Local Rayleigh Number Research Articles

A possible role of weak zone at plate margin on secular mantle cooling

We study a simple 2‐D model of secular mantle cooling by convection in which the viscosity depends exponentially on the temperature. To understand the effects of weak zones near the sinking and rising areas, their position and viscosity are imposed. The heat transport efficiency is measured by introducing local Rayleigh (Ral) and Nusselt (Nul) numbers defined at each thermal boundary layer. We find that the Nul ‐ Ral relationship for the bottom boundary layer is similar to that of the steady‐state convection with constant viscosity in a rectangular box. The existence of weak zones predicts a higher Nul for the top thermal boundary layer, suggesting that this will enhance the heat transport of the top layer. If we change Ral of the top layer into the higher of the two Rals, which is usually that of bottom layer, the resultant Nul ‐ Ral approaches that of steady‐state constant viscosity convection. This may imply that the heat transport of convection with secular cooling may be controlled by the boundary layer which has higher Ral. We also suggest that the commonly used parameterization, in which the convecting medium is treated as having a constant viscosity equal to that of its isothermal core, may overestimate the heat transport efficiency.

Local Rayleigh and Nusselt Numbers for Cartesian convection with temperature‐dependent viscosity

We present a local treatment of the heat transport efficiency of the base‐heated cartesian convection with a temperature‐dependent viscosity, which allows us to give a clearer interpretation of Nusselt (Nu)‐Rayleigh (Ra) numbers relation. The “local” Nusselt (Nul) and Rayleigh (Ral) numbers are defined by the local values at each boundary layer except the length scale which is the total depth. We find that Nul‐Ral relation obtained for the convection with constant viscosity and free surfaces is nearly coincident with that of the bottom boundary layer. For top boundary layer, we can treat it as a constant viscosity convection with rigid surface for large enough viscosity contrast. This interpretation can be made by choosing an appropriate temperature drop and viscosity for the definitions of Nul and Ral.

Effect of natural convection on laminar pipe flow solidification

This paper investigates numerically the effect of natural convection on the solidification of laminar fluid flow in the thermal entrance region of a horizontal isothermally cooled tube. The theoretical solution assumes that the Prandtl number is large, and the variation of the liquid-solid interface is gradual in the axial direction. For the liquid phase, a vorticity stream function equation is formulated and solved by a boundary vorticity method. The significance of the natural convection effect is found to depend on the local Rayleigh number. A circular liquid-solid interface is assumed with its center moving up gradually away from the tube axis. The growth of the solid shell is strongly influenced by the superheat ratio λ, the Rayleigh number and the axial position. The theoretical analysis yields the profiles of liquid-solid interface, pressure drop and heat transfer coefficient with varying axial location for the cases of 0 ⩽ Ra ⩽ 10 7 and 0.1 ⩽ λ ⩽ 10. The numerical predictions agree fairly well with the existing experimental data for the heat transfer rate, the pressure drop and the liquid-solid interface.

The sublithospheric mantle as the source of continental flood basalts; the case against the continental lithosphere and plume head reservoirs

Continental flood basalt (CFB) provinces have been attributed to a variety of sources. Among these are: (1) ancient continental lithosphere (CL); (2) newly arrived plume heads from the core-mantle boundary; (3) steady-state plumes coincident with CL rifts; (4) fossil plume heads; and (5) contamination of deep upwellings (passive or active) by enriched sublithospheric shallow mantle. The criticism that CL is too cold to provide extensive magmatism has been countered by the proposal that the CL is wet, thereby lowering the melting point. This also lowers the viscosity and increases the local Rayleigh number. The calculated seismic velocities and viscosities in this proposed CL source show that it has a low velocity and is weak and has asthenosphere-like physical properties. It is apparent that what has been called ‘continental lithosphere’ is actually asthenosphere or the lower part of the thermal boundary layer (TBL) and is unlikely to be a long-lived part of the plate. However, a low density and low viscosity region of the sublithospheric mantle (the perisphere) is a suitable reservoir for the enriched component of CFB. It helps to isolate the deeper depleted reservoir from contamination due to recycling at subduction zones. Lithospheric pull-apart at cratonic boundaries, rather than stretching of uniform lithosphere, is suggested as the trigger for extensive continental magmatism.

