Quantitative Modeling of Granitic Melt Generation and Segregation in the Continental Crust

[1] We present a new quantitative model of granitic (in a broad sense) melt generation and segregation within the continental crust. We assume that melt generation is caused by the intrusion of hot, mantle-derived basalt, and that segregation occurs by buoyancy-driven flow along grain edges coupled with compaction of the partially molten source rock. We solve numerically the coupled equations governing heating, melting, and melt migration in the source rock, and cooling and crystallization in the underlying heat source. Our results demonstrate that the spatial distribution and composition of the melt depends upon the relative upward transport rates of heat and melt. If melt transport occurs more quickly than heat transport, then melt accumulates near the top of the source region, until the rock matrix disaggregates and a mobile magma forms. As the melt migrates upward, its composition changes to resemble a smaller degree of melting of the source rock, because it thermodynamically equilibrates with rock at progressively lower temperatures. We demonstrate that this process of buoyancy-driven compaction coupled with local thermodynamic equilibration can yield large volumes of mobile granitic magma from basaltic and meta-basaltic (amphibolitic) protoliths over timescales ranging from ∼4000 years to ∼10 Myr. The thickness of basaltic magma required as a heat source ranges from ∼40 m to ∼3 km, which requires that the magma is emplaced over time as a series of sills, concurrent with melt segregation. These findings differ from those of previous studies, which have suggested that compaction operates too slowly to yield large volumes of segregated granitic melt.


Introduction
[2] Many granitic rocks (in a broad sense) originate as partial melts of rocks in the mid-to lower-crust [e.g., Chappell, 1984;Atherton, 1993;Brown, 1994].The melt segregates from its partially molten host rock and accumulates to form a magma, which migrates upward through the crust before being emplaced at higher structural levels [e.g., Miller et al., 1988;D'Lemos et al., 1992;Atherton, 1993;Brown, 1994].Melting is caused by the advection of heat from the mantle to the crust, either by the intrusion of hot mantle-derived magma, asthenospheric upwelling following delamination, or a combination of these processes [e.g., Hodge, 1974;Bird, 1979;Huppert and Sparks, 1988;Kay andKay, 1991, 1993;Nelson, 1991;Bergantz, 1989;Bergantz and Dawes, 1994].During melting, the solid fraction of the source rock (the restite) maintains an interconnected matrix unless the melt volume fraction reaches the Critical Melt Fraction (CMF), which is the fraction at which the solid matrix disaggregates [Arzi, 1978;van der Molen and Paterson, 1979].Geochemical evidence suggests that the degree of melting required to yield granitic partial melt fractions is small for most source rock compositions, so static melt volume fractions are significantly less than the CMF [e.g., Clemens, 1989Clemens, , 1990;;Vielzeuf et al., 1990;Beard and Lofgren, 1991;Rushmer, 1991;Rapp and Watson, 1995].Consequently, the granitic partial melt fraction must flow relative to the solid matrix and accumulate until it forms a magma, which is mobile and which can migrate away from the source region [e.g., Wickham, 1987;Brown, 1994;Sawyer, 1994Sawyer, , 1996]].This process is termed segregation.The problem is that the physical mechanism by which segregation occurs is poorly understood.Note that we define ''melt'' to be crystal free liquid, and ''magma'' to be melt plus suspended crystals, although we recognize that, depending upon the efficacy of the segregation process, a magma may contain a negligibly small proportion of crystals.
[3] Melt segregation requires that the partially molten rock matrix is permeable, that there is a pressure gradient to drive melt flow relative to the matrix, and that there is space available to accommodate the melt.All of the available theoretical and experimental evidence suggests that the matrix of most partially molten crustal source rocks will be permeable even at low (<0.04)porosities [e.g., von Bargen and Waff, 1986;Cheadle, 1989;Wolf andWyllie, 1991, 1995;Vicenzi et al., 1988;Wark and Watson, 1998;Lupulescu and Watson, 1999], and it is difficult to envisage how segregation could otherwise occur.However, the origin of both the pressure gradient required to drive flow, and the space to accommodate the melt, is still poorly understood.Several studies have suggested that if the source rock is layered and the layers have different rheological properties, then tectonically driven deformation can provide both a pressure gradient and an accommodation space: Because the layers respond differently to deformation, pressure gradients form between them, and space is created at dilatant sites such as boudins and fractures [e.g., Sawyer, 1991Sawyer, , 1994Sawyer, , 1996;;Brown, 1994;Brown et al., 1995].However, these models are largely qualitative or semiquantitative; no quantitative model has yet been developed which predicts the mobilization of large quantities of granitic melt by deformation-enhanced melt segregation.Moreover, they are based upon observations of exposed migmatite terrains, yet it is not clear that migmatite terrains represent the source regions of mobile granitic magmas: migmatites can rarely be genetically linked to granitic plutons at higher structural levels [Wickham, 1987], and have been described as ''failed'' granites because the granitic partial melt fraction has been retained in the source region as leucosome [e.g., White and Chappell, 1990;Clemens and Mawer, 1992].
[4] Other studies have suggested that melt migration is driven by buoyancy, either along grain boundaries or through a network of fractures [e.g., McKenzie, 1985;Wickham, 1987;Petford, 1995;Petford and Koenders, 1998].The problem with these models is that they do not provide space to accommodate the melt.The flow behavior of fractured oil reservoirs demonstrates that flow occurs through fractures only if the fractures are connected to a well, through which fluid is extracted from the reservoir [e.g., Aguilera, 1995].By analogy, in a partially molten source region, flow will occur through fractures only if the fractures provide a path out of the source region to higher structural levels, where melt extracted from the source region can be accommodated.Yet the work of Rubin [1993Rubin [ , 1995] ] suggests that fractures filled with granitic melt find it difficult to propagate out of a partially molten source region unless the surrounding rock has been warmed, within a localized region, by repeated dyke intrusion.Repeated dyke intrusion requires an abundant source of segregated melt to be available within the source region.Consequently, we argue that fractures are unlikely to transport significant quantities of melt during segregation, although we agree that they may subsequently provide a mechanism for segregated melt (magma) to ascend through the crust.
[5] If flow occurs along grain boundaries, then meltenhanced diffusional creep processes provide a mechanism for changing the morphology of the grains [Pharr and Ashby, 1983;Cooper andKohlstedt, 1984, 1986;Karato et al., 1986;Kohlstedt and Chopra, 1994], so the partially molten source rock can compact in response to melt flow [McKenzie, 1984].Compaction provides space to accommodate the melt.However, applications of McKenzie's characteristic compaction length-and time-scales suggest that compaction cannot accommodate significant quantities of granitic melt in the crust within geologically reasonable timescales [Wickham, 1987;Petford, 1995].Yet these length-and time-scales are derived from a model in which the processes of melt generation and segregation are decoupled [Richter and McKenzie, 1984].In crustal melting zones, melt generation and segregation are complimentary, coupled processes which occur simultaneously; the results of Jackson and Cheadle [1998] demonstrate that the interplay between heat and mass transport within a compacting melting zone has a significant impact on the distribution and compositional evolution of the melt.However, their model is general and they did not apply it specifically to melting within the crust.
[6] The aim of this paper is to present a new quantitative model of granitic melt generation, coupled with buoyancydriven melt segregation and compaction, within the continental crust.Melting is assumed to be caused by the intrusion of hot mantle-derived basaltic magma.We present the conservation equations governing cooling and crystallization in the underlying basaltic heat source, and melting coupled with buoyancy-driven melt segregation and compaction in the overlying source rock.To simplify the problem, we assume that only heat is transported from the cooling magma into the melting source rock; in reality, mass transport may also occur as buoyant melt migrates upward from the magma into the source rock.
[7] The model formulation is quite general; however, in this paper we do not attempt to apply it to all granite source rock compositions and tectonic settings.Rather, we investigate melting and melt segregation from homogenous basaltic and meta-basaltic (amphibolite and eclogite) source rocks in settings where high heat fluxes and thickened crust promote partial melting.In continental arc environments, partial melting of basalt has been invoked as the source of Na-rich granitic batholiths which comprise the large Mesozoic Cordilleran complexes of the western Americas, Antartica, and New Zealand [e.g., Tepper et al., 1993;Leat et al., 1995;Atherton and Petford, 1996;Petford and Atherton, 1996;Petford et al., 1996;Wareham et al., 1997;Muir et al., 1998].High heat fluxes and crustal thickening result from basaltic underplating, which acts both as a heat source for partial melting and as a material source for granitic melt [e.g., Atherton andPetford, 1993, 1996;Tepper et al., 1993;Petford and Atherton, 1996].Basaltic and meta-basaltic rocks are also classically regarded as protoliths for Archean trondjhemite-tonalite-granodiorite (TTG) suites [e.g., Rapp et al., 1991;Rapp and Watson, 1995;Martin, 1999] and our results may contribute to the debate concerning their petrogenesis [Atherton and Petford, 1993;Martin, 1999].
[8] We solve the governing equations numerically, and demonstrate that buoyancy-driven melt migration coupled with local thermodynamic equilibration of the melt and matrix during segregation can yield large volumes of mobile granitic melt from a basaltic protolith over timescales ranging from $4000 years to $10 Myr.The thickness of magma required as a heat source ranges from $40 m to $3 km, which implies that the magma is most likely emplaced over time as a series of sills, concurrent with melt segregation in the overlying source rock.Our model predictions are qualitatively similar, but quantitatively very different to those of Fountain et al. [1989], who also presented a numerical model of melting and buoyancy-driven melt migration in a compacting rock.This is principally because their expressions for conservation of energy and mass are different to those presented in this paper: they do not describe phase change in the correct moving reference frame [Jackson and Cheadle, 1998].Consequently, their model neglects melting and freezing due to the migration of melt and matrix, which has a significant impact on the solutions obtained.

