Ross C. Kerr

Fluid‐mechanical models of crack propagation and their application to magma transport in dykes

The ubiquity of dykes in the Earths crust is evidence that the transport of magma by fluid-induced fracture of the lithosphere is an important phenomenon. Magma fracture transports melt vertically from regions of production in the mantle to surface eruptions or near-surface magma chambers and then laterally from the magma chambers in dykes and sills. In order to investigate the mechanics of magma fracture, the driving and resisting pressures in a propagating dyke are estimated and the dominant physical balances between these pressures are described. It is shown that the transport of magma in feeder dykes is characterized by a local balance between buoyancy forces and viscous pressure drop, that elastic forces play a secondary role except near the dyke tip and that the influence of the fracture resistance of crustal rocks on dyke propagation is negligible. The local nature of the force balance implies that the local density difference controls the height of magma ascent rather than the total hydrostatic head and hence that magma is emplaced at its level of neutral buoyancy (LNB) in the crust. There is a small overshoot beyond this level which is calculated to be typically a few kilometres. Magma accumulating at the LNB will be intruded in lateral dykes and sills which are directed along the LNB by buoyancy forces since the magma is in gravitational equilibrium at this level. Laboratory analogue experiments demonstrate the physical principle of buoyancy-controlled propagation to and along the LNB. The equations governing the dynamics of magma fracture are solved for the cases of lithospheric ascent and of lateral intrusion. Volatiles are predicted to be exsolved from the melt at the tips of extending fractures due to the generation of low pressures by viscous flow into the tip. Chilling of magma at the edges of a dyke inhibits cross-stream propagation and concentrates the downstream flow into a wider dyke. The family of theoretical solutions in different geometries provides simple models which describe the relation between the elastic and fluid-mechanical phenomena and from which the lengths, widths and rates of propagation can be calculated. The predicted dimensions are in broad agreement with geological observations.

Dike transport of granitoid magmas

Thermal and fluid-dynamical analyses suggest that for viscosities and density contrasts spanning the range considered typical for many calc-alkalic granitoids, dike ascent is a viable mechanism for the transport of large volumes of granitoid melt through the continental crust. We present calculations showing that a granitoid melt with calculated viscosity of the order of 10 6 Pa ⋅ s and a density contrast between magma and crust of 200 kg/m 3 can be transported 30 km through the crust in ∼1 month, corresponding to a mean ascent velocity of 1 cm/s. Using analysis modified from numerical studies of the flow of basaltic magmas in dikes, we also present an expression that allows the calculation of the critical (minimum) dike or fault width required for granitic magma to ascend without freezing. For all reasonable estimates of Cordilleran granitoid viscosity and density contrast, the critical dike width is determined to be between ∼2 and 7 m. Calculated peak batholith-filling rates are orders of magnitude greater than mean cavity-opening rates based on estimated fault slippage, which is consistent with chemical evidence for intermittent supply of magma pulses.

Postcumulus processes in layered intrusions

During the postcumulus stage of solidification in layered intrusions, fluid dynamic phenomena play an important role in developing the textural and chemical characteristics of the cumulate rocks. One mechanism of adcumulus growth involves crystallization at the top of the cumulate pile where crystals are in direct contact with the magma reservoir. Convection in the chamber can enable adcumulus growth to occur to form a completely solid contact between cumulate and magma. Another important process may involve compositional convection in which light differentiated melt released by intercumulus crystallization is continually replaced by denser melt from the overlying magma reservoir. This process favours adcumulus growth and can allow adcumulus growth within the pore space of the cumulate pile. Calculations indicate that this process could reduce residual porosities to a few percent in large layered intrusions, but could not form pure monomineralic rocks. Intercumulus melt may also be replaced by more primitive melt during episodes of magma chamber replenishment. Dense magma, emplaced over a cumulate pile containing lower density differentiated melt may sink several metres into the underlying pile in the form of fingers. Reactions between melt and matrix may lead to changes in mineral compositions, mineral textures and whole rock isotope compositions. Another important mechanism for forming adcumulate rocks is compaction, in which the imbalance of the hydrostatic and lithostatic pressures in the cumulate pile causes the crystalline matrix to deform and intercumulus melt to be expelled. For cumulate layers from 10 to 1000 metres in thickness, compaction can reduce porosities to very low values (< 1 %) and form monomineralic rocks. The characteristic time-scale for such compaction is theoretically short compared to the time required to solidify a large layered intrusion. During compaction changes of mineral compositions and texture may occur as moving melts interact with the surrounding matrix. Both compaction and compositional convection can be interrupted by solidification in the pore spaces. Compositional convection will only occur if the Rayleigh number is larger than 40, if the residual melt becomes lower in density, and the convective velocity exceeds the solidification velocity (measured by the rate of crystal accumulation in the chamber). Orthocumulates are thus more likely to form in rapidly cooled intrusions where residual melt is frozen into the pore spaces before it can be expelled by compaction or replaced by convection.

