J. S. Turner

Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows

The entrainment assumption, relating the inflow velocity to the local mean velocity of a turbulent flow, has been used successfully to describe natural phenomena over a wide range of scales. Its first application was to plumes rising in stably stratified surroundings, and it has been extended to inclined plumes (gravity currents) and related problems by adding the effect of buoyancy forces, which inhibit mixing across a density interface. More recently, the influence of viscosity differences between a turbulent flow and its surroundings has been studied. This paper surveys the background theory and the laboratory experiments that have been used to understand and quantify each of these phenomena, and discusses their applications in the atmosphere, the ocean and various geological contexts.

The Fluid Dynamics of Evolving Magma Chambers [and Discussion]

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.

Turbulent fountains in an open chamber

The flow and density distribution produced by injecting dense fluid upwards at the bottom of a homogeneous fluid have been investigated experimentally and theoretically. Both axisymmetric and line sources have been studied using small-scale laboratory experiments in which salt water is injected into a tank of fresh water. The turbulent fountain formed in this way rises to a maximum height which can be related to the Froude number of the inflow, and then falls back and spreads out along the floor. Continuing the inflow builds up a stable stratification in a similar manner to that discussed earlier for the ‘plume filling box model’ of Baines & Turner (1969) which is complementary to the present work. The fountain flows considered here have the important new feature that the volume of the inflow is significant, so the total volume of fluid in the ‘open’ container increases with time. The evolution is determined by the rate of entrainment into the fountain from its surroundings, which is found directly by experiment. Re-entrainment of fluid into the fountain continually changes the density profile in the mixed fluid collecting at the bottom of the chamber below the level of the fountain top, and controls the rate of rise of a ‘front’ of marked fluid. The top of the fountain rises linearly in time, at a rate which, for axisymmetric fountains, has been shown both experimentally and theoretically to be close to half the rate of rise of the free surface due to the inflow. Thus at a certain time the front rises above the top of the fountain. Once the mixed fluid at the bottom of the chamber has risen above the fountain its density profile remains unchanged. The front velocity, the fountain height and the density profile have all been obtained as functions of time using a theory which is in good agreement with the experimental results for a large range of input Froude numbers. For line fountains the results are less precise owing to an instability which causes the flow to switch irregularly from a symmetrical state to one in which the downflow occurs on one side only, and with a smaller maximum height. In concluding we discuss the applications which motivated the work, particularly the development of a stratified hybrid layer in magma chambers replenished from below, and the dynamically identical, but inverted problem of heating large buildings through ducts located near the roof.

The formation of ‘optimal’ vortex rings, and the efficiency of propulsion devices

The formation of an axisymmetric vortex ring by forcing uid impulsively through a pipe is examined. An idealized model of the circulation, impulse and energy provided by the injected plug is developed, and these quantities are equated to the corresponding properties of the class of rings with finite cores described by Norbury (1973). It is shown that, as the length-to-diameter aspect ratio L / D of the plug increases, the size of the core increases in comparison with all the fluid carried along with the ring, until the limiting case of Hills spherical vortex is reached. For aspect ratios larger than a certain value it is not possible to produce a single ring while conserving circulation, impulse, volume and energy. This implies that the limiting vortex is ‘optimal’ in the sense that it has maximum impulse, circulation and volume for a given energy input. While this matching calculation makes the physical mechanism clear, the L / D ratio that can be achieved in practice is more appropriately taken from the direct experimental measurements of Gharib et al . (1998) who concluded that the limiting value is L / D = 4. This is close to the value found in our calculation.

'Optimal' vortex rings and aquatic propulsion mechanisms.

Fishes swim by flapping their tail and other fins. Other sea creatures, such as squid and salps, eject fluid intermittently as a jet. We discuss the fluid mechanics behind these propulsion mechanisms and show that these animals produce optimal vortex rings, which give the maximum thrust for a given energy input. We show that fishes optimize both their steady swimming efficiency and their ability to accelerate and turn by producing an individual optimal ring with each flap of the tail or fin. Salps produce vortex rings directly by ejecting a volume of fluid through a rear orifice, and these are also optimal. An important implication of this paper is that the repetition of vortex production is not necessary for an individual vortex to have the ‘optimal’ characteristics.

