American Mineralogist, Volume 80, pages 1188-1207, 1995
Simulations of convectionwith crystallization in the systemKAISiTO6-CaMgSirOu: Implications for compositionallyzoned magma bodies Fuux J. Srrnl,, Cunrrs M. Or,nnxrunc, CoNsr,l,NcrCnmsrENsEN, Mrcor, Toorsco DepartmentorGeologicar ttn::';liiff orcarirornia' 3"'1,'#*,l;,:ff:t:t:"*'universitv
Ansrnlcr A model has been developedand applied to study the origin of compositional and phase heterogeneityin magma bodies undergoing simultaneous convection and phase change. The simulator is applied to binary-component solidification of an initially superheated and homogeneousbatch of magma. The model accounts for solidified, mushy (two- or three-phase),and all-liquid regions self-consistently,including latent heat effects,perco- lative flow of melt through mush, and the variation of systementhalpy with composition, temperature, and solid fraction. Phaseequilibria and thermochemical and transport data for the system KAlSirOu-CaMgSirOuwere utilized to addressthe origin of compositional zonation in model peralkaline magmatic systems.Momentum transport is accomplished by Darcy percolation in solid-dominated regions and by internal viscous stressdiffusion in melt-dominated regions within which relative motion between solid and melt is not allowed. Otherwise, the mixture advectsas a pseudofluid with a viscosity that dependson the local crystallinity. Energy conservation is written as a mixture-enthalpy equation with subsidiary expressionsthat are based on thermochemical data and phase relations that relate the mixture enthalpy to temperature, composition, and phase abundancesat each location. Speciesconservation is written as the low-density component (KAlSirOu) and allows for advection and diffirsion as well as the relative motion betweensolid and melt. Systematic simulations were performed to assessthe role of thermal boundary condi- tions, solidification rates, and magma-body shape on the crystallization history. Exami- nation of animations showing the spatial development of the bulk (mixture) composition (C), melt composition (C,), temperature (7), solid fraction ({), mixture enthalpy (h), and velocity (V), reveals the unsteady and complex nature of convective solidification owing to nonlinear coupling among the momentum, energy,and speciesconservation equations. A consequenceof the coupling includes the spontaneousdevelopment of compositional heterogeneityin terms of the modal abundancesof diopside and leucite in the all-solid parts of the domain (i.e., modal mineralogical heterogeneity)as well as spatial variations in melt composition particularly within mushy regions where phase relations strongly couple compositional and thermal fields. Temporal changesin the rate of heat extraction becauseof bursts of crystallization and concommitant buoyancy generationare also found. Crystallization of diopside, the liquidus phasein all cases,enriches residual melt in low- density K-rich liquid. The upward flow of this material near the mush-liquid interface leads to the development of a strong vertical compositional gradient. The main effect of magma-body shapeand different thermal boundary conditions is in changing the rate of solidification; in all casescompositional heterogeneitiesdevelop. The rate of formation of the compositional stratification is highest for the sill-like body becauseof its high cooling rate. Compositional zonation in a fully solidified body is found to be both radial and vertical. The most salient feature of this simple model is the spontaneousdevelopment of large-scalemagma heterogeneityfrom homogeneousand slightly superheatedinitial states, assuminglocal equilibrium prevails during the course of phasechange.
INrnouucrroN the discipline itself, and it would be difficult to review the subject in a single contribution. An adequatecritique One of the most important problems in igneous pe- would encompassthe entire field of igneous petrology, trology is elucidation of the mechanismsthat give rise to include the history of planetary differentiation and crustal magmatic diversity. This problem is practically as old as growth (e.9., Galer and Goldstein, l99l; Taylor and 0003-004x/95/rtt2-1 I 88$02.00 I 188 SPERA ET AL.: SIMULATION OF CONVECTION AND CRYSTALLIZATION I 189
Mclennan, 1985),and draw upon geochemicaldata from evidence that separation of crystals from melt had oc- hundreds if not thousands of ancient and modern mag- curred, although the mechanism and rate by which this matic centers. was accomplishedare not constrainedby the geochemical Sincethe pioneering work of Bowen ( I 928), it has been data. Evidently, the lack of interphenocryst chemical generally acceptedthat crystal fractionation is one of the variation and correlation between phenocryst composi- most effective magmatic differentiation processes.By tion and enclosingmelt (glass)composition arguesagainst some means, crystals become separatedfrom the melts simple crystal settling. Similar conclusionswere reached in which they crystallize. Despite the generalsignificance by Dunbar and Hervig (Dunbar and Hervig, 1992; Her- of this problem, the mechanism of crystal fractionation vig and Dunbar, 1992) in their detailed study of melt- and the associatedquestion ofthe vigor ofconvection in inclusion traceelement abundances and comparisonswith magma bodies remain controversial(e.g., Marsh, 1989; enclosingpumice and whole-rock compositions. Martin and Nokes, 1989; Gibb and Henderson, 1992., A popular scenario whereby compositionally evolved Huppert and Worster, 1992). The problem is challenging melt becomes separated from associatedcrystals is by mainly becauseof the enormous changesin magma vis- maryinal upflow, a mechanismfirst hypothesizedby Shaw cosity that occur in the crystallization interval. Although (1974) and extensively investigated theoretically, exper- adequatemodels exist for the calculation of melt viscos- imentally, and by numerical simulation since then (e.g., ity as a function of temperature and composition (e.g., Turner, 1980;Chen and Turner, 1980;Turner and Gus- Shaw,1972;Bottinga and Weill, L972;Urbainet al., 1982; tafson,198 l; Speraetal., 1982,1984,1989; Huppert and see also Stein and Spera, 1993), accurate description of Sparks,1984; Turner, 1985; Nilson et al., 1985; Mc- the complex rheological behavior of magmatic mixtures Birney et al., 1985;Lowell, 1985;Huppert et al., 1986; is a far more difrcult task (Lejeune and Richet, 1995). Bergantzand Lowell, 1987;Trial and Spera,1988; kitch, Because,in general,there can be relative motion between 1989;Langmuir, 1989;Huppert, 1990).The essentialidea the constituent solid, melt, and vapor phasesand because is that in systemswhere phasechange occurs along a sub- component phases have such widely different physical vertical sidewall (e.g., solidification or anatexis),the ini- properties, the characterization of the rheology of mag- tial downward flow of cooled marginal magma can be matic mixtures presents a formidable problem. Briefly reversed if phase change enriches melt in low-density stated, it is difficult to describe simply and'accurately components(e.g., SiOr, NarO, KrO, and HrO) sufficiently both the rheological properties of magma and the rela- to overcome negative thermal buoyancy effects. In this tionship between phase fraction (e.9., fraction solid, f,) model, accumulation by advective transport of the cooler and permeability of magmatic multiphasemixtures across but chemically enriched (and buoyant) melt at the top of the wide range of conditions relevant to magma transport the magma body produces a thermally and chemically and evolution. The problem is particularly acute in silicic zoned layer. Additional in situ crystallization within the and intermediate-composition systemsfor which there is roofward layer then producesa magmatic mixture with a overwhelming and incontrovertible evidence that large phenocryst content inconsistent with any single-stage, volumes of evolved magma, sometimesexceeding several closed-systemfractionation scheme,but consistent with thousand cubic kilometers, exist within the crust. Fur- trends expected from crystal-fractionation processesas thermore, differentiation may occur at geologicallyrapid commonlynoted (Michael, l9B3; Wolffand Storey,1984). rates(e.g., lO-'zkm3/yr) prior either to solidifyingto form If HrO is significantly enriched upward, then the local mineralogically zoned or layered plutons or other com- (compositionally dependent)liquidus to solidus temper- posite batholiths or to erupting to form compositionally ature interval, as well as the liquidus temperature itself, zoned ignimbrites or lava flows (Smith, 1979; Hildreth, varies with depth within the magma body. This leads to l98l; Trial and Spera,1990; Wiebe, 1993). A classicex- variation in phenocryst content with location within the ample is the 600 km3 Bishop Tuff in eastern California body independent of the effectsof crystal settling. (Hildreth, 1979) emplaced as a plinian pumice fall, fol- Although the simple version of the marginal upflow lowed by a sequenceof pyroclastic flows. Hildreth con- hypothesisis conceptually pleasingand there is no doubt cluded that no mixing of crystalsby gravitational settling that marginal upwelling occursin laboratory experiments had occurred prior to the eruption. His conclusion was with aqueous solutions, its relevance to magma differ- basedprincipally on the observationthat individual hand- entiation has been questioned (see review by Trial and sized pyroclasts had compositionally uniform popula- Spera, 1990). To develop an evolved roofward layer, tions of phenocrysts.Recently, Lu et al. (1992)compared chemically induced buoyancy within the very thin chem- the amount of crystallization in a single Bishop Tuffpyr- ical boundary layer