Claude Jaupart

Transient high-Rayleigh-number thermal convection with large viscosity variations

The characteristics of thermal convection in a fluid whose viscosity varies strongly with temperature are studied in the laboratory. At the start of an experiment, the upper boundary of an isothermal layer of Golden Syrup is cooled rapidly and maintained at a fixed temperature. The fluid layer is insulated at the bottom and cools continuously. Rayleigh numbers calculated with the viscosity of the well-mixed interior are between 10 6 and 10 8 and viscosity contrasts are up to 10 6 . Thermal convection develops only in the lower part of the thermal boundary layer, and the upper part remains stagnant. Vertical profiles of temperature are measured with precision, allowing deduction of the thickness of the stagnant lid and the convective heat flux. At the onset of convection, the viscosity contrast across the unstable boundary layer has a value of about 3. In fully developed convection, this viscosity contrast is higher, with a typical value of 10. The heat flux through the top of the layer depends solely on local conditions in the unstable boundary layer and may be written [Q_{rm s} = - CK_{rm m} (alpha g/kappa nu_{rm m})^{frac{1}{3}} Delta T^{frac{4}{3}}_{rm v}] , where k m and ν m are thermal conductivity and kinematic viscosity at the temperature of the well-mixed interior, κ thermal diffusivity, α the coefficient of thermal expansion, g the acceleration due to gravity. Δ T v , is the ‘viscous’ temperature scale defined by [Delta T_{rm v} = - frac{mu (T_{rm m})}{({rm d}mu /{rm d}T)(T_{rm m})}] where μ( T ) is the fluid viscosity and T m the temperature of the well-mixed interior. Constant C takes a value of 0.47 ± 0.03. Using these relations, the magnitude of temperature fluctuations and the thickness of the stagnant lid are calculated to be in excellent agreement with the experimental data. One condition for the existence of a stagnant lid is that the applied temperature difference exceeds a threshold value equal to (2Δ T v ).

The generation and collapse of a foam layer at the roof of a basaltic magma chamber

Basaltic volcanoes erupt in several different regimes which have not been explained. At Kilauea (Hawaii), eruption can take the form of either fire fountaining, where gas-rich jets propel lava clots to great heights in the atmosphere, or quiet effusive outflow of vesicular lava. Another regime is commonly observed at Stromboli, where large gas slugs burst intermittently at the vent. In an attempt to provide a unifying framework for these regimes, we investigate phenomena induced by degassing in a reservoir which empties into a small conduit. Laboratory experiments are done in a cylindrical tank topped by a thin vertical tube. Working liquids are silicone oils and glycerol solutions to investigate a range of viscosity and surface tension. Gas bubbles are generated at the tank bottom with known bubble diameter and total gas flux. The bubbles rise through the tank and accumulate in a foam layer at the roof. Depending on the behaviour of this foam layer, three different regimes can be distinguished: (i) steady horizontal flow of the foam leading to bubbly flow in the conduit; (ii) alternating regimes of foam build-up and collapse leading to the eruption of a single, large gas pocket; (iii) flow of the foam partially coalesced into larger gas pockets leading to intermittent slug flow in the conduit. These regimes have natural counterparts in basaltic volcanoes. A simple theory is proposed to explain regimes (i) and (ii). The bubbles in contact with the roof deform under the action of buoyancy forces, developing flat contact areas whose size increases as a function of foam thickness. Maximum deformation corresponds to a critical thickness hc = 2σ/eρ l g R, where σ is the coefficient of surface tension, ρ l the liquid density, g the acceleration due to gravity, R the bubble radius and e the gas volume fraction in the foam. The foam thickness is determined by a balance between the input of bubbles from below and the output into the conduit, and is proportional to (μ l Q /e 2 ρ l g) ¼ , where μ l is the liquid viscosity and Q the gas flux. A necessary and sufficient condition for collapse is that it exceeds the critical value h c . In a liquid of given physical properties, this occurs when the gas flux exceeds a critical value which depends on viscosity, surface tension and bubble size. Experimental determinations of the critical gas flux and of the time between two events of foam collapse are in agreement with this simple theory.

