3.1 Stagnant mode I

A small plate is easily trapped when it encounters a downwelling and a counter-rotating convection cell, as the opposing flows of the two cells converge beneath the plate (figure 2 a,b). The opposing forces exerted by these opposing flows on the bottom of the plate tend to balance each other by adjusting the plate position. This is a stable mode, since the plate will be pulled back if it is moved slightly away from the balanced position. As a result, the plate is trapped for a long time, which we term stagnant mode I (SM I). For a small plate of $L=0.25$ , it never shows significant displacement after being arrested by the downwelling (figure 2 c) and the underlying flow structure remains approximately the same. Nevertheless, the system shows an alternation of quiescence (figure 2 a) and activity (figure 2 b). The former is characterized by a more regular flow structure (figure 2 a) and smaller fluctuations of plate velocity and flow properties with time (figure 2 c). Furthermore, during the quiescent duration, the thermal plumes are striking almost vertically towards the opposing boundaries (figure 2 a). In contrast, they show significant bifurcation and variation during the active duration (figure 2 b).

Figure 2.

Stagnant mode I (SM I) for $L=0.25$ at $Ra=10^{6}$ , $h=0$ . (a,b) Snapshots of streamlines and temperature for the quiescent duration and the active duration, respectively, with a sketch of the flow structure of SM I on the right-hand side. The corresponding times are marked in (c) with red circles and cyan lines. The solid and dashed streamlines represent anticlockwise and clockwise flows, respectively. (c) Time series of plate centre location  $x$ , plate velocity  $u$ , the average temperature $T$ and average normal stress $\unicode[STIX]{x1D70E}_{yy}$ at plate bottom.

Figure 3.

Alternation of stagnant mode I (SM I) and velocity bursts for $L=0.5$ at $Ra=10^{6}$ , $h=0$ . (a–g) Snapshots of streamlines and temperature with sketches on the right-hand side. (h) Time series of plate centre location  $x$ , plate velocity  $u$ , average temperature  $T$ and average normal stress $\unicode[STIX]{x1D70E}_{yy}$ at plate bottom.

For a small plate of $L=0.25$ , the two convection cells underlying the plate do not show significant variation in size over time. As the plate becomes larger, its capability to modify the flow increases. As a result, as $L$ increases to 0.5, the two convection cells underneath shrink gradually as the two associated hot plumes approach the plate, which is evident by comparing figure 3(a) with 3(b), and figure 3(d) with 3(e) (supplementary movie 1 available at https://doi.org/10.1017/jfm.2019.189). Such an evolution of flow pattern is due to the thermal blanket effect as the plate reduces the local heat loss and weakens the downwelling. As the two cells shrink, one of them eventually vanishes and the remaining one merges with the neighbouring cell of the same sign (this evolution is evident by comparing figure 3 b with 3 c, and figure 3 f with 3 g), thus the plate rides on a single convection cell and exhibits a burst in velocity (supplementary movie 1). This process of cell vanishing and merging is accomplished when one of the two associated hot plumes reaches the plate. As a result, the bursts in plate velocity are accompanied by evident bursts in temperature $T$ at plate bottom and alleviations of the extensional (positive) normal stress $\unicode[STIX]{x1D70E}_{yy}$ there (figure 3 h).

3.2 Stagnant mode II

Another stagnant mode, i.e. stagnant mode II (SM II) (supplementary movie 2), starts to appear at $L\sim 0.75$ and gradually becomes dominant at $L\sim 1.5$ . For $L=1.5$ , the plate rides on a single convection cell most of the time (for example, figure 4 a,e,h). However, plate motion can be severely hindered when the leading edge of the plate encounters a counter-rotating cell and the associated downwelling. As the plate slows down and becomes stagnant, a new downwelling starts to form at the other end of the plate and a hot plume develops underneath, initiating two counter-rotating convection cells underneath and a divergent flow at plate bottom (figure 4 b,f, supplementary movie 2). The two cells underlying the plate together with the neighbouring cells on each side of the plate form a flow structure that traps the plate, which we term stagnant mode II (SM II). This is a stable structure, since a minor shift of the plate position will be inhibited by the opposing flow of the neighbouring cell.

Figure 4.

The evolution of stagnant mode II (SM II) for $L=1.5$ at $Ra=10^{6}$ , $h=0$ . (a–h) Snapshots of streamlines and temperature with sketches of flow structure on the right-hand side. (i) Time series of plate centre location  $x$ , plate velocity  $u$ , average temperature $T$ and average normal stress $\unicode[STIX]{x1D70E}_{yy}$ at plate bottom.