must overcome negative buoyancy oclast inferred from trace element abundancepatterns with within the thicker sidewall thermal boundary layer. De- that observed in the host pyroclast utilizing melt (glass) tailed numerical simulations and scaleanalysis were used inclusionswithin quartz phenocrysts.They concludedthat by Speraet al. (1989) to show that roofward enrichment the trace elementcompositions of the melt inclusionsspan of light component occurs only if a critical ratio of com- a range larger than explicable by in situ phenocryst crys- positional to thermal buoyancy is exceeded.They found tallization within the pyroclast. The larger extent of crys- that this critical buoyancy ratio (Rp") dependson the Lewis tallization inferred from trace element abundancesis good number (Ie), defined as the ratio of thermal to chemical I 190 SPERA ET AL.: SIMULATION OF CONVECTION AND CRYSTALLIZATION diffusivity. For roofward enrichment to occur by sidewall ics of realistic silicate systems,including the rheological upflow, the condition in a two-component single-phase and permeability propertiesof mush, should also be made melt is for the reasonscited earlier. If the solidus to liquidus temperature interval is not Rp" > T'Ld/' (l) smatt (the usual case), a significant thickness of mush where Rp : Bj*k/aq*pD, Le = x/D, and B, j*, k, o, q*, p, would develop along marginal cooling surfaces(Bergantz D, and,c represent the compositional expansivity of the and Lowell, 1987; Bergantz, 1992). This invalidates the melt, mass flux of rejected light component becauseof a criteria written as Equations I and 2. If positive com- phase change along the sidewall, magma thermal con- positional buoyancy exceedsnegative thermal buoyancy, ductivity, isobaric thermal expansivity of the melt, heat then upward-flowing, chemically enriched interstitial melt flux into the country rock, magma density, chemical dif- in a marginal two-phasemush zone may exist. The thick- fusivity of light component, and magma thermal diffu- nessof this mush is delimited by the liquidus and solidus sivity, respectively. This criterion is valid for a strictly isotherms. The vigor of the potential upflow that may one-phasefluid subject to maryinal boundary conditions occur depends on many factors. It is important to note on mass (l*) and heat (q*) flux. If sidewall conditions are that the upward flow of buoyant and chemically evolved set on composition (C*) and temperature(Z*) rather than melt is not difusion limited in this casebecause buoy- on their fluxes, the appropriate inequality is ancy is generatedby phasechange everywhere within the mush. In this version of the marginal upflow hypothesis, Rp. > l.2Le" (2) one which is somewhatmore realistic, the mush thickness wherein this caseRp is defined Rp: gA,C/aAL Because and interstitial velocity of melt through the mush by per- the Lewis number for a typical oxide component in a colative flow and rheological properties of the magmatic silicatemelt is >10a,Rp valueson the order of 10-100 mixture are the critical factors. Diftrsion through the melt are necessaryto enable upflow in magmatic situations. is of minor importance because the thickness of the Application of these criteria suggeststhat the SiO, com- chemically enriched region is governed by the compli- ponent, becauseof its small on-diagonal chemical difrr- cated feedback between the rate of marginal crystalliza- sivity (D r 10-16m2ls), would not be significantlyen- tion and phase relations, particularly the relationship riched upward, whereasHrO possibly could be. Details, among the mixture enthalpy, temperature, and compo- including numerical examples,are provided in the stud- sition. This feedback includes latent heat effects associ- iesofTrial and Spera(1988, 1990) and Speraet al. (1989). ated with phase change and the differential velocity of Although Equations I and 2 are valid criteria govern- solid and melt within the mushy region. The rheological ing the conditions for upflow, the limitations of the model transition between solid-dominated and melt-dominated must be kept in mind if one seeks to explain magma magma is also critically important. differentiation. Perhapsthe most significant limitation is With the above comments in mind, the purpose of this that phasechange (e.g., solidification or assimilation)along study can be given. We have developedan algorithm (Ol- the margin is not explicitly taken into account. In simple denburgand Spera,1990,1991,1992a,1992b) that con- models (Shaw, 1974; Spera et al., 1984; Nilson et al., siders simultaneous phase change (melting and solidifi- 1985;Lowell, 1985;McBirney et al., 1985),one merely cation) and convection in two-component multiphase specifiesthat the wall temperature (Z*) and composition systemsto model convection in a binary eutectic system (C-) or the boundary (wall) flux of light component (f) undergoingcrystallization. This model accountsfor solid- and heat flux (q*) are time-independent, uncoupled ified, mushy, and all-liquid regions in a self-consistent boundary conditions. This is a useful procedure but is manner, including latent heat effects,flow of melt through only an approximation. Heat and chemical transport permeable mush, and the variation of system (melt plus through a marginal mush are not explicitly consideredin solid) enthalpy with composition, temperature, and frac- thesestrictly one-phasemodels. Consequently,the effects tion solid ([). We used phaseequilibria, thermochemical of latent heat, the generation of compositionally distinct data, and transport data for the binary system KAlSirO. melt resulting from constraints imposed by phase rela- (leucite)-CaMgSirOu (diopside) in the simulations. Al- tions, and the percolation of melt through the sidewall though we do not intend the calculations to be relevant mush are not taken into account. to a particular ignimbrite, the results are relevant to the The single-phaseapproximation and hence the condi- origin of compositionally zoned peralkalinemagma (Wolff tions for upflow representedby Equations I and 2 are and Storey, 1984).As an example,one may cite the series valid provided the thickness of the multiphase region is of eight compositionally zoned phonolitic to trachytic ig- small in comparison with the thickness of the thermal nimbrites erupted from the Somma-Vesuvius eruptive boundary layer, which is on the order of several meters. centerin the last 2500 yr. Somma-Vesuviusis a classic But it must be emphasized that firm quantitative con- example of a single magmatic center that has repeatedly straints on the flux of mass 0*) and heat (4*) along the emitted zoned magma (Joron et al., 1987).Elementary contacts between country rock and magma (or C* and mass-balancecrystal-fractionation models for these Z*) are difficult to establishand probably highly variable. chemically zoned ignimbrites involve removal of leucite A proper accountingof the energeticsand thermodynam- and clinopyroxene. The main purpose of the simulations SPERA ET AL.: SIMULATION OF CONVECTION AND CRYSTALLZATION I l9l is to assessthe likelihood that a boundary-layer flow can 1992b)we presenteda two-dimensional continuum mod- causecompositional zonation in the system KAlSirO6- el of convection with phasechange in which a single set CaMgSirOuunder boundary and initial conditions appro- of conservation equations for mass, momentum, energy, priate for the crystallization of phonolitic magma within and speciesis applicable to all regions (solid, mush, and the crust starting with a homogeneous,slightly superli- melt) in a binary eutecticsystem. The model is a modified quidus melt of appropriate bulk composition. We also version of the formulation developedby Bennon and In- investigatedthe role of convective macrosegregationwith cropera(1987a, 1987b)and is closelyrelated to similar respectto the formation of a mineralogically zoned intru- formulations (Voller et al., 1989; Beckermann and Vis- sive body following complete solidification. kanta, 1988;Ni and Beckermann,l99l). This approach The remainder of this paper is organizedas follows. In integratessemiempirical laws, a microscopic approach to the next section, a brief description of the model is pre- transport processesin multiphase regions, and classical sented in nonmathematical terms. This is followed by macroscopicmixture theory to achievea quantitative de- presentation of the mathematical model, including rele- scription ofcrystallization and convection in binary sys- vant initial and boundary conditions, phase equilibria, tems. The model has been validated by comparison of and melt-density relations in the system KAlSirO6- simulations with laboratory experiments for both single- CaMgSirOu.The results of the simulations, including ex- component and binary-component systems(e.g., Bennon amination of the effectsof magma-chamberaspect ratio, and Incropera, 1987a, 1987b; Christensonand Incropera, rate of cooling, and thermal boundary conditions, on the 1989;Neilson and Incropera, l99l; Oldenburgand Spera, development of chemical differentiation are then given. 1991,1992a,1992b;Yoo and Viskanta, 1992).In partic- The consequencesof macrosegregation,especially with ular, important characteristics of solidification such as regards to mineralogical zonation in plutons, are dis- remelting of earlier formed solid, the spatial irregularity cussedin the penultimate section. In the last section, the of the liquidus isotherm, the spontaneousdevelopment results are summarized and some speculationson the rel- of high-permeability regions (i.e., channels)within mush, evance of the simulations to the origin of magmatic di- and the development of macroscopic compositional zo- versity are offered. nation (macrosegregation)in the solidified products are faithfuly captured by the model.
