3.1 Spatial structure of sidewall modes

Figure 1 displays snapshots of the velocity field structure of the magnetoconvection flows. Isosurfaces of the vertical velocity component and field lines of the velocity field are shown for runs at $Ha\geqslant 500$ . For small external magnetic field strength, a cellular structure of up- and downwelling flows is observed that fills the entire cell. A sufficiently strong external magnetic field, which corresponds here to $Ha\geqslant 1000$ , expels convective motion from the interior of the cell, where heat is then transported solely by diffusion. We note that the critical Hartmann number, which corresponds to the Chandrasekhar linear stability limit, is given by $Ha_{c}=\sqrt{Ra}/\unicode[STIX]{x03C0}\approx 1007$ for the present Rayleigh number of $Ra=10^{7}$ . Figure 1(b–d) displays the structure of the wall modes, which consist of alternating up- and downwelling flow regions attached to the sidewalls. They correspond to neighbouring circulation rolls which do not move along the sidewalls or oscillate as in rotating convection for the total integration times that we could run the simulations. The wall modes are increasingly closely attached to the sidewalls as $Ha$ increases from $1000$ to $2000$ . Interestingly, vertical velocity maxima reach out into the bulk in the form of tongue-like filaments, which might be a relic from the turbulent flow pattern, a point that will be analysed more closely in subsection § 3.3.

Figure 2.

Global transport of heat and momentum. (a) Nusselt number $Nu$ normalized with respect to the Nusselt number value at $B_{0}=0$ versus $Ha/\sqrt{Ra}$ . Present data are given by red filled circles. A subset of the data from the experiment by Cioni et al. (Reference Cioni, Chaumat and Sommeria2000) is added (open circles). (b) Mean profiles of temperature $T(z)=\langle T(z)\rangle _{A,t}$ . (c) Mean profiles of convective heat current, $j_{c}(z)$ (solid lines), and diffusive heat current, $j_{d}(z)$ (dash-dotted lines), for various Hartmann numbers. Both currents are defined in (3.1). The dotted lines in (c) correspond to $Nu(z)=j_{c}(z)+j_{d}(z)=\text{const}.$ (d) Mean profile of the root mean square (r.m.s.) for the full velocity field (solid line) and the vertical velocity component (dashed line).

3.2 Global and near-sidewall transport of heat and momentum

Table 1 shows that the Nusselt number is still greater than unity for runs 4–6 beyond the Chandrasekhar limit. With increasing $Ha$ , the global transport converges to the pure diffusion case at $Nu=1$ . The strong temporal velocity fluctuations of the highly inertial low- $Pr$ turbulence for absent or weak magnetic fields then decrease to vanishingly small values and result in a practically quasisteady laminar wall-mode flow. The global analysis of transport is refined in figure 2 by vertical mean profiles taken with respect to the horizontal plane $A=\unicode[STIX]{x1D6E4}^{2}$ and time, $\langle \cdot \rangle _{A,t}$ . Figure 2(a) compares our Nusselt numbers directly against the experiments in a closed cylinder by Cioni et al. (Reference Cioni, Chaumat and Sommeria2000). It can be seen that their data record coincides well with the DNS below the Chandrasekhar limit. Deviations are observed for the highest magnetic fields, where the data points of the simulation branch off due to the existence of wall modes. Figure 2(b) shows the temperature profiles for all six data sets. The well-mixed bulk region for the turbulent cases at $Ha<1000$ changes to an almost linear diffusion-dominated profile for $Ha\geqslant 1000$ . The Nusselt number is decomposed into two terms that stand for the convective and diffusive heat fluxes across the layer,

(3.1) $$\begin{eqnarray}\displaystyle & \displaystyle Nu(z)=j_{c}(z)+j_{d}(z)=\sqrt{RaPr}\,\langle u_{z}T\rangle _{A,t}-{\displaystyle \frac{\unicode[STIX]{x2202}\langle T\rangle _{A,t}}{\unicode[STIX]{x2202}z}}. & \displaystyle\end{eqnarray}$$

