3.1. Variety of initial number of rolls at high $Q$

The application of a magnetic field restricts flow structures to quasi-2-D shape by Joule dissipation on the fluid motion parallel to the magnetic field (Davidson Reference Davidson1995). For RBC with a horizontal magnetic field, this provides convection rolls arranged parallel to the magnetic field, and also modifies the number of convection rolls (i.e. wavenumber of the structure) depending on $Ra$ and $Q$ (Yanagisawa et al. Reference Yanagisawa, Hamano, Miyagoshi, Yamagishi, Tasaka and Takeda2013; Tasaka et al. Reference Tasaka, Igaki, Yanagisawa, Vogt, Zuerner and Eckert2016). Unlike our earlier studies conducted at smaller $Q$ (Yanagisawa et al. Reference Yanagisawa, Yamagishi, Hamano, Tasaka and Takeda2011, Reference Yanagisawa, Hamano, Miyagoshi, Yamagishi, Tasaka and Takeda2013; Tasaka et al. Reference Tasaka, Igaki, Yanagisawa, Vogt, Zuerner and Eckert2016), a notable feature of the present experiments performed at relatively high $Q$ is that several repetitions of the experiments do not produce exactly the same number of convection rolls. Depending on the $Ra$ number, different probabilities are found for the occurrence of an initial number of rolls $n$. Figures 2(a)–2( f) show histograms of the probability density on $n$ observed at a fixed $Q$ of $2.0 \times 10^5$, and variable $Ra$ values ($2.3 \times 10^4$ to $1.5 \times 10^5$), where solid circles with bars represent the averages and corresponding standard deviations. The number of rolls was identified using the spatio-temporal velocity maps, $u_y(y, t)$, measured by the ultrasonic transducer (UV1, figure 1). An example of a map is shown in figure 2(g), where the seven white and black stripes represent individual convection rolls and the adjacent rolls show opposite rotation directions. The identification of the number of rolls based on velocity measurements along the measurement line perpendicular to the magnetic field is quite straightforward, because a sufficiently high field strength stabilizes the rolls and aligns them along the field lines. The seven-roll regime ($n = 7$) shown in figure 2(g) was first observed in this study, whereas previous studies assumed this number of rolls for sufficiently high $Q$ from the observation that $n$ increases with $Q$ (Yanagisawa et al. Reference Yanagisawa, Hamano, Miyagoshi, Yamagishi, Tasaka and Takeda2013; Tasaka et al. Reference Tasaka, Igaki, Yanagisawa, Vogt, Zuerner and Eckert2016).

Figure 2.

Probability density function for the initial number of convection rolls observed at $Q = 2.0 \times 10^5$ and at several $Ra$ numbers: (a) $2.3 \times 10^4$, (b) $3.8 \times 10^4$, (c) $7.7 \times 10^4$, (d) $9.2 \times 10^4$, (e) $1.2 \times 10^5$, and (f) $1.5 \times 10^5$, where solid circles with bars represent the average and standard deviations. (g) An example of a spatio-temporal velocity map for $n = 7$.

The initial number of convection rolls obtained in the present experiments ranges from $n = 3$ to 8 as shown in the histograms (figure 2a–f). In these, $n = 3$, 4 and 8 are unstable states, where the initial roll number changed over time for the same $Ra$ and $Q$ numbers. With three stable states, $n = 5$, 6 and 7 occur with bistability that has not been previously reported for smaller $Q$. This may be due to a reduced horizontal size of the stable convection rolls at larger $Q$ relative to the vessel size, $\varGamma = 5$ in aspect ratio. The experimental procedure in the present study, giving initially a certain temperature difference for the set $Ra$ number, may also affect the bistability. The average and standard deviation of $n$ in the histograms show no significant tendency with respect to $Ra$ number in the range examined here.