Moonr, Assumptions and limitations Overview The most fundamental assumption invoked is that of Unlike solidification of a single-component liquid, local thermodynamic equilibrium. Equal phase temper- crystallization of magma is not characterizedby the pres- atures and chemical potentials for components in coex- ence of a distinct front separating regions of solid and isting solid and melt phasesare assumedto be valid lo- liquid (melt). Crystallization of multicomponent silicate cally. Phase relations are uniquely determined by the melts gives rise to a permeabletwo-phase mushy region equilibrium phasediagram. When equal temperaturesand characterizedby a microstructure that dependsupon the equal chemical potentials are assumed,then temperature, conditions ofsolidification (e.g.,rapid chilling, slow cool- melt composition, and solid fraction are related to the ing, etc.) and the thermodynamic and transport proper- mixture enthalpy and composition through the phasedi- ties of the material undergoing phase change.The thick- agram and to the thermodynamic properties of relevant nessof the mushy zone, where f" varies betweenzero and phases.In the present case,simple thermodynamic rela- one, is delimited by the liquidus and solidus isotherms, tions applicable to binary eutectic systems provide the respectively.In a binary system at fixed pressure,the li- means of relating temperature, enthalpy, and the com- quidus temperature depends on the local bulk composi- positions and amounts of all phasesin each, arbitrarily tion, whereasthe solidustemperature is constantand equal small, portion of the magma body (control volume). Be- to the eutectic temperature. During phase change,latent causesolid-state diffusivities are tiny relative to those in heat releasedalong the microscopic crystal-melt interface melts, it is assumedthat diffusion in the solid is negligible is redistributed within the mush by conduction and, when in comparison with diffirsion in the melt (i.e., D. << D,) significant relative motion exists between solid and melt, (Voller et al., 1990; Rappazand Voller, 1990).Because by melt advection. Compositional variations arise be- of the extremely slow nature of solid-state diffusion it is causeof differencesin solubilities of components in co- unlikely that, on the scale of a single crystal (millimeter existing melt and solids. Becauseof thermodynamic con- to cantimeter size), perfect equilibrium will prevail, es- straints on solidification as embodied by the equilibrium pecially in rapidly chilled systems.Although it is possible phasediagram, selectiverejection of componentsalso oc- to relax the assumption of local equilibrium and incor- curs along phase interfaces.This results in differencesin porate nonequilibrium effects (e.g., allow for effects of melt density, which set up local buoyancy flows of ther- supercooling),the lack ofa comprehensiveset ofexper- mal and compositional origin and, becausegradients in imental data for crystal nucleation and growth rates in chemical potential exist, concommitant diffusive flows magma makesthis attempt premature, although some ex- (e.g.,Morse, 1988). istent models do incorporate nonequilibrium effectsas- In previous work (Oldenburg and Spera, 1991, 1992a, sociated with undercooling (e.g., Brandeis and Jaupart, tl92 SPERA ET AL.: SIMULATION OF COIWECTION AND CRYSTALLIZATION
1987; Spohn et al., 1988; Kerr et al., 1989).However, 9rrt*v.(pv):o (3) none of these models accounts self-consistentlyfor vari- dt"' ations in melt composition and for associatedchanges in DudPn enthalpy, temperature, solid fraction, and magma rheo- o,fi: - + Y'(ltYu)- - u') (4) logical properties as in the model presentedhere. dx V@ It is assumed that there is no volume change upon DvdPn- v'(avv)-iO-v")- (5) crystallization and that solids that form are not acted upon ,"1fi: ay+ Ps by deviatoric stresses.Without this assumption, forma- tion ofvoid spacecould not be avoided upon solidifica- Dh + v.[xV(/r.- /r)] tion; additionally, the generationof internal stresseswith- ;:v.(Kvh) in the solid would occur during phase change. This - - - (6) approximation implies that density currents driven by a v.I(h, frxv v")l crystal-laden parcel of magma are ignored (i.e., constant phasedensities except for variations associatedwith tem- o.troc) + v.tDV(c, - C)l 5:DI perature and composition in the melt). The multiphase region is viewed as a porous solid with -vKC,-CXv-vJl Q) an isotropic permeability (K). Although the study by Yoo and Viskanta (1992) illustrates that anisotropy ofthe per- where all symbols are as defined in the Appendix Table. meability is relatively easyto incorporate into the present The Boussinesqequation of state is employed so that the formulation, experimental information on the perme- melt density obeys the relation ability anisotropy of silicate mush is nonexistent and p: p,ll - a(T - I"") - F(C,- C)l (8) therefore neglected. Although restrictions are not placed on isobaric specific where p. is the density of melt of bulk composition at the heat and thermal conductivities ofdifferent phases,each eutectic temperature, and C is the mass fraction of is assumedconstant in the present version of the model KAlSirOu in the liquid. The domain and boundary con- (i.e.,c. : q and k": h). Becausethe isobaricspecific heat ditions and phaserelations are depicted in Figures I and of silicate melts (l) and solids (s) are within a few percent 2 as is the initial state in temperature-compositionspace. of one another, this assumption seemsreasenable. Sim- At the beginning of the simulation, the melt is homoge- ilarly, unlike those of salt-HrO and liquid metals, the neous, isothermal, and superheatedabout 40 "C above thermal conductivities of silicate liquids and solids are the liquidus temperature of 1356 .C. Note that lines of not appreciably different at magmatic temperatures. constant melt density slope away from the KAlSirOu side Momentum transport is accomplished by Darcy per- of the diagram. As diopside crystals are extracted from colation in solid-dominated mush ([ > 0.5) and by in- melt, the density of residual liquid decreasesalong the ternal viscous stressesin melt-dominated mush (t < 0.5). liquidus as the melt composition moves toward the eu- That is, in the melt-dominated, two-phaseregion, no rel- tectic at C,' : 0.38. All the simulations reported here ative motion exists between solid and melt; crystals are were accomplished using bulk compositions on the di- simply advected along with melt. In regions where { > opside-rich side of the phasediagram. 0.5, we assumethat solid is static (V": 0) and melt per- The mixture continuum properties (velocity, enthalpy, colatesthrough the solid matrix-supported crystal plexus. composition, density, and chemical diffusivity) are de- The critical isofrac (q : 0.5) is chosen on the basis of fined as available experimentaland theoreticalinvestigations (e.g., + (l - (9) Arzr, 1978;-van der Molen and Patterson,1979; Marsh, d: td, [)d' 198l; Furman and Spera, 1985; Marsh, 1989; Lejeune where @is a general field variable that could be u, v, h, and Richet, 1995). C, or D. Note that the symbol Cdenotes the massfraction In summary, the main limitations of the model are as of leucite component in the mixture, and C, denotes the follows: (l) it assumeslocal thermodynamic equilibrium; correspondingquantity in the liquid phase. (2) the densities of solid and melt are solely dependent The hybrid formalism of Oldenburg and Spera(1992a) on temperature and composition, and solid and melt at is usedto account for momentum transport. That is, per- the same temperature and composition have the same meability and viscosity terms in the momentum equation density (3) the porous region is isotropic; (4) the calcu- are switched on dependingon the local value ofthe solid lations are conducted in a two-dimensional domain; and fraction (t). For t < 0.5, where melt is the connected (5) the calculations are limited to binary-component sys- phase, no relative motion between melt and solid is al- tems with a simple eutectic topology. lowed. In this case,the viscosity of the mixture depends on the local value of { (e.g.,see Metzner, 1985, for a review). In regions where f > 0.5, the only relevant vis- Mathematical model cosity is that of the liquid, and the local permeability is The continuum conservation equations for mass, mo- assumedto vary with f" accordingto the Kozeny-Carman mentum, and energy may be expressedas expressron SPERA ET AL.: SIMULATION OF COIIVECTION AND CRYSTALLZATION I 193
