Taxonomy Of Magma Mixing Imagma.geol.ucsb.edu/library/posters/AGU_2013.Poster 4.pdf ·...
1
FIGURE 2a: Temperature vs. Enthalpy Crystal-laden Suspension M Converted to Superheated Liquid by Addion of Hot R f o = 0.5 f o = 0.5 f o = 0.2 f o = 0.65 f o = 0.66 f o = 0.83 f o = 0.75 FIGURE 3: Generalized Toy Inial Condions & Soluon FIGURE 1: Binary Eutecc Phase Diagram FIGURE 2b: Temperature vs. Enthalpy Taxonomy Of Magma Mixing I: Magma Mixing Metrics And The Thermochemistry Of Magma Hybridizaon Illuminated With A Toy Model Frank Spera 1 , Jason Schmidt 1 , Wendy Bohrson 2 , and Guy Brown 3 QUANTITY DEFINITION X Mass fraction component B Y Mass fraction component A X e Eutectic composition T e Eutectic temperature T m.p. α Melting point phase α T m.p. β Melting point phase β ∆h α Fusion enthalpy phase α ∆h β Fusion enthalpy phase β C S Isobaric specific heat of solid C L Isobaric specific heat of liquid ∆C C L -C S X o M Mass fraction of component B in M magma X o M Mass fraction of component B in M magma liquid X o R Mass fraction of component B in R magma X o R Mass fraction of component B in R magma liquid T o M Initial T of M magma T o R Initial T of R magma f o Mass fraction of M magma h Lα M Specific enthalpy M magma contributes to hybrid magma when M magma is single phase liquid of bulk composition X o M < X e h Lβ M Specific enthalpy M magma contributes to hybrid magma when M magma is single phase liquid of bulk composition X o M > X e h α+L M Specific enthalpy M magma contributes to hybrid magma when M magma is L+ α mixture of bulk composition X o M < X e h β+L M Specific enthalpy M magma contributes to hybrid magma when M magma is L+ β mixture of bulk composition X o M < X e h α+β M Specific enthalpy M magma contributes to hybrid magma when M magma is α + β crystal mixture h Lα R Specific enthalpy R magma contributes to hybrid magma when R magma is single phase liquid of bulk composition X o R < X e h Lβ R Specific enthalpy R magma contributes to hybrid magma when R magma is single phase liquid of bulk composition X o R > X e h α+L R Specific enthalpy R magma contributes to hybrid magma when R magma is L+ α mixture of bulk composition X o M < X e h β+L R Specific enthalpy R magma contributes to hybrid magma when R magma is L+ β mixture of bulk composition X o M > X e h α+β R Specific enthalpy R magma contributes to hybrid magma when R magma is α + β crystal mixture TABLE 1: Nomenclature Speci�ic enthalpy Fields Separated Expressions for Speci�ic enthalpy for X H <X e and X H >X e cases h max L and α+L or β+L �ield C S (T e − T m.p. α ) X H X e ( ) + C S T m.p. α +∆h α + X H (∆h β −∆h α ) +∆CX H (T m.p. α − T m.p. β ) + X H X e (T e − T m.p. α ) ( ) C S (T e − T m.p. β ) Y H Y e ( ) + C S T m.p. β +∆h β + Y H ( ∆h α −∆h β ) +∆CY H (T m.p. β − T m.p. α ) + Y H Y e (T e − T m.p. β ) ( ) h mid α+L or β+L and L e +α+β �ield C S T e + X H X e ( ) ∆h α + X H (∆h β −∆h α ) +∆CX H (T m.p. α − T m.p. β ) + X H X e ( ) (T e − T m.p. α ) ( ) C S T e + Y H Y e ( ) ∆h β + Y H (∆h α −∆h β ) +∆CY H (T m.p. β − T m.p. α ) + Y H Y e ( ) (T e − T m.p. β ) ( ) h min L e +α+β and α+β �ield C S T e TABLE 3: Specific Enthalpy Boundary Values Separang Phase Assemblages 1 Earth Science, UC Santa Barbara, Santa Barbara, CA, USA; 2 Geological Sciences, Central Washington University, Ellensburg, WA, USA; 3 Yardi Systems Inc, Santa Barbara, CA, USA I: Introducon The mixing of two thermally and composionally disnct magmas (resident magma, M and Recharge magma, R) each containing disncve crystal and melt populaons plays a major role in modern petrogenec theory. The study of the composional diversity of igneous rocks observed on spaal scales from micrometers to kilometers and on me scales from seconds to millions of years has a long history dang back over 160 years. Abundant evidence of magma mixing includes mafic enclaves in plutonic rocks, syn-plutonic mafic or silicic dikes, and from cumulate rocks in layered intrusions and other intrusive bodies. Mixed-pumice erupons and crystal-scale heterogeneity of phenocrysts and cumulate minerals provide prima facie evidence for magma mixing. It is not surprising that this evidence has lead to the view that magma mixing along with fraconal crystallizaon is a first-order petrogenec process acve in all petrotectonic environments throughout geologic me. When two magmas mix, there are a wide variety of potenal end states depending on myriad details. Indeed the fluid dynamics of mixing of magmas (a heterogeneous suspensions of crystals, supercrical fluid bubbles