3.1. Global heat and mass transfer

With the increasing of magnetic field imposed on RBC, the electromagnetic force gives rise to a transition from a 3-D convective roll pattern (called cell structures by Vogt et al. (Reference Vogt, Yang, Schindler and Eckert2021); see figure 1a) to a quasi-2-D flow pattern (see figure 1d) in a way that convective rolls become more and more aligned with the magnetic field (see figures 1b,c). Such an evolution has been observed already in the previous investigations by Burr & Müller (Reference Burr and Müller2002), Tasaka et al. (Reference Tasaka, Igaki, Yanagisawa, Vogt, Zuerner and Eckert2016) and Vogt et al. (Reference Vogt, Yang, Schindler and Eckert2021). Generally speaking, the quasi 2-D roll-like structure ensures a direct heat transfer between the bottom and the top, and extends the uniform bulk central region of the convection, thinning the thermal boundary layers and causing an increase of heat transfer coefficient for moderate $Q$ as shown in figure 2(a).

Figure 1. The isosurfaces of $\varOmega = 0.5$ are shown, coloured with vertical velocity $u_{z}$ for $Ra=1 \times 10^{6}$. The identified method of the vortex structures ($\varOmega$) applied here is defined by Liu et al. (Reference Liu, Wang, Yang and Duan2016). Isosurfaces are for (a) $Q/Ra=0$, (b) $Q/Ra=0.16$, (c) $Q/Ra=0.64$, and (d) $Q/Ra=2.25$.

Figure 2. (a) Relative deviations of the heat transfer from the reference state of RBC (without magnetic field), $(Nu-Nu_{0})/Nu_{0}$, as a function of $Q$ for three different $Ra$ values. Data from Vogt et al. (Reference Vogt, Yang, Schindler and Eckert2021) are used for comparison. (b) The corresponding values of the compensated $Re$ number, $Re\,Pr/Ra^{2/3}$, plotted versus $Q$, where $Re$ from Yang et al. (Reference Yang, Vogt and Eckert2021) is based on $\max (u_{x})$. The $Q$ values in the simulation are listed in ascending order as follows: $2.5 \times 10^3$, $10^4$, $2.25\times 10^4$, $4\times 10^4$, $9\times 10^4$, $1.6\times 10^5$, $2.5\times 10^5$, $6.4\times 10^5$, $10^6$, $1.5625\times 10^6$, $2.25\times 10^6$, $3.0625\times 10^6$, $4\times 10^6$, $6.25\times 10^6$ and $9\times 10^6$.

However, prior to the augmentation of heat transfer, it is noteworthy that $Nu$ exhibits a decline under relatively weak magnetic fields, particularly in cases of higher $Ra$, which is consistent with the findings of the experiments conducted by Burr & Müller (Reference Burr and Müller2002) and Vogt et al. (Reference Vogt, Yang, Schindler and Eckert2021). The reduction can be attributed to the suppression of small-scale convective vortices by the Lorentz force, which results in a decrease in the injection of plumes, the primary agents of heat transfer. It is also important to mention that $Re$ follows a similar trend to the Nusselt number with the increase of $Q$, indicating that convective heat conduction is always dominant within our parameter space. Moreover, this trend exhibits the intricate way in which a magnetic field affects a turbulent flow. For small $Q$, the velocity in the magnetic field is inhibited, and damping becomes the primary effect on flow, resulting in a decrease in $Re$. However, for moderate $Q$, the enhancement of flow perpendicular to the magnetic field dominates, leading to an increase in both $Nu$ and $Re$. This phenomenon has also been observed in duct flow (Sukoriansky, Zilberman & Branover Reference Sukoriansky, Zilberman and Branover1986; Zikanov et al. Reference Zikanov, Krasnov, Boeck and Sukoriansky2019), electromagnetically driven square cavity flow (Pothérat & Klein Reference Pothérat and Klein2017) and vertical convection flow in a square container under the influence of magnetic fields (Chen, Liu & Ni Reference Chen, Liu and Ni2018).