Under the insulating plate, the hot plume grows and the two convection cells expand. Meanwhile, the neighbouring hot plume on each side of the plate moves towards the plate. As a result, the two underlying convection cells merge respectively with the nearest cell of the same sign on each side of the plate at the cost of a counter-rotating cell, which is evident by comparing figure 4(c) with 4(d), and figure 4(f) with 4(g) (supplementary movie 2). This is contrary to the shrinkage of the underlying cells in SM I. Nevertheless, SM II becomes unstable when the two underlying cells merge with the neighbouring cells and become large (figure 4 d,g), since a small shift of the plate position will be amplified rather than inhibited as in SM I. One of the two cells soon becomes dominant in moving the plate and thus the plate rides on a single cell again (figure 4 e,h), ending its stagnant state. Since the downwelling which initially slows down the plate tends to drag the plate downwards, the decrease in plate speed $u$ owing to the encounter of downwelling is accompanied by an increase in the extensional stress $\unicode[STIX]{x1D70E}_{yy}$ at the plate bottom, and a decrease in the average temperature $T$ there (figure 4 i). Conversely, during the stagnant duration, the growth of hot plumes underneath the plate leads to a gradual increase of $T$ and a decrease of $\unicode[STIX]{x1D70E}_{yy}$ (figure 4 i). As the insulating effect of the plate increases with its size, a larger plate needs a shorter time to warm up the underlying flow and modify the underlying flow structure. Therefore, SM II is much shorter-lived than SM I.

3.3 Unidirectional moving mode

Figure 5.

The unidirectional moving mode (UMM) for $L=2.5$ at $Ra=10^{6}$ , $h=0$ . (a–g) Snapshots of streamlines and temperature with sketches of flow structure on the right-hand side. (h) Time series of plate centre location  $x$ , plate velocity  $u$ , average temperature $T$ and average normal stress $\unicode[STIX]{x1D70E}_{yy}$ at plate bottom. Except moments of forced stop in (d), a single large convection cell is always identified underlying the plate.

As the plate size increases to $L=2.5$ , the aforementioned two stagnant modes disappear and the unidirectional moving mode (UMM) emerges (supplementary movie 3). In this mode, the plate rides most of the time on a single large convection cell and moves unidirectionally until it encounters the end walls (figure 5), where it is forced to stop until hot plumes rise up underneath and the net underlying horizontal flow reverses (figure 5 d), similar to that observed by Whitehead & Behn (Reference Whitehead and Behn2015). the leading edge of the plate encounters a cold downwelling, instead of being stopped as in the stagnant modes, the plate slows down transiently but continues moving forwards, taking the cold plume along with it (supplementary movie 3). Meanwhile, the underlying convection cell merges with the frontal cell of the same sign, at the cost of a counter-rotating cell, which is evident by comparing figure 5(a) with 5(b), and figure 5(e) with 5(f). On the other hand, when the underlying convection cell becomes large through merging, it disintegrates by forming new hot upwelling near the trailing edge of the plate, which is evident by comparing figure 5(b) with 5(c), and figure 5(f) with 5(g). Such a process of cell merging near the leading edge and disintegration near the trailing edge continues, resulting in the propagation of a large convection cell with the moving plate (supplementary movie 3).

A comparison of the average normal stress $\unicode[STIX]{x1D70E}_{yy}$ at the plate bottom among different cases suggests that as plate size increases, the value of $\unicode[STIX]{x1D70E}_{yy}$ changes from generally positive to generally negative (figures 2–5), suggesting that the plate transfers from being dragged down by cold downwelling to being uplifted by hot upwelling. This is consistent with the generally larger number of hot plumes identified underneath the larger plate (figures 2–5), which embodies the increase of the thermal blanket effect with plate size.

4.1 Variation of plate speed with plate size

It is clear now that plate motion varies significantly with its size. Very small plates ( $L\leqslant \sim 0.25$ ) are permanently trapped in SM I with no significant displacement once being trapped (figure 6, $L=0.25$ ). As plate size increases to $L=0.5$ , the plate shows long durations of stagnancy punctuated by short episodes of velocity burst (figure 6). Excluding the forced stop at the end walls, the duration of stagnancy decreases with increasing plate size (figure 6). For $L\geqslant 2.5$ , the plate shows unidirectional movement until it encounters the end walls (figure 6 a, $L=2.5,4.0$ ).

Figure 6.

(a) Plate centre position $x$ and (b) plate velocity $u$ for $Ra=10^{6}$ , $h=0$ . Time starts from when the plate is freed. The red horizontal lines denote the maximum range of plate centre position  $x$ .

