Simulations of crustal anatexis: Implications for the growth and differentiation of continental...

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. B10, PAGES 22,629-22,648, OCTOBER 10, 1997

Simulations of crustal anatexis: Implications for the growth and differentiation of continental crust

Federica Raia and Frank J. Spera Department of Geological Sciences and Institute for Crustal Studies, University of California, Santa Barbara

Abstract. Them is overwhelming and incontrovertible petrological and geophysical evidence for the significant role played by mantle-derived mafic magma in the generation and growth of continental crust. Likewise, intrusion of mafic magmas beneath or into continental crust is very likely the major source of enthalpy that drives intracrustal differentiation. A simple dynamical model has been constructed to examine the critical factors that govern the evolution and timescale of crustal anatexic events when driven by basaltic magma underplating or injection into crust. Critical factors include the intensity of the enthalpy input (i.e., the power dissipated) and the rheological propertie.s, bulk composition and compositional structure of the source rocks undergoing partial fusion. We have performed numerical simulations to evaluate these factors using phase equilibria and thermochemical and transport property data applicable to the binary eutectic system CaA12Si2Oa-CaMgSi20 6 as a rough analog to study anatexis of mafic lower crust. The role of enthalpy power is tested by varying the enthalpy (or temperature) along the base of the crustal block while maintaining a fixed temperature at the top. As ratio rbot/rto p increases modestly from 1.05 to 1.15, the average fraction of melt at steady state in the anatexic region increases from 50% to 75%. Time to attain steady state scales inversely with rbot/rtop with an increase by a factor of 2 for a 10% decrease in rbot/rtop. Typical anatexic timescales are in the range 103 to 105 years for length scales in range 102 to 103 m. The consequences of different rheological models, especially the importance of Darcy percolative flow relative to en masse flow within the partial melt region was also investigated. A value of the solid fraction called the critical value (fScrit) is set such that for 0 < fs < fScrit no relative motion is allowed between solid and melt. The relevant viscosity is the viscosity of the magmatic suspension which is a function of the local value offs. In contrast, for values offs such that fScrit <fs < l, momentum transport is accomplished by Darcy flow so that solid is considered immobile and melt is free to percolate through the solid matrix of varying composition. Numerical experiments have been performed for fScrit of 0.7, 0.5, 0.3 and 0.0 to assess the importance of the rheological model. At high values of fScrit (e.g., > 0.5) relatively large volumes of nearly homogeneous melt are rapidly generated. In distinction, forfscrit small, more enthalpy is stored in the solid and less melt is produced. However, the compositional spectrum of liquids generated within the anatexic region is significantly larger for smallfscrit. Because of the sensitive dependence of solidi and liquidi on bulk composition, melt productivity is related to bulk composition given a particular enthalpy power input and rheological model. There is a factor of 2 difference in volume of melt generated when the bulk composition of the source is 10 modal percent less refractory. Compositionally zoned melts form by melting of either homogeneous or layered sources, although composition- frequency relations are sensitive to the initial (subsolidus) compositional structure. The effects of anatexic events are to fundamentally reorganize the pattern of compositional structure within the crust. Intracrustal differentiation is a long-term inevitable process associated with the underplating or injection of mafic magma into preexisting crust.

Introduction

An understanding of the origin and mechanisms of continental crust growth is an outstanding geological problem. Geochronological data suggest crustal formation rates have been nonuniform, perhaps quasiperiodic, throughout geologic time [Moorbath, 1977, 1978; Taylor and McLennan, 1981, 1985, 1995; Anderson, 1994; Condie, 1993]. Large regions, such as the Superior Province in North America, have been assembled in rather short periods of time (circa 150 Ma [Hoffman, 1988]). It

Copyright 1997 by the American Geophysical Union

Paper number 97JB01589. 0148-0227/97/97JB-01589509.00

has been argued that geologic history is punctuated by episodes of enhanced exchange between the upper (d < 670 km) and lower (d > 670 km) mantle during which rapid continental growth takes place [Stein and Hofmann, 1994]. A model involving quasiperiodic upper/lower mantle exchange is consistent with the results of three-dimensional mantle convection simulations of a

variable-property fluid subject to the olivine-spinel and spinel- perovskite transitions near 410 km and 670 km, respectively [Christensen and Yuen, 1984; Machetel and Weber, 1991; Solheim and Peltier, 1994; Tackley et al., 1993; Lay, 1994; Steinbach and Yuen, 1994]. Presently, however, the history of mass exchange between the upper and the lower mantle, especially the mass flux between these reservoirs and its fluctuation on short and long timescales, is unclear [Christensen,

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22,630 RAIA AND SPERA: CRUSTAL ANATEXIS SIMULATIONS

1995]. Rare gas elemental and isotopic considerations impose restrictions on the extent and frequency of upper/lower mantle mass exchange and are only partly in agreement with dynamical simulations [O'Nions and Tolstikhin, 1996]. A reasonable working hypothesis is that a relationship exists between the net growth and in situ differentiation of continental crust [Wolde et al., 1996], the assembly, uplift [Veevers, 1995] and dispersal of supercontinents and the history of convective mass exchange between the upper and lower mantle. The important point is that mafic magma, generated within the upper mantle, is plausibly the critical agent providing the connection between mantle and continental crustal evolution.

The apparent rate of net growth of continental crust throughout geologic time can be approximated by examination of the geological record coupled with precise geochronological and isotopic data [Taylor and McLennan, 1995; McCulloch and Bennett, 1995; Bowring and Housh, 1995; Sylvester et al., 1997]. There is some consensus that the crust grows episodically and that a significant fraction of its present-day volume of 7.5 x 109 km 3 was in place by the late Arcbean (2.7 Ga). It appears unlikely that continental crust preservation was significant in the interval between the end of Earth accretion (-4.5 Ga) and the termination of intense meteoritic bombardment around 3.9 to 3.8

Ga [Taylor, 1992], although a few scraps of very ancient crust have been found [Bowring et al., 1990]. Crustal volume data give an apparent average rate of net crustal growth of about 3.5 km3/yr for the post-Hadean Archean between 4.0 and 2.5 Ga. This rate is significantly greater than the net apparent rate of crustal growth within the last billion years of about 0.8 km3/yr [Nakamura, 1974; Fujii, 1975; Crisp, 1984]. Indeed, Nb/U ratios in 2.7 Ga basalts from the Yilgarn Craton, Australia, have been used by Sylvester et al. [1997] to argue that a large amount of continental crust had already formed during the first- 2 Ga of Earth history. This is in accord with the Armstrong [1968, 1991] model of rapid and early (Hadean) crustal growth with later new crustal growth balanced by mantle recycling. In this view, the apparent net growth of continental crust is a preservation artifact. Although details remain obscure, the rate of crustal growth was probably significantly greater in the first half of geologic history compared to more recent times. Formation of a now destroyed basaltic protocrust from the Hadean mantle before formation of the continental crust as suggested by Galer and Goldstein [ 1991] serves only to amplify this conclusion.

There are two basic means whereby continental crust may grow: lateral and vertical accretion. Subduction of hydrothermally altered oceanic crust gives rise to arc magmatism. Andesites, island arc tholeiites and other subduction-related rocks

are mainly derived from the mantle wedge above subduction zones. A contribution due to sediments cannot be ruled out and

indeed in some cases can be demonstrated by Be isotopy [Brown et al., 1982; Tera et al., 1986]. Melting of the slab itself may have been important in the Arcbean [Defant and Drummond, 1990, 1993; Martin, 1994; Pearce and Peate, 1995] when steeper geothermal gradients prevailed at least in some spatially avero. ged sense. Thickened crustal sections of island arc type are ultimately laterally accreted onto preexisting crust due to plate convergence. In the oceanic realm, piles of thickened mafic crust from plume-generated (?) oceanic island basalt provinces (e.g., Hawaii) and oceanic plateaux (e.g., Ontong-Java) can be added by lateral tectonic accretion onto preexisting cratons generating new, predominantly mafic, continental crust. Lateral accretion relies ultimately on plate tectonic kinematics to amalgamate thickened sections of nascent crust onto preexisting older cratons.