On the stability of mean-field models of the solar convection zone

The stability of the solutions of the mean-field theories of turbulent media is questioned. It is done here for the model equations for the solar convection zone which have been used, in particular, to explain the differential rotation. We present an approximation valid for axisymmetric, short-wave disturbances. A critical local Rayleigh number can be defined - involving eddy diffusivities - above which the stratification becomes unstable. For mixing-length models of the solar convection zone we always find sub-critical Rayleigh numbers. One must be careful, however, with other theoretical models. Those we have considered do not reach sufficiently high surface pressure values so that there the associated Rayleigh numbers exceed their critical limits. In the outermost layers in such models, therefore, the solutions could really be unstable.

Collisions between pulses of traveling-wave convection.

The first nonlinear state of traveling-wave convection in binary fluids with moderate negative separation ratio in a narrow geometry consists of ``pulses'': localized patches of convection whose spatial shape is fixed, and that drift at a velocity that depends on the local Rayleigh number. I present the results of two kinds of experiments on traveling-wave pulses. First, I study the behavior of pulses as they drift past narrow, fixed peaks in Rayleigh number. The pulses are sensitive to the Rayleigh number in a spatial domain that extends far ahead of the main body of the pulse. Second, I study collisions between pairs of counterpropagating pulses as a function of the velocity with which they approach one another. At high approach velocity, only one pulse survives the collision. At low approach velocity, a persistent double-peaked structure is formed. Under certain circumstances, this structure can be interpreted as a weakly bound state of two pulses.

Perturbation analysis of convective instability of oceanic lithosphere and initiation of subduction zones

A linear perturbation analysis is made on convective instability of the oceanic lithosphere to understand the occurrence of subduction on the Earth. The oceanic lithosphere is modeled by the upper part of top thermal boundary layer in a convecting Newtonian temperature‐dependent viscosity fluid. Thickening of oceanic lithosphere with its age due to cooling at the surface is modeled by thickening of the thermal boundary layer with time; its thickening rate is a measure of the rate with which temperature in the thermal boundary layer decreases with time and hence is a measure of the rate with which viscosity in the thermal boundary layer increases with time. The linear perturbation analysis shows the possible excitation of plural eigenmodes to give rise to an instability in the thermal boundary layer depending on a threshold value Rc of the local Rayleigh number R of the thermal boundary layer. Further, Rc is shown to be a linearly increasing function of , which is treated as a free parameter. From estimates of R and appropriate for the major oceanic lithosphere (denoted as Robs and obs, respectively), I find that Rc( = 0) < Robs < Rc( = obs). The flow pattern of mantle convection depends strongly on which eigenmodes are excited by the convective instability in thermal boundary layer: (1) When R is lower thanRc, only the fundamental and the first higher modes at most are excited. The resulting mantle convection manifests itself as the secondary convection. (2) When R is higher than Rc, the modes higher than the first higher mode are also excited. The resulting mantle convection involves lithospheric instability, leading to large local deformation in the oceanic lithosphere and finally to subduction of the oceanic lithosphere. This result implies that (1) subduction does not begin newly in the major oceanic lithosphere at present, since lithospheric instability is suppressed by thickening of the major oceanic lithosphere, neither by its high viscosity nor by its buoyancy, (2) subduction will newly begin in the major oceanic lithosphere due to lithospheric instability in future when the oceanic lithosphere becomes so old that its thickening rate decreases to sufficiently small value. The latter result suggests that lithospheric instability is an important agent that maintains multiplate tectonics on the Earth.

Flow at the Onset of Traveling Wave Convection

We studied steady and transient mass flow in a localized traveling wave state of convection of binary fluid mixture in an annular cell by use of a photochromic dye. The results indicate that no mass flow occurs across the self-organized boundary in a steady state, but that a large-scale flow exists at the transient stage to establish a localized traveling state. The transient flow was found to occur in the direction to lower the local critical Rayleigh number and to stabilize the localized convection state.