The present study focuses on the onset of convection roll oscillations with decreasing $Q$ at a given $Ra$ number by velocity mapping and temperature fluctuation measurements. The thermocouple positions with respect to the convection rolls must be taken into account to evaluate the temperature fluctuation because, in the case of oscillations, their amplitude should be greater if the measuring position is in between two adjacent rolls. In the stable states, five- and six-roll states are sustained with decreasing $Q$ whereas the seven-roll state shows a transition to the six-roll state with slight decrease of $Q$. We therefore investigated the onset of the oscillations for the five- and six-roll states. In figure 3, the examined $Ra$ and $Q$ values are displayed with the corresponding roll number state and flow regimes with different symbols. Light circular symbols with a central mark and bold circular symbols with no central mark indicate five- and six-roll states, respectively, and bold circular symbols with a central mark indicate that both states are observed for the same set parameters; and open and red-filled symbols represent non-oscillation and oscillatory regimes, respectively. The recognition of an oscillatory or non-oscillation regime was tentatively given from the spatio-temporal velocity map.

Figure 3.

Parameter range examined in the present experiments. Open and red-filled circles represent non-oscillation and oscillatory states, respectively. The thickness of the circle and central mark in the circles indicates the corresponding number of rolls.

3.2. Onset of oscillatory convection

Each measurement series performed for a $Ra$ number starts with sufficiently large $Q$ values at which distinct oscillatory motion is not observed. With the stepwise reduction of the magnetic field, oscillations occur at certain $Q$, whose properties change with further reduction of $Q$. Figure 4 shows an example for an oscillating five-roll state at $Ra = 1.2 \times 10^5$ and $Q = 5.0 \times 10^4$. The spatio-temporal velocity map in figure 4(a) was measured by UV1 and the corresponding temperature fluctuation in figure 4(b) was measured by the temperature probe $T$ (figure 1). In the velocity map, oscillations are visible as periodic changes in the boundaries between the neighbouring convection rolls, with the three inner rolls showing greater fluctuations than those on the vessel sidewalls. The main features are similar to those observed in the 2-D oscillations (Vogt et al. Reference Vogt, Ishimi, Yanagisawa, Tasaka, Sakuraba and Eckert2018, figures 4 and 10), which are the main subject for investigation in the present study. The two-dimensionality of the oscillations is further evaluated below. The corresponding temperature fluctuation shows clearer periodic oscillations than in the velocity map; in the five-roll state, the position of probe $T$ is at the boundary between the second and third rolls and can effectively capture the oscillation because of the relatively large oscillation amplitude expected at the boundary.

Figure 4.

Sample of (a) spatio-temporal velocity map measured by UV1 and (b) corresponding temperature fluctuations measured by temperature probe $T$ (figure 1) in the oscillation regime at $Ra = 1.2 \times 10^5$ and $Q = 5.0 \times 10^4$.

The power spectra from a Fourier transformation of the fluctuations of velocity and temperature shown in figure 4 are plotted in figure 5. The velocity fluctuation for the spectra in figure 5(a) is extracted from figure 4(a) at $y = 120.56\ \textrm {mm}$ and corresponds to the boundary between the main convection rolls, where the root-mean-square (r.m.s.) value of the velocity fluctuations reaches the maximum. Both spectra have a sharp peak corresponding to the roll oscillation at $f = 0.086\ \textrm {Hz}$ ($\,f_{OS}$). The critical Chandrasekhar number for the onset of oscillations, $Q_{OS}$, is determined from the velocity fluctuations. Taking into account the occurrence of the five- or six-roll state, the location to extract the velocity fluctuations is always chosen as the boundary between the two central convection rolls. The zero-crossing point from the linear extrapolation of the spectral peak values from the velocity fluctuations gives the critical value $Q_{OS}$ (see schematic diagram in figure 5c). The spectral peak values of the temperature fluctuations are also used to cross-check the $Q_{OS}$ values estimated from the velocity fluctuations.

Figure 5.

Power spectra for (a) velocity (at $y = 120.56\ \textrm {mm}$) and (b) temperature fluctuations, both with the data shown in figure 4. (c) Schematic diagram for the variation of peak power of the frequency component of the velocity fluctuation at $f = f_{{OS}}$.