An analysis on the time series of plate speed $|u|$ reveals that as a plate becomes larger, it is less likely to be stagnant and its speed becomes increasingly concentrated on high values (figure 7 a), suggesting its increasing capability to overcome impeding flow structures and maintain a steady motion. As $L$ increases to 2.5, the probability of a plate to be stagnant (low speed, for example, $|u|<100$ ) approaches zero (figure 7 a,b), corresponding to the UMM of large plates. The enhanced moving capability of larger plates is also manifested in the increase of the average plate speed  $|u|_{mean}$ with plate size (figure 7 c). The standard deviation of plate speed reaches a maximum at a plate size of $L=1.5$ –2 (figure 7 c). This is because towards the lower end of plate size, the plate shows an increasing tendency to be stagnant, and thus the standard deviation decreases. Towards the upper end, the increasing steadiness of the unidirectional plate motion also results in a decrease of the standard deviation.

Figure 7.

Analysis of plate speed $|u|$ for different plate size $L$ at $Ra=10^{6}$ , $h=0$ (a) Probability density of plate speed $|u|$ . (b) Percentage of low plate speed $|u|$ in the time series of $|u|$ . (c) Mean plate speed $|u|_{mean}$ with the standard deviation denoted by the upper and lower bars, and the maximum plate speed $|u|_{max}$ . Dashed vertical lines indicate the rough boundaries between different modes ( $\text{SM I}+\text{II}$ denotes a transition from SM I to SM II, where both modes exist). (d) Streamlines and temperature for $L=3$ (upper panel) and 4 (lower panel) when the plate starts to ride on a single convection cell again after the forced stop at the end wall. The two red lines at the top denote the range of the end-wall cell. Plate speeds affected by this cell are excluded from the analysis.

The maximum plate speed is reached at a plate size of $L\sim 0.75$ (figure 7 c). Since plate speed reflects the average flow speed at the plate–fluid interface, an increase in plate size has the effect of smoothing out local fluctuations and results in more steady plate movement. Therefore, as plate size increases, it is less susceptible to local velocity pulses caused by local flows and thus its maximum speed decreases. On the other hand, a very small plate (e.g. $L=0.25$ ) is always trapped in SM I, and therefore its maximum speed is also small. It is worth noting that data at the very initial stage when the plate is just freed are excluded from the above analysis in order to exclude the start-up effect. Furthermore, plate speed affected by the artificially forced stop at the end walls is also excluded, starting from when the plate is forced to stop until it returns back and rides on a single convection cell again (figure 7 d). A comparison of the single convection cell underlying different plates suggests that it becomes larger as plate size increases (figure 7 d). For a large plate of $L=4$ , there is not much room left for the development of impeding flow structures for the plate (figure 7 d). Therefore, for $L=4$ , the distribution of plate velocity is much more concentrated on relatively high values (figure 7 a), showing a rather steady plate movement.

4.2 Comparison of $Nu$ with fixed-plate cases

Figure 8.

(a) Time series of $Nu$ at the bottom of the flow domain for $Ra=10^{6}$ and $h=0$ with fixed and freely moving plates. (b) Time-averaged $Nu$ during the quasi-steady-state at $Ra=10^{6}$ for cases with fixed and free plates and their difference ( $\unicode[STIX]{x0394}Nu$ ). (c) Temperature and streamlines at the quasi-steady-state for cases with a fixed plate. (d) Time-averaged $Nu$ during the quasi-steady-state for $L=2.5$ at different $Ra$ .

Compared to cases with a fixed plate, the Nusselt number $Nu$ , increases when the plate is free to move (figure 8 a,b). The value of this increase, $\unicode[STIX]{x0394}Nu$ , depends on plate size  $L$ , increasing dramatically with $L$ up to $L=0.75$ , staying approximately constant for $L$ ranging from 0.75 to 2.5, and decreasing gradually with $L$ for $L>2.5$ (figure 8 b). Compared to heat transfer by conduction, the thermal plumes that bring fluid from one horizontal boundary straightforwardly towards the opposing boundary present a more efficient means of heat transfer. These thermal plumes are henceforth termed striking thermals since they strike the opposing boundary, in contrast to those short thermals that move fast horizontally with the convective flow until uniting with the striking thermals. Therefore, the presence of a larger number of striking thermals should be associated with a higher efficiency of heat transfer, and hence larger  $Nu$ .

