John R. Lister

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.

Coalescence of liquid drops

When two drops of radius R touch, surface tension drives an initially singular motion which joins them into a bigger drop with smaller surface area. This motion is always viscously dominated at early times. We focus on the early-time behaviour of the radius rm of the small bridge between the two drops. The flow is driven by a highly curved meniscus of length 2rm and width rm around the bridge, from which we conclude that the leading-order problem is asymptotically equivalent to its two-dimensional counterpart. For the case of inviscid surroundings, an exact two-dimensional solution (Hopper 1990) shows that / r 3 m and rm (t= )l n [t=(R)]; and thus the same is true in three dimensions. We also study the case of coalescence with an external viscous fluid analytically and, for the case of equal viscosities, in detail numerically. A signicantly dierent structure is found in which the outer-fluid forms a toroidal bubble of radius / r 3=2 m at the meniscus and rm (t=4 )l n [t=(R)]. This basic dierence is due to the presence of the outer-fluid viscosity, however small. With lengths scaled by R a full description of the asymptotic flow for rm(t) 1 involves matching of lengthscales of order r 2 m , r 3=2 m , rm, 1 and probably r 7=4 m .

Particle-driven gravity currents.

Gravity currents created by the release of a fixed volume of a suspension into a lighter ambient fluid are studied theoretically and experimentally. The greater density of the current and the buoyancy force driving its motion arise primarily from dense particles suspended in the interstitial fluid of the current. The dynamics of the current are assumed to be dominated by a balance between inertial and buoyancy forces; viscous forces are assumed negligible. The currents considered are two-dimensional and flow over a rigid horizontal surface. The flow is modelled by either the single- or the twolayer shallow-water equations, the two-layer equations being necessary to include the effects of the overlying fluid, which are important when the depth of the current is comparable to the depth of the overlying fluid. Because the local density of the gravity current depends on the concentration of particles, the buoyancy contribution to the momentum balance depends on the variation of the particle concentration. A transport equation for the particle concentration is derived by assuming that the particles are vertically well-mixed by the turbulence in the current, are advected by the mean flow and settle out through the viscous sublayer at the bottom of the current. The boundary condition at the moving front of the current relates the velocity and the pressure head at that point. The resulting equations are solved numerically, which reveals that two types of shock can occur in the current. In the late stages of all particle-driven gravity currents, an internal bore develops that separates a particle-free jet-like flow in the rear from a dense gravity-current flow near the front. The second type of bore occurs if the initial height of the current is comparable to the depth of the ambient fluid. This bore develops during the early lock-exchange flow between the two fluids and strongly changes the structure of the current and its transport of particles from those of a current in very deep surroundings. To test the theory, several experiments were performed to measure the length of particle-driven gravity currents as a function of time and their deposition patterns for a variety of particle sizes and initial masses of sediment. The comparison between the theoretical predictions, which have no adjustable parameters, and the experimental results are very good.

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.

On the thermal evolution of the Earth's core

The Earths magnetic field is sustained by dynamo action in the fluid outer core. The energy sources available to the geodynamo are well established, but their relative importance remains uncertain. We focus on the issue of thermal versus compositional convection, which is inextricably coupled to the evolution of the core as the Earth cools. To investigate the effect of the various physical processes on this evolution, we develop models based on conservation of energy and the assumption that the core is well mixed by vigorous convection. We depart from previous numerical studies by developing an analytical model. The simple algebraic form of the solution affords insight into both the evolution of the core and the energy budget of the geodynamo. We also present a numerical model to compare with the quantitative predictions of our analytical model and find that the differences between the two are negligible. An important conclusion of this study is that thermal convection can contribute significantly to the geodynamo. In fact, a modest heat flux in excess of that conducted down the adiabatic gradient is sufficient to power the geodynamo, even in the absence of compositional convection and latent-heat release. The relative contributions of thermal and compositional convection to the dynamo are largely determined by the magnitude of the heat flux from the core and the inner-core radius. For a plausible current-day heat flux of Q = 3.0 × 1012 W and the current inner-core radius, we find that compositional convection is responsible for approximately two thirds of the ohmic dissipation in the core and thermal convection for the remaining one third. The proportion of ohmic dissipation produced by thermal convection increases to 45% with an increase in Q to 6.0 × 1012 W. In the early Earth, when the inner core was smaller and the heat flux probably greater than the present values, thermal convection would have been the dominant energy source for the dynamo. We also calculate the history of inner-core growth as a function of the heat flux. For example, the inner core would have grown to its present size in 2.8 × 109 years if the average heat flux was Q = 4.0 × 1012 W. The model does not require the heat flux to be constant.

Buoyancy-driven fluid fracture: the effects of material toughness and of low-viscosity precursors

When buoyant fluid is released into the base of a crack in an elastic medjura the crack will propagate upwards, driven by the buoyancy of the fluid. Viscous fluid flow in such a fissure is described by the equations of lubrication theory with the pressure given by the sum of the hydrostatic pressure of the fluid and the elastic pressures exerted by the walls of the crack. The elastic pressure and the width of the crack are further coupled by an integro-differential equation derived from the theory of infinitesimal dislocations in an elastic medium. The steady buoyancy-driven propagation of a two-dimensional fluid-filled crack through an elastic medium is analysed and the governing equations for the pressure distribution and the shape of the crack are solved numerically using a collocation technique. The fluid pressure in the tip of an opening crack is shown to be very low. Accordingly, a region of relatively inviscid vapour or exsolved volatiles in the crack tip is predicted and allowed for in the formulation of the problem. The solutions show that the asymptotic width of the crack, its rate of ascent and the general features of the flow are determined primarily by the fluid mechanics; the strength of the medium and the vapour pressure in the crack tip affect only the local structure near the advancing tip of the crack. When applied to the transport of molten rock through the Earths lithosphere by magma-fracture, this conclusion is of fundamental importance and challenges the geophysicists usual emphasis on the controlling influence of fracture mechanics rather than that of fluid mechanics.