In distinction, vertical accretion involves the intrusion of

magma beneath, into or onto preexisting continental (or oceanic) crust. Decompression partial fusion of a peridotitic source generates basaltic magma with a density that depends upon composition, pressure, crystallinity and temperature. A typical value is 2600 kg/m 3. In contrast, crustal densities typically lie in the range 2600 kg/m 3 < p < 2900 kg/m • although, of course, the crust is very heterogeneous and values outside this range are not uncommon [Glazner and Usslet, 1988]. The heterogeneity of continental crust is consistent with the stagnation of mafic magmas at virtually any depth or eruption upon the surface. Continental flood tholeiite provinces, with sustained eruption rates as large as several km 3 yr -1 are examples of the latter [Peng et al., 1994]. Perhaps more commonly, rising magma is trapped within the crust at depths corresponding to neutral buoyancy. Buoyant trapping of magma near the Moho is referred to as underplating. Ryan [1987] has suggested that the Moho is a likely place for trapping dense primitive magma due to the large- density contrast between crust and mantle. The level at which mafic magma rises depends most critically upon its density and the density and thermal structure of the crust into which it is emplaced. Underplating of the crust by ponded mafic magmas is an important source of enthalpy power needed to accomplish intracrustal differentiation. As is well known, upper crust (-66% SiO2) is significantly depleted in Mg, Ca, Ti and Fe and enriched in Na, Si, and K compared to more mafic lower crust (-53% SiO2) [Taylor and McLennan, 1995]. Derivation of the upper crust by lower crustal anatexis almost certainly is a major mechanism of intracrustal differentiation. Fractional

crystallization of mantle-derived magmas generated by partial t'usion of peridotite can produce both evolved compositions and gabbroic source materials which can then undergo yet additional phases of anatexis driven by renewed or additional mafic magma input.

Considerable effort has been made to understand crustal

magmatism driven by basaltic injection and underplating [Bergantz, 1989; Bergantz and Dawes, 1994; Huppert and Sparks, 1988]. Its characterization is difficult because of the variability of temporal and spatial scales involved and the variety of controlling factors. Perhaps one of the most critical factors is the thermorheological relationship of the relevant materials [Sparks and Marshall, 1986]. The most significant material parameters are the solidus and liquidus temperatures and the variation of viscosity in the melting interval for both the intruded magma and the crustal source undergoing anatexis. These characteristic temperatures are composition and pressure dependent and can sometimes be nearly identical (e.g., intrusion of basaltic magma into preexisting gabbro) or quite different. In some situations, such as injection of mafic, primitive melts into granitic crust, the melting intervals may not even overlap, although fluid-mediated mass exchange may occur. An extreme case which maximizes magma production occurs when the solidus of intruded mafic magma exceeds the liquidus of its crustal host. Large-scale melting of the crust above the trapping level will then occur and presumably lead to further differentiation and the development of density-stratified upper crust [e.g., Mueller, 1995].

Additional evidence for the commonality of the underplating mechanism comes from examination of the thermal evolution of

granulite facies terranes from the vantage of geothermometric and geobarometric studies [e.g., Bohlen, 1987; Harley, 1989]. An important class of granulite terrains are those which exhibit anticlockwise pressure-temperature-time (PT-t) paths. These

RAIA AND SPERA: CRUSTAL ANATEXIS SIMULATIONS 22,631

paths are indicative of prograde metamorphism with an early rapid increase in temperature at relatively low pressures succeeded by pressure increase and ultimately by cooling. Such paths evidently demand magmatic heat input early in the evolution of the process. More directly, field studies of both volcanic and plutonic rocks have long pointed to a strong connection between mafic magma intrusion, magma commingling and hybridization and the genesis of intermediate to silicic magmas [e.g., Wiebe, 1973; McGarvie, 1984; Furman and Spera, 1985; Sparks and Marshall, 1986; Bacon, 1986; Elburg, 1996]. The extremely common occurrence of mafic enclaves within granitic plutonic rocks as well as the great abundance of geochemical data from volcanic terrains for magma mixing is prima facie evidence for the connection between intrusion of mafic magmas and anatexis.

In light of the important role played by underplating and injection of mantle-derived basaltic magma in the development and differentiation of continental crust, a simple computational magma dynamical model is explored in this paper to reveal critical factors governing the evolution and timescale of crustal anatexis. This is an area of rapid recent progress, and the development of more sophisticated modeling efforts are actively underway. The studies of Bergantz and coworkers [Barboza and Bergantz, 1996, 1997; G.W. Bergantz and J. Ni, A numerical study of crystal sedimentation in basaltic magma chambers, submitted to International Journal of Multiphase Flow, 1997] are exemplary in this regard.

Crustal Anatexis: An Overview

A number of questions arise when one considers the dynamics and energetics of anatexis. How efficient is the process of partial melting? How important is the convection process during the melting episode? What are the scales of the processes that enable one to predict heat transfer and rates of melting? What role is played by the theological properties of the region undergoing partial fusion? How does the bulk composition of the crustal source influence the vigor and style of convection? How long are anatexic events and how does the duration of one depend on the rate of supply of enthalpy? In this paper we address some of these issues using a fluid dynamical model previously developed [Oldenburg and Spera, 1990, 1991, 1992a, 1992b; Spera et al., 1995]. A cartoon of the scenario investigated is shown in Figure 1; boundary and initial conditions are portrayed in Figure 2. A region of mafic crust, either homogeneous or compositionally layered is modeled by specifying a composition Co in the system CaMgSi206-CaA12Si208 (hereinafter Di-An). Initially, the temperature of this crustal section T o is fixed at some constant value below that of the eutectic (solidus) T,, < Tso 1. At t > 0, the bottom of the domain (y = 0) is set at some temperature greater than the eutectic (or solidus) temperature Tso 1. The temperature at the top (y = 1) is maintained at the initial value Tto p = T o. Given these thermal boundary conditions, the steady state condition will be one of a partially molten domain since the condition T < Tso • at y = 1 precludes total melting throughout the domain. The simulations enable one to follow the detailed history of a partial melting episode in terms of the solid fraction fs, temperature T, bulk mixture composition C, liquid or melt composition C•. and velocity (both mixture and liquid, V and Vt., respectively) and other parameters through time within the anatexic region. The boundary condition set along the top at y = 1 implies that this surface is a perfect conductor. This was chosen so that total melting would not occur throughout the domain.

Figure 1. Composite cartoon cross section of underplating scenario. Magma stagnation is depicted for two likely places: the Moho and the upper/lower crust boundary. Horizon of stagnation depends most critically upon buoyancy/pressure forces and thermal effects. As heat initially stored in underplated magma is delivered to the crust, anatexis can occur. Heat transfer by both conduction and advection takes place in the source region of newly generated anatexic magma. No specific scale is intended by this diagram.

The critical factors that govern the course of crustal anatexis include the intensity of the heat power input, the rheological properties of the source rocks and the bulk composition and compositional structure of the source materials undergoing partial fusion. We have performed simulations to evaluate these factors using phase equilibria, thermochemical and transport property data applicable to the binary eutectic system CaAI2Si208- CaMgSi206 at 0.1 MPa as a model to study anatexis of mafic (nonpelitic) crust. The effects of pressure on the composition of eutectic melt and on the temperature of the eutectic are not taken into account in this simple model. A binary isobaric eutectic system is clearly a simplification of complex geologic reality [Barboza and Bergantz, 1997]. Simulation results, however, are broadly compatible with natural patterns of magma production and compositional evolution observed for many continental igneous provinces and provide a flamework for further thought, a focus for new observations and enable one to grasp the essential mechanics of the process in a semiquantitative way.

The role of heat input (or enthalpy) power is tested by varying the enthalpy (or temperature) along the base of the crustal block while maintaining a fixed temperature at the top. The ratio Tbot/Tto p is a proxy for the rate of which heat is delivered to the anatexic region by latent and sensible effects associated with the crystallization and cooling of mafic magma emplaced beneath the region undergoing partial fusion. Relatively small differences in this ratio lead to significantly different geological scenarios because the location of magma stagnation by heat death sensitively depends on the thermal structure of the host crust. Results show that a 10% difference in the ratio T•ot/Tto p has a demonstrable effect on the rate and style of crustal anatexis with implications for the composition of anatextic melts, the duration of anatexic events, and the eventual (subsolidus) pattern of crustal-scale compositional zonation.

