1. metamorphic Geol., lW, 8, 6834%

A model for garnet and plagioclase growth in pelitic schists: implications for thermobarometry and P-T path determinations F. S . SPEAR, M. J . KOHN, F . P. FLORENCE AND T. MENARD Department of Geology, Rensselaer Polytechnic Institute, Troy, New York, lZlOS, USA

ABSTRACT Numerical models of the progressive evolution of pelitic schists in the NCMnKFMASH system with the assemblage garnet + biotite + chlorite f staurolite + plagiedase + muscovite + quartz + H,O are pre- sented with the goal of predicting compositional changes in garnet and plagioclase along different P-T paths. The numerical models support several conclusions that should prove useful for interpreting the P-T paths of natural parageneses: (i) Garnet may grow along P-T vectors ranging from heating with decompression to cooling with compression. P-T paths deduced from garnet zoning that are inconsistent with these growth vectors are self-contradictory. (ii) There is a systematic relation between garnet and plagioclase composition and growth such that for most P-T paths, garnet growth requires plagioclase consumption. Furthermore, mass balance in a closed system requires that as plagioclase is consumed the remaining plagioclase becomes increasingly albitic. Inclusions of plagioclase in the core of garnet should be more anorthitic than those near the rim and zoned matrix plagioclase should have rims that are more albitic than the cores. Complex plagioclase textures may arise from the local variability of growth and precipitation kinetics. (iii) A decrease of Fe/(Fe + Mg) in a garnet zoning profile is a reliable indicator of increasing temperature for the assemblage modelled. However, there is no single reliable AP monitor and inferences about A P can only be made by considering plagioclase and garnet together. (iv) Consumption of garnet during the production of staurolite removes material from the outer shell of a garnet and may make recovery of peak metamorphic compositions and P-T conditions impossible. Low ‘peak’ temperatures typically recorded by staurolite-bearing assemblages may reflect this phenomenon. (v) Diffusional homogenization of garnet af€ects the computed P-T path and results in a clockwise rotation of the computed P-T vector relative to the true P-T path.

Key words: barometry; crystal growth; garnet; pelitic schists; thermometry; zoned garnets.

I N T R O D U C T I O N

Essential to the determination of metamorphic P-T paths is an understanding of the reaction history that a metamorphic rock has experienced. This reaction history will, in general, be the result of the interaction of P-T history, strain history, infiltration history and kinetics. It is difficult for the petrologist to decipher this reaction history from observation alone because of the complex interplay between phase equilibria and mass balance that dictates the reactions that involve the whole rock.

Forward modelling using the Gibbs method can help in this endeavour (e.g. Spear, 1988b) because it is possible to consider all of the reactions that occur in the rock simultaneously. Models can be constructed that predict the mineral composition (X) and mineral abundance (M) along any prescribed P-T path and for any mineral assemblage, using any arbitrary assumptions about equilibrium or fractional crystallization, and open or closed system behaviour. The effects of diffusion can also be considered (e.g. Florence & Spear, 1989). The results

of such modelling can then be compared with natural samples to help interpret complex reaction textures.

The purpose of this paper is to present the results of such modelling for a simple pelite with the assemblage garnet + biotite + chlorite f staurolite + plagioclase + muscovite + quartz + HzO in the system SiO2-AI2O3-Mg0- FeO- MnO-CaO-Na,O- K,O-H,O (NCMnKFMASH) . A similar experiment was described by Spear (1988b) but did not include Na,O or CaO and phase relations involving plagioclase were not considered. Furthermore, the thermobarometric implications of garnet resorption during staurolite growth and the effects of diffusional relaxation of garnet zoning on calculations of P-T paths were not assessed. In the present paper, particular attention is directed towards plagioclase-garnet phase relations be- cause of the importance of these minerals in determining metamorphic pressures and P-T paths (e.g. Crawford, 1974, 1977).

In the first part of the paper, equilibrium P-T-X-M phase relations in the assemblage garnet + biotite+ chlorite + plagioclase + muscovite + quartz + H,O are pre-

663

684 F. S . SPEAR ET A l .

sented; secondly, garnet and plagioclase zoning are compared along different P-T paths; thirdly, the effect on thermobarometry calculations of staurolite introduction into this assemblage is analysed; and fourthly, the effect of intracrystalline d i h i o n in garnet on P-T conditions and P-T paths inferred from garnets grown from this assemblage is evaluated. Finally, the model results are examined in light of published studies of natural parageneses.

METHOD

The numerical modelling experiments were performed using the techniques of differential thermodynamics in a manner similar to that described by Spear (1988a,b; for a summary see Spear, 1989). This technique involves performing a Jacobian transformation of the thermo- dynamic variables pressure (P), temperature (T). com- position (X) and mineral abundance (M) so that the changes in dependent (P-T-X-M) variables may be calculated from changes in the independent variables. Calculation of the prograde reaction history of a rock involves solving for changes in the dependent variables X and M as functions of an arbitrarily chosen P-T path, where P and T a r e the independent variables.

