R. Stephen J. Sparks

Bio: R. Stephen J. Sparks is an academic researcher from University of Bristol. The author has contributed to research in topics: Volcano & Magma. The author has an hindex of 39, co-authored 93 publications receiving 5745 citations.

The Generation of Granitic Magmas by Intrusion of Basalt into Continental Crust

Abstract: 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 time-scales for melting and crystallization in basalt-induced crustal melting (10—10 y) are very short compared to the lifetimes of large silicic magma systems (>10 y) or to the timescale for thermal relaxation of the continental crust (> 10 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 time-scales of order 10—10 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 would ascend-from the source region and form magmas ranging from those with high temperature and low crystal content to those with high crystal content and a significant proportion of restite. I N T R O D U C T I O N One of the central questions in igneous petrology concerns the generation of silicic magmas. There is now convincing evidence that most of the large plutonic complexes of granite in the continental crust are the result of crustal anatexis (Pitcher, 1987). There is also [Journal of Petrologf, Vol. 29, Ptn 3, pp 599-«24, 1988] © Oxford Umvcroty Prcu 19S8 600 HERBERT E. HUPPERT AND R. STEPHEN J. SPARKS widespread evidence that basaltic magma from the mantle is often intimately associated with the generation of silicic magmas (Hildreth, 1981). This association of mafic and silicic magmas can occur in orogenic belts above subduction zones, in continental hot-spots, and in regions of crustal extension. In plutonic complexes, mafic and intermediate igneous activity are recorded in contemporaneous dyke swarms, small satellite intrusions, and in mafic enclaves within the granites (Vernon, 1983; Pitcher, 1986, 1987). In silicic volcanic centres, evidence of basaltic magmatism is found in satellite lava fields and cinder cones, early lava shields and stratovolcano complexes prior to the main silicic volcanism (Lipman, 1984), and as mafic inclusions and bands within the silicic volcanic rocks (Smith, 1979; Bacon, 1986). Petrological and geochemical features of many silicic igneous rocks are also convincingly explained by admixture of a mantle-derived (mafic) component with a crustal melt. Regions of high temperature and low pressure metamorphism are commonly associated with granite plutonism and a plausible explanation of this association is that basalt is intruded into the crust, causing melting and high heat flow. Indeed basalt underplating of the crust is a currently popular idea to explain both large scale crustal melting and the strongly layered character of the lower crust. While there may be some silicic magmas that are generated by processes without the aid of basaltic input, such as tectonic thickening of radioactive crust (England & Thompson, 1984; Pitcher, 1987), this paper takes the position that in many cases the additional thermal energy of basalt is essential. The continental crust is strongly layered in terms of its composition, density, and mechanical behaviour. The upper crust is cold and brittle whereas the lower crust is hotter, has a higher density, deforms in a ductile manner, and is commonly characterized by prominent horizontal layering. Basalt magma can be emplaced into the continental crust as dykes and sills and, in some cases where the rate of magma input is high, these intrusions can coalesce to form larger magma chambers. Dyke emplacement does not seem an efficient way of generating large volumes of silicic magma, because dykes are usually small in width and much of the potential heat for melting will not be utilized if the mafic magma erupts. Sills provide a more promising situation in which extensive crustal melting can occur. Horizontal intrusions concentrate their heat at a particular level in the crust and do not dissipate their heat over a large depth range. Sills are intrinsically more efficient than dykes in this respect. Dykes may play an important role in heating up the crust to initiate melting. However, once a region of the crust has become hot, ductile, and partially molten, conditions for dyke propagation become less favourable. A layer or region of partially molten crust provides an effective density barrier and we suggest that basalt magma reaching such a level will spread out as horizontal intrusions. An additional factor which promotes sill formation in the lower parts of the crust is its strongly layered character providing a structural environment in which horizontal intrusions are favoured. For these reasons this paper is concerned principally with the heat transfer and fluid dynamics of sills intruded into hot continental crust. We consider the cooling and crystallization of basaltic sills emplaced into the continental crust. In particular, we emphasize the situation where the roof of the sill is composed of rock which has a fusion temperature that is lower than the magma temperature and the roof rock consequently melts. This is likely to be the normal situation where basalt intrudes into the typical rock types of mature and ancient middle and upper crust which are already at high temperature. However, the concepts developed in this paper are also likely to be applicable to conditions in immature continental crust such as in island arcs, to more refractory lower crust and to lower crust formed by slightly older or even contemporaneous episodes of basalt underplating. In each of these latter cases, lithologies which have relatively low fusion temperatures can form by differentiation processes and can be remelted by further intrusion THE GENERATION OF GRANITIC MAGMAS 601 of basalt. Thus the model is not confined to the origin of granites, but should be relevant to the origins of intermediate rocks such as tonalites and evolved alkaline rocks such as syenite. We present experimental studies on the melting of the roof of a sill. We develop a quantitative model of the melting process at the roof, which describes the rates at which a new layer of roof melt forms and the rates at which the underlying liquid layer solidifies. We discuss possible mechanisms by which the melts can be mixed together and also their implications for magma genesis within the continental crust. A companion paper (Huppert & Sparks, 1988a) describes the melting of the roof of a chamber from a detailed fluid mechanical point of view. Throughout this paper the magma will be considered to be Newtonian. Although magma in reality can be non-Newtonian, especially when it is rich in crystals (McBirney & Murase, 1984) its nonlinear Theological properties and the consequences of its non-Newtonian rheology are poorly understood. Two effects may be evident: there may exist a yield strength, so that for a sufficiently low applied stress the magma will not move; and the nonlinear viscosity may alter the heat flux transferred by a convecting magma. Because of the relatively large values of the Rayleigh number that result in most of our calculations, we anticipate that the yield strength will be exceeded by quite a margin. The alterations in the heat flux are at the moment difficult to anticipate and we suggest that the reader views our quantitative results as an indication of the calculated quantity rather than as a precise value. It may be possible to examine non-Newtonian effects with greater insight in the future, but a Newtonian description illuminates many of the fundamental effects and is a necessary first step in order to form the basis for any comparison. EXPERIMENTAL STUDIES The geological problem i