Heat flow and thickness of the lithosphere in the Canadian Shield

Heat flow and radioactive heat production data were obtained in the Canadian Shield in order to estimate the crustal heat production and the mantle heat flow. Several methods have been used to determine radioactive heat production in the crust. The analysis yields values for the mantle heat flow in the craton that are consistently between 7 and 15 mW m -2 . Assuming that the lithosphere is in thermal equilibrium, we investigate the conditions for small-scale convection to supply the required heat flux through its base. For a given creep raw, the thickness of the lithosphere, the temperature at the base of the lithosphere, and the effective viscosity of the mantle are determined from the value of the mantle heat flow beneath the shield. The viscosity of the mantle depends on the creep mechanism and on the fluid content. Wet diffusion creep implies a viscosity between 10 20 and 10 21 Pa s, corresponding to a mantle temperature of 1620 K at a depth of 250 km. The other creep mechanisms can be ruled out because they imply values for viscosity and temperature inconsistent with geophysical data. For a given creep raw, there is a minimum mantle temperature below which equilibrium cannot be reached. For wet diffusion creep, this minimum mantle temperature (1780 K at 280 km depth) is close to that of the well-mixed (isentropic) oceanic mantle at the same depth. For a thermally stable lithosphere, out model requires the mantle heat flow to be at least 13 mW m -2 and the compositional lithosphere to be less than 240 km.

On the effect of continents on mantle convection

At the Earths surface, continents and oceans impose different thermal boundary conditions at the top of the mantle. Laboratory experiments are used to investigate the consequences of this for mantle convection. The upper boundary of the experimental tank was made of copper plates enforcing a fixed temperature and had a conductive lid of finite width in the middle. Beneath this lid, the thermal boundary condition was of the “mixed” type, with a Biot number depending on the dimensions and thermal conductivity of the lid. Experimental values of the Biot number were scaled to Earth values. Experiments were run for a large range of Rayleigh numbers, from 104 to 107, and for several lid widths. The effects of temperature-dependent viscosity and of the shape of the lid were investigated. At steady state, in all cases, there is an upwelling beneath the conductive lid, which feeds two symmetrical and elongated convective cells. Three different dynamic regimes were identified as a function of Rayleigh number, independently of the lid width. At Rayleigh numbers lower than 1.2 105, the upwelling is steady both in geometry and temperature structure. At Rayleigh numbers between 1.2 105 and 2 106, this central upwelling is intermittent. At larger values of the Rayleigh number, there is no longer a simple upwelling structure, but a set of small plumes rising together and distorted by a cellular circulation of large horizontal extent. Thus the conductive lid always imposes a large-scale flow pattern. The length of these convective cells is a function of lid width. It is equal to the lid width at large values and decreases to the Rayleigh-Benard value as the lid width decreases to zero. A fluid loop model explains the most important features of this form of convection. The cell length is such that the upwelling temperature is minimized for a given Rayleigh number and lid width and is an increasing function of lid width and a decreasing function of Rayleigh number. Transient experiments demonstrate that the large-scale flow structure develops rapidly with even small horizontal temperature differences. Implications for the Earth are that large-scale convection cells exist in conditions which, in the absence of continents, would probably lead to a chaotic convection pattern dominated by plumes. At high Rayleigh number, continental breakup is effected by a large-scale line upwelling structure which includes a number of individual plumes.

