Boundary layer interaction in plume-based variable-viscosity thermal convection

Transcription

1 Magistère des Sciences de la Matière STAGE École Normale Supérieure de Lyon SCHAEFFER Nathanaël Université Claude Bernard Lyon1 2 ème année Option Physique Boundary layer interaction in plume-based variable-viscosity thermal convection Résumé : Nous avons mesuré la fréquence de formation de panaches dans une expérience de convection thermique à Pr >>1. Comme la viscosité du fluide utilisé varie fortement avec la température, les couches limites chaude et froide ont différentes épaisseurs et viscosités. Ceci entraîne des fréquences différentes pour ces couches. Il semblerait qu il y ait une interaction directe entre les couches limites, par l intermédiaire des panaches. En particulier, s il y a une fréquence principale bien définie pour l apparition de panaches froids, il y a en a plusieurs pour les panaches chauds, dont l une est celle des panaches froids. Ces mesures suggèrent donc que des «panaches» froids dans le manteau (les pans de croûte plongeant dans les zones de subduction) peuvent indure des panaches chauds. Mot-clés : convection, viscosité variable, couche limite, panaches. Department of Geological Sciences 1272 University of Oregon Eugene OR USA Advisor : Pr. Michael Manga May to August 2000

2 Thanks : I d like to thank Michael Manga for his help and support, and for his interest in my work; David Senkovich for his mine of ideas and tricks; Alison Rust for sharing her office and much more; the geology department and especially the graduate students for all the fun we had together. 2

3 Table of content Table of content... 3 Introduction... 4 Analysis of thermal convection... 5 Experimental setup... 7 Convection experiments... 7 Viscosity measurements... 9 Values of the physical properties used for our dilute corn syrup (10%vol of water) Results Rayleigh and Nusselt numbers Middle temperature Time series analysis and boundary layer interaction Frequency of plumes, a scaling approach Comparison with experimental data How it might work Visualization Further investigations Conclusion & applications References

4 Introduction Thermal convection can be found in many settings. For example heating water in a sauce pan, as well as movement in the Earth s mantle are convection phenomena. Thermal convection can be driven by several forces. The most common one is gravity, which tends to move the hot and thus lighter fluid to the top. But surface tension variations with temperature, or even other quantities varying with temperature (such as solubility), can induce thermal convection. Here we will focus on gravity-induced thermal convection, as it occurs in the Earth s mantle, which involves large viscosity variations between the bottom and hot thermal boundary layer, and the top and colder thermal boundary layer. In very viscous fluids (such as the mantle, or the corn syrup studied here), flow is not turbulent. However, in order to achieve higher heat transfer, rising and sinking plumes may form, carrying a large fraction of the heat. These thermal plumes are believed to be important in some geological phenomena, such as hot spots (volcanic activity outside plate boundaries). Understanding how these plumes form and how they evolve is thus of great interest. It has been shown that the boundary conditions at the top and bottom are critical parameters for thermal convection. In this study, we will restrain to no-slip boundaries, which is not the case for earth (plate tectonics), but may be appropriate for planets such as Venus and Mars. Geological implications are discussed in detail in Schaeffer & Manga (2000). 4

5 Analysis of thermal convection In fluid mechanics, it is convenient to work with dimensionless numbers, so that results can easily be transposed and compared to other experiments. There are two key parameters for thermal convection. First, the Rayleigh number Ra, which characterizes the ratio of diffusive to advective timescales, is defined by 3 ρgα TD Ra=, µκ where ρ is the density of the fluid, α and µ are respectively its thermal expansivity and viscosity, T is the temperature difference across the layer, g the acceleration of gravity, κ the thermal diffusivity, and D the depth of the layer. A high Rayleigh number means that density differences between cold and hot fluids are sufficiently large for the fluid to move fast enough to transport heat more efficiently than thermal diffusion does. Thus, above a certain critical Rayleigh number, the fluid will convect. The second number is the Prandtl number Pr, which compares the diffusion of heat (κ) and the diffusion of momentum (ν): ν Pr =. κ If the Prandtl number is larger than one, then heat diffuses slower than momentum. The Reynolds number Re tells us if the flow is turbulent or not, and is thus of major importance ud Re =, ν where u and d are respectively characteristic speed and length for the flow, and ν is the kinematic viscosity. A high Pr is often associated with low Re numbers and thus non-turbulent flow. To characterize the heat transfer, the Nusselt number Nu is defined as the ratio of the actual heat transfer compared to the conductive heat transfer in the absence of convection: Nu= k HD, T 5