Vertically extensive and unstable magmatic systems: A unified view of igneous processes

Abstract: Volcanoes are an expression of their underlying magmatic systems. Over the past three decades, the classical focus on upper crustal magma chambers has expanded to consider magmatic processes throughout the crust. A transcrustal perspective must balance slow (plate tectonic) rates of melt generation and segregation in the lower crust with new evidence for rapid melt accumulation in the upper crust before many volcanic eruptions. Reconciling these observations is engendering active debate about the physical state, spatial distribution, and longevity of melt in the crust. Here we review evidence for transcrustal magmatic systems and highlight physical processes that might affect the growth and stability of melt-rich layers, focusing particularly on conditions that cause them to destabilize, ascend, and accumulate in voluminous but ephemeral shallow magma chambers.

Axisymmetric collapses of granular columns

Abstract: 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 = hi/ri ,w herehi and ri 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∞ = ri(1 + 1.24a); while for 1.7

Collapses of two-dimensional granular columns.

Abstract: The first detailed quantitative observations of the two-dimensional collapse of a granular column along a horizontal channel are presented for a variety of materials. Together with the complementary study for the axisymmetric situation, we conclude that for granular collapses the generally accepted approaches, that are highly dependent on frictional parameters, do not describe the main flow phenomena. The motion divides in two main flow regimes at a∼1.8, where the aspect ratio a=hi∕di and hi and di are the initial height and width of the column. We describe the details of collapse by emphasizing the sequential occurrence of a main spreading followed by a final avalanching phase. For the low a regime, a 1.8, we determine that nearly all details of the collapse, including the position of the flow front as a function of time, the emplacement time, the self-similar final profiles, and especially their maximum vertical and horizontal extension, are established during the spreading phase and can be expressed in terms of the initial geometrical parameters but are independent of basal and internal friction parameters.

