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The Volume and Composition of Melt Generated by Extension of the Lithosphere by D. McKENZIE AND M. J. BICKLE Department of Earth Sciences, Bullard Laboratories, Madingley Road, Cambridge CB3 OEZ (Received 21 July 1987; revised typescript accepted 14 January 1988) ABSTRACT Calculation of the volume and composition of magma generated by lithospheric extension requires an accurate initial geotherm, and knowledge of the variation and composition of the melt fraction as a function of pressure and temperature. The relevant geophysical observations are outlined, and geotherms then obtained from parameterized convective models. Experimental observations which constrain the solidus and liquidus at various pressures are described by simple empirical functions. The variation in melt fraction is then parameterized by requiring a variation from 0 on the solidus to 1 on the liquidus. The composition of the melts is principally controlled by the melt fraction, though those of FeO, MgO, and SiO2 in addition vary with pressure. Another simple parameterization allows the observed compositions of major elements in 91 experiments to be calculated with a mean error of 11%, and those of TiO 2 and Na 2 O to 0 3 % . These expressions are then used to calculate the expected compositions of magma produced by adiabatic upwelling. The mean major element composition of the most magnesium-rich MORB glasses resemble the mean composition calculated for a mantle potential temperature Tf of 1280°C. Adiabatic melting during upwelling of mantle of this temperature generates a melt thickness of 7 km. The observed variations of the MgO and TiO 2 concentrations in a large collection of MORB glass compositions suggest that extensive low pressure fractional crystallization occurs, but that its effect on the concentrations of SiOj, A12O3, and CaO is small. There is no evidence that normal oceanic crust is produced from magmas containing more than 11% MgO. The mantle potential temperature within hot rising jets is about 1480°C and can generate 27 km of magma containing 17% MgO. Extension of the continental lithosphere generates little melt unless /?>2 and Tr> 138O°C. The melts generated by larger values offtor of Tr are alkali basalts, and change to tholeiites as the amount of melting increases. Large quantities of melt can be generated, especially at continental margins, where estimates of ft obtained from changes in crustal thickness will in general be too small. 1. I N T R O D U C T I O N "In summary, then, the formation of magma remains a geophysical enigma, and much work, particularly experimental work, on the melting behaviour of possible mantle materials remains to be done. . . . What is there in the thermal regions of the earth that makes it impossible for magma to reach the surface at temperatures greater than about 1200°C? Geophysical and geochemical implications of this observation have, to our knowledge, never been fully explored, even though they could provide important constraints on temperature distribution in the mantle and on the very mechanism of magma formation" (Carmichael et al., 1974, p. 359). Though there is now an extensive body of experimental work on the melting of mantle materials, and the existence of peridotitic komatiites as lava flows suggests that some melts at some times reach the surface at temperatures considerably higher than 1200°C, the physical processes associated with magma formation within the earth are not much better understood now than they were in 1974. The principal purpose of [Journal of Petrology, Vol 29, P«rt 3, pp. 623-679, 1988] © Oxford Umvcrsty Pros 1988 626 D. McKENZIE AND M. J. BICKLE this paper is to combine ideas derived from the geophysical observation of spreading ridges and of extension of the continental lithosphere to determine the P, T paths of mantle material in such environments, and hence to calculate the volume and composition of the magma generated. Magma is also produced beneath island arcs, and by hot jets rising beneath plate interiors which need not cause extension. But the thermal structure of both these regions is less well known than is that of ridges and for this reason they will not be considered further. Furthermore all such sources of magma are small compared with spreading ridges, which produce about 20 km 3 of melt a year. Though the aim of this investigation is simple and few of the ideas original, a number of the arguments on which it rests are either controversial, or not widely known, or spread across the rather extensive relevant literature or suffer from all three problems! It therefore seemed to us worthwhile to provide non-technical explanations, illustrated by sketches, of some of the more important of these arguments where necessary. The new results will be found in section 4 which is concerned with spreading ridges and the origin of oceanic basalts, especially tholeiitic basalts, and in section 5 which is concerned with extension in continental areas and the origin of alkaline melts. 2. PRELIMINARIES Geotherms Figure l(a) illustrates two commonly used geotherms, taken from Green & Ringwood (1967a), together with the dry solidus of garnet peridotite. Many authors illustrate the melting process with isothermal upwelling lines from the geotherm to the solidus, followed by melt separation and isothermal upwelling to the surface. As Verhoogen (1973) has remarked, it is essential to take into account the latent heat of melting. The effect of doing so and of the adiabatic gradient in the solid and melt considerably decreases the melt temperature. Another problem concerns the geotherms themselves, which must be the result of physical processes within the mantle. Many of the geotherms which are commonly drawn predate our understanding of mantle convection and do not consider the processes by which FIG. 1. (a) Geotherms proposed by Green & Ringwood (1967a) for ocean basins and shields, principally on the basis of pressure and temperature estimates from experimental petrology, (b) Convective geotherms with a potential temperature of I28O°C and a viscosity of 4x 1 0 " m 2 s ~ ' , calculated for old ocean basins with mechanical boundary layer thicknesses of 100 km and 200 km respectively. The temperature of the ridge axis does not take into account melting MELT GENERATED BY LITHOSPHERIC EXTENSION 627 they are maintained. Those shown in Fig. l(b) are obtained from the expressions in Appendix B for a vigorously convecting constant viscosity fluid overlain by a solid slab 100 km thick if melting is ignored. The solidus is that obtained below, and differs little from Green & Ringwood's. But the geotherms are different because they all tend to the same adiabatic curve in the convecting interior. Beneath a ridge axis the upwelling of material is rapid and dominates the geotherm, because the solid carries heat with it as it moves upwards. Heat transport of this type is known as advective (or convective) transport, as distinct from the conductive heat transport, which occurs in the absence of movement. The question of which type of transport dominates in any region is controlled by the thermal Peclet number Pex /* (1) where v is the material velocity, / a length scale and K the thermal diffusivity. If Pel»\ advective transport of heat dominates, whereas if Pet« 1 conduction dominates. Beneath a spreading ridge i>~ 10 mm a" 1 and therefore Pet^3xl0-4/ (2) 5 where / is in metres. The lithosphere is about 100 km or 10 m thick, giving Pet = 30. Therefore the temperature beneath a spreading ridge will be dominated by the advection of heat and conduction can be ignored. Equation (2) can also be used to discover how thick the conductive layer will be beneath the ridge axis by finding the value of / which satisfies (2) when Pet = 1. This value is 3 km, or less than the thickness of the oceanic crust. Therefore the temperature variation which controls melting beneath ridges is entirely governed by advection. The same argument can also be used to discover how small must v be before the melting processes are affected by heat loss to the earth's surface. As we shall show most of the melting occurs within 30 km of the surface. Setting / = 30 km and Pet =1 in (1) gives v ~ 1 mm a ~ '. This velocity is smaller than the spreading rate on most ridge axes, and is less than the upwelling rate beneath many (though not all) continental regions where the lithosphere is stretching. Therefore it is clearly a good first approximation to neglect the conduction of heat when considering melting. It is then necessary to determine both the geotherm and melt fraction which is generated as part of the same calculation. Geotherms cannot be imposed, since they are entirely controlled by the melting process itself. One important result follows at once from these arguments. Because the interior of the mantle is everywhere hotter than the mantle solidus at atmospheric pressure, mantle material will always melt when it is brought up to depths shallower than about 40 km. The only exceptions will be regions both whose size is small and where the material moves slowly, such as the upwelling regions on slowly spreading ridges within a few kilometres of the older cold plate, where a ridge abuts a fracture zone. Samples of mantle undepleted by melting will therefore reach the surface only rarely and in special tectonic situations. The second problem concerns the temperature of the magma as it upwells. If it does so quickly (Pe,»l) then it will neither gain nor lose heat to its surroundings as it moves to regions of lower pressure. It therefore undergoes adiabatic decompression, and will do so at constant entropy to a good approximation. Then the adiabatic temperature gradient (dT/dz)s is 628 D. McKENZIE AND M. J. BICKLE where g is the acceleration due to gravity, af the thermal expansion coefficient of the magma, 7" the absolute temperature and CF the specific heat at constant pressure. Using the values of af and CP given by McKenzie (1984a) and 7 = 1500 K gives '. (4) Therefore if magma is generated near the base of the lithosphere it may cool by as much as 100 °C as it travels to the surface, even though it loses no heat. The corresponding gradient in the solid material below the melting zone is also adiabatic, and given by (3) but with the thermal expansion coefficient of the solid not the magma. The gradient in the solid region is smaller, and is about CH^Ckm" 1 . It is commonly necessary to compare the heat content of material at different depths. Such a comparison is straightforward if the material is incompressible, since the difference in heat content is then simply proportional to the difference in temperature. But this simple result fails when the material is compressible. Those who ascend mountains are well aware of the adiabatic temperature gradient which exists because air is compressible. The heat content of two air masses is only proportional to the temperature difference between them if they have been brought to the same pressure, which must be done reversibly to conserve entropy. In meteorology and oceanography this problem has been well understood for many years. Rather than using the entropy of a mass of fluid to define its heat content, it is common practice in these subjects to define a new temperature, called the potential temperature, which is the temperature the fluid mass would have (hence the term 'potential') if it were compressed or expanded to some constant reference pressure. A similar concept is very useful in discussion of mantle dynamics, and the relationship between the actual temperature T at a depth z and the potential temperature Tf is easily obtained by integrating (3) where a = a, within the solid part of the mantle. The reference depth at which T= TP is the earth's surface. Adiabatic upwelling leaves TP unchanged. TP only changes when the entropy of the material changes. Though there is a simple relationship between the change in entropy AS and the associated change in potential temperature, the concept of entropy is less familiar than that of temperature: hence the usefulness of the potential temperature. The reason why it is necessary to take account of the compressibility of the mantle is that the horizontal temperature differences within the convecting mantle are not likely to exceed 200 CC. The interior of the upper mantle is likely to have a temperature gradient which differs little from the adiabatic gradient and hence material will increase in temperature by 200 °C on sinking 300 km (equivalent to a change in pressure of about lOGPa). Hence, if substantial vertical movements occur, the temperature differences are not a good guide to differences in heat content. Such differences are, however, clearly reflected in differences of the potential temperatures which are therefore used throughout this paper. For the same reason it is necessary to use potential temperature when discussing mantle convection if the lateral temperature differences within the convecting system are comparable to those produced by adiabatic compression between the top and bottom of the convecting layer. In the mantle the two temperature differences are of similar magnitude. Interestingly such compressibility has little effect on the dynamics of the convection. The reason why it concerns us here is that we are interested in the actual temperature variations which exist within the convecting system, and because the temperature, not the potential temperature, is measured in the laboratory. MELT GENERATED BY LITHOSPHERIC EXTENSION <•) Axis 1400" 1500° 1500° 1400° (b) Axis 200 100 1 Zone of extensive silicate melting / okm Ml /100 n£- •-— 200 '/////////ft///"////, p-___ Trace amounts of melting 100 1350° km 1400° 200 Tp =1280°C FIG. 2. (a) A sketch of the temperature distribution beneath a spreading ridge axis (from Oxburgh, 1980) when it coincides with a hot rising jet in the mantle For reasons discussed in the text few spreading ndges are now believed to coincide with such jets, and must instead be passive features underlain by mantle of constant potential temperature, (b) Shows a sketch of the resulting temperature structure. Mantle circulation beneath ridges Most textbook illustrations of spreading ridges, such as that shown in Fig. 2(a) from Oxburgh (1980), show a hot rising jet in the convecting mantle below. Such a close association between the convective geometry of the mantle and the movement of plates was a feature of much of the early work on continental drift and sea floor spreading (see for instance, Hess, 1962), and caused major conceptual problems. It was difficult to understand how Africa and Antarctica could be surrounded by spreading ridges. Where did the upwelling material go, and, if the ridges migrated, how did the convective system below move with them? What happens where a ridge is offset by a transform fault? Is the convective 630 D. McKENZIE AND M J. BICKLE system offset in the same way and if so how? What happens when a ridge jumps and leaving a fossil ridge and starting a new one, sometimes thousands of kilometres away? Does the hot sheet move also, and if so how? These and other problems with the ideas in Fig. 2(a) suggested that they should be examined carefully. In particular all these difficulties disappear if ridges are simply passive features where two plates are separating, and mantle material upwells simply because of this separation, rather than because there is a hot jet in the deeper convecting part of the mantle. This line of argument suggested that the logical consequences of the most extreme model, in which the ridge is underlain by a horizontal isotherm at some depth (Fig. 2(b)), should be examined carefully. This reasoning was the motivation behind the simple model of plate creation (McKenzie, 1967), which has successfully accounted for the variation of depth and oceanic heat flow with age (see Sclater & Francheteau, 1970; Parsons & Sclater, 1977; Sclater et al., 1980). This success has diverted attention from the original motivation, which is what is of concern here. The important result from the point of view of magma generation is that the simple model in Fig. 2(b) succeeds in accounting for the observations in considerable detail, and yet it contains a horizontal isotherm beneath the lithosphere. It was the high heat flow and shallow bathymetry of ridges which led to the idea that they