6 where H is the total heat flux, and k is the thermal conductivity. Thus, Nu = 1 means that there is no convection and only diffusive heat transport. Nu measures the efficiency of the heat transfer. We expect Nu to increase as Ra increases. In fact, Nu and Ra follow a very simple power relation. This can be understood in the following way : while increasing the depth of the layer, and keeping all the other parameters constant, we expect the boundary layers to remain unaffected, and thus the heat transfer to be roughly the same, whereas the corresponding conductive transfer in absence of convection drops, scaling like 1/D. So the Nusselt number should scale like D. Since the Rayleigh number increases like D 3, we expect the following scaling : Nu = a.ra 1/3. Experimentally, the powers vary between 1/4 and 1/3 (Grossmann and Lohse 2000). Other specific scaling theories give results like 2/7 for turbulent convection (Castaing et al 1989), but there is no theoretical result for large Pr flows that gives this value. The dimensionless temperature θ corresponding to the temperature T, is used to describe interior temperature, and is defined by T T cold θ =, Thot Tcold where T hot and T cold are the hot and cold boundary temperature respectively. Finally, when the viscosity varies with temperature, we can introduce the viscosity contrast λ µ cold λ=. µ hot This one parameter is suffucient if the temperature-dependence of the viscosity is simply exponential. 6

7 Experimental setup Convection experiments To perform experiments on convection, we build a tank (see Figure 1) of aspect ratio 2. The bottom is a square aluminum plate (33 cm wide), which is heated by an electric heater, the top is a hollow aluminum plate, in which there is water flowing to keep it at a constant temperature. The side walls (16.5 cm high) are made of glass, so that we can look inside. The entire apparatus is insulated with five-centimeter-thick styrofoam. Through the bottom plate, 3 thermocouples record the temperature at the bottom, and 2 others measure the temperature at different heights in or near the lower boundary layer (4 and 9 mm). +ROORZDOXPLQXPSODWH 7KHUPRFRXSOHV )ORZRXW :DWHU )ORZLQ &RUQV\UXS ' *ODVV 3ODWLQXPZLUH ' FP,QVXODWLRQ $OXPLQXPSODWHV 7KHUPRFRXSOHV 5HVLVWDQFHKHDWHU Figure 1. Schematic illustration of the tank used for our convection experiments. The tank is filled with corn syrup, and connected to a reservoir to avoid complications from thermal expansion or contraction of the fluid. On the top, we have 3 thermocouples, one near the water input, another at the output, and the last one at one of the remaining corners. Similar to the bottom, we also have two thermocouples to record the temperature at different depths (8 and 15 mm) in the cold boundary layer. A few thermocouples were also placed in the insulation and in the room, for a total of 19 thermocouples. 7

8 In addition, we also stretched a platinum wire, at a depth D/2, between two opposite corners, supported by two nearby thermocouples, in the corners. However, the device used to record the resistance of the platinum wire was not adapted, and no useful data could be retrieved. All these temperatures are recorded digitally every 5 to 20 seconds (depending on the expected value of Ra) by a scanning unit. Since the fluid is very viscous, the fluid moves very slowly. Each experiment ran for at least 24 hours, and as long as 3 days. This is small compared to the diffusive 2 time scale D (almost 3 days), but because the fluid is convecting, time to reach equilibrium is κ much shorter, of order 2 Nu D, which is always less than 24 hours..κ With no-slip boundary conditions at both the top and bottom of the tank, the temperature profile consists basically of a steep temperature gradient at the top and at the bottom, and a constant averaged temperature between these boundary layers. Since the velocity of the fluid at the very bottom and very top is zero, heat is thus transferred only by conduction, and measuring the temperature gradient at these places will give us the total heat transfer. The measurement of the gradient is sometimes used to determine the Nusselt number (Manga & Weeraratne 1999). However, this is not very accurate since the layer is often only a few millimeters thick, compared to the one millimeter spatial resolution of our thermocouples. By using a resistance heater, instead of a temperature controlled bottom, we can determine exactly the power added to the system, by simply reading the voltage input. There is still a problem with the glass side walls. In fact glass conducts heat 3 times better than corn syrup. This, combined with the 6 mm side walls, leads to as much as 21% of the heat being conducted by the side walls, if we assume the temperature gradient in the glass walls is close to the one in the interior fluid. However, when convecting, this heat comes back in the tank at a height where the interior fluid is convecting, (because the vertical temperature gradient vanishes on the inside of the side walls). This gives us an upper limit to the uncertainty on the Nusselt number (21%). In fact all the Nusselt numbers we give in this paper are corrected. We have simply subtracted 0.21 from the actual value. This is then close to the Nusselt number we could measure via the temperature gradient. 8