How volcanoes work: a 25 year perspective

Abstract: Over the past 25 years, our understanding of the physical processes that drive volcanic eruptions has increased enormously thanks to major advances in computational and analytical facilities, instrumentation, and collection of comprehensive observational, geophysical, geochemical, and petrological data sets associated with recent volcanic activity. Much of this work has been motivated by the recognition that human exposure to volcanic hazard is increasing with both expanding populations and increasing reliance on infrastructure (as illustrated by the disruption to air traffic caused by the 2010 eruption of Eyjafjallajokull volcano in Iceland). Reducing vulnerability to volcanic eruptions requires a thorough understanding of the processes that govern eruptive activity. Here, we provide an overview of our current understanding of how volcanoes work. We focus particularly on the physical processes that modulate magma accumulation in the upper crust, transport magma to the surface, and control eruptive activity.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

The Genesis of Intermediate and Silicic Magmas in Deep Crustal Hot Zones

Abstract: A model for the generation of intermediate and silicic igneous rocks is presented, based on experimental data and numerical modelling. The model is directed at subduction-related magmatism, but has general applicability to magmas generated in other plate tectonic settings, including continental rift zones. In the model mantlederived hydrous basalts emplaced as a succession of sills into the lower crust generate a deep crustal hot zone. Numerical modelling of the hot zone shows that melts are generated from two distinct sources; partial crystallization of basalt sills to produce residual H2O-rich melts; and partial melting of pre-existing crustal rocks. Incubation times between the injection of the first sill and generation of residual melts from basalt crystallization are controlled by the initial geotherm, the magma input rate and the emplacement depth. After this incubation period, the melt fraction and composition of residual melts are controlled by the temperature of the crust into which the basalt is intruded. Heat and H2O transfer from the crystallizing basalt promote partial melting of the surrounding crust, which can include meta-sedimentary and meta-igneous basement rocks and earlier basalt intrusions. Mixing of residual and crustal partial melts leads to diversity in isotope and trace element chemistry. Hot zone melts are H2O-rich. Consequently, they have low viscosity and density, and can readily detach from their source and ascend rapidly. In the case of adiabatic ascent the magma attains a super-liquidus state, because of the relative slopes of the adiabat and the liquidus. This leads to resorption of any entrained crystals or country rock xenoliths. Crystallization begins only when the ascending magma intersects its H2O-saturated liquidus at shallow depths. Decompression and degassing are the driving forces behind crystallization, which takes place at shallow depth on timescales of decades or less. Degassing and crystallization at shallow depth lead to large increases in viscosity and stalling of the magma to form volcano-feeding magma chambers and shallow plutons. It is proposed that chemical diversity in arc magmas is largely acquired in the lower crust, whereas textural diversity is related to shallow-level crystallization.

The physical character of subaqueous sedimentary density flows and their deposits

