Adding moving boundaries to convective fluids is known to result in non-trivial and surprising dynamics, leading to spectacular geoformations ranging from kilometre-scale karst terrains to planetary-scale plate tectonics. On the one hand, the moving solid alters the surrounding flow field, but on the other hand, the flow modifies the motion and shape of the solid. This leads to a two-way coupling that is significant in the study of fluid–structure interactions and in the understanding of geomorphologies. In this work, we investigate the coupling between a floating plate and the convective fluid below it. Through numerical experiments, we show that the motion of this plate is driven by the flow beneath. However, the flow structure is also modified by the presence of the plate, leading to the ‘thermal blanket’ effect where the trapped heat beneath the plate results in buoyant and upwelling flows that in turn push the plate away. By analysing this two-way coupling between moving boundary and fluid, we are able to capture the dynamical behaviours of this plate through a low-dimensional stochastic model. Geophysically, the thermal blanket effect is believed to drive the continental drift, therefore understanding this mechanism has significance beyond fluid dynamics.

In numerical studies of thermal convection that includes a layer of lighter surface fluid, the light fluid naturally forms clusters that bulge downward at downwelling sites. A curious result is that in some cases, the clusters have maximum bulging downward near the sides of the cluster instead of a single bulge downward centred above the downwelling. The fluid mechanics leading to this ‘double bulge’ formation is analysed. To accomplish this, a simplified model replaces the thermally driven convection cells with driving cells with a fixed speed. Adding a layer of dense fluid on the bottom to the previous configuration leads to bulges along the top and bottom. More importantly, this allows a new scaling that reduces the number of governing parameters from four to three and even to two in this study. The mechanism for the double bulges comes from buoyancy of the clusters. This produces localized vorticity at the sides of the cluster that has the opposite sign of the driving cells. When this vorticity is approximately the same order of magnitude as the driving cell vorticity, a divergence in the middle of each cluster leads to the double bulges. The effect can be so great that the underlying flow cells are tilted so that vertical motion is reversed under the middle of each bulge.

Convection within planetary bodies is often modelled using a temperature-dependent rheology which, when cooled from the surface, naturally leads to the formation of a so-called stagnant lid at the cold outer surface. However, for sufficiently large planets the phase diagram describing the partially molten system may depend significantly on pressure in addition to temperature, leading to significant variations in solid fraction. The aggregate rheology may therefore exhibit significant dependence on both the temperature and pressure, and hence may exhibit marked dependence on depth in addition to the dependence on the thermal structure due to convection. Here, we consider the growth and stability of a planetary stagnant lid. We first characterise the effect of a pressure- and temperature-dependent rheology on the evolution of a symmetric, planetary stagnant lid. This analysis further suggests that the pressure dependence of the rheology may lead to an instability of the growing stagnant lid which, importantly, may lead to asymmetric lid growth. We find that the most unstable mode is at the longest wavelengths, and discuss the implications for stagnant-lid convection and the growth of asymmetric surfaces of planetary bodies. In particular, we discuss the possibility that this instability has implications for the formation of the crustal dichotomy found on the Moon.

Floating rather stably above the convective mantle, continents exert a thermal blanket effect on the mantle by locally accumulating heat and altering the flow structure, which in turn affect continent motion. This thermal-mechanic feedback has been captured by a simplified model―a thermally insulating plate floating freely over a bottom-heated convecting fluid. Previous studies have revealed that, as plate size increases, plate motion transitions from long-term stagnancy over cold downwelling to short-term stagnancy over hot upwelling, termed stagnant modes (SMs) I and II, respectively, and eventually to a unidirectionally moving mode (UMM) with no stagnancy. During SM I/II, the adjacent hot/cold plumes are observed to move toward/away from the plate. The present study quantifies these plume motions through analysis on a partially insulated loop model established for SM I and SM II, respectively. The model predictions are well verified by results of numerical simulation. Moreover, the dominant frequency of plate speed is shown to increase linearly with plate size, with an abrupt jump as the coupling mode transitions from SM I to SM II. This suggests that the enhanced thermal blanket effect with plate size enables an increasingly fast feedback between the slowdown of a plate and the strengthening of hot upwelling which boosts plate motion. When plate motion is boosted before it stops, the UMM emerges. The predicted critical plate size for the UMM is well validated by simulation results. Geological implications of the results for continent motion, continent–mantle coupling and particularly the migration of plume and subduction sites are discussed.