q=j=v=0 '/ ,;t , or ,f ;, T=Tw,j=v=0
i/,i,,/ T =Tw initialconditions /,"t/,F Q=0 /,/ i./ i j=o j=o ./ ,, ,i ,' V=0 T =To u=0 ,o.oi1',i"13'fl3,., C=Co Ev i /h //j2 dx V=0
0 02 KAlSip.KAtSip6 CaMgSi.Ou
Fig.2. Equilibrium phaserelations for the systemKAlSirO6- q=j=v=0 CaMgSirOuafter Schairer and Bowen (1938). Densities of su- perliquidus melts were computed from Lange and Carmichael Fig. 1. Domain, initial, and boundary conditions for the sim- (1990). All simulations presentedin this study have initial con- ulations.The physicaldomain is either 4 x 1,2 x 1, or I x I ditions for temperature (%) and mixture composition (C") in- to simulate dikelike, boxlike, or sill-like magma bodies, respec- dicated on the diagram. Note that diopside is the liquidus phase tively. The aspectratio ofthe body is defined,4R : d/L, where and that residual melts become less dense (more K-rich) along d is the depth and I, the width of the body. In all casesthe right the liquid line ofdescent. Lc : leucite, Di: diopside. side of the computational domain is a mirror plane (i.e., the midpoint of a magma body). The left side is a rigid wall set at a fixed temperature either 20 or 100 "C below the eutectic tem- heat (i.e., c" : ct), the expressions for the solid, liquid, perature with the flux oflight component and velocity the equal and mixture enthalpies are given respectively by to zero. Along the upper boundary (roof), either conditions are identical to those set along the left wall (i.e., 7* : subsolidus h":cT for O
TABLE2. Relations among fraction solid (f"), enthalpy (lt), and A temperature (D for binary eutectic systems h"o,+ah+c(li;o'T.o1) Region h- Lh 0
Ir( - Lf' B rootof Eq.17 1- t" c (hi + hd- h)lhi r- n D '|
Nofe.' Regions are depicted in Fig. 3. Additionaldefinitions are given in D the Appendix. T Fig. 3. Generalizedenthalpy-temperature diagram for binary SubstitutionofEquations ll, 12, and 17,into l3 givesa eutectic systems.The amount of melt formed at the solidus de- quadratic expressionfor the mixture enthalpy as a func- pends upon the bulk composition, as reflected by the liquidus tion of f, valid within regron B as shown in Figure 3: temperature, ?iiq, and the fusion enthalpies for leucite and di- opside. The /rf is the heat of fusion of solid of eutectic compo- . : / A \^ - (z,l+cr^\",cT,q+Ar' + (I 8) sition (C".' : 0.38). The enthalpy term denoted Aft is a fictive , |,r- r"---1lt '-1- quantity that correspondsto the heat neededto transform solid \ffii of bulk composition (C: Co)at temperature I*, to liquid of the where same composition and temperature. The enthalpy of the two- f /r -_ - \-r /^ - \ phasemixture at the boundary betweenregions B and C depends - Lh= A: Ir (+" !-)lrr + {*,1:- :'')r.0,.(rs) on 2,,0,a function only of the bulk composition under isobaric L \i ,",- T^/)'-' \f"., f-/ conditions. Note that the quantity of Aft depends implicitly on the initial composition of the system (Co) becausethe liqui- respectively. The hl represents the enthalpy offusion for dus temperature (2,,o)depends on composition. Mixing diopside-leucite mixture of eutectic composition (C,.r ). enthalpies are almost certainly a small fraction of fusion For the system KAlSirOu-CaMgSirOu, this quantity is ap- enthalpies in this system. Therefore, ideal mixing be- : proximately defined bV hl 0.38ft." + 0.62h.dj because tween KAlSirOu and CaMgSi'Ou liquids has been as- eutectic liquid has a mass fraction of KAlSirOu of 0.38 sumed. Newton-Raphson iteration of Equation l8 is used (see Fig. 2).That is, ftf is a eutectic-weighted linear com- to determine { with the mixture enthalpy (ft). Tempera- bination offusion enthalpies for leucite and diopside. ture, I, is found from Equation 17. In region B of Figure 3, the lever rule implies that Finally, at the solidus temperature (region C, Fig. 3), the fraction solid (l) varies continuously in the range (16) Tuo- T*, < I < I, T: T*, (20) - T^- T*r- for f" in the range 0 < t < [(?""" - Tno)/(T*t Z-)]. This expressionmay be inverted to give an explicit relation- and the enthalpy-t relationship is ship between temperature and fraction solid: (2r) T"ot Thermodynamic properties of leucite and diopside used in the calculations are listed in Table l. The relations properties TABLE1. Thermodynamic of diopsideand leuciteused between f, and h are summarized in Table 2. in the simulations Property Diopside Methodology 't942 ofthe coupled partial differential L (K) 1665 To facilitate solution 7:.,(K) 1573 1573 equations and the supplementaryrelations, a nondimen- hl (10sJ/kg) 3.574 1.642 sionalization was performed. The dimensionless vari- p, (at 7-[kg/ms) 2624 2236 expressions c [J/(ks. K)] 1100 1100 ables and the final form of the conservation are presented in the Appendix. A few other details are Nofe.' The fusion enthalpy for leucite was approximated using the de- pression of the freezing point of pure KAlSirO6at 105 Pa and standard also presentedin the Appendix to define the model. thermochemicalrelations based on the experimentalphase diagram (Schairer The physical significanceof the dimensionlessparam- and Bowen, 1938). In fact, leucite does not melt congruently. eters of this problem must be interpreted with care and SPERA ET AL.: SIMULATION OF CONVECTION AND CRYSTALLIZATION I 195 not confused with their classical interpretation. For ex- described, and this coupling is part ofany phase-change ample, in a single-phasefluid with simple boundary con- system, regardlessof length scale. ditions and no phasechange, the melt convective velocity Each conservation equation can be cast into a general is a simple function of a dimensionlessgroup known as form consisting of an unsteady term, an advective term, the Rayleigh number, Ra, a measure of the importance diffusion-like terms, and generalized source terms (Ol- of thermal buoyancy forces relative to the impeding ef- denburg and Spera, 1990). Once in this generalform, the fects of viscous drag. In the presentproblem, the irregular equations are amenable to solution by a primitive vari- time-dependent configuration of the solid, melt, and able, control-volume based,iterative finite-diference al- mushy regions and the dependenceof the fluid enthalpy gorithm. Details of the implementation used are present- on temperature, solid fraction, and composition pre- ed in Oldenburgand Spera(1990, 1991,1992a,1992b). cludes the establishment of a single set of simple (uni- versal) scalesfor length, temperature, and composition. Rnsur,rs The requirement of thermodynamic equilibrium and the use of a phase diagram for the system precludes a Introduction straightforward analysis such as that commonly done in Solidification of melt in the binary system KAlSirO6- simpler problems (e.g.,Trial and Spera, 1990).For ex- CaMgSirOuwas simulated for variously shaped,two-di- ample, in the two-phase region the thermally and com- mensional domains with boundary and initial conditions positionally driven flows are not independentbut instead as depicted in Figure l. In all casesthe temperaturealong are linked through the phase diagram and the thermo- the left side is set at some constant value below that of chemicalproperties of the appropriatephases. In the mush the solidus (eutectic)temperature (2.") to simulate a cold region, the ratio of chemical to thermal buoyancy de- wall (i.e., T*: T*, - X, where X : 20 or 100 'C). The pends on the slope of the liquidus curve. However, in the upper boundary is either adiabatic or set at the same all-liquid region, the buoyancy ratio is not explicitly fixed subsolidus temperature as the left wall. All boundaries by phase relations. Becauseof these complexities, it is are rigid (V : 0) and impermeable except the midplane not particularly useful to discussthe evolution ofthe sys- of the domain, which is a mirror plane, with appropriate tem in terms of simple classicaldimensionless numbers. boundary conditions. The computation is