and silicate or carbonac melt) is complex due to the influence of oſten ill-defined boundary and inial condions and the relevant mul-level scales of heat, momentum and material transport. At zeroth–order, one can consider that magma mixing occurs by two end-member processes: hybridizaon and mingling. Magma mingling produces a heterogeneous mixture containing discrete porons of the end-member magmas, (e.g., M and R), due to the incomplete blending of the different magmas. In this case, the final product comprises spaally discrete porons of each lithic type and is very spaally heterogeneous. The ‘goodness’ of mixing has previously been defined in terms of a length scale (L) and an intensity (I) of mixing (Oldenburg and Spera, 1989, JGR, 94, p.9215) and depends on both the space correlaon (L) and the second moment (I) of the composion field. The space correlaon gives informaon on the probability that a posive (negave) composion fluctuaon at one point is correlated with a posive (negave) anomaly a distance r away. In any flow with nonzero diffusion effects, as since normal and shear strains associated with mixing diffusively reduce a composional anomaly to a length of order ; anomalies are essenally erased at macroscopic scales. As , as well; however, the mescale for reducon of L only weakly relates to molecular diffusivity and depends, instead, on flow dynamics, specifically the temporal and spaal evoluon of the velocity field, a complicated issue not subject to soporific generalizaons on an AGU poster. Magma hybridizaon involves the homogeneous mixing of two unlike magmas (M and R), forming a chemically and physically homogeneous mixture of intermediate bulk composion and enthalpy (but NOT necessarily temperature!). Unlike the case of magma mingling the end-member process of magma hybridizaon can be studied thermodynamically since inial and final states are uniquely defined according to thermodynamic dictates. In magma hybridizaon, two inially disparate magmas, each in internal equilibrium, thoroughly mix (hybridize) to form a new magma at thermodynamic (chemical potenal, temperature, pressure) equilibrium. The hybrid magma is a mulphase mixture of homogeneous melt, unzoned crystals and possibly bubbles of a supercrical fluid; all coexisng phases are at a unique and idencal temperature and pressure. The goal of this study is to examine the thermodynamics of magma hybridizaon by studying a model binary eutecc system (ideal mixing, no crystalline soluon) and then apply the principles gleaned to mulcomponent-mulphase systems using the Magma Chamber Simulator (See adjacent POSTER). The thermodynamics of the toy model are transparent and hence the sensivity of hybridizaon outcome on the composion, phase state, temperature and mixing proporons of the mixing magmas is easy to grasp. The toy model enables rapid development of intuive basis for understanding magma hybridizaon in natural systems. Magma mixing is very different than melt mixing to form a third blended liquid. When two melts mix to form a third, the final temperature and composion of the hybrid liquid is the mass- weighted average of the mixing melts. This trivial case is not very important since superheated magmas are rare. t ∞ I 0 l ~ t ∞ L 0 II: Thermodynamics of Hybridizaon Illustrated via Toy Model The phase diagram of the toy system is depicted in Figure 1. Pressure is constant. Table 1 gives the variable idenficaons. The toy system topology of is that of a binary component eutecc phase diagram not unlike CaMgSi 2 O 6 - CaAl 2 Si 2 O 8 , with no crystalline soluon. There are three phases in this system: crystals of phase α, crystals of phase β or a liquid phase. The two components of this system are A and B with α phase solely constuted by component A and phase β solely constuted by component B. For the ease of explanaon, and without loss of any generality, liquidii are rendered as straight lines in Figure 1. There are eight thermodynamic parameters needed to define the topology of the toy system: the eutecc composion, X e , eutecc temperature, T e , melng temperature of α and β ( and ) and the specific fusion enthalpies of α and β ( , ). In the toy model, two isobaric specific heats are also introduced: one for all solid phases, C S and one for the liquid, C L . These isobaric specific heats are taken as constants independent of temperature and composion. At igneous temperatures, C P varies remarkably lile with temperature composion to first order. In the toy model, heats of mixing associated with non-ideality are