were underlain by a hot rising sheet below the lithosphere. The success of the model in Fig. 2(b) in accounting for these phenomena shows clearly that no such sheet is required by the observations. The resulting freedom to move ridges, irrespective of convective geometries in the mantle below, removed one of the major difficulties faced by the early concepts of sea floor spreading. From the point of view of magma generation on ridges these results are of great importance because they lead to a natural explanation of why the oceanic crust is of such a uniform thickness (see section 3). Two more subtle questions have recently been raised about the relationship between ridges and the convective circulation of the upper mantle below. Though for the reasons discussed above there is in general no observational evidence for an association between mantle circulation and spreading ridges, under some circumstances it is possible that the movement of the plates may trap an upwelling hot jet or sheet in the mantle below simply because the plates are moving apart. The question of whether such trapping can occur is controlled by how fast the plates and the ridge are moving. Houseman (1983) carried out a number of numerical experiments and showed that attachment was only possible if the ridge was moving at a velocity of less than a few millimetres a year. The question of whether ridges are underlain by hot rising sheets can now be examined directly by using satellite altimeters to map the gravity field with wavelengths between 500 and 4000 km (Watts et al, 1985). These signals are the direct expression of the mantle circulation, and, though a variety of rising and sinking jets have now been mapped in oceanic regions, they have no obvious association with spreading ridges, which in places lie above regions where the mantle below is sinking. The lithosphere Exactly what different authors mean by the word 'lithosphere' is of central importance to igneous petrology. At present the most commonly used definition is that obtained from the thermal model of plate formation: the depth to the horizontal isotherm. Though this depth can now be determined with considerable accuracy, it is not obvious that the thickness of the lithosphere defined in this way has any significance for igneous processes. From the point of view of isotopic studies what is of concern is that part of the plate which is isolated from mantle convection, which vigorously stirs the upper mantle. Only in this part of the upper MELT G E N E R A T E D BY L I T H O S P H E R I C EXTENSION 631 mantle can radioactive decay produce distinctive isotopic anomalies. It is not obvious that these two definitions of the lithosphere are the same. To understand why they need not be, and why the thermal thickness is in general greater than that of interest to petrologists, it is necessary to examine the processes which control the temperature structure of old plate far from plate boundaries. Near spreading ridges the temperature of the upper mantle is controlled by the upwelling temperature and the thermal conductivity of rock, and is independent of the depth of the horizontal isotherm (Parsons & Sclater, 1977). However as the plate ages its thermal structure becomes more and more dependent on the depth / and temperature 7\, of this isotherm, until in steady state it becomes independent of all other parameters. The time x taken to approach the steady state is given by T (6) ^ n K and can be estimated directly from the observations of ocean depth (Parsons & Sclater, 1977). Because T depends on the square of the depth to the horizontal isotherm, equation (6) provides the most accurate estimate available for /. Parsons & Sclater obtained a value of 125 km. Recent studies of Pacific bathymetry (Watts et al., 1985) suggest that this value is slightly too large, though the depth is not likely to be less than 100 km. The plate model assumes that heat is transported by conduction only above this isotherm. If the same is true below it then this isotherm will not remain at constant depth, but continue to become deeper. Parsons & Sclater (1977) demonstrated that the horizontal isotherm did not continue to deepen with age, and estimated that Tx = 1333°C and / = 125 km. For some time this explanation of the depth-age curve was controversial and a variety of other suggestions were offered. Recently, however, a new test of these ideas has been provided by the satellite altimeters. The magnitude of the geoid step across fracture zones depends principally on the thermal structure of the plate, and the observations have strongly confirmed the plate model (Cazenave et al., 1983). There are, however, still some technical difficulties in the data reduction which cause the estimates of / obtained from such observations to be less accurate than those from the bathymetry. Unfortunately the success of the plate model has obscured its fundamental weakness: it provides no mechanism by which the horizontal isotherm can be maintained at constant depth. Conductive heat loss to the earth's surface will cause all isotherms to move downward unless heat is supplied by some means, and this means will in turn affect the temperature structure. This problem lead to the suggestion that trfere was a convective instability which removed cold material from the base of old plate and replaced it with hotter material (Parsons & McKenzie, 1978). A considerable amount of theoretical work has now been carried out on this suggestion, and no obvious inconsistencies have yet been discovered. The major virtue of this scheme is that it provides a natural explanation for the constant depth of the horizontal isotherm. Unlike the plate model, this model provides no natural definition of the lithosphere. The boundary layer at the earth's surface consists of two parts. The upper part is rigidly attached to the surface, and moves with the magnetic lineations. This part is referred to as the mechanical boundary layer, and in steady state is perhaps 100 km thick. Only its upper part can maintain elastic stresses for geological times, so estimates of the elastic thickness of old oceanic plate are even smaller. The mechanical boundary layer is underlain by a thermal boundary layer, which loses heat by conduction to the earth's surface. It episodically becomes unstable and is replaced by hotter mantle material. 632 D. McKENZIE AND M. J. BICKLE fi Depth km Depth km FIG. 3. (a) The horizontally averaged temperature for a potential temperature of I28O°C, a thickness of the mechanical boundary layer of 100 km and a viscosity o f 2 x l 0 1 7 m 2 s ~ ' , obtained from the expression in Appendix B. The corresponding adiabatic upwelling curve is shown dashed. The elevation of ridge axes above the surface of old plate is controlled by the area between the geotherm and the dashed line. The 'lithospheric' thickness is obtained by requiring the corresponding area for the plate model to be the same as that obtained from the convective model. The temperature gradients at the surface in the two cases are not identical, though they are indistinguishable in (a) (b), (c) Enlargements of the geotherm at the base of the thermal boundary layer for two geotherms with interior viscosities of 2x 1017 m 2 s ~ \ (b), and 4 x 1 0 " m2 s " ' , (c), both with mechanical boundary layer thicknesses of 100 km and potential temperatures of 1280°C. It is clear that the thickness of the iithosphere' in Fig. 3 will depend on the method used to obtain the estimate. The oceanic bathymetry will provide an estimate of the mean temperature difference between the ridge axis and old plate, or the area between the geotherm and the adiabatic upwelling curve in Fig. 3(a). The corresponding 'lithospheric' thickness is obtained by calculating a depth on the adiabatic geotherm with the same area. The lower boundary of the 'Iithosphere' defined in this way is within the convecting region, and is therefore not simply related to the thickness of the mechanical boundary layer in which isotopic anomalies can be generated on time scales of 10 8 -10 9 y. The convective geotherms in Figs. 1 and 3 from the expressions in Appendix B are everywhere within 100°C of those from the plate model (Fig. 3(b) and (c)), and only differ by this value at the base of the 'Iithosphere'. It is presumably for this reason that the plate models are so useful. Though the upper part of the thermal boundary layer has a linear MELT GENERATED BY LITHOSPHERIC EXTENSION 633 temperature gradient because it is moving slowly relative to the mechanical boundary layer, it is nonetheless moving and will not accumulate isotopic variations. In studies of melting by lithospheric extension it is of considerable importance to use an accurate initial temperature profile because most of the melting occurs in material which initially is within or just above the thermal boundary layer. The total volume of melt generated and the amount of melt produced within the mechanical boundary layer are sensitive to small changes in the temperature structure. All the discussion in this section has been concerned with the oceanic lithosphere, which, in contrast to that of the continents, is now relatively well understood. The question of whether the continental lithosphere is thicker than that of the oceans is still controversial. There seems no reason to believe that all continental crust is underlain by thick lithosphere. A powerful argument against such a proposal is the thermal subsidence of sedimentary basins such as the Michigan Basin (Sleep, 1971) and the North Sea (Barton & Wood, 1984) and its explanation in terms of lithospheric extension (McKenzie, 1978). The thermal time constant derived from the subsidence is indistinguishable from that obtained from ridge subsidence, and therefore requires the same lithospheric thickness. In continental areas which have not undergone extension the thermal structure must be estimated from the surface heat flux, which is a very uncertain enterprise. Sclater et al. (1980), argue that the continental lithospheric thickness need be no thicker than that of old oceans. Pollack & Chapman (1977) disagree, and believe the lithospheric thickness beneath shields is considerably greater. The other type of observation which can be used is seismic velocity profiles, though it is not straightforward to relate either VP or Vs to the temperature. A variety of studies over the last 20 y have concluded that lateral velocity variations exist which extend to depths of at least 200 km and which correlate with the surface geology of continents. Convincing lateral variations at depths greater than 100 km are confined to shields, which have higher V? and Vs velocities than surrounding regions (Grand & Helmberger, 1984; Rial et al., 1984). It is not yet clear whether the mantle velocity structure of Archaean and Proterozoic Shields are the same. There is no doubt that lateral variations of velocity exist throughout the upper mantle to depths of 700 km. Jordan (1978) has repeatedly argued that these require the existence of a lithosphere as thick as 400 km beneath some continental regions. In the light of the thermal evolution of sedimentary basins it seems improbable that such a thickness is associated with all continental areas. Even beneath shields such a great thickness is not easily reconciled with conductive heat transport. The best evidence for a thickness of at least 180 km for the mechanical boundary layer beneath Archaean Shields comes from isotopic studies on diamond inclusions (Richardson et al., 1984). Mantle convection As already explained, the principal aspect of mantle convection which is of concern to us is the magnitude and length scale of the variations of potential temperature within the convecting upper mantle. More complicated questions, such as the planform of the circulation, certainly affect the distribution of intraplate vulcanism. Our need here is, however, simpler, because we are concerned only with melting resulting from plate tectonics. The convecting mantle can then be regarded as a large source of material of constant potential temperature. The length scale of the variations is not of interest for ridge upwelling. Beneath old lithosphere, however, the vertical length scale of the potential temperature variations within the convecting region is the thickness of the thermal boundary layer, which must be estimated if accurate initial temperature profiles are to be calculated. 634 D. McKENZIE AND M. J. BICKLE The magnitude Ad and length scale <5 of variations of potential temperature can be estimated using a modification of the convective boundary layer theory of Turcotte & Oxburgh (1967). 2=AR-v* (7) where the Rayleigh number R is (9) A and B are constants which are best determined by numerical experiments. In these expressions d is the thickness of the convecting layer, K the thermal diffusivity, k the thermal conductivity, v the viscosity of the solid material of the upper mantle. F is the heat flux/unit area through the layer. Substitution of (9) into (7) and (8) shows that both d and Ad are independent of d. With the exception of v, the appropriate values of the variables in equations (7)-(9) are reasonably well known. It is therefore useful to rewrite (7) and (8) as /v Y A0=F (11) V'o/ where vo = 2x 1017 m2 s~' and A and B'do not depend on v. Though (10) and (11) show that the dependence of <5 and AQ on v is weak, estimates for the value of v vary between 2 x 1017 and 4 x 1015 m2 s~' and correspond to a variation in v1/4 of about a factor of 3. Furthermore (7) and (8) are only valid if the viscosity within the convecting region is constant: a condition which is certainly not satisfied. Unfortunately, despite several attempts, no satisfactory theory yet exist which can be used to obtain expressions like (7) and (8) when the viscosity is a function of temperature. A variety of arguments suggest that the viscosity of the thermal boundary layer is at least two orders of magnitude less than that of the principal part of the upper mantle (see Craig & McKenzie, 1986). A reasonable estimate for this viscosity is 4 x 1 0 " m 2 s ~ ' leading to values of <5 and A9 of about 30 km and 200°C respectively. These are probably appropriate for the thermal boundary layer at the upper boundary of the upper mantle. The viscosity of the bulk of the upper mantle must be considerably greater and there has been general agreement for more than 50 y that a value of around 2 x 1017 m 2 s~' is more appropriate. Substitution into (7) and (8) leads to estimates of S and Ad of 80 km and 400 °C. These values are sufficient to account for the magnitude and extent of the bathymetric and gravity anomalies associated with features such as the Hawaiian Swell and the Cape Verde rise (Courtney & White, 1986), which are believed to be the surface expression of a hot jet of rising mantle material. Such jets will control the potential temperature of mantle material entering the upwelling region beneath spreading ridges as ridges pass across the convecting system. These variations of potential temperature then produce variations in the quantity of magma generated by the upwelling. In the case of sedimentary basins the quantity of melt generated is determined by the initial geotherm. The principal region of interest lies within the thermal boundary layer, MELT GENERATED BY LITHOSPHERIC EXTENSION 635 which, as Fig. 3(b) shows, will be the first to reach the solidus. A suitable parameterization of the temperature structure of the thermal boundary layer was proposed by Richter & McKenzie (1981). The modifications required to match this parameterization to the rigid mechanical boundary layer above and the adiabatic interior are described in Appendix B. Average melt composition O'Hara (1985) in particular has emphasized that the composition of magma erupted at the surface is a rather complicated weighted average of the melt produced at depth. A useful approximation is to imagine the melting and extraction processes beneath a ridge as occurring in two steps. A vertical prism of mantle of the same thickness as the oceanic lithosphere is first brought up beneath a ridge so that its top is at the sea floor. During this step the material is allowed to melt, but no movement is allowed between the melt and matrix. Then all the melt is removed to make the oceanic crust. This scheme is not realistic, because the melt fraction