performed in In fact, such a simple analysiswould be misleading. This one-half of the physical domain (i.e., the magma chamber is not to say that systematic relationships do not exist per se)taking advantageof the symmetry of the problem. between field variables and the parametersof the prob- The results presented here are portrayed in one-half of lem, but only that theserelations are not simple functions the physical domain. A few caseswere computed for the of a singledimensionless parameter. By performing a suf- entire domain, and the solution was found to be essen- ficient number of numerical simulations one may devel- tially symmetric for all field variables. op insight into the phenomena and parametersthat are By far, the most informative method of analysis is by most important during the evolution of magma solidifi- examination of two-dimensional animations depicting the cation in this model system. temporal evolution of field variables such as the bulk It must be emphasizedthat the presentmodel describes composition (C), melt composition (C,), temperature(T), some very complicated processes.The phenomena de- fraction solid ([), mixture enthalpy (h), ard the flow field scribed by the model may occur over the entire range of (V). Animation has revealed the dynamic and unsteady macroscopiclength scales.These can be as small as crys- nature of convective solidification, including the devel- tals (millimeters to centimeters) or as large as linear di- opment of heterogeneityin both melt composition and mensions of magma bodies (meters to kilometers). So- the fraction solid, the nonmonotonic behavior of the ki- lution ofthe equationsdescribing the processofconvective netic energy (velocity) of magma, and the variation of heat and mass transport becomescomputationally more heat extraction rates through time. Complex, time-de- difficult as the length scaleof the system becomeslarger. pendent flow structuresdevelop during solidification be- Such calculations of convection are limited to values of causeof the strong effectsof compositional buoyancy in the Rayleigh number that are small relative to those of the system modeled (Rp - 3). Results of this study are natural magma bodies. Except for the length scale,all the portrayed here by use of time-seriesplots of spatially av- parameters in Ra are frxed in terrestrial systemsby the eragedquantities and freeze-framesnapshots ofparticu- choice of the components of the phase diagram and the lar field variables. Video animations portraying the evo- general range of temperaturesof the system. Employing lution of all field variables are available and provide a the correct thermophysical parametersfor the leucite-di- very informative view of the evolution of convective so- opside binary systemin Ra, we find the length scaleI to lidification. be on the order of centimeterseven for the largestvalues In the next section, a detailed description of a typical of Ra for which calculations can be performed. This lim- simulation is given. This is followed by a discussion of itation to small systemsshould not be seen as a serious the sensitivity of system behavior on (1) magma-body defect. In fact, the model describesprocesses that would shape,(2) location of cooling surfaces(i.e., marginal cool- occur at larger length scalesand larger values ofRa. Fur- ing or simultaneous marginal and roof cooling), and (3) thermore, the crucial coupling between the processesis the solidification rate. Then the results are shown at an Lt96 SPERA ET AL.: SIMULATION OF CONVECTION AND CRYSTALLIZATION (a) 2 0 2u l/' tc FS Y 10 00r 00 000 050 000 0.50 X X ft) 2.O FS Y r.o 00r 0.0 000 050 000 0.50 X X (c) FS Y 10 050 050 X X SPERA ET AL.: SIMULATION OF COIWECTION AND CRYSTALLZATION tr97 (d) FS Y 10 050 X (e) FS Y 10 0.00 0.50 1 00 000 050 I00 X X Fig. 4. Development of flow field (V), liquid composition tectic melt-enriched porosity waves that travel vertically. These (C,), and fraction solid (t) for solidification of magma from a are especially striking in two-dimensional animation and rep- vertical sidewall in a boxlike domain. Note that Crwas not nor- resent an interesting phenomenon in porous media geochemical malized; it representsthe mass fraction of KAlSirO. component dynamics. Evolved melt is delivered episodically (but continu- in the liquid phase.Roofand bottom boundaries are adiabatic ously) to the top of the magma body by porosity-wavetransport. : and rigid. Left margin simulates a cold country rock-magma (c) Nondimensional time t 0.03. Isocomps and isofracs as in contact at temperature T* : T*, - 100 €, where T*r is the b. Note strong development of countercurrentflow with leucite- eutectic temperature in the system KAlSizOo-CaMgSLOu.The enriched melt accumulatingroofward to form a compositionally flow field for the mixture velocity is shown by nondimensional zoned region within a roofuard-thickened mushy region. (d) : maximum mixture-velocity vectors. Note how the mush thick- Nondimensional time 7 0.04. Isocomps and isofracs as in b. ens with time through the sequencea-e. (a) Nondimensional Flow is strongly defined by clockwise rotation resulting from time / : 0.01. At 7 : 0.01, the effectsof crystallization-induced buoyancy torque associatedwith generation of chemical buoy- buoyancy are important at the left bottom corner of the magma ancy within the mush along the left sidewall. The roofuard mushy body, where KAlSirOu-enriched melt is in clockwisecirculation. zone is stratified in KAlSirOu component in the upper central : Solid lines are isocornps. O) Nondimensional time 7 : 0.02. part of the magma body. (e) Nondimensional time t 0.05. Note the enhanceddevelopment of clockwise flow in the mushy Isocomps and isofracs as in b. About 30o/oof the magma body region along the solidification front, where KAlSi'Ou-enriched has solidified by this time. The magma body is vertically strat- melt forms by fractionation. In melt-dominated regions(t = 0), ified with respect to melt composition, temperature, and phe- the circulation remains counterclockwise.Note existenceof eu- nocryst content (i.e., fraction solid). I 198 SPERA ET AL.: SIMULATION OF CONVECTION AND CRYSTALLZATTON TreLE3. Parametersheld constant and applicablefor all simulationsreported upon in this study Co:0.15 C- = 0'38 I,q : 1356'C %: 1400.c 7d: 1300.C hi :2.84 x 1tr J/kg Lh:3.28 x 105J/k9 a:5x105K-t = 5 kg/(m's) ( : ?o 10-6 m2ls p:0.07 Da:5.3 x 10 8 Pr: 1946 Fa: 1.52x 10s Ps:4.67 x 105 sr: 0.338 Rp= RgRa:3.2 Nofe; Additional thermodynamicproperties are given in Table 1. advancedtime when most of the magma has crystallized, about l0 m/yr based on a magma-body thickness of I depicting the extent of mineralogical zonation within the km and the parametersgiven in Table 3. For comparison, crystallized magma body. the Stokes settling rate of a 2 mm diameter spherical olivine crystalis about I m/yr. Simulation overview At t : 0.01 (Fig. 4a), a counterflow with a clockwise A typical result is discussed here for crystallization senseofrotation nucleatesnear the left bottom corner of within a boxlike magma body. To focus on the sidewall the magma body as crystallization of diopside enriches processes,the roof and floor of the body are adiabatic residual melt in low-density leucite component. This sets (seeFig. l) and the left margin is a vertical magma-coun- up a chemically driven flow that deflectsthe isocomps; a try rock wall along which the temperature is fixed at 100 region of enrichedmush moves upward as a porosity wave "C below that of the eutectic temperature in the system (cf. Fig. 4a-4c). Ati :0.01, the flow is dominated by KAlSirOu-CaMgSirOu.Note that the computational do- thermal effects throughout most of the domain, except main is one-half of the cross-sectionalarea of the two- near the crystallization front where chemical buoyancy is dimensional magma body. Parametersused to define the being vigorouslygenerated. Note the developmentof small model are collected in Table 3. Figure 4 shows the de- clockwise vortices at the top and bottom of the magma velopment of the velocity, liquid composition, and solid body along the mirror plane (right side). fraction fieldsat nondimensionaltimes 0.01 < t < 0.05 At later times (Fig. 4b and 4c), the thickened mushy in increments of 0.01. Recall (see the Appendix) that region begins to dominate the flow along the solid-mush real time (l) is related to nondimensional time (r) lV r : and mush-liquid transition regions.Although flow within - t L'Ah/Kc(To 4") or t : rt , where z is a characteristic the liquid portion ({ : Ol is still thermally driven, an scalefor time for this problem. Note that the initial tem- increasinglysmaller (but still dominant) region of the do- perature (f0), the solidus (or eutectic) temperature ({'), main is affected.AtT : 0.02,7^ is two to three times and the characteristicfusion enthalpy (seeEq. 