neglected. Although excess enthalpies are generally not idencally zero, they are relavely small compared to fusion enthalpies and, when translated into temperature differences, are of order 10-40 K. For example, in the system CaMgSi 2 O 6 - CaAl 2 Si 2 O 8 the maximum excess enthalpy associated with mixing is ~60 kJ/kg. In comparison, the fusion enthalpies of diopside and anorthite are 636 kJ/kg and 478 kJ/kg, respecvely, about 10 mes larger. Similarly, if the excess enthalpy (heat of mixing) is couched in terms of a temperature difference then ~ 35 K. In addion to the aforemenoned thermodynamic parameters, five addional parameters are needed. These parameters are: the inial temperatures ( , ) and bulk composions ( , ) of both M and R magmas and the mass fracon of M magma (f o ) of the M+R mixture. Because the process of magma mixing is closed and adiabac, the final bulk composion (X H ) and specific enthalpy (h H ) of the hybrid magma are known. That is, the T, bulk composion and phase assemblage (proporons and liquid composion) of the final hybridized magma are uniquely determined. In Figure 3, an example of a hybridizaon soluon is given along with the inial condions for a toy system with thermodynamics very similar to the 1-bar system CaMgSi 2 O 6 - CaAl 2 Si 2 O 8 . A downloadable spreadsheet is available at hp://magma.geol.ucsb.edu/index.html. Insights gained from analysis of the toy model pertaining to the thermodynamics of magma hybridizaon are discussed in the adjacent poster. T α m.p . T β m.p. Δh α Δh β ΔT T M 0 T R 0 X M 0 X R 0 TABLE 2: Enthalpy Expressions of M and R Magmas h Lα M f o C S T o M +∆h α + X o M ( ∆h β −∆h α ) +∆CX o M (T m.p. α − T m.p. β ) + (T o M − T m.p. α ) ( ) [ ] h Lβ M f o C S T o M +∆h β + Y o M ( ∆h α −∆h β ) +∆CY o M (T m.p. β − T m.p. α ) + (T o M − T m.p. β ) ( ) [ ] h α+L M f o C S T o M + X o M X o M ( ) ∆h α + X o M ( ∆h β −∆h α ) +∆C X o M (T m.p. α − T m.p. β ) + X o M X o M ( ) (T o M − T m.p. α ) ( ) [ ] h β+L M f o C S T o M + Y o M Y o M ( ) ∆h β + Y o M ( ∆h α −∆h β ) +∆CY o M (T m.p. β − T m.p. α ) + Y o M Y o M ( ) (T o M − T m.p. β ) ( ) [ ] h α+β M f o C S T o M [ ] h Lα R (1 − f o )C S T o R +∆h α + X o R ( ∆h β −∆h α ) +∆CX o R (T m.p. α − T m.p. β ) + (T o R − T m.p. α ) ( ) [ ] h Lβ R (1 − f o )C S T o R +∆h β + Y o R ( ∆h α −∆h β ) +∆CY o R (T m.p. β − T m.p. α ) + (T o R − T m.p. β ) ( ) [ ] h α+L R (1 − f o )C S T o R + X o R X o R ( ) ∆h α + X o R ( ∆h β −∆h α ) +∆C X o R (T m.p. α − T m.p. β ) + X o R X o R ( ) (T o R − T m.p. α ) ( ) [ h β+L R (1 − f o )C S T o R + Y o R Y o R ( ) ∆h β + Y o R ( ∆h α −∆h β ) +∆CY o R (T m.p. β − T m.p. α ) + Y o R Y o R ( ) (T o R − T m.p. β ) ( )] [ h α+β R (1 − f o )C S T o M [ ] • By choosing a temperature range, an enthalpy range is implicitly defined • Enthalpy of M, R and H is a funcon of temperature, phase abundance and melt (if present) composion. • A global enthalpy maximum and minimum (h GMAX and h GMIN ) are also defined (see Fig 2) • Once IC’s have been defined, PAO’s can be determined explicitly (refer to Fig 2) • define Δh = h GMAX - h GMIN • The probability of a parcular of Phase Assemblage Outcome is proporonal to the fracon of hybrid system enthalpies that fall within a given increment based on Figure 2. Specifically, PAO1 fracon = (h GMAX - h max ) / Δh (PAO2 + PAO3) fracons = (h max - h mid ) / Δh PAO4 fracon = (h mid - h min ) / Δh PAO5 fracon = (h min - h GMIN ) / Δh ∑ TABLE 5: Thermodynamics of Toy binary eutecc system. Parameters closely follow those in system CaMgSi 2 O 6 - CaAl 2 Si 2 O 8 at 10 5 Pa. Thermodynamic parameter Symbol Value Units Eutectic composition, mass fraction component A X e 0.42 Eutectic temperature T e 1547 K Melting point of α crystals T m.p. α 1665 K Enthalpy of fusion of α crystals at T m.p. α ∆h α 636000 J/kg Melting point of β crystals T m.p. β 1830 K Enthalpy of fusion of β crystals at T m.p. β ∆h β 478000 J/kg Crystal speci�ic isobaric heat capacity C S 1400 J/kg·K Liquid speci�ic isobaric heat capacity C L 1600 J/kg·K X M 0 X R 0 T M 0 T R 0 III: Analysis For fixed thermodynamic parameters, the five inial condions of , , , , and f 0 determine the state of hybridized magma (H). There are five possible mutually exclusive Phase Assemblage Outcomes (PAO’s) (see Fig 1) for any set of inial condions: L only (PAO1), L + α (PAO2), L + β (PAO3), L eutecc + α + β (PAO4) or α + β (PAO5) QUESTION: What is the probability distribuon of Phase Assemblage Outcomes given a reasonable range for each of the five inial condions? SOLUTION: Apply a