The determined $Q_{OS}$ values obtained at both the five- and six-roll states and at different $Ra$ numbers are plotted in figure 6, with the five- and six-roll states shown by solid squares and open circles, respectively. The symbols for the two states appear to overlap, which suggests only a small influence of the number of rolls on the onset of oscillation. For comparison, the critical values obtained in similar experimental work by Burr & Müller (Reference Burr and Müller2002) are plotted as crosses in figure 6. The difference between the present results and data obtained in Burr & Müller (Reference Burr and Müller2002) is apparent, even if different criteria and procedures were applied to determine the critical values. For a given $Q$ value, the critical $Ra$ number for the onset of oscillatory convection is one order of magnitude smaller than the present results. The most significant difference between the experiments is the size of the fluid layer; the present study uses a square vessel with $\varGamma = 5$ aspect ratio, whereas Burr & Müller (Reference Burr and Müller2002) used a larger rectangular vessel with sides of $10L$ and $20L$ in directions, respectively, parallel and perpendicular to the magnetic field. Simply stated, this difference is from a magnetohydrodynamics (MHD) effect due to the vessel size and differences in the oscillatory convection criteria. A detailed investigation of the corresponding flow structures in § 3.4 elucidates that a different oscillatory convection type emerges even below the boundary determined here. These points are further discussed below.

Figure 6.

Critical $Q$ for the onset of oscillation, $Q_{OS}$, obtained at different $Ra$ values with decreasing $Q$. Solid squares and open circles represent $Q_{OS}$ for $n = 5$ and 6, respectively. The crosses indicate critical $Ra$ numbers for the onset of oscillation obtained at different $Q$ values from Burr & Müller (Reference Burr and Müller2002).

3.3. Development of oscillatory convection

This section considers the development of oscillation intensity with decreasing magnetic field strength beyond the onset of oscillations at $Q = Q_{OS}(Ra)$. Temperature fluctuations are used to monitor the development instead of the spatio-temporal velocity maps. This decision is justified by the long measurement times of approximately 4 h, during which a noticeable separation of reflecting particles occurs due to the small difference in density between the liquid metal and particle. This leads to a deterioration of the measured signals by the increasing noise level. The variations of r.m.s. of the temperature fluctuations, $T_{rms}$, with decreasing $Q$ are shown in figure 7(a) for different $Ra$ values, in a five-roll state (i.e. $n = 5$). The grey band at the bottom of the figure indicates the noise level below which no temperature fluctuations can be reliably detected. The points at which the curves reach the noise level correspond fairly well with the onset of oscillations determined by analysing the velocity data (see figure 5).

Figure 7.

(a) Variations of the r.m.s. value of the temperature fluctuations $T$ (figure 1) with decreasing $Q$ at different $Ra$ values, where the grey hatched area represents data lower than the noise level. (b) Variations normalized by the maximum r.m.s. values, with respect to $Q/Ra$, where the grey symbols (right side) represent data points lower than the noise level. The corresponding number of convection rolls is $n = 5$ (five-roll state).

The variations of $T_{rms}$ versus $Q$ can be merged into a single curve by normalizing $T_{rms}$ with the maximum r.m.s. value, $T_{rms,max}$, for the respective measurement series at a given $Ra$ (figure 7b) and $Q$ with $Ra$ values. This suggests that developments in the oscillation are organized by $Ra/Q$, such as the organization of different flow regimes determined in our previous studies (Yanagisawa et al. Reference Yanagisawa, Hamano, Miyagoshi, Yamagishi, Tasaka and Takeda2013; Tasaka et al. Reference Tasaka, Igaki, Yanagisawa, Vogt, Zuerner and Eckert2016; Vogt et al. Reference Vogt, Ishimi, Yanagisawa, Tasaka, Sakuraba and Eckert2018). It may be expected from the unified curves that $T_{rms}$ decreases with a further reduction of $Q$ at lower $Ra$ conditions than $Ra = 1.8 \times 10^5$. This might be caused by 3-D developments of the oscillatory motion of a convection roll (Vogt et al. Reference Vogt, Ishimi, Yanagisawa, Tasaka, Sakuraba and Eckert2018), and is discussed further in § 3.5 from the viewpoint of the oscillation frequency and characteristic velocity of the convection roll.