Model Formulation
[9] Consider a region of homogenous, isotropic hydrated basaltic rock in the lower crust, into which hot mantlederived basaltic magma, initially at temperature T sill , is intruded as a horizontal layer of thickness z sill (Figure 1).Let the spatial origin of the model (z = 0) denote the position of the magma/rock contact, and the temporal origin denote the time at which the magma is ''instantaneously'' emplaced.For convenience, we describe the layer of basaltic magma as a sill, although we do not expect that all of the magma within the sill is intruded as a single event; rather, we envisage that intrusion is continuous over the timescale of melting and melt segregation in the source rock, and eventually yields the sill thickness initially assumed in the model.
[10] Before magma emplacement (t < 0), we assume that the temperature profile in the solid rock is dictated by the current lower crustal geotherm, the rock is at its solidus (T sol ) at z = 0, and there is no melt present (Figure 1a).These initial conditions are consistent with our application of the model to tectonic settings where high heat fluxes and thickened crust yield high lower crustal temperatures [e.g., Atherton and Petford, 1993;Tepper et al., 1993].At time t = 0, magma emplacement in the region z < 0 causes the temperature at the contact (z = 0) to be increased from T sol to T c (Figure 1b), which leads to heating and partial melting of the rock in the region z > 0 and conjugate cooling and crystallization of the magma in the region z < 0. We assume that the thermodynamic properties of the basaltic source rock and magma heat source are identical.
[11] At times t > 0, the model can be divided into three distinct regions: solid rock in the region z > z sol , partially molten rock in the region 0 z z sol , and magma in the region z < 0. In the region z > z sol , heat transport occurs by conduction only; mass and momentum transport are zero because there is no melt present.In the region z < 0, heat transport is also assumed to occur only by conduction, based upon the work of Marsh [1989] and Bergantz and Dawes [1994] which suggests that although the magma in the sill convects, the transport of heat from the magma to the adjacent country rock occurs ''as if the magma were cooling by conduction only'' [Bergantz and Dawes, 1994].Their findings are supported by observational data from cooling basaltic lava lakes [e.g., Helz et al., 1989].However, the assumption of conductive cooling is not valid if the magma is superheated [Huppert and Sparks, 1988], so we assume this is not the case.In the region 0 z z sol , heat, mass, and momentum transport occur as buoyant melt is generated and migrates upward along grain edges through the solid rock matrix, and the matrix compacts or dilates in response.These transport processes within a melting, deformable mush have been described in general terms by Jackson and Cheadle [1998].For simplicity, we follow their approach and consider the problem in one dimension only (1-D), invoking the Boussinesq approximation throughout (i.e., assuming r m % r s = r except in the buoyancy term).

Governing Equations
[12] The 1-D conservation equation governing the conductive transport of heat in the solid rock above the partially molten rock (i.e., in the region z > z sol ) may be written as (see Table 1 for an explanation of the nomenclature) The 1-D conservation equations governing the transport of heat, mass, and momentum in the partially molten rock (i.e., in the region 0 z z sol ) may be written as [Jackson and Cheadle, 1998] Figure 1.Model formulation: (a) initially, the temperature in the region z > 0 is dictated by an initial temperature gradient, the rock is at its solidus (T sol ) at z = 0, and there is no melt present.At t = 0, a sill of thickness z sill , containing basaltic magma at a uniform temperature T sill , is instantaneously emplaced in the region z < 0. (b) Heat transport from the magma in the region z < 0 to the rock in the region z > 0 causes heating and melting of the rock and conjugate cooling and crystallization of the magma.
with the supplementary relations where equation (2) describes conservation of heat, equation (3) describes conservation of mass, equation (4) describes conservation of momentum, equation ( 5) relates the melt and matrix velocities, equation ( 6) describes the rate of melt production in the correct reference frame, in terms of the equilibrium melt volume fraction (also termed the ''degree of melting''), and equation ( 7) is a simple expression relating the matrix porosity and permeability.
[13] Given that the thermodynamic properties of the source rock and magma are assumed to be identical, the 1-D conservation equation governing the conductive transport of heat in the magma (i.e., in the region z < 0) may be expressed as [Bergantz, 1989] where À c denotes the rate of production of crystals (rate of crystallization).If the degree of melting in the source rock and the degree of crystallization in the magma are identical functions of temperature, then the rate of crystallization may be expressed as The initial conditions are given by where T geo is a linear initial temperature gradient.The boundary conditions are given by The boundary condition (11c) follows from the assumption that conductive heat transport in the sill is symmetric about a horizontal plane at its center.This leads to an underestimate of the heat transported from the sill to the overlying rock.