19) are all smaller than the peak7^^-, and7-.* is again smaller at factors. Systemsinitially close to their solidus (small su- t : 0.03. At t : 0.03, about 20o/oof the magma has perheat) and with large fusion enthalpiesevolve lessrap- crystallized. Notice by comparing Figure 4b and 4c Ihat idly than those with larger amounts of superheat.For the alarge amount of KAlSirOu-rich liquid is delivered to the particular model system studied here, a nondimensional top of the magma reservoir when the porosity wave that time of 0.01 correspondsroughly to 120 yr and 3000 yr contains melt of eutectic composition (C, : 0.38) reaches for bodies of I and 5 km thickness,respectively. the roof. This causesthe incipient development of a strat- For a magma-body thicknessof 3 km the time interval ified region at the top of the magma body. Note also that betweensuccessive snapshots is about 103yr. During this the isofracs (lines of constant fraction solid) within the time interval (Fig. 4a-4e), about 300/oof the magma, ini- mush remain subparallelto the C, isocomps,an effectthat tially superheatedby 40 "C, crystallizes.In Figure 5, time- reflects the attainment of local thermodynamic equilib- series plots that depict the variation of the maximum rium within the mushy two-phase region. mixture velocity (7-",) and spatially averaged fraction At t : 0.05 (Fig. 4e), about 300/oof the magma has solid ([,"") are shown for the evolution portrayed in Fig- crystallized (Fig. 5). Stratification of I and C, is well de- ure 4. veloped in the middle of the magma body, especiallyin At the very earliesttimes (r < 0.01, not shown),the the upper 40o/oof the chamber. The flow now is essen- cold left wall sets up a descending (counterclockwise) tially chemically driven becauseof solidification effects. thermally driven flow throughout the entire domain. Melt Although irregular in plan, the circulation is mainly velocities reach their greatestvalues (7.". * 250) at very clockwise, with maximal mixture velocities along the early times (Fig. 5), at which the maximum value forT-". mush-liquid interface (i.e., near the f" = 0 isofrac). is close to the value predicted using the scale relation Perhaps the most salient feature of the evolution por- given by Trial and Spera (1990) for a thermally driven trayed in Figures 4 and 5 and noted in all the numerical single-phaseflow: experimentsis the relationship betweenthe development V^,* x 3fRah. (22) of compositionally zoned melt near the top of the magma body and the changein dynamic regime. At early times, This velocity corresponds to a dimensional velocity of the flow is thermally driven, and KAlSirO.-rich melt does SPERA ET AL.: SIMULATION OF COI'{VECTION AND CRYSTALLIZATION I 199 not accumulateroofuard. At later times, chemical buoy- 250 ancy plays an increasingly important role, and melts be- 0.3 come more evolved during the course of solidification. The transition from thermally driven to chemically driv- 0.2s en flow occurs becauseofthe relatively large value ofRp o.2 (-3) characteristic of this particular binary system. The buoyancy ratio Rp depends on the eutectic composition 0.15 (C".,) and temperature (f*r), the initial conditions (Co and Io), and the temperature and composition depen- 0.1 dence of the density of KAlSirOu-CaMgSirOumixtures. 0.05 As magma undergoesprogressive solidification, the frac- tional volume of the body within the crystallization-tem- 0 perature interval increases.This is noted by examination 0.ol o.o2 0.03 0.05 ofthe fraction solid ([) field plotted in Figure 4. It is in this volumetrically expanding mushy region that chemi- Fig. 5. Time-seriesplots for the simulationsillustrated in cal buoyancy is generated. Fig. 4. The nondimensionalmaximum velocity (V-..) and spa- In Figure 6, a sequenceofsnapshots is shown depicting tially averagedfraction solid ({,.") areplotted against time. The the evolution of the temperature field superimposedon nonmonotonicbehavior ofV^* is correlatedto qualitativevari- the flow field for the time interval 0.01 < / < 0.05. Up- ationsin the flow field,as portrayed in Fig. 4. The rateofsolid- for 7 > 0.01.About 300/oof flow is most rapid in the melt-dominated (l < 0.5) por- ification(d{/dr) is nearlyconstant the magmabody has solidifiedat 7 : 0.05.This corresponds tion of the two-phase region. The solidus isotherm re- roughlyto 5000yr for a 3 km thick magmachamber. mains nearly subparallelto the { : 0.5 isofrac throughout the courseof solidification. The fact that the most vigorous upflow occursin a part of the domain where no relative motion between solid (t < 0.01), motion near the cooled, vertical left wall is and melt exists illustrates the critical nature of the as- dominated by negative thermal buoyancy (Fig. 8a). How- sumptions made to characterizethe rheological and per- ever, unlike the casefor the adiabatic roof(Fig. 4a) when meability properties of magmatic mush. This is the crux cooling out the top takes place,strong downwelling owing of the magma-diversity problem by convective fraction- to roofward heat loss occurs(cf. Figs. 4a, 4b,8a,8b). At ation. Additional experimental work on the rheology of 7 : 0.02, upward wall flow developsalong the entire left magma and the relationship between permeability and sidewall, and K-rich melt accumulatesunder the down- fraction solid is neededfor further progress.The kinetics ward-growing crystalline top. By comparing Figures 4 and and dynamics of the microinterface betweencrystals and 8, it is noted that thermal and compositional stratifica- melt and how theseprocesses can be modeled at the mac- tion are enhanced by cooling through the roof. At the roscopic level is a challenging problem for magma dy- latest time, t : 0.03 (Fig. 8c), the flow is chemically driv- namicists. en throughout the entire domain, and the solidified roof zone is substantially thicker than that along the sidewall. Roof and sidewall cooling Perhaps the most significant feature of this caseis that, A numerical experiment was performed for conditions as in the adiabaticroof case,a chemicallystratified mushy identical to the last example exceptthat both the sidewall region develops in the upper part of the domain during and the roof were cooledto I-, : T*^r: 1'".,- 100 'C. convective solidification. In the previous casethe roofwas treated as an adiabatic surfaceto isolate the effectsof sidewall crystallization. Magma-body shape The overall dynamics are broadly similar to the side- Magrna bodies undoubtedly come in a variety of shapes wall cooling case;as time progressesthe effectsof chem- and sizes.To assessthe sensitivity of magma-body shape ical buoyancy come to dominate the flow. There are some on the evolution of solidification, a seriesof simulations significant differences,however. Roofcooling createsad- were performed with identical initial and boundary con- ditional thermal instabilities associatedwith cooling from ditions but ditrering shapes.Bodies with aspectratio (,4R) above. This triggers small-scale horizontal convection, equal to 2 (dike), I (box), and 7z (sill) were cooled from - 'C). more vigorous convection early on, and, ofcourse, higher above and from the sides(f.., : T*: Td 20 The solidification rates. This may be seenby comparison of rate of heat extraction, which is proportional to the Figures 5 and 7. Note, in particular, the higher V ^ ^ and boundary temperature, is somewhat smaller than in the its more rapid decay in the roof-cooled case,as well as case examined in Figures 4-6, where the wall tempera- the greater extent of crystallization (-450lo) in compari- ture was 100 "C below the eutectic temperature and the son with the adiabatic roofcase in Figures 4 and 5. roofwas taken as an adiabatic surface. Figure 8 is a snapshot sequencefor the roofand side- In Figure 9, the maximum convective velocity (Z--) wall-chilled box. Frame by frame comparison of Figures and the fraction solid ([) are plotted against time for a 4 and 8 revealssome interestingdifferences. At early times sill, dike, and box magma body. As expected, the sill 1200 SPERA ET AL.: SIMULATION OF CONVECTION AND CRYSTALLIZATION 2.0 2.0 2.0 (a) /--\\\\\-./ / (b) (c) / /-.t\\\\\-- //1\\\\\\-- i,-:)))::r l 1.5 I I I I I Y 1.0 I I I NYiiii il:::;l:-->11. 