For cases with a fixed plate, $Nu$ decreases dramatically with $L$ up to $L=0.75$ and less abruptly as $L$ continues to increase (figure 8 b, blue line). The dramatic decrease of $Nu$ corresponds to the dramatic decrease of convection cells from five to two, or the associated striking thermals from six to three as $L$ increases from 0.25 to 0.75 (figure 8 c). As $L$ continues to increase, the number of convection cells (or striking thermals) becomes constant (figure 8 c) and therefore the decrease of $Nu$ with $L$ becomes less abrupt. For $0.75\leqslant L\leqslant 2.5$ , only two convection cells, or three associated striking thermals are present for cases with a fixed plate (figure 8 c). However, the corresponding cases with a freely moving plate have evidently more convection cells (or striking thermals) (figures 3–5), which lead to a higher efficiency of heat transfer and therefore higher $Nu$ . For large plates of $L>2.5$ , the convection cell underneath the moving plate occupies a large portion of the flow domain (for example, figure 7 d). In this case, the space left for the development of other convection cells (or the associated striking thermals) becomes quite limited and it decreases with increasing $L$ (figure 7 d). Therefore, as $L$ increases, the number of striking thermals for fixed-plate cases and that for free-plate cases gradually becomes close, and thus $\unicode[STIX]{x0394}Nu$ , the difference of $Nu$ between cases with free and fixed plates, decreases with $L$ for $L>2.5$ . It is expected that $\unicode[STIX]{x0394}Nu$ approaches zero as $L$ approaches the length of the fluid domain.

For a unit square box, the boundary layer theory has predicted a power-law relation of $Nu$ being linearly proportional to $Ra^{1/3}$ (Turcotte & Schubert Reference Turcotte and Schubert2002). Nevertheless, the present configuration is significantly different from a unit square box, not only in the horizontal size of the flow domain but also in the top boundary condition. Therefore, to obtain the $Nu$ – $Ra$ power-law relation ( $Nu=\unicode[STIX]{x1D6FC}\,Ra^{\unicode[STIX]{x1D6FD}}$ ) for the case of $L=2.5$ , both the exponent $\unicode[STIX]{x1D6FD}$ and the coefficient $\unicode[STIX]{x1D6FC}$ are set as the fitting parameters. Our results suggest that the average $Nu$ over the quasi-steady-state scales with $Ra^{0.2831}$ for the fixed-plate cases, and $Ra^{0.2565}$ for the free-plate cases, and therefore the exponents are much smaller than the theoretical value of $1/3$ predicted for a unit square box.

4.3 Variation of plate motion with Ra

For $L=2.5$ , the plate moves unidirectionally under different Rayleigh numbers until it encounters the end wall (figure 9 a). Nevertheless, for $Ra=2\times 10^{5}$ , the plate motion can still be severely hindered when the plate encounters impeding flow structures (figure 9 b). As $Ra$ increases, the probability of low plate speed (for example, $|u|<100$ ) decreases significantly, approaching zero for $Ra\geqslant 10^{6}$ (figure 9 c). Therefore, apart from plate size, the increase of $Ra$ also increases plate capacity to overcome impeding flow structures. Excluding data affected by the end-wall effect, the mean plate speed scales relatively well with $Ra^{2/3}$ , which is consistent with the scaling prediction for the mantle flow (Turcotte & Schubert Reference Turcotte and Schubert2002; Grigné, Labrosse & Tackley Reference Grigné, Labrosse and Tackley2005). However, for $Ra=2\times 10^{5}$ , the mean plate speed is evidently lower than the scaling prediction (figure 9 d). This is because for this lower $Ra$ , plate movement is often significantly hindered by impeding flow, which counteracts the main flow that drives the plate (figure 9 b), and therefore the plate speed is lower than the speed of the underlying flow predicted by the scaling.

Figure 9.

Variation of plate motion with $Ra$ for $L=2.5$ and $h=0$ . (a) Time series of plate centre location $x$ for $Ra=2\times 10^{5}$ , $5\times 10^{5}$ , $10^{6}$ , $2\times 10^{6}$ , $5\times 10^{6}$ from top to bottom. (b) Time series of the corresponding plate velocity  $u$ . (c) Probability density of plate speed $|u|$ . (d) Time-averaged plate speed $|u|_{mean}$ during the quasi-steady-state, the straight line is a fit to the value at $Ra=5\times 10^{6}$ and is given by $|u|_{mean}=0.0627\,Ra^{2/3}$ .

4.4 The correlation between plate movement and thermal signals of the flow

The presence of the plate affects the underlying thermal structure, as shown by the isotherms and streamlines in § 3. Further, plate movement or stagnancy can affect the thermal signals of the flow. Some of the thermal signals, including the spatial root mean square temperature $T_{rms}$ , the average temperature $\overline{T}$ over the flow domain, the average temperature $\overline{T}_{i}$ at the plate–fluid interface and the Nusselt number $Nu$ are shown in figure 10.