3.4. Structures and source of the 2-D oscillations

To verify the two-dimensionality of the oscillatory motions from phase differences of the local velocity fluctuations, we simultaneously measured velocity profiles along three different measurement lines at UV1, UV2 and UV3 (figure 1). The velocity fluctuations corresponding to $f = f_{OS}$ were extracted by discrete Fourier transformation from the time series of the velocity signals for the position $y = 120.56\ \textrm {mm}$, where two neighbouring rolls encounter each other. Figure 8 presents the velocity time series of all three sensors, showing distinct oscillations at $f = f_{OS}$. The signals of sensors UV1 and UV3, which are mounted at the same height ($z = 10\ \textrm {mm}$) and different $x$-positions ($x = 160$ and 100 mm), are in phase, whereas the signals recorded by UV1 and UV2 mounted at the same $x$-position at the bottom ($z = 10\ \textrm {mm}$) and top ($z = 30\ \textrm {mm}$) of the fluid layer oscillate in antiphase. These results coincide with the basic features of the 2-D flow structure and thus fundamentally differ from the 3-D travelling wave oscillations in conventional Rayleigh–Bénard convection (Clever & Busse Reference Clever and Busse1987).

Figure 8.

Velocity fluctuations extracted at $f = f_{OS}$ by Fourier transformation from the original velocity fluctuations measured by different ultrasonic transducers, UV1 to UV3 (figure 1) at $y = 120.56\ \textrm {mm}$, and at $Ra = 1.2 \times 10^5$ and $Q = 5.0 \times 10^4$.

The reason that 2-D oscillatory motion appears at the onset of oscillatory convection rather than the 3-D travelling waves, as predicted in previous studies (Busse & Clever Reference Busse and Clever1983; Clever & Busse Reference Clever and Busse1987), requires further explanation. Stability analysis performed by Busse & Clever (Reference Busse and Clever1983) indicated that the stable region boundary of the 2-D convection rolls in ordinarily occurring oscillatory instability widens by applying a horizontal magnetic field. Our measurements suggest that another type of instability occurs in the case of the magnetic field. This section discusses the mechanics of these 2-D oscillations, together with the details of roll motion corresponding to the 2-D oscillations obtained in the present experimental results.

The mean flow structure in the case where the 2-D oscillations occur at $Ra = 1.2 \times 10^5$ and $Q = 5.0 \times 10^4$ ($Q/Ra = 0.42$) is presented in figure 9 as profiles of the time-averaged velocity $U_y$ (solid line) measured at the bottom (black, UV1) and top (grey, UV2) (see figure 1). The mean velocity profile exhibits five conversions of flow directions corresponding to a five-roll state. However, the profile shape is not simply sinusoidal but shows substantial asymmetric deformations (i.e. humps). Furthermore, neighbouring positive and negative parts of the profile with a hump seem to appear in the point symmetry, and the profiles measured by the top and bottom line are also related by point symmetry. A profile of the standard deviation of the velocity fluctuations measured along the bottom line (UV1), $u_{y,rms}$, is also plotted in figure 9 (broken line). The standard deviation assumes sharp peaks around positions with $U_y = 0$, corresponding to the boundaries between rolls and large values around the deformation area of the $U_y$ profiles.

Figure 9.

Profiles of mean velocity $U_y$ and standard deviation of velocity fluctuations $u_{y,rms}$ measured at $Ra = 1.2 \times 10^5$ and $Q = 5.0 \times 10^4$ ($Q/Ra = 0.42$) by ultrasonic transducers at the bottom and top positions, UV1 and UV2 (figure 1) (the corresponding spatio-temporal velocity map for UV1 is shown in figure 4a).

We performed additional numerical simulations to obtain more details of the structures corresponding to the typical velocity profiles. The simulation parameters were set close to the experimental conditions in figure 9, $Ra = 1.0 \times 10^5$ and $Q = 3.2 \times 10^4$ ($Q/Ra = 0.32$). The isosurface of the $Q_{3D}$ value, the second invariant of the velocity gradient tensor calculated from the simulation results, displays convection rolls. A quasi-2-D shape of the rolls in the oscillatory condition provides evidence of 2-D oscillatory motion (figure 10a). We also calculated a spatio-temporal velocity map from the simulation results along a line corresponding to the present UV1 line measurement ($z = 0.25L$; dark red line in figure 10a). This map shows finer structures of the oscillatory convections (figure 10b). The corresponding mean velocity profile shown in figure 10(c) also has humps that reflect modifications of the main rolls. Figures 10(d)–10(f) show sequential snapshots of the vorticity field on a vertical cross-section of the fluid layer perpendicular to the magnetic field ($\omega _x$). Red and blue contours in the various panels represent positive and negative values of vorticity in the direction of the applied magnetic field ($x$-axis), respectively. The broken line in figure 10(d) indicates the measurement line corresponding to the ultrasonic transducer (figure 1). At the places corresponding to the humps, there is an additional recirculation vortex pair between the main rolls. The vortices seem to be a separation vortex pair from the main rolls and appear in between the roll flows detached from the wall, not approaching or adhering to the wall.