Monte Carlo (MC) method to study the stascs of the five possible outcomes given a realisc range of possible values for each of the five inial parameters , , , , f 0 • Table 6 provides the distribuon (mean, one sigma, max and min) of the input parameters • Use Monte Carlo (MC) approach to generate several million soluons (realizaons) • Perform stascal analysis of outcomes to seek preferred states X M 0 X R 0 T M 0 T R 0 Input Mean Value 1 σ Minimum Maximum X o M 0.5 ± 0.3 > 0 < 1 X o R 0.5 ± 0.3 > 0 < 1 T o R If X o R < X e , then mean = average of T m.p. α and 0.98 T e ± 150 0.98 T e T m.p. α T o R If X o R > X e , then mean = average of T m.p. β and 0.98 T e ± 150 0.98 T e T m.p. β T o M If X o M < X e , then mean = average of T m.p. α and 0.98 T e ± 150 0.98 T e T m.p. α T o M If X o M > X e , then mean = average of T m.p. β and 0.98 T e ± 150 0.98 T e T m.p. β f o 0.5 ± 0.3 f o > 0 f o < 1 TABLE 6: Distribuon of Inial Condion variables , , , ,f 0 X M 0 X R 0 T M 0 T R 0 Eutecc “Aractor” - the tendancy for the hybridized magma H to be at the invariant point f o = 0.5 f o = 0.5 Magma Cooling Effect - T of H is LESS than inial T of both M and R • Some realizaons produce a thermal outcome of special interest when the temperature of the equilibrated hybrid (H) magma is lower than EITHER of the starng magmas (M and R). I.e., T H < and T H < • This thermal condion occurs in 18% of the M (melt) + R (magma) realizaons (~90,000 cases) • This thermal condion occurs in 9% of the M (magma) + R (magma) realizaons (~405,000 cases) T M 0 T R 0 IV: Results • Stascal analyses were conducted on a populaon of over 5 million MC realizaons in which M was melt and R magma (crystal + liquid) or both M and R were magmas ( i.e., both M and R were crystal + liquid, respecvely). The stascal results were: • melt (M) + magma (R) mixing: • magma (M) + magma (R) mixing: L = 46.3% L = 6.5% L + α = 11.3% L + α = 8% L + β = 31.4% L + β = 18.5% L eutecc + α + β = 10.8% L eutecc + α + β = 44% α + β = 0.2% α + β = 23% • Note proclivity toward the eutecc condion L eutecc + α + β in magma - magma mixing Suppression of α Crystallizaon of Previously α-Saturated M f o = 0.4 f o = 0.4 f o = 0.6 f o = 0.75 Loss of ‘memory’ of Liquidus Phase of R Magma Due to Mixing Variable Effects of Stoping Cold Blocks of R into M In order to bring a melt M below the solidus, a large stoped block (33% total system mass) is needed, whereas the same result is accomplished for a magma M with a much smaller stoped block (17% total system mass) even though the starng M temperatures only differ by < 20K and is the same in both cases T R 0 A cool R block dropped into a melt M promotes cyrstallizaon, but only of the α phase; crystallizaon of the β phase (50% of R) is suppressed TABLE 4: Hybrid magma bulk composion less enriched in component A than eutecc composion (X H < X e ) NOTE: analagous tables exist for (X H > X e ) and (X H = X e ) Specific enthalpy range Hybrid system state h H > h max T H = h o −∆h α − X H (∆h β −∆h α ) +∆CT m.p. α − X H (T m.p. α − T m.p. β ) ( ) C S +∆C X H = X H h mid < h H < h max Simultaneous solution of the following expressions gives X H and T H : C S T H + X H X H ( ) ∆h α + X H (∆h β −∆h α ) +∆CX H (T m.p. α − T m.p. β ) + X H X H ( ) (T H − T m.p. α ) [ ] T H = (T e − T m.p. α ) X H X e + T m.p. α mass fraction α crystals =1 − X H X H mass fraction liquid = X H X H h min < h H < h mid T H = Te X H = X e mass fraction of liquid ( θ ) of eutectic composition θ = h H − C S T e ∆h α + X e (∆h β −∆h α ) +∆CX e (T m.p. α − T m.p. β ) + (T e − T m.p. α ) ( ) mass fraction a phase , w α H =1 − θ l − w β H mass fraction b phase, w β H = X H − (θ l X e ) h H < h min T H = h H C S w α H = (1 − X H ) w β H = X H V: Conclusions • In many magma mixing scenarios, the T of the hybrid magma (H) is less than the T of the magmas (M and R) that mix. This can lead to some interesng phenocryst composional profiles. • The tendency for hybrid magma (H) to be “aracted” to the invariant point is common, parcularly in magma-magma mixing • Expression of crystallizaon in H can be highly variable and is sensive to the mixing condions (i.e. , , , , and f 0 ) • Experimentaon with stoping blocks of different masses & into M suggests a sensive dependence on mass, bulk composion and T of the stoped rocks • Toy model provides valuable insights into complex mul-component, mul-phase systems (see next poster, MCS) T M 0 T R 0 X R 0 X M 0 T R 0
Transcript of Taxonomy Of Magma Mixing Imagma.geol.ucsb.edu/library/posters/AGU_2013.Poster 4.pdf ·...