To understand how a spherical geometry influences the dynamics of gravity-driven subduction of the oceanic lithosphere on Earth, we study a simple model of a thin and dense axisymmetric shell of thickness $h$ and viscosity $\eta _1$ sinking in a spherical body of fluid with radius $R_0$ and a lower viscosity $\eta _0$. Using scaling analysis based on thin viscous shell theory, we identify a fundamental length scale, the ‘bending length’ $l_b$, and two key dimensionless parameters that control the dynamics: the ‘flexural stiffness’ $St = (\eta _1/\eta _0)(h/l_b)^3$ and the ‘sphericity number’ $\varSigma = (l_b/R_0)\cot \theta _t$, where $\theta _t$ is the angular radius of the subduction trench. To validate the scaling analysis, we obtain a suite of instantaneous numerical solutions using a boundary-element method based on new analytical point-force Green functions that satisfy free-slip boundary conditions on the sphere's surface. To isolate the effect of sphericity, we calculate the radial sinking speed $V$ and the hoop stress resultant $T_2$ at the leading end of the subducted part of the shell, both normalised by their ‘flat-Earth’ values (i.e. for $\varSigma = 0$). For reasonable terrestrial values of $\eta _1/\eta _0$ ($\approx$ several hundred), sphericity has a modest effect on $V$, which is reduced by $< 7\,\%$ for large plates such as the Pacific plate and by up to 34 % for smaller plates such as the Cocos and Philippine Sea plates. However, sphericity has a much greater effect on $T_2$, increasing it by up to 64 % for large plates and 240 % for small plates. This result has important implications for the growth of longitudinal buckling instabilities in subducting spherical shells.

As a simplified model to study continental drift over the convective mantle, a thermally insulating plate is introduced to float freely over a bottom-heated thermally convecting fluid. Results of numerical simulations reveal that plate size plays an essential role in plate motion and the flow--structure interaction. The insulating plate exerts a thermal blanket effect on the flow underneath by locally accumulating heat and modifying the flow structure. As the thermal blanket effect enhances with plate size, the plate becomes increasingly capable of overcoming the impeding flow and continue its movement. As plate size increases, plate motion transitions from long durations of stagnancy over cold downwelling, to short-term stagnancy over newly formed hot upwelling and eventually to a unidirectionally moving mode (UMM) with no stagnancy. For the UMM, a hot ‘propelling plume’ is always observed near the trailing edge of the plate, moving with the plate and generating a convection cell that propels the plate. However, it moves toward the plate centre as the plate extends to cover almost the entire flow domain. The average plate speed is revealed to be proportional to the multiplication of the average plume distance and the open surface area. The critical plate sizes for the UMM predicted by a simplified model agree well with results of the full numerical simulations. Moreover, spectra of the streamfunction field show that an increase in plate size induces longer-wavelength structure, and an increase in plate speed is accompanied by an enhanced dominance of longer-wavelength structure.

We numerically simulate three-dimensional Rayleigh–Bénard convection, the flow in a fluid layer heated from below and cooled from above, with inhomogeneous temperature boundary conditions, to explore two distinct regimes described in recent literature. We fix the non-dimensional temperature difference, i.e. the Rayleigh number, to $Ra=10^8$, and vary the Prandtl number between $1$ and $100$. By introducing stripes of adiabatic boundary conditions on the top plate, and making the surface of the top plate only $50\,\%$ conducting, we modify the heat transfer, the average temperature profiles and the underlying flow properties. We find two regimes: when the pattern wavelength is small, the flow is barely affected by the stripes. The heat transfer is reduced, but remains a large fraction of that of the unmodified case, and the underlying flow is only slightly modified. When the pattern wavelength is large, the heat transfer saturates to approximately two-thirds of the value of the unmodified problem, the temperature in the bulk increases substantially, and velocity fluctuations in the directions normal to the stripes are enhanced. The transition between the two regimes happens at pattern wavelength around the distance between the two plates, with different quantities transitioning at slightly different wavelength values. This transition is approximately Prandtl-number-independent, even if the statistics in the long-wavelength regime slightly vary.