For a very small plate of $L=0.25$ , the quiescent duration (highlighted by the cyan bands in figure 10 a) is characterized by lower plate velocity  $u$ , lower $T_{rms}$ and higher $Nu$ than the active duration (figure 10 a). The lower $T_{rms}$ corresponds to the more regular flow structure during the quiescent duration (figure 2 a). The higher $Nu$ during the quiescent duration corresponds to the more straightforward movement of the thermals towards the opposing boundaries. The vertically moving thermals during the quiescent duration bring colder/warmer material to the lower/upper boundary than the bifurcating and inclining thermals during the active duration (figure 2 a,b), resulting in a higher efficiency of heat transfer, and therefore higher $Nu$ . Over the quiescent duration, the time series of the average temperature $\overline{T}$ over the flow domain shows a general trend of slow increase and is much smoother than over the active duration (figure 10 a). On the other hand, the average temperature $\overline{T}_{i}$ at the plate–fluid interface exhibits higher values (suggesting weaker downwellings underneath the plate) during the active durations, consistent with the snapshots of temperature (figure 2 a,b).

Figure 10.

The response of thermal signals to plate movement for different plate sizes at $Ra=10^{6}$ , $h=0$ : time series of plate centre location  $x$ , plate velocity  $u$ , the spatial root mean square temperature $T_{rms}$ , the average temperature of the flow domain $\overline{T}$ , the average temperature $\overline{T}_{i}$ over the plate–fluid interface, and the average Nusselt number $Nu$ at the bottom boundary. (a)  $L=0.25$ , the cyan bands highlight the quiescent durations, the red line in the time series of $Nu$ shows the result of $Nu$ after the plate is removed at $t=0.47$ . (b)  $L=0.5$ , the magenta bands highlight the velocity bursts. (c)  $L=1.5$ , the cyan bands highlight the durations when the plate slows down and becomes stagnant. (d)  $L=2.5$ , the cyan bands highlight the durations when the plate slows down at first but retains or even increases its speed afterwards.

To investigate the effect of a very small mobile plate ( $L=0.25$ ) on the flow pattern underneath, we remove the plate (at time $t=0.47$ ) after the coupled system reaches a quasi-steady-state. It is found that the quasi-steady-state flow pattern for the no-plate case is similar to the flow pattern for the case with a small plate of $L=0.25$ , shown in figure 2(a), in which the flow domain is occupied by five convection rolls of similar size. Nevertheless, for the no-plate case, the differentiation between the active and the quiescent duration is less noticeable and the fluctuation of flow properties with time is much smaller, as illustrated by the time series of $Nu$ (red line in figure 10 a). Therefore, rather than being a completely passive tracer, the small plate magnifies the fluctuation of flow properties during the active duration and lowers the average  $Nu$ .

As plate size increases to $L=0.5$ , the plate experiences long durations of stagnancy on top of cold downwelling until hot upwelling arrives underneath and brings it a velocity burst. As the plate alternates between stagnancy and velocity burst, the thermal signals, including $T_{rms}$ , $\overline{T}$ , $\overline{T}_{i}$ and  $Nu$ , exhibit consistent trend of variation over time (figure 10 b). Plate velocity burst is associated with a general increase of $T_{rms}$ , a maximum of $\overline{T}$ , a burst of $\overline{T}_{i}$ , and a decrease of $Nu$ (highlighted by the magenta bands in figure 10 b). Since plate velocity bursts happen when hot upwelling arrives underneath, bursts of plate velocity correspond to bursts of temperature $\overline{T_{i}}$ at the interface. When the plate experiences a velocity burst, the position of the plate changes abruptly. Corresponding to this abrupt change, the underlying convection cells undergo a process of reorganization. During this process, the plume pattern becomes less regular and the number of convection cells decreases (figure 3 b,c). As a result of flow reorganization, bursts of plate velocity correspond to general increases of $T_{rms}$ and general decreases of  $Nu$ , as highlighted by the magenta bands in figure 10(b). Shortly after the velocity burst, the plate arrives at another downwelling and becomes stagnant again. The thermal plumes and the convection cells reorganize to adjust to the changed plate position. As the process of reorganization goes on, the spatial pattern of convection cells and the associated plumes becomes more regular and the number of convection cells increases (figure 3 d,e), resulting in a decrease/increase of $T_{rms}/Nu$ . This process of variation in $T_{rms}$ and $Nu$ persists as the plate alternates between stagnancy and velocity burst. Since a stagnant plate accumulates heat underneath more efficiently than a fast moving plate, the burst in plate velocity is associated with the start of a sharp decrease in the average temperature $\overline{T}$ (figure 10 b). This decrease persists for a while after the plate reaches a new downwelling. After the plate sits on top of the downwelling for a sufficiently long time, its thermal blanket effect becomes evident and the value of $\overline{T}$ begins to increase with time. As a result, the average temperature $\overline{T}$ reaches a maximum around the time when the plate shows a velocity burst, as highlighted by the magenta bands in figure 10(b).