The kinetics of nucleation and crystal growth and scaling laws for magmatic crystallization

Magmatic crystallization depends on the kinetics of nucleation and crystal growth. It occurs over a region of finite thickness called the crystallization interval, which moves into uncrystallized magma. We present a dimensional analysis which allows a simple understanding of the crystallization characteristics. We use scales for the rates of nucleation and crystal growth, denoted by Im and Ym respectively. The crystallization time-scale τc and length-scale dc are given by (Ym3/Im)−1/4 and (κ·τ)m1/2 respectively, where κ is thermal diffusivity. The thickness of the crystallization interval is proportional to this length-scale. The scale for crystal sizes is given by (Ym/Im)1/4. We use numerical calculations to derive dimensionless relationships between all the parameters of interest: position of the crystallization front versus time, thickness of the crystallization interval versus time, crystal size versus distance to the margin, temperature versus time. We assess the sensitivity of the results to the form of the kinetic functions. The form of the growth function has little influence on the crystallization behaviour, contrary to that of the nucleation function. This shows that nucleation is the critical process. In natural cases, magmatic crystallization proceeds in continously evolving conditions. Local scaling laws apply, with time and size given by τ =(Y3/I)−1/4 and R=(Y/I)1/4, where Y and I are the rates at which crystal are grown and nucleated locally. τ is the time to achieve crystallization and R the mean crystal size. We use these laws together with petrological observations to infer the in-situ values of the rates of nucleation and growth. Two crystallization regimes are defined. In the highly transient conditions prevailing at the margins of basaltic intrusions, undercoolings are high and the peak nucleation and growth rates must be close to 1cm−3· −1 and 10−7cm/s, in good agreement with laboratory measurements. In quasi-equilibrium conditions prevailing in the interior of large intrusions, undercoolings are small. We find ranges of 10−7 to 10−3 cm−3 s−1 and of 10−10 to 10−8cm/s for the local rates of nucleation and growth respectively.

Steady-state operation of Stromboli volcano, Italy : constraints on the feeding system

Stromboli volcano has been in continuous eruption for several thousand years without major changes in the geometry and feeding system. The thermal structure of its upper part is therefore expected to be close to steady state. In order to mantaim explosive activity, magma must release both gas and heat. It is shown that the thermal and gas budgets of the volcano lead to consistent conclusions. The thermal budget of the volcano is studied by means of a finite-element numerical model under the assumption of conduction heat transfer. It is found that the heat loss through the walls of an eruption conduit is weakly sensitive to the dimensions of underlying magma reservoirs and depends mostly on the radius and length of the conduit. In steady state, this heat loss must be balanced by the cooling of magma which flows through the system. For the magma flux of about 1 kg s-1 corresponding to normal Strombolian activity, this requires that the conduits are a few meters wide and not deeper than a few hundred meters. This implies the existence of a magma chamber at shallow depth within the volcanic edifice. This conclusion is shown to be consistent with considerations on the thermal effects of degassing. In a Strombolian explosion, the mass ratio of gas to lava is very large, commonly exceeding two, which implies that the thermal evolution of the erupting mixture is dominated by that of the gas phase. The large energy loss due to decompression of the gas phase leads to decreased eruption temperatures. The fact that lava is molten upon eruption implies that the mixture does not rise from more than about 200 m depth. To sustain the magmatic and volcanic activity of Stromboli, a mass flux of magma of a few hundred kilograms per second must be supplied to the upper parts of the edifice. This represents either the rate of magma production from the mantle source feeding the volcano or the rate of magma overturn in the interior of a large chamber.

Thermal convection in lava lakes

In magma reservoirs, large temperature contrasts imply large variations of viscosity. We determine the characteristics of thermal convection in the laboratory for viscosity ratios of up to 106. In a fluid layer cooled from the top, convection develops below a stagnant lid. Plumes generate temperature fluctuations whose magnitude, θmax, is proportional to the temperature contrast across the unstable region, ΔTe. Scaling analysis and experimental data show that both temperature scales depend solely on the local function describing the variation of viscosity µ at temperatures close to that of the layer interior, Tm, and are equal to: In the Makaopuhi lava lake (Hawaii), temperature fluctuations were recorded below the growing crust. For the viscosity function of the Makaopuhi magma, their magnitude is predicted to be 18°C, in agreement with the observations.