9 Viscosity measurements Since the viscosity of corn syrup varies strongly with temperature, we need to measure this viscosity at different temperatures. To do so, we used a rotating viscometer, which measures the torque needed to rotate a cylinder in corn syrup. The temperature of the fluid was controlled with a water flow. However, evaporation of water at high temperatures changes the composition of our fluid and thus its viscosity. At low temperatures, condensation acted in the opposite way, adding water to the corn syrup and making it less viscous. To limit evaporation, we wrapped some plastic film around the viscometer, and to prevent condensation, we used anhydrous calcium sulfate which absorbed air moisture as temperature went down. These techniques allowed us to measure the viscosity of our dilute corn syrup for temperatures ranging from 5 to 80 degrees celsius, with 5 degree steps, assuming the temperature of the corn syrup was the one of the circulating bath, which appeared not to be true. A sample of the fluid was also sent to the University of Berkeley, for improved measurements performed by Helge Gonnermann (using a better viscometer). These measurements give slightly lower viscosities, especially at high temperatures. This effect appears to be partly due to temperature differences between the circulating water and the sample, in our setup measures best fit Viscosity (Pa.s) Temperature (Celsius) Figure 2. Viscosity of our dilute corn syrup (about 10% water). Measurement for temperatures ranging from 2.5 to 100 degrees are shown, as well as the fit used to calculate viscosity at any temperature. We choose to use a phase-transition like function, which fits quite well to the data, and illustrate the fact that the corn syrup will eventually solidify (µ diverges at T c ): T T µ = µ 0 Tc c β 9

10 The least square best fit, performed on log(µ) gives the following values: µ 0 = Pa.s T c = K β = 4.85 All these results are summarized in Figure 2. Theoretical investigations usually assume an exponential temperature-dependence. This is not the case for the corn syrup (Figure 2). We thus expect some systematic deviations between experimental data and theoretical predictions. Values of the physical properties used for our dilute corn syrup (10%vol of water) The thermal conductivity and the heat capacity are calculated from water and pure corn syrup values, assuming they are additive. The density was measured with glass hydrometers (the calculated value 1388 kg.m -3 lies in the error range). The corn syrup values used to interpolate are obtained from previous measurements (Giannandrea and Christensen 1993). Thermal conductivity (interpolated) k = W.m -1.K -1 Heat capacity (interpolated) C p = J.K -1.m -3 Thermal diffusivity (k/c p ) κ = m 2.s -1 (± 10%) Density (measured) ρ = 1387 (± 4) Thermal expansivity (interpolated) α =

11 Results When the system has reached a steady state, that is when there are no long term changes in the recorded temperatures, we averaged the temperatures of the bottom, top and middle thermocouples, to compute the dimensionless numbers discussed previously, as shown in Table 1. The missing Nu have been lost. The Prandtl number is always larger than 10 3, and the calculated Reynolds number based on different types of flow is always smaller than 0.5. index Ra Nu Pr λ θ m T (ºC) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E Table 1. List of experiments done, with their characteristic numbers. λ is the viscosity across the whole tank, θ m is the dimensionless middle temperature. Highest and lowest values are bold. Rayleigh and Nusselt numbers Since the viscosity variations through the whole tank can be a factor of a few hundred, the Rayleigh number no longer has a single definition. In Figure 3, the Nusselt number has been plotted versus different definitions of the Rayleigh number, based on the viscosity in the middle of the tank (which is the most straightforward), on the bottom of the tank, and on the average temperature of both top and bottom. The measurements of Giannandrea and Christensen (1993) are shown for reference. For the relationship between Nu and Ra to be still valid with variable viscosity, there is only one choice that appears to work well: the Rayleigh number defined with the viscosity at the average 11