Furthermore, we note that $Re\,Pr/Ra^{2/3}$ scales well with $Q^{-n}$ ($n=0.35$) for cases at sufficiently large $Q$, where quasi-2-D roll structures have been well developed. Now we incorporate Grossmann–Lohse (GL) theory (Grossmann & Lohse Reference Grossmann and Lohse2000) into the MHD-RBC flow, and provide an explanation for this scaling relationship. Regarding a regular flow, the GL theory takes into account the spatial- and time-averaged viscous and thermal dissipation rates, $\varepsilon _{\nu }$ and $\varepsilon _{\kappa }$, in convective heat transfer:

(3.1)$$\begin{gather} \varepsilon_{\nu} = \frac{\nu}{2}\,\langle(\partial_{i}u_{j} + \partial_{j}u_{i})^{2}\rangle, \end{gather}$$

(3.2)$$\begin{gather}\varepsilon_{\kappa} = \kappa \langle (\partial_{i}T)^{2} \rangle. \end{gather}$$

Here, the Einstein summation convention is used over the coordinates $i, j=x, y, z$, and $\partial _{i} = \partial / \partial x_{i}$ is a short notation for the spatial partial derivatives.

As suggested by Zürner (Reference Zürner2020), under the influence of a magnetic field, the additional Joule dissipation rate

(3.3)\begin{equation} \varepsilon_{\eta} = \frac{\eta}{2}\,\langle(\partial_{i}b_{j} + \partial_{j}b_{i})^{2}\rangle \end{equation}

has to be considered. Here, the secondary magnetic field ${{\boldsymbol {b}}} = b_i{{\boldsymbol {e}}}_i$ is induced by the interaction of ${{\boldsymbol {u}}}$ and the imposed magnetic field ${{\boldsymbol {B}}}_0$, and $\eta$ is the the magnetic diffusivity. When turbulence reaches a relatively steady state, the average dissipation rate satisfies the following rigorous relations, which are derivable easily from (2.2), (2.3), the transport equation for magnetic induction, and the definition of $Nu$ (see Shraiman & Siggia Reference Shraiman and Siggia1990; Chakraborty Reference Chakraborty2008; Zürner et al. Reference Zuerner, Liu, Krasnov and Schumacher2016):

(3.4)$$\begin{gather} \varepsilon_{\nu} + \frac{\varepsilon_{\eta}}{\mu \rho_{0}} = \frac{\nu^{3}}{H^{4}}\,\frac{(Nu-1)\,Ra}{Pr^{2}}, \end{gather}$$

(3.5)$$\begin{gather}\varepsilon_{\kappa} = \kappa\,\frac{(\Delta T)^{2}}{H^2}\,Nu, \end{gather}$$

where $\mu$ represents the the magnetic permeability. Based on the GL theory, the contributions of the dissipation rate from the bulk region and the boundary layer are separated:

(3.6)$$\begin{gather} \varepsilon_{\nu} = \varepsilon_{\nu,Bulk} + \varepsilon_{\nu,BL}, \end{gather}$$

(3.7)$$\begin{gather}\varepsilon_{\eta} = \varepsilon_{\eta,Bulk} + \varepsilon_{\eta,BL}, \end{gather}$$

(3.8)$$\begin{gather}\varepsilon_{\kappa} = \kappa\,\frac{(\Delta T)^2}{{H}^2} + \varepsilon_{\kappa,Bulk} + \varepsilon_{\kappa,BL}. \end{gather}$$

Zürner (Reference Zürner2020) made estimates for the different terms in the above equations, based on the following assumptions: (1) the Prandtl number of liquid metals is sufficiently low, $Pr\ll 1$; (2) $Q^{1/2}$ is sufficiently large, and the viscous boundary layers on the walls perpendicular to the magnetic fields are replaced by the Hartmann layer; (3) the magnetic Reynolds number is low enough, $Rm \ll 1$; and (4) additional regime transitions, e.g. transition concerning the onset of convection under the effects of magnetic fields. He obtained the following relationships:

(3.9)$$\begin{gather} \varepsilon_{\nu,Bulk} \sim \frac{{\nu}^{3}}{H^{4}}\,Re^{3}, \end{gather}$$

(3.10)$$\begin{gather}\varepsilon_{\nu,BL} \sim \frac{{\nu}^{3}}{H^{4}}\,Re^{2}\,Q^{1/2}, \end{gather}$$