As plate size increases to $L\geqslant 1.5$ , the overall shape of thermal plumes becomes more irregular and their movement towards the opposing boundary becomes less straightforward (figures 4 and 5). The alternation between stagnancy (SM II) and movement becomes less distinct and less differentiable. Further, the duration of stagnancy (SM II) is much shorter than that of SM I for smaller plates, leaving a very limited time for the fluid to reorganize and respond to the newly settled plate position. As a result, the variation of the thermal signals of the fluid with plate movement becomes less obvious (figure 10 c). Nevertheless, a decrease in plate velocity $u$ and its subsequent stagnation are associated with an evident decrease and then a subsequent increase in the average temperature $\overline{T}_{i}$ at the interface (highlighted by the cyan bands in figure 10 c). This corresponding variation in plate velocity $u$ and $\overline{T}_{i}$ is consistent with the temporal evolution of the stagnant mode II. The encounter of cold downwelling at the leading edge of the plate initiates a slowdown of plate velocity and a decrease in $\overline{T}_{i}$ , which is evident by comparing figure 4(a) with 4(b), and figure 4(e) with 4(f). As the plate slows down, it gradually enters into the stagnant mode II and nurtures the growth of hot upwelling underneath the plate (figure 4 b,c), resulting in an increase of $\overline{T}_{i}$ at the interface. This variation of $\overline{T}_{i}$ after the slowdown of plate velocity is also observed for the even larger plate case of $L=2.5$ . As the thermal blanket effect is enhanced with increasing plate size, the decrease and the subsequent increase in $\overline{T}_{i}$ (highlighted by the cyan bands in figure 10 d) is completed in shorter durations for the case of $L=2.5$ than for the case of $L=1.5$ (figure 10 c,d). Further, the large plate of $L=2.5$ slows down when it encounters a cold downwelling, but the plate soon warms up the downwelling, resulting in the decrease of the spatial root mean square temperature $T_{rms}$ with decreasing  $u$ . Conversely, an increase of plate velocity from zero, after the forced stop at the end wall, is associated with the generation of hot thermal plumes underneath the plate (figure 5 d), which is reflected in the increase of $T_{rms}$ as $u$ increases from zero (figure 10 d).

4.5 The effect of internal heating on plate movement

Apart from bottom heating at the core–mantle boundary, radioactive decay provides an essential heating source for the interior of the Earth (Turcotte & Schubert Reference Turcotte and Schubert2002). To investigate the effect of internal heating on plate movement, simulations were conducted by applying different intensities of internal heating for the case of $L=2.5$ (figure 11). Time series of the plate centre position $x$ and plate velocity $u$ show increasing steadiness of plate movement as the intensity of internal heating $h$ increases (figure 11 a,b). As  $h$ increases, the distribution of plate speed $|u|$ becomes increasingly concentrated on high values (figure 11 c), suggesting an increasing steadiness of plate movement and an increasing overall plate speed. Meanwhile, as $h$ increases, the probability and the percentage of low plate speed decrease (figure 11 c,d), suggesting a decreasing likelihood of plate movement being severely hindered by impeding flow structures. The average plate speed $|u|$ increases with increasing $h$ (figure 11 e). Therefore, plate mobility or its capability to overcome impeding flow structures increases with increasing internal heating intensity  $h$ .

Figure 11.

The effect of internal heating on plate movement illustrated with a parameter setting of $L=2.5$ , $Ra=10^{6}$ , $h=0$ , 2, 4, 8. (a) Plate centre position $x$ and (b) plate velocity $u$ for $Ra=10^{6}$ and $L=2.5$ with different internal heating rates, $h=0$ , 2, 4, 8. Time starts from when the plate is freed. The red horizontal lines denote the maximum range of plate centre position  $x$ . (c) Probability density of plate speed $|u|$ . (d) Percentage of low plate speed $|u|$ in the time series of $|u|$ . (e) Time-averaged plate speed $|u|$ during the quasi-steady-state. Data shown in (d,e) exclude those when plate movement is affected by the artificially forced stop at the end wall.