present in the mantle is never likely to exceed 2 or 3%. But it is the only scheme which allows the melt composition to be calculated from the laboratory experiments. To understand the approximations involved in such a scheme requires certain quantities to be defined. The most basic of these is the composition of the melt which is added to increase the fraction of melt from X to X + dX by transfer of material from the solid to the melt phase. This composition will be referred to as the instantaneous melt composition and will be written c. If c(X, P) is known, the average composition C of all the melt which has been generated from any particular element of solid can be obtained by integration along the melting path C(*) = i | c(X')dX' A Jo (12) or c = ~(XC). (13) dX When melting occurs at constant pressure C will be referred to as the point average composition, because it is the average composition of the melt generated from any point as the temperature increases (Fig. 4{a)). Finally the average composition of all the melt generated depends on the weighted average of C over the melting interval 0->h V=\ XCdzl I Xdz = \ dz\ Jo ciX')dX' Jo Xdz. (14) / Jo # will be referred to as the point and depth average composition. Any average constructed using (12) will be referred to as the point average, and using (14) as the point and depth average. The relationship between these compositions for the melting scheme described above are illustrated in Fig. 4(a). The compositions <<?, C, and c all depend on the way in which melting occurs. The difference between them can be examined using the standard expressions for the trace element concentration in the melt for a material undergoing batch or Rayleigh melting CB= DD(\+X(}r-l))2 °5) (b) (a) 0-3 c c \ n _g 0-2 a Na,0 \ c 2 -c 0-1 a 2 n 50 Depth km 100 m Z N m 40% z a 2 30% 22% X 0-2 CD 20% o rm 18% 0-1 16% 40 60 SO 100 Depth km FlG. 4. (a) The instantaneous melt composition c(X, z) is that of the melt which is added to increase the melt fraction from X to X + dX. The point average C(z) is the average melt composition produced from one element of solid at a depth z. The point and depth average is the average melt composition generated in the melting region, (b) The instantaneous melt composition for Na 2 O calculated using D = 0169 and c 0 = 0-547 from Table A1 (a) for Rayleigh and batch melting, (c) Contours of cB(X, z) for MgO calculated using Table Al(a), with Xt(z) for two potential temperatures, (d) A sketch of contours of c(X, z) for MgO for Rayleigh melting with 7"P= 1480°C. Both the contours and X(z) assume cst = cB and Xt = XB and are therefore not accurate. Typical regions from which melt is extracted are shown dotted. MELT GENERATED BY LITHOSPHERIC EXTENSION 637 and „ 0/, v\(i/D~D i\a\ where c 0 is the bulk concentration of the trace element and D is the distribution coefficient. A subscript B denotes batch and R denotes Rayleigh melting. The usual expression for CB can be obtained by integrating (15). Curves for cB and cR are illustrated in Fig. 4{b) for Na 2 O. The melting processes beneath ridges are not likely to leave more than 2-3% melt in contact with the residue at any time (Ahern & Turcotte, 1979; McKenzie, 1985), and therefore resemble Rayleigh rather than batch melting. Because cR # cB, batch melting experiments will not in general provide accurate estimates of the melt composition. A further difficulty concerns the calculation of (€. No problem arises for the scheme in which batch melting occurs with no separation, followed by separation with no interaction with the matrix. Then, as Fig. 4(c) shows for MgO, cB = cB(X, z), where z is the depth, and is independent of the melting path. Two curves are shown for X(z) for two potential temperatures, and calculation of l€B is straightforward. If, however, extraction of melt occurs during the melting process, the problem of calculating "<f is more difficult because the integral (14) must be carried out along each matrix stream line. This difference is illustrated in Fig. 4(d) where cR has been assumed equal to cB in Fig. 4(c). The dotted regions show typical regions from which melt is extracted. Clearly the point average is now unrelated to any melt which might be produced. The last difficulty concerns X{z). The discussion in the next section is concerned with estimating XB(P, T). It is, however, likely that the melt fraction produced will be strongly affected by whether or not the melt is extracted, since the change of the activity of some component in the melt produced by the addition of a fixed mass of solid must depend on the melt fraction present. It is therefore unlikely that (xB-xR)<<L (17) But it is not obvious how XR can be estimated from existing batch melting experiments. Parameterization All the calculations described below require continuous descriptions of temperature, melt fraction and composition, and considerable effort is involved in obtaining suitable functions. The most obvious and least useful method of proceeding is to use linear interpolation between experimental results. To obtain a general description of any physical process it is important to use an understanding of the physics of the problem to suggest a functional form for the parameterization, and that the function used should contain some free parameters which can be adjusted to fit the experiments. A well known example of this process is the use of an activation energy and a frequency factor to describe the rate of a chemical reaction. The functional form is soundly based on statistical mechanics, but, except in the simplest cases, the two constants must be obtained from experiments. A similar approach is adopted here. For instance the variation of melt fraction X with temperature and pressure must clearly change from 0 on the solidus to 1 on the liquidus. Any functional form must satisfy these conditions. Imposing this constraint leads to a polynominal form for X{P, T) and all the experiments at different pressures and temperatures can be fitted with the simple polynominal with only two constants. In all cases the constants were determined by minimizing the absolute value of the difference between the observed and 638 D. McKENZIE AND M. J. BICKLE calculated values, rather than its square. Minimization of the square gives greatest weight to the most discrepant, and therefore least relevant, experimental results. The implementation of Powell's algorithm given by Press et al. (1986) was used throughout. The main results of this section Much of the previous discussion is only indirectly relevant to the melting problem beneath regions undergoing extension, and it therefore is worthwhile to collect the results we need later. Perhaps the principal one is that upwelling is generally sufficiently fast for heat conduction to be neglected. It is therefore essential to calculate both the melt fraction and the geotherm together. To a good approximation this can be achieved by maintaining the entropy of the melting system constant during the upwelling. This approximation reduces the somewhat intractable general problem to the integration of a nonlinear ordinary differential equation, which can be integrated by standard methods (see Appendix D of McKenzie, 1984a). The melt fraction generated is then controlled by the heat content of the solid mantle entering the upwelling zone. Because the solid mantle is slightly compressible its temperature is not uniquely related to its heat content. It is therefore convenient to define a potential temperature, the temperature the material would have at the surface if melting does not occur, which does satisfy this condition. Variations in potential temperature are controlled by convection and are probably as large as 200 °C between the hot rising jets and the surrounding upper mantle. These variations must be reflected in variations in the amount of melting produced by the upwelling as ridges wander across the convecting system. The ridge motion has little influence on the convective geometry and ridges are not geometrically related to convective upwellings. The melt generated during continental extension depends on the initial temperature distribution, established over hundreds of millions of years. Like that beneath ridges, the upwelling is often adiabatic to a good approximation. Hence it is important to start with the best possible estimate of the initial temperature variation. It is therefore necessary to understand the physical processes which control steady state continental geotherm, and in particular the distinction between the mechanical boundary layer, in which large scale isotopic heterogeneities can be produced over geologic time, and the thermal boundary layer, where they cannot. The combined thickness of both boundary layers is about 100 km beneath the North Sea and Michigan Basins, whereas the thickness of the mechanical boundary layer alone must be more than 180 km beneath Archaean Shields. This variability must influence the melting processes. Finally the average composition of the magma produced in any environment is a complicated function of the melt compositions which exist at depth. 3. MELTING UNDER PRESSURE Calculations of the volume and composition of melt require knowledge of the variation of the melt fraction with pressure and temperature. The simple functional forms used by McKenzie (1984a) were not based on a detailed analysis of the experimental results, and such a study is required if detailed comparisons are to be carried out. No general theory of melting exists which can be applied to the melting of a multiphase multicomponent mantle material at variable pressure in the presence of variable amounts of solid solution between different phases. The traditional approach to such problems uses chemical thermodynamics and the concept of activity, and represents the results in terms of MELT GENERATED BY LITHOSPHERIC EXTENSION 639 multidimensional phase diagrams. A great effort has been devoted to this problem, but the information required to deal with a system as complicated as the mantle does not yet exist. We therefore adopted a different approach which is both more empirical and more quantitative than that commonly used, since we needed to be able to calculate both the quantity of melt produced and its composition. For numerical convenience we wished to use analytical expressions wherever possible. In the absence of any general theory of melting all we could hope to do was to parameterize the results of experiments carried out on rocks of given compositions. The approach was only possible because of the number of careful experiments which have now been carried out on ocean ridge basalts, garnet peridotites and rocks of similar composition. Though the expressions obtained in Appendix A are restricted to the melting of peridotites, the approach is more general, and can be used wherever suitable experimental results of sufficient quality are available. The approach we adopted consists of three steps. We first obtained analytic expressions for the variation of the solidus T, and liquidus TY temperatures with pressure which agreed with the experimental observations to within reasonable estimates of the likely experimental errors (Fig. 5). We then obtained the melt fraction, X, as a function of pressure and temperature. Finally we parameterized the melt composition as a function of X and pressure. utdus 1900 1700 T ° C 1500 A liquid o solid & liquid • »olld 1300 1100 4 PGPa FIG. 5 Experimental determinations of the solidus and liquidus of garnet peridotite. The curves were determined by minimizing | rob> — Talc| and are generated from equations (18) and (19). The data points for the solidus are from Ito& Kennedy (1967), (0,—, 1150), (2.04, 1300, 1350), (218, 1200,—), (2-31, 1320,— \ (2-58, 1350,—I (2-99, 1400, 1500), (408,1550,1600), Green & Ringwood( 19676), (1-8,1300,1360), (2-25, 1400,15OO),(2-7,1450,1500), (2-9,1500, —),(31, 1500, 1550), (3-6, 1570, 1660) Jaques & Green (1980), (0, —, 1170), (0-25,—, 1100), (0-2, —, 1150), (0-5, —, 1200), (0-675,—, 1200), (0-9,—, 1200), (1,—, 1250), (15,—, 1350), (0-2,—, 1200), (0-5,—, 1200), (1,—, 1250), (1-5,—, 1350), Stolper (1980), (1, —, 1250), (1-5, —, 1350), (2, —, 1400), Harrison (1981), (3 5, 1575, 1580), Takahashi & Kushiro(1983),(0, 1100, 115OX(O-5, 1175, 1200), (0-8, 1200, 1225), (11, 1200, 123OH15, 1275, 1300), (2-, 1350,1375), (2-5, 1375, 1400), (3, 1475, 1500), and Takahashi (1986), (0, 1100, 1150), (0-5, 1175, 1200), (1, 1250, 1275), (1-25, —, 135O),(l-5, 1350,1400U2,1350,1400M3,1500,1550), (3-5,1600,—),(5,1600,1700),(6,1750,1850), (7,—, 1800), (7-5, 1800, 1900). The first entry inside the brackets is the pressure in GPa, the second the highest temperature in °C at which no melt was present, and the third the lowest temperature at which melt was present. If either the second or the third entry is marked with —, the appropriate temperature bound cannot be determined from the experiment. The data points for the liquidus are from Takahashi (1986) (0,1600, —), (1 -5,1800,1850), (3,1900,1950), (5,2000, —), (7-5, 1900, 2000), where the second entry is the highest temperature at which solid was still present, and the third entry the lowest temperature at which there was no solid. 640 D. McKENZIE AND M. J. BICKLE The solidus temperature is both better known and more important than 7j, and different groups have obtained results which agree well. The expression used was -968xl(T4exp(l-2xl(r2(7;-1100)) (18) C where P is the pressure in GPa and T, the solidus temperature in C. Because the expression gives P(T,\ T,(P) is obtained by numerical iteration. This expression fits the observations reported by the authors listed in the caption to Fig. 5 with a mean error of 6°C, and it is not likely that the temperature measurements themselves are of this accuracy. The expression used for the liquidus temperature Tt in °C was T, = 1736-2 + 4-343P + 180 tan-'(P/2-2169). (19) This expression fits the liquidus temperatures with a mean error of 7°C. The observations and empirical expressions are shown in Fig. 5. The next question concerns the fraction by weight A" of a rock which melts at a given pressure and temperature. To determine X{ T, P) directly is difficult. It is easier first to use the requirement that X = 0 when T=T, and =1 when 7"= 7j, by defining a dimensionless temperature 7" r Then from the definitions T, and Tt, X(T') must pass through ( - 0 5 , 0 ) and (0-5,1). The general polynominal which satisfies these conditions is X-0-5 = T + {T'2-0-25)(ao + a,7" + a2T'2. . .). (21) The observed values are plotted in Fig. 6. With the exception of Mysen & Kushiro's (1977) results, which were affected by quench overgrowth (Takahashi & Kushiro, 1983) and were not used or plotted, there is good agreement between different authors. What is surprising is that there is no evidence of any variation of X(T') with pressure. Using two coefficients only gave ao= 0-4256 a, =2-988 (22) with a mean error of 3%. The inclusion of more coefficients and linear variation with pressure did not produce an appreciable improvement. Since it is not likely that the observations themselves are accurate to 3 % no such improvement would be expected. The curve calculated from equation (21) using the constants (22) illustrated in Fig. 6 has the general form expected for the melting of garnet peridotite. The rapid increase of X with increasing T near A" = 0 corresponds to the cotectic melting of olivine and two pyroxenes. As will be shown below, the clinopyroxene is exhausted when X ~0-25. The rapid increase as 7" ->0-5 corresponds to the melting of olivine. The absence of any pressure effect means that X depends only on 7". Of course X = X(T, P) because the solidus and liquidus temperatures are pressure dependent. The parameterization of T,(P), Tt(P) and X(T') are simple and convenient, and easily included in calculations like those of McKenzie (1984a) and Furlong & Fountain (1986). The results obtained by integrating the equations in Appendix D of McKenzie (1984a) are illustrated in Fig. 7, but are labelled with the potential temperature of the geotherms rather than with the solidus intersection temperature used in McKenzie (1984a). The principal uncertainty in these calculations is still the entropy change AS on melting. A value of 250 J k g " ' °C~' was used to generate the curves in Fig. 7. The principal difference between MELT GENERATED BY