FIGURE 2a: Temperature vs. Enthalpy
Crystal-laden Suspension M Converted to Superheated Liquid by Addition of Hot R
fo = 0.5 fo = 0.5
fo = 0.2fo = 0.65
fo = 0.66 fo = 0.83
fo = 0.75
FIGURE 3: Generalized Toy Initial Conditions & Solution
FIGURE 1: Binary Eutectic Phase Diagram
FIGURE 2b: Temperature vs. Enthalpy
Taxonomy Of Magma Mixing I:Magma Mixing Metrics And The Thermochemistry Of Magma Hybridization Illuminated With A Toy Model
Frank Spera1, Jason Schmidt1, Wendy Bohrson2, and Guy Brown3
QUANTITY DEFINITION X Mass fraction component B Y Mass fraction component A Xe Eutectic composition Te Eutectic temperature Tm.p.
α Melting point phase α
Tm.p.β Melting point phase β
∆hα Fusion enthalpy phase α ∆hβ Fusion enthalpy phase β CS Isobaric specific heat of solid CL Isobaric specific heat of liquid ∆C CL-CS Xo
M Mass fraction of component B in M magma
XoM
Mass fraction of component B in M magma liquid
XoR Mass fraction of component B in R magma
XoR
Mass fraction of component B in R magma liquid
ToM Initial T of M magma
ToR Initial T of R magma
fo Mass fraction of M magma hLα
M Specific enthalpy M magma contributes to hybrid magma when M magma is single phase liquid of bulk composition Xo
M < Xe hLβ
M Specific enthalpy M magma contributes to hybrid magma when M magma is single phase liquid of bulk composition Xo
M > Xe hα+L
M Specific enthalpy M magma contributes to hybrid magma when M magma is L+ α mixture of bulk composition Xo
M < Xe hβ+L
M Specific enthalpy M magma contributes to hybrid magma when M magma is L+β mixture of bulk composition Xo
M < Xe hα+β
M Specific enthalpy M magma contributes to hybrid magma when M magma is α+β crystal mixture
hLαR Specific enthalpy R magma contributes to hybrid magma when R
magma is single phase liquid of bulk composition XoR < Xe
hLβR Specific enthalpy R magma contributes to hybrid magma when R
magma is single phase liquid of bulk composition XoR > Xe
hα+LR Specific enthalpy R magma contributes to hybrid magma when R
magma is L+ α mixture of bulk composition XoM < Xe
hβ+LR Specific enthalpy R magma contributes to hybrid magma when R
magma is L+ β mixture of bulk composition XoM > Xe
hα+βR Specific enthalpy R magma contributes to hybrid magma when R
magma is α+β crystal mixture
TABLE 1: Nomenclature
Speci�ic enthalpy
Fields Separated Expressions for Speci�ic enthalpy for XH<Xe and XH>Xe cases
hmax L and α+L or β+L �ield CS(Te − Tm.p.
α ) XH
Xe( )+ CS Tm.p.
α + ∆hα + XH (∆hβ − ∆hα )
+∆C XH (Tm.p.α − Tm.p.
β )+ XH
Xe
(Te − Tm.p.α )( )
CS(Te − Tm.p.β ) YH
Ye( ) + CS Tm.p.
β + ∆hβ + YH (∆hα − ∆hβ )
+∆C YH (Tm.p.β − Tm.p.
α ) +YH
Ye
(Te − Tm.p.β )( )
hmid α+L or β+L and Le+α+β �ield
CSTe + XH
Xe( )∆hα + XH (∆hβ − ∆hα )
+∆C XH (Tm.p.α − Tm.p.
β )+ XH
Xe( ) (Te − Tm.p.
α )( )
CSTe + YH
Ye( )∆hβ + YH (∆hα − ∆hβ )
+∆C YH (Tm.p.β − Tm.p.
α )+ YH
Ye( ) (Te − Tm.p.
