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 tα, 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 ] t1, when α < αc and for t [Lt ] t1 when α > αc, where t1, is the transition time at which the inertial and viscous forces are equal, with $\alpha_{\rm c} = \frac{7}{4}$ for a two-dimensional current and αc = 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.

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 fluid into 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. Relationships 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.

When basalt magmas are emplaced into continental crust, melting and generation of silicic magma can be expected. The fluid dynamical and heat transfer processes at the roof of a basaltic sill in which the wall rock melts are investigated theoretically and also experimentally using waxes and aqueous solutions. At the roof, the low density melt forms a stable melt layer with negligible mixing with the underlying hot liquid. A quantitative theory for the roof melting case has been developed. When applied to basalt sills in hot crust, the theory predicts that basalt sills of thicknesses from 10 to 1500 m require only 1 to 270 y to solidify and would form voluminous overlying layers of convecting silicic magma. For example, for a 500 m sill with a crustal melting temperature of 850 °C, the thickness of the silicic magma layer generated ranges from 300 to 1000 m for country rock temperatures from 500 to 850 °C. The temperatures of the crustal melt layers at the time that the basalt solidifies are high (900-950 °C) so that the process can produce magmas representing large degrees of partial fusion of the crust. Melting occurs in the solid roof and the adjacent thermal boundary layer, while at the same time there is crystallization in the convecting interior. Thus the magmas formed can be highly porphyritic. Our calculations also indicate that such magmas can contain significant proportions of restite crystals. Much of the refractory components of the crust are dissolved and then re-precipitated to form genuine igneous phenocrysts. Normally zoned plagioclase feldspar phenocrysts with discrete calcic cores are commonly observed in many granitoids and silicic volcanic rocks. Such patterns would be expected in crustal melting, where simultaneous crystallization is an inevitable consequence of the fluid dynamics. The timescales for melting and crystallization in basalt-induced crustal melting (10 2-10 3 y) are very short compared to the lifetimes of large silicic magma systems (>10 6 y) or to the timescale for thermal relaxation of the continental crust (> 10 7 y). Several of the features of silicic igneous systems can be explained without requiring large, high-level, long-lived magma chambers. Cycles of mafic to increasingly large volumes of silicic magma with time are commonly observed in many systems. These can be interpreted as progressive heating of the crust until the source region is partially molten and basalt can no longer penetrate. Every input of basalt triggers rapid formation of silicic magma in the source region. This magma will freeze again in timescales of order 10 2-10 3 y unless it ascends to higher levels. Crystallization can occur in the source region during melting, and eruption of porphyritic magmas does not require a shallow magma chamber, although such chambers may develop as magma is intruded into high levels in the crust. For typical compositions of upper crustal rocks, the model predicts that dacitic volcanic rocks and granodiorite/tonalite plutons would be the dominant rock types and that these wo...

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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.

The motion of instantaneous and maintained releases of buoyant fluid through shallow permeable layers of large horizontal extent is described by a nonlinear advection–diffusion equation. This equation admits similarity solutions which describe the release of one fluid into a horizontal porous layer initially saturated with a second immiscible fluid of different density. Asymptotically, a finite volume of fluid spreads as t1/3. On an inclined surface, in a layer of uniform permeability, a finite volume of fluid propagates steadily alongslope under gravity, and spreads diffusively owing to the gravitational acceleration normal to the boundary, as on a horizontal boundary. However, if the permeability varies in this cross-slope direction, then, in the moving frame, the spreading of the current eventually becomes dominated by the variation in speed with depth, and the current length increases as t1/2. Shocks develop either at the leading or trailing edge of the flows depending upon whether the permeability increases or decreases away from the sloping boundary. Finally we consider the transient and steady exchange of fluids of different densities between reservoirs connected by a shallow long porous channel. Similarity solutions in a steadily migrating frame describe the initial stages of the exchange process. In the final steady state, there is a continuum of possible solutions, which may include flow in either one or both layers of fluid. The maximal exchange flow between the reservoirs involves motion in one layer only. We confirm some of our analysis with analogue laboratory experiments using a Hele-Shaw cell.