Simplification and Nondimensionalization of the Governing Equations
[14] Following the approach of Jackson and Cheadle [1998], the degree of melting of the rock and the underlying magma heat source will be described as a continuous, linear function of temperature and furthermore, latent heat will be assumed to be absorbed/ released linearly as melting/crystallization proceeds.This approach is reasonable for hydrated basaltic and metabasaltic rocks in which melting occurs due to the breakdown of hydrous minerals such as amphiboles, under water absent or undersaturated conditions [e.g., Clemens and Vielzeuf, 1987;Beard and Lofgren, 1991;Rapp and Watson, 1995;Petford and Gallagher, 2001] (see also Figure 2).The assumption of linear melting means that the dimensionless temperature may be written as [Jackson and Cheadle, 1998] in which case it is numerically equivalent to the degree of melting v m and will be represented by a new variable q; i.e., q = T 0 = v m .We now wish to normalize the porosity (f) and the dimensionless degree of melting (q).In previous models of compaction, the porosity has typically been normalized against a uniform ''background'' or ''early formed'' porosity upon which the solutions are superimposed [e.g., Richter and McKenzie, 1984;Barcilon and Richter, 1986;Spiegelman, 1993;Wiggins and Speigelman, 1995].However, in this model there is no background or early formed porosity, so a new scaling factor is required.A convenient factor is the initial degree of melting at the contact between source rock and underlying magma (at t = z = 0 in Figure 1) defined by [Jackson and Cheadle, 1998] The porosity (f) and dimensionless degree of melting (q) may then be normalized by writing The remaining variables are nondimensionalized by writing [Jackson and Cheadle, 1998] where the characteristic lengthscale d is identical to the compaction length of McKenzie [1984McKenzie [ , 1985]].Note that the characteristic permeability is defined in terms of the initial degree of melting at the contact (j).
[15] Substituting for the degree of melting (v m ) in equations ( 6) and ( 9), substituting for the rate of melt production in equations ( 2) and ( 3), substituting for the rate of crystallization in equation ( 8), substituting for the permeability (k) in equation ( 4), substituting the scaled and dimensionless variables in equations ( 15a) -( 19), simplifying, and dropping primes, yields the dimensionless governing equations Figure 2. Empirically derived degree of melting (melt volume fraction) as a function of temperature, at 8 kbar, for a meta-basalt (amphibolite) source rock.Data from Rushmer [1991].
Note that equation ( 25) governing conductive heat transport in the magma heat source is valid only if cooling of the magma is always accompanied by crystallization (with associated latent heat exchange), which is consistent with our requirement that the magma is not superheated.
[16] In dimensionless terms, the initial conditions become and the boundary conditions become where z 0 denotes the dimensionless position of the q = 0 isotherm, and z sill denotes the dimensionless half thickness of the underlying magma.
[17] Simplifying and nondimensionalizing the governing equations has reduced them to a system of six equations (equations ( 20) -( 25)) governed by six externally prescribed dimensionless parameters: the effective thermal diffusivity in the partial melt zone and in the crystallizing sill (k eff ); the Stefan number (Ste); the exponent in the permeability relation (n); the initial degree of melting at the contact (j); the ratio of the thermal diffusivity in the solid rock to the effective thermal diffusivity in the partial melt zone (c); and the initial dimensionless temperature gradient (y geo ).We will explore the system in terms of these dimensionless parameters.In the next section, we discuss their likely values.

Dimensionless Parameters
[18] The values of the dimensionless governing parameters k eff , y geo , Ste, c, and j depend upon the dimensional variables which appear in their defining equations ( 14), ( 26), ( 27), (29), and (31).These variables are listed, along with suitable values for basaltic source rocks, in Table 2, and are discussed further in the Appendices.Substituting this range of dimensional variables yields the values for the dimensionless parameters given in Table 2.As we demonstrate in the next section, the key dimensionless parameter governing melting and melt segregation is the effective thermal diffusivity (k eff ).The predicted value of this parameter for a basaltic protolith is very uncertain.The key variables which contribute to this uncertainty are the melt shear viscosity (m m ), the matrix bulk and shear viscosities (x s , m s ), and the characteristic matrix permeability (K).These variables govern the values of the characteristic timeand lengthscales (t and d), which appear in the definition of k eff (equations ( 16), ( 17), and ( 26)).
[19] The characteristic matrix permeability is poorly constrained because it depends strongly upon the matrix grain size (k eff $ a 2 ) and the initial degree of melting at the contact (k eff $ j 3 ; j is used as the porosity scaling factor in the defining equation ( 19)).Consequently, small uncertainties in each yield large uncertainties in permeability.It also contains a scaling factor (b), which must be estimated on the basis of very limited experimental and theoretical data (Appendix A).Predicted values of the matrix grain size vary by an order of magnitude, while values of the permeability scaling factor vary by nearly 2 orders of magnitude (Table 2).The value of j depends upon the solidus, liquidus, and initial temperatures of the protolith, the initial temperature of the intruded magma, and the kinetics of heat transfer in the magma.It cannot exceed 0.5 because the thermodynamic properties of the source rock and magma heat source are assumed to be identical, and the magma is not superheated.It must be less than the CMF for the model to be valid.The parameter j also corresponds to the maximum degree of melting, and to yield reasonable volumes of granitic melt, it cannot be very small; in this paper we assume it approaches 0.5 and calculate the characteristic permeability on this basis.Significantly smaller values of j will yield only small volumes of granitic melt which are unlikely to become mobilized.
[20] The other significant dimensionless parameter governing melting and melt segregation is the initial temperature gradient (y geo ), the predicted value of which is also uncertain.This uncertainty results from the scaling of length using the characteristic lengthscale (d) rather than from uncertainty in the initial dimensional temperature gradient (T geo ).
[21] None of the governing dimensionless parameters j, c, Ste, or y geo , are independent of k eff (equations ( 26) -( 29)).However, for fixed values of Ste and c, k eff may still vary over effectively its entire range; consequently, we assume Ste and c are independent of k eff .In contrast, Ste and c cannot be assumed to be independent; however, numerical experiments demonstrate that varying the values of c and Ste has negligible effect on the results, and we fix their values at c = 2.67 and Ste = 0.625.More significantly, k eff and y geo cannot be assumed to be independent, and both vary over a wide range of values.Substituting the dimensional parameters given in Table 2 yields the permissible combinations of k eff and y geo shown in Table 3.

Results
[22] The governing equations ( 20) -( 25) were approximated using finite difference schemes, following the ap-proach of Jackson and Cheadle [1998], and solved numerically.We are interested in four key issues: (1) whether melt will segregate; (2) the thickness of intruded basalt required for segregation; (3) the time required for segregation; and (4) the composition of the segregated melt.Our numerical experiments were designed to investigate these issues in terms of the dimensionless governing parameters.

