Herbert E. Huppert

The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface

The viscous gravity current that results when fluid flows along a rigid horizontal surface below fluid of lesser density is analysed using a lubrication-theory approximation. It is shown that the effect on the gravity current of the motion in the upper fluid can be expressed as a condition of zero shear on the unknown upper surface of the gravity current. With the supposition that the volume of heavy fluid increases with time like t α , where α is a constant, a similarity solution to the governing nonlinear partial differential equations is obtained, which describes the shape and rate of propagation of the current. The viscous theory is shown to be valid for t [Gt ] t 1 , when α c and for t [Lt ] t 1 when α > α c , where t 1 , is the transition time at which the inertial and viscous forces are equal, with

The slumping of gravity currents

\alpha_{\rm c} = \frac{7}{4}

Double-diffusive convection

for a two-dimensional current and α c = 3 for an axisymmetric current. The solutions confirm the functional forms for the spreading relationships determined for α = 1 in the preceding paper by Didden & Maxworthy (1982), as well as evaluating the multiplicative factors appearing in the relationships. The relationships compare very well with experimental measurements of the axisymmetric spreading of silicone oils into air for α = 0 and 1. There is also very good agreement between the theoretical predictions and the measurements of the axisymmetric spreading of salt water into fresh water reported by Didden & Maxworthy and by Britter (1979). The predicted multiplicative constant is within 10% of that measured by Didden & Maxworthy for the spreading of salt water into fresh water in a channel.

The fluid dynamics of a basaltic magma chamber replenished by influx of hot, dense ultrabasic magma

Experimental results for the release of a fixed volume of one homogeneous fluid into another of slightly different density are presented, From these results and those obtained by previous experiments, it is argued that the resulting gravity current can pass through three states. There is first a slumping phase, during which the current is retarded by the counterflow in the fluidinto which it is issuing. The current remains in this slumping phase until the depth ratio of current to intruded fluid is reduced to less than about 0,075. This may be followed by a (previously investigated) purely inertial phase, wherein the buoyancy force of the intruding fluid is balanced by the inertial force. Motion in the surrounding fluid plays a negligible role in this phase. There then follows a viscous phase, wherein the buoyancy force is balanced by viscous forces. It is argued and confirmed by experiment that the inertial phase is absent if viscous effects become important before the slumping phase has been completed. R’elationships between spreading distance and time for each phase are obtained for all three phases for both two-dimensional and axisymmetric geometries. Some consequences of the retardation of the gravity current during the slumping phase are discussed.

Cooling and contamination of mafic and ultramafic magmas during ascent through continental crust

In this paper we present a rather personal view of the important developments in double-diffusive convection, a subject whose evolution has been the result of a close interaction between theoreticians, laboratory experimenters and sea-going oceano-graphers. More recently, applications in astrophysics, engineering and geology have become apparent. In the final section we attempt to draw some general conclusions and suggest that further progress will again depend on a close collaboration between fluid dynamicists and other scientists.

The Fluid Dynamics of Evolving Magma Chambers [and Discussion]

This paper describes a fluid dynamical investigation of the influx of hot, dense ultrabasic magma into a reservoir containing lighter, fractionated basaltic magma. This situation is compared with that which develops when hot salty water is introduced under cold fresh water. Theoretical and empirical models for salt/water systems are adapted to develop a model for magmatic systems. A feature of the model is that the ultrabasic melt does not immediately mix with the basalt, but spreads out over the floor of the chamber, forming an independent layer. A non-turbulent interface forms between this layer and the overlying magma layer across which heat and mass are transferred by the process of molecular diffusion. Both layers convect vigorously as heat is transferred to the upper layer at a rate which greatly exceeds the heat lost to the surrounding country rock. The convection continues until the two layers have almost the same temperature. The compositions of the layers remain distinct due to the low diffusivity of mass compared to heat. The temperatures of the layers as functions of time and their cooling rate depend on their viscosities, their thermal properties, the density difference between the layers and their thicknesses. For a layer of ultrabasic melt (18% MgO) a few tens of metres thick at the base of a basaltic (10% MgO) magma chamber a few kilometres thick, the temperature of the layers will become nearly identical over a period of between a few months and a few years. During this time the turbulent convective velocities in the ultrabasic layer are far larger than the settling velocity of olivines which crystallise within the layer during cooling. Olivines only settle after the two layers have nearly reached thermal equilibrium. At this stage residual basaltic melt segregates as the olivines sediment in the lower layer. Depending on its density, the released basalt can either mix convectively with the overlying basalt layer, or can continue as a separate layer. The model provides an explanation for large-scale cyclic layering in basic and ultrabasic intrusions. The model also suggests reasons for the restriction of erupted basaltic liquids to compositions with MgO<10% and the formation of some quench textures in layered igneous rocks.