A comparison was also made between one case with $h=0$ (no internal heating) and another one with $h=8$ for a small plate size of $L=0.5$ . Time series of plate centre position $x$ and plate velocity $u$ show that plate mobility increases significantly as internal heating is switched on (figure 12 a). For $h=8$ , as a result of the enhanced plate mobility, the distinction between stagnation and velocity burst becomes less differentiable in the times series of plate velocity $u$ , and the plate is able to reach the end walls (figure 12 a). Consequently, the probability of plate stagnation (low plate speed $|u|$ ) decreases significantly as $h$ increases from 0 to 8 (figure 12 b).

Figure 12.

The effect of internal heating on plate movement illustrated with a parameter setting of $L=0.5$ , $Ra=10^{6}$ , $h=0,8$ . (a) Plate centre position $x$ and plate velocity $u$ for $Ra=10^{6}$ with $h=0$ (black line) and $h=8$ (red dashed line). Time starts from when the plate is freed. (b) Probability density of plate speed  $|u|$ .

4.6 The presence of different modes in three dimensions

To illustrate the presence of different modes in three dimensions, 3-D simulations were conducted in a $8\times 8\times 1$ ( $8\times 8$ is the horizontal dimensional size and 1 is the depth) convective domain with a square plate of different sizes floating on top of it. Although the plate is set free to move from the beginning when the entire flow is isothermal and stationary, it is observed that all of the plates remain stagnant at the initial stage and only begin to move at a more developed stage.

The addition of a new dimension in space introduces much richer dynamics in the fluid domain and more possibilities in plate movement. Nevertheless, the three different coupling modes (SM I, SM II and UMM) revealed in two dimensions are all identified in three dimensions, and the basic characteristics of the different modes are consistent with those observed in two dimensions.

For a small plate with a horizontal size of $0.5\times 0.5$ , the plate shows a short episode of velocity burst when its underlying flow is coherently flowing in one direction, from a site of hot upwelling towards a site of cold downwelling (figure 13 a,d). When the plate arrives at the site of downwelling, the underlying flow converges underneath the plate and flows downwards towards the bottom of the fluid domain. The convergence of flow underneath the plate involves flows in two opposing directions for the 2-D cases (figures 2 a, 3 a,e), but involves flows in all horizontal directions for the 3-D case (figure 13 b,c). The coupled system achieves a balance by adjusting the plate position so that the forces exerted by the underlying flow on the plate are balanced. This is a stable mode since the plate will be pulled back to the balanced position if it is moved slightly away from it. Figure 13(b) shows the snapshot of two isothermal surfaces and streamlines of SM I at an early stage of flow development, while the presence of SM I at a more developed stage is illustrated in figure 13(c). The convergence of flow underneath the plate is shown by the streamlines there, and the presence of cold downwelling at the plate position is demonstrated by the translucent blue isothermal surface.

Figure 13.

The evolution of stagnant mode I in three dimensions with a plate size of $0.5\times 0.5$ , $Ra=10^{6}$ and $h=0$ . The translucent blue and red iso-surfaces represent isothermal surfaces of 0.25 and 0.8, respectively. The grey lines represent streamlines around the plate. (a–c) are snapshots of isothermal surface and streamlines. (a) Corresponds to a time when plate shows velocity burst. (b,c) Correspond to the time when the plate shows SM I at an early and a more developed stage respectively. (d) Time series of plate velocity components, $u$ , $w$ and its horizontal speed $(u^{2}+w^{2})^{1/2}$ . The corresponding times of (a–c) are marked with cyan dots and lines in (d).

Figure 14.

The evolution of stagnant mode II in three dimensions with a plate size of $1.5\times 1.5$ , $Ra=10^{6}$ and $h=0$ . The translucent blue and red iso-surfaces represent isothermal surfaces of 0.25 and 0.8, respectively. The grey lines represent streamlines around the plate. (a–d) Snapshots of isothermal surface and streamlines. (a) Corresponds to the time when the plate shows velocity burst. (b) Corresponds to the time when plate begins to show SM II. (c) Corresponds to the time when the SM II is about to end. (d) Corresponds to the time when the plate shows velocity burst again. (e) Time series of plate velocity components, $u$ , $w$ and its horizontal speed $(u^{2}+w^{2})^{1/2}$ . The corresponding times of (a–d) are marked with cyan dots and lines in (e). The grey bands mark the duration when plate velocity is affected by the end wall, i.e. the plate is not allowed to penetrate the end wall.