Figure 2(c) displays the profiles of $j_{c}(z)$ and $j_{d}(z)$ for the three runs with $Ha\geqslant 1000$ as well as their sum (which has to be constant and equal to $Nu$ ). It is seen that $j_{d}(z)>j_{c}(z)$ . The increasing suppression of fluid turbulence is also demonstrated in figure 2(d) by the root-mean-square (r.m.s.) velocity profiles, which are given for the quasisteady cases for $Ha\geqslant 1000$ by $u_{rms}(z)=\langle u_{x}^{2}+u_{y}^{2}+u_{z}^{2}\rangle _{A}^{1/2}$ and $w_{rms}(z)=\langle u_{z}^{2}\rangle _{A}^{1/2}$ . It is not only that the total fluctuation level decreases as a whole, but also the vertical velocity fluctuations provide an increasing fraction of the total fluctuation magnitude for $Ha\geqslant Ha_{c}$ . Furthermore, we find that the ratio of the root-mean-square values, taken over the full cell volume, is $w_{rms}/u_{rms}=0.55,0.78,0.90,0.71,0.82$ , and 0.83 for the six simulation runs.

The importance of the wall modes for the transport of heat and momentum beyond the Chandrasekhar limit is highlighted in figure 3, where we determine the Nusselt number and root-mean-square velocities over successively smaller concentric cross-section areas $S=[r,\unicode[STIX]{x1D6E4}-r]\times [r,\unicode[STIX]{x1D6E4}-r]$ , with $r$ being the sidewall-normal distance. These averages are indicated by $\langle \cdot \rangle _{S}$ . As visible in figure 3, the transport decreases significantly within a few Shercliff layer thicknesses $\unicode[STIX]{x1D6FF}_{Sh}=1/\sqrt{Ha}$ , thus further supporting the hypothesis that the transport for $Ha>Ha_{c}$ is connected to the wall modes. It is also seen that the heat transfer drops to the diffusive lower bound of $Nu=1$ for $r\gtrsim 2\unicode[STIX]{x1D6FF}_{Sh}$ for the highest Hartmann number. All velocity profiles drop significantly towards the centre of the cell.

Figure 3.

Near-sidewall transport of heat and momentum beyond the Chandrasekhar stability limit. Nusselt numbers and root-mean-square velocities are determined over successively smaller horizontal cross-section areas $S$ . Profiles are plotted versus a sidewall-normal distance $r$ in units of the Shercliff layer thickness $\unicode[STIX]{x1D6FF}_{Sh}$ . (a) Nusselt number $\langle Nu\rangle _{S}$ . (b) Root-mean-square velocity $\langle u_{i}^{2}\rangle _{S}^{1/2}=\langle u_{x}^{2}+u_{y}^{2}+u_{z}^{2}\rangle _{S}^{1/2}$ . (c) Root-mean-square vertical velocity component $\langle u_{z}^{2}\rangle _{S}^{1/2}$ .

Table 2.

Summarizing list of different boundary layer thicknesses which can be obtained in the magnetoconvection flow in a closed cell. We list thermal BL thicknesses at top/bottom and sidewalls as well as Hartmann and Shercliff layer thicknesses. For the runs with wall modes we also list the ratios $r_{cr}/\unicode[STIX]{x1D6FF}_{Sh}$ and $\unicode[STIX]{x1D6FD}/N$ to quantify a two-layer structure. Here, $r_{cr}$ is the crossover width of the two spatial decay laws of convective heat flux of the wall modes. The exponential decay of the inner layer is measured by the exponent $\unicode[STIX]{x1D6FD}$ , which is found in a fixed ratio to the corresponding interaction parameter (or Stuart number) $N=Ha^{2}/\sqrt{Ra/Pr}$ .

3.3 Wall-mode structure together with thermal and Shercliff boundary layers

We proceed with an analysis of the viscous and thermal boundary layers in conjunction with the wall modes. In table 2, we list all important BL thicknesses. The thermal boundary layer thickness $\unicode[STIX]{x1D6FF}_{T}=1/(2Nu)$ at the top and bottom approaches 0.5, in agreement with $Nu\rightarrow 1$ as $Ha$ increases. The mean thermal BL thickness at the sidewall, $\unicode[STIX]{x1D6FF}_{T}^{(sw)}$ , is obtained from profiles of the temperature fluctuations, $\unicode[STIX]{x1D703}(\boldsymbol{x},t)=T(\boldsymbol{x},t)-\langle T(z)\rangle _{A,t}$ , with respect to the sidewall-normal coordinate $r$ . These profiles are calculated for $Ha\geqslant 1000$ only, i.e. when the dominantly vertical up- and downflows are attached to the sidewalls. They are always obtained as an average over all four sidewalls. The value of $\unicode[STIX]{x1D6FF}_{T}^{(sw)}$ is determined by a standard slope method, i.e. as the intersection point of the horizontal line drawn through the mean value of $\langle \unicode[STIX]{x1D703}(r)\rangle _{sw}$ in the bulk and a tangent which is fitted to the same profile very close to the sidewall. The corresponding values decrease as $Ha$ increases and are given in table 2.