The consequences of different rheological models, especially the importance of Darcy percolative flow relative to solid plus crystal en mass (pseudofluid) flow within the anatexic region was investigated by implementation of the hybrid momentum (HM) transport model of OMenburg and Spera [1992a]. In the HM model a value of the solid fraction called the critical value OCScrit)

22,632 T = Tto p 3C/3y = u = v = 0

y=l [

•v/•x = 0 [ T(t=O) = T O u =0 0.5 •

•T/•x = 3C/•x = 0 C(t=0) = C O or C(t---0) = fly)

T = Tbo t 3t2/3y = u = v = 0

x=l

8v/3x = 0

u=0

•T/•x = 3C/•x = 0

Figure 2. Physical domain, coordinate system, initial and boundary conditions used in the simulations are portrayed. The domain is infinite in the horizontal direction (reflective boundary conditions) with rigid and impermeable top and bottom boundaries at y = 1, 0. The top boundary temperature Tt,,p is held at a fixed value, equal to the initial temperature of the crustal block To which is always less than the solidus (eutectic) temperature of the system CaAI2Si208-CaMgSi206. The lower boundary temperature T,•,t is set at a value greater than the solidus depending on the specific simulation. At t = 0 the crustal block is isothermal (T = T,,), completely solid (fs = 1.0) and motionless (u = v = 0). The initial composition of the crustal block is either the same everywhere (homogeneous case) with C = Co or possesses some a priori imposed structure C = C(y). When subblock heterogeneity is imposed, the compositional boundary occurs at y = 0.5, the half-height of the domain.

is set such that for 0 < J3 <fScrit no relative motion between solid and melt is allowed. That is, solid is entrained by the melt and the relevant viscosity is the viscosity of the magmatic suspension which depends on the local value of fs through some empirical relation [e.g., Metzner, 1985; Ryerson et al., 1988; Lejeune and Richet, 1995]. For values of fs such that fScrit < fs < 1, momentum transport is accomplished by Darcy flow: solid is considered immobile and melt is free to percolate through the solid matrix in response to local pressure, buoyancy and viscous forces. Numerical experiments have been performed for critical solid fractions of 0.7, 0.5, 0.3 and 0 to explicitly test the importance of the rheological model on the evolution of anatexis. The case with fscrit set equal to 0 is called the relative motion (RM) model since relative motion between solid and melt is

allowed throughout the melting interval, clearly an extreme case. Results of simulations with different values of the critical solid

fraction vary substantially. Simulations utilizing fs - 0.5 _+ 0.15 appear most consistent with geological and laboratory observations and the naturally observed timescales for anatexic events.

The final factor investigated is the dependence of partial melting evolution on the bulk composition and compositional structure of the region undergoing anatexis. Because of the dependence of solidi and liquidi on bulk composition, melt productivity ought to be sensibly related to the initial (solid) composition of the source region given a particular heat power input and rheological model. This also has implications on "self- organization" of an initially homogeneous mafic crust into a heterogeneous one as observed on Earth.

The remainder of this paper is organized as follows. A nonmathematical description of the fluid dynamic model is presented in the next section. Results of simulations performed to study the effects of heat power input, magma rheology and source bulk composition are then presented along with some experiments on intracrustal differentiation. In the last section a summary is presented.

Model

The model is based on the assumption of local thermodynamic equilibrium and accounts for solidified, mushy (two or three

phase) and all-liquid regions self-consistently. Latent heat effects, percolative flow of melt through a variable-permeability mush and the variation of system enthalpy with composition, temperature and solid fraction are taken into account. Momentum transport is accomplished by Darcy percolation in solid-dominated regions where the local solid fraction fs exceeds a critical value denoted f$crit and by internal viscous stress diffusion in melt-dominated regions where the local solid fraction is less than f$crit. Physically, in melt-dominated regions, relative motion between solid and melt is not allowed; the mixture advects as a single fluid (pseudofluid) with a viscosity that depends on the local crystallinity which in turn is linked nonlinearly to the time-dependent enthalpy field. In regions with f$ > f$crit, it is assumed that solid is immobile and melt percolates through the matrix-supported crystal network at a rate determined by the balance of Darcian drag and buoyancy forces as enforced by momentum conservation. Energy conservation is written in terms of a mixture enthalpy equation with subsidiary expressions, based on thermochemical data and phase relations, that relate the mixture enthalpy to temperature, composition and phase abundances at each location. Species conservation is written in terms of the low-density component CaAI2Si208 and allows for advection and diffusion as well as the relative motion between

solid and melt. The nondimensional form of the conservation

equations and parameters of this problem are given in the Appendix along with some auxiliary information. Details of the numerical methods are given by Oldenburg and Spera [1990, 1991, 1992a, b] and Spera et al. [1995] and are not repeated here.

The physical significance of the dimensionless parameters of this problem must be interpreted with care and not confused with their classical interpretation. For example, in a single-phase fluid with simple boundary conditions and no phase change, the melt convective velocity is a simple function of a dimensionless group known as the Rayleigh number, Ra, a measure of the importance of thermal buoyancy forces relative to the impeding effects of viscosity. In the present problem, the irregular time-dependent configuration of the solid, melt, and mushy regions and the dependence of the fluid enthalpy on temperature, solid fraction, and composition precludes the establishment of a single set of simple (universal) scales for length, temperature, and composition. The assumption of local thermodynamic


equilibrium and the use of a phase diagram for the system precludes a straightforward analysis such as that commonly done in simpler problems [e.g., Trial and Spera, 1990]. For example, in the two-phase region the thermally and compositionally driven flows are not independent but instead are linked through the phase diagram and the thermochemical properties of the appropriate phases. In the mush region, the ratio of chemical to thermal buoyancy depends on the slope of the liquidus curve. However, in the all-liquid region, the buoyancy ratio is not explicitly fixed by phase relations. Because of these complexities, it is not particularly useful to discuss the evolution of the system in terms of simple classical dimensionless numbers. In fact, such a simple analysis would be misleading.

It must be emphasized that the present model describes some very complicated processes. The phenomena described by the model may occur over the entire range of macroscopic length scales. These can be as small as crystals (millimeters to centimeters) or as large as linear dimensions of anatexic regions (meters to kilometers). Solutions of the equations describing the process of convective heat and mass transport becomes computationally more difficult as the length scale of the system becomes larger. Such calculations of convection are limited to values of the Rayleigh number that are small relative to those of anatexic regions. Except for the length scale, all the parameters in Ra are fixed in terrestrial systems by the choice of the components of the phase diagram and the general range of temperatures of the system. Employing the correct thermophysical parameters for the CaAI2Si208-CaMgSiO 6 binary system in Ra, we find the length scale L to be of the order of centimeters even for the largest values of R a for which calculations can be performed. This limitation to small system should not be seen as a serious defect. In fact, the model

describes processes that would occur at larger length scales and large values of Ra. Furthermore, the crucial coupling between the processes is described, and this coupling is part of any phase change problem, regardless of length scale [Spera et al., 1995].

Although extensions to the continuum model have been discussed to account for non-equilibrium effects, solid transport (crystal floatation and settling) and stress effects due to shrinkage or expansion due to phase change [Ni and lncropera, 1995a, b], these phenomena are not modeled in the present simulations in order to focus attention on the major factors governing the anatexic process, that is, the role of enthalpy power, magma rheology and source composition. The recent work of Barboza and Bergantz [1997] allows more complex phase relations such as multiple invariant points to be included. This is especially relevant for anatexis of pelitic bulk compositions where phase relations are more complicated than for anatexis of mafic crust.

The computational domain, initial and boundary conditions are portrayed in Figure 2. The crustal source region is infinite in the horizontal direction with reflective boundary conditions along the vertical boundaries at x = 0 and 1. The lower and upper boundaries of the domain at y = 0 and 1, respectively, are impermeable, rigid and isothermal. Along the upper (y = 1) boundary, temperature is held constant at a value below that of

the solidus (Tto p < rsol) and equal to the initial temperature set throughout the crustal block at0<x < 1 and0<y < 1. An upper surface of fixed enthalpy corresponds to a perfectly conducting boundary. One way this could occur in nature is if a rigorous meteoric-hydrothermal system lies above the anatexic region such that heat transported to the top of the anatexic region is rapidly advected away by meteoric fluids [e.g., see Spera et al., 1982, p. 8759]. Presumably, this would be more likely in regions of the crust undergoing extension [Taylor, 1987]. The

temperature along the lower boundary (y = 0) is set at a temperature greater than the solidus (Tbot > Tso 0 at t > 0 and held at that wilue. The initial composition of the crustal block is set at Di80 (Co = 0.2) except for the cases studied to examine the role of compositional layering within the source. The liquidus temperature of Di80 is 1620 K. The phase diagram used in the simulations is shown in Figure 3. The dependence of density on melt composition is also shown there. Note that the density of melt decreases along an isotherm as melt becomes enriched in CaA12Si20 8, the so-called light component. The salient parameters for all of the simulations presented here are gathered in Tables 1 and 2. A uniform grid of 40 x 40 nodes and a nondimensional time step of 10 -4 is used in all of the calculations. Several calculations using nonuniform 60 x 60 and 70 x 70 grids and a time step of 5 x 10 -5 produced results which differed only marginally from the 40 x 40 grid results.

