall modes are dense and uniformly distributed along the sidewalls for $\delta =9$. However, for $\delta =1$, they are visibly thinner, with isosurfaces clustering more towards the corners instead of being uniformly distributed along the sidewalls. The clustering occurs because of weaker Lorentz force in the corners, which, in turn, results due to the electric current getting further constrained by the two insulated sidewalls in proximity to each other. The above trend in the local differences of the wall modes is also observed for $\delta =0.01$, 0.3 and 3, as shown later in this section.

For each sidewall in the strong magnetic flux region, we compute the mean distance $\bar {\delta }_w$ of the wall modes from the sidewall. This distance is computed as

(3.17)\begin{equation} \bar{\delta}_w = \frac{\displaystyle\int {\delta_w\,u_{z,max}}\,{\rm d}l}{\displaystyle\int {u_{z,max}}\,{\rm d}l}, \end{equation}

where ${\rm d}l = {{\rm d}y}$ for the longitudinal sidewall, and ${\rm d}l = {{\rm d} x}$ for the lateral sidewall. In the above, $\delta _w$ is the shortest distance between the sidewall and the point of maximum absolute velocity $u_{z,max}$ adjacent to the sidewall, and the summations are over the regions where ${{Ha}}_z(y)>{{Ha}}_{z,c}$. Thus $\bar {\delta }_w$ is the weighted average of the shortest distance between the point of maximum absolute velocity and the sidewall, with the maximum absolute velocity being the weights. We also compute the mean distance averaged over all the sidewalls. We plot the mean distance for each sidewall, as well as the overall mean distance versus $\delta$, in figure 14. The figure shows that the wall modes are in general more attached to the lateral sidewall ($y=32$) compared to the longitudinal sidewalls ($x=0$ and $x=16$). There is, however, no clear trend regarding the variation of $\bar {\delta }_w$ with the fringe width.

Figure 14.

The average distance between the wall modes and the sidewalls versus $\delta$ in the regions ${{Ha}}_z(y) > {{Ha}}_{z,c}$. The wall modes are more attached to the lateral sidewalls than to the longitudinal sidewalls.

We further analyse the amplitudes of the wall modes along the three sidewalls for our runs. Towards this objective, we measure the vertical velocity field at every point on the horizontal midplane at distance $\bar {\delta }_w$ from the sidewalls. We plot this vertical velocity profile along every sidewall in figure 15. The figure shows that with the exception of $\delta =9$, the amplitudes of the wall modes are stronger near the corners, and become weaker as one moves away from the corners. This behaviour in the amplitudes is consistent with figure 13(a), which shows that the wall modes are clustered in the corners. As explained earlier, these local differences in amplitudes occur due the current being more constrained in the corners, resulting in weaker convection-suppressing Lorentz forces in these regions. Figure 15 also shows that near the corners, the amplitudes of the wall modes grow with an increase of $\delta$. However, in the regions away from the corners, the amplitudes decrease as $\delta$ is decreased from 9 to 1, and then increase with a further decrease of $\delta$. For $\delta =0.3$ and 1, the amplitudes become negligibly small away from the corners. However, for $\delta =0.01$, although the amplitudes of the modes away from the corners are smaller than those near the corners, they are larger than the amplitudes of wall modes away from the corners for $\delta =0.3$ and 1.

Figure 15.

Vertical velocity $u_{z,w}$ measured on the horizontal midplane at distance $\bar {\delta }_w$ from the sidewalls in the regions ${{Ha}}(y)>{{Ha}}_{z,c}$ (non-shaded regions) for (a) $\delta =0.01$ (black curve), (b) $\delta =0.3$ (purple curve), (c) $\delta =1$ (blue curve), (d) $\delta =3$ (green curve), and (e) $\delta =9$ (red curve). The shaded regions correspond to ${{Ha}}_z(y) < {{Ha}}_{z,c}$, where the flow is not fully suppressed in the bulk. The amplitudes of the wall modes in the regions away from the corners exhibit a non-monotonic dependence on $\delta$. Note that these plots are resolved by a total of at least 9000 grid points in the horizontal direction.

The reason for the non-monotonic behaviour of wall modes with $\delta$ away from the corners is not yet clear. A possible explanation is that in addition to the vertical component of the magnetic field $B_z$, the horizontal component $B_y$ also plays a role in stabilizing convection near the sidewalls. From figure 2(b), it can be seen that for small $\delta$ or fringe width, although $B_y$ is strong in $y=l_y/2$ midplane, it falls sharply and becomes very small as one moves away from the midplane. Thus for $\delta =0.01$, $B_y$ is too weak to stabilize convection near the sidewalls at $y \gg l_y/2$, resulting in sustained wall-attached convection. On the other hand, for large $\delta$, $B_y$ in the regions away from $y=l_y/2$ midplane is not as weak as that for small $\delta$. Thus as $\delta$ is increased up to $\delta =1$, $B_y$ becomes stronger and suppresses wall-attached convection. On further increasing $\delta$, although $B_y$ grows only marginally, $B_z$ becomes too weak to suppress wall-attached convection. Hence we get prominent wall modes again for $\delta =3$ and 9. The non-monotonic behaviour of wall modes will be explored in detail in a future study.