Compositional convection in viscous melts

DURING solidification of multi-component melts, gradients in temperature and composition develop on different scales because of the large difference between their respective molecular diffusivities. Two consequences are the development of double-diffusive convection1 and the creation of mushy zones in which solid and liquid intimately coexist with a complex small-scale geometry2,3. Theoretical analysis requires simplifying assumptions that must be verified by laboratory experiments. Hitherto, experiments have been carried out with aqueous solutions which do not accurately represent the dynamics of melts with high Prandtl numbers, such as magmas. Here we describe the characteristics of compositional convection using a new experimental technique which allows the viscosity of the solution to be varied independently of chemical composition and liquidus temperature. A supereutectic melt was cooled from below, causing the growth of a horizontal layer of crystals. Convective instability occurred when the local solutal Rayleigh number of the compositional boundary layer ahead of the advancing crystallization front attained a value of ∼3 on average. We observed a novel regime of convection in which the thermal boundary layer above the crystallization front was essentially unmodified by the motion of the plumes. The plumes carried a small heat flux and did not mix the fluid to a uniform temperature.

Natural Convection Heat Transfer From a Discrete Thermal Source on a Channel Wall

This paper presents the results of an experimental study of natural convection heat transfer from a discrete thermal source located on the wall of a vertical channel. This configuration has both an unheated inlet and outlet region and produces a confined wall plume, which rises between the parallel plates that form the channel walls. In the present study, a thermal source of length was located on one wall of a vertical channel formed by two plates of length L separated a distance b. The thermal source was located a distance x along the plate was measured from the leading edge of the source. A dimensional analysis for this problem indicates that the local Nusselt number Nu{sub x} can be correlated in terms of the local Rayleigh number Ra{sub x}, the Prandtl number Pr, a dimensionless distance x/l, and three geometric ratios L/l, d/l, and b/l.

Viscosity and thickness of the sub-lithospheric low-viscosity zone: constraints from geoid and depth over oceanic swells

The medium-wavelength geoid to depth anomalies ratio (GDR) at oceanic hotspot swells has been found to increase from ∼ −0.5 m/km to ∼ 5 m/km according to the age of the lithosphere they occur on. In order to interpret this trend, the geoid and topography anomalies associated with mantle convective plumes crossing a sublithospheric low viscosity zone (LVZ) have been derived from numerical models and a systematic investigation of the GDR dependence on the viscosity and depth extent of the LVZ, on the thickness and thermal structure of the lithosphere and on the Rayleigh number has been conducted. It is shown that, for viscosity drops across the base of the LVZ, greater than one order of magnitude, the GDR is strongly dependent on the depth of shallow interfaces such as the lithosphere/ athenosphere boundary and on the LVZ's thickness. Consequently, the empirical trend can be accounted for by the thickening of the lithosphere with age provided it occurs at the expense of a LVZ whose base is at a fixed depth (around 200 km). In such a frame, no significant variation with age of the LVZ's viscosity is required by the GDR data. Best fit with the empirical trend is found for a LVZ about 50 times less viscous than the underlying mantle. The mantle flow starts to fluctuate when the local Rayleigh number of the low-viscosity layer exceeds the Rayleigh number of the underlying mantle. The fluctuations are initiated in the upper boundary layer, in the diverging part of the plume, at a distance of a few hundreds of kilometers from the main ascending current. For viscosity contrasts in the range of 40–60, deduced from the present study, the conditions for the development of these small-scale instabilities are realized only where the lithosphere has not yet grown significantly downwards (ages < 50 Ma). These fluctuations are tentatively related to the small wavelength (∼ 200 km) geoid anomalies pattern observed in a young and hot area of the southwestern Pacific Ocean. This pattern could be the surface expression of a mantle plume rising just beneath the East Pacific Rise axis.

The effect of a shallow low viscosity zone on the apparent compensation of mid-plate swells

A simple model for mid-plate swells is that of convection in a fluid which has a low viscosity layer lying between a rigid bed and a constant viscosity region. Finite element calculations have been used to determine the effects of the viscosity contrast, the layer thickness and the Rayleigh number on the flow and on the perceived compensation mechanism for the resulting topographic swell. As the viscosity decreases in the low viscosity zone, the effective local Rayleigh number for the top boundary layer of the convecting cell increases. Also, because the lower viscosity facilitates greater velocities in the low viscosity zone, the low viscosity layer produces proportionally greater horizontal flow near the conducting lid, causing the base of the conducting lid to appear like a free boundary. The change in the local Rayleigh number and in the effective boundary condition both cause the top boundary layer to thin. Through a Green's function analysis, we have found that the low viscosity zone damps the response of the surface topography to the temperature anomalies at depth, whereas it causes the gravity and geoid response functions to change sign at depth counteracting the positive contributions from the shallower temperature variations. By increasing the viscosity contrast, the conbined effects of the thinning of the boundary layer and the behaviour of the response functions allow the apparent depth of compensation to become arbitrarily small. Therefore, shallow depths of compensation cannot be used to argue against dynamic support of mid-plate swells. Furthermore, we compared the distribution of the effective compensating densities, which is used to obtain the geoid, to that of Pratt compensation, which is often used to calculate the depth of compensation from geoid and topography data for mid-plate swells. For all of our calculations including those with no low viscosity layer, the effective gravitational mass distribution is more complex than assumed in simple Pratt models, so that the Pratt models are not an appropriate gauge of the compensation mechanism.