(3.11)$$\begin{gather}\varepsilon_{\eta,Bulk} \sim \mu \rho_{0}\,\frac{{\nu}^{3}}{H^{4}}\,Re^{2}\,Q, \end{gather}$$

(3.12)$$\begin{gather}\varepsilon_{\eta,BL} \sim \mu \rho_{0}\,\frac{{\nu}^{3}}{H^{4}}\,Re^{2}\,Q^{3/2}, \end{gather}$$

(3.13)$$\begin{gather}\varepsilon_{\kappa,Bulk} \sim \kappa\,\frac{(\Delta T)^{2}}{H^{2}}\,Re\,Pr, \end{gather}$$

(3.14)$$\begin{gather}\varepsilon_{\kappa,BL} \sim \kappa\,\frac{(\Delta T)^{2}}{H^2}\,\sqrt{Re\,Pr}. \end{gather}$$

When $N$ ($=Q/Re$) is large enough, the bulk flow region approaches a quasi-2-D state, where the viscous dissipation should be dominated by $\varepsilon _{\nu,BL}$. To substantiate this viewpoint more rigorously, we have tabulated the time-averaged values of viscous and Joule dissipation in the bulk flow and boundary layer regions, respectively, when $Ra=1\times 10^6$, as shown in table 2. It is worth noting that when considering dissipation, especially in the bulk flow region, forces other than the Lorentz force, such as the inertial force, must also be taken into account (Pothérat & Klein Reference Pothérat and Klein2017). Therefore, $N$ is used as a control parameter here. The results demonstrate strongly that when $N \geq 253.46$, the viscous dissipation in the bulk flow region is much smaller than that in the boundary layer, almost $\varepsilon _{\nu,Bulk}\approx 0.1\varepsilon _{\nu,BL}$. Therefore, it can be concluded that the main contribution to viscous dissipation comes from the boundary layer, i.e. $\varepsilon _{\nu } \sim \varepsilon _{\nu,BL}$. This approximation is evidently unsuitable for cases where $N$ is relatively small. For instance, when $N = 75.60$, the viscous dissipation in the bulk flow region is similar to that in the boundary layer. Meanwhile, the thermal dissipation for convective liquid metals with $Pr \ll 1$ satisfies $\varepsilon _{\kappa } \sim \varepsilon _{\kappa,BL}$.

Table 2. Comparison of the volume- and time-averaged viscous dissipation $\varepsilon _{\nu }$, the Joule dissipation rates $\varepsilon _{\eta }$, and the averaged number of the roll structures at $Ra=1\times 10^6$.

For the Joule dissipation, when the magnetic field is strong enough, it can be inferred from (3.11) and (3.12) that the contribution from the boundary layer should be greater than that from the bulk region. This phenomenon can be explained easily from a physical perspective. Under a strong magnetic field, the flow becomes more regular, and the Joule dissipation in the bulk region is small. The electric current concentrates mainly in the Hartmann layer, thus resulting in greater Joule dissipation in the boundary layer. The computational results also confirm this viewpoint. When the magnetic field is weak, such as $N=75.60$, $174.69$ and $253.46$, the Joule dissipation in the bulk region is roughly equivalent to that in the boundary layer. However, as the magnetic field continues to increase and $N$ reaches $530.34$ and $1045.96$, the Joule dissipation in the bulk region is significantly lower than that in the boundary layer.

It is difficult to obtain the relative magnitudes of viscous dissipation and Joule dissipation from the approximate equations (3.9)–(3.12) under a strong magnetic field. Based on the calculation results, it can be observed that when the magnetic field is weak, such as $N < 253.46$, the contributions of viscous dissipation and Joule dissipation are relatively close. When $N\geq 253.46$, viscous dissipation will be much greater than Joule dissipation, and as the magnetic field continues to increase, the difference between the two will become more significant. Therefore, under a strong magnetic field, we can conclude that

(3.15)\begin{equation} \varepsilon_{\nu}+ \varepsilon_{\eta}\sim \varepsilon_{\nu} \sim \varepsilon_{\nu,BL}. \end{equation}