Melt Segregation
[23] Figure 3 shows a selection of normalized spatial porosity (melt volume fraction) and degree of melting (dimensionless temperature) distributions after 30 time units have elapsed.The most significant feature of the solutions is the presence of a ''porosity wave,'' the amplitude of which is particularly high for the solutions with k eff = 1 and k eff = 100 (Figures 3b-3e).The leading wave may be trailed by several smaller waves, the amplitude of which decreases with increasing depth (e.g., Figure 3b) [see also Jackson and Cheadle, 1998].
[24] The development of these porosity waves depends upon the relative upward transport rates of heat and melt, which is predominantly governed by the magnitude of k eff  [1995], valid over the temperature range 600 -1300 K. Thermal conductivities vary by less than 25% over this temperature range.The experimentally determined values presented by Murase and McBirney [1973] for basalt, andesite, and rhyolite melts also lie within this range.b Calculated using data for individual minerals (albite/anorthite, diopside) from Robie et al. [1978], valid over the temperature range 600 -1300 K. Specific heat capacities of minerals vary by less than 20% over this temperature range, and by less than 25% between minerals and their melts [Richet and Bottinga, 1986;Neuville et al., 1993].c Calculated using enthalpies of fusion of individual minerals (albite/anorthite, diopside) from Richet and Bottinga [1986].d Data from Clemens and Vielzeuf [1987], Beard and Lofgren [1991], Rushmer [1991], Rapp and Watson [1995], and Petford and Gallagher [2001].e Data from Pollack and Chapman [1977] and Uyeda and Watanabe [1970].We apply the model to tectonic settings with high heat flow and corresponding steep geothermal gradients (e.g., continental arc environments such as the western Cordillera of Peru).
f Rock densities calculated using mineral modal analysis from Beard and Lofgren [1991], Rushmer [1991], and Rapp and Watson [1995], and density data for individual minerals [Deer et al., 1992].The values presented by Turcotte and Schubert [1982] lie within the range of calculated values.g Rock densities calculated using mineral modal analysis from Beard and Lofgren [1991], Rushmer [1991], and Rapp and Watson [1995].Densities for granitic melt compositions calculated using the model of Lange and Carmichael [1987], and compositional data from Beard and Lofgren [1991], Rushmer [1991], Rapp and Watson [1995], and analysis of granitic dykes from the Rosses pluton, Co. Donegal, Eire.The experimentally determined values presented by Murase and McBirney [1973] for rhyolite melt lie within the range of calculated values.h Typical grain size for high grade metamorphic (granulite facies) rocks, many of which are interpreted to be the residues left after the extraction of a granitic partial melt fraction.Grainsize measurements from Vernon [1968], Spry [1969], Yardley et al. [1990], and sections of granulite facies rocks from the Ivrea Zone, northwest Italy.i In the original derivation of the Blake-Kozeny-Carman permeability equation, n = 2 for a network of randomly oriented tubes of constant cross section, while n = 3 for a bed of packed spheres [Scott and Stevenson, 1986].The experimental results of Wark and Watson [1998] and Zhang et al. [1994], and the theoretical data of Cheadle [1989] indicate that a value of n = 3 is the most suitable for both texturally equilibrated and unequilibrated rocks (see also Figure A1).The degree of melting at the contact depends upon the solidus, liquidus, and initial temperatures of the crust, the initial temperature of the intruded magma, and the kinetics of heat transfer in the magma.It cannot exceed 0.5 because the thermodynamic properties of the source rock and magma heat source are assumed to be identical, and the magma is not superheated.It must be less than the CMF for the model to be valid.See text for details.o As the degree of melting at the contact tends toward zero, k eff tends toward infinity.The maximum value of k eff given in Table 1 is obtained assuming that j = 0.5.p As the degree of melting at the contact tends toward zero, y geo tends toward zero.The minimum value of y geo given in Table 1 is obtained assuming that j = 0.5.[Jackson and Cheadle, 1998].High amplitude porosity waves are observed for k eff in the range $10 À2 to $10 4 .Porosity waves do not form for large values of k eff (>10 4 ) because these correspond to conditions which promote slow melt transport and rapid heat transport; consequently, the porosity distribution is similar to that obtained from purely conductive heating with no melt migration [see Figure 3 of Jackson and Cheadle, 1998].As k eff decreases, the rate of melt transport increases relative to the rate of heat transport, until the melt migrates upward more quickly than the top of the partial melt zone (the position of the solidus isotherm), at which the porosity and permeability fall to zero.The upward migrating melt accumulates below the top of the zone, the matrix dilates to accommodate it, and a porosity wave forms (Figure 3).If the compaction rate exceeds the melting rate in the region immediately below a porosity wave, then the porosity decreases and a local restriction to melt migration forms below which melt accumulates, leading to the formation of a trailing wave.High-amplitude porosity waves do not form for small values of k eff (<10 À2 ) because melt transport occurs so rapidly compared to heat transport that freezing of the melt during ascent keeps the porosity too low for a wave to form [Jackson and Cheadle, 1998].
[25] The porosity distribution also depends to a lesser extent upon the value of the dimensionless initial temperature gradient y geo ; for a given value of k eff the effect of increasing the temperature gradient is to increase the amplitude of the leading porosity wave, decrease the spatial extent of the partial melt zone, and move the position of the porosity maximum upward relative to the top of the melt zone (compare Figures 3c and 3e, and Figures 3b and 3d).The extent of the melt zone decreases because the thermal perturbation cannot propagate as far from the sill; this decreases the distance over which the melt must migrate before it reaches the top of the melt zone, thus enhancing the accumulation of melt.
[26] If the amplitude of the leading porosity wave continuously increases with time, then the local melt volume fraction will eventually exceed the CMF, in which case the contiguity of the solid grains breaks down and a mobile magma forms [Jackson and Cheadle, 1998].Numerical experiments to track the increase in porosity with time suggest that if sufficient enthalpy is available, and if the value of the CMF does not greatly exceed the initial degree of melting at the contact (j $ 0.5), mobile magma formation is possible for k eff approximately in the range 10 À2 < k eff < 10 4 and y geo approximately in the range 10 À4 < y geo < 1 (Figure 4).This corresponds to the majority of available values of k eff and y geo (Table 3).If the value of the CMF greatly exceeds the initial degree of melting at the contact, it takes longer for the amplitude of the porosity wave to reach the CMF and the range of k eff for which segregation is predicted decreases slightly to 1 < k eff < 10 4 .The key finding of this section is that, if sufficient enthalpy is made available, melt segregation is predicted for the majority of values of the key dimensionless parameter governing melting and melt migration: the effective thermal diffusivity k eff .We investigate the enthalpy required for segregation in the following section.

Thickness of Basalt Heat Source (Sill) Required for Melt Segregation
[27] If the contact temperature begins to fall before a porosity wave has developed, the source rock may cool too quickly for magma mobilization to occur.The timescale over which the contact temperature begins to fall is primarily governed by the thickness of the basalt sill which provides the enthalpy for melting, and the values of k eff and y geo .For given values of k eff and y geo , we can estimate the minimum dimensionless sill thickness required for granitic melt mobilization.Figures 5a and 5c show the minimum sill half thickness (z min ) as a function of k eff , for y geo = 10 À4 and 10 À2 , respectively, and for CMF $0.5.In both cases, the minimum thickness increases with increasing k eff .This is partly because the time required for segregation varies as a function of k eff (see following section), and partly because the position of the porosity maximum, and hence of incipient magma formation, moves closer to the sill/rock contact [Jackson and Cheadle, 1998; see also Figure 3].Figures 5b and 5d show the corresponding dimensionless temperature at the sill/rock contact (q c ) and the center of the sill (q s ), at the time of granitic melt mobilization.In both cases, q c and q s increase with increasing k eff .The generally low dimensionless temperatures at the time of magma mobilization indicate that mobilization can occur while the source region as a whole is cooling.

Time Required for Melt Segregation
[28] We have found that the dimensionless thickness of basalt heat source required for segregation increases as a function of k eff .Another important quantity is the time required for segregation.We can estimate this by recording the time required for the porosity at any point to reach the CMF. Figure 6 shows the dimensionless time required for segregation obtained using the minimum sill thickness derived in the previous section, as a function of k eff , with CMF $ 0.5.Also shown is the segregation time obtained The dimensionless initial temperature gradient y geo and the dimensionless effective diffusivity k eff are not independent; for a given value of k eff , the range of available values for y geo is restricted.''Yes'' denotes values that are available.The range of values of k eff and y geo shown in bold denotes the range of values for which the formation of a mobile granitic (in a broad sense) magma is predicted; see text for details.
Figure 3. Normalized dimensionless spatial porosity (f) and temperature/degree of melting (q) distributions in the sill (z < 0) and partially molten rock (z > 0), after 30 time units, with: (a) k eff = 10 À3 , y geo = 1; (b) k eff = 1, y geo = 10 À2 ; (c) k eff = 1, y geo = 10 À4 ; (d) k eff = 100, y geo = 10 À2 ; and (e) k eff = 100, y geo = 10 À4 .In all cases, j = 0.5, Ste = 0.625, and c = 2.67.The half thickness of the sill is given by z sill = 2(k eff t) 1/2 ; note that the initial dimensionless temperature of the sill is 2, and the initial dimensionless contact temperature between the underlying sill and the overlying rock is 1.The temperature profile is shown only in the upper half of the sill.Note that both ordinate and abscissa axis scales differ between plots.assuming a constant contact temperature; this is equivalent to assuming an infinite sill thickness.Segregation times are generally shortest for k eff in the range 1 < k eff < 10 4 for y geo = 10 À4 , and 1 < k eff < 10 2 for y geo = 10 À2 , and then increase with both increasing and decreasing values of k eff .There is clearly a range of optimum values of k eff for which the dimensionless segregation time is minimized.Segregation times are similar regardless of whether the minimum or infinite sill thickness is used, which suggests that, so long as sufficient enthalpy is made available for segregation to occur, the dynamics of melting and melt migration in the source rock are not significantly affected by the thickness of basaltic magma which is ultimately emplaced.