LITHOSPHERIC EXTENSION 641 10r 0-8 0-6 0-4 + 0<SP Z 0-5 0-2 O 0 - 5 < P « 1-5 * 1-5 < P 0-0 -0-5 -0-3 00 0-3 0-5 r FIG. 6. Melt fractions of a rock with the composition of a garnet peridotite plotted as a function of T = T,-T, where 7J and T, are the hquidus and sohdus temperatures. Data points are from Bickle et al. (1977), Arndt (1977), Bickle (1978), Green et al. 1(979), Stolper (1980), and Jaques & Green (1980). The melt fractions corresponding to Stolper's experiments were calculated from the composition of the olivine and orthopyroxene by requiring the bulk composition to contain 44-48% SiO 2 and 39-22% MgO. Where the bulk composition differed from this composition in the other experiments olivine and orthopyroxene were added or subtracted to adjust the MgO and SiOj compositions to these values. The curve was obtained by minimizing \Xalc — Xob, | and calculated using equation (21) with the constants (22). Fig. 7 and the earlier results of McKenzie (1984a) concerns the behaviour between X = 0-3 and X = O6 (see his Fig. 13). This difference has little effect on the total amount of melt generated below a given depth, shown in Fig. 7(b), which therefore is now well determined. In order to generate a melt thickness of 7 km, corresponding to the average thickness of the oceanic crust, the potential temperature must be 1280°C. This calculation assumes that all the melt is extracted, and that adiabatic melting continues to the surface rather than to the Moho. Once AS is better known more accurate calculations will be possible. The temperatures of the melt generated in this way range from 1300°C on the solidus at a depth of 45 km to 1200°C at the surface, and average 1232°C. The average depth of generation is 15 km and melt fraction is 0135. These numbers are somewhat dependent on the choice of AS. A value of 400 J k g " ' °C~' requires a potential temperature of 1300°C to generate 7 km of melt and the melt temperatures range from 1325°Cat 51 km to 1191 °C at the surface, and the average melt fraction is 0114. These estimates are in general agreement with experimental determinations of the Hquidus temperatures of oceanic basalts (Tilley et al, 1972; Bender et al, 1978; Fujii & Bougault 1983). The answer to one of the questions posed by Carmichael et al (1974), quoted in the introduction, is now clear. Magmas are not in general erupted at temperatures much greater than 1200°C because the average potential temperature of the mantle is 1280°C. Because the thickness of the oceanic crust varies little, most of the upper mantle apart from the hot rising jets must have approximately the same temperature. Hence the constancy of the eruption temperature. The potential temperature of the hot jets is probably about 200 °C D. McKENZIE AND M J. BICKLE 642 Ib) Depth km 100 2 4 Pressure GPi FIG 7 (a) Adiabalic decompression paths calculated using the equations given by McKenzie(l984a) Appendix D, a fourth order Runge-Kutta scheme and A5=250Jkg-'°C"'. The curves are labelled with their potential temperatures, and entropy is conserved to I part in 10* during the numerical integration. The curves between the solidus and the liquidus are labelled with the melt fraction by weight. (b)The total thickness of melt present below a given depth plotted as a function of depth, calculated by integrating the volume of melt present in (a). greater than tjiat of the mantle interior, or about 148O°C. As Fig. 7(b) shows, the resulting melt thickness is 27 km, in agreement with the crustal thicknesses of the aseismic ridges. These structures are produced when a hot jet coincides with a spreading ridge. Iceland is the best known example of such a coincidence. The measurement of crustal thickness is at present the most accurate method of mapping variations in mantle potential temperature. Originally the principal purpose of this project was to use convective geotherms calculated for plate interiors and X = X(T') to calculate the melt volumes generated by lithospheric stretching. But the unexpectedly simple relationship between X and 7", and the absence of any pressure effect, suggested that it might be possible to calculate the composition of the melt as well as its volume. It seemed likely that the melt composition would be principally controlled by the quantity produced. Since X = X{T'), this suggestion was easily tested by plotting the abundance of various oxides as functions of 7". The results of so doing are illustrated in Fig. 8. The plots labelled (b) are those for which the melt fraction is known, either because it was determined in the original experiment or could be calculated from the published information. These experiments are the 38 used in Fig. 6. The plots labelled (a) contain results from the larger number of experiments which determine the melt composition as a function of T and P. Since 7" can easily be calculated, these experiments can also be used to test whether the composition is a function only of X and hence of 7". The results are striking. With the exception of SiO 2 , FeO, and MgO, the melt compositions are functions only of 7" MELT GENERATED BY LITHOSPHERIC EXTENSION 643 to within experimental error. SiO 2 shows a systematic decrease and both MgO and to a lesser extent FeO an increase in concentration with increasing pressure. Most experimental results are available for values of T' between —0-5 and —02, which, as Fig. 7 shows, is the region of most interest from the point of view of MORB generation. Despite the obvious variation between the oxide concentrations in the melt and T', the experimental results were not easily parameterized. Appendix A contains description of how this was achieved. The melting region O ^ X ^ l was divided into three regions with boundaries at Xx and X2, where the composition CB (but not cB) of the batch melt was required to be continuous. In each region the oxide composition of the instantaneous melt was required to satisfy Rayleigh's Law or to be zero. Hence the total mass of any oxide in the melt could never decrease with increasing X. For all oxides except FeO, MgO, and SiO 2 , the concentration in the instantaneous melt was taken to be zero when X > X l and no pressure dependence was allowed. The concentration of MgO and FeO was required to vary linearly with pressure at constant A" up to a pressure of 3-5 GPa. The SiO 2 concentration was obtained by subtracting that of all the other oxides from 100. Those oxides, such as • o < P < o-s O 0-5 < P < 1-5 A 15 < P -0-2 00 0-2 0-4 T' (b) * 0 < P « 0-5 O 0-5 < P * 54 » 1-5 < P 50 SIO 2 % 46 -0-4 -0-2 0-0 0-2 T' 0-4 1-5 644 D. McKENZIE AND M. J. BICKLE MELT GENERATED BY LITHOSPHERIC EXTENSION CO CJ | S* O , 645 646 D. McKENZIE AND M J. BICKLE 2 # 2 a? o (- MELT GENERATED BY LITHOSPHERIC EXTENSION 647 (a) 08 K2O 0-2 -0-4 00 -0-2 0 2 0-4 r (b) 0-8 0-6 K2O % 0-4 0-2 00 -0-2 r FIG. 8. Compositions of melts as functions of 7" given by equation (20). (b) shows the compositions of 38 experiments for which the melt fraction was measured (Fig. 6), with the exception of K 2 O for which only Jaques & Green's (1980) results are plotted, (a) shows results from these 38 experiments together with those from Takahashi & Kushiro (1983), Takahashi (1986), and Fujii & Scarfe (1985), for which X was not determined, 91 in total. Crosses mark compositions from experiments carried out with P<,Q-5GVa-, circles those with Q-5<P<, 1-5 GPa, and triangles those with P > 1-5 GPa. Where three curves are shown they are the calculated compositions at pressures 0 (not labelled), I and 2 GPa. The parameterization takes C to be a continuous function of pressure and melt fraction (see Appendix A). All the results for K 2 O are shown in (a). The solid lines are calculated from the two parameterizations obtained by minimizing \Cob, —C^ | see text and Appendix A. The curve in series (b) for K 2 O was obtained from fitting Jaques & Green's (1980) results only. MnO, Cr 2 O 3 , and P 2 O 5 , for which the data were insufficient were included as a single concentration independent of melt fraction and pressure called 'the rest'. The constants of the parameterization were then obtained by minimizing G C = X>JC ob ,.-C c . lc .| (23) n where the sum was taken over SiO 2 , TiO 2 , A12O3, FeO, MgO, CaO, Na 2 O, K 2 O, and 'the rest'. Values of wH of 1 were used for the major elements, iron and 'the rest', 20 for TiO 2 , 10 648 D. McKENZIE AND M. J. BICKLE for Na 2 O, and 30 for K 2 O. These weights were used to force the adjustments required to match the overall concentrations into the parameters of the major elements, and so to fit those of the minor elements as well as possible. The concentration of TiO 2 in Jaques & Green's experiments is about a factor of four greater than that of all other experiments because they used a Hawaiian basalt to construct their pyrolite. Their results for this oxide were therefore not included (but see Appendix A). Three sets of constants were then determined (Appendix A, Table Al), two ((a) and (c)) for the experimental results shown in Fig. 8(a), and one, (b), for the results in Fig. 8(b). Where X was not given it was obtained from (21) and (22). The minimization run for 8(a) included all the K 2 O observations rather than just those of Jaques & Green (1980). After some experimentation 22 parameters were used for the minimizations, and the concentrations calculated for the two cases (a) and (b) are shown in Fig. 8. The difference between the two minimisations (a) and (b) is not large, and is most obvious near T' = —0-5. Few of the experiments in Fig. 8(b) involve small melt fractions, whereas those in 8(a) include a number of runs made by Takahashi & Kushiro (1983) where the calculated melt fraction was 10% or less. These results constrain the behaviour as T' -<• — 0-5, and therefore the parameters from 8(a) are to be preferred. They also involve more than twice the number of experiments. The values of A^ and X2 were two of the parameters determined, giving values of 0-245 and 0438. Xt corresponds to the removal of all oxides except FeO, MgO, and SiO 2 from the residual solid, and hence to the elimination of clinopyroxene. It is shown as 'cpx-out' on later plots. X2 has no physical significance. It is required simply to provide enough variables to fit the observations and should not be interpreted as 'opx-out'. When more experimental results are available with small values of X it will probably be necessary to add a third boundary X3 at which the aluminous phase disappears, with X2>X^. However, as Fig. 8 illustrates, the scatter in the observations is still too large at present for such an effect to be resolved. The validity of the parameterization does not depend on the assumptions underlying the derivation of Rayleigh's Law being satisfied. The expressions used are simply convenient functional forms for parameterization, and their validity should be judged only by how well they fit the observations. Since 22 parameters were used tofit750 observations, the system is strongly overdetermined. Formally the bulk distribution coefficient, D, is given by D = n/{n + 1), but this expression should only be used as an estimate of D for TiO 2 , Na 2 O, and K 2 O. The values in Table Al suggest TiO 2 is the least compatible. Figure 9 shows a detailed comparison of the error distributions for the major and two of the minor elements (the data for K 2 O are inadequate, see Fig. 8(a)), together with normal distributions with the same mean absolute error. In the case of the major elements the mean difference between the observed and calculated compositions is 106%. Since the compositions were determined using electron probes to a relative accuracy of 2%, or 50 ± 1% in the case of SiO 2 , the difference between the calculated and observed compositions is probably comparable to the observational error. The agreement is less good for TiO 2 and Na 2 O because the bulk concentrations of these elements were intentionally varied in some of the experiments. Nevertheless the mean error was only 027%. It is also of interest to compare the melt composition obtained from the parameterization when X = X i with that estimated by Stolper (1980) for cpx-out. His estimates are shown in Figure 10(b) in the projection onto the Ol-Di-Qz plane from PI suggested by Walker et a\. (1979). The calculated compositions in Fig. 10(a) agree excellently with those of Stolper, and confirm his argument that cpx-out does not lie within the MORB field at pressures of between 1 and 2 GPa (see section 4). O'Hara (1968a,b) argued that cpx-out would reenter the tholeiitic field in Fig. 10 as the pressure increased beyond 3 GPa. No such tendency is MELT GENERATED BY LITHOSPHERIC EXTENSION 649 9 O 2 , AI 2 O 3 , FeO, MgO, CaO mean error 1-06% '•xp* (b) • TK> 2 , Na2O 30 mean error 0-27% 20 • • 10 • • • n 0-2 0-4 0-6 08 FIG. 9. Distribution of errors for major and minor elements obtained from the parameterization (a) and experiments shown in 8(a). 399 observations were used in (a) and 26 were outside the range of the plot. The corresponding figures for (b) were 145 and 4. The bin sizes s were 0 2 % for (a) and 006% for (b). The Gaussian distributions were calculated from Aexp(-x2f2o2) where A = INsjnx F and N is the number of points used whose mean value is x. The values of .x were 1-06% for (a) and 0 2 7 % for (b). observed in the experiments discussed here, but few of them were carried out at such high pressures. Tliese tests show that the parameterization is a compact and accurate description of the experimental data. However we wish to use it to calculate mantle melt compositions, and D. M c K E N Z I E A N D M. J. B I C K L E 650 a FIG. 10. (a) Experimental results for which the observed or calculated A'^A r 1 =O245 (cpx-out) plotted in the manner proposed by Walker et al. (1979, caption to their fig. 2). When the points fall inside the triangle this plot is identical to one produced from the CIPW norm if Or, Ab, and An are plotted as PI. The curves are calculated from the parameterization (a) and are marked with the pressure in GPa and end at X =0-35. (b) is taken from Stolper (1980) for comparison. The symbols correspond to ranges of pressure in GPa The ellipses show the projections of the 2<r 8 dimensional ellipse of the error of the mean for the parameterization, (oblique lines), and for an individual electron probe analysis of the Smithsonian data set (Presnall & Hoover, 1984). therefore desire to test whether the parameterization is also satisfactory for this purpose. Though the most convincing test is the agreement between the MORB compositions and those calculated from the parameterization which is discussed in detail in the next section, a simpler test can also be used for CaO, A1 2 O 3 and TiO 2 whose abundance ratios can be estimated using cosmochemical arguments (Ganapathy & Anders 1974; O'Nions et al, 1981) to be CaO = 0-806 A12O3 and CaO = 15 9. TiO 2 These oxides are all in the melt when X> Xu and their concentration ratios then are CaO = 0781 A12O3 and CaO = 15-2. TiO 2 Since these element oxides are among the least volatile, their abundances are believed to be constant in meteorites and the silica rich planets. Because A12O3 and CaO are very abundant in the mantle, and TiO 2 is not concentrated in the continents, their abundance in the mantle has scarcely been affected by continent formation. Another test of the parameterization is the ratio (MgO/FeOJi^^MgO/FeO),^,,, which lies between 0 2 and 0-3 for the conditions under which the experiments were conducted. M E L T G E N E R A T E D BY L I T H O S P H E R I C E X T E N S I O N 651 We did not expect to be able to produce a single set set of parameters that agreed with all experimental observations. Since we have made little use of the conventional phase diagram approach to such problems, solid solution and other difficulties which complicate its use do not affect our parameterization. We will now use this approach to calculate the expected compositions of MORBs. 