From this, it can be derived that the momentum dissipation (including both the viscous dissipation and Joule dissipation) and thermal dissipation for $Pr \ll 1$ (Grossmann & Lohse Reference Grossmann and Lohse2000) satisfy approximately

(3.16)$$\begin{gather} \frac{(Nu-1)\,Ra}{Pr^{2}} \sim Re^{2}\,Q^{1/2}, \end{gather}$$

(3.17)$$\begin{gather}Nu-1 \sim \sqrt{Re\,Pr}. \end{gather}$$

Regarding the flow of liquid metal studied in this paper, with $Pr$ being unchanged, it can be obtained that

(3.18)\begin{equation} (\sqrt{Re\,Pr})Ra \sim Re^{2}\,Q^{1/2} \rightarrow Re\,Pr/Ra^{2/3} \sim Q^{-1/3}. \end{equation}

As shown in figure 2(b), the numerical results exhibit a scaling law of 0.35 for cases at different $Ra$, which agrees well with the derived scaling law of $1/3$.

3.2. Flow reversal and turbulence intermittency

The phenomenon of flow reversal, which was observed previously in experimental studies at $Ra\sim 10^5$ (Yanagisawa et al. Reference Yanagisawa, Yamagishi, Hamano, Tasaka and Takeda2011, Reference Yanagisawa, Hamano, Miyagoshi, Yamagishi, Tasaka and Takeda2013) and numerical studies at $Ra \leq 10^6$ (Vogt et al. Reference Vogt, Ishimi, Yanagisawa, Tasaka, Sakuraba and Eckert2018; Liu Reference Liu2019), has been rediscovered in the present work at $Ra=10^7$ with a larger aspect ratio $\varGamma =5$. Our investigations have revealed that this reversal phenomenon is an inherent characteristic of MHD-RBC flows, and is associated with specific parameter combinations. Furthermore, we intend to provide a detailed depiction of the evolution process of flow reversals based on the integrated kinetic energy equation

(3.19)\begin{equation} \dfrac{{\rm d}}{{\rm d}t}\int\dfrac{u^{2}}{2}\,{\rm d}V =-\int\varepsilon_{\nu} \,{\rm d}V-\int\varepsilon_{\eta} \,{\rm d}V+\int u_{z}T \,{\rm d}V, \end{equation}

where $V$ represents the entire domain. Presented here are the developments in the flow field resulting from a singular flow reversal procedure, as well as the subsequent evolutions of the kinetic energy $u^{2}$, with regard to $u_{x}^{2}$, $u_{y}^{2}$ and $u_{z}^{2}$, the rate of viscous dissipation $\varepsilon _{\nu }$, the rate of Joule dissipation $\varepsilon _{\eta }$, and the thermal flux $u_{z}T$.

The exemplification of temporal progression manifests in figures 3 and 4, where turbulent bursts alternate with periods of quasi-2-D behaviour. As delineated in figure 4(a), the nascent phase ($t_{1}$) manifests as quasi-2-D, comprising four quasi-2-D rolls labelled as 1, 2, 3 and 4, correspondingly. At this juncture, the vertical energy diminishes substantially but does not reach null, while the energy within the plane perpendicular to the magnetic field attains its zenith, illustrated in figure 3(c). Progressing from time $t_{1}$ to $t_{2}$, the rolls undergo curvature where the latter halves of rolls 1 and 2 contract, whereas rolls 3 and 4 expand, exhibiting an evolution contrary to that near the frontal wall. Throughout this process, depicted in figure 3, both $u^{2}_{y}$ and the Joule dissipation $\varepsilon _{\eta }$ escalate, reaching their peak at $t_{2}$, signifying that the restoration of 3-D turbulent flow occurs. Following this, from $t_{2}$ to $t_{3}$, as the rolls persist in their curvature, rolls 5 and 6 materialize at the extremities of the vessel, as portrayed in figure 4(c), while the roll-like configurations adjacent to the remaining corners almost dissipate. Concurrently, rolls 2 and 3 undergo displacement in the centre (refer to figure 4). During this phase, $u^{2}_{y}$ declines, and the predominant contributions of kinetic energy, $u^{2}_{x}$ and $u^{2}_{z}$, also dwindle further as $\varepsilon _{\nu }+\varepsilon _{\eta }$ continues to surpass $u_{z}T$. When the intersection of $\varepsilon _{\nu }+\varepsilon _{\eta }$ with $u_{z}T$ occurs at $t_{3}$, $u^{2}_{x}$ and $u^{2}_{z}$ attain their minimum values.