The Earth’s mantle is chemically heterogeneous and probably includes primordial material that has not been affected by melting and attendant depletion of heat-producing radioactive elements. One consequence is that mantle internal heat sources are not distributed uniformly. Convection induces mixing, such that the flow pattern, the heat source distribution and the thermal structure are continuously evolving. These phenomena are studied in the laboratory using a novel microwave-based experimental set-up for convection in internally heated systems. We follow the development of convection and mixing in an initially stratified fluid made of two layers with different physical properties and heat source concentrations lying above an adiabatic base. For relevance to the Earth’s mantle, the upper layer is thicker and depleted in heat sources compared to the lower one. The thermal structure tends towards that of a homogeneous fluid with a well-defined time constant that scales with $Ra_{H}^{-1/4}$, where $Ra_{H}$ is the Rayleigh–Roberts number for the homogenized fluid. We identified two convection regimes. In the dome regime, large domes of lower fluid protrude into the upper layer and remain stable for long time intervals. In the stratified regime, cusp-like upwellings develop at the edges of large basins in the lower layer. Due to mixing, the volume of lower fluid decreases to zero over a finite time. Empirical scaling laws for the duration of mixing and for the peak temperature difference between the two fluids are derived and allow extrapolation to planetary mantles.

Continents exert a thermal blanket effect to the mantle underneath by locally accumulating heat and modifying the flow structures, which in turn affects continent motion. This dynamic feedback is studied numerically with a simplified model of an insulating plate over a thermally convecting fluid with infinite Prandtl number at Rayleigh numbers of the order of $10^{6}$. Several plate–fluid coupling modes are revealed as the plate size varies. In particular, small plates show long durations of stagnancy over cold downwellings. Between long stagnancies, bursts of velocity are observed when the plate rides on a single convection cell. As plate size increases, the coupled system transitions to another type of short-lived stagnancy, in which case hot plumes develop underneath. For an even larger plate, a unidirectional moving mode emerges as the plate modifies impeding flow structures it encounters while maintaining a single convection cell underneath. These identified modes are reminiscent of some real cases of continent–mantle coupling. Results show that the capability of a plate to overcome impeding flow structures increases with plate size, Rayleigh number and intensity of internal heating. Compared to cases with a fixed plate, cases with a freely drifting plate are associated with higher Nusselt number and more convection cells within the flow domain.

Solid-state convection can take place in the rocky or icy mantles of planetary objects, and these mantles can be surrounded above or below or both by molten layers of similar composition. A flow towards the interface can proceed through it by changing phase. This behaviour is modelled by a boundary condition taking into account the competition between viscous stress in the solid, which builds topography of the interface with a time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ , and convective transfer of the latent heat in the liquid from places of the boundary where freezing occurs to places of melting, which acts to erase topography, with a time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D719}}$ . The ratio $\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D719}}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ controls whether the boundary condition is the classical non-penetrative one ( $\unicode[STIX]{x1D6F7}\rightarrow \infty$ ) or allows for a finite flow through the boundary (small $\unicode[STIX]{x1D6F7}$ ). We study Rayleigh–Bénard convection in a plane layer subject to this boundary condition at either or both its boundaries using linear and weakly nonlinear analyses. When both boundaries are phase-change interfaces with equal values of $\unicode[STIX]{x1D6F7}$ , a non-deforming translation mode is possible with a critical Rayleigh number equal to $24\unicode[STIX]{x1D6F7}$ . At small values of $\unicode[STIX]{x1D6F7}$ , this mode competes with a weakly deforming mode having a slightly lower critical Rayleigh number and a very long wavelength, $\unicode[STIX]{x1D706}_{c}\sim 8\sqrt{2}\unicode[STIX]{x03C0}/3\sqrt{\unicode[STIX]{x1D6F7}}$ . Both modes lead to very efficient heat transfer, as expressed by the relationship between the Nusselt and Rayleigh numbers. When only one boundary is subject to a phase-change condition, the critical Rayleigh number is $\mathit{Ra}_{c}=153$ and the critical wavelength is $\unicode[STIX]{x1D706}_{c}=5$ . The Nusselt number increases approximately two times faster with the Rayleigh number than in the classical case with non-penetrative conditions, and the average temperature diverges from $1/2$ when the Rayleigh number is increased, towards larger values when the bottom boundary is a phase-change interface.

Natural convection is omnipresent on Earth. A basic and well-studied model for it is Rayleigh–Bénard convection, the fluid flow in a layer heated from below and cooled from above. Most explorations of Rayleigh–Bénard convection focus on spatially uniform, perfectly conducting thermal boundary conditions, but many important geophysical phenomena are characterized by boundary conditions which are a mixture of conducting and adiabatic materials. For example, the differences in thermal conductivity between continental and oceanic lithospheres are believed to play an important role in plate tectonics. To study this, Wang et al. (J. Fluid Mech., vol. 817, 2017, R1), measure the effect of mixed adiabatic–conducting boundary conditions on turbulent Rayleigh–Bénard convection, finding experimental proof that even if the total heat transfer is primarily affected by the adiabatic fraction, the arrangement of adiabatic and conducting plates is crucial in determining the large-scale flow dynamics.