Predicted Melt Compositions
[29] We now investigate the composition of the segregated melt.If the melt and matrix remain in local thermodynamic equilibrium during melting and melt segregation, the equilibrium composition of each may be predicted in terms of the local temperature and bulk composition.Figure 7 shows a plot of dimensionless temperature at the point of incipient magma formation for the same system shown in Figure 6.This temperature controls the initial composition of the segregated melt.In general, the temperature increases with increasing k eff because the position of the porosity maximum, and hence of incipient magma formation, moves closer to the sill/rock contact [Jackson and Cheadle, 1998; see also Figure 3].The anomalous decrease in temperature with increasing k eff observed in Figure 7b, for k eff > 10 2 , occurs because the leading porosity wave bifurcates, and with increasing k eff the position of the porosity wave moves closer to the top of the source region.
[30] If a suitable phase diagram was available, melt and matrix compositions could be predicted directly.Indeed, if the number of components were not too large, an alternative method of handling mass conservation would be to track the movement of components rather than phases, and to use the phase diagram to determine the equilibrium porosity (melt volume fraction) corresponding to the predicted local bulk and phase compositions.Unfortunately, for the complex rock types we are attempting to model, the number of components is large and a suitable phase diagram is not available.Consequently, we will estimate melt compositions using the results of equilibrium melting experiments.The compositions we predict will not be exactly correct as the local bulk composition has changed, but the approach is reasonable for major elements due to their close to eutectic behavior.[31] Figure 8 shows empirically derived major element melt compositions, as a function of temperature and degree of melting, obtained during dehydration partial melting of meta-basaltic rocks at various pressures [Beard and Lofgren, 1991;Rapp and Watson, 1995].Regardless of the source rock and pressure, the composition of the melt is granitic (in a broad sense) for degrees of melting up to $0.4.For example, melting at high pressures, at which garnet is a residual phase, yields small melt fractions which are initially trondjhemitic in character, and change to tonalitic-granodioritic as the degree of melting increases [Rapp and Watson, 1995].Melt compositions may be estimated using Figure 8 in conjunction with the appropriate dimensionless temperature curve.For example, consider the melt composition in the leading porosity wave shown in Figure 3c.The porosity wave is located at a dimensionless height of $16, and the melt at this location has a dimensionless temperature of $0.08, which for j = 0.5 corresponds to a dimensional degree of melting of 0.04 (equation 15).If the source rock were amphibolite melting at lower crustal depths and pressures, then the melt composition would be trondhjemitic (Figure 8).In like fashion, it may be deduced from Figure 7b that the composition of the segregated melt in a system with k eff = 10 and y geo = 10 À4 corresponds to a dimensional degree of melting of $0.15 (a dimensionless temperature of 0.3) and would be tonalitic-granodioritic.
[32] As the melt migrates upward through the partial melt zone, its composition changes because it thermodynamically equilibrates with partially molten rock at progressively lower temperatures [Jackson and Cheadle, 1998].For a given source rock composition, the composition of the melt therefore depends upon the position of the melt relative to the position of the solidus isotherm (the top of the partial melt zone).The melt in a porosity wave may occupy a large volume fraction of the source rock, yet its major element composition resemble only a small degree of melting of the rock.This is important, as the degree of melting required to yield granitic partial melt fractions is typically small.Our results suggest that basaltic and meta-Figure 5. Dimensionless minimum sill half thickness (z min ) required for magma mobilization in the overlying rock, and dimensionless sill temperature at the time of mobilization (contact q c and center q s ), against k eff , for the case CMF % j % 0.5, with (a and b) y geo = 10 À4 ; (c and d) y geo = 10 À2 .In all cases, Ste = 0.625 and c = 2.67.The initial dimensionless temperature of the sill is 2, and the initial dimensionless contact temperature between the underlying sill and the overlying rock is 1.Curves denote best fit polynomials.basaltic protoliths will yield segregated melt compositions ranging from granitic/trondjhemitic to tonalitic-granodioritic. Jackson and Cheadle [1998] argue that the assumption of local thermodynamic equilibrium is justified in the lower crust, but it remains one of the more stringent limits on the validity of the model predictions.

Melting and Buoyancy-Driven Compaction in the Continental Crust
[33] We have investigated melting and buoyancy-driven melt migration and compaction in the continental crust in terms of six dimensionless parameters.We have found that the spatial distribution of porosity (melt volume fraction) within a partial melting zone depends upon the relative rates of upward transport of heat and buoyant melt, which is predominantly governed by the magnitude of a dimensionless parameter which we term the effective thermal diffusivity k eff .This is the key parameter governing melt segregation.So long as sufficient enthalpy is made available, then in melt zones characterized by values of k eff approximately in the range 10 À2 < k eff < 10 4 , the rate of melt transport is rapid compared to the rate of heat transport, so the melt migrates upward faster than the position of the solidus isotherm, which acts as a ''lid'' on top of the zone.Consequently, the melt accumulates and a porosity wave forms, the amplitude of which increases with time until the Figure 6.Dimensionless segregation time (t seg ) against k eff , obtained using the minimum sill thickness presented in Figure 5, and an ''infinite'' sill thickness (constant contact temperature), for the case CMF % j % 0.5, with (a) y geo = 10 À4 ; (b) y geo = 10 À2 .In all cases, Ste = 0.625 and c = 2.67.Curves denote best fit polynomials.The segregation time is obtained by recording the time required for the porosity at any point to reach the CMF.
Figure 7. Dimensionless segregation temperature (q seg ) against k eff , obtained using the minimum sill thickness presented in Figure 5, and an ''infinite'' sill thickness (constant contact temperature), for the case CMF % j % 0.5, with (a) y geo = 10 À4 ; (b) y geo = 10 À2 .In all cases, Ste = 0.625 and c = 2.67.Curves denote best fit polynomials.The segregation temperature is defined as the temperature at the position of incipient magma formation.local porosity exceeds the CMF.This yields a mobile magma which may migrate away from the partial melt zone.Because the melt has thermodynamically equilibrated with cool solid matrix, its major element composition resembles a small degree of melting of the source rock despite occupying a large volume fraction of the rock.In partially molten basaltic and meta-basaltic source rocks the predicted composition of the melt fraction of the mobile magma is granitic (in a broad sense).
[34] This progressive evolution of the melt composition to reflect smaller degrees of melting of the source rock is similar in some respects to the process of zone refining, in which a melt body moving through rock by melting at the leading edge and freezing at the trailing edge becomes enriched in the more incompatible elements [Harris, 1957].As yet, we have investigated the evolution only of major elements; however, it is well known that melt-matrix interaction during melt migration can give rise to characteristic trace element signatures [e.g., Richter, 1986;Navon and Stolper, 1987;Kenyon, 1990].Work is underway to predict these signatures and use them as a diagnostic test.