β )( )
hmin Le+α+β and α+β �ield CSTe
TABLE 3: Specific Enthalpy Boundary ValuesSeparating Phase Assemblages
1Earth Science, UC Santa Barbara, Santa Barbara, CA, USA; 2Geological Sciences, Central Washington University, Ellensburg, WA, USA; 3Yardi Systems Inc, Santa Barbara, CA, USA
I: IntroductionThe mixing of two thermally and compositionally distinct magmas (resident magma, M and Recharge magma, R) each containing distinctive crystal and melt populations plays a major role in modern petrogenetic theory. The study of the compositional diversity of igneous rocks observed on spatial scales from micrometers to kilometers and on time scales from seconds to millions of years has a long history dating back over 160 years. Abundant evidence of magma mixing includes mafic enclaves in plutonic rocks, syn-plutonic mafic or silicic dikes, and from cumulate rocks in layered intrusions and other intrusive bodies. Mixed-pumice eruptions and crystal-scale heterogeneity of phenocrysts and cumulate minerals provide prima facie evidence for magma mixing. It is not surprising that this evidence has lead to the view that magma mixing along with fractional crystallization is a first-order petrogenetic process active in all petrotectonic environments throughout geologic time. When two magmas mix, there are a wide variety of potential end states depending on myriad details. Indeed the fluid dynamics of mixing of magmas (a heterogeneous suspensions of crystals,supercritical fluid bubbles and silicate or carbonatitic melt) is complex due to the influence of often ill-defined boundary and initial conditions and the relevant multi-level scales of heat, momentum and material transport. At zeroth–order, one can consider that magma mixing occurs by two end-member processes: hybridization and mingling. Magma mingling produces a heterogeneous mixture containing discrete portions of the end-member magmas, (e.g., M and R), due to the incomplete blending of the different magmas. In this case, the final product comprises spatially discrete portions of each lithic type and is very spatially heterogeneous. The ‘goodness’ of mixing has previously been defined in terms of a length scale (L) and an intensity (I) of mixing (Oldenburg and Spera, 1989, JGR, 94, p.9215) and depends on both the space correlation (L) and the second moment (I) of the composition field. The space correlation gives information on the probability that a positive (negative) composition fluctuation at one point is correlated with a positive (negative) anomaly a distance r away. In any flow with nonzero diffusion effects, as since normal and shear strains associated with mixing diffusively reduce a compositional anomaly to a length of order ; anomalies are essentially erased at macroscopic scales. As , as well; however, the timescale for reduction of L only weakly relates to molecular diffusivity and depends, instead, on flow dynamics, specifically the temporal and spatial evolution of the velocity field, a complicated issue not subject to soporific generalizations on an AGU poster. Magma hybridization involves the homogeneous mixing of two unlike magmas (M and R), forming a chemically and physicallyhomogeneous mixture of intermediate bulk composition and enthalpy (but NOT necessarily temperature!). Unlike the case of magma mingling the end-member process of magma hybridization can be studied thermodynamically since initial and final states are uniquely defined according to thermodynamic dictates. In magma hybridization, two initially disparate magmas, each in internalequilibrium, thoroughly mix (hybridize) to form a new magma at thermodynamic (chemical potential, temperature, pressure) equilibrium. The hybrid magma is a multiphase mixture of homogeneous melt, unzoned crystals and possibly bubbles of a supercritical fluid; all coexisting phases are at a unique and identical temperature and pressure. The goal of this study is to examine the thermodynamics of magma hybridization by studying a model binary eutectic system (ideal mixing, no crystalline solution) and then apply the principles gleaned to multicomponent-multiphase systems using the Magma Chamber Simulator (See adjacent POSTER). The thermodynamics of the toy model are transparent and hence the sensitivity ofhybridization outcome on the composition, phase state, temperature and mixing proportions of the mixing magmas is easy to grasp. The toy model enables rapid development of intuitive basis forunderstanding magma hybridization in natural systems. Magma mixing is very different than melt mixing to form a third blended liquid. When two melts mix to form a third, the final temperature and composition of the hybrid liquid is the mass-weighted average of the mixing melts. This trivial case is not very important since superheated magmas are rare.
t ∞I 0l ~ t ∞ L 0
II: Thermodynamics of Hybridization Illustrated via Toy ModelThe phase diagram of the toy system is depicted in Figure 1. Pressure is constant. Table 1 gives the variable identifications. The toy system topology of is that of a binary component eutectic phasediagram not unlike CaMgSi2O6 - CaAl2Si2O8, with no crystalline solution. There are three phases in this system: crystals of phase α, crystals of phase β or a liquid phase. The two components of this system are A and B with α phase solely constituted by component A and phase β solely constituted by component B. For the ease of explanation, and without loss of any generality, liquidii arerendered as straight lines in Figure 1. There are eight thermodynamic parameters needed to define the topology of the toy system: the eutectic composition, Xe, eutectic temperature, Te, meltingtemperature of α and β ( and ) and the specific fusion enthalpies of α and β ( , ). In the toy model, two isobaric specific heats are also introduced: one for all solid phases, CS and one for the liquid, CL. These isobaric specific heats are taken as constants independent of temperature and composition. At igneous temperatures, CP varies remarkably little with temperature composition to first order. In the toy model, heats of mixing associated with non-ideality are neglected. Although excess enthalpies are generally not identically zero, they are relatively small compared to fusionenthalpies and, when translated into temperature differences, are of order 10-40 K. For example, in the system CaMgSi2O6 - CaAl2Si2O8 the maximum excess enthalpy associated with mixing is ~60 kJ/kg. In comparison, the fusion enthalpies of diopside and anorthite are 636 kJ/kg and 478 kJ/kg, respectively, about 10 times larger. Similarly, if the excess enthalpy (heat of mixing) is couched in terms of a temperature difference then ~ 35 K. In addition to the aforementioned thermodynamic parameters, five additional parameters are needed. These parameters are: the initialtemperatures ( , ) and bulk compositions ( , ) of both M and R magmas and the mass fraction of M magma (fo) of the M+R mixture. Because the process of magma mixing is closed andadiabatic, the final bulk composition (XH) and specific enthalpy (hH) of the hybrid magma are known. That is, the T, bulk composition and phase assemblage (proportions and liquid composition) of the final hybridized magma are uniquely determined. In Figure 3, an example of a hybridization solution is given along with the initial conditions for a toy system with thermodynamics very similar to the 1-bar system CaMgSi2O6 - CaAl2Si2O8. A downloadable spreadsheet is available at http://magma.geol.ucsb.edu/index.html. Insights gained from analysis of the toy model pertaining to thethermodynamics of magma hybridization are discussed in the adjacent poster.