Capillary breakup of a viscous thread surrounded by another viscous fluid

Previous long-wavelength analyses of capillary breakup of a viscous fluid thread in a perfectly inviscid environment show that the asymptotic self-similar regime immediately prior to breakup is given by a balance between surface tension, inertia, and extensional viscous stresses in the thread. In contrast, it is shown here that if viscosity in the external fluid, however small, is included then the asymptotic balance is between surface tension and viscous stresses in the two fluids while inertia is negligible. Scaling estimates for this new balance suggest that both axial and radial scales decrease linearly with time to breakup, so that the aspect ratio remains O(1) with time but scales with viscosity ratio like (μint/μext)1/2 for μint≫μext, where μint and μext are the internal and external viscosities. Numerical solutions to the full Stokes equations for μint=μext confirm the scalings with time and give self-similar behavior near pinching. However, the self-similar pinching region is embedded in a logari...

Viscous Control of Peeling an Elastic Sheet by Bending and Pulling

Propagation of a viscous fluid beneath an elastic sheet is controlled by local dynamics at the peeling front, in close analogy with the capillary-driven spreading of drops over a precursor film. Here we identify propagation laws for a generic elastic peeling problem in the distinct limits of peeling by bending and peeling by pulling, and apply our results to the radial spread of a fluid blister over a thin prewetting film. For the case of small deformations relative to the sheet thickness, peeling is driven by bending, leading to radial growth as t(7/22). Experimental results reproduce both the spreading behavior and the bending wave at the front. For large deformations relative to the sheet thickness, stretching of the blister cap and the consequent tension can drive peeling either by bending or by pulling at the front, both leading to radial growth as t(3/8). In this regime, detailed predictions give excellent agreement and explanation of previous experimental measurements of spread in the pulling regime in an elastic Hele-Shaw cell.

Sediment-laden gravity currents with reversing buoyancy

There are many natural occurrences of sediment-laden gravity currents in which the density of the interstitial fluid is less than that of the ambient fluid, although the bulk density of the current is greater. Such currents are driven by the excess density of suspended particles. However, after sufficient particles have sedimented, the current will become buoyant, cease its lateral motion and ascend to form a plume. Examples of such currents include brackish underflows in deltas, turbidity currents and pyroclastic flows. Experimental studies are described which show that, due to sedimentation, sediment-laden gravity currents decelerate more rapidly than saline currents of the same density. There is little difference in the experiments between a sediment-laden current with neutrally buoyant interstitial fluid and one with buoyant interstial fluid until sufficient sediment has been lost to cause the latter kind of current to lift-off. A marked deceleration is then observed and a plume is generated, with lift-off occurring along the length of the current. The resulting buoyant plume then generates a gravity current below the upper surface of the fluid in the tank. The deposit from a current with buoyant fluid shows a fairly abrupt decrease in thickness beyond the lift-off distance and has a flatter profile than that from a simple sediment current. A theoretical model is presented, which is based on the two-layer shallow-water equations and incorporates a model of the sedimentation in which particles are assumed to be uniformly suspended by the turbulence of the current. The model shows good agreement with the observed lengths of the experimental currents as a function of time and predicts the lift-off distance reasonably well. These processes have implications for the behaviour of turbidity currents, the interpretation of turbidites, mixing processes in the oceans and the lift-off of pyroclastic flows.

The strength and efficiency of thermal and compositional convection in the geodynamo

Abstract The evolution of the Earths outer core is calculated theoretically by assuming that the fluctuations about a vigorously convecting, well-mixed, isentropic and hydrostatic mean state are small. The global energy equation describes the evolution of the mean state, but contains no information about the division between convective and diffusive transport of heat or composition within the core itself. As the diffusive transport of either heat or composition makes no contribution to the dynamo, estimates of the dynamo power must be derived by explicit consideration of the correlation of small density fluctuations with the large convective velocities. By such consideration, the dynamo power is shown to be equal to the integrated thermal and compositional buoyancy flux, which is calculated from the requirement that the mean state remain isentropic and well mixed. The resulting new and simple expression for the dynamo efficiency has an easily understood physical interpretation in terms of Carnot-style redistribution of heat and mass in the core, though it takes full account of both thermal and compositional effects and the evolution of the mean state. Once the equation of state and thermodynamic parameters are given, the dynamo power can be evaluated readily and explicity. Analytic solutions based on a simple example equation of state show that the relative importance of thermal and compositional convection depends on the size of the inner core and the amount by which the heat flux across the core-mantle boundary (CMB) exceeds that which can be conducted up the core adiabat. Thermal convection was dominant in the early Earth when the inner core was small and the Earth was probably cooling rapidly. Estimates of the present-day heat loss across the CMB suggest that thermal convection now contributes about 20% of the dynamo power and compositional convection about 80%. As a result of the release of latent heat at the inner-core boundary, thermal convection can make a net positive contribution to the dynamo power even if the heat flux at the CMB is subadiabatic.