Figure 10.

Results of numerical simulation performed for $Ra = 1.0 \times 10^5$ and $Q = 3.2 \times 10^4$ ($Q/Ra = 0.32$). (a) The isosurface of $Q_{3D}$ represents the roll structures, where the dark red line indicates the place of the UV1 measurement line. (b) Spatio-temporal velocity map extracted along a line corresponding to the UV1 measurement line (see figure 1). (c) Time-averaged velocity profile of the velocity map. (d–f) Instantaneous contour maps of vorticity on the vertical cross-section at $x = 160\ \textrm {mm}$, where red and blue contours represent positive and negative values of the vorticity, respectively (see also supplementary movie 1).

Sequential snapshots shown in figure 10(d–f) also represent the motion of the rolls corresponding to the 2-D oscillations as a periodic distortion of the rolls into elliptic shapes. A remarkable feature of this distortion is that the roll axes are almost fixed at both horizontal and vertical positions. The distortion is accompanied by entraining the vorticity from the recirculation vortex pair; the entrainment occurs periodically and alternately between neighbouring main rolls, and deformation of the rolls behaves synchronously (see also supplementary movie 1 available at https://doi.org/10.1017/jfm.2020.1047). By the entrainment, the main rolls seem to be surrounded by an annular layer of opposite-sign vorticity. In this oscillation mode, fluid particles may follow an elliptic particle path in a $y$–$z$ cross-section different from the 3-D travelling wave oscillations. Along the periodic distortion, the cross-sectional area of the rolls periodically expands and shrinks similar to ‘breathing’. This behaviour of the rolls with rotation of the major axis of the elliptic shape is expressed by diagonal lines on the space–time velocity map (figure 10b). Similar diagonal stripe patterns are also observed in the velocity map measured by UVP (figure 4a). This supports the reliability of this numerical simulation.

No oscillatory convection is detected at $Q > Q_{OS}(Ra)$, according to the present criteria. We investigated the mean velocity profile taken at this condition of the experiment to highlight the differences in the convection roll structures (figure 11). Here the profile was measured along UV1 at $Ra = 7.9 \times 10^4$ and $Q = 2.0 \times 10^5$ ($Q/Ra = 2.5$). The profile still has small but distinct humps and is asymmetric, representing elliptic main rolls with small recirculation vortex pairs. The corresponding r.m.s. values of the velocity fluctuations are small but significantly distributed. This condition therefore cannot be distinguished as a steady-state regime, even though a distinct peak frequency is not detected on the velocity fluctuation spectra. Under considerably smaller $Ra$ conditions, the velocity profile assumes a sinusoidal shape with horizontal symmetry in each roll. An example of these profiles is shown in figure 12 as the mean velocity profiles, where the original velocity data for the profile is from Vogt et al. (Reference Vogt, Ishimi, Yanagisawa, Tasaka, Sakuraba and Eckert2018) measured at $Ra = 1.7 \times 10^4$ and $Q = 8.0 \times 10^3$ ($Q/Ra = 0.47$) by UV2. Even in this condition, the profile of the r.m.s. value shows small, local maxima at the boundaries between the main rolls. A numerical simulation was also performed at similar parameters, $Ra = 1.4 \times 10^4$ and $Q = 5.7 \times 10^3$ ($Q/Ra = 0.41$), to determine the corresponding motion of the main rolls. A snapshot of the isosurface of the $Q_{3D}$ value also represents quasi-2-D rolls along the direction of the magnetic field (figure 13a).

Figure 11.