The Effects of Shape on Crystal Settling and on the Rheology of Magmas

The effects of the shape of phenocrysts on their settling velocity and on the rheology of magmas are discussed. Spherical crystals fall at a velocity given by Stokes law and need to be present in large concentrations before they have a marked effect on the viscosity of a suspension. Elongate crystals fall at a much smaller velocity, which may be determined from the formulas and graphs given herein. Such elongate crystals can have a dramatic effect on the viscosity of a suspension even at low volume fractions. Suspended crystals and bubbles will not impart a yield stress to the magma unless they form a touching network across the entire suspension. It is suggested, therefore, that most previous inferences of a yield stress in subliquidus magmas are erroneous. Measurements of the subliquidus rheology of magmas need to be accompanied by careful characterization of the microstructure and distribution of the solid phase, on which such rheologies critically depend.

The ascent of felsic magmas in dykes

Abstract The results of a thermal and fluid-dynamical analysis are presented that allow geologists to estimate the critical dyke or fracture width ( w c ) necessary for the transport of felsic magma through the crust without control by solidification. The principal geological variables required to estimate w c are the magmatic viscosity (μ), the density and temperature contrasts ( Δϱ and ΔT ) between magma and country rock, and the height of ascent H . The results show that for typical values of μ and Δϱ , and for transport distances of 20 to 30 km, the critical dyke width is constrained to lie between about 2 and 20 m.

A theoretical model of a turbulent fountain

A theoretical model of axisymmetric turbulent fountains in both homogeneous and stratified fluids is developed. The model quantifies the entrainment of ambient fluid into the initial fountain upflow, and the entrainment of fluid from both the upflow and environment into the subsequently formed downflow. Four different variations of the model are considered, comprising the two most reasonable formulations of the body forces acting on the ‘double’ structure and two formulations of the rate of entrainment between the flows. The four model variations are tested by comparing the predictions from each of them with experimental measurements of fountains in homogeneous and stratified fluids.

Emplacement and erosion by Archean komatiite lava flows at Kambalda: Revisited

We have developed a mathematical model to evaluate the flow and erosional potential of submarine, channelized komatiite lavas at Kambalda, Western Australia. Field data from Kambalda were used to constrain the choice of important input parameters, and model results were compared with data from field studies and geochemical analyses. Our results suggest that thermal erosion is strongly dependent upon the nature and behavior of the substrate. If the substrate is treated as an unconsolidated, hydrous sediment that can be fluidized by vaporized seawater, then our model predicts that an initially 10-m-thick basal Kambalda komatiite lava could have produced very high thermo-mechanical erosion rates (∼23–10 m/day), crustal thicknesses of ∼5–20 cm at distances of ∼5–35 km from the source, and a high degree of lava contamination (∼3–12%). In contrast, if the substrate is treated as a more consolidated, anhydrous sediment that could not be fluidized, then our model predicts that a Kambalda komatiite flow would have had much lower thermal erosion rates (∼1.2–0.4 m/day) and degrees of contamination (∼1–3%), and would have had crustal thicknesses of ∼5–20 cm at longer flow distances of ∼30–165 km. Field constraints are generally consistent with our predictions for a non-fluidized substrate. The reentrant embayments at Kambalda are thought to form from either erosion of deep channels in a flat basaltic seafioor [Huppert et al., 1984], or erosion of a thin (<5 m) sediment with minor undercutting of basalt in pre-existing topography [Lesher et al., 1984]. Our modeling indicates that the former was possible for long eruption durations (months), whereas the latter was possible for short eruption durations (<2 weeks). As the latter hypothesis is more consistent with the existing field evidence for thermal erosion at Kambalda, we believe it is the preferred interpretation.