Length-and Time-Scales of Melt Segregation
[35] We have predicted that melt segregation due to buoyancy-driven compaction can occur in a partial melting zone in the crust.However, predicting the length-and timescales over which segregation will occur is complicated because of the large number of dimensional variables which dictate the behavior of the system.It is also complicated by the wide range of uncertainty in the values of some of these variables, particularly the matrix bulk and shear viscosities (x s and m s ), the melt viscosity (m m ), the grain size (a), and the permeability constant (b) (Table 2).The dimensional results are difficult to generalize and we have not attempted to exhaustively investigate them in terms of the dimensional governing variables.However, we have estimated the timescale of segregation and the lengthscale of intruded basaltic magma required as a heat source.We choose a value of k eff and an associated value of y geo , and calculate the dimensionless time and lengthscales for segregation (e.g., Figures 5 and 6).We then permute the values of the dimensional variables which yield this value of k eff , calculating the dimensional time and lengthscales for each combination.Repeating this for the full range of k eff for which segregation is predicted (approximately 10 À2 k eff < 10 4 ) yields segregation times ranging from $4000 years to $10 Myr.These are rapid compared to previous estimates; for example, Wickham [1987] predicts that segregation due to buoyancy-driven compaction occurs at the kilometer scale over timescales >10 8 years.
[36] The corresponding thickness of intruded basaltic magma required for segregation ranges from $40 m to $3 km.This requires that the magma is emplaced over time as a series of sills, concurrent with melt segregation in the overlying source rock.Short segregation times indicate that batches of granitic magma may be generated by only a few underplating events, so repeated underplating will yield numerous individual batches.Longer segregation times indicate that repeated underplating is required to yield a single, large batch of granitic magma.A pulsed magma supply is consistent with the internal contacts and compositional zonation observed in many granitic bodies [Pitcher, 1979[Pitcher, , 1993]].
[37] We illustrate our findings by considering a few specific dimensional results.Consider a partial melt zone with matrix bulk and shear viscosities of x s $ m s $ 10 15 Pa s, melt viscosity of m m $ 10 4 Pa s, grain size of a $ 2.5 mm, permeability constant of b $ 1/500, density contrast of (r s À r m ) $ 600 kg m À3 , initial melt fraction at the contact of j $ 0.5, and values of k, c p , L, (T liq À T sol ), T geo , and r given in Table 2 (in the column labeled "Values Used in Figure 9").Substituting these values into equations ( 14), ( 16), ( 17), (19), and (26) yields a value of k eff $ 36.The resulting porosity distribution is shown in Figure 9a at the time of 8. Melt composition in terms of component oxide mass fractions, as a function of the degree of melting (v m ) and temperature (T), during partial melting of amphibolite: (a) data from Beard and Lofgren [1991], obtained at 1 kbar; (b) data from Rapp and Watson [1995], obtained at 16 kbar.In all cases, plain lines plot on the left-hand ordinate axis; dashed lines plot on the right-hand ordinate axis.incipient magma formation (with CMF $ 0.5).The time required for segregation is $20,000 years, the thickness of the partial melt zone is $900 m and the sill thickness required for segregation is $500 m.The melt in the leading porosity wave has a composition corresponding to a degree of melting of $0.14, which for the amphibolite shown in Figure 8b, melting at 16 kbar, would be trondjhemitic.Increasing the matrix bulk and shear viscosities yields smaller values of k eff , longer segregation times, and a thicker partial melt zone at the time of incipient magma formation (Figures 9b and 9c).Note how much cooler the sill is in Figure 9c when compared to Figure 9a; the reason the minimum sill thickness required for segregation increases so slowly compared to the thickness of the partial melt zone is that the melt segregates much closer to the top of the source region for smaller values of k eff .Consequently, the sill can cool further before inhibiting melt segregation in the overlying rock.
[38] Previous estimates of the time required for buoyancy-driven compaction and melt segregation in the crust have been based on applications of the characteristic compaction time of McKenzie [1985McKenzie [ , 1987]].
in which the porosity f is equated with the degree of melting of the source rock, and h denotes the source region thickness [Wickham, 1987;Wolf andWyllie, 1991, 1994;Petford, 1995].Predicted segregation times are generally very large.  2 for the values of the other governing dimensional variables.Porosity (f) is plotted against the lower abscissa axis; temperature (T) is plotted against the upper abscissa axis.The curve which denotes the temperature (T) also denotes the degree of melting (v m ) if read from the lower abscissa axis.The temperature profile in the upper half of the sill only is shown; in both cases CMF % j % 0.5.In all cases, Ste = 0.625 and c = 2.67.The solidus and liquidus temperatures of both source rock and magma heat source are 1160 and 1560 K, respectively.
For example, consider the time required for melt segregation in the partial melt zone shown in Figure 9b.The model presented in this paper yields a time of $170,000 years to segregate a granitic melt fraction with a composition corresponding to a small degree of melting of $0.02.In contrast, setting f = 0.02, assuming a source region thickness of h $ 1.8 km, and substituting for the values of m m , a, b, and (r s À r m ) in equation ( 33), yields a time of $20 Myr.

Residues of Partial Melting in the Lower Crust
[39] Many exposed lower crustal granulite terrains exhibit geochemical and mineralogical characteristics which indicate that they may be residues left after the in situ extraction of a granitic partial melt fraction, although the evidence is rarely unambiguous [e.g., Fyfe, 1973;Clifford et al., 1981;Clemens, 1989Clemens, , 1990;;Pin, 1990;Vielzeuf et al., 1990].The model presented in this paper predicts that the solid matrix of the partially molten rock will not be disrupted during segregation except in the localized region of magma mobilization, and will remain in the source region as a ''restitic'' residue.The mineralogy of the residue following partial melting of basaltic and meta-basaltic protoliths corresponds to that of many granulite terrains (pyroxene ± plagioclase, quartz, olivine, garnet, magnetite, ilmenite) [Beard and Lofgren, 1991;Rushmer, 1991;Rapp and Watson, 1995].This residue may become gravitationally unstable and undergo delamination, detaching from the crust and sinking into the underlying mantle [e.g., Bird, 1979;Houseman et al., 1981;Kay andKay, 1991, 1993;Nelson, 1991].Basaltic underplating, segregation of a low density partial melt, and delamination of the high density residue, are important processes in the evolution of the continental crust and crust-mantle recycling [e.g., Arndt and Goldstein, 1989;Kay andKay, 1991, 1993;Nelson, 1991].
[40] Many granitic rocks appear to contain restitic material which has been transported with the magma from the source region to the emplacement level [e.g., White andChappell, 1977, 1990;Chappell, 1984;Chappell et al., 1987;Pitcher, 1993].The model predicts that the mobile magma will initially contain a high proportion of restite, but the amount which will become entrained within the magma and subsequently leave the source region is poorly constrained, because it depends upon factors such as the rate at which restite settles out from the magma, the time the magma spends in the source region, and the ascent mechanism.If the restite settles quickly and the magma has a long residency in the source region prior to ascent, then the restite fraction could be very small.Alternatively, if the restite settles slowly and the magma ascends soon after segregating, then the restite fraction could be high.In the following section, we argue that flow through dykes is the most likely ascent mechanism; further work is required to clarify whether dykes are likely to transport large volumes of restite from the source region to the emplacement level.