Figure 3. Results for $Q=2.25\times 10^6$, $Ra=1 \times 10^{7}$. (a) Time-space maps of vertical velocity along line $(2.5,y,0.75)$. (b) Time variations of energy $u_{x}^{2}$, $u_{y}^{2}$, $u_{z}^{2}$. (c) Time variations of the volume-averaged heat transport $u_{z}T$, viscous dissipation rate $\varepsilon _{\nu }$, and Joule dissipation rate $\varepsilon _{\eta }$.

Figure 4. Flow evolution during one reversal procedure with $Q=2.25\times 10^6$, $Ra=1 \times 10^{7}$. Six snapshots are shown, denoted by the six vertical dashed lines in figure 3. The streamlines of 3-D velocity field are coloured with vertical velocity: red for upwards motion, and blue for downwards motion.

From the interval between $t_{3}$ and $t_{5}$, the fragmented convection rolls begin a process of reintegration. Roll 5, newly formed, inclines towards the posterior section of roll 2, while the leading half of roll 3 appears to merge with the recently generated roll 6. Within the central zone, the trailing half of roll 3 endeavours to connect with the leading half of roll 1, while the frontal section of roll 2 seeks to bond with the rear part of roll 4. Throughout this phase, the magnitude of $u_{z}T$ surpasses that of $\varepsilon _{\nu }+\varepsilon _{\eta }$, consequently initiating an ascent in both $u_{x}^{2}$ and $u_{z}^{2}$. As alignment between the rolls and the magnetic field occurs at $t_{4}$, the values of $u_{y}^{2}$ and $\varepsilon _{\nu }+\varepsilon _{\eta }$ descend to their minimal levels. By $t_{6}$, $\varepsilon _{\nu }+\varepsilon _{\eta }$ exceeds $u_{z}T$, leading to a surge in $u_{x}^{2}$ and $u_{z}^{2}$. A novel quasi-2-D structure emerges, comprising four rolls rotating in the opposite direction to those observed at $t_{1}$.

Let us now examine the possible physical mechanisms of flow reversal, i.e. the skewed-varicose instability. As mentioned by Greenside, Cross & Coughran (Reference Greenside, Cross and Coughran1988) and Busse, Kropp & Zaks (Reference Busse, Kropp and Zaks1992), the skewed-varicose instability is crucial for understanding the transition of fluid from small to medium $Pr$ ($Pr\leq 1$) towards chaos. One of its key elements is the mean flow, which refers to the flow with a larger length scale compared to the wavelength of convective eddies. Yanagisawa et al. (Reference Yanagisawa, Hamano, Miyagoshi, Yamagishi, Tasaka and Takeda2013, Reference Yanagisawa, Hamano and Sakuraba2015) and Vogt et al. (Reference Vogt, Ishimi, Yanagisawa, Tasaka, Sakuraba and Eckert2018) explained the process of flow reversals in RBC as the creation of new rolls and the following skewed-varicose instability (instability in both directions perpendicular and parallel to the axis of roll). In this process, the horizontal circulation plays an important role, which can be quantified by introducing the global angular momentum. For our research, the global angular momentum along the $z$ direction around the centre point $C(x_c, y_c, z_c)$ of the container can well reflect the mean flow. This is defined as $L(z) = \langle (x-x_c)u_y-(y-y_c)u_x \rangle _{V}$, where the brackets denote the spatial average.