4. MELTING BENEATH RIDGES Adiabatic decompression Seafloor spreading generates about 20 km 3 a" 1 of MORB, which is the dominant volcanic rock type on the earth. The most obvious interpretation of the observations is that this melt is a primary melt generated directly from the mantle by adiabatic decompression, and only modified by fractional crystallization within the crust. We demonstrate below that this model can account for almost all the observations, and discuss the problems with the views of O'Hara and others below. The term 'primary' will be used here to describe any melt that is in chemical equilibrium with the solid with which it is in contact. Since melts must move with respect to the matrix from which they are separating, the distinction between primary and fractionated melt is only meaningful when melt and solid residue cease to interact chemically. Such interaction must cease when the melt crosses the Moho, and therefore only at shallower levels is there necessarily a distinction between primary and fractionated magmas. The melting process must be able to generate enough melt to produce the average thickness of oceanic crust of about 7 km, increasing to about 25 km where the ridge crosses a hot rising jet. These thicknesses are produced by adiabatic upwelling of the mantle with potential temperatures of 1280°C and 1480 °C respectively. The mean composition of this melt should agree with that of the oceanic crust, as should also the range of compositions. The calculated compositions of the point averages for a potential temperature of 1280°C are shown as stereographic pairs in Fig. 11, together with a variety of dredged glass compositions which are either ne-normative or have an MgO concentration > 9 % . These glasses are least likely to have undergone low pressure fractional crystallization. Figures 12 and 13 show projections from the plagioclase and diopside apices of the point averages corresponding to several potential temperatures, together with those of the glass compositions selected by Elthon. The ellipses are projections of the error ellipses for the observations and for the mean composition from the parameterization in Fig. 8(a). Curves from the parameterization in Fig. 8(a) are shown in Fig. 12(a) and 13(a), those from 8(b) in 12(b) and 13(b). The agreement between the curves for a potential temperature of 128O°C in 12(a) and 13(a) and the glass compositions is good. Indeed it is probably better than would be expected from the size of the error ellipses. No extensive olivine fractional crystallization is required to generate these glass compositions from the point average melt compositions. An important feature of the 1280°C melting curves is that they do not quite exhaust the clinopyroxene in the matrix. The result is in excellent agreement with Dick & Fisher's (1984) work on abyssal peridotites. They found that the enstatite in these rocks was saturated with diopside, but that there was little free diopside. They argued that calcium had been removed by melt separation. Since the potential temperature of 1280°C was chosen to generate 7 km of melt, it is encouraging that the resultant depletion is in agreement with Dick & Fisher's observations. Table 1 shows the point and depth average for a potential temperature of 1280°C, together with various average compositions. 652 D. M c K E N Z l E AND M. J. BICKLE (a) (b) n t (C) FIG. 11. Stereographic pairs with rotation angles of 6°, showing Elthon's glass analyses (1987 and pers. comm, 108 analyses) and those from Melson et al. (1977 and pers. comm.) that either have MgO>9% or are ne-normative, plotted (a) using Walker et al.'s (1979) projection with tetrahedron faces being 01=0, Di = 0, Pl = 0, and Si = O, (b) using O'Hara's (1968a) CM AS with faces C = 0-15, M = 0-10, A = 0-15, and S = 0-45, and (c) Walker el al.'s (1972) projection, equation (4.1), with faces An =0-20, Di = 0, Ol = 0-20, and Si = 0-30. The line is calculated for adiabatic melting with Tr= 128O°C. These plots were produced by M. Spiegelman. 653 M E L T G E N E R A T E D BY L I T H O S P H E R I C E X T E N S I O N GlasMs FIG. 12. Projections from PI into the Ol-Di-Qz plane of Walker el a/'s (1979) tetrahedron, using molar concentrations. The crosses represent glass compositions collected by Elthon (1987) with MgO>9-0 %. The curves in (a) are point average compositions calculated from the parametenzation shown in Table Al(a), whereas those in (b) are from that in Table Al(b). The dashed line marks a line of constant melt fraction corresponding to 0245 (cpxout). In (b) cpx-out occurs at X = 0265. The small open circles on the curves mark melt fractions of 005, O15, and 035, with the arrows pointing in the direction of increasing X. The solid dots inside larger circles show the point and depth averages. The ellipses are as in Fig. 10. FIG 13. As for Fig. 12 but projected from Di. TABLE 1280°C Mean Elthon's glasses SAVE (Pallister, 1984) Mean Melson el al.'s glasses, M g O > 8 % Mean of all of Melson et al.'s glasses Mean IGCP basalts Cumulus gabbro (Pallister, 1984) SiO2 TIO2 51-89 49-37 511 49-87 5113 49-64 49-9 O92 084 060 1-01 1 69 1-44 031 1 14-57 16-31 16-6 1617 1508 15-48 16-9 FeO MgO CaO Na20 K20 8-53 8-85 7-2 9-23 1050 9-77 6-0 1027 9-50 9-2 8-78 7-09 7-65 105 11O1 12-38 12-8 12-20 I1O8 11-67 14-6 216 2-15 2-3 2-26 2-67 2-62 1-5 025 O08 012 009 022 021 003 654 D. McKENZIE AND M. J. BICKLE The average composition of the glasses analysed by Melson et al. (1977) and the mean IGCP oceanic basalt composition probably contain more FeO and TiO 2 and less MgO than does the 1280°C point and depth average because considerable quantities of the primary magma have been removed by fractional crystallization. Pallister's (1984) estimate of the average composition of the Semail Ophiolite, SAVE, and the average of Elthon's (1987) MgO-rich glass compositions are more similar to the 1280 °C point and depth average. The observed MgO concentrations are, however, about 1 % less than that calculated. The removal of about 2% olivine from the melt by fractional crystallization could account for this difference. The observed concentrations of SiO 2 , A12O3, and CaO differ from those calculated by between 1 and 2%. The differences are probably just greater than the experimental and parameterization errors. A simpler comparison between the observed and calculated compositions than those illustrated in Fig. 11-13 is obtained by using oxide-oxide plots. Five of these are illustrated in Fig. 14 for Melson et al.'s glasses containing more than 8% MgO. All oxides except K 2 O are plotted as functions of Na 2 O, which varies inversely with the melt fraction. The listed concentrations of K 2 O are not accurate enough to be useful. The usual 1280°C reference line shows that the concentrations of FeO and TiO 2 vary in the way expected from the melting experiments, but the concentration of A12O3 is approximately constant, that of CaO increases as the melt fraction increases and that of Na 2 O decreases. This behaviour suggests that the aluminous phase is the first to melt, followed by diopside. A more complicated parameterization was tried in which the concentration of CaO and A12O3 in the experiments was modeled in the same way as that of FeO and MgO by allowing the compositions to depend on pressure (Table Al(c)). The fit was scarcely improved, with the mean error for major elements remaining at 106% and that for Na 2 O and TiO 2 decreasing to 0-26%. The pressure dependence introduced was, however, considerable and agrees with that discussed by Jaques & Green (1980). Curves calculated from this parameterization are illustrated in Fig. 14 as dashed lines, and agree better with the observed trends. Instead of excluding the glass analyses which have been affected by fractional crystallization, Klein & Langmuir (1987) have taken account of such effects by estimating the composition that a parental magma would have had when the concentration of MgO was 8%. They argued that the observed regional variations in the parental composition result from variations in the mantle potential temperature, and correspond to variations in the residual depth of the ridge axis. Figure 15 shows their estimates of the composition when MgO = 8%, together with curves showing the point and depth averages calculated from the parameterization (a) in Table A1, except in the case of CaO/Al 2 O 3 , where both (a) and (c)are shown. The depths in Fig. 15 were calculated by isostatic compensation of a column 150 km deep against a standard ridge axis with a water depth of 2-8 km, a crustal thickness of 7 km and a mantle potential temperature of 1280 °C. The density of the crust was calculated from the CIPW norm of the point and depth average, with the opx distributed between olivine and quartz. If the orthopyroxene is not distributed in this way the calculated water depth decreases by 100m for r P =1160°C and increases by 270m when r P =1480°C. The agreement between the calculated and observed variations is in general good, even though the point and depth average contains considerably more than 8% MgO when TP is greater than 1280 c C. However Fig. 15(d) suggests that the observed concentration of Na 2 O may increase more rapidly than would be expected as the potential temperature decreases below 1250°C and the crustal thickness below 5 km. This difference could result from errors in the calculated melt compositions, since the accuracy of the parameterization becomes poor when the melt fraction is small. It is, however, also possible that it is caused by the generation of Na 2 O-rich melts at temperatures below the onset of extensive silicate melting at Ts. MELT GENERATED BY LITHOSPHERIC EXTENSION 655 o o co CM 2 o o * * D. McKENZIE AND M. J. BICKLE 656 15 O MgO 10 NB2O % FIG. 14 (aH e ) oxide-oxide plots for all Elthon's glasses, and those of Melson et al. (1977) with MgO > 8%. The l280°C lines are calculated from the parameters in Fig. 8(a). The arrows show the direction in which X increases, with open circles marking values of 0-05 and Ol5 The dot inside a circle shows the point and depth average. The ellipses show estimates of the 2a error ellipses for individual probe analyses and for the error in the mean from the parameterization. The ellipse for the parameterization is marked with oblique lines. The dashed lines in (b) and (d) show compositions calculated with Tr = 128O°C and a parameterization of the CaO and AI 2 O 3 experimental compositions with C o = C0(P) (Table Al(c)). A further condition which must be satisfied is that the primary melt from the mantle must be able to generate the observed range of compositions of oceanic basalts and gabbros by low pressure fractional crystallization. The most sensitive measure of fractional crystallization available from the major and minor oxide compositions is the MgO/TiO 2 ratio, since MgO is the most and TiO 2 the least compatible oxide. Three plots showing oxide concentrations as a function of TiO 2 are shown in Fig. 16. Plots of MgO and FeO against TiO 2 are to be preferred to one of TiO 2 against m#-number because they do not confuse MgO depletion with FeO enrichment. Also plotted in Fig. 16 are the 0 GPa fractional crystallization trends given by Biggar (1983), obtained from the experiments of Shibata et al. (1979), Walker et al. (1979) and of Biggar & Kadik (1981). There is good agreement between the trends shown by the glass data and those derived from the experimental results. Furthermore it is clear that the removal of the cumulus gabbro can in a general way account for the observed trends. As Biggar (1983) has pointed out, the point average melt compositions first produce olivine as they cool. After the loss of a few percent olivine, plagioclase, and clinopyroxene join olivine and thereafter the removal of large melt fractions have little effect on the major element compositions. This behaviour is clearly illustrated by the small change in MgO concentration which is associated with a change in TiO 2 concentration by a factor of 3 in Fig. 16(b). Pallister(1984) gives the mean composition of the Semail cumulus gabbros as containing 105% MgO and 0 3 % TiO 2 (Table 1). This point is marked in Fig. 16 by a cross inside a circle. If MORBs result from the removal of this gabbro from the primary melt, then the mean gabbro composition, the point and depth average, and the average MORB composition should all lie on a straight line in Fig. 16, and the amount of crystallization calculated from the point and depth average should be the same for all oxides. This calculation gives the MELT GENERATED BY LITHOSPHERIC EXTENSION 657 fraction of gabbro required as 057 and 044 for TiO 2 and FeO respectively to produce the mean glass composition, and 0-80 and 078 to produce the mean of the basalts from the Galapagos which are particularly rich in FeO and TiO 2 , the so-called Fe-Ti basalts. Cumulus gabbros make up a fraction of 066 of Pallister's (1984) measured section. If about half of the primary melt crystallizes as gabbro to make the oceanic layer 3, its expected thickness is between 3 and 4 km. The thickness of the dykes and pillow lavas which make layer 2 should also be between 3 and 4 km. These numbers are in satisfactory agreement with each other and with the geophysical observations. The Fe-Ti basalts are the extreme products of fractional crystallization, requiring as they do removal of 80% of the primary melt by crystallization. If the oceanic crust was usually produced in this way, many of the (a) Na2O % (b) 0 8 - CaO 0 7 - 0 6 2 4 Depth km ' (c) FeO % FIG. 15. (aHc). 658 D M c K E N Z I E A N D M J. B I C K L E 3-5 FIG. 15. Estimates of the Na 2 O, (a), CaO/Al 2 O 3 , (b), and FeO, (c), concentrations in basalts containing 8% MgO plotted against the average depth to the ndge axis, (d) Seismically determined crustal thickness as a function of the Na 2 O concentration in basalts with 8% MgO. The curves show the point and depth average compositions for the potential temperatures marked in (a) and (d), obtained from the parameterization in Table Al(a), shown as continuous curves, and from Al(c), shown as the dashed curve in (b). The depths were calculated by isostatic compensation against a ridge axis depth of 2-8 km, a crustal thickness of 7 km and TP = 1280 °C with a 'lithospheric' thickness of 150 km (see text). The points are taken from Klein & Langmuir (1987), but the point corresponding to the Cayman Trough in (d) is not plotted. erupted magmas would have this composition. Mass balance would then require the production of 5-6 km of gabbro to every 1 -4 km of Fe-Ti basalts. Their rarity in Melson et al.'s data set implies that the proportion of the oceanic crust which consists of gabbro is on average closer to 50 than to 80%. This conclusion is also in agreement with the seismically determined thickness of about 4 km for the oceanic layer 3, which is believed to consist of gabbro (White, 1984). The same calculation is not successful for MgO. Though the crystallization trend is clear, and is consistent with Pallister's mean and with Biggar's (1983) experimental estimates, the point and depth average contains about 0-8% more MgO than is required by the gabbro and MORB compositions. It is not clear whether this difference is due to errors in the calculations, or is an indication of some physical process which has not been taken into account. If, for instance, the value of the entropy of melting is increased from 250 to 400 J ° C ~ ' k g " 1 , a change which is within the present experimental uncertainty, the difference between the required and calculated MgO concentrations is reduced to 0-4%. The case of Na 2 O is more complicated. Pallister's (1984) estimate of the mean composition of the cumulus gabbro contains less Na 2 O than does the point and depth average, and removal of 45% of this gabbro from the primary melt produces a melt with the mean composition of Melson et a/.'s glasses. Biggar's (1983) estimates of the Na 2 O concentration in the residual melts illustrated in Fig. 16 are not likely to be as accurate as are those for MgO, FeO, and TiO 2 because Shibata et al.'s (1979) experiments produced zoned, nonequilibrated feldspars. Biggar & Kadik's (1981) experiments suffered from the same problem and in addition their charges lost Na through evaporation (Biggar, 1983). Nonetheless there is reasonable agreement between the expected and observed trends in Fig. 16(c). Though the removal of the cumulus gabbros can account for the average Na 2 O concentrations in MELT GENERATED BY LITHOSPHERIC EXTENSION MORB glasses, it cannot account for the absence of any continued Na 2 O enrichment at the larger degrees of fractional crystallization required to produce the Fe-Ti basalts. Mass balance calculations are less useful for CaO and A12O3 because the changes in composition produced by fractional crystallization