Profiles of mean velocity $U_y$ and standard deviation of velocity fluctuations $u_{y,rms}$ measured at $Ra = 7.9 \times 10^4$ and $Q = 2.0 \times 10^5$ ($Q/Ra = 2.5$) by UV1.

Figure 12.

Profiles of mean velocity $U_y$ and standard deviation of velocity fluctuations $u_{y,rms}$ using data from Vogt et al. (Reference Vogt, Ishimi, Yanagisawa, Tasaka, Sakuraba and Eckert2018) at $Ra = 1.7 \times 10^4$ and $Q = 8.0 \times 10^3$ ($Q/Ra = 0.47$) by UV2.

Figure 13.

Results of numerical simulation performed for $Ra = 1.4 \times 10^4$ and $Q = 5.7 \times 10^3$ ($Q/Ra = 0.41$). (a) Isosurface of the $Q_{3D}$ value representing roll structures, where the dark red line indicates the UV2 measurement line. (b) Spatio-temporal velocity map extracted along a line corresponding to the UV2 measurement line (figure 1). (c) Time-averaged velocity profile of the velocity map. (d,e) Instantaneous contour map of vorticity on the vertical cross-section at $x = 160\ \textrm {mm}$, where red and blue contours represent positive and negative values of the vorticity, respectively (see also supplementary movie 2).

The spatio-temporal velocity map shown in figure 13(b) extracted from the simulation results along the line corresponding to the UV2 measurement line (dark red line in figure 13a) displays longer-period periodic oscillation with substantially smaller amplitude than in the oscillations shown in figure 10(b). The mean velocity profile calculated from the map also has a quasi-sinusoidal shape with small deformation around the boundaries between the main rolls (figure 13c). Contour maps of the instantaneous vorticity field given from the simulation show the existence of small recirculation vortex pairs (figures 13d and 13e; see also supplementary movie 2). The numerical simulation elucidates that the vortex pair oscillates periodically and locally, and the distinct peaks of the r.m.s. values shown in figure 12 may represent such local oscillations. Such small vortex pairs and oscillations may not be distinguished by UVP measurements in the present experiment because of the arrangement of the measurement lines relative to the vortex pair and the inability to resolve small velocity fluctuations. The results summarized above suggest that there are two different types of 2-D oscillatory convection: weak and local oscillations on the recirculation vortex pair (corresponding to figures 11–13), and more dynamic and global elliptic oscillations of the main rolls accompanied by vorticity entrainment from the recirculation vortex pairs (figures 9 and 10).

Previous studies that assumed a wake of straight cylinders reported that such recirculation vortex pairs show absolute instability at relatively small Reynolds numbers, $Re = O(10)$, and display periodic oscillations (e.g. Drazin & Reid Reference Drazin and Reid1998). With increasing $Re$ number, $O(10^2)$, the oscillations propagate from the recirculation vortex pair downstream. Here, unlike the wake, the main rolls are in tight interaction with the recirculation vortex pairs via vorticity entrainment and elliptic deformation of the roll in the 2-D oscillation with increasing velocity of the main rolls. In the next section, we investigate the convective flow velocity and dimensionless frequency of the oscillatory motion to characterize the 2-D oscillations.

3.5. Frequency and flow velocity on the 2-D oscillatory convection

This section discusses the characteristic frequency of the oscillatory convection $f_{OS}$ and the local maximum value of the time-averaged velocity profile at the central convection rolls $U_{y,max}$ (figure 9), which are the basis for calculating the dimensionless frequency and $Re$ numbers. Variations in these values with respect to $Q$ are shown in figure 14. The $f_{OS}$ and $U_{y,max}$ values for the plots were obtained from spatio-temporal velocity maps measured by UV1 at different $Ra$ numbers for the five-roll state via the spatially averaged power spectrum of the velocity fluctuations and time-averaged velocity profile, respectively. The $f_{OS}$ detected by UV3 placed at the centre of the vessel parallel to UV1 are also plotted to evaluate the range of the 2-D oscillations in figure 14(a). The frequency gradually increases with decreasing $Q$ at all of the $Ra$ conditions studied here, with the largest increases observed for small $Q$ and the highest $Ra$ number of $Ra = 1.8 \times 10^5$. There is a very close coincidence of the $f_{OS}$ obtained by UV1 and UV3 at relatively large $Q$ values within the frequency resolution as expressed by the symbol overlap. At smaller $Q$, however, distinct peaks corresponding to $f_{OS}$ in UV3 cannot be distinguished and are buried in multiple peaks or noise, whereas the spectra obtained by UV1 show clear peaks. The coincidence suggests the existence of 2-D oscillations, whereas the disappearance may indicate a transition from the 2-D oscillatory convection regime to a 3-D flow regime. Vogt et al. (Reference Vogt, Ishimi, Yanagisawa, Tasaka, Sakuraba and Eckert2018) suggested that a transition from quasi-2-D convection to a 3-D flow regime appears as a meandering structure along the centreline of the vessel perpendicular to the magnetic field while fine structures move along the roll axis. The disappearance of a distinct peak in the spectra obtained by UV3 may reflect this; however, further detailed investigation is beyond the scope of the present study.