Convective instabilities in a variable viscosity fluid cooled from above

Whether in the mantle or in magma chambers, convective flows are characterized by large variations of viscosity. We study the influence of the viscosity structure on the development of convective instabilities in a viscous fluid which is cooled from above. The upper and lower boundaries of the fluid are stress-free. A viscosity dependence with depth of the form ν 0 + ν 1 exp(−γ.z) is assumed. After the temperature of the top boundary is lowered, velocity and temperature perturbations are followed numerically until convective breakdown occurs. Viscosity contrasts of up to 10 7 and Rayleigh numbers of up to 10 8 are studied. For intermediate viscosity contrasts (around 10 3), convective breakdown is characterized by the almost simultaneous appearance of two modes of instability. One involves the whole fluid layer, has a large horizontal wavelength (several times the layer depth) and exhibits plate-like behaviour. The other mode has a much smaller wavelength and develops below a rigid lid. The “whole layer” mode dominates for small viscosity contrasts but is suppressed by viscous dissipation at large viscosity contrasts. For the “rigid lid” mode, we emphasize that it is the form of the viscosity variation which determines the instability. For steep viscosity profiles, convective flow does not penetrate deeply in the viscous region and only weak convection develops. We propose a simple method to define the rigid lid thickness. We are thus able to compute the true depth extent and the effective driving temperature difference of convective flow. Because viscosity contrasts in the convecting region do not exceed 100, simple scaling arguments are sufficient to describe the instability. The critical wavelength is proportional to the thickness of the thermal boundary layer below the rigid lid. Convection occurs when a Rayleigh number defined locally exceeds a critical value of 160–200. Finally, we show that a local Rayleigh number can be computed at any depth in the fluid and that convection develops below depth z r (the rigid lid thickness) such that this number is maximum. The simple similarity laws are applied to the upper mantle beneath oceans and yield estimates of 5 × 10 15−5 × 10 16 m 2 s −1 for viscosity in the thermal boundary layer below the plate.

The effect of a shear flow on convection in a layer heated non-uniformly from below

In an earlier paper (Walton 1982) we showed that, when a fluid layer is heated non-uniformly from below in such a way that the vertical temperature difference maintained across the layer is a slowly varying monotonic function of a horizontal coordinate x, then convection occurs for x &gt; xc, where xc is the point where the local Rayleigh number is equal to the critical value for a uniformly heated layer. Furthermore, the amplitude of the convection increases smoothly from exponentially small values for x [Lt ] xc and asymptotes to a value given by Stuart–Watson theory for x [Gt ] xc.At the present time no solutions of this kind are available for a class of problems in which the onset of instability is heavily influenced by a shear flow (e.g. Görtler vortices in a boundary layer on a curved wall, convection in a heated Blasius boundary layer). In a first step to bridge the gap between these problems and in order to elucidate the difficulties associated with the presence of a shear flow, we investigate the effect of a (weak) shear flow on our earlier convection problem. We show that the onset of convection is delayed and that it appears more suddenly, but still smoothly. The role of horizontal diffusion is shown to be of paramount importance in enabling a solution of this kind to be found, and the implications of these results for instabilities in higher-speed shear flows are discussed.