are so small, as Biggar (1983) has remarked. The complementary relationship between the MORB glass compositions and that of the gabbro of the Semail Ophiolite supports the commonly held view that at least this ophiolite is a reasonably representative piece of oceanic crust. The simple model of primary melt segregation followed by low pressure fractional crystallization which accounts for the major and minor element concentrations in MORB can also account for those of some compatible trace elements like Ni. The point and depth average Ni concentration calculated from Hart & Davis's (1978) distribution coefficient is 245 ppm for a bulk Ni concentration of 0-23%, and is little changed if the distribution coefficient of Takahashi (1978) is used instead. Hart & Davis's (1978) distribution coefficient £>(ol/melt) depends inversely on the MgO concentration in the melt, and may be too large (a) 15 FeO © 10 20% 40% .1 1 60% 80% 2 TiO 2 % (bi 15 1280° MgO % 1 2 TK> 2 % Fio. 16. (a>-(b) I D McKENZIE AND M. J. BICKLE :6«Ci Ic) Na,0 1280 '-#<>> 0 1 2 3 4 TiO 2 % FIG. 16. Oxide-oxide plots of all of Elthon's and Melson et a/.'s (1977 and pers comm., 1092 analyses) glass data. The solid dots inside circles mark the point and depth averages, solid circles the compositions of the basalts nch in Fe and Ti from the Galapagos Spreading Centre at 0-71 °N, 85-5O°W, and heavy crosses within circles the average composition of layered cumulus gabbro from the Semail Ophiohte (see Table 1, taken from Pallister, 1984). The heavy dashed lines show the 0 GPa fractional crystallization trends from Biggar (1983, Table 1), with an initial MgO concentration decreasing in the direction of the arrow from 9%, 7% where the lines intersect, to 5% at the end. The percentage of crystallization required to generate a particular TiO 2 concentration by the separation of cumulus gabbro from the point and depth average is marked in (a) along the .x-axis. See Fig. 14 for the meaning of the error ellipses. (Clarke & O'Hara, 1979; Elthon & Ridley, 1979). But Fig. 17 shows that the fractional crystallisation of 5% olivine from the 1280°C point and depth average can produce the average observed Ni concentration in the IGCP basalts in which the Ni concentration was determined. The only other trace elements for which a clear trend was visible in the IGCP data when their concentration was plotted as a function of MgO were Cr and V. The first behaves like Ni, presumably because chrome spinel crystallises at the same time as does olivine. The V concentration, like that of TiO 2 , increases by a factor of 2 as the MgO concentration decreases from 10 to 5%. The MORB primary magma debate The calculations above show that the compositions of the most MgO-rich glasses closely resemble that of the primary point average compositions generated by adiabatic decompression. These calculations therefore support the views of Green & Ringwood (1967a), Kushiro (1973), Presnall et al. (1979), Fujii & Bougault (1983), and Takahashi & Kushiro (1983). The depthaverageofthepressure(equation(14))for7 P =1280 o CisO48GPaandfor^isl3-5%. The point average magnesium concentration ranges from 9-3 to 10-4%, and the maximum amount of melt is not quite sufficient to eliminate clinopyroxene. The calculated MgO composition of the instantaneous melt is illustrated in Fig. 4, but it is not clear that there is any relationship between the instantaneous batch melting and the instantaneous Rayleigh melting compositions. A quite different origin for MORB has been proposed by O'Hara (1968a, b), Green et at. (1979), Jaques & Green (1980), Stolper (1980), and Elthon & Scarfe (1980, 1984), who have argued that a major fraction of the primary magma from which MORB is generated is MELT GENERATED BY LITHOSPHERIC EXTENSION 661 (a) 1380° Melting 400 1330 200 . * • ••• 5 (b) 10 15 MgO% 400 1380° Fractional crystallisation 200 10 15 Mgo % FIG. 17 Ni concentration as a function of MgO in the basalt of the IGCP data set resulting from batch melting during adiabatic upwelling, (a), and fractional crystallization of olivine, (b) The solid dots in (a) and (b) mark the point and depth averages for the potential temperatures shown, and the dashed line shows the melt composition after the separation of 5% olivine. The large circle enclosing a cross in (b) shows the mean composition of the basalt analyses in (a). produced at pressures of between 2 and 3 GPa, and contains 15% or more MgO. The following arguments have been used to support this view. The composition of some MgO-rich glasses is not in equilibrium with orthopyroxene and olivine at any pressure (O'Hara 1968a, b; Green el al., 1979; Elthon & Scarfe, 1984; Elthon, 1987). One method of producing such compositions is to generate picritic melts with MgO concentrations of 15-20% at pressures of 2-3 GPa, then to remove olivine by fractional crystallization. But this proposal suffers from major geophysical difficulties. If a large quantity of the melt which is to form the oceanic crust is produced at a depth of at least 45 km then extensive melting must occur at such depths. Reference to Fig. 7(b) shows that 662 D. McKENZIE AND M. J. BICK.LE the potential temperature required is about 1480cC, or 1600°C at a depth of 200 km (Fig. 7(a)). This temperature is hotter, though probably not much hotter, than most proposed geotherms. A more serious problem is how melting at depths shallower than 45 km can be prevented. As was shown in section 2, conductive cooling is quite unable to remove heat from such great depths. Unless the heat is removed, another 20 km of melt will be added on the way to the surface, and the composition of the MORB which forms the oceanic crust will be dominated by this melt generated at lower pressure. The only escape seems to be to circulate sea water to depths of 40 km, even beneath rapidly spreading ridges like the East Pacific Rise. Such a proposal generates more problems than it solves. A possible solution to these difficulties was proposed by Klein & Langmuir (1987), who remarked that there was no reason why a melt generated from a range of depths should be in equilibrium with opx at any particular depth. Only if the compositions generated by adiabatic decompression lie on straight lines will mixtures between melts from different depths also lie on the same lines, and thus remain on the opx-bearing cotectic. (see Walker et al., 1979). This condition is, however, approximately satisfied during adiabatic decompression at melt fractions less than cpx-out at 24-5%, and is therefore probably not the explanation. The point and depth average in Table 1 also does not support Klein & Langmuir's suggestion. It is constructed by averaging point averages, all of which are in equilibrium with opx, and is about 2% richer in SiO 2 than is the mean of Elthon's MgO-rich glasses. Presnall & Hoover (1984) have argued that such differences may result from analytical errors. But there is no evidence from the present study that their argument is correct, since the parameterization of a large number of different experiments agrees excellently with the results from individual laboratories. Therefore, though the difference between the calculated SiO 2 concentration and the average from MgO-rich glasses is small, it appears to be real. It is, however, not clear how representative the compositions of the rare MgO-rich glasses are of the mean composition of the primary magma. A better estimate of the mean composition of the oceanic crust may be Pallister's (1984) estimate, SAVE, from Oman, and it has essentially the same SiO 2 concentration as the point and depth average (Table 1). Another argument used by O'Hara (1968a, b), Jaques & Green (1980) and Stolper (1980) in favour of a picritic parent for MORB is concerned with the trend in Figs. 11-13. Walker el al. (1979) argued that low pressure fractional crystallization of olivine, diopside and plagioclase moves the residual melt composition towards the quartz apex. This behaviour is illustrated in Fig. 10. If it is to account for the trend in Figs. 11-13 then the initial composition should be close to the olivine-diopside-plagioclase plane, which is a thermal divide. Fractional crystallization of olivine from a melt generated at the cpx-out point at 2 G P a in Fig. 10 can generate such a melt. But is the trend due only to fractional crystallization? Its presence in the MgO-rich glasses, which can only have crystallized olivine, suggests that it is not. Plots of m^-number and TiO 2 concentration should decrease and increase respectively as the normative quartz increases. Elthon (1983) found no such tendency in projections of MORB compositions using Walker et a/.'s (1979) projection, and therefore modified the method of projection. Presnall & Hoover (1984) plotted the mgnumber against normative quartz and only found a clear decrease when Q z > 2 5 % . An alternative explanation of the trend in Figs. 11-13 is that it results from different amounts of melting. This explanation can account for the good agreement between the observed trend and that of the calculated point average compositions, and the extension of both across the thermal divide into the ne-normative region. 1-3% of the analyses of Melson et al. (1977) and 10% of those of Elthon (1987) are ne-normative. As was recognized by Elthon (1983), Takahashi & Kushiro (1983), and Fujii & Scarfe (1985), the observed trend MELT GENERATED BY LITHOSPHERIC EXTENSION 661 results from variations in the concentration of Na 2 O, not SiO 2 . This proposal is easily tested by recalculating the locations of the points within the tetrahedron of Walker et al. (1979) without the contributions from the alkali metal oxides. An = Al 2 O 3 Di = C a O - A l 2 O 3 Ol = (FeO + MgO + A12O3 - CaO)/2 Qz = SiO 2 - (A12O3 + FeO + MgO + 3CaO)/2 (24) where the symbols on the right hand side of equations (24) refer to the molar concentrations. This tetrahedron was used by Walker et al. (1972) in their discussion of lunar basalts. The resulting plots are simple and straightforward to interpret. Figure 18 shows the calculated and experimental compositions at various pressures. What was before a dispersed cloud of points in Fig. 10 has concentrated in a small region whose composition moves towards the Ol corner as the pressure increases. Figure 19 shows the data sets projected onto two planes, with the 1280°C curve and Biggar's (1983) fractional crystallization lines for reference. Crystallization has little effect in these plots because the concentration of MgO + FeO is scarcely changed by the removal of as much as 80% of the melt as cumulate. The trend observed in Figs. 11-13 is no longer present, and therefore it was indeed produced by variations in the alkali oxide concentrations. The compositions now cluster about the 0 GPa cotectic. The plots show several other Major Element Oxides only (a) FIG . 18. The normative projection of Walker et al. (1972) using only the major element concentrations expressed in mol. % (see equation (24)). (a) is a projection from An, (b) from Di. The upper triangles show the paths calculated from the parameterization shown in Fig. 8(a), with the circles in (b) and 5% and 15% melting by weight. Diopside disappears at 24-5% melt, shown as a dashed line. The arrows point in the direction of increased melting, and the numbers show the pressure in GPa. The lower triangles show the experimental observations, with crosses for experiments between 0 and 0 5 GPa, open circles O5 to 1-5 GPaand filled tnangles for higher pressures. Only those results for which the observed or calculated melt fraction was less than 24-5% are plotted. The ellipses show the projections of the la errors for the mean composition from the parameterization (oblique lines) and the probe analyses (open). 664 D. McKENZIE AND M. J BICKLE interesting features. The plots of glasses with MgO greater than the average value of 7 1 % are clearly more dispersed than are those with MgO < 7-1 %. This behaviour is that expected if the initial differences in melt composition are removed by low pressure fractional crystallization which concentrates all compositions at the cotectic, where olivine, clinopyroxene and plagioclase all crystallize together. The ne-normative melts are shown separately as crosses in Fig. 19 because they are the most extreme representatives of the trends in Figs. 11-13. Therefore, if they plot in the same group as do the other analyses, the observed trend is entirely produced by the alkali oxides. In Fig. 19(b) it is clear that this condition is satisfied. In 19(a) however the ne-normative points are all below the 1280 °C reference line whereas most of the glasses with MgO > 7-1 % lie above it. This observation suggests that the melt fraction formed by small degrees of partial melting is depleted in CaO relative to MORB. The resulting variation in the CaO/Al 2 O 3 ratio leads to a dispersion in the plane containing the diopside corner and the cotectic. It is this dispersion which causes the trend to be less clear in Fig. 12 than it is in Fig. 13. It is now straightforward to understand how the trends in Figs. 11-13 arise. The initial melt fraction generated by small amounts of melting is a ne-normative basalt which plots well to the left of the Ol-Di-Pl join in Fig. 20 and contains a large concentration of Na 2 O. As the melt fraction increases it is diluted by melt which contains less Na 2 O. As the melt fraction increases, the path of the melt composition in any tetrahedron is a straight line if the distribution coefficient of Na 2 O between melt and solid is constant. The composition of the melt therefore lies on the line joining the initial melt composition to that of the cotectic. The trend is clearer in the projection from Di than from PI because the CaO content of the initial melt differs from that of the cotectic. This simple explanation of the observations is consistent with physical models of ridge axis processes, with modern ideas of melt extraction and with the geochemical observations. FIG. 19. (a) M E L T G E N E R A T E D BY L I T H O S P H E R I C E X T E N S I O N FIG. 19. Projections from An (a) and Di (b) of the tetrahedron calculated from the major elements using equation (24). The points from Elthon's collection are shown in the centre, with the data set of Melson el at. arbitrarily separated into two parts, depending on whether the MgO concentration was greater or less than the average of all the analyses. Those analyses which are ne-normative and the Fe-Ti rich basalts from the Galapagos Ridge are also shown below the central triangle as crosses and solid dots respectively. The boundaries of the enlarged triangles are (a) 20% Ol, 0% Di, 50% Qz giving a variation of 30% from apex to base, and (b) 20% Ol, 20% An, 38% Qz giving a variation of 22% from apex to base. The heavy cross within a circle marks the average composition of layered cumulus gabbro from the Semail Ophiolite (Pallister, 1984). The heavy dashed lines show the OGPa fractional crystallization trend in the direction of the arrows from Biggar( 1983, table 1), starting from an MgO concentration of 9%, decreasing to 7% where the lines intersect and to 5% at their ends. The large ellipses show the errors for the enlarged triangles, the small ellipses those for the full triangle, using the same convention as in Fig. 18. Picritic basalts and ophiolites If the primary melts which generate typical oceanic crust are not strongly picritic, what is the origin of those picritic magmas which are produced, and of the thick olivine accumulations commonly found in ophiolites, both of which have been used to support O'Hara's ideas? Picritic magmas are not common, and some of those that do occur are related to hot jets in the mantle. A potential temperature of 1480°C produces 27 km of a melt whose point and depth average contains 17% MgO. The composition of two well known picritic basalts from Baffin Bay analysed by Clarke (1970) are listed in Table 2, together with the 1480°C point and depth average. The agreement is good. The existence of a hot jet in the mantle beneath Baffin Bay at the time of eruption is also suggested by the shallowness of the region between Disko and Baffin Islands. It is important to determine whether other examples of picritic magmas are also the surface expression of mantle upwelling. Melts of such extreme composition are not likely to be common because the thickness of the