Results

Role of Heat Power Input

In the first set of experiments, denoted H70, H140 and H210

(see Table 1), all parameters are identical except for Tbo t and Tto p, or equivalently, the fixed value of the enthalpy along the lower and upper boundaries. A useful way to view these differences is

by comparison of the ratio Tbo t/Tto p which varies from 1.05 to 1.10 to 1.15 for H70, H140 and H210, respectively. In experiment H70 there is a 70 K difference in temperature across the region; this corresponds to a lower boundary temperature 60 K above the eutectic. For H140 and H210 the lower boundary is held at temperatures 100 K and 130 K above the solidus. In all cases, the enthalpy of the top boundary is chosen to correspond to a subsolidus temperature. This insures that in the final state a rigid solid lid of pristine Di80 exists throughout the course of the

T (K)

Di 20 40 60 80

? /

1670 --

* •'ø/r'•/o•9 An+Liq 1620

1570

An

An + Di 1520 I I I I I I I I I

D i 20 40 60 80 An

CaMgSi206 CaAI2 Si208

Figure 3. Phase diagram of system CaAI2Si2Os-CaMgSi20 6 at 0.1 MPa after Osborn [1942]. Isopycnals were computed from data of Lange and Carmichael [1990]. In the melting interval, temperatures are continuous, but the equilibrium solid fraction is discontinuous. For example, for bulk composition Di80 the value of fs changes from unity at T < Tso I to 0.55 at T = Tsm when sufficient latent heat is supplied.


Table 1. Parameters for First Set of Experiments

T,,, K C o, % An Tb,,t, K Tto p, K Tb,,t/Tt,,p H•o t, J/kg Hto p, J&g H70 1508 20.0 !608 1538 1.05 !.96 X 10 6 1.55 X 10 6 HI40 1508 20.0 1648 !508 1.10 2.00 X 10 6 1.52 X 10 6 H210 1468 20.0 !678 1468 1.15 2.04 X 10 6 1.48 X 10 6 C35 !538 27.5 1608 1538 1,05 1.96 x !06 1.55 x 106 C42 1538 31.0 1608 1538 1.05 1.96 X 10 6 1.55 X 10 6 C20 1538 31.0 1608 1538 1.05 1.96 x 106 1.55 x 106

•,,•, K •,q, K 1548 1620 1548 1620

1548 1620 1548 1603

1548 1598 1548 1598

T m, K Ra Rr St

1665 4.45 x !04 2.31468 2.09x 104 1665 4.45 x 104 2.31468 2.09x 10 '• 1665 4.45 x 104 2.31468 2.09x 10 '• 1665 3.4 x 104 3.02941 1.61 x 104 1665 3.9x 104 3.33117 1.47x 10 '• 1665 3.9x 104 3.33117 1.47 x 104

anatexic event. The critical solid fraction (fScrit) is equal to 0.5 in all three cases. The point of this series of numerical experiments is to assess the role of heat or enthalpy intensity (or enthalpy power) on the evolution of anatexis. Boundary conditions were chosen so that the steady (final) state is one of partial, not total, melting.

In Figure 4 the time series of the average solid fraction and averaged specific kinetic energy of the mixture (solid plus melt) is shown. The average melt fraction at steady state increases from 50% to about 75% as Tbot/Tto p increases from 1.05 to 1.15, although it does so nonlinearly. The evolution of the specific kinetic energy (KE) for the three cases is also quite distinct. There is a factor of about 10 2 difference in average mixture velocities (plotted in terms of the spatially average specific kinetic energy) between the most sluggish case (H70) and the most energetic one (H210). Note that the temperature of the lower boundary in case H70 is below that corresponding to the liquidus temperature of the bulk composition (Di80). In this case, a region of solid-free (superheated) liquid never develops. Motion is very sluggish and accomplished mainly by Darcy percolative flow. Melting is practically one dimensional for numerical experiment H70 because little heat, momentum or light-component is carried about by fluid advection. In contrast, for H140 and H210, the basal temperature exceeds the Di80 liquidus temperature by 28K and 58K, respectively, and a significant portion of the domain becomes totally molten in the steady state. The marked effect of small changes in the average solid fraction on the velocity of convection can be discerned by examination of Figures 4 and 5.

The time for the system to reach steady state is inversely proportional to the temperature difference across the region. For example, for H70, H 140 and H210 the times to attain steady state decrease from nondimensional times 0.5, 0.35 and 0.2,

respectively. Nondimensional times are related to dimensional ones by the relation • = ttc St/L 2 where t, tc, St and L represent the dimensional time, thermal diffusivity, Stefan number and thickness of the crustal source region, respectively (see Appendix and notation for definition of parameters). Based on a length scale of 1 km, a nondimensional time of œ = 1.0 corresponds to approximately 200 kyr for the appropriate value of the Stefan number (see Table 1). This order of magnitude estimate shows how quickly geological melting events can occur. For example, if the basal temperature of a region undergoing anatexis is 130 K above the local crustal solidus temperature then 50% of the domain can become molten within 40 kyr. This is quite rapid by geologic standards and corresponds well with the few successful attempts made to date melting events by U-Th disequilibrium

chronological isotopy (see review by Macdougall [1995]) and precision 4øAr/39Ar dating [Renne et al., 1992; Baksi, 1994; Baker et al., 1996].

In Figure 5 the evolution of solid fraction fs, mixture composition C and liquid compositionC t are shown superimposed upon the mixture velocity field V for simulation H140. The region of partial melting lies between the fs = 1 (solidus) and fs = 0 (liquidus) isofracs. Up until a nondimensional time of about 0.05, heating (and melting) is essentially one-dimensional and little distortion of isofracs, isotherms (not shown) and liquid isocomps is evident. At greater times, however, advective transport becomes significant and the tell-tale signature of multiphase convection (isofrac curvature) becomes evident. The closer spacing of isofracs in the range 0.5 <fs < 1 is a consequence of the phase diagram, the initial bulk composition and the assumption of local equilibrium. It is an elementary exercise to obtain a plot offs or melt productivity versus temperature for Di80 based on the phase diagram. One notes that unlike the temperature, the fraction solid fs is discontinuous in the melting interval. For Di80, fs = 1 for all T < Tso I. However, at r,•o•, fs jumps discontinuously from unity to a value, based on the lever rule, of about 0.55. Then at T > Tso•,fs decreases eventually becoming equal to 0 at locations where the liquidus temperature of Di80 is reached.

Inspection of the liquid isocomps for case H 140 clearly shows that a consequence of partial fusion is to produce a thick region of compositionally graded melt within the mushy region. Initial liquid compositions are constrained by the phase diagram to be eutectic (Ct = 0.42). As isotherms move upward due to heat transport, the C e = 0.42 eutectic isocomp follows along. As temperatures exceed the solidus, melt compositions move along the steep diopside liquidus and melts become depleted in An component relative to Di component. At [ = 0.5 (steady state) the all-liquid part of the domain has a composition close to that of the bulk composition ( Ce= 0.2) and is practically homogeneous due to convective mixing. In the mushy region, however, especially where mixture (or melt) velocities are low, melt is distinctly compositionally stratified. Both horizontal and vertical gradients in liquid composition exist, although the tendency toward vertical zonation becomes more pronounced in that part of the domain closest to the solidus where fs approaches unity. Recall thatfscrit '-- 0.5 for these numerical experiments. Note that mixture velocities are quite small within that part of the domain where fs > 0.5. Darcy percolation, although clearly present, is feeble compared to the more vigorous flow elsewhere, especially in the simply connected region of total melting (rs = 0).