Mixing and compositional stratification produced by natural convection: 1. Experiments and their application to Earth's core and mantle

An extensive series of laboratory experiments is used to quantify the circumstances under which fluids can be mixed by natural convection at high flux Rayleigh number. A compositionally buoyant fluid was injected at a fixed rate into an overlying layer of ambient fluid from a planar, horizontally uniform source. The nature of the resulting compositional convection was found to depend on two key dimensionless parameters: a Reynolds number Re and the ratio U of the ambient fluid viscosity to the input fluid viscosity. Increasing the Reynolds number corresponded to increasing the vigor of the convection, while the viscosity ratio was found to determine the spacing between plumes and whether buoyant fluid rose as sheets (U 1). From measurements of the final density profile in the fluid after the experiments we quantified the extent to which buoyant liquid was mixed in terms of a thermodynamic mixing efficiency E. The mixing efficiency was found to be high (E > 0.9) when either the Reynolds number was large (Re > 100) or the viscosity ratio was small (U 200. The amount of mixing was related to whether ascending plumes generated a large-scale circulation in the ambient fluid. When our results are applied to the differentiation of the Earths core, we suggest that the convection resulting from the release of buoyant residual liquid into the liquid outer core due to crystallization at the boundary between the inner and the outer core will probably lead to nearly complete mixing. In the dynamically very different context of the mantle, mantle plumes are predicted to ascend through the mantle and pond beneath the lithosphere, whereas convection driven by the subduction of oceanic lithosphere is expected to produce moderate to extensive mixing of the mantle. When the competing plate and plume modes of mantle convection are considered together, we find that owing to a larger driving buoyancy flux, the plate-scale flow will destroy any stratification at the top of the mantle produced by mantle plumes. Applying our results to the “stagnant lid” style of thermal convection predicted to occur in the mantles of the Moon, Mercury, Mars, Venus, and pre-Archean Earth, we expect the respective flows to produce minor thermal stratification at the respective core-mantle boundaries. In part 2 of this study [Jellinek and Kerr, this issue] we apply our results to the differentiation of magma chambers and komatiite lava flows.

Mixing and compositional stratification produced by natural convection: 2. Applications to the differentiation of basaltic and silicic magma chambers and komatiite lava flows

The petrogenesis of igneous rocks can be controlled significantly by the mixing of dissimilar magmas. Within the contexts of basaltic and silicic magma chambers and komatiite lava flows we identify circumstances in which the extent to which contrasting magmas are mixed by natural convection potentially controls their differentiation. To evaluate the amount of mixing in each context, we apply the experimental results from part 1 [Jellinek et al., this issue] of this study, in which we quantified the conditions under which fluids could be mixed by convection at large Rayleigh numbers (> 1011). When our laboratory results are applied to basaltic magma chambers, we find that convection driven by compositionally buoyant magma released during floor crystallization or floor dissolution will produce partial to nearly complete mixing of the ascending fluid in chambers that are tens of meters to kilometers high, respectively. We also conclude that substantial floor melting (with extensive mixing) is expected only for basaltic chambers emplaced in the deep crust. During the turbulent flow of Archean komatiites, underlying sediments melted by forced convective heat transfer are predicted to have been mixed nearly completely into the overriding lavas. During the replenishment of silicic magma chambers by basaltic magmas we predict that the convection of buoyant silicic magma overrun by a spreading injection of denser basalt will cause little mixing. However, after emplacement, heat transfer from a basalt layer will gradually melt and mobilize its felsic floor, producing a small flux of buoyant felsic liquid that will be mixed extensively.

Solidification of an alloy cooled from above Part 1. Equilibrium growth

The interaction between the solidification and convection that occurs when a melt is cooled from above is investigated in a series of three papers. In these papers we consider a two-component melt that partially solidifies to leave a buoyant residual fluid. The solid forms a mushy layer of dendritic crystals, the interstices of which accommodate the residual fluid. The heat extraction through the upper boundary, necessary to promote solidification, drives convection at high Rayleigh numbers in the melt below the mushy layer. The convection enhances the heat transfer from the melt and alters the rate of solidification. In this paper the various phenomena are studied in a series of laboratory experiments in which ice is frozen from aqueous solutions of isopropanol. The experiments are complemented by the development of a general theoretical model in which the mush is treated as a continuum phase with thermodynamic properties that are functions of the local solid fraction. The model, which is based upon principles of equilibrium thermodynamics and local conservation of heat and solute, produces results in good agreement with the experimental data. Careful comparisons between this theory and experiments suggest the need to explore non-equilibrium effects, which are investigated in Parts 2 and 3.