The Brinkman model for natural convection about a semi-infinite vertical flat plate in a porous medium

The Brinkman model is used for the theoretical study of boundary effects in a natural convection porous layer adjacent to a semi-infinite vertical plate with a power law variation of wall temperature, i.e. T ̄ wα x ̄ λ . It is shown that the dimensionless governing equations based on this model contain two parameters ϵ, = (Ra) − 1 2 and σ = (Da/φ) 1 2 where Ra and Da are the Rayleigh number and the Darcy number based on a reference length and φ is the porosity. For the limit of ϵ → 0, σ → 0 and σ ⪡ ϵ, a perturbation solution for the problem is obtained based on the method of matched asymptotic expansions. The physical problem can then be visualized to be consisting of three layers : an inner viscous sublayer, adjacent to the heated surface, with a thickness of the order of 0(σ) ; a middle thermal layer of thickness of the order of 0(ϵ) ; and the outer potential flow region with thickness of the order of 0(1). It is found that the first-order problem of the thermal layer is identical to that based on Darcy's law with slip flow, whose solution was obtained previously. Composite solutions for stream function and temperature, uniformly valid everywhere in the flow field, are constructed from the solutions for the thermal layer and the viscous sublayer. A new parameter Pn x , defined as Pn x = (Ra xDa x) 1 2 with Ra x and Da x denoting the local Rayleigh number and the local Darcy number based on x̄, is found to be a measure of the boundary effect. It is shown that the viscous effect on the boundary has a drastic effect on the streamwise velocity component near the wall with a lesser effect on heat transfer characteristics. The boundary effect slows down the buoyancy-induced flow with a resulting decrease in heat transfer. The local Nusselt number is found to be of the form Nu x(Ra x) 1 2 = C 1− C 2Pn x where the values of C 1 and C 2 depending on λ. For an isothermal vertical plate (λ = 0), the first-order correction to the local Nusselt number is identically zero, i.e. C 2 = 0. In general, the boundary effect on the local Nusselt number becomes more pronounced as the value of Pn x , Ra x or Da x is increased.

Experimental Investigation of Natural Convection Losses From Open Cavities

Experimental results for natural convection in a cavity are reported. Both constrained and unconstrained cavity geometries were studied. Detailed velocity profiles were obtained using Laser doppler velocimetry for Rayleigh numbers between 3 × 1010 and 2 × 1011, corresponding to a constant elevated wall temperature boundary condition. Characteristics of two-dimensional and three-dimensional flows obtained with dye flow visualization are discussed, including boundary layer transition to turbulence, flow patterns in the cavity, and flow outside of the cavity. Local Nusselt number is correlated with local Rayleigh number for constrained and unconstrained cavities.

Matched asymptotic expansions for free convection about an impermeable horizontal surface in a porous medium

The problem of steady free convection in a porous medium adjacent to a horizontal impermeable heated surface, with wall temperature distribution w= ∞+ A λ(0≤λ<2), for ≥0 and = ∞ for <0, is investigated by the method of matched asymptotic expansions. The small parameter in the perturbation series is found to be the inverse one-third power of the Rayleigh number. For the first-order inner problem the governing equations reduced to the boundary layer approximations which have been solved previously. The effects of fluid entrainment, streamwise heat conduction and upward-drift induced friction are taken into consideration in the second and third-order theory for which similarity solutions are obtained. Numerical results for temperature and streamwise velocity profiles as well as the local Nusselt number at different local Rayleigh numbers and different prescribed wall temperature distributions are presented. It was found that the local Nusselt numbers as obtained from the boundary layer theory for λ=0.5 are accurate to the third-order, while those for λ=0 are accurate to the second-order. For other values of λ, the boundary layer theory underestimates the local Nusselt number slightly; the accuracy of the boundary layer theory decreases as the Rayleigh numbers decrease and as λ increases from λ=0.5.