The effects of insulating lids on the convection beneath were investigated experimentally using rectangular convection cells in the flux Rayleigh number range $2.3\times 10^{9}\leqslant Ra_{F}\leqslant 1.8\times 10^{11}$ and cylindrical cells in the range $1.4\times 10^{10}\leqslant Ra_{F}\leqslant 1.2\times 10^{12}$ with the Prandtl number $Pr$ fixed at 4.3. It is found that the presence of the insulating lids leads to reduction of the global heat transfer efficiency as expected, which primarily depends on the insulating area but is insensitive to the detailed insulating patterns. At the leading-order level, the magnitude of temperature fluctuations in the bulk fluid is, again, found to be insensitive to the insulating pattern and mainly depends on the insulating area; while the temperature probability density function in the bulk is essentially invariant with respect to both the insulating area and the spatial pattern of the lids. The flow dynamics, on the other hand, is sensitive to both the covering area and the spatial distribution of the lids. At fixed $Ra_{F}$ , the flow strength is found to increase with increasing insulating area so as to transfer the same amount of heat through a smaller cooling area. Moreover, for a constant insulating area, a symmetric insulating pattern results in a symmetric flow pattern, i.e. double-roll structure; whereas an asymmetric insulating pattern leads to asymmetric flow, i.e. single-roll structure. It is further found that the symmetry breaking of the insulating pattern leads to a stronger flow that enhances the horizontal velocity more than the vertical velocity.

Thermal evolution of telluric planets is mainly controlled by secular cooling and internal heating due to the decay of radioactive isotopes, two processes that are equivalent from the standpoint of convection dynamics. In a fluid cooled from above and volumetrically heated, convection is dominated by instabilities of the top boundary layer and the interior thermal structure is non-isentropic. Here we present innovative laboratory experiments where microwave radiation is used to generate uniform internal heat in fluids at high Prandtl number (${>}300$) and high Rayleigh–Roberts number (ranging from $10^{4}$ to $10^{7}$), appropriate for planetary mantle convection. Non-invasive techniques are employed to determine both temperature and velocity fields. We successfully validate the experimental results by conducting numerical simulations in three-dimensional Cartesian geometry that reproduce the experimental conditions. Scaling laws relating key characteristics of the thermal boundary layer, namely its thickness and temperature drop, to the Rayleigh–Roberts number have been established for both rigid and free-slip boundary conditions. A robust conclusion is that for rigid boundary conditions the internal temperature is significantly higher than for free-slip boundary conditions. Our scaling laws, coupled with plausible physical parameters entering the Rayleigh–Roberts number, enable us to calculate the mantle potential temperature for the Earth and Venus, two telluric planets with different mechanical boundary conditions at their surface.

The structure of a steady axisymmetric thermal plume rising through a very viscous fluid with strongly temperature-dependent viscosity of the form $\mu \propto \exp (-\gamma T)$ is investigated. An analytic asymptotic solution is derived for the fast-flowing core of the plume, which predicts that the excess centreline temperature decays exponentially as $\exp \{ - 12 \pi \kappa z/(\gamma A) \}$, where $\kappa $ is the thermal diffusivity, $z$ the height and $A$ the vertical heat flux. This rate of decay, which is found to be in good agreement with numerical simulations of the boundary-layer equations, is three times faster than that predicted by the oft-quoted model of Olson, Schubert and Anderson (J. Geophys. Res., vol. 98 (B4), 1993, pp. 6829–6844).

An upper bound on the rate of vertical heat transport is established in three dimensions for stress-free velocity boundary conditions on horizontally periodic plates. A variation of the background method is implemented that allows negative values of the quadratic form to yield ‘small’ ($O\left(1/ \mathit{Pr}\right)$) corrections to the subsequent bound. For large (but finite) Prandtl numbers this bound is an improvement over the ‘ultimate’ $R{a}^{1/ 2} $ scaling and, in the limit of infinite $Pr$, agrees with the bound of $R{a}^{5/ 12} $ recently derived in that limit for stress-free boundaries.