For a larger plate with a horizontal size of $1.5\times 1.5$ , the presence of SM II is illustrated in figure 14. Figure 14(a) shows the snapshot of isothermal surfaces and streamlines when the fluid underneath the plate flows almost coherently in one direction, nearly parallel to the $z$ axis. It is about to encounter flows in different directions at its leading edge. As the plate moves ahead, the coherent flow structure underneath the plate disappears and the plate slows down. This slowing down of plate enhances the thermal blanket effect of the plate to the underlying flow and nurtures the growth of hot thermal plumes underneath the plate. Consequently, a hot plume starts to rise up underneath the plate (figure 14 b), similar to the 2-D case (figure 4 b,f). The presence of the hot plume underneath the plate gives a divergent flow structure there, as shown by the streamlines (figure 14 b), analogous to the 2-D case (figure 4 b,f). As the plume grows and the underlying flow evolves with time, the flow in the direction parallel to the positive $x$ axis starts to become dominant (figure 14 c), and the plate is about to move in the same direction as the dominant flow. Figure 14(d) shows the termination of SM II, in which the plate rides on a coherent flow again, almost parallel to the positive $x$ axis. During this evolution of SM II, the plate changes its direction from originally moving parallel to the $z$ axis before the stagnation mode to moving parallel to the $x$ axis after the stagnation mode, as illustrated by the direction of streamlines in figure 14(a,d). This change of direction is also evident in the time series of velocity components, in which the $w$ component decreases and the $u$ component increases during the evolution of SM II (figure 14 e).

For an even larger plate with a horizontal size of $2.5\times 2.5$ , the presence of the unidirectional moving mode is illustrated in figure 15. The plate is observed to move almost straightforwardly until it encounters the end wall. Figure 15(a) shows the snapshot when the plate rides on a coherent flow structure and is about to encounter impeding flow structure at its leading edge. Hot thermals are observed to rise up at its trailing edge while cold downwellings occur close to its leading edge, similar to the pattern of a single underlying convection cell in the 2-D case. When the plate encounters the downwelling and the associated impeding flow structure at its leading edge, it slows down but continues its movement forwards, taking the cold downwelling along with it (figure 15 b). As a result, the cold downwelling is observed to dip at an angle beneath the moving plate, similar to the 2-D case (figure 5 b). This slowing down of plate enhances the thermal blanket effect of plate to the underlying flow and the underlying flow quickly readjusts by forming new uprising hot plumes which are observed at a later stage (figure 15 c). The underlying flow quickly reorganizes by merging with the frontal flow of the same direction and disintegrating at the trailing edge of the plate. As a result of this reorganization, the plate rides on top of a strong coherent flow again and resumes its fast motion (figure 15 c). This process of flow merging at the leading edge and disintegrating near the trailing edge is consistent with that observed in the 2-D case (figure 5).

Figure 15.

The unidirectional moving mode (UMM) in three dimensions with a plate size of $2.5\times 2.5$ , $Ra=10^{6}$ and $h=0$ . The translucent blue and red iso-surfaces represent isothermal surfaces of 0.25 and 0.8, respectively. The grey lines represent streamlines around the plate. (a–c) Snapshots of isothermal surface and streamlines. (a) Corresponds to the time when the plate is about to encounter impeding flow. (b) Corresponds to the time when plate slows down after it encounters downwelling. (c) Corresponds to the time when the underlying convective flow reorganizes and the plate resumes its velocity. (d) Time series of plate velocity components, $u$ , $w$ and its horizontal speed $(u^{2}+w^{2})^{1/2}$ . The corresponding times of (a–c) are marked with cyan dots and lines in (d). The grey bands mark the duration when plate velocity is affected by the end wall, i.e. the plate is not allowed to penetrate the end wall.

The time series of plate centre position is shown in figure 16. While a small plate ( $0.5\times 0.5$ ) remains almost in the same position after a velocity burst, larger plates ( $1.5\times 1.5$ , $2.5\times 2.5$ ) show increased mobility. As the flow develops to a relatively mature stage, the largest plate ( $2.5\times 2.5$ ) shows almost a unidirectional movement until it encounters the end wall. Whereas the medium plate ( $1.5\times 1.5$ ) detours during its movement, since it is easily affected by the flow structure it encounters and changes its direction afterwards (figure 14). These characteristics of plate movement agree with the time series of velocity components (figures 13 d, 14 e, 15 d) and are consistent with the characteristics of plate motion in the corresponding 2-D cases.

Figure 16.

Time series of plate centre position $x$ , $z$ with three plate sizes of $0.5\times 0.5$ , $1.5\times 1.5$ , $2.5\times 2.5$ , $Ra=10^{6}$ and $h=0$ .