The evolution of anatexis for the more energetic case H210,

Table 2. Parametersfor Second Set of Experiments

T,,,K C,,, %An T%,K Ttop, K HI40* 1508 20 1508 1508 C20' 1538 31 1538 1538

, .

T•,,, t ITt,,p Ht,,, ,, J/kg H,"p, J&g I 1.52 x 106 1.52 x 10 •' I 1.55 x 106 1.55 x 106


• , , , ß .t, 6 6 • 6 6 6 • 6 6 6 • 6 6 6 1

100.0 -- x x x x x x x x x x

0.8

1.0

06

0o0 '* 04

0.0 02

0.0 0

0.0 0.2 0.4 0.6 0.8 1.0

time

Figure 4. Time series of spatially averaged solid fraction and spatially averaged specific kinetic energy of the magmatic mixture. Note sharp increase in the vigor and extent of partial fusion in the sequence H70 (triangles), H140 (crosses) and H210 (circles). Convective velocities are about 102 times faster for the case with a bottom temperature 130øC above the solidus (case H210) compared to 60øC above the solidus (H70). Advective transport of heat in H210, H140 and H70 is rapid and allows attainment of steady state at circa t ~ 0.2, 0.35 and 0.4, respectively. As ratio T, ofTbo, increases from 1.05 to 1.15, the average melt fraction (1 -fs) at steady state increases from 50% to 75%. Time to attain the steady state scales inversely with T,,,,/T,•. a measure of the enthalpy power delivered by underplating magma.

where the basal temperature is maintained 130 K above the Di- An solidus (eutectic) is shown in Figure 6. Comparison between H140 and H210 reveals an overall similarity but with some revealing differences. As already noted, the average fraction of melt is greater in H210 compared to H140. Flow is more vigorous in H210 with the development of smaller-scale flow in the all-liquid part of the domain. This insures good mixing in the all-liquid part of the domain. A larger region of homogeneous liquid develops in the steady state for H210 compared to H140. Convective velocities are high enough in H210 to insure mixing and homogenization of melts even within a portion of the mushy region (,rs < 0.10). However, in the more viscous part of the mush, once again the development of compositionally zoned melt takes place and advective transport of heat and matter is small. Finally, the progression of anatexis toward the steady state is faster for H210 compared to H140. This suggests that simply from the anatextic point of view, high apparent rates of magma eruption (or generation) ought to positively correlate with degree of chemical homogeneity. Put in other terms, rapid melting tends to produce large volumes of nearly constant composition melt, although some compositionally distinct and more evolved (lower density) melts will always be present.

Role of Magma Rheology and Mush Permeability

Although moderately successful predictive models for estimating viscosity of silicate melts as a function of temperature, composition and (less precisely) pressure exist [Shaw, 1972; Bottinga and Weill, 1972; Urbain et al., 1982; Ryan and Bevins, 1987; Scarfe et al., 1987;Richet and Bottinga, 1996], knowledge regarding the complex rheological properties of magmatic mixtures remains incomplete. The presence of suspended crystals and bubbles imparts significant non-Newtonian flow

properties to magma. This induces behavior related to nonzero yield strengths, nonzero normal stress coefficients, viscoelasticity, shear-rate dependence of the viscosity and thixotropy [Shaw, 1965, 1969; Shaw et al., 1968; Murase and McBirney, 1973; Spera, 1980; Marsh, 1981; Spera et al., 1982; McBirney and Murase, 1984; Ryerson et al., 1988; Webb and Dingwell, 1990; Bagdassarov and Dingwell, 1992, 1993; Stein and Spera, 1992; Kohlstedt and Zimmerman, 1996].

Despite this complexity, there is an overriding characteristic of suspension rheometry which has been obvious since the first measurements by Sakuma [1953] nearly a half century ago. This feature is the abrupt and drastic decrease in magma viscosity in a small temperature range that itself lies within the melting (solidus to liquidus) interval. This concept has been addressed by many workers since that time and is the dominant influence affecting momentum transport in magmatic systems [Shaw, 1969; Arzi, 1978; van der Molen and Paterson, 1979; Paquet and Francois, 1980; Marsh, 1981; Pharr and Ashby, 1983; Bergantz, 1989; Vielzeuf et al., 1990; Lejeune and Richet, 1995]. The essential point is that between a crystal content of roughly 30 to 70 vol % (i.e., 0.3 < fs < 0.7), the viscosity of basaltic magma varies by at least 10 orders of magnitude! Although factors such as the modal mineralogy and the details of the size and shape distributions of suspended crystals exert an influence [Cashman and Bergantz, 1991], the overwhelming physical effect is the rheological transition between melt-dominated and solid-dominated mixtures

that occurs around some critical fraction solid (fScrit). The temperature at which the critical solid fraction is attained depends on bulk composition and one other independent thermodynamic parameter such as pressure or enthalpy.

In order to assess the role played by the rheological properties of crystal-liquid mushes, a systematic series of numerical experiments was carried out using the HM model for momentum

0.0 1.0 .20 An •42

liquid:

[ --• Vmax = 35.0

Figure 5. Snapshots depicting the evolution of the__ (left) solid fraction fs, (middle) melt composition C t and (right) mixLure velocity V (-- (u, v)) at specific times of t = 0.05, 0.20, 0.30 and 0.50 (steady state) for simulation H140. At t < 0.05 the melting along the bottom boundary (y = 0) is practically one dimensional. Upon further melting, eute_ctic melt ( C t = 0.42) forms near the bottom boundary and rises to the top of the domain by advective transport. At t = 0.3, convection has intensified and exerts a pronounced effect on the isopleths and isofracs. Impingement of hot melt along the fs = 0 isofrac along the right-hand side of the domain erodes the interface locally. At t = 0.5, steady conditions prevail. The lower part of the domain (fs = 0) is compositionally homogeneous although not isothermal (temperature field not shown). Distinctively, the mushy (two-phase) region is occupied by a melt graded in composition such that eutectic melt ( Ct.= 0.42) which is intrinsically buoyant lies above more CaMgS120 6- enriched melt.

transport. In this formulation, a particular value of the fraction solid (called the critical solid fraction, fScrit ) separates two conceptually distinct flow regimes. For fs between 0 and fScrit (melt dominated regime), no relative motion between solid and melt is allowed to occur. The viscosity of the solid-melt mixture (suspension) depends upon fs according to a relation such as •1/•1o = (1 -fs/fs,,,,) [Metzner, 1985; Leujeune and Richet, 1995] where fScrit is set to some appropriate value. In contrast, for fs between fScrit and unity (solid-dominated regime), the solid plexus is a rigid (stationary) framework and melt is free to percolate through the interconnected porosity of the isotropically permeable mush. This is the Darcy flow regime, and the permeability is taken according to a modified Kozeny-Carman relationship K = K0(1 -fs) 2 ! fs • where K0 is a constant, based on available experimental data, which provides a scale for the relative permeability of the two-phase region.

The simulations conducted to test the importance of variations of the critical solid fraction OCScrit) are labeled FS00, FS03 and FS07. Parameters for these runs are identical to those of the

H 140 simulation except that fScrit has been set to 0.0, 0.3 and 0.7, respectively; recall that in the H140 run (results portrayed on Figures 4 and 5) fScrit equals 0.5. Experiment FS00 is the extreme case of purely Darcy flow or relative motion (RM).

Figure 7 shows the variation of the specific averaged kinetic

energy of the flow (KE) and the average fraction solid fs as a function of nondimensional time for all runs including H140 OeScrit equals 0.5). Although at very small times, the average rates of melting are similar, rates are lowest and decrease most dramatically for the Darcy dominated cases (FS00 and FS03). The total amount of melt generated also is smaller for the f$crit = 0.3 case. For example, the steady state integral fraction of melt is 0.53 and 0.70 for cases FS00 and FS07, respectively. Active advective transport of heat and material obviously enhances the rate and quantity of melt produced. In this framework we expect that allowing bulk convection in the partially molten region for higher solid fraction and thus earlier in the melting evolution would facilitate heat and momentum transfer, enhance the vigor of convection and, consequently, the rate of melt productivity. However, comparing the average specific kinetic energy and the average solid fraction for the FS00, FS03, FS05 and FS07 systems (Figure 7) shows that the relation is not so direct. While in the case of FS05 both the convection and the melt productivity are definitely enhanced, in the FS07 system this holds true only for the early phase of the system's evolution. In fact, the lighter melt (An enriched) generated at the solidus accumulates roofward, atop the more heavy component (Di). The fluid motion intensifies until •-0.2, then strong chemical buoyancy maintains a segregated compositional field, which reduces the


liquiff. ......

o.o 1 .o .42 I --• Vmax = 35.0

Figure 5. (continued)

kinetic energy of the system and prevents hot melt from reaching the melting front near the "roof' of the 'domain. Examination of Figures 8, 9 and 10 reveals that the region of mush with interstitial eutectic melt is twice as thick in FS07 compared to FS05.