As an example, consider the bulk composition listed in Table 1 with the modal proportions and compositions of phases as listed in Tables 1 & 2 at the P-T conditions of 500" C, 6 kbar. A system of simultaneous equations is set up consisting of (i) the total differential of the thermodynamic equations describing the P and T dependence of the equilibrium constant for each linearly independent reaction in the assemblage and (ii) the total differential of the mass-balance constraints for each system component. These equations are then inverted with d T and d P as independent variables to obtain the Jacobian, as shown in Table 3. For any arbitrary change in P and T, the change in mineral composition (AX) and abundance (AM) can be calculated by multiplying the Jacobian by AT and AP. A new set of mineral compositions and abundance is then computed using M, = Md + AM and X,, = Xold + AX at the new T and P conditions (Tmw = Td + AT and

Tnbk 1. Bulk compositions and modal proportions of rocks at reference conditions (500" C, 6 kbar).

Weight % oxides

Volume proportions Molar

of proportion of minerals minerals

62.78 17.40 3.46 7.00 0.15 0.85 2.00 3.18 3.18

100.00

Quanz (Qtz) 35 Muscovite (Ms) 20 Plagioclase (PI) 20 Garnet (Grt) 0 Biotite (Bt) 10 Chlorite (Chl) 15 H2O 0

100

1.542 0.143 0.199

0 0.066 0.070

0

2.020

Table 2. Compositions of minerals at reference conditions (500" C. 6 kbar). Mined Component Mole fraction

1 .o 1 .o

Muscovite (Ms) Musmvite (ms) 0.85 Paragonite (pg) 0.15

Plagioclase (PI) Albite (ab) Anorthite (an)

0.79 0.21

Garoct (Grt) pyropc (PrP) 0.0587 ALmaodine (ah) 0.5774 Spessartine (sps) 0.2267 Grossular (grs) 0.1372

Biotite (Bt) Phlogopite (phl) Annite (am) Mn-Biotite (nmbt)

0.4m 0.5704 0.0088

chlorite (all) Mg-Chlorite (mgchl) 0.4893 FeChlorite (fcchl) 0.4981 Mo-Chlorite (mochl) 0.0126

P, = Pa + AP). A new Jacobian is then calculated at the new T and P conditions using the new mineral compositions and abundances and the process repeated until the entire P-T path is modelled. Typically steps of A T s 1 " C and bP1100bar are used to preserve numerical precision.

All calculations were performed in the chemical system SiO, - AlzOJ - MgO- FeO - MnO - G O - NazO-KzO-H20 (NCMnKFMASH). Thermodynamic data are modified and augmented from Berman (1988), as reported in Spear & Cheney (1989). The following assumptions are made concerning the phases: quartz and H20 are pure; white mica is a binary solution between muscovite and paragonite components; plagioclase is a binary solution between albite and anorthite; chlorite, biotite, and staurolite are ternary Mg-Fe-h4n solid solutions; garnet is a quaternary Mg-Fe-Mn-Ca solution. Solution models for garnet, plagioclase. chlorite, biotite and staurolite are ideal ionic models. Muscovite-paragonite solutions are modelled according to Chatterjee & Flux (1986). The fluid is modelled as pure HzO with Pad = pt0,.,.

It is important to stress that the results of any forward modelling calculation depend on the bulk composition being modelled. In this study the bulk commit ion was inferred from the compositions and modal abundances of minerals from natural data and was chosen to be similar to the composition of a typical shale (Table 1) (e.g. Pettijohn, 1957, Table 61). The reference mineral compo- sitions are representative of natural data at the garnet isograd (500°C. 6kbar) with the assemblage garnet+ biotite + chlorite + plagioclase + muscovite + quartz (Table 2). Also note that the bulk composition in these calculations is fixed, except for the removal of H20 and the fractional crystallization of garnet. In particular, for tbe discussion that follows it is important to remember that the only phases that contain Na are plagioclase and white mica and the only phases that contain Ca are garnet and plagioclase.

MINERAL COMPOSITIONAL CHANGE ALONG P-T PATHS 66!i

Tmbk 3. Partial derivatives of mineral abundance and composition with respect to P and Tat 500°C. 6kbar (the Jacobian).