Upon examining figure 5, it is apparent that under the condition $Ra=1\times 10^6$, when $Q=2.5\times 10^5$, the flow pattern exhibits an irregular three-roll structure, accompanied by a flow reversal. When $Q=6.4\times 10^5$, the flow displays a relatively regular four-roll structure, with a flow reversal occurring periodically. Conversely, when $Q=2.25\times 10^6$, the flow maintains a stable five-roll structure, without any instances of reversal, as shown in figure 1(d). Figure 6 illustrates the global angular momentum $L(z)$ corresponding to these three conditions. When $Q=2.5\times 10^5$, the global angular momentum $L(z)$ reaches its peaks at $t=176$ and $t=221$, respectively, and these peaks are greater than those observed under $Q=6.4\times 10^5$. However, as shown in figure 5, the flow does not undergo a rapid reversal after these two peaks, but rather gradually reverses at $t=230$. In contrast, for $Q=6.4\times 10^5$, the flow experiences rapid reversals after the global angular momentum $L(z)$ reaches its peaks at $t=46$, $102$, $160$ and $220$. It is worth noting that a change in the direction of flow does not necessarily imply a change in the sign of the global angular momentum. The two are not directly related, as both positive and negative global angular momentum can lead to the same result in terms of flow reversal. When $Q=2.25\times 10^6$, the flow does not undergo any reversal, resulting in the global angular momentum $L(z)$ almost entirely equalling 0. Therefore, the global angular momentum $L(z)$, which reflects the mean horizontal circulation, can explain to some extent the occurrence of a flow reversal. Additionally, when the magnetic field is weak, the many small-scale, 3-D vortices attached to the primary convective roll contribute to an increase in the global mean horizontal circulation. As a result, $L(z)$ becomes less useful in explaining a flow reversal under these conditions.

Figure 5. Time–space maps of vertical velocity along line $(2.5,y,0.75)$ for the case with $Ra=1\times 10^6$ and (a) $Q=2.5\times 10^5$, (b) $Q=6.4\times 10^5$.

Figure 6. Global angular momentum $L(z)$ along the $z$ direction at the central point of the vessel for three typical flow regimes with $Ra=1\times 10^6$ and $Q=2.5\times 10^5$, $6.4\times 10^5$, $2.25\times 10^6$.

In fact, it is not difficult to observe that this periodic reversal bears resemblance to the large-scale temporal intermittency observed in channel flow with a spanwise magnetic field (Boeck et al. Reference Boeck, Krasnov, Thess and Zikanov2008), forced turbulence in a periodic box (Zikanov & Thess Reference Zikanov and Thess1998), and inviscid flow in a triaxial ellipsoid (Thess & Zikanov Reference Thess and Zikanov2007). In these cases, the flows experience prolonged periods of nearly steady laminar states, punctuated by brief turbulent bursts. And the typical duration of the large-scale intermittent cycle is linked intricately to both driving strength and magnetic field intensity. The evolution of all the phenomena also appears to be analogous. When $Q$ is moderate, the magnetic field is not potent enough to suppress 3-D secondary instabilities, but is strong enough to impede their transition to fully developed turbulence. After a brief outburst, the flow reverts to a nearly laminar base state, where the Joule dissipation is low, and instabilities can subsist. However, with increasing $Q$, the strength of the turbulent burst diminishes, and the duration of the instability growth phase extends, resulting in a longer reversal period.

3.3. Three-dimensionality

As described above, a horizontal magnetic field of sufficient strength can convert convective motion into quasi-2-D rolls that align parallel to the magnetic field. Additionally, the derivation of the scaling relationship for the cutoff frequency $f_{c}$ (or cutoff length scale) that distinguishes high-frequency fluctuations in three dimensions from low-frequency fluctuations in two dimensions relies solely on the true interaction parameter Sommeria & Moreau (Reference Sommeria and Moreau1982). Hence we expect the scaling law between the interaction parameter and cutoff frequency to apply in the case of turbulent thermal convection influenced by a horizontal magnetic field, as observed in turbulence with a strong background flow (Klein & Pothérat Reference Klein and Pothérat2010; Pothérat & Klein Reference Pothérat and Klein2014; Baker et al. Reference Baker, Pothérat, Davoust and Debray2018) and shear turbulence (Chen et al. Reference Chen, Pothérat, Ni and Moreau2021).