The development of layering, fluxes through double-diffusive interfaces, and location of hydrothermal sources of brines in the Atlantis II Deep: Red Sea

The brines in the Atlantis II Deep of the Red Sea occur in horizontally uniform, well-mixed layers, with the hottest and saltiest water at the bottom, separated from the successively cooler and fresher layers above by very sharp vertical temperature and salinity gradients. Data acccumulated over 3 decades are used to test the widely accepted hypothesis that all heat and salt for the brine layers are supplied from below and that the layered brine structure is the result of double diffusion. Using the changes in temperature and salinity in each layer over successive time intervals, one can deduce the corresponding fluxes of heat and salt across the interfaces. It is found that the required flux of salt cannot be sustained by double diffusion alone. An alternative calculation shows that most of the salt in the successively forming upper layers must have been injected directly from the bottom of the deep through one or more vents located above the level of the lowest brine interface. For the bottom layer, however, it is not possible to obtain the observed salinity and temperature changes unless hot saline water is injected directly into that layer and some of the heat and a smaller fraction of the salt are transferred upward through the interface. This process will also maintain convection in each of the layers and keep them well mixed, as is observed. The new interpretation in terms of separate inputs at various levels in the Atlantis II Deep is also supported by recent geochemical evidence.

Laboratory studies of double-diffusive sources in closed regions

Various observations of layering and intrusions in the ocean strongly suggest that such structures and motions are produced and driven by horizontal and vertical gradients of temperature and salinity, i.e. by double-diffusive processes. Much of the laboratory work in this field has concentrated on one-dimensional problems, with the neglect of two-dimensional phenomena. The latter are addressed explicitly in the present paper, using the salt–sugar analogue system in a simple geometry, but with the aim of establishing some more widely applicable general principles. Two sources of salt or sugar solution were fed in at opposite ends of a 750 mm long tank, with an overflow tube drawing fluid from a point at the centre of the tank. With two salt sources of different concentrations and densities, a stratification built up through the ‘filling box’ process, and the total density range lay within that of the input solutions. For one salt and one sugar source, a much larger density gradient could be set up, with the range lying outside that of the inputs. The flows were monitored using various experimental techniques: photographs of dye streaks with still and video cameras; a polarimeter to monitor sugar concentration; and the withdrawal of samples for the measurement of density and refractive index, from which the separate contributions of salt and sugar to the density could be calculated. Three related experiments with simple input conditions were particularly instructive, and these will be described first. Both inputs and the withdrawal tube were located at mid-depth, and the tank fluid and the salt and sugar supplies had the same density. The only difference between runs was the initial composition of the solution in the tank: pure salt, pure sugar, and a 50[ratio ]50 mixture of the two. Following an initial transient response which was different in the three experiments, they all tended to the same asymptotic distributions of salt, sugar and density after about 100 h, with a sharp central interface and weakly stratified upper and lower layers. This state corresponded approximately to the one-dimensional ‘rundown’ of a layer of salt solution above sugar solution, with a slightly higher, unstable concentration of salt in the top layer compared to the bottom and a very stable sugar distribution, with a much larger concentration in the bottom layer than in the top one. This distribution cannot be produced by ‘finger’ rundown, and it corresponds to the maximum release of potential energy. It was, however, achieved through the action of many intrusions, which remained active in the dynamic final state, and maintained a strong communication between the two ends of the tank. A comparable experiment was carried out using a tank 1820 mm long. With this larger aspect ratio there was a predominantly local influence of the sources at each end of the tank. Other runs have explored a variety of geometries of the sources and sinks, and the final state has been shown to be sensitive to these boundary conditions.