12 temperature between top and bottom. It is this Rayleigh number which is shown in Table 1, and used by Giannandrea and Christensen. This result is quite surprising as we could expect the relevant Rayleigh number to be the one defined with the actual viscosity of the interior convecting fluid (small squares). This issue is well known and has already been investigated, but there is no satisfying explanation (Morris and Canright 1994). One rough explanation could be that the Rayleigh number is somehow a measure of the distance in instability space. Instabilities occur when the fluid is not convecting. The location where instabilities can grow easiest, is far from top and bottom which is best in the middle of the tank. There, the viscosity is the one we found relevant, when there is no convection. We may also notice the slight shift between our measurements and the data given by Giannandrea & Christensen. This is due to the quite important uncertainty on the other parameters of the corn syrup (each corn syrup might be different). 100 Giannandrea and Christensen (1993) 1/2 Ra Tm Ra Th Ra Tc Nu 10 1 y = x R 2 = E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 Figure 3. The dark blue diamonds represent other independent measurements, and the other symbols are our own measurement, represented using different definition of the Rayleigh number: Tm, Th, Tc and ½ are based respectively on the viscosity at the middle, hot, cold and mean temperature. It is quite obvious that the Rayleigh number based on the viscosity at the average of both top and bottom temperatures is the relevant one. Ra The best fit on our data with a power law gives the following results: Nu = a.ra β with a = 0.13 and β = ± This is close to 2/7 and even closer to other results. In a turbulent convection experiment with liquid helium, performed at the University of Oregon, for Ra ranging from 10 6 to (Niemela et al 2000), a power-law with an exponent of is obtained. This similarity between two very 12

13 different exponents is striking, but might only be a coincidence. While the turbulent convection regime has some theoretical support for scaling (Grossmann and Lohse 2000), there is no really satisfying one for the very large Pr regime. This is partly due to the complex nature of boundary layer instabilities, which produce the thermal plumes and interact with a possible convecting flow. Middle temperature When the viscosity is constant, symmetry considerations tell us that the middle temperature should be the mean of both top and bottom temperatures. When viscosity varies with temperature, this is no longer true, because the symmetry is broken. Theory suggests that the middle temperature θ m depends only on the viscosity ratio λ (Manga and Weeraratne 1999; Zhang et al 1997): θ m 1 = 1+ λ This result assumes an exponential viscosity profile, and thus deviations for high viscosity ratios might be expected. When compared to experimental results (Figure 4), this theory works surprisingly well for the whole range of viscosity ratios we studied (up to 285). 1/ middle temperature set 1 set 2 theory my mes Viscosity ratio Figure 4. The middle temperature dependence on the viscosity ratio follows the theoretical curve quite well for the range of viscosity ratios we have observed. For bigger viscosity ratios, deviations appear. We can also notice that the temperatures we measured are systematically higher by a small amount than the theoretical predictions. This is consistent with the actual temperature-dependence of the viscosity. 13

14 Time series analysis and boundary layer interaction. 2 In this section, time is made dimensionless using the diffusive timescale D. κ Frequency of plumes, a scaling approach Assume that a thermal boundary layer grows by thermal diffusion only. Once the local Ra (for the boundary layer itself) exceeds a critical value Ra *, the boundary layer becomes unstable and forms a plume (Howard 1966). In more detail, the time scale for heat to diffuse across a layer of thickness δ is The critical thickness and Ra are defined by δ 2 t. κ Ra αg T δ =. νκ * layer The time needed to reach the critical thickness is then t * Ra νκ κ αg * 1 Now, if the subscripts c and h are used to refer to the cold and the top boundary layer respectively, 14 T layer and assuming Ra * is the same for both layers, we get the following scaling: t t h c µ h T µ c T c h 2 / 3 * 3 2 / 3 and hence for frequencies:. 2/3 fh µ c Th f c µ h Tc where µ is the viscosity at the average temperature of top and bottom of the boundary layer, since it has been shown that this is the relevant viscosity for the Rayleigh number (see above). Thus, Ti + Tm µ i = µ and Ti = Ti Tm, 2 with T m being the temperature in the middle of the tank. Note also that there is no Ra-dependence in the frequency ratio. If we assume an exponential-dependence of the viscosity with temperature, we could write the frequency ratio as a function of the total viscosity ratio λ only. This is approximation has been tested, but is not good in that case, for the predicted frequencies ratios are overestimated by a factor of 2 for large viscosity ratios.,