0.0 E 0"0 0.0 0.00 1.00 0.00 1.00 0.00 0.50 2"Q 2.0 X (d) (e) T 082 0.64 Y 1.0 046 028 0.09 -0 09 -o.27 -0 45 -0 63 -0.81 0.0 0.0r 0.00 0.50 1"00 0.00 0"50 X X Fig. 6. Evolution of the nondimensional temperature field along the entire sidewall. (c) Nondimensional time a : 0.03. for the simulations presentedin Figs. 4 and 5. The solidus iso- Evolved melt is being delivered to the top of the magma body : therm is 7 0; initially T : I throughout the entire domain. at this time. (d) Nondimensionaltime T :0.04. Note develop- (a) Nondimensionaltime t : 0.01. Note nucleationof a clock- ment of thermal zoning within the magma body. (e) Nondimen- wise circulation in the bottom left part of the domain. (b) Non- sional time t : 0.05. For characteristic dimension d : L : 3 : dimensional time I 0.02. Compositional buoyancy effects km, this snapshotcorresponds to roughly 5000 yr after the be- overcome negative thermal buoyancy, and local upflow occurs ginning of crystallization. SPERA ET AL.: SIMULATION OF COIWECTION AND CRYSTALLIZATION t20l 400 0.5 to becomesluggish as crystallization occurs.For a magma 350 body of I km width, V^ : 100 correspondsroughly to a convective velocity of severalmeters per year, which is 300 the same as the settling rate of a millimeter-sized phe- 250 nocryst. Solidification times are roughly 30, 20, and 15 ka for the sill, box, and dike, respectively (Z : 3 km). r>- ?oo In Figure 10, a series of snapshotsare shown for the 150 variously shaped bodies at equal time intervals in the range0.01 < 7 < 0.05. In eachcase the flow is thermally 100 dominated at early times and becomesmore chemically 50 driven as solidification proceeds.Because the sill cools fastest, stratification develops quickly, and at / : 0.05 0 the upper half of the sill is either completely solidified or 0 0.0r 0.02 0.03 0.04 0.05 mush that includes zoned melt. tlDG occupied by chemically For all shapes,thermal and chemical stratification be- Fig. 7. Time-series plots for the simulations illustrated in comes well developed near the roof at later times (e.g., Fig. 8. The nondimensional maximum velocity (V*) and spa- cf. Fig. l0g, l0h, and lOi). tially averagedfraction solid ([."") are plotted againsttime. This Careful examination of Figure l0 reveals that the ten- presented caseis identical to that in Fig. 5, except that cooling dency to form compositionally stratified melt near the takes place through the roofas well as the sides.About 45oh of top of the magma body occurs regardlessof body shape. the body has solidified at 7 : 0.05, which correspondsto roughly The key point is form 5000 yr for a 3 km thick magma body. that the tendency to a composi- tionally layeredsystem is most sensitivelydependent upon the thermodynamic and transport properties of the mag- cools the fastest and the dike the slowest. At / : 0.05, ma rather than the shape of the body per se. The main more than 350/o of the sill has solidified, whereas only role of body shape is in controlling the rate at which about 22o/oof the dike has crystallized. Maximum veloc- compositional zonation develops as a result of the inter- ities are greatest for the dike because ofits vertical extent. action between the surfacearea of magma body-country In all cases,T** is nonmonotonic, although the flow tends rock, rate of solidification, rate of generationof chemical 2.0 2.0 2.O (a) i (b) (c) I I t.c t.J I I 1 1, \. 034 E 030 o26 1.0 Y -tffio23 - 019 0.0 rr 0.0 0.0 E 0.00 0.50 1.00 0.00 0.50 1.00 0.00 0.50 X X X Fig. 8. Development of flow field (V) and liquid composition well developed along the bottom of the sidewall. Note the growth (C ) for solidification of magma from the side and roof of a I x of a solidifred zone along the roof. (b) Nondimensional time I I magma body. Roof and left sidewall have I..", : 7* : 7"., - : 0.02. Significantupward flow extendsalong the entire sidewall, 100'C. Bottom boundary is rigid and adiabatic; right side is a and the downward thickening mush is strongJyzoned with re- mirror plane. Plotting symbols are identical to those described spect to KAlSirO6 component. (c) Nondimensional time 7 : in legendto Fig. 4. (a) Nondimensional time t : 0.01. In com- 0.03. Note the well-developed,liquid compositional stratifica- parison with Fig. 4a, the compositionally driven flow is not as tion within the roofirard mush zone. r202 SPERA ET AL.: SIMULATION OF CONVECTION AND CRYSTALLIZATION 0.4 Initially, homogeneous melt of bulk composition C : 0.3s 0.15 fills the magma chamber. This composition corre- sponds to 850/oby mass of CaMgSirO6 component and 0.3 I 5oloby mass of KAlSirOu component. At I : 0.3 I , more 250 0.25 than 900/oof the melt has crystallized, and substantial I variations in modal abundancesare clearly present. Di- ,oE 2oo 0.2 ! opside is concentrated near the floor of the body (C = 150 0.15 0. 14) and is also enriched, relative to the initial compo- sition along the left (cooled) vertical wall. In contrast, 100 0.1 leucite is enriched in the upper part of the magma body 50 0.0s and reachesa maximum value of about 0. 17 in the mid- dle ofthe body. 0 0 The mineralogic zonation establishedin the intrusive 0.01 0.o2 0.03 0.04 0.o5 magma body has both a horizontal and vertical compo- Fig. 9. V-." and f,.""for dike, box, and sill magmabodies. nent. For example, a horizontal slice through the body just Coolingis from theroof andsidewall (f,*r: Z* : f"., - 20 "C). above its midpoint would reveal a radially zoned Thecurves for thebox canbe comparedwith thecase illustrated pluton. Similarly, a vertical profile taken along the right in Fig. 7, which is identicalexcept that T,a: T-: fd - 100 side of the domain would show a sandwich horizon of 'C. Thermalbuoyancy efects are maximized in thedike because KAlSirOu-enriched material. We emphasizethat this zo- of the greaterextent of coolingsurface. These greater thermal nation resulted from an initially homogeneousmelt. buoyancyforces show up asa morevigorous flow comparedwith In the particular model shown in Figure I l, solidifi- (sill) the shallowenclosure solution.The fastercooling ofthe sill cation is rapid becausethe boundary temperature, along is alsoseen by thegreater value for the spatiallyaveraged func- both the roof and the right side,is set to Z*, - 100'C. tion solidified(t,.") at a given time for the sill relativeto the undercooling and an dike. In other simulations with a smaller adiabatic roof, even greatermodal variations are found becauseof a more protracted cooling history. buoyancy, and generation of latent heat. These interac- tions are highly nonlinear and give rise to unsteadyflow, Sumrnary including reversals in flow directions that occur during Compositionally zoned ignimbrites and plutonic bod- the course of solidification. ies are exceedingly common. In extrusive deposits, evolved magma commonly erupts early. As eruption con- Mineralogic zonation tinues, magma compositions become more mafic. In in- Radially zoned granitic (sensulato)bodies are exceed- trusive bodies, it is common to find normally zoned bod- ingly common in the geologic record and include both ies with more evolved bulk compositionsinward and more normally zoned (mafic margin, silicic core) and reversely mafic compositions toward the magma-country rock zonedplutons (e.g.,Bateman and Chapple,1979; Ayuso, contact. In regions characterizedby considerable relief, 1984). Large-volume layered intrusions are almost al- bulk compositions generallybecome more mafic with in- ways vertically zoned in terms of modal abundancesand creasingdepth. When roof zonesof plutons or intrusions bulk composition. It seemslogical, if not obvious, that a are exposed,it is not uncommon to find sandwich hori- connection ought to exist betweenzoned extrusive bodies zons. These extremely well-known attributes of igneous (ignimbrites, lava flows, and domes) and plutonic bodies. systemscan be simulated by the convective solidification To study the implications of the model presented here model illustrated in this study. Although there are signif- with respectto the origin of modally zoned intrusive bod- icant limitations of this model in its presentform, incor- ies, a seriesof simulations was perfonned until essentially poration of more accurate macroscopic physics on the all the magma crystallized (t - unity). In Figure I l, re- basis of a deeperunderstanding of microscopic interfacial sults from a typical caseare shown for the spatial varia- dynamics and extension to three dimensions and multi- tion in modal leucite fraction. In this example, cooling component systemsis presently in sight becauseof sig- from both the roof and sides was allowed, with ?"