Experimental observations of the collapse of initially vertical columns of small grains are presented. The experiments were performed mainly with dry grains of salt or sand, with some additional experiments using couscous, sugar or rice. Some of the experimental flows were analysed using high-speed video. There are three different flow regimes, dependent on the value of the aspect ratio a = h i /r i , where h i and r i are the initial height and radius of the granular column respectively. The differing forms of flow behaviour are described for each regime. In all cases a central, conically sided region of angle approximately 59• , corresponding to an aspect ratio of 1.7, remains undisturbed throughout the motion. The main experimental results for the final extent of the deposit and the time for emplacement are systematically collapsed in a quantitative way independent of any friction coefficients. Along with the kinematic data for the rate of spread of the front of the collapsing column, this is interpreted as indicating that frictional effects between individual grains in the bulk of the moving flow only play a role in the last instant of the flow, as it comes to an abrupt halt. For a < 1.7, the measured final runout radius, r ∞ , is related to the initial radius by r ∞ = r i (1 + 1.24a); while for 1.7 < a the corresponding relationship is r ∞ = r i (1 + 1.6a 1/2 ). The time, t ∞ , taken for the grains to reach r ∞ is given by t ∞ = 3(h i /g) 1/2 = 3(r i /g) 1/2 a 1/2 , where g is the gravitational acceleration. The insights and conclusions gained from these experiments can be applied to a wide range of industrial and natural flows of concentrated particles. For example, the observation of the rapid deposition of the grains can help explain details of the emplacement of pyroclastic flows resulting from the explosive eruption of volcanoes.

[1] Geological carbon dioxide (CO 2 ) storage is a means of reducing anthropogenic emissions. Dissolution of CO 2 into the brine, resulting in stable stratification, increases storage security. The dissolution rate is determined by convection in the brine driven by the increase of brine density with CO 2 saturation. We present a new analogue fluid system that reproduces the convective behaviour of CO 2 -enriched brine. Laboratory experiments and high-resolution numerical simulations show that the convective flux scales with the Rayleigh number to the 4/5 power, in contrast with a classical linear relationship. A scaling argument for the convective flux incorporating lateral diffusion from downwelling plumes explains this nonlinear relationship for the convective flux, provides a physical picture of high Rayleigh number convection in a porous medium, and predicts the CO 2 dissolution rates in CO 2 accumulations. These estimates of the dissolution rate show that convective dissolution can play an important role in enhancing storage security. [2] The storage of carbon dioxide (CO 2 ) in geological formations has been proposed as a technological means to reduce anthropogenic emissions of this greenhouse gas [Orr, 2009;Benson and Cook, 2006]. The positive buoyancy of supercritical CO 2 relative to the ambient brine filling the pore spaces may lead to leakage along imperfections in the geological seal, which is of considerable concern for the security of long-term storage [Gasda et al., 2004;Pruess, 2005;Neufeld et al., 2009]. One of the primary mechanisms for stable long-term geological storage of CO 2 is the dissolution of injected CO 2 within ambient brine. Under typical conditions injected CO 2 dissolves into the ambient brine thereby increasing the density of the brine [Teng et al., 1997]. This layer of dense, saturated brine forms by the processes of diffusion, dispersion and mechanical mixing during injection and, once it has reached sufficient thickness, becomes rapidly unstable to convective overturning [Ennis-King et al., 2005;Riaz et al., 2006]. The process of convective dissolution of CO 2 has recently been imaged at ambient conditions in a Hele-Shaw cell [Kneafsey and Pruess, 2009], and enhanced mass transfer has been measured at reservoir conditions [Yang and Gu, 2006;Farajzadeh et al., 2007]. Convection is therefore expected in most sequestration sites, and controls the dissolution rate and hence the long-term risk of leakage. Geochemical observations in natural CO 2 reservoirs require large amounts of CO 2 dissolution into the ambient brine and provide field evidence for sustained convective transport of dissolved CO 2 [Gilfillan et al., 2008[Gilfillan et al., , 2009. Convective dissolution of CO 2 is therefore expected in most natural and anthropogenic CO 2 reservoirs, and controls the mobility of carbon in the subsurface. It is therefore an important mechanism in the deep carbon cycle [Sherwood and Ballentine, 2009], and controls the long-term risk of leakage of CO 2 from geological storage.[3] Despi...