oceanic crust rarely exceeds 10 km. D. McKENZIE AND M. J. BICKLE (a) (b) N* normative initial rrwlt FIG 20. Sketches to illustrate how the trends in Fig 11—13 arise by mixing variable proportions of the nenormative basalt with the cotectic melt TABLE Clarke (1970) no. 3 Clarke (1970) no. 4 1480° 2 SiO2 TiO1 Al>03 FeO MgO NiO CaO Na2O K2O Total 451 44-4 49-34 076 118 065 108 102 11-70 102 109 9-75 197 18-6 17-27 012 013 006 9-2 9-7 9-04 1-04 1-37 1 63 008 013 021 97-0 96-6 99-65 The composition marked 1480° is the point and depth average, the greatest NiO concentration in the instantaneous melt for TT= 1480° is 0 23%. Considerable thicknesses of olivine cumulates are often present in ophiolites, which are widely believed to represent pieces of oceanic crust and lithosphere. But many ophiolites were generated by some form of back-arc spreading. As Thompson (1987) has pointed out, the presence of water in such environments is likely to give rise to boninites, and these will produce large quantities of dunite by fractional crystallization. Thompson also emphasizes that the mean composition of ophiolites can rarely be determined with confidence. The main results of this section Mantle melting by adiabatic upwelling beneath ridges can produce the mean oceanic crustal thickness of 6-7 km, and the primary melts have the range of concentrations of MgO, CaO, A12O3, and SiO 2 observed in MgO-rich MORBs. In the basalt tetrahedron the calculated compositions form a band pointing at the Qz apex of the tetrahedron. This trend results from variations in the Na 2 O concentration in the melts, not from low pressure fractional crystallization. About half the melt is removed by low pressure fractional crystallization to generate the cumulus gabbros. This process accounts for the range in the observed MORB compositions. The other half is erupted as pillow lavas, or intruded as dykes. There is no reason to suppose that MORBs are generated by the fractional crystallization of a picritic parent, as has often been proposed. Picritic magmas are, however, produced by melting above hot rising jets in the mantle. Variations in the potential MELT GENERATED BY LITHOSPHERIC EXTENSION 667 temperature of the converting upper mantle can also account for the correlation between ridge depth and composition discussed by Klein & Langmuir (1987). The results are in excellent agreement with Thompson's (1987) arguments, who also demonstrated that the trend seen in MgO-rich glasses in Walker et al.'s (1979) projections must be due to variable amounts of melting, and not crystal fractionation. A feature of the method used to calculate the melt compositions is that it makes comparatively little use of phase diagrams, and so is unaffected by solid solution. The parameterization allows the composition of the primary magmas to be obtained without understanding any of the chemistry involved in the melting process! It is, however, only possible to be so dumb if there is an excellent experimental base. For this reason a similar approach cannot yet be used when the melt fractions are small or when the pressure is greater than 4 GPa. In one way the success of the batch melting calculations is disappointing. From the discussion in section 2 it is clear that they cannot be correct, yet they reproduce the melt compositions very well. Therefore the melt compositions cannot be very sensitive to the fluid mechanics of the melting process. The small differences between the observed and calculated compositions of SiO 2 , A12O3, and CaO in Table 1 may result from the naivety of the melting model, but they could also be caused by errors in the assumed mantle composition and in the parameterization. 5. MELTING PRODUCED BY EXTENSION OF THE CONTINENTAL LITHOSPHERE The original purpose of the work described in this paper was to test the suggestion (McKenzie 1984£>) that significant volumes of basaltic melt can be generated when the continental lithosphere is stretched. This question has previously been studied by Foucher et al. (1982), who used geotherms obtained from the plate model and Ahern & Turcotte's (1979) expression for X(P, T) to calculate the thicknesses of melt produced. Though the parameterization used here is likely to be the more accurate, in practice the resulting differences are small. As in the case of ridges, the total melt thickness is not sensitive to the details of the calculation, and the results obtained below agree with those of Foucher et al. (1982) for corresponding values of TP. Their calculations were, however, concerned with the northern margin of the Bay of Biscay, which was underlain by mantle whose potential temperature was similar to that of the mean mantle temperature at the time of stretching, and produced little melt. The assumption that crust is conserved during stretching is therefore reasonably accurate. But the results are very different for a potential temperature of 1480° C. The other difference arises because Foucher et al. (1982) assumed that about 10% of melt must be generated before melt and matrix could separate. It now seems likely that this value is nearer 2 than 10%. Therefore for geophysical calculations melt separation can be regarded as complete and as occurring in times of 1 Ma or less. Because so little melt is present even during'extension, its influence on the mean density can be neglected. The most important effect of this melt is therefore the resulting increase in crustal thicknesses. If the melt is emplaced into the base of the crust (Cox, 1980; Herzberg et al, 1983; McKenzie, 1984b), the amount of extension will be underestimated if it is determined from the change in the crustal thickness. There is an important difference between the behaviour of continental and oceanic lithosphere when it is stretched. In the oceans stretching leads to a spreading ridge, with very localized deformation. The same is rarely true of the continents, where the extension is generally distributed over a considerable region such as the Basin and Range province in the 668 D. McKENZIE AND M J. BICKLE western U.S.A., or the Aegean Sea. The cause of this difference is not properly understood, but must be taken into account in any melting calculations. The amount of extension is determined by the areal extension which some part of the lithosphere has undergone, and is generally expressed in terms of /?, the ratio of the final to the initial surface area. In the North Sea the values of P obtained from refraction and thermal subsidence are less than 1-5 for the Jurassic-Cretaceous stretching event (Barton & Wood, 1984). In the Basin and Range P is probably as large as 2 over considerable regions and may locally exceed 3 (Gans & Miller, 1983). Even larger values of P have resulted from the stretching of continental margins (White et al., 1987). To calculate the amount of melt produced by extension requires an initial geotherm. Since most of the melt is generated from the material which is initially within and below the thermal boundary layer, the initial geotherm must be chosen with care. The method used to calculate the initial temperature variation is described in Appendix B and makes use of a rather complicated parameterization obtained from numerical experiments. The behaviour is illustrated in Fig. 21, which closely resembles a sketch of Dixon et al. (1981) and the calculations of Foucher et al. (1982), and shows the melt volumes generated for various values of)? when the interior potential temperature of the mantle is 1480°C. Such high temperatures are probably restricted to hot rising jets. Figure 22 (a) shows that the amounts of melt generated are considerable. An alternative method (McKenzie, 19846) of representing the results is in terms of the apparent value of P, p,pp, determined from the change in crustal thickness resulting from stretching when all the melt thickness tm is assumed to be added to the crust of thickness tc. Then Cm + tJ (25) Values of P,PP(P) in Fig. 22(b) were calculated for tc = 30 km and show that melting can have a large influence on estimates of fi made from the change in crustal thickness. For instance when the mechanical boundary layer is 100 km before extension, and the potential temperature of the upper mantle is 1480°C, /? app < 14 no matter how large P may be. The melt thickness generated in this way reaches 7 km when ^ = 2 and, because of its density, it is likely to be emplaced in the lower crust (Herzberg et al., 1983). Such volumes of melt may well account for the crustal thickness beneath the Basin and Range province ofthe western U.S., and may also be the heat source which generated the widespread rhyolites by crustal melting. The volume of melt generated by extension depends strongly on both the thickness ofthe mechanical boundary layer and on the potential temperature of the interior of the upper mantle. When the potential temperature is 1280°C even a mechanical boundary layer as thin as 70 km only leads to the production of 2 km of melt when P = 3. Most sedimentary basins have a value of P<2 and are not above subduction zones or rising jets. Presumably for this reason changes in crustal thickness often provide satisfactory estimates of p. These are the conditions which Foucheref al. (1982) used, and they also found the effect of melting to be small. However the conditions beneath the Basin and Range province are different, because it is still about 1 km above sea level despite having been stretched by a factor of 2 over large areas. Both its elevation and crustal thickness of 25-30 km are probably caused by a rising jet in the mantle, which can maintain 1 km of uncompensated topography and add large volumes of basalt during the extension (Gans, 1987). In contrast little melt has been added to the crust beneath the North Sea, where changes in crustal thickness provide an accurate measure of the amount of extension. MELT GENERATED BY LITHOSPHERIC EXTENSION 669 Depth km 0 50 2100 100 I • 1S0 1 f- 1800 1700 1500 / / 1300 i m n Wftf 2 Thermal boundary layer /-- """ Mechanical boundary layer _i i 3 Pressure GPa (b) Ic) 100km 200km 100 50 Depth km 50 100 Depth km FIG. 21. (a) Adiabatic upwelling due to stretching of convective geotherm generated from a mechanical boundary layer thickness of 100 km, an interior potential temperature of 1480°C and a viscosity of 4 x 1 0 " m2 s'1. The numbers against the geotherms give the values of /?, the stretching factor. The curves between the solidus and liquidus show the melt fraction by weight, and the clashed line marks the path of the lower boundary of the mechanical boundary layer as 0 increases, (b) The melt fraction as a function of depth generated in (a). The numbers against the curves give the values of p. (c) as for (b) but with a mechanical boundary layer thickness of 200 km. 670 1480'C 1380 C 2 3 4 STRETCHING FACTOR P (b) 1280°C P.pp » 1380"C 1480"C Ptrue FIG. 22. (a) Total melt thicknesses generated by various amounts of extension, /?. The numbers against the curves give the thickness of the mechanical boundary layer, and the temperatures on the right are the interior potential temperatures, (b) The apparent extension /), pp as a function of the true extension 0lnit for the examples in (a), calculated from equation (25). As is clear from Fig. 22, estimates of p from the change in crustal thickness produced by the extension are likely to be underestimates, especially if the mechanical boundary layer is thin and the potential temperature high. This effect may account for the difference between the values of/? obtained from crustal thinning and from thermal subsidence by Royden & Keen (1980) and by Royden et al. (1980). An important difference between the me/t produced by continental extension and that from ridges is that the former is added to the continental crust. The commonly accepted view MELT GENERATED BY LITHOSPHERIC EXTENSION 671 is that new continental crust is largely generated at subduction zones by partial melting of the oceanic crust and overlying mantle. Figure 22 shows that this is not the only possible mechanism, and that large volumes of new continental crust can be generated by continental extension. However, unlike island arc vulcanism, this process requires the existence of continental crust to operate, and generates basaltic, not granitic, crust. Otherwise the extension and the vulcanism will be confined to a ridge axis. It is therefore more likely to be important when an appreciable fraction of the earth has already been covered by continental crust. It will then become an efficient method of increasing the continental volume. The association of alkali basalts with continental rifting and extension is well known, and it is of interest to discover the extent to which the observed compositions can be accounted for by simple extension. It is clear from Figs. 11-13 that melt fractions generated at considerable depths will have the composition of alkali basalts. Figure 21 shows that the melt generated during extension does indeed come from the necessary depth, and can be expected to consist of alkali basalts. This qualitative argument was used by Dixon et al. (1980) to account for the composition of North Sea basalts, and is strongly supported by the present calculations. Quantitative estimates of the composition can easily be made from the empirical expressions. It is important, however, to realize that they are not likely to be as reliable as those for MORB because the compositions are controlled by those of the small melt fractions until the thickness of melt generated exceeds a few kilometres. There is little experimental control on the melt compositions until the melt fraction exceeds about 10%. The point average compositions for two examples are illustrated in Fig. 23 together with the point and depth averages. The arrows on the curves show the compositional trend as the depth of the point averages decreases. The boundaries of the magma types are taken from Cox et al. (1979,fig.2.2) except for the boundary between alkali basalts and olivine tholeiites from McDonald & Katsura (1964) (see Carmichael et al., 1974, p. 413). The onset of extensive silicate melting occurs when /?=l-5 when 7p=l480°C, and produces 2-4 km of alkali basalts. When the amount of extension exceeds yS = 2 0 the point and depth average composition crosses into the olivine tholeiite field. As the melt fraction increases further the melts become picritic. Though Fig. 23 implies that the point average melt fractions produced when /? = 50 are basaltic andesites, their true composition contains up to 16% MgO. The effect of increasing potential temperature is to move the curves downwards in the alkali-silica plot. If the value of ft is kept fixed they also move to the right. The composition of two North Sea basalts from the most extended region of the basin (Fig. 23) plot near the extension of the curve for T P = 1280°C. They therefore result from small amounts of partial melting, probably smaller than can be accurately modelled with the parameterization discussed here. When the mantle potential temperature is 1480 °C stretching by a factor of 2 generates 7 km of melt whose composition is on the line separating alkali basalts from olivine tholeiites, in agreement with the compositions of the large eruptions of plateau basalts (Norry et al., 1980; Mohr, 1983). As the stretching increases further the composition of the melt changes to tholeiitic. A detailed comparison of the crustal structure and vertical motions (White et al., 1987; White & McKenzie, in prep.) shows that the melt generated by adiabatic decompression can produce the observed lower crustal velocities of 7-2 k m s ' 1 if the large quantities of hot picritic melt containing an average of 16% MgO are underplated beneath the original continental crust in the manner discussed by Herzberg et al. (1983). The P-wave velocities of 7-2 k m s " 1 are then the result of the increased MgO content, rather than due to phase changes as Foucher et al. (1982) and Furlong & Fountain (1986) argued. A potential temperature of about 1480 °C is required to generate the melt volumes observed. Little subsidence then occurs until stretching ceases, but thereafter the thermal subsidence is very rapid. Figure 21 shows that most of the melt D. McKENZIE AND M. J. BICKLE 672 / Basaltic B A S A L T S / and* sites afcal / Ni 2 O / thotofitlc - (b) Na2O / A*—Hortt\ Sea ,'•• K2O 40 44 48 52 SJOj % FIG. 23. (a) Various basalt types in an alkali-silica diagram from Cox et al. (1979). (b) and (c) melt compositions generated by extension of a lithosphere with a mechanical boundary layer thickness of 100 km, a viscosity of 4 x 1 0 " m2 s" ', and the potential -temperature shown. The curves are marked with the value of/? and represented point averages. The solid dots are point and depth averages. The two points marked 'North Sea' show the compositions of 9B and 9C from Gibb & Kanans-Sotiriou (1976) MELT GENERATED BY LITHOSPHERIC EXTENSION 673 comes from below the mechanical boundary layer, and therefore will have the asthenospheric isotopic composition. Any isotopic signature from the mechanical boundary layer is likely to occur through interaction with melt as it moves upwards. All these results are in agreement with the observations. A simple extensional model can therefore account for the principal geochemical and geophysical features. This agreement suggests that the melt distribution and composition can be used to map the vertical movement of the base of the 'lithosphere', in the same way as crustal thinning is used to map crustal extension. The eruption of alkaline magmas within the area of the North Sea which has undergone most crustal extension suggests that the stretching of the mantle part of the 'lithosphere' is also greatest beneath the graben. Further tectonic applications of these ideas depend on understanding what controls the compositions of small melt fractions. Few relevant experiments have yet been carried out, but it may be possible to exploit knowledge of the erupted magma compositions to provide the constraints. 