15 Comparison with experimental data The frequency analysis is done using Fast Fourier Transform (and autocorrelation as well) on two horizontally close thermocouples (but not exactly on the same vertical line). A typical power spectrum is shown in Figure 5. 1.E-04 1.E-05 1.E bottom top Power 1.E-07 1.E-08 1.E-09 1.E-10 1.E-11 1.E-12 Frequency Figure 5. A typical power spectrum for the boundary layer thermocouples. From experiment 7. Even if the active part of the spectrum is wide, several individual peaks can be identified. These peaks coincide with autocorrelation peaks as well. In collecting the main peaks out of each time series, we find that the main frequency of the top and bottom layer are similar, unlike the predictions of our simple boundary layer growth model. It appears that something else is occuring Main frequency (cold) Frequency Secondary frequency (hot) top bottom Figure 6. The Fourier Transform of the sample of Figure 8 shows the relative importance of the different frequencies. 15

16 Taking a closer look at the power spectra graphs, and picking the secondary peaks as well, we find out that, most of the time, the frequency ratio of the main and one of the secondary frequencies present in the hot layer, matches quite well the predicted frequency ratio, as shown in Figure 7. measured hot/cold frequency ratio predicted frequency ratio exp 15 exp 11 exp 6 exp 7 exp 4 exp 8 Figure 7. The measured frequency ratios for different experiments with plumes. There is always a frequency very close to the cold one (blue line), and one of the other frequencies is close to the predicted one (red line). If the cold boundary layer imposes its frequency (ratio = 1), the hot boundary layer can also exhibit its own frequency. Looking at the time series themselves (Figure 8), we effectively find both frequencies in the hot boundary layer, and the amplitude spectra (Figure 6) confirms that the ratio is close to the predicted one. Temperature Cold frequency Hot frequency hot layer 0.65 cold layer 0.55 middle time (dimensionless) Figure 8. At the beginning of the sample, we can see the hot and cold boundary layers releasing plumes at the same frequency. Later in the sample, the hot boundary layer release plumes at its own frequency, probably because of the lower activity level of the cold boundary layer at that time. This sample was taken from experiment 7. One could argue that the cold frequency appearing in the hot boundary layer could simply be cold plumes coming down through the much less viscous hot boundary layer. But looking carefully at 16

17 the time series, the fluctuations are hot peaks (upwards) for the hot boundary layer, and cold peaks (downwards) for the cold one. All these facts tend to suggest that there is a strong interaction between boundary layers. It appears that the cold boundary layer somehow imposes its plume release frequency to the hot boundary layer. How it might work This is the mechanism we propose, to explain our observations. A cold plume is forming at the top, consisting of cold and viscous fluid, flowing slowly down through the interior fluid. Since the hot boundary layer is much less viscous and quite thin, the cold plume, when hitting the hot boundary layer continues and goes through. This amount of fluid added to the hot boundary layer can trigger the release of a new hot plume. Then, the cooled down hot boundary layer starts to warm up again, but may take longer, since a non-negligible amount of cold fluid has to be heated up as well. If the growth of a new hot boundary layer takes then longer than the cold one, another cold plume will hit the hot boundary layer, which may release another plume if its state is close to critical. If the cold plume activity is dropping in this particular region (we believe these plume interactions to have only local impact), the own frequency of the hot boundary layer can show up again. 17

18 Visualization In order to see such an interaction occurring, we removed the insulation, and used a bright projector light to obtain shadowgraphs. The refraction index of a fluid varies with density. Because the density of corn syrup varies with temperature, light will act differently if it encounters a hot or a cold plume. In fact, hot plumes are seen as dark zones, and cold plumes as bright, as illustrated on Figure 9. /LJKW KRW 'HQVLW\ JUDGLHQW 'DUN %ULJKW /LJKW FROG Figure 9. Principle of shadowgraphs: plumes act like lenses, focusing or diffusing light. One of these shadowgraphs is reproduced on Figure 10. Thermal structure of the flow is visible, and we can identify a cold and a hot plume. +RWSOXPH &ROGSOXPH Figure 10. Shadowgraph showing a cold and a hot plume. 18