Tαm.p . Tβ
m.p. Δhα Δhβ
ΔTTM
0 TR0 XM
0 XR0
TABLE 2: Enthalpy Expressions of M and R MagmashLα
M fo CSToM + ∆hα + Xo
M (∆hβ − ∆hα ) + ∆C XoM (Tm.p.
α − Tm.p.β ) + (To
M − Tm.p.α )( )[ ]
hLβM fo CSTo
M + ∆hβ + YoM (∆hα − ∆hβ ) + ∆C Yo
M (Tm.p.β − Tm.p.
α ) + (ToM − Tm.p.
β )( )[ ]
hα+LM
fo CSToM +
XoM
XoM( ) ∆hα + Xo
M (∆hβ − ∆hα ) + ∆C XoM (Tm.p.
α − Tm.p.β ) +
XoM
XoM( ) (To
M − Tm.p.α )( )[ ]
hβ+LM
fo CSToM +
YoM
YoM( ) ∆hβ + Yo
M (∆hα − ∆hβ ) + ∆C YoM (Tm.p.
β − Tm.p.α ) +
YoM
YoM( ) (To
M − Tm.p.β )( )[ ]
hα+βM fo CSTo
M[ ]
hLαR (1− fo ) CSTo
R + ∆hα + XoR (∆hβ − ∆hα ) + ∆C Xo
R (Tm.p.α − Tm.p.
β ) + (ToR − Tm.p.
α )( )[ ]
hLβR (1− fo ) CSTo
R + ∆hβ + YoR (∆hα − ∆hβ ) + ∆C Yo
R (Tm.p.β − Tm.p.
α ) + (ToR − Tm.p.
β )( )[ ]
hα+LR
(1− fo ) CSToR +
XoR
XoR( ) ∆hα + Xo
R (∆hβ − ∆hα ) + ∆C XoR (Tm.p.
α − Tm.p.β ) +
XoR
XoR( ) (To
R − Tm.p.α )( )[ ]
hβ+LR
(1− fo ) CSToR +
YoR
YoR( ) ∆hβ + Yo
R (∆hα − ∆hβ ) + ∆C YoR (Tm.p.
β − Tm.p.α ) +
YoR
YoR( ) (To
R − Tm.p.β )( )][
hα+βR (1− fo ) CSTo
M[ ]
• By choosing a temperature range, an enthalpy range is implicitly defined • Enthalpy of M, R and H is a function of temperature, phase abundance and melt (if present) composition. • A global enthalpy maximum and minimum (hGMAX and hGMIN) are also defined (see Fig 2)• Once IC’s have been defined, PAO’s can be determined explicitly (refer to Fig 2) • define Δh = hGMAX - hGMIN
• The probability of a particular of Phase Assemblage Outcome is proportional to the fraction of hybrid system enthalpies that fall within a given increment based on Figure 2. Specifically, PAO1 fraction = (hGMAX - hmax) / Δh (PAO2 + PAO3) fractions = (hmax - hmid) / Δh PAO4 fraction = (hmid - hmin) / Δh PAO5 fraction = (hmin - hGMIN) / Δh
∑
TABLE 5: Thermodynamics of Toy binaryeutectic system. Parameters closely followthose in system CaMgSi2O6 - CaAl2Si2O8
at 105 Pa.Thermodynamic
parameter Symbol Value Units Eutectic
composition, mass fraction component
A
Xe 0.42
Eutectic temperature Te 1547 K
Melting point of α crystals
Tm.p.α 1665 K
Enthalpy of fusion of α crystals at Tm.p.
α ∆hα 636000 J/kg
Melting point of β crystals
Tm.p.β 1830 K
Enthalpy of fusion of β crystals at Tm.p.
β ∆hβ 478000 J/kg
Crystal speci�ic isobaric heat
capacity CS 1400 J/kg·K
Liquid speci�ic isobaric heat
capacity CL 1600 J/kg·K
XM0 X
R 0 T
M0 T
R 0
III: AnalysisFor fixed thermodynamic parameters, the five initial conditions of , , , , and f0 determine the state of hybridized magma (H). There are five possible mutually exclusive Phase Assemblage Outcomes (PAO’s) (see Fig 1) for any set of initial conditions:
L only (PAO1), L + α (PAO2), L + β (PAO3), Leutectic + α + β (PAO4) or α + β (PAO5)
QUESTION: What is the probability distribution of Phase Assemblage Outcomes given a reasonable range for each of the five initial conditions?
SOLUTION: Apply a Monte Carlo (MC) method to study the statistics of the five possible outcomes given a realistic range of possible values for each of the five initial parameters , , , , f0
• Table 6 provides the distribution (mean, one sigma, max and min) of the input parameters • Use Monte Carlo (MC) approach to generate several million solutions (realizations) • Perform statistical analysis of outcomes to seek preferred states
XM0 X
R 0 T
M0 T
R 0
Input Mean Value 1σ Minimum Maximum Xo
M 0.5 ±0.3 > 0 < 1 Xo
R 0.5 ±0.3 > 0 < 1
ToR
If XoR < Xe, then
mean = average of Tm.p.