-.r : nificant advancesin high-performancecomputing. At the I"io. : Z"o,- 100 "C. The parameters of the simulation same time, the thermochemical databasefor oxide and are otherwise identical to those illustrated in Fieure 8. silicate phasesrelevant to magmatic rocks is expanding ---J Fig. I 0. Snapshotsshowing a comparison of the evolution of Also note the common occutrenceof vertically migrating poros- melt composition (C) and flow field (velocity vectors) for mag- ity waves.(a) Dike at t : 0.0t . (b) box at I : 0.01. (c) sill at I ma crystallization in a dike, box, and sill. Cooling occurs along : 0.01, (d) dike at I : 0.03, (e) box at 7 :0.03, (f) sill at t : the sidewall and roof, where the temperatureis I,*r: T-: T*, 0.03, (g) dike at 7 : 0.05, (h) box at 7 : 0.05,and (i) sill at 7 : - 20 "C. Note ditrerent x- and y-axis scalesamong the figures. 0.05. SPERA ET AL.: SIMULATION OF CONVECTION AND CRYSTALLIZATION t203 40 20 (a) (b) 35 30 25 20 Y 1.0 Y 050 15 - \ \\\\-- \ \\\\\- 10 UC 0.0 00r 000 000 000 050 100 000 X 40 (d) 35 3.0 25 20 Y 10 '1.5 1.0 05 00 0.00 0.50 1 00 X (s)40 35 30 25 2.O 1-5 10 0.5 0.0 0.00 050 X r204 SPERA ET AL.: SIMULATION OF CONVECTION AND CRYSTALLIZATION 2.0 layered intrusions that show strong evidence of multiple intrusion and fractional crystallization. Transiently per- meable magma-body floors need to be incorporated into the next generation of the simulator. Acxxowr,rncMENTs We gratefully acknowledge financial support from the NSF Geochem- istry) and DOE. Computational work was performed at the NASA Ames Research Crnter in Moffet Field, California, the San Diego Supercom- puter Center at UCSD, and the Magma Dynamics Iaboratory at UCSB. Comments from B.D. Marsh and J. Longhi improved the manuscript significantly and are greatly appreciated. 0.168 RnrnnBxcps crrED 0.1.61 Arzi, A.A. (1978) Critical phenomenain the rheology of partially melted rocks. Tectonophysics,44, 173-184. Y A1tso, R. (l 984) Field relations,crystallization, and petrology ofreverse- ly zoned granitic plutons in the Bottle Lake Complex. U.S. Geological o.147 0.154 Survey,1320, 64 p. Bateman, P.C., and Chapple, B.W. (1979) Crystallization, fractionation, and solidification of the Toulumme Intrusive Series,Yosemite Nation- al Park, California. GeologicalSociety ofAmerica Bulletin, 86(5), 563- 579. Beckermann, C., and Viskanta, R. (1988) Double-diftrsive convection during dendritic solidification ofa binary mixture. Physiochemical Hy- drodynamics,10, 195-213. Bennon, W.D., and Incropera, F.P. (1987a)A continuum model for mo- mentum, heat and speciestransport in binary solidJiquid phase change systems:1. Model formulation. International Journal of Heat and Mass Transfer, 80, 2161-2170. - (l 987b) The evolution ofmacrosegregationin staticallycast binary ingots. Metallurgical Transactions,l 88, 61 l-6 16. 0.140 Bergantz,G.W. (1992) Conjugatesolidification and melting in multicom- ponent open and closed systems. International Journal of Heat and 0.0 MassTransfer, 35, 533-543. Bergantz,G.W., and Lowell, R.P. (1987) The role ofconjugate convection 0.0 1.0 in magmatic heat and mass transfer. In D.E. Loper, Ed., Structureand dynamics of partially solidified systems,NATO ASI Series,series E, X no. 125, 370-382. Fig. 11. Depiction of the modal amount of leucite in equant Bottinga, Y., and Weill, D.F. (1972) The viscosity of magmatic silicate magma body (lR : 1) at t : 0.31 when the averagefraction liquids: A model for calculation. American Journal of Science,272, solidified is >90ol0.The magma body is cooled from rhe right 438-47 5. Bowen, N.L. (1928) The evolution ofthe igneousrocks, 332 p. Dover, side and from the roof (7.*, : Z"rc : f".r - 100 'C). Other New York. parametersare collected in Table 3. Initially, magma is super- Brandeis,G., and Jaupart,C. (l 987) The kinetics ofnucleation and crystal heatedby 40 and is homogeneouswith C: 0.15 (i.e., l5olo "C $owth and scaling laws for magmatic crystallization. Contributions to KAlSirO6 component in the melt). Contours are drawn for the Mineralogy and Petrology, 96, 24-34. modal amount of solid leucite in the two-phase (diopside + Chen, C.F., and Turner, J.S. (1980) Crystallization in a double-diff.rsive leucite) mixture. Note the modal enrichment of leucite near the system.Joumal of GeophysicalResearch, 85, 2573-2593. center of the body (C : 0. I 68). The most mafic part of the body Christenson,M.S., and Incropera,F.P. (l 989) Solidiflcation ofan aqueous is along the floor midway betweenthe left sidewall and the center ammonium chloride solution in a rectangular cavity: I. Experimental of the model pluton. The minimum and maximum values of C study. International Journal of Heat and Mass Transfer, 32, 47-68. 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A ldentical to enthalpy scale Ah defined in Fig. 3 and Eq. 19 M,c,Nuscnrpr REcETvEDSsprerusen l, 1994 in text v KAlSir06 Mnxuscnrr.r AcCEPTEDJvw 26,1995 mixture mass fraction isobaric specific heat U/(kg' K)l cl depth of domain (m) chemicalditfusivity (m'/s) AppnNorx Da Darcy number f" fraction solid To facilitate solution ofthe equationsofchange, a nondimen- g accelerationof gravity sionalization was performed by introduction of the following h specific enthalpy (J/kg) dimensionlessvariables: hi specific fusion enthalpy of eutectic mixture (J/kg) Ah enthalpy scale defined in Fig. 3 (J/kg) x-y-uL-vLL2P species flux (kg/m'?s) x:L u:- v:- r:- I ,:L k thermalconductivity U(m s.K)l K permeability(m'?) L width of domain (m) r : rf st , :' ,":0"#^ Le Lewis number of^ P pressure (N/m'z) Pr Prandtl numb€r -;- hr-h,", a T-T-, q heat flux (J/m'?s) "' ' ah To-T*, Ra thermal Rayleighnumber Fs compositionalRayleigh number 7 c"-c" v Cr-Co Rp buoyancy number (= Rs/Fa) number C*' -Co sr Stetan Cr-G' T temperature(K) (K) When thesevariables are substitutedinto Equations 3-7 the con- r*, eutectic temperature f'o liquidustemperature (K) servation equations take the form T^ melting temperature of diopside (K) To initial temperature (K) v.v:0 (Al) time (s) u horizontal velocity (m/s) v v,,):-{*r.o,-fir"-""t (A'2) vertical velocity (m/s) *(',#+ v velocity vector (m/s) x,y Cartesian coordinates vvr: -finr.r,+Rar Greek symbols *("#+ (K ') a isobaric thermal expansivity, isothermal,isobaric compositionalexpansivity : + RsC,- - ,", (A3) K thermalditfusivity klpc(m2lsl *, p density (kg/m3) \ viscosity (kg/ms) st! * v.vh : v2h+ v,(h"- h) o generalcontinuum variable T characteristictime - v'l(h, - ,xv - v,)l (A4) Sub3criPts 0 initial conditions di diopside s,ff *v.vc: v j|,vc * o - ct f fusion [],o The quantity ftf is the fusion enthalpy of eutecticsolid (seeTable causeliquidus temperaturesin the systemKAlSirOu-CaMgSi.Ou l). The quantity Aft is the amount ofenergy neededto trarrsform depend on the initial bulk composition. The ftr* representsthe solid of original bulk composition (a mixture of diopside and fusion enthalpy of diopside crystals,and 7. is the melting tem- leucite crystals) into a fictive melt of bulk cornposition at the perature of diopside. I,,o is ihe liquidus temperature for the ini- solidus temperature. Note that A,h depends on Co implicitly be- tial bulk composition.