6. DISCUSSION AND CONCLUSIONS The most important result is the demonstration that the range of composition observed in MgO-rich MORBs is very similar to that of the primary melts produced during adiabatic decompression of a rock with the composition of a garnet peridotite. The compositional variations in these glasses result from differing degrees of melting, and not from either variations of the source rock composition or fractional crystallization. Beneath a normal ridge axis the melting starts at a depth of about 40 km and generates a basalt containing about 10% MgO. The greatest degree of melting occurs at the shallowest depths, and does not exceed 24%. A small amount of diopside is therefore left in the residue. These results depend on an accurate parameterization of experimental batch melt compositions in terms of the melt fraction present. It is surprising that the agreement between the calculated and observed compositions is so good, because batch melting is likely to be a poor approximation to the melting processes beneath ridges. It is not yet clear whether the small differences between the calculations and the observations are due to differences in the bulk compositions, errors in the parameterization, or fluid dynamical and chemical processes involved in the generation and separation of the melt that have not been taken into account in this discussion. The trend seen in the MORB compositions when plotted in either Walker et a/.'s (1979) or O'Hara's (1968a, b) projections has been interpreted as being due to fractional crystallization. But the primary melt compositions calculated from the parameterization of the experiments show exactly the same trend, which in both cases clearly extends into the nenormative field. That this behaviour is produced by variations in Na 2 O concentrations was recognised by Elthon (1983), Takahashi & Kushiro (1983), and Fujii & Scarfe (1985). It results from the dilution of the ne-normative basalts generated by small amounts of melting by a cotectic melt. This simple result is not evident when all the normative plagioclases are combined together, but is obvious when the alkalis are omitted from such plots. Comparison of the calculated FeO, TiO 2 , and Na 2 O concentrations with the mean value of Melson et a/.'s glasses shows clear evidence of low pressure fractional crystallization, and require on average about half of the primary melt to be removed as gabbro to produce a cumulus thickness of between 3 and 4 km. This result agrees well with the measured sections in the Semail Ophiolite, and with the thickness of layer 3 of the oceanic crust determined by seismic refraction. These conclusions are in excellent agreement with those of Thompson (1987) and of Klein & Langmuir (1987). 674 D. McKENZIE AND M. J. BICK.LE The extension of the calculations to basins formed by stretching supports the arguments of Dixon et al. (1981) and shows how this process can generate considerable thicknesses of alkali basalts. The thickness, location, and composition of the melt produced can provide estimates of the potential temperature of the upper mantle and the shape of the base of the 'lithosphere' resulting from the extension. Such applications are likely to be of considerable importance, especially when combined with studies of thermal subsidence, but will require the work discussed here to be extended to allow the accurate calculation of the composition of melt fractions of less than 10%. The calculations also show that crustal conservation can be used to estimate the amount of extension ft which has occurred if fl < 2 and TP < 1380 °C, in agreement with Foucher et a/.'s (1982) results. These conditions are satisfied by many stretched basins. Many continental margins, however, have undergone larger amounts of extension, and the value of P estimated from the change in crustal thickness or the initial subsidence will then be less than the true value. Uniform stretching of the crust and lithosphere can easily account for Royden and Keen's (1980) observations, and is especially important where the potential temperature is greater than average. ACKNOWLEDGEMENTS This project was only possible because of the generosity of W. Melson, who collected and supplied the data set on which most of the arguments in this paper depend. We would also like to thank G. Biggar, P. Browning, K. Cox, J. Durham, D. Elthon, P. England, G. Fitton, T. Holland, E. Klein, C. Langmuir, R. K. 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APPENDIX A The compositions of melts generated from a rock with the composition of a garnet peridotite The functional form chosen for the parameterization was based on Rayleigh's law. The instantaneous melt composition c was required to satisfy X)ll") (Al) M E L T G E N E R A T E D BY L I T H O S P H E R I C E X T E N S I O N 677 in each of the intervals 0<A" <X,, X ,<X <X2 and X2<X <\ denoted by subscripts 1, 2 and 3 respectively for all oxides except SiO 2 , which was calculated by requiring the sum of all concentrations to equal 100%. The values of a, b, and n(D = nj(n+ 1) where D the distribution coefficient) are required to be constant in each interval, and c is required to be ^ 0 for all values of X. This condition prevents any component returning to the residue as X increases. The relationship between c and the batch melt (or point average) composition C is given by equation (13), and integration gives C{X) = a+ ( l - ( l - X ) ( 1 " " + 1 ) + —. (A2) Since C(0) is finite, d must be zero in the first interval for all oxides. Therefore (A3) C (but not c) was required to be continuous on X = X,, X2. For the parameterizations (a) and (b), a was taken to be zero in all intervals, and b to be zero in all except the first interval, for all oxides except FeO, MgO, SiO 2 , and 'the rest'. Those melt compositions where X had also been determined were then plotted as functions of 7". For (c) A12O3 and CaO were treated in the same way as FeO and MgO in the first interval, but a and b were set to zero when X > X t. Now Fig. 6 shows that X = X (7"). Therefore if a, b, and d are constant in each interval, C = C( 7"') and all points should lie on a single curve if the same bulk composition was used in all experiments. This condition is approximately satisfied for the major elements. Of these SiO 2 , FeO and MgO show a pressure dependence, the first decreasing and the second two increasing with pressure. This behaviour was taken into account by requiring C(0) and C(X,) to be linear functions of pressure for these oxides. The additional conditions that C(l) must correspond to the bulk composition, C be continuous on Xu X2, and the value of c(l) should correspond to the melting of forsterite provides six equations in the six unknowns at, b,, i= 1, 2, 3 for each of FeO and MgO which were solved analytically, d was set to zero in all intervals for these two oxides. After a number of numerical experiments it became clear that the values of n for FeO, MgO and CaO were poorly constrained if 0<X <XU and for FeO and MgO if Jf j <A'<A' 2 . This behaviour presumably occurs because these oxides are essential components of the phases present. These values of n were therefore fixed at unity, giving a linear dependence of C on X. The concentrations of the minor elements TiO 2 , Na 2 O, and K 2 O in the melts were more variable. Jaques & Green's (1980) pyrolite was produced from a Hawaiian basalt and contained about three times more TiO 2 than any of the other melts. Their results for this oxide were therefore not used in the determination of the best parameters. A separate determination of n for this oxide using their results alone gave a value of 0-114, in excellent agreement with that of 0-115 from all other results. All measurements of Na 2 O and K 2 O were used together, and those for Na 2 O are likely to be reliable, though absolute concentration may be too low because of losses due to heating by the electron probe. Since the same difficulty occurs in the MORB glass analyses it should not affect the comparison between the two. The tabulated abundance of K 2 O in most of the melts was too low to allow accurate parameterization. However, because Jaques & Green's (1980) K 2 O concentration was a factor of three greater than that in most of the other experiments, their results allow the value of n to be determined, and were best fit with n = 0-200. This value is surprisingly similar to that of 0-203 obtained for Na 2 O. Since SiO 2 was calculated by subtraction, all oxides present in the melt had to be included in the parameterization. The concentrations of those other than SiO 2 , TiO 2 , A12O3, FeO, MgO, CaO, Na 2 O, and K 2 O were lumped together and fit with a single constant independent of melt fraction and pressure. This constant is included as 'the rest' in Table Al. Though the value obtained by the fitting program for this constant is only 0-41 %, a considerable reduction in the mean error resulted when it was included. The parameters were determined by minimizing the absolute value of the difference between the observed and calculated concentrations, weighted in the manner described in the text. This procedure avoids giving undue weight to the observations which fit the worst. It does, however, have one disadvantage. Any curve which is fitted to two points with the same value of A" (or 7") has the same value of the misfit, provided it passes between them. This effect leads to the minimum being shallower than it is for the least squares condition. Twenty-two parameters were determined to produce the fits shown in Fig. 8(a). Three determinations of the parameters were carried out. The first, shown as (b) in Table Al and Fig. 8, used only those experiments listed in the caption to Fig. 8 -for which X was 678 D. M c K E N Z I E A N D M. J. B I C K L E TABLE C(0) SiO 2 (a) (b) (c) TiO 2 (a) (b) (c) AI 2 O 3 (a) (b) (c) FeO(a) (b) (c) MgO(a) (b) (c) CaO(a) (b) (c) Na 2 O(a) (b) (c) K,O(a) (b) (c) The rest (a) (b) (c) fdC\ ( —) VdP/,,0 C(XX) Al (AC\ Mean I—) \dPj . , Qi) "2 error x t 1-77 1 38 1-77 17-26 1796 18-89 0 0 0 0 0 — 1 65 0-115 0-162 0-116 0-521 0-354 1-0 860 611 7 58 4 61 2-04 361 12-05 12-20 12-02 3 24 518 314 0-26 0-16 0-27 0-41 0-27 0-37 0-59 3-05 1-82 3-01 2-29 4-76 0 0 -1-61 0 0 0 0 0 0 1-0 1-0 1-0 1-0 1-0 10 1-0 0-905 10 0-203 0114 0-227 1-777 0-278 10 1314 818 8O0 7 88 10-53 12-34 11-58 0-36 0-82 0-25 312 2 29 3-20 11 16 10 10 10 10 10 10 1-38 1 45 1-83 0-60 0-60 0-60 45-53 45-35 4412 017 017 018 309 3-23 3 82 8-50 8-50 850 39-20 39-20 39 20 2-59 2-75 3-25 044 050 O49 O06 003 007 041 027 037 1 15 022 118 098 082 1 19 029 023 015 C(X) is given in %. The values of X, and X2 were O245 and 0438 for (a), 0263 and 0438 for (b), and 0291 and 0439 for (c), respectively. c(l) for FeO was taken to be 0%, and for MgO to be 57%. (a) and (b) refer to the plots in Fig. 8, and the fit labelled (a) is to be preferred, (c) is used in Figs 14 and 15. measured or could be calculated. It excluded Stolper's (1980) results for AI 2 O 3 because they were affected by the presence of spinel in the olivine and orthopyroxene, and Jaques & Green's (1980) results for TiO 2 . The second determination, shown as (a) in Table Al, used a large number of experiments listed in the caption to Fig. 8 for which 7" but not X was available, and obtained X from T' in these cases. Stolper's (1980) results for AI 2 O 3 were included but not those for TiO 2 of Jaques & Green (1980). The distribution of errors for this fit is shown in Fig. 9. In the third determination, (c), the concentrations of AI 2 O 3 and CaO were allowed to vary with pressure in the interval 0 < X < A", in the similar way to those of FeO and MgO. Outside this interval c for these elements was taken to be 0. The care with which the experiments have been carried out is demonstrated by the agreement between different studies and by the surprising success of the parameterization using the single bulk composition given in Table Al(a). A listing of the FORTRAN subroutine used to calculate Cand X as functions of depth for adiabatic upwelling is available on request, or on an IBM PC floppy disc for a small fee. APPENDIX B Steady state geotherms Richter & McKenzie( 1981) give parameterizations for three types of constant viscosity calculation; with free boundaries on which the heat flux or the temperature are given, and for rigid boundaries at constant temperature. To generate geotherms for the thermal boundary layer beneath an old plate the MELT GENERATED BY LITHOSPHERIC EXTENSION 679 heat flux through the mechanical boundary layer must match that out of the top of the thermal boundary layer. It is therefore necessary to convert the expressions Richter & McKenzie (1981) give for constant temperature rigid boundary conditions to constant flux conditions. The Rayleigh number for fixed temperature boundaries RaT is RaT=- gad1 AT (Bl) KV where g is the acceleration due to gravity, a the thermal expansion coefficient, d the depth of the layer, AT the temperature difference between top and bottom, JC the thermal diffusivity and v the viscosity of the fluid. The corresponding expression RaT when the heat flux F is given is RaF=- gad'F (B2) kxv where k is the thermal conductivity of the fluid. The Nusselt number Nu defined as Nu= Fd (B3) kAT and an empirical expression which agrees with laboratory experiments is Nu = 0-184Ka° 2 8 1 (B4) substitution of (B3) and (B4) into (B2) gives Ra F = 0-184Raj 2 8 1 (B5) RaT = 315Ra%lsl. (B6) or If the origin is taken to be at the base of the mechanical boundary layer and the positive direction is downwards, the appropriate stretched variable z is Q (B7) and the increase in temperature ATfrom the base of the mechanical boundary is given by AT= — Kaf 0 2 1 9 1-84 + e - z £ aj") (B8) with ao=-l-84 a2=-O22 a, = — 1-18 a3 The value given for the thickness of the thermal boundary layer is the value of z for which z = 5. It is then necessary to find the value of AT(0) from the requirement that the heat transported by convection and conduction down the adabatic gradient at the top of the thermal boundary layer is the same as that conducted through the mechanical boundary layer when the interior potential temperature and the thickness of the mechanical boundary layer are given. The problem is solved by iteration, the potential temperature ATP(z) then determined from (B8) and converted to the actual temperature. The temperature structure depends weakly on the depth of the convecting layer, which was taken to be 700 km, and strongly on the viscosity through the Rayleigh number. Two geotherms for different viscosities are illustrated in Fig. 3(b) and (c). The maximum in TP is a feature of all horizontally averaged convective temperatures, and is produced by the shape of the isotherms in Fig. 2(a).