Implications for Magma Ascent
[41] The magma which forms in the source region must subsequently ascend through the crust to the emplacement level.However, the ascent mechanism is still a source of controversy; the orthodox view is that granitic magmas ascend through the crust as diapirs [e.g., Ramberg, 1970;White andChappell, 1977, 1990;Bateman, 1984;Chappell, 1984, Weinberg andPodladchikov, 1994;Weinburg, 1996], yet magma ascent via diapirism is a slow process limited by the high viscosity of the surrounding crust, and theoretical studies indicate that diapirs cool and crystallize before reaching upper crustal levels [e.g., Marsh, 1982;Mahon et al., 1988].An alternative is that granitic magmas ascend through dykes, faults, or fractures in a manner analogous to that of basaltic magmas [Clemens and Mawer, 1992;Petford et al., 1993Petford et al., , 1994;;Petford, 1996;Clemens et al., 1997;Clemens, 1998].In contrast to diapirism, magma ascent via dykes is a rapid process limited only by the viscosity of the magma [Petford et al., 1993[Petford et al., , 1994;;Petford, 1996].
[42] The ascent mechanism may ultimately be governed by the rheology and geometry of the magma body which forms in the source region.The results presented in this paper are 1-D and cannot be used to quantitatively predict the 3-D geometry of the magma body.Jackson and Cheadle [1998] argue that in a homogeneous, isotropic rock heated constantly from below, the 1-D results may be robust in 3-D at early times, because the rate of upward migration of the leading porosity wave is limited by the rate of upward migration of the solidus isotherm; the lengthscale for flow is constrained by the thermal lengthscale rather than the compaction lengthscale.Connolly and Podladchikov [1998] found that porosity waves are stabilized in 3-D when deformation is controlled by thermally activated creep; in this case, the lengthscale for flow is constrained by the activation energy for deformation rather than the compaction lengthscale.They concluded that lateral flow may occur on greater lengthscales than previously thought.In our model, the mobilized granitic magma would initially form a ''sill'' like layer which may act as a reservoir from which magma is extracted by dykes/fractures (Figure 10).The results of Rubin [1993Rubin [ , 1995] ] indicate that ''isolated,'' granitic magma filled dykes propagating out of a partial melt zone are rapidly halted by freezing of the magma at the tip; however, the presence of a magma reservoir facilitates ''repeated'' dyke intrusion which warms the surrounding rock and promotes the propagation of subsequent dykes.Repeated, localized dyke intrusion would lead to the formation of a localized ascent zone, to which granitic magma could migrate laterally through the ''sill.''As the source region will be laterally extensive, this migration of granitic magma is likely to be significant.Such an ascent zone is similar to that proposed by Ryan et al. [1981] and Ryan [1988] for the ascent of basaltic magma beneath Kilauea Volcano, Hawaii; moreover, granitic plutons are often associated with granitic dyke swarms [e.g., Pitcher, 1979Pitcher, , 1993;;Scaillet et al., 1995], which may represent the surface manifestation of the ascent zone.At later times the magma layer is likely to become unstable, and form a more complex structure in 3-D.This may provide a further mechanism for channeling the melt into a localized ascent zone.

Conclusions
[43] We have presented a new quantitative model of granitic (in a broad sense) melt generation and segregation within the continental crust, assuming that melt generation is caused by the intrusion of hot mantle-derived basalt, and that segregation occurs by buoyancy-driven flow along grain edges coupled with compaction of the partially molten source rock (Figure 10).We applied the model to tectonic settings where high heat fluxes and/or thickened crust promote partial melting, with basaltic underplating acting as both a heat source for partial melting and as a material source for granitic melt.
[44] Our results suggest that significant volumes of granitic partial melt may segregate from a basaltic protolith within geologically reasonable timescales, and that underplating can sustain melting for the length of time required for segregation and mobilization to occur.These findings differ from those of previous studies, which have suggested that compaction operates too slowly in the crust to yield large volumes of segregated granitic melt.However, the results of the model are sensitive to the values of the variables which govern the dynamics of melt segregation; in particular, the bulk and shear viscosities of the partially molten protolith, the melt viscosity, and the grain size.These are poorly constrained, and the limited experimental and theoretical data has been used to estimate suitable values (Table 2).The model predicts that melt segregation and mobilization will occur in a basalt/amphibolite protolith over $4000 years to $10 Myr, that underplate thicknesses of $40 m to $3 km are required to provide sufficient heat, and that the composition of the melt which becomes mobilized ranges from granitic/trondjhemitic to tonaliticgranodioritic.
[45] These results suggest that partial melting of basaltic underplate in continental arc environments may yield large volumes of Na-rich granitic magma within reasonable timescales.This magma can ascend through the crust, perhaps through dykes, faults, or fractures, and be emplaced to form the Na-rich granitic batholiths found in the western Americas, Antartica, and New Zealand.The composition of the predicted residue left in the lower crust corresponds to that of many exposed granulite terrains; this residue may become gravitationally unstable and undergo delamination, detaching from the crust and sinking into the underlying mantle.These processes contribute to the evolution of the continental crust and crust-mantle recycling.
[46] The model results also suggest a possible mechanism for generating the large volumes of TTG rock found within Archean crust [e.g., Jahn et al., 1981;Martin et al., 1983].The origin of these rocks is controversial, with some workers suggesting that they are formed by partial melting of basalt within subducting oceanic crust and others suggesting that they are formed by partial melting of underplated basalt [e.g., Martin, 1999 and references therein].As yet no physical model has been presented to describe the generation and segregation of Archean slab melts.Here we suggest a mechanism for the formation of Archean TTG melts through partial melting of underplated basalt.
[47] Despite its relatively simple nature, this model represents a significant advance on the qualitative and semiquantitative models of granitic melt generation, segregation, and mobilization which have previously been developed, and lays the foundation for more sophisticated models.