Abstract: The complexity of flow and wide variety of depositional processes operating in subaqueous density flows, combined with post-depositional consolidation and soft-sediment deformation, often make it difficult to interpret the characteristics of the original flow from the sedimentary record. This has led to considerable confusion of nomenclature in the literature. This paper attempts to clarify this situation by presenting a simple classification of sedimentary density flows, based on physical flow properties and grain-support mechanisms, and briefly discusses the likely characteristics of the deposited sediments. Cohesive flows are commonly referred to as debris flows and mud flows and defined on the basis of sediment characteristics. The boundary between cohesive and non-cohesive density flows (frictional flows) is poorly constrained, but dimensionless numbers may be of use to define flow thresholds. Frictional flows include a continuous series from sediment slides to turbidity currents. Subdivision of these flows is made on the basis of the dominant particle-support mechanisms, which include matrix strength (in cohesive flows), buoyancy, pore pressure, grain-to-grain interaction (causing dispersive pressure), Reynolds stresses (turbulence) and bed support (particles moved on the stationary bed). The dominant particle-support mechanism depends upon flow conditions, particle concentration, grain-size distribution and particle type. In hyperconcentrated density flows, very high sediment concentrations (>25 volume%) make particle interactions of major importance. The difference between hyperconcentrated density flows and cohesive flows is that the former are friction dominated. With decreasing sediment concentration, vertical particle sorting can result from differential settling, and flows in which this can occur are termed concentrated density flows. The boundary between hyperconcentrated and concentrated density flows is defined by a change in particle behaviour, such that denser or larger grains are no longer fully supported by grain interaction, thus allowing coarse-grain tail (or dense-grain tail) normal grading. The concentration at which this change occurs depends on particle size, sorting, composition and relative density, so that a single threshold concentration cannot be defined. Concentrated density flows may be highly erosive and subsequently deposit complete or incomplete Lowe and Bouma sequences. Conversely, hydroplaning at the base of debris flows, and possibly also in some hyperconcentrated flows, may reduce the fluid drag, thus allowing high flow velocities while preventing large-scale erosion. Flows with concentrations <9% by volume are true turbidity flows (sensuBagnold, 1962), in which fluid turbulence is the main particle-support mechanism. Turbidity flows and concentrated density flows can be subdivided on the basis of flow duration into instantaneous surges, longer duration surge-like flows and quasi-steady currents. Flow duration is shown to control the nature of the resulting deposits. Surge-like turbidity currents tend to produce classical Bouma sequences, whose nature at any one site depends on factors such as flow size, sediment type and proximity to source. In contrast, quasi-steady turbidity currents, generated by hyperpycnal river effluent, can deposit coarsening-up units capped by fining-up units (because of waxing and waning conditions respectively) and may also include thick units of uniform character (resulting from prolonged periods of near-steady conditions). Any flow type may progressively change character along the transport path, with transformation primarily resulting from reductions in sediment concentration through progressive entrainment of surrounding fluid and/or sediment deposition. The rate of fluid entrainment, and consequently flow transformation, is dependent on factors including slope gradient, lateral confinement, bed roughness, flow thickness and water depth. Flows with high and low sediment concentrations may co-exist in one transport event because of downflow transformations, flow stratification or shear layer development of the mixing interface with the overlying water (mixing cloud formation). Deposits of an individual flow event at one site may therefore form from a succession of different flow types, and this introduces considerable complexity into classifying the flow event or component flow types from the deposits.