Figure 14.

Variations of (a) the main frequency on the oscillatory convection, $f_{OS}$, obtained by UV1 and UV3, and (b) the maximum time-averaged velocity of the central convection rolls, $U_{y,max}$, in the five-roll state with respect to $Q$ obtained at different $Ra$. In (a), the frequencies obtained by UV3 at most of the $Q$ values agree with those by UV1 within the frequency resolution as overlaps of the solid and open symbols.

The dependence of $U_y$ on $Q$ shown in figure 14(b) is similar to the behaviour of the temperature oscillations in figure 7(a): the mean velocity initially slightly increases with decreasing magnetic field, reaches a maximum and then decreases. These observations can also be understood as growth of the 2-D oscillations and subsequent transition to a 3-D flow regime. At the opposite viewpoint, $U_{y,max}$ increases with increasing magnetic field strength in the range of small $Q$. This effect can likely be attributed to the stabilization and strengthening of the mean flow by the magnetic field. Burr & Müller (Reference Burr and Müller2002) reported an increase in heat transfer in RBC with the application of a horizontal magnetic field over a specific parameter range for Hartmann numbers $Ha = Q^{1/2} < 400$. As an explanation for the increase of flow transport properties, the authors suggested an enhancement of the quasi-2-D roll-type convection. A damping of the flow intensity occurs with further increases in $Q$ due to the emerging magnetic damping. This point is important in the discussion of flow transition and is further addressed in the following section.

The dimensionless frequency reduced by the circulation time (turnover time) of the main convection rolls, $f^* = f_{OS}\tau _{TO}$, is plotted as a function of $Q/Ra$ in figure 15, where the circulation time is given as $\tau _{TO} = 2{\rm \pi} \Delta z/U_{y,max}$, with the displacement of the measurement line from the centre of the rolls, $\Delta z = 10\ \textrm {mm}$ (figure 1). At sufficiently high magnetic fields ($Q/Ra \gtrsim 0.3$) the values obtained by UV1 and UV3 collapse into a unified curve (i.e. the symbols overlap) and $f^*$ assumes a nearly constant value around 0.6. The supplemental numerical simulations shown in figures 10 and 13 provide a similar dimensionless frequency, $f^* \sim 0.6$. As mentioned, the finding of a consistent and constant dimensionless frequency can be interpreted as an indication of a 2-D flow structure, whereas the scattering of the data for $Q/Ra \lesssim 0.3$ can be understood as the transition from the 2-D oscillatory convection regime to a 3-D flow regime. The transition point at $Q/Ra \approx 0.3$ corresponds to the one at which the r.m.s. value of the temperature fluctuation, $T_{rms}$, reaches the maximum (figure 7b). Furthermore, the $Re$ number also reaches a maximum here as discussed in § 4.1. Together these results indicate that the decrease in $T_{rms}$ and flow velocity are both caused by the transition to a 3-D flow regime and that the transition can be organized by $Ra/Q$.

Figure 15.

Dimensionless frequencies determined by UV1 and UV3 plotted at different $Ra$ versus $Q/Ra$, where the frequencies obtained by UV3 at most of the $Q/Ra$ values agree with those by UV1 within the frequency resolution as overlaps of the solid and open symbols. There exist no symbols for UV3 for small $Q/Ra$ values because of unclear frequencies by UV3.