Figure 8 shows the development of the fraction solid, melt composition and mixture velocity fields at the steady state time • = 0.5 for the limiting case FS00 (fScrit ---- 0). This case is a purely relative motion (RM) case with Darcy flow everywherefs > 0. The solution is practically identical to the one-dimensional Stefan solution because advective transport of material and heat is minimal. Note that within the all-liquid part of the domain (fs = 0) convection occurs with mild deflection of isofracs and isocomps. Within the mushy part of the domain, melt velocities are small of order several meters per year. Melt is compositionally graded from eutectic (C•= 0.42) along the solidus to bulk composition ( C•= 0.2) along the liquidus which shows minimal distortion from the horizontal. This is a strongly gravitationally stable system.

In Figure 9 the fs, Ctand V fields are shown for simulation FS03. Note that fields are significantly less one dimensional because that part of the mush with fs < 0.3 convects as a single fluid (i.e., a solid-melt suspension). Finally, in Figure 10 results are shown for FS07. In this case, Darcy flow is restricted to the region in which fs > 0.7. Hence the mush is reasonably well stirred although mixing is not sufficiently vigorous to homogenize it completely. There is a fairly large region of mush characterized by a nearly constant composition of melt (C t- 0.3 +_ 0.05).

Examination and comparison of Figures 5, 8, 9 and 10 clearly shows the sensitive role played by the parameter fScrit. For high values offScrit, large volumes of nearly homogeneous melt can be produced on a relatively rapid timescale. Although complete homogenization of melt is difficult to achieve due to Darcian behavior in high solid fraction regions, significant volumes of melt can be generated which span small ranges in composition. The situation is radically different for small values of fScrit . In this case, melting is practically one-dimensional, smaller amounts of melt are generated (more enthalpy is stored in the solid) and the melt that does form spans a wide range of compositions, and the interstitial melt is strongly compositionally graded (stable).

Role of Source Composition and Structure

Because crust is a very heterogeneous entity, it was thought worthwhile to conduct experiments to assess the role of the bulk composition and large-scale compositional layering within the crust on the evolution of anatexis. Three experiments, labeled C35, C42 and C20 were performed; parameter values are given in Table 1 and a picture showing the initial compositional structure is given in Figure 11. For all experiments, the hybrid momentum formulation was used with fScrit set equal to 0.5. A 70 K temperature difference was maintained between the lower and upper boundaries in all cases. The goal was to investigate the effects of different source region initial compositions on the rate of melting, total melt productivity and detailed geochemical and dynamical evolution of the magma and resite. In both the C35 and C42 experiments the initial compositional layering of the


solid ......... •'•

•0.2 O--

0.0 1.0 .20 An .42

Figure 6. Snapshots depicting the evolution of (left) fraction solid fs, (middle) melt composition C e and (right) mixture velocity V at t = 0.05 and t = 0.30 (steady state) for H210 simulation where the bottom boundary is maintained 130 ø above the Di-An eutectic (solidus) temperature. Note the vigor of this case compared to H70 and H140 (Figure 5); the liquidus temperature of Dis0 is 1620 K, and T, .... exceeds this value by almost 60 K in this case. The all-liquid region is significantly more voluminous compared to the H 140 case. Within the mushy zone where 0 < fs < 1, the melt is strongly graded in composition.

protolith is gravitationaly stable. In contrast, run C20 is unstable in the sense that initially dense solid material (C = 0.20) lies above eutectic composition solid (C = 0.42). Note that the bulk composition of the upper layer in C42 as well as bottom layer in C20 is eutectic (C = 0.42).

Time series plots showing the variation of KE (a measure of the vigor of convection) and average solid fraction fs for simulations C35, C42 and C20 are depicted in Figure 12. Data for run H70, identical in every aspect except that the protolith is homogeneous (C = 0.20), are also shown for comparison. Because all three layered systems are less rel¾actory in total bulk composition than the H70 case, convection is more vigorous at any given time because of higher melt production. The C20 case, which has a lower half of eutectic bulk composition, shows the most vigorous flow (highest KE) as expected. Note that a solid of eutectic composition melts like a single component congruently melting solid (i.e., at a fixed temperature under isobaric conditions).

Figure 13 depicts the evolution of the solid fraction fs, melt composition Ct. and mixture velocity V fields as the anatexic event evolves to the steady state (• = 0.70) for simulation C42 (see Figure 11 for the initial state). At very early times (7 < 0.05), melting is conduction dominated and approximately

one dimensional. At intermediate times (7 - 0.10 ) the volume of the mushy region expands and regions with fs < 0.5, the value of the critical solid fraction in this experiment, experience convective circulation. At •' = 0.15 the solidus isotherm enters

into the top layer of the C42 configuration which is of eutectic composition. A large region of essentially eutectic melt is rapidly generated within the mush and remains, because of its low density, at the top of the mushy zone. The steady state system at •' = 0.70 is characterized by a thick region of partial melt of eutectic composition stably residing on a subequal volume of mush within which melt is compositionally graded ( Ce= 0.35 to G. =0.22).

The steady state fields for solid fraction, melt composition and velocity are shown at the steady state time (7 = 0.70) for simulation C35 in Figure 14 (see Figure 11 for initial state). The final state is similar to run C42 except that the compositional gradient within the melt-dominated mush with 0 _<fs _< 0.3 is somewhat steeper in the lower part of the mushy region. This has the effect of damping convection somewhat (e.g., the maximum mixture velocity is reduced by about 50%).

Finally, in Figure 15 the steady state fields of solid fraction fs, melt composition ( C e) and velocity V are shown for simulation C20. In this case the melt productivity is high because the


1 0 3

101

10 -1

1 0 '3

1 0 's

/

0.8

0.6

0.4

0.2

0.0 0.2 0.4 0.6 0.8 1.0

time

Figure 7. Time series of spatially averaged solid fraction (fSav e) and spatially averaged specific kinetic energy (KE) showing the effects upon changing the critical solid fraction. Triangles denote the FS03 case, Circles denote the H 140 case 0rScrit -- 0.5), Crosses denote the FS07 case, and Diamonds denote the FS00 (RM) case. A comparison between the evolution of FS03 and H140 simulations shows that allowing bulk convection in the partially molten region for higher solid fraction facilijates more vigorous convection and higher melt productivity. Experiment FS07 attains a steady state quite rapidly ( t = 0.2), and a large volume of nearly homogeneous melt is produced. Note that melt productivity is highest for large fScrit experiments.

bottom half of the domain has an initial composition equal to the eutectic. Melting proceeds rapidly, and nearly the entire mushy domain undergoes convection because rapid advection of heat and material pushes the fs = 0.5 isofrac to the very top of the domain. The lid of entirely solid crust, of composition C - 0.2 is very thin in the steady state. Compositional gradients within the melt are supressed in this case because the production of buoyant eutectic melt near the bottom induces relatively strong convective mixing throughout most of the domain.

Intracrustal Differentiation

As is well-known, present-day upper crust of average thickness of 10 _+ 3 km is significantly more silica rich and depleted in Mg, Ca, Ti and Fe compared to the thicker (circa 30 km) and more mafic lower crust. To what extent can anatexis

drive the process of intracrustal differentiation? To address this problem several numerical experiments were performed. In each of the experiments, the block of crust undergoing compositional

.,..,..,..,..,..,..,..,..,..,..,..,..,..,..,..,..,..,..,..,. a. a. a. a. a. a.

Figure 8. Snapshots of (left) fraction solid fs, (middle) liquid composition Ct and (right) mixture velocity V at ] = 0.50 (steady state) for FS00 simulation where the RM model is used. Note that melting is essentially one dimensional with weak convection only where the system is totally molten. Darcy flow prevails everywhere within the mushy region even when fs is small in this end-member model.