Temperature derivatives (K-’) Pressure derivatives (bar-’)

(aM,,/ar), = 0.10069 x I O - ~ (aM,,/dP), = 0.29178 x lo-’ ( ~ M , , / J T ) , = - 0 . ~ 7 6 9 x 1 0 - ~ ( d M , / a P ) , = -0.67669 x ( ~ M , , , I ~ T ) , = 0.21344 x I O - ~ ( a M H f l / 3 P ) , = 0.65031 x lo-’ (aM,/ar), = 0.41794 x 1 0 - ~ ( a M , / J P ) , = 0.16878x lo-’ (~M,/BT), = -0.97405 x 1 0 - ~ ( a M , / a P ) , = -0.37761 x lo-’ (aM,,/ar), = 0.14159 x 1 0 - ~ ( a M , , / a P ) , = 0.88701 x 10-6 (aM,,/aT), = -0.67349 X (aM, , /aP) . = -0.22695 x 10-5

(ax,/ar), = 0.34658 x 1 0 - ~ (dX,, /dT), = -0.10536 X (aX, , , /aT) , = 0.38267 x (aX,/dT), = -0.25626 x lo-’ (dX , /aT) , = -0.18939 X (aX,, , /aT), = -0.13701 X (a~,,/a~), = - 0 . ~ ~ 4 x lo-‘ (a~,=,,/ar), = -0.65060 x 10 -~ (a~,,,/a~), = -0.14481 x lo-)

(dX,/aP), = 0.161% x lo-‘ (aX,/aP), = -0.20486 x 1 0 - 5 (ax,,/a~j,. = 0 . 1 ~ 0 0 ~ x 1 0 - ~ (dX, /af ) , = -0.65219 x (aX , /aP) , = O.WM x 1 0 - ~ (aX , , /dP) , = -0.30904 x 10-5 (aXmnb,/~P), = -0.28824 x lo-’ (aX,,,, ,/aP), = -0.27581 x lo-’ (aX,,/af), = -0.41286 x lo-’

M = moles of mincral/100 g of rock; X is mole fraaion.

Two different types of models are discussed. (i) In the ‘equilibrium crystallization’ model all phases are composi- tionally homogeneous at all times. This is equivalent to a closed system in which all phases have infinite diffusivities. (ii) In the ‘fractional crystallization’ model some phases are permitted to undergo fractional crystallization. Fractionating phases are compositionally heterogeneous and have vanishing masses such that only the infinitesimal rim of the phase is in equilibrium with the other phases in the system. Phases that are not undergoing fractional crystallization have non-zcro effective masses and are compositionally homogeneous as in the equilibrium crystallization model. This model is equivalent to a system in which fractionating phases either (i) have vanishing difisivities or (ii) leave the system. The thermodynamic system is not closed to the exchange of matter if material is fractionating. In the models presented below. both garnet and H,O are allowed to fractionate.

RESULTS : E Q U I L I B R I U M C R Y S T A L L I Z A T I O N M O D E L S

The P-T-X-M phase relations assuming equilibrium crystallization are shown in Fig. 1. Isopleths are shown for garnet composition (a-e), plagioclase composition (f) and garnet and plagioclase abundance (g. h). The large dot shows the reference P-T conditions (Tables 1 & 2). This diagram predicts the composition and abundance of garnet and plagioclase at any P-T conditions, as well as the change in garnet and plagioclase modal abundance assuming the equilibrium crystallization model. This diagram does not predict chemical zoning, however, because all phases are assumed to be homogeneous.

As expected from analysis of T - X equilibria in this assemblage (Thompson, 1976). X, increases (Fig. la) and Fe/(Fe + Mg) (Fig. le) decreases monotonically with increasing T . X,,, (Fig. lb) goes through a maximum at around 560 “C, reflecting the fact that TMn < TFc < TM, for the end-member reactions in this assemblage (cf.

Thompson, 1976). X , , also decreases with increasing T (Fig. lc) and, because of the strong partitioning of Mn into garnet relative to chlorite and biotite, the X, contours are very nearly coincident with the Ma contours (Fig. lg). That is, the decrease in spessartine component in garnet is nearly perfectly correlated with the growth of garnet. X , contours have positive slopes (Fig. Id) with X , decreasing with increasing T and decreasing P . The slopes ofmntours of X, (Fig. If) are highly variable with P and T and are roughly parallel to the MPlw isopleths (Fig. lh). The near parallelism of the plagioclase composition isopleths (Fig. If) and plagioclase abundance isopleths (Fig. lh) signifies that, for most P-T paths, as plagioclase is consumed the remaining plagioclase will become more albitic. This result is a consequence of mass balance in the system; that is, the only Na-bearing phases in the assemblage are plagioclase and white mica and the system is closed to Na. This result has some important implications for the interpretation of plagioclase textures and zoning, as will be discussed below.

RESULTS : F R A C T I O N A L C R Y S T A L L I Z A T I O N M O D E L S

Garnet-plagioclase relations

The P-T-X-M isopleths presented in Fig. 1 are constructed assuming a closed system and that all phases are homogeneous over all of P-T space. Clearly, this latter assumption is not warranted for garnet at P-T conditions in the greenschist or lower-amphibolite facies, nor for plagioclase at P-T conditions below the liquidus. Therefore, the evolution of garnet and plagioclase compositions has been studied using a model in which garnet undergoes fractional crystallization.