α and 0.98 Te ±150 0.98 Te Tm.p.
α
ToR
If XoR > Xe, then
mean = average of Tm.p.
β and 0.98 Te ±150 0.98 Te Tm.p.
β
ToM
If XoM < Xe, then
mean = average of Tm.p.
α and 0.98 Te ±150 0.98 Te Tm.p.
α
ToM
If XoM > Xe, then
mean = average of Tm.p.
β and 0.98 Te ±150 0.98 Te Tm.p.
β
fo 0.5 ±0.3 fo > 0 fo < 1
TABLE 6: Distribution of Initial Condition variables , , , ,f0 XM0 X
R 0 T
M0 T
R 0
Eutectic “Attractor” - the tendancy for the hybridized magma H to be at the invariant point
fo = 0.5fo = 0.5
Magma Cooling Effect - T of H is LESS than initial T of both M and R• Some realizations produce a thermal outcome of special interest when the temperature of the equilibrated hybrid (H) magma is lower than EITHER of the starting magmas (M and R). I.e., TH < and TH < • This thermal condition occurs in 18% of the M (melt) + R (magma) realizations (~90,000 cases) • This thermal condition occurs in 9% of the M (magma) + R (magma) realizations (~405,000 cases)
TM 0 T
R0
IV: Results• Statistical analyses were conducted on a population of over 5 million MC realizations in which M was melt and R magma (crystal + liquid) or both M and R were magmas ( i.e., both M and R were crystal + liquid, respectively). The statistical results were: • melt (M) + magma (R) mixing: • magma (M) + magma (R) mixing: L = 46.3% L = 6.5% L + α = 11.3% L + α = 8% L + β = 31.4% L + β = 18.5% Leutectic + α + β = 10.8% Leutectic + α + β = 44% α + β = 0.2% α + β = 23% • Note proclivity toward the eutectic condition Leutectic + α + β in magma - magma mixing
Suppression of α Crystallization of Previously α-Saturated M
fo = 0.4fo = 0.4
fo = 0.6 fo = 0.75
Loss of ‘memory’ of Liquidus Phase of R Magma Due to Mixing
Variable Effects of Stoping Cold Blocks of R into MIn order to bring a melt M below the solidus, a large stoped block (33% total system mass) is needed, whereas the same result is accomplished for a magma M with a much smaller stoped block (17% total system mass) even though the starting M temperatures only differ by < 20K and is the same in both cases
T R0
A cool R block dropped into a melt M promotes cyrstallization, but only of the α phase; crystallization of the β phase (50% of R) is suppressed
TABLE 4: Hybrid magma bulk composition lessenriched in component A than eutectic composition(XH < Xe) NOTE: analagous tables exist for (XH > Xe) and(XH = Xe)Specific enthalpy
range Hybrid system state
hH > hmax
TH =ho − ∆hα − XH (∆hβ − ∆hα )+ ∆C Tm.p.
α − XH (Tm.p.α − Tm.p.
β )( )CS + ∆C
XH = XH
hmid < hH
< hmax
Simultaneous solution of the following expressions gives XHand TH :
CSTH + XH
XH( ) ∆hα + XH (∆hβ − ∆hα )+ ∆C XH (Tm.p.α − Tm.p.
β )+ XH
XH( ) (TH − Tm.p.α )[ ]
TH = (Te − Tm.p.α ) XH
Xe
+ Tm.p.α
mass fraction α crystals =1− XH
XH
mass fraction liquid = XH
XH
hmin < hH < hmid
TH = Te
XH = Xe mass fraction of liquid ( θ
) of eutectic composition
θ = hH − CSTe
∆hα + Xe(∆hβ − ∆hα )+ ∆C Xe (Tm.p.α − Tm.p.
β )+ (Te − Tm.p.α )( )
mass fraction a phase , wα
H = 1 − θl − wβH
mass fraction b phase, wβ
H = XH − (θl Xe) hH < hmin
TH =hH
CS
wα
H = (1− XH ) wβ
H = XH
V: Conclusions• In many magma mixing scenarios, the T of the hybrid magma (H) is less than the T of the magmas (M and R) that mix. This can lead to some interesting phenocryst compositional profiles.• The tendency for hybrid magma (H) to be “attracted” to the invariant point is common, particularly in magma-magma mixing• Expression of crystallization in H can be highly variable and is sensitive to the mixing conditions (i.e. , , , , and f0)• Experimentation with stoping blocks of different masses & into M suggests a sensitive dependence on mass, bulk composition and T of the stoped rocks• Toy model provides valuable insights into complex multi-component, multi-phase systems (see next poster, MCS)
TM 0 TR
0XR0XM
0
TR0
![Transport and mixing by artificial cilia · showing different motion. This leads to a phase difference between the two cilia, which has been shown to enhance mixing [9]. The mixing](https://static.fdocuments.in/doc/165x107/5fdc7dbb014bfd7cf4592891/transport-and-mixing-by-artificial-cilia-showing-diierent-motion-this-leads-to.jpg)