19 Since shadowgraphs are a two-dimensional projection of the whole tank, even though we saw what could have been cold plumes triggering hot plumes, the lack of spatial resolution prevent us from bringing undeniable proof. Further investigations Since the frequency data is not entirely understood, we suggest some further investigations. In a steady-state convective experiment, suddenly suppressing the heating should stop the natural formation of hot plumes, so that the new hot plumes are more likely to be triggered by cold ones. This experiment should make it easier to decouple the hot and cold frequencies. Putting more thermocouples in a vertical row, should make it easier to identify rising and sinking plumes, to obtain velocities and correlations. Recording both temperatures and shadowgraph at the same time, in order to be able to determine if the correlations are really cause and effect. 19

20 Conclusion & applications Our study of thermal convection, focused on time series analysis and boundary layer interaction, shows that the hot and cold boundary layers are not independent in plume-dominated flow. Figures 6 to 8 show that hot plumes can form with more than one frequency and size, and thus hot plumes presumably form by different processes. Specifically, our measurements indicate that hot plumes can form either naturally as boundary layer instabilities once the local Rayleigh number exceeds a critical value or indirectly through the arrival of a cold plume in the lower thermal boundary layer. To apply these results to Earth-mantle convection, we have to keep in mind that the model problem considered here was specifically designed to allow us to focus on the temporal interaction of plumes formed from thermal boundary layers. Several potentially important aspects of planetary interiors are thus not accounted for in the experimental model, including internal heating, the motion of surface plates, and the possible influence of chemical heterogeneity (Davaille 1999). However, the fundamental physical processes that result in the formation and interaction of plumes should still apply to more complicated systems. There is observational evidence from both Venus and the Earth that mantle plumes with different sizes and properties can coexist in the same system (Herrick 1999). The interaction between descending plumes and the hot boundary layer is one way to produce hot plumes with different sizes and frequencies. On the Earth, a spatial correlation between the locations of hotspots and subduction is wellestablished (Weinstein and Olson 1989). It has also been argued that the accumulation of slabs at the base of the mantle can create the large viscosity contrasts required to make hotspot-forming plumes (Nataf 1991). Unfortunately, establishing a correlation between the initiation of individual plumes and the arrival of a slab at the base of the mantle is hampered by the long time lag between subduction and plume eruption relative to the time period over which plate motion histories can be reconstructed. Of course, correlations do not prove causality. Nevertheless, the measurements shown in Figures 6 to 8 provide further evidence that cold sinking plumes in the mantle (e.g., slabs) may have the ability to induce hot rising plumes. 20

21 References Castaing, B., G. Gunaratne, F. Hesloy, L. Kadanoff, A. Libchaer, S. Thomae, X.Z. Wu, S. Zaleski, and G. Zanetti, Scaling of hard turbulence in Rayleigh Bénard convection, J. Fluid Mech., 204, 1-30, Davaille, A., Simultaneous generation of hotspots and superswells by convection in a heterogenous planetary mantle, Nature, 402, , Giannandrea, E. and U. Christensen, Variable viscosity convection experiments with a stress-free upper boundary and implications for the heat transport in the Earth's mantle, Phys. Earth Planet. Int., 78, , Grossmann, S. and D. Lohse, Scaling in thermal convection: a unifying theory, J. Fluid. Mech., 407, 27-56, Herrick, R.R., Small mantle upwellings are pervasive on Venus and Earth, Geophys. Res. Lett., 26, , Howard, L.N., Convection at high Rayleigh number, in Proc. 11 th Intl. Congr. Applied Mech., edited by H. Görtler, Springer, Berlin, , Manga, M. and D. Weeraratne, Experimental study of non-boussinesq Rayleigh-Bénard convection at high Rayleigh and Prandtl numbers, Phys. Fluids, (1999). Nataf, H.C., Mantle convection, plates, and hotspots, Tectonophysics, 187, , Niemela, JJ; Skrbek, L; Sreenivasan, KR; Donnelly, RJ., Turbulent convection at very high Rayleigh numbers. Nature, 2000 april 20, V404 N6780: Schaeffer, N. and M. Manga, Interaction of rising and sinking mantle plumes, Geophys. Res. Lett., submitted August Weinstein, S.A. and P.L. Olson, The proximity of hotspots to convergent and divergent plate boundaries, Geophys. Res. Lett., 16, , Zhang, J., S. Childress and A. Libchaber, Non-Boussinesq effect: Thermal convection with broken symmetry, Phys. Fluids, 9, ,