The generation of gas overpressure in volcanic eruptions

Abstract Observations of natural eruption products show that different parts of a single magma batch may experience different degassing histories during ascent in a volcanic conduit. In non-explosive eruptions, lava issuing from a volcanic vent may contain overpressured gas bubbles. These important features of volcanic eruptions cannot be accounted for by existing flow models, which rely on simplifying hypotheses for the relationship between pressures in the gas phase and in the bulk flow. Volcanic flows involve highly compressible material which undergoes large viscosity variations as degassing proceeds. We show that these properties may lead to large gas overpressures in erupting lava. The magnitude of this overpressure depends on the initial volatile content of magma and is largest for relatively volatile-poor magmas, due to the extreme viscosity variations at water contents less than 1 wt%. We develop a simple analytical model to illustrate the main features of compressible viscous flows: (1) at any level, gas pressure is larger near the conduit axis than at the walls, (2) gas overpressure is an increasing function of mass discharge rate.

Influence of cooling on lava-flow dynamics

Experiments have been carried out to determine the effects of cooling on the flow of fluids with strongly temperature dependent viscosity. Radial viscous-gravity currents of warm glucose syrup were erupted at constant rate into a flat tank filled with a cold aqueous solution. Cold, viscous fluid accumulates at the leading edge, altering the flow shape and thickness and slowing the spreading. The flows attain constant internal temperature distributions and bulk viscosities. The value of the bulk viscosity depends on the Peclet number, which reflects the advective and diffusive heat transport properties of the flow, the flow skin viscosity, which reflects cooling, and the eruption viscosity. Our results explain why most lava flows have bulk viscosities much higher than the lava eruption viscosity. The results can be applied to understanding dynamic lava features such as flow-front thickening, front avalanches, and welded basal breccias.

Instability of a chemically dense layer heated from below and overlain by a deep less viscous fluid

Near the threshold of stability, an intrinsically denser fluid heated from below and underlying an isothermal fluid can undergo oscillatory instability, whereby perturbations to the interface between the fluids rise and fall periodically, or it can be mechanically stable and in thermal equilibrium with heat flux extracted by small-scale convection at the interface. Both the analysis of marginal stability and laboratory experiments in large-Prandtl-number fluids show that the critical Rayleigh number, scaled to parameters of the lower fluid, depends strongly on the buoyancy number, B, the ratio of the intrinsic density difference between the fluids and the maximum density difference due to thermal expansion. For small buoyancy number, B xs223C 0.5 and RaC > xs223C1100, a second form of instability develops, in which convection is confined to the lower layer. The analysis of marginal stability for layers with very different viscosities shows further that two modes of oscillatory instability exist, depending on the value of B. For B 0.275, only the bottom of the lower layer is buoyant, and instability occurs by its penetrating the upper part of the lower layer; the wavelengths of the perturbations that grow fastest are much smaller than those for B < 0.275, and the maximum frequency of oscillatory instability is much larger than that for B < 0.275. Oscillations in the laboratory experiments show that the heights to which plumes of the lower fluid rise into the upper one increase with the Rayleigh number. Moreover, in the finite-amplitude regime, the oscillation is not symmetrical. Plumes that reach maximum heights fall quickly, folding on themselves and entraining some of the upper fluid. Hence oscillatory convection provides a mechanism for mixing the fluids. Applied to the Earth, these results bear on the development of continental lithosphere, whose mantle part is chemically different from the underlying asthenosphere. As shown by the laboratory experiments and stability analysis, the lithosphere can be mechanically stable and in thermal equilibrium such that heat supplied by small-scale convection at the top of the asthenosphere is conducted through it. The lithosphere seems to have developed in a state near that of instability with different thicknesses depending on its intrinsic buoyancy. It may have grown not only by chemical differentiation during melting, but also by oscillatory convection entraining chemically denser material from the asthenosphere.