I 0.0 1.0 .20 %An .42

...... liquid

--• Vmax = 15.0 I

Figure 9. Snapshots of (left) fraction solid fs, (middle) liquid composition Ct and (right) mixture velocity V at • = 0.50 (steady state) for FS03 simulation with fScrit equal to 0.3. Note that melting occurs mostly by conduction forfs > 0.3, while significant convection forfs < 0.3 causes rapid propagation of heat and a more expanded mushy region, now sufficiently thick to be gravitationally unstable and to convect. Compare to the FS00 case illustrated in Figure 8.

solid .......... ,'

0.0 1.0 .20 An .42

Figure 10. Time t evolution of fields for the FS07 simulation, where the critical fraction solid equals 0.7. (left) Solid fractionfs and (middle) isopleths of liquid composition C e contours interval at 0.25 and 0.05, respectively. (right) The region of the mushy two-phase mixture, defined by the 0 and I isofracs, is significantly more extended than in HI40 (fScrit = 0.5) (Figure 5).


% % % % % % % % % % % % % ",l % % % % % % % % % % % % % ',,,

,r' ,r' ,r' ,r' ,r' ,r' ,r' ,g ,g ,g ,g ,g ,g ,g % % % % % % % % % % % % %

, % % % % • .... % % % %

, % % % % % % % % % % % % %

Figure 11. (left) C35, (middle) C42 and (right) C20 initial compositional fields for simulation in which the source region undergoing anatexis is initially heterogeneous. At t = 0 the domains are isothermal (T = To), solid (fs = 1), and motionless (u = v = 0), and the same heat power input and rheological model are used as for case H70 (see Table 1). Note that the C35 and C42 domains are gravitationally stable, while C20 is unstable.

reorganization due to the waxing and waning of an anatexic event is treated as a closed system.

In the first experiment (H140*) the steady state of a waxing anatectic event (peak melting state) is used as the initial condition. The bottom boundary temperature is set equal to the top boundary temperature at the start of this new simulation. This experiment is labeled H140* and relevant parameters are given in Table 2. Details of the initial fields at the start of cooling cycle are given in Figure 5 at • = 0.50 for the HI40 simulation. In essence, the temperature of the bottom boundary was lowered to simulate the end of an underplating event. The aim was to "turn off" the heat source and allow the block of partially molten crust to solidify completely and to study the resultant compositional structure as monitored by the composition of the mixture upon complete solidification. The initial state at the instant the bottom boundary temperature is dropped from 1608 to 1538 K is shown on the left-hand panel of Figure 16. The mixture composition (solid plus melt) is plotted here. Recall that the block of crust

was of composition C = 0.20 everywhere before the anatexic event. The right-hand panel of Figure 16 shows the situation after the crustal block has cooled to below the solidus. The upper 10% of the region is of composition C = 0.20 since no melting has occurred there. Below this cap is a region enriched in pyroxene at the expense of plagioclase; the depletion approaches 3 modal percent. The pyroxene-enriched region is rather thin and occupies about 10% of the crustal block. Below this region there is a large volume of plagioclase-enriched crust with a thin layer near the base of slightly less enriched material.

The overall pattern is consistent with the notion of intracrustal differentiation being driven by anatexiso Although the overall bulk composition of the block remains the same by constraint, significant reorganization has taken place. Interestingly, and somewhat counterintuitively, the lower half of the crustal block is enriched in light component compared to the upper part. This is because although An enriched melt accumulates upward, the melt productivity, as gauged by the local solid fraction, also strongly

20.0 1.0

0.8

15.0 06

lO.O O4

5.0

0.0 ,0 0

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 time time

Figure 12. Time series of spatially averaged solid fraction (fSave) and spatially averaged specific kinetic energy (KE) to study the effects of different source bulk compositions and layering. Triangles indicate the C42 case, Crosses indicate C35 case, Circles indicate the C20 case, and diamonds indicate the H70 case. As a consequence of variable bulk composition, for the same heat power input and rheological model the average KE of the systems is raised by a factor of 102 for C20, when the bottom bulk composition is eutectic compared to the case of Di80 (case H70 with C = 0.2). Note the nonmonotonical behavior of KE resulting from the competition between thermally and compositionally driven flows for C42 and C35 and the heightened convection due to the coupling of the thermal and compositional buoyancy.


0.0 1.0 20 %An 42

Figure 13. Snapshots of (left) fraction solidfs, (middle) melt composition C•. and (right) mixture velocity V at • = 0.05, 0.10, 0.15, and 0.70 (steady state) for the C42 simulation. Note the decrease in the vigor of flow when the upper layer begins to melt at 7 --- 0.15. This is also shown in Figure 12 by a drop of KE by a factor of 2 after a quasi-monotonical increase.

decreases upward. The antisympathetic variation of these two complexly coupled variables is responsible for the post-melting pattern of bulk composition profiles.

In Figure 17 the "before and after" images of a waxing-waning anatexic event are shown for the C20 simulation. Recall that in

run C20, the lower half of the crustal block was enriched in

CaAI2Si208 (C = 0.42), whereas the upper half has a composition C = 0.20 (see Figure 11). First note that all semblance of the initial (preanatexic) compositional structure is lost at the peak of the anatexic event (compare Figure 11 (C20) and Figure 15 at [ = 0.50). Overall, the mixture has been considerably homogenized. In experiment C20' the initial state is the peak anatexic condition (left-hand panel Figure 17). Thereafter the bottom temperature is set equal to the top one (T = 1538 K; see Table 2), and the system is allowed to cool and solidify. The end state, in terms of the mixture (or bulk) composition, is shown in the right-hand panel of Figure 17. In this case the upper crust, which initially was rather mafic (C = 0.20), is decidedly more enriched in CaA12Si208 component. The lower crust, which started with the eutectic bulk composition (C = 0.42) is CaMgSi206 enriched with respect to its initial state. Once again, the effects of an anatexic event are to fundamentally reorganize the pattern of compositional structure relative to the preanatexic state.

Conclusions

The critical factors that govern the timescale and style of anatexis when driven by either mafic magma underplating or injection of magma into the crust include the intensity of the enthalpy supply, the rheological properties of the source materials during anatexis and the bulk composition and compositional structure of the source region undergoing partial fusion. The following points are recognized as particularly salient ones:

1. Greater quantities of nearly constant composition melt are correlated with high rates of enthalpy supply (enthalpy power) and high melting rates. Anatexic timescales lie in the range 10 3 to 10 5 years for length scales of order 10 2 to 10 3 m. Long timescale anatexic events are associated with a wider composition span of generated partial melts compared to otherwise similar but more intense melting episodes.

2. Much of the variation of rheological properties occurs at the transition between a solid-dominated and a liquid-dominated region within the melting interval. The critical fraction solid OCScrit) is the parameter of the transition. For small values offScrit, melting is practically one dimensional, and the small amount of melt that is produced is strongly compositionally zoned, preserving a wide range of compositions. For higher values of fScrit, larger amounts of nearly homogeneous melt can be


0.0 1.o 20 %An 42 • Vmax = 14.0 I

Figure 13. (continued)

produced on a relatively rapid timescale. However, if the rate of melt generation is lower than that of redistribution within the anatexic region, rapid development of compositional stratification may limit the vigor of convection and thus may hinder the formation of an extended homogeneous molten region. This leaves the premelting structure of the source region dramatically modified.

3. Because of the presence of less refractory materials in heterogeneous crust, melt productivity may be heightened, which allows for more vigorous convection. Even in the simplest case of an upper granodioritic and lower gabbroic crust, the gravitationally stable compositional and density structure may counteract convection and therefore prevent further mixing and hybridization. Nonetheless, the formation of compositionally

0.0

•.•.,..:,- ..................

,.2o ::: ::: : ---l-,i-,•,u• d .......... ._

.... ?•'??•ii""'-'"'"•=•'••••:•••J-. '-- ...... I --• Vmax = 10.0 1.0 .20 An .42

Figure 14. Snapshots of (left) fraction solid fv, (middle) melt composition C• and (right) mixture velocity V at • = 0.70 (steady state) for the C35 simulation. Note the somewhat steeper melt compositional gradients in this case compared to C42.


t =.50

'0 0'3 C

0.0 1.0 20 %An 42

Figure 15. Snapshots of (left) fraction solidfs, (middle) melt composition Ce. and (right) mixture velocity V at • = 0.50 (steady state) for the C20 simulation. The initial state is gravitationally unstable with dense crust above lighter crust. This unstable arrangement leads to a convection field which is very well developed throughout the domain at steady state shown here. A more voluminous mush region characterized by less compositional variation of the melt characterizes the final state of this simulation.

graded mush and the absence of any remnants of the original compositional discontinuity leave the crust substantially modified by anatexis.