Particle-driven gravity currents.

When magma ascends turbulently through continental crust, heat transfer can be rapid and the wall rocks of the conduit can melt and be assimilated into the magma. Calculations are presented for cooling, crystallization and contamination during the turbulent ascent of a komatiite, a picritic basalt and a tholeiitic basalt. Primitive magmas, like komatiite and picritic basalt, are predicted to erupt with moderate to large amounts of olivine phenocrysts except at very high flow rates. Little crystallization takes place in a basalt on ascent, and any phenocrysts are likely to be inherited from the magma chamber. The erosion rate of the conduit walls and amount of contamination are greatest in primitive magmas and least for cool, fractionated magmas. Contamination is also affected by flow rate. For low flow rates, where the Reynolds number is significantly less than 2000, movement is laminar and the magma is likely to solidify against the dyke walls and so the amount of contamination is negligible. Maximum contamination will occur for flow rates at Reynolds number around 2000 and the total amount of contamination will decrease as the Reynolds number increases above this value. This kind of contamination can produce trends on geochemical diagrams which are opposite to those produced by assimilation and fractional crystallization processes in magma chambers. Indices of crustal contamination such as 87Sr, REE, K2O and other incompatible elements can be greatest in magmas with high values of Mg/(Mg+Fe) and low SiO2. Both highly incompatible and highly compatible trace elements can show positive correlation with one another (for example, Ni and K2O). These features are shown by the Plateau lavas of Skye, Scotland, and some lava groups of the Deccan Traps. Curved trends produced in this way on many types of geochemical diagrams are not mixing hyperbolas and do not necessarily point towards contaminant compositions. Thermal erosion rates are proportional to the difference between the magma temperature and the fusion temperature of the wall rocks. Contamination will thus tend to be selective towards rocks of low fusion temperature. Because of their high temperatures, komatiites are relatively indiscriminate in what they assimilate, while basalts are highly selective. The calculations show that komatiites are highly susceptible to contamination by both continental and oceanic crustal components. Under suitable flow conditions they can be contaiminated with up to 30% of crustal material. Contamination could result in spurious conclusions about their age and mantle-source characteristics. The geochemical differences between komatiites and closely associated basaltic komatiites can often not be attributed to fractionation of olivine. These basaltic komatiites may represent highly contaminated komatiite rather than an unrelated magma type derived from a different mantle source.

On the thermal evolution of the Earth's core

Recent developments in petrology indicate that fluid dynamic effects are of fundamental importance in controlling magma genesis. The forms of convection in magma chambers arise from compositional variations caused by processes such as fractional crystallization, partial melting and contamination, as well as from thermal effects. These processes, together with phase changes such as volatile exsolution, generally cause much larger density changes in magmas than the thermal effects arising from associated temperature changes. Magmas exhibit a wide range of convective phenomena not encountered in one-component fluids that are due to these compositional changes and to the differences between the diffusivities of chemical components and heat. When crystallization occurs in such multi-component systems, fluid immediately adjacent to the growing crystals is generally either depleted or enriched in heavy components and can convect away from its point of origin. Experimental studies of convection in crystallizing systems together with theoretical analyses suggest that convective separation of liquid from crystals is the dominant process of fractionation in magmas. This paper provides a synopsis of these new ideas on convection in magmas and their application to the interpretation of igneous rocks. Crystal settling is shown to be an inadequate and, in many situations, improbable mechanism for fractional crystallization. The convective motions in chambers are usually sufficiently vigorous to keep crystals in suspension, although settling can occur from thin fluid layers and within the boundary layers at the margins of a magma chamber. We propose that convective fraction, a term introduced to embrace a wide variety of convective phenomena caused by crystallization, is the dominant mechanism for crystal fractionation. The process enables compositional and thermal gradients to be formed in magma chambers both by closed-system crystallization and by repeated replenishment in open systems. During crystallization along the margins of a chamber, highly fractionated magmas can be generated without requiring large amounts of crystallization, because the removal and concentration of chemical components affects only a small fraction of the total magma. These convective effects also give insights into many features observed in layered intrusions, including the various types of layering and the formation of different kinds of cumulate rock.

Convective dissolution of carbon dioxide in saline aquifers.

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.

Axisymmetric collapses of granular columns

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.