Appendix A: Characterization of Permeability
[48] The simple permeability-porosity equation used in this paper (equation 7) is based upon the semiempirical Blake-Kozeny-Carman equation, in which the constant b is chosen to fit experimentally derived permeability-porosity data for a given material [Bear, 1972;Dullien, 1979].This equation is a reasonable fit for permeability measurements on both texturally equilibrated and unequilibrated systems, so long as the percolation threshold f c is accounted for [McKenzie, 1984;Cheadle, 1989;Wark and Watson, 1998] Figure 10.Schematic of a granite source region in the lower crust, which is produced by heating from below following basaltic underplating.The thickness of the partial melt zone increases with time as heat migrates from the underlying basalt magma into the overlying source rock.The porosity distribution in the partial melt zone is governed by the relative upward transport rates of heat and melt; if melt migrates upward more quickly than the top of the source region (defined by the position of the solidus isotherm) then it accumulates at the top of the source region and a porosity wave forms.Eventually, this local porosity exceeds the Critical Melt Fraction yielding a mobile magma.As the melt migrates upward its major element composition evolves to resemble a smaller degree of melting of the source rock (granitic in a broad sense for basaltic and meta-basaltic (amphibolite) protoliths).The accumulated granitic magma may be tapped by dykes and ascend to higher crustal levels.As the source region will be laterally extensive this magma migration to a localized ascent zone is likely to be significant.
(Figure A1).Estimates of the percolation threshold vary: Zhang et al. [1994] estimated a threshold of $0.04 in a texturally unequilibrated calcite aggregate, Wark and Watson [1998] estimated thresholds ranging from $0.005 to 0.027 in texturally equilibrated quartz and calcite aggregates, while Wolf and Wyllie [1991] estimated a threshold of <0.02 in a texturally unequilibrated, partially molten hydrated basalt.Numerical experiments reveal that the effect of a percolation threshold of 0.04 is negligible on solutions of the governing equations for which magma mobilization is observed, because the porosity at the leading edge of the leading porosity wave does not fall to the percolation threshold until late times, by which time the wave is well developed.Based on this, all solutions presented in this paper are obtained using the permeabilityporosity equation ( 7) over the entire porosity range.The maximum and minimum values of b given in Table 2 are based upon the permeability-porosity relations shown in Figure A1, and the range of values include that proposed by McKenzie [1984] of b = 1/1000, based upon permeability-porosity data obtained experimentally by Maaloe and Scheie [1982].and Tullis, 1988].We assume that deformation in the partially molten lower crust occurs by melt-enhanced diffusion creep, although the available data is very limited.The experimental work of Kohlstedt and Chopra [1994] indicates that melt-enhanced diffusion creep governs the rheology of olivine aggregates containing $9% basaltic melt, at temperatures of $1573 K, for strain rates of $10 À10 s À1 and a grain size (diameter) of 1 mm.Given that maximum strain rates predicted by the model are $10 À13 -10 À20 s À1 , up to 7 orders of magnitude less than the strain rate used by Kohlstedt and Chopra [1994], the assumption of meltenhanced diffusion creep appears reasonable, although an olivine-basalt system is clearly not an ideal analogue for partially molten basalt/amphibolite.
[50] Melt-enhanced diffusion creep is a form of grain boundary (Coble) creep, in which the microscale diffusion rate of components along solid-solid grain boundaries governs the macroscale creep rate [Pharr and Ashby, 1983;Cooper and Kohlstedt, 1984;Ranalli, 1987].The rheology of an aggregate undergoing steady state deformation by diffusion creep is Newtonian [Cooper and Kohlstedt, 1984;Ranalli, 1987].For the simplest case of a single component, equigranular aggregate deforming by grain boundary creep, the shear viscosity may be expressed as [Ranalli, 1987] where k b is Boltzmann's constant (= 1.381 Â 10 À23 J K À1 ), is the atomic volume of the material, and Dg is the effective grain boundary diffusivity.If the aggregate contains a small fraction (<0.1) of spherical pores, the bulk viscosity may be expressed as [Arzt et al., 1983] x [51] Experimentally measured values of Dg are >10 À21 m 2 s À1 for calcium in anorthite at a temperature of 1273 K and atmospheric pressure [Farver and Yund, 1995], but have not been measured for other species or for the other major minerals found in basalt/amphibolite.Using the value of for anorthite [6.69 Â 10 À28 m 3 ; Lide and Frederiske, 1998], assuming a temperature of 1273 K, and assuming Dg $ 10 À21 , equation (B1) yields a maximum shear viscosity of $10 18 Pa s for a grain size (radius) of 2.5 mm, and a minimum shear viscosity of $10 15 Pa s for a grain size (radius) of 0.25 mm.Equation (B2) yields a maximum bulk viscosity of $10 19 Pa s for an aggregate with a grain size (radius) of 2.5 mm and a porosity of f = 0.01, and a minimum bulk viscosity of $10 15 Pa s for an aggregate with a grain size (radius) of 0.25 mm and a porosity of f = 0.1.
[52] It is not clear how these calculated viscosities relate to the viscosities of partially molten basalt/amphibolite as the diffusion controlled creep of polymineralic aggregates is poorly understood.Ranalli [1987] states that when more than one diffusing component is present, the slowest moving controls creep, yet Wheeler [1992] argues that components may mix and interact during diffusion, leading to complex behavior which may significantly enhance diffusional creep rates.Neither of the equations for the shear and bulk viscosity includes the effect of melt in the pore spaces, which should enhance the creep rate [Cooper and Kohlstedt, 1984;Dell'Angelo et al., 1987], although during compaction the melt must be expelled from the pores and the effect of this on the bulk viscosity is unknown [Dingwell et al., 1993].Moreover, the spherical pores assumed in the derivation of equation (B2) are not likely to be a good approximation of the interconnected pores of a partially molten rock [McKenzie, 1984].However, in the absence of experimental data for either the bulk and shear viscosities of partially molten basaltic rock, our calculated range of shear and bulk viscosities will be used provide order of magnitude estimates.The values of the shear and bulk viscosities given in Table 2 are based upon these estimates.

Appendix C: Critical Melt Fraction
[53] The CMF is important because it dictates the porosity at which a partially molten rock disaggregates and forms a mobile magma.If the rock is in textural equilibrium, then the value of the CMF depends only upon the contiguity of the matrix [Miller et al., 1988].Contiguity is a quantitative measure of the solid grain-grain interconnectivity, and depends upon the fluid volume fraction, the microscopic distribution of fluid throughout the mush, and the solid grain size distribution.The contiguity of a texturally equilibrated, monomineralic, equigranular mush of isotropic grains as a function of porosity and dihedral angle has been derived theoretically by Cheadle [1989], and for appropriate dihedral angles of 30°-60°, yields values for the CMF of 0.37-0.46.These represent ''minimum'' values, because at porosities greater than the minimum energy porosity the melt becomes inhomogeneously distributed and does not all contribute to reducing the contiguity.For partially molten rocks which are not in textural equilibrium, experimentally and theoretically obtained estimates of the CMF range from 0.2 to 0.7 [e.g., Arzi, 1978;van der Molen and Paterson, 1979;Philpotts and Carroll, 1996].The values of the CMF given in Table 2 are based upon these estimates for both equilibrated and unequilibrated rocks, with the exception of the lowest estimate of 0.2 ± 0.1 presented by Arzi [1978], which was obtained theoretically using simple arguments based upon the Einstein-Roscoe equation for rigid particles suspended in a viscous, Newtonian melt.This is a poor approximation of a partially molten rock.Also, the highest estimate of $0.7 obtained experimentally by Philpotts and Carroll [1996] for a partially molten tholeiitic basalt has been reduced to 0.65.They performed their experiments at compressive stresses of the order of kilopascals, yet compressive stresses in the lower crust are of the order of gigapascals, and it is unlikely that their sample would have maintained its strength at such high melt fractions for stresses of this magnitude.

Figure 9 .
Figure 9. Porosity (f) and temperature (T) against vertical distance (z) at the time of incipient magma formation, for (a) m s $ x s $ 10 15 Pa s (k eff = 35.6,y geo = 1.67 Â 10 À3 ); (b) m s $ x s $ 10 17 Pa s (k eff = 3.56, y geo = 1.67 Â 10 À2 ); (c) m s $ x s $ 10 19 Pa s (k eff = 0.356, y geo = 0.167).See Table2for the values of the other governing dimensional variables.Porosity (f) is plotted against the lower abscissa axis; temperature (T) is plotted against the upper abscissa axis.The curve which denotes the temperature (T) also denotes the degree of melting (v m ) if read from the lower abscissa axis.The temperature profile in the upper half of the sill only is shown; in both cases CMF % j % 0.5.In all cases, Ste = 0.625 and c = 2.67.The solidus and liquidus temperatures of both source rock and magma heat source are 1160 and 1560 K, respectively.

Table 2 .
Summary of the Parameters Used in the Model, With Suitable Values for Hydrated Basalt and Meta-Basalt (Amphibolite and Eclogite) Source Rocks a Data from Clauser and Huenges

Table 3 .
Range of Available Values for the Dimensionless Initial Temperature Gradient y geo , and the Dimensionless Effective Diffusivity k eff , Obtained Using Equations (26) and (31) and Values of the Constituent Variables (x s , m s , etc.) From Table1 a