Numerical simulation of thermohaline convection in the upper ocean

The intradiurnal heating and cooling cycle of the mixed layer of a tropical ocean is investigated through the use of a ‘pseudo-two-dimensional’ numerical model. Particular emphasis is given to two-component diffusion resulting from dynamic instabilities in the water column. The conservation equations for salt and heat include the effects of solar heating, horizontal advection and turbulent fluxes at the sea surface, while wind mixing enters through the use of depth-dependent eddy diffusion coefficients resulting from the wave-orbital shear model of Kitaigorodskiy (1961). All inputs are treated as functions of time of day, or calculated via the bulk aerodynamic method. The entrainment fluxes of salt and heat due to the mechanical stirring of the wind and the fluxes due to molecular diffusion are treated as separate, their respective contributions being added to form the diffusion coefficients used in an alternating-direction explicit scheme to integrate the heat and salt equations. Near the surface, in the absence of strong solar heating (i.e. during the night), these two fluxes alone are insufficient to remove the near-surface static instabilities; thus the presence of some additional process is suggested. A dynamic stability analysis is carried out, based on the temperature and salinity gradients. The resulting Rayleigh numbers indicate the possibility of double-diffusive convection, whereby the vertical transfers of salt and heat may proceed at rates far greater than can be accounted for by molecular diffusion alone. Therefore, the molecular diffusions in the model are increased by a factor roughly proportional to the one-third power of the ratio of the local effective Rayleigh number to a critical Rayleigh number. The modified molecular diffusivities are then added to the eddy diffusion coefficients due to the wind, to form the total diffusion coefficients used in the numerical integrations. Comparisons are made between the model-generated profiles of temperature and the profiles observed in the ocean. The comparisons show reasonable agreement in the diurnal cycle of the heat wave at 1 m vertical resolution (except for the model-generated surface layer being too deep during the late afternoon hours). (Previous models typically predict only the temperature and thickness of a homogeneous layer.) The results obtained with the model are instructive in estimating the relative importance of the various mixing processes in the upper ocean.

Moving lithospheric plates and mantle convection

The coupling between a rigidly moving lithospheric plate and a convecting mantle is investigated using a simple two-dimensional numerical model that incorporates a horizontally moving upper boundary, simulating the effect of a moving plate, over a fluid layer heated from below. The moving boundary strongly controls the horizontal length scale of convection cells when its velocity is greater than the free convective velocity (i.e. the velocity with which the fluid would convect under a stationary boundary). In a box of aspect ratio 4 (width/depth), a transition in flow structure occurs from several equidimensional convection cells under a slowly moving boundary to a single long convection cell under a rapidly moving boundary. The flow structure transition occurs approximately when Pe/Ra2/3= 0.04, where the Peclet number, Pe, measures the (prescribed) velocity of the upper boundary, and the Rayleigh number, Ra, measures the heating of the fluid layer. Near the transition, the flow tends to be unsteady; this behaviour can be well understood in terms of the instability of the thermal boundary layers, which can be characterized by a local Rayleigh number. Using conventional estimates of mantle parameters, the mantle is either near or above the transition to single-cell convection, whether upper-mantle or whole-mantle convection is assumed. The net tangential force exerted by the fluid on the upper boundary varies approximately linearly with the boundary velocity above the transition, and it is positive (driving) for boundary velocities ranging from the value at the transition to about three times the transition value. Boundary velocities corresponding to zero net force are close to those expected for convection under a free-slip boundary. When scaled to whole-mantle convection, the shear stress on the upper boundary is of the order of 10 bar, and above the transition to a single cell the net force on a large plate is sufficient to overcome even kilobars of frictional resistance at plate boundaries. These results indicate that mantle convection and lithospheric plate motions are likely to be strongly coupled, with the rigid moving plates strongly controlling the mantle flow structure, with single cells under the faster plates, but with the mantle convection cells possibly exerting large driving forces on the plates.

Mantle convection and the thermal structure of the plates

Though a simple thermal model of plate creation matches the observed bathymetry and heat flow with considerable accuracy, it only does so because a constant temperature is imposed as a lower boundary condition. A more realistic model is developed here, based on the idea that the thermal structure of the plate becomes unstable and leads to the development of small‐scale convection. Convective heat transport then supplies the heat flux needed to match the observations rather than an artificial constant temperature boundary condition. The temperature dependence of the rheology is represented in a simple manner. Below a given temperature the material is assumed to move rigidly, defining an upper mechanical boundary layer. Beneath this rigid layer, where the temperatures are greater, the material is assumed to have a constant Newtonian viscosity. The part of the viscous region where there are significant vertical temperature gradients, immediately below the mechanical boundary layer, forms a thermal boundary layer. As the plate cools, both the mechanical and thermal boundary layers increase in thickness. A local critical Rayleigh number criterion is used to test the stability of the thermal boundary layer. On this basis a convective instability is predicted, its occurrence coinciding with the breakdown of the linear dependence of the depth of the ocean floor on the square root of age. Though the small‐scale convection which develops from this instability modifies the thermal structure, the basic observational constraints are shown to be satisfied. The stability criterion is further tested in two different laboratory experiments. These experiments also illustrate a possible form for the instability, with cold dense material breaking away from the base of the plate and being replaced by hotter material from below.