In the presence of a strong $B_{0}$ , the standard viscous boundary layer thickness $\unicode[STIX]{x1D6FF}_{v}$ has to be substituted by the Hartmann layer thickness $\unicode[STIX]{x1D6FF}_{Ha}=1/Ha$ at the top and bottom. As seen in table 2, the Hartmann layers become extremely thin, and their appropriate resolution makes these DNS very demanding. The viscous sidewall layers are also affected by the external magnetic field. Here the fluid motion in the horizontal ( $x$ , $y$ )-directions, i.e. transverse to the sidewall-parallel magnetic field, is affected and Shercliff layers with thickness $\unicode[STIX]{x1D6FF}_{Sh}$ are formed. The Shercliff thickness will be chosen as the length scale by which the wall modes are measured. The latter ones establish a complex flow structure at the sidewalls, which is quantified in figure 4 for the highest external field at $Ha=2000$ . We observe alternating up- and downflows of warmer and colder fluid, respectively (see figure 4 a,b). The tongue-like structure consists of three thin counterflowing jets (up–down–upwelling or down–up–downwelling) which arise due to the incompressibility condition (see figures 4 b and 1). The velocity amplitude inside the modes is still remarkably large, with a maximum of $u_{z}\sim 0.1$ , as seen in figure 4(d).

Figure 4.

Detailed structure of the wall modes at $Ha=2000$ . Horizontal cross-section of the time-averaged (a) temperature field $T$ , (b) vertical velocity component $u_{z}$ and (c) heat transport $u_{z}T$ at the midplane ( $z=0$ ). The violet horizontal line indicates the Shercliff boundary layer thickness $\unicode[STIX]{x1D6FF}_{Sh}$ . The green horizontal line indicates $r_{cr}=3.47\unicode[STIX]{x1D6FF}_{Sh}$ . The sidewall distance $r_{cr}$ is determined in figure 5. (d) Profiles of the vertical velocity component versus the wall $y$ -distance are taken at seven $x$ -positions indicated by the cyan vertical lines in (b). Profiles from red to blue via grey correspond to the cyan lines from left to right. The vertical velocity component is averaged over the whole cell height in these profiles.

Figure 4(b,c) shows a two-layer structure of the wall modes on the basis of the vertical velocity component and heat transport. Figure 5 supports this observation further by an analysis of the convective heat transfer $\langle Nu\rangle _{S}-1=\sqrt{RaPr}\langle u_{z}T\rangle _{S}$ (see also figure 3 a). For all runs at $Ha\geqslant 1000$ , two exponential decays laws can be observed. These decays separate two different layers: the bulk region dominated by diffusive heat transport and the inner near-wall region with residual convective flow motion, particularly well observable at the highest $Ha=2000$ . We have verified that the pronounced minimum at this largest Hartmann number persists for finer computational grids by shorter test reruns at higher resolutions. The crossover distance $r_{cr}$ , identified as the intersection point of both exponential fits, is found at a fixed ratio to the Shercliff layer thickness $\unicode[STIX]{x1D6FF}_{Sh}$ for the three runs, as given in table 2. This result implies that the inner section of the sidewall layer is on average of the Shercliff-type despite the alternating pattern of horizontal up- and downflows. The distance $r_{cr}$ matches the point where the thin tongue-like vertical flows in figure 4(b,c) appear. Interestingly, the exponential fit $Nu-1\approx A\times \exp (-\unicode[STIX]{x1D6FD}r)$ of the inner sublayer for $r\leqslant r_{cr}$ results in spatial decay rates $\unicode[STIX]{x1D6FD}$ being a fixed ratio to the interaction parameter (or Stuart number) $N=Ha^{2}/\sqrt{Ra/Pr}$ . This parameter relates Lorentz to inertial forces and, in the present DNS series, underlines the dominance of Lorentz forces at $Ha\geqslant 1000$ .

Figure 5.

Two-layer structure of the wall modes. The global convective heat flux $Nu-1$ taken over cross-sections $S$ is plotted with respect to the sidewall-normal coordinate $r$ for $Ha\geqslant 1000$ . The spatial exponential decay with $r$ is fitted by two separate scaling laws which intersect at $r=r_{cr}$ .