A comparison between the equilibrium and fractional crystallization models can be Seen in Fig. 2. Consider the assemblage modelled in Fig. 1. At approximately 575°C and 6 kbar the rock contains 5 vol.% garnet (0.039 moles)

686 F . S. SPEAR ET AL.

I I I I 1 I I I

pb. 1. Equilibrium crystallization model. Isopleths of garnet composition (a-e), plagioclasc composition (0, garnet abundance (g). and plagioclasc abundance (h) for the asscrnblagc garnet + biotite + chlorite + muscovite + plagioclasc + quartz + H,O in thc NCMnKFMASH system. The large dot shows the reference P-Tconditions (Tables 1 & 2).

M I N E R A L C O M P O S I T I O N A L C H A N C E A L O N G P-r PATHS sn

10 I I I I I I I

L

( 0 ) L

I - x o I I - 2 I 10 I I I I I I Q

2

0 un 500 600 700

r ("C)

Ftg. 2. P-T diagrams showing the relationship between garnet and plagioclasc growth in the equilibrium (a) and fractional crystallization (b) models. In each diagram ao isopleth of garnet abundance is indicated (dM,, = 0). In (b) several isopleths of plagioclase abundance are indicated (labelled 0% Mica, 5% Mica, etc.) representative of different bulk compositions. The large arrows indicate P-T paths along which garnet and plagioclase will be either produced (+Grt. +Plg) or consumed (-Grt, -Plg).

with the compositions of phases as indicated in Fig. 1. In Fig. 2(a) lines of constant moles of plagioclase (dM@,,=O) and garnet ( d M - = O ) are drawn, assuming equilibrium crystallization. As can be seen, these contom divide P-T space into four quadrants in which garnet and plagioclase are either grown or consumed.

Now consider the fractional crystallization model. In this model, it is assumed that the dihivi ty of Fe, Mg, Mn and Ca in garnet is negligible such that only the outermost rim of the garnet is in equilibrium with the matrix. It is also assumed that H20 fractionates by leaving the system as won as it is produced. That is, the effective mass of garnet and H,O in the assemblage is zero, even though there is 5% (modal) garnet in the rock. In this case, the grossular in the core of the garnet is unavailable for reaction to form an anorthite component, and the growth of garnet requires the consumption of anorthite. Whether plagioclase grows or is consumed depends, in addition, upon reactions involving muscovite, and the isopleths of plagioclase

growth are a function of the ratio of the amount of muscovite to plagioclase. Figure 2(b) shows five curves representing plagioclase abundance isopleths for different modal amounts of muxovite (0-30%mica). It is not strictly correct to draw isopleths as is done in Fig. 2(b) in a system where phases are fractionating because the effective bulk composition is changing over the entire diagram. Nevertheless, the slopes of the isopleths at the reference conditions are correct, which is adequate for this discussion. The isopleth for 0% mica is coincident with the isopleth for garnet abundance (Won = 0). In assemblages with significant quantities of mica there are P-T paths involving heating with decompression along which garnet and plagioclase will both grow, although for most P-T paths, garnet growth requires plagioclase consumption. It is important to note that in all cases the new plagioclase will be more albitic than the old plagioclase, provided there are no additional Ca-bearing phases present and that the system is closed to Na and Ca. This result leads to an important generalization: over most P-T paths '0s gurnet grows, plugioclase must be consumed and it must become more ulbitic.'

The above results have some important implications for the interpretation of plagioclase textures and zoning. The diffusivity of albite and anorthite components in plagioclase are quite small at conditions of the greenschist and amphibolite faaes, largely because of the difficulty in breaking Si-0 and AI-0 bonds (Grove er ul., 1984). For example, at 500°C, the length scale for difFusion in plagioclase is several orders of magnitude less than the diameter of an electron microprobe beam for metamorphic events lasting even hundreds of millions of years and is therefore unmeasurable using this technique. This slow diffusivity in plagioclase means that newly formed plagioclase cannot equilibrate through intracrystalline difiision. Therefore, the only effective means of equilibration is for old plagioclase to dissolve partially or completely and new plagioclase of a merent composition to grow.

Consider the implications for a rock undergoing progres- sive metamorphism. As garnet grows, plagioclase must become increasingly albitic as the anorthite component reacts to form grossular and is fractionated into garnet. Old plagioclase must dissolve and re-precipitate in order to maintain equilibrium with the coexisting garnet. Possible textures that might arise from this paragenesis are shown in Fig. 3. In sites of plagioclase dissolution, 'old' plagioclase with the initial composition is progressively dissolved as garnet grows (time 1-3). In sites of plagioclase precipitation, plagioclase may precipitate continuously, producing a 'normally' zoned crystal (An-rich core, Ab-rich rim) or may precipitate discontinuously, producing a sequence of discrete overgrowths that become increasingly albitic. Inclusions of plagioclase within garnet might be of the original composition (An.,,), or might display a core-to-rim progression of increasing albite. The specific texture produced is difficult to predict, but should be a function of kinetics and perhaps the local strain rate. For example, plagioclase grains in high strain areas of a

F. S . SPEAR E T A L .

Time I

Time 2

Time 3

Site of plogioclose dissolution

Site of continuous plogioclose

precipitotlon

albitic towards the rim

rock are more likely to dissolve whereas those in low strain areas are likely to precipitate (e.g. Bell et al., 1986).