In summary, even in the simplified models considered here, the complex coupling between convective dynamics and melt productivity can variably reorganize the pattern of compositional structure within the crust relative to the preanatexic state. This is

in accord with the recognition of the very complex character of the crust in which refractory materials, variably depleted and undepleted terrains spatially coexist [Jahn, 1988]. In the lower crust, for example, granulites terrains, dominated by more evolved rock types with high SiO2, high K, Th and U content and low Mg numbers, coexist with lower crust xenoliths that are more mafic and variably depleted in incompatible elements [Rudnick et

0.18 0.19 0.20 %An 0.18 0.19 0.20

Figure 16. An example of intracrustal differentiation is examined by modeling the waning of an anatextic event. (left) The final state of simulation H140 in terms of the mixture composition is depicted. Recall that the initial composition of H140 was C = 0.20 (20 wt % CaA12Si2Os) everywhere. At the steady state, regions enriched and depleted in CaA12Si20 8 component develop, depicted in Figure 5. To model the waning of anatexis, the temperature of the bottom boundary is set equal to that at the top. The parameters of this experiment H140* are given in Table 2. This causes the anatexic region to solidify, (right) the mixture composition once solidification has been completed is shown.


...

,:,...:..:.<...: ..: ..-:• . ., ,...-.--?. !:: ,:-...- "' -.:. :?*'"" "'":"•':::'::':::'<•"'-.-.---:-'-- "'"": :-'•:-:i'.:::.:... "" --:--' '-•'"": ---::' '":'":':' '" ::." " '-'::::,""--•;:., :;-:::<•½:::--.'":':':'•:':S--4--

-:...:... :4.{: ..: ß - -•:.•..: --.: .....:.•.; .:.

'•i: :%::/?: ':'"'

...

.....

..

0.24 0.28 0.24 0.28 0.20 0.30 0.20 0.30

Figure 17. Before and after images of a waning anatexic event: (left) Partial melting and (right) solidification. The initial conditions for this experiment C20' are identical to the final state of run C20 depicted in Figure 15 (steady state at • = 0.50). The initial state for C20 is shown in Figure 11. Note that the lower crust becomes significantly more mafic than its starting condition. The effect of this anatexic event is to reorganize the pattern of compositional structure rather dramatically consistent with the concept of intracrustal differentiation in response to magma underplating or injection into previously existing crust.

al., 1988]. Both show a similar Th/U ratio and a common depletion in large-ion lithophile elements, enriched in the upper crust, which suggests a common process is involved [Roberts and Ruiz, 1989; Rudnick and Taylor, 1987; Rudnick and Fountain, 1995]. This process, we believe, is crustal anatexis driven by the upward ascent and eventual stagnation of mantle-derived mafic magma. If recent interest in this problem is a measure of progress [e.g., see Brown et al., 1995], we can expect to learn a great deal more about the differentiation and growth of continental crust and its relationship to mantle dynamics in the next decade.

Appendix

In order to facilitate solution of the conservation equations, a nondimensionalization was performed by introduction of the following dimensionless variables: - x - - uL - vL x=-- y=Y u=• v= m

L L •c •c

- L2p • t•St • h-hsø! rlo•C Ah - - - h e - hso I - T- •ol hs = hs hsøl he = T = •

Ah Ah Tli q - Tso !

C-Co G-Co Q-Co C,o -Co C,o -Co C,o -Co

When these variables are substituted into (2) - (6) given by Spera et al., [ 1995], the conservation equations take the dimensionless form

V. v =0 (Al)

l istOn ] 3p , p-•. -•t-t +v. Vu =---+V-Vu - (u-us) (A2) 8x Da

I [St3½+v. Vv]= 3p 1 p"•' -•yy + V. Vv + RaT + RsC e •a (V- v.,. ) (A3)

St Oh = V 2 V2(hs-h)-V-[(h I -h)(V-Vs) ] (A4) +v.Vh h+ 8t

8C+ 1 D St-•t t v. VC = V-••VC Le D e 1 D

+V.Tee V(C - C)- - C)(v- v, )] where all bars have been suppressed. The parameters appearing in (A 1)-(A5) are defined according to

ga --

Rs --

PrO•g(rliq -- :o,)L 3 KT[ o

prg•(Csol -Co)L 3 KT[o

Pr= 11o Da = Kø •Cpr L 2

St= Le=• (A6) Ah Dœ

where Ra is the Rayleigh number, Pr is the Prandtl number, Da is the Darcy number, Rs is the solutal Rayleigh number, St is the Stefan number and Le is the Lewis number. The convention is

adopted that the compositional variable C is expressed in terms of the component that is rejected by the liquidus phase. For example, in all of the calculations discussed in this paper, diopside is the liquidus phase and so C refers to the mass fraction of CaAI2Si20 8 (the so-called light component) in the melt, solid or mixture. The equation of state used is

D=Dr(l-o•(r-rsol)-•(eg-c•..o)) (A7) where Pr is a reference density for melt of composition C o (the bulk composition) at the eutectic temperature (T = Tso I ). The isobaric expansivity ct and its analog for composition [3 are defined, respectively'

(AS) p74/

All parameters are defined in Table 1. Note that [3 for the light component in a binary system is a positive number. The fictive


quantity Ah, used to provide an appropriate scale for energy in the relationship between fs, T and C t , is defined

Tsøl Tliq /l;. + hf,di (A9) Ah = 1- Tsol _ rm Tsol _ rm The quantity hf is the fusion enthalpy of eutectic solid (see Table 2). The quantity Ah is the amount of energy needed to transform solid of original bulk composition (a mixture of diopside and anorthite crystals) into a melt of the same bulk composition at the liquidus temperature. Note that Ah depends on the initial composition implicitly because the liquidus temperature depends on the initial bulk composition. Variable hf,di, represents the fusion enthalpy of diopside crystals, and T m is the melting temperature of diopside. Tli q is the liquidus temperature for the initial bulk composition. Additional details may be found within the papers cited in the text. The Darcy number Da has been set equal to 10 -lø in all cases.

Notation

c

L

D

Da

h

Ah

J q

k

K

Ra

Rs

St

Pr

Rp T

Tsol rliq rm ro p

t

v

x,y

fs g

K

P

mixture mass fraction CaA12Si20 8. isobaric specific heat capacity (J / kg K). depth of domain (m). chemical diffusivity (m2/s). Darcy number. specific enthalpy (J / kg). specific fusion enthalpy of eutectic mixture (J / kg). enthalpy scale defined in (A9) (Appendix). species flux (kg / m 2 s). heat flux (J / m 2 s). thermal conductivity (J / m s K). permeability (m2). Lewis number.

thermal Rayleigh number. compositional Rayleigh number. Stefan number.

Prandtl number.

buoyancy number (-- Rs/Ra). temperature (K). eutectic temperature (K). liquidus temperature (K). melting temperature of diopside (K). initial temperature (K). pressure (N / m2). time (s). horizontal velocity (m/s). vertical velocity (m/s). velocity vector (m/s). Cartesian coordinates.

fraction solid.

acceleration of gravity. isobaric thermal expansivity (K-•). isothermal, isobaric compositional expansivity. thermal diffusivity -- k / pc (m7s). density (kg / m3). viscosity (kg / m s).

Subscripts

sol solidus (eutectic). liq liquidus. s solid.

œ liquid (melt).

m melting point, pure phase. f fusion. Di diopside. An anorthite.

o initial conditions.

w wall.

Acknowledgements. This work was supported by the U.S. Department of Energy grant DE-FG03-91ER 14211 and National Science Foundation grants EAR93-03906 and OCE93-02058. Reviews by G.Bergantz and K.Condie improved this work and are gratefully acknowledged. Contribution 0275-45CM of the Institute of Crustal Studies.

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F. Raia and F. J. Spera, Department of Geological Sciences and Institute for Crustal Studies, University of California, Santa Barbara, CA 93106. (e-mail: [email protected])

(Received September 19, 1996; revised April 28, 1997;

accepted May 27, 1997.)

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