The Generation of Granitic Magmas by Intrusion of Basalt into Continental Crust

Abstract: 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 time-scales for melting and crystallization in basalt-induced crustal melting (10—10 y) are very short compared to the lifetimes of large silicic magma systems (>10 y) or to the timescale for thermal relaxation of the continental crust (> 10 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 time-scales of order 10—10 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 would ascend-from the source region and form magmas ranging from those with high temperature and low crystal content to those with high crystal content and a significant proportion of restite. I N T R O D U C T I O N One of the central questions in igneous petrology concerns the generation of silicic magmas. There is now convincing evidence that most of the large plutonic complexes of granite in the continental crust are the result of crustal anatexis (Pitcher, 1987). There is also [Journal of Petrologf, Vol. 29, Ptn 3, pp 599-«24, 1988] © Oxford Umvcroty Prcu 19S8 600 HERBERT E. HUPPERT AND R. STEPHEN J. SPARKS widespread evidence that basaltic magma from the mantle is often intimately associated with the generation of silicic magmas (Hildreth, 1981). This association of mafic and silicic magmas can occur in orogenic belts above subduction zones, in continental hot-spots, and in regions of crustal extension. In plutonic complexes, mafic and intermediate igneous activity are recorded in contemporaneous dyke swarms, small satellite intrusions, and in mafic enclaves within the granites (Vernon, 1983; Pitcher, 1986, 1987). In silicic volcanic centres, evidence of basaltic magmatism is found in satellite lava fields and cinder cones, early lava shields and stratovolcano complexes prior to the main silicic volcanism (Lipman, 1984), and as mafic inclusions and bands within the silicic volcanic rocks (Smith, 1979; Bacon, 1986). Petrological and geochemical features of many silicic igneous rocks are also convincingly explained by admixture of a mantle-derived (mafic) component with a crustal melt. Regions of high temperature and low pressure metamorphism are commonly associated with granite plutonism and a plausible explanation of this association is that basalt is intruded into the crust, causing melting and high heat flow. Indeed basalt underplating of the crust is a currently popular idea to explain both large scale crustal melting and the strongly layered character of the lower crust. While there may be some silicic magmas that are generated by processes without the aid of basaltic input, such as tectonic thickening of radioactive crust (England & Thompson, 1984; Pitcher, 1987), this paper takes the position that in many cases the additional thermal energy of basalt is essential. The continental crust is strongly layered in terms of its composition, density, and mechanical behaviour. The upper crust is cold and brittle whereas the lower crust is hotter, has a higher density, deforms in a ductile manner, and is commonly characterized by prominent horizontal layering. Basalt magma can be emplaced into the continental crust as dykes and sills and, in some cases where the rate of magma input is high, these intrusions can coalesce to form larger magma chambers. Dyke emplacement does not seem an efficient way of generating large volumes of silicic magma, because dykes are usually small in width and much of the potential heat for melting will not be utilized if the mafic magma erupts. Sills provide a more promising situation in which extensive crustal melting can occur. Horizontal intrusions concentrate their heat at a particular level in the crust and do not dissipate their heat over a large depth range. Sills are intrinsically more efficient than dykes in this respect. Dykes may play an important role in heating up the crust to initiate melting. However, once a region of the crust has become hot, ductile, and partially molten, conditions for dyke propagation become less favourable. A layer or region of partially molten crust provides an effective density barrier and we suggest that basalt magma reaching such a level will spread out as horizontal intrusions. An additional factor which promotes sill formation in the lower parts of the crust is its strongly layered character providing a structural environment in which horizontal intrusions are favoured. For these reasons this paper is concerned principally with the heat transfer and fluid dynamics of sills intruded into hot continental crust. We consider the cooling and crystallization of basaltic sills emplaced into the continental crust. In particular, we emphasize the situation where the roof of the sill is composed of rock which has a fusion temperature that is lower than the magma temperature and the roof rock consequently melts. This is likely to be the normal situation where basalt intrudes into the typical rock types of mature and ancient middle and upper crust which are already at high temperature. However, the concepts developed in this paper are also likely to be applicable to conditions in immature continental crust such as in island arcs, to more refractory lower crust and to lower crust formed by slightly older or even contemporaneous episodes of basalt underplating. In each of these latter cases, lithologies which have relatively low fusion temperatures can form by differentiation processes and can be remelted by further intrusion THE GENERATION OF GRANITIC MAGMAS 601 of basalt. Thus the model is not confined to the origin of granites, but should be relevant to the origins of intermediate rocks such as tonalites and evolved alkaline rocks such as syenite. We present experimental studies on the melting of the roof of a sill. We develop a quantitative model of the melting process at the roof, which describes the rates at which a new layer of roof melt forms and the rates at which the underlying liquid layer solidifies. We discuss possible mechanisms by which the melts can be mixed together and also their implications for magma genesis within the continental crust. A companion paper (Huppert & Sparks, 1988a) describes the melting of the roof of a chamber from a detailed fluid mechanical point of view. Throughout this paper the magma will be considered to be Newtonian. Although magma in reality can be non-Newtonian, especially when it is rich in crystals (McBirney & Murase, 1984) its nonlinear Theological properties and the consequences of its non-Newtonian rheology are poorly understood. Two effects may be evident: there may exist a yield strength, so that for a sufficiently low applied stress the magma will not move; and the nonlinear viscosity may alter the heat flux transferred by a convecting magma. Because of the relatively large values of the Rayleigh number that result in most of our calculations, we anticipate that the yield strength will be exceeded by quite a margin. The alterations in the heat flux are at the moment difficult to anticipate and we suggest that the reader views our quantitative results as an indication of the calculated quantity rather than as a precise value. It may be possible to examine non-Newtonian effects with greater insight in the future, but a Newtonian description illuminates many of the fundamental effects and is a necessary first step in order to form the basis for any comparison. EXPERIMENTAL STUDIES The geological problem i