Figures 7(c–f) display pairs of spectra obtained by applying different filter frequencies to the frontal velocity signal. It is noteworthy that, according to the discussion of Kumar & Verma (Reference Kumar and Verma2018), the Taylor assumption is applicable to turbulent convection. The original signals were collected at positions on the Hartmann walls that aligned precisely with the magnetic field lines (i.e. at $y=0$ and $y=\varGamma$) but at the same coordinates $(x,z)$, and the probes were located in the central square region to minimize the influence of the sidewalls. Moreover, it should be noted that only the velocity signals recorded by probes positioned in the middle region of adjacent convection rolls are utilized. In the case where $Ra=10^6$ and $Q=2 \times 10^5$, the frontal and back spectra are partially identical in figure 7(c), which is expected since the flow tends to exhibit quasi-2-D behaviour at most scales. However, the spectra differ significantly at higher frequencies, indicating that small-scale structures are more sensitive to three-dimensionality. As $Q$ decreases (with $Ra$ fixed), the frequency range affected by three-dimensionality extends to lower frequencies, eventually contaminating the entire spectrum at the lowest $Q$ values (e.g. $Q=0$). By comparing pairs of spectra, the theory of Sommeria & Moreau (Reference Sommeria and Moreau1982) and experiments conducted by Baker et al. (Reference Baker, Pothérat, Davoust and Debray2018) have identified a cutoff length scale $k_{c}$ (corresponding to $f_{c}$) that separates quasi-2-D structures from 3-D structures.

Figure 7. (a) Variations of partial correlation $c(\,f)$ with the cutoff frequency of filter $\mathscr {B}_{f}$ applied to the velocity signal measured at the frontal wall. Curves are obtained for $Ra =10^{6}$ and $Q=2 \times 10^{5}$ for a pair of probes within the centre square. Curves obtained for different parameters exhibit the same features. (b) The cutoff frequency $f_{c}$, separating quasi-2-D large structures from the small 3-D ones with the interaction parameter $N$. (c–f) Power spectral density (PSD) of the filtered frontal and back signals plotted in terms of the frequency.

Here, we applied a partial correlation function between signals on the front and back Hartmann walls (see Pothérat & Klein Reference Pothérat and Klein2014) to quantify the variation of $f_{c}$ with $N$ ($=Q/Re$):

(3.20)\begin{equation} c(\,f)= \dfrac{\sum_{t=0}^{Ti} \mathscr{B}_{f}\,u_{front}(t)\,u_{back}(t)}{\sqrt{\sum_{t=0}^{Ti} \mathscr{B}_{f}\,u^{2}_{front}(t)\,u^{2}_{back}(t)}}. \end{equation}

Here, Ti is the duration of the recorded signals. The signal obtained from the frontal wall is replaced with a filtered counterpart, processed through an eighth-order low-pass filter of cutoff frequency denoted as $\mathscr {B}_{f}\,u_{front}$, as shown in figure 7(a). Generally, the large frequency displayed in the spectrum corresponds to a small flow structure in physical space. As the filter frequency $f$ is increased from 0, more and more fluctuation scales are added to the filtered frontal signal, resulting in better correlation with the back signal in the quasi-2-D flow field. However, when $f$ is increased beyond a critical value $f_{c}$ (refer to figure 7f), the small scales, where three-dimensionality is still present, will disrupt the correlation between the two signals, leading to a decrease in $c(\,f)$. Hence the position of the maximum $f_{c}$ provides an excellent measure of the separation between large-scale 2-D structure and small-scale 3-D structure.

Figure 7(b) depicts the variations of the spatially averaged cutoff frequency $f_{c}$ with $N$, indicating that the changes in $f_{c}$ for all the analysed cases converge onto a single curve. This result implies that $f_{c}$ follows a scaling law that is determined by $N$:

(3.21)\begin{equation} f_{c} \sim N^{0.36}. \end{equation}

This general scaling law provides a precise estimate for the minimum frequency of vortices that are influenced by 3-D inertial effects. Thus, similar to shear turbulence or turbulence with a strong background flow, the MHD-RBC turbulence also exhibits the same scaling law for the cutoff frequency $f_{c}$ (Klein & Pothérat Reference Klein and Pothérat2010; Pothérat & Klein Reference Pothérat and Klein2014; Baker et al. Reference Baker, Pothérat, Davoust and Debray2018).