Garnet zoning profiles

The development of chemical zoning in garnet and plagioclase has been modelled using a forward approach in which the P-T path is specified and the compositions (X) and abundance ( M ) of phases computed along this path. Five different P-T paths were chosen for the forward models, as shown in Fig. 4(a). Note that the garnet molar isopleth (MGn = 0) has a negative slope at the reference P-T conditions. This means that garnet can grow over P-T paths that range from heating and decompression ( + T - P : Fig. 4b) to heating with compression ( + T + P : Fig. 4d) to cooling with loading (-T + P: Fig. 4f).

Zoning profiles for garnets and associated plagioclase compositions, grown along the five P-T paths, are shown in Fig. 4(b-f). The size of the garnets produced range from approximately 0.85 cm radius ( - T + P : Fig. 4f) to 1.2 cm ( + T + P : Fig 4d) for a starting rock volume of 100cm'. Of course, the absolute size of the garnets depends on the nucleation density, which was chosen arbitrarily to be one crystaI/lCQcm' of rock, but the relative scaling between different P-T paths will remain the same irrespective of the nucleation density.

The systematics of the zoning predicted for garnet are worth examining for the insights they might provide to the interpretation of natural parageneses. The Fe/(Fe + Mg) decreases along the + T - P , isobaric, and + T + P paths

Slte of discontinuous

plogiociase precipi totion

Site of garnet growth

-@

~

Fig. 3. Cartoon showing a possible sequence of plagioclase consumption. growth and inclusion textures. Plagioclase dissolves at some locations while new, more albitic plagioclase grows in other locations. In some locations. plagioclasc growth may bc continuous whereas in others it may be discontinuous. Inclusions within garnet may preserve the original plagioclase compositioo (An,) or become more

(Fig. 4b, c & d), is very nearly constant along the isothermal path (Fig. 4e) and increases along the - T + P path (Fig. 40. Fe/(Fe + Mg) is therefore the most reliable indicator of temperature changes in this assemblage. X m I , and X,, increase along all paths except - T + P (Fig. 49. X, decreases along every P-T path and is therefore not a very useful path discriminator. The reason spessartine decreases systematically is a function of the strong partitioning of h4n into garnet relative to chlorite and biotite, As mentioned above, comparison of Fig. l(c) and (g) reveals that isopleths of spessartine and moles of garnet are very nearly coincident, reflecting the fact that if garnet grows, spessartine must decrease.

X, decreases along paths + T - P, isobaric and + T + P (Fig. 4b, c & d) and increases along paths isothermal and - T + P (Fig. 4e & f; see also Fig. Id). Change of grossular alone is not a good monitor of how pressure is changing. X, is also plotted against garnet radius and it can be seen that X.. decreases over all paths. This is the result of the fractional crystallization of garnet, as discussed above with respect to Fig. 2. The only reliable indicator of pressure change is the relative change of anorthite and grossular, as dictated by the equilibrium

It should be noted in this regard that the use of equilibrium constants to infer metamorphic P-T paths from the chemical zoning recorded in minerals such as garnet and plagioclase must be done with caution. T h i s is illustrated in Fig. 5 , on whicb are superimposed composition isopleths for grossular and anorthite as well as

partitioning.

MINERAL C O M P O S I T I O N A L C H A N G E A L O N G P - T PATHS 689

' '.I 1 A P - T path vector:

c

0.5 I , , , I , , , , , , I .n

+ r - P

. 4 0.9 - c

0 0 0.2 0.4 0 6 0.8 I .o I 2

r

( I I

(1

Radius (cm)

I ~ " " " " " I ( I 5

0.1

( I I

( I

t i ' O

I I I I I I I I I I I I I (1 0 . 2 (1.1 (1.6 I1.U I .n I ?

Radiuh (cm)

+ T f P 0 4 - - 0.9 Fen k M g 1

- 0 3 -

- XiSP.) 1

0 2 -

0 0 2 0 4 0 6 0 8 10 Radius (cm)

I I I I I I I

0.5 10

I I I I I I I I 1 . 1 I I J 0 0.2 0.4 0.6 0.8 I .o I 2

Radius (crn)

FQ. 4. Comparison of garnet zoning profiles (b-9 produced along five different P-Tpaths. shown in the P-Tdiagrm in (a). Starting conditions are 500" C. 6 kbar. with the assemblage garnet + biotite + chlorite + muscovite + plagrodasc + quartz + H,O io the NCMnKFMASH system (Tabla 1 & 2). Fractional crystallization is assumed in every case. Nucleation density is 1 garnet crystal/100cm3. No modification of garnet zoning by diffusion.

lines of constant K, for the reaction 3 anorthite + annite = grwular + aimandine + muscovite,

Note that the slopes of the respective lines are not the Same and therefore K-7 x.. and x, may increase Or decrease depending on the P-T path. In fact, there are six different possible sets of changes for K,. X,", and X,, depending on the P-T path, as shown by the P-T vectors on Fig. 5. It is not possible, therefore, to infer

where: ( l )

(2 ) a#rs~.lrn~rns K, =

ah."" .

690 F. S. SPEAR E T A L .

10

8

- 6 n 0

1 - Q

4

T (OC)

m. 5. Relation between the equilibrium constant (K,) for the reaction 3anorthite + annitc = grossular + almandine + muscovite and isopleths of X and X,. There are 6 possible P-T vectors along which K,, ,e", and X , may either increase ( + ) or decrease ( - ).

unambiguously the P-T path based on K, lines and zoning in a single mineral.

Whole rock reactions

The fact that different P-T paths give rise to different garnet zoning profiles (Fig. 4) implies that the whole rock reaction is different along each P-T path. Whole rock reactions are the total reaction that occurs in a rock and these can be easily computed for any P-T path from the partial derivatives of mineral composition and abundance with respect to T and P. As an example, the stoichiometric coefficients for the whole rock reactions along each of the five P-T vectors in Fig. 4(a) have been computed from the Jacobian in Table 3 and the values of AT and AP along each path. The results are given in Table 4, normalized to

1.0 moles of garnet. Note that the partial derivatives in Table 3, and therefore the whole rock reactions in Table 4, are only valid at a point, in this case the reference conditions of the garnet isograd (500" C, 6 kbar). Also note that, in order to compute the coefficients of the exchange vectors (Table 4), it is necessary to multiply the partial derivative of the exchange vector (Table 3) by the moles of the phase in the rock.

The whole rock reaction for the reference assemblage is dominantly

chlorite = garnet + H20, but the side of the reaction on which quartz, muscovite, plagioclase and biotite fall depends on the path. For example, for the + T - P path the reaction is

0.114chlorite + 0.692biotite + 0.417muscovite + 1.892quartz + 1.046NaK-,(Ms) + 0.558CaAINa-,Si-,(PI) + 0.191FeMg-,(Bt) + 0.054MnMg_,(Bt) + O.llOMnMg-l(Chl)

= 1 .Ogarnet + 0.697plagioclase + 1 .565H20

+ 0.062FeMg-,(Chl), (3) whereas for the - T + P path the reaction is l.CY20chlorite + 2.022plagioclase + O.MSFeMg-,(Bt)

+ 0.069MnMg-,(Bt) + 0.114FeMg-,(Chl) + O.O%MnMg-,(Chl)

= l.0garnet + 0.375quartz + 0.942muscovite + 0.668biotite + 2.471H20 + 1.469NaK1(Ms)

+ 0.013CaAINa- ,Si- l(Pl). (4) In each reaction the composition of the phase is that at the reference condition (Table 2). There is even a very restricted range of P-T paths (increasing T, decreasing P) where chlorite and garnet are both produced. If the reaction facing can be inferred based on textural criteria, this may help to constrain the P-T path.

T.Me 4. Whole rock reaction

computed at 500" C, 6 kbar.

+ T-P Isobaric + T + P Isothermal - T + P stoichiornetria along P-Tvectors

-2.4 AT ("C) 2.6 1 .o 2.3 0.0 - 75

1 .am -1.892

1.565 -0.417 0.697

-0.692 -0.114 -1.046 -0.558

0 0 0

-0.191 -0.054

0.062 -0.110

0

1 .am -0.504

2.120 0.415

-0.967 0.141

-0.669 0.493

-0.209 0 0 0

-0.089 -0.063 -0.046 -0.101

90 1 .am

-0.360 2.178 0.502

-1.141 0.227

-0.727 0.654

-0.172 0 0 0

-0.079 -0.064 -0.057 -0.101

50 1 .m

-0.232 2.229 0.578

- 1.294 0.304

- 0 . m 0.7%

-0.140 0 0 0

-0.070 -0.065 -0.067 -0.100

120 1 .ooo 0.375 2.471 0.w2

-2.022 0.668

- 1 . m 1.469 0.013 0 0 0

-0.025 -0.069 -0.114 -0.096

M I N E R A L C O M P O S I T I O N A L C H A N G E ALONG P - T P A T H S 651

Thennobarometry in staurolite-bearing assemblages

The reaction relating garnet + chlorite assemblages to staurolite + biotite assemblages in the SiO2-AI2O3-Mg0- FeO-K,O-H20 (KFMASH) system is

garnet + chlorite + muscovite = staurolite + biotite + quartz + H,O. ( 5 )

Using the method described in the preceding section, it is possible to derive the whole rock reaction responsible for the production of staurolite in the more complex chemical system under consideration in this paper. The computed compositions and abundances of minerals at the staurolite isograd are given in Table 5 , along with the Jacobian. From these data the following reaction can be derived for

the staurolite isograd along an isobaric heating path:

1.0gamet + 1.278chlorite + 2.539muscovite + 0.OlONaK-,(Ms) + 0.031FeMg-,(Bt) + 0.023FeMg-,(Chl)

= 0.590staurolite + 2.082biotite + 0.603plagioclase + 1.478quartz + 4.389H20 + 0.033CaAINa-,Si .‘(PI)

+ 0.002MnMg-,(Bt) + O.OOlMnMg-,(Chl). (6)

The exact stoichiometry of the reaction depends on the P-T trajectory and using the Jacobian in Table 5 the stoichiometry along any P -T path may be calculated. Note that this reaction consumes garnet, which will destroy a part of the zoning record and hence P-T path information. Therefore it is important to evaluate quantitatively the

T d k 5. Mineral compositions and abundances and the Jacobian at the Mineral Moles Volume (cm’) Component Mole fraction staurolite isograd ( 5 7 p C. 6 kbar). Quartz 1.5100 34.26 9h 1 .m

Water 0.0 0.0 H2O 1 .om Muscovite 0.1478 20.59 ms 0.8241

Pg 0.1759

Plagioclase 0.1882 18.88 ab 0.8128 an 0.1872

Garnet 0.0 0.0 Prp 0.1395 alm 0.8081 sps 0.0037 grs 0.0486

Biotite 0.0658 9.98 PM 0.4986 ann 0.5013 ’

mnbt O.OOO2

fechl 0.4338 mnchl O.ooo2

Staurolite 0.0 0.0 m e t 0.1583 f a t 0.8412 rnnst o.Ooo5

Pressure derivatives (bar-’)

Chlorite 0.0509 10.80 mgchl 0.5660

Temperature dervatives (K-’) ~~~~~

(aM,,/ar), = -0.19061 x lo-’ ( a M , , / a P ) , = 0.23073 X lo-’ (aM,/aT), = 0.28180 x lo-’ ( a M , / a P ) , = -0.33371 X

(aM,,,/aT), = 0.83661 x lo-’ (aMH,,JdP), = -0.89334X lo-’ (aM, , /dT) , = -0.48387 x lo-’ (dM, /aP) , = 0.54395 X lo-’ (aM,,/aT), = 0.39676 x lo-: ( a M , , / a P ) , = -0.44331 X lo-’ (aM,/aT), = 0.11486 x lo-’ (aM,IaP), = - 0 . 1 3 4 ~ x 10-3

(aM,,/aT), = -0.24361 X 10- ( a ~ , , / a ~ ) , = 0.26112~ lo-’ (aM,,/aT), = 0.11245 x lo-’ (aM,,iaP), = - 0 . 1 ~ 1 3 ~ x lo-’

(aX , /aT) , = -0.13233 X lo-* (ax,/aP), = 0.33208 x 1 0 - ~

(aX,/aT), = 0.42486 X lo-’ (ax,iaf), = 0.43744 x 1 0 - ~

(ax,,/ar), = -0.88811 x lo-:

(dX,,/dT), = 0.33417 x ( a X , , / a P ) , = -0.45233 X (3X,,,,,/3T), = -0.12019 X lo-’ (dX, , , /aP) , = 0.10659x lo-’ (aX,/aT), = 0.83167 x ( J X , / a P ) , = -0.10115X lo-’

( ~ x , , , / J T ) , = 0 . 4 ~ 0 1 x lo-’ (JXmnbI/af), = -0.54752 X

(aX,,/aT), = 0.44952 x 10- (JXfe,,/aT)p = -0.5%76 X (aX-,/JT), = 0.11303 x

(aX,,/aT), = -0.91135 X lo-’ ( J X , , , / a P ) , = 0.51338 X

(aX,,,/aP), = 0.50113 X (aX, , , /aP), = -0.54% x

(JX, , , /aP) , = -0.13717 X ( a X , , , / W ) , = 0.37443 X

692 F . S . SPEAR ET A l .

1 , 1 1 1 1 1 1 0 1

effect of garnet consumption on the zoning profile and the implications of this consumption on the estimation of P-T conditions and paths.

It should be mentioned that reactions that produce both garnet and staurolite have been proposed for the production of staurolite in pelitic schists (e.g. Grambling, 1986; personal communication, 1990). Given the assump tions of the model in this paper, all reactions we computed resulted in the production of staurolite at the expense of garnet. However, other reactions are possible under different assumptions. For example, metasomatism involv- ing the flux of garnet components into the rock could result in production of both garnet and staurolite. A possible way to achieve this would be for volume diffusion in garnet to produce a significant flux of Mn into the matrix at the conditions of the staurolite isograd. Simultaneous growth and diffusion' is beyond the scope of the present paper, but will be addressed in a future study.

-

-

-

-

0.9 - r" 9

- 4

+ 0 8

\

07 F

< - 4

0.6

t--

40

- 0.8 +

G c \

- 0.7 E

- 06

I I I I I I I I I I

Quanz/_- -

- 0.8 +

9 - \

- 0.7 9 a

%- - 0 6

(b) - 0.5

I1 0 . 2 (1 4 0.6 0.8 I .o I .2 Radius (crn)

The effect of garnet consumption by reaction (6) is shown in Fig. 6. In this model garnet grows initially in the assemblage garnet + biotite + chlorite + plagioclase + muscovite+quartz (Tables 1 & 2) along an isobaric heating path at 6kbar from 500 to 5 T C (Fig. 6a). Staurolite enters the assemblage at S T C at which point reaction (6) proceeds until all the chlorite is exhausted at approximately 582°C (Fig. 6b). In this model, approxi- mately 0.35 cm of the outer radius of garnet is consumed. The consumption of garnet produces a step function in the zoning profile in the absence of diffusional homogeniza- tion, the size of which depends on the volumetric amount of garnet consumed by reaction (6), which depends principally on the amount of chlorite in the rock.

Conditions of equilibrium on the rim of the garnet are maintained throughout garnet consumption. Hence Fe/(Fe + Mg), X.,,, and X, decrease on the rim, whereas X,, increases. X, is interesting because the phase

0.5 I .o I I I I I I I I I I I

- 4 $- 0.3 a ln

g 0 2

< 01

/ 6OoC Final

(C)

Fi. 6. (a-c) Chemical zoning profiles in garnet and corresponding plagioclase composition produced along an isobaric heating path (see Fig. 4c) and using a fractional crystallization model for garnet and H,O. The starting conditions are shown in Tables 1 & 2. The letters (A)-(6) correspond to points in the evolution of the garnet and are discussed further in Fig. 7: (A) is the garnet isograd (sooOC); (B) is the staurolite isograd ( 5 T C); (C) is the chlorite-out isograd (582" C); (D) is the peak metamorphic condition (600" C); (E) is the shoulder on the garnet zoning profile caused by resorption of garnet as staurolite grows; (F) is the trough in the zoning profile caused by diffusional relaxation on cooling; (G) is the rim garnet composition after cooling. (a) Chemical zoning at 57" C, just prior to the nucleation of staurolite. (b) Chemical zoning in garnet at 58T C at which point staurolite has grown and the moles of chlorite = 0 in the assemblage. X, and X, have been offset slightly for clarity. (c) Chemical zoning in garnet at 600" C (solid lines) and schematic modification by diffusive processes on cooling (dotted lines). (d) Plot of mineral volumes versus Tshowing the evolution of mineral volumes over the P-Tpath.

MINERAL C O M P O S I T I O N A L CHANGE A L O N G P - T PATHS 693

equilibria constraints dictate that it increares on the rim of the garnet relative to the rim value before garnet resorption (cf. X, at 577 and 582°C). However, dissolution of the garnet rim exposes the interior of garnet that is richer in X, than the rim value. The net effect on the zoning profile is to steepen the zoning of spessartine at the rim. X.lm shows the same phenomenon in the opposite sense, but not as pronounced (see Spear, 1988b, for further discussion).

Once chlorite is removed from the assemblage, garnet continues to grow by a new set of reactions until the final conditions are attained (Fig. 6c). Figure 6(c) shows both the chemical zoning at the 'peak metamorphic' conditions (600" C, 6 kbar) assuming no diffusional re-equilibration (solid lines) and a schematic representation of the zoning after partial diffusional re-equilibration during cooling (dotted lines).

Now consider the problem of estimating P-T conditions from the garnet shown in Fig. 6(c) using conventional thermobarometry. Points (A)-(D) represent different stages in the compositional evolution of the garnet as shown on the AFM diagram in Fig. 7(a). Garnet (A) with biotite (A) gives the P-T conditions of the garnet isograd (SWC, 6 kbar; Fig. 7b). Garnet (D) with biotite (D) (the matrix biotite composition at the peak of metamorphism) would give the peak metamorphic conditions (600" C, 6 kbar). However, garnet point (D) is no longer available for measurement because of diffusional re-equilibration on cooling. The garnet rim composition (G) with the matrix biotite (D) will give a 'closure temperature' for garnet Fe-Mg exchange of approximately 525°C (see Spear, 1989). A likely garnet analytical p i n t to choose for thermobarometry would be the 'trough' point (F) (Fig. 6c), which is the lowest value of Fe/(Fe+Mg) on the zoning profile. Garnet (F) with matrix biotite (D) gives a 'presumed' peak metamorphic temperature that is somewhat lower than the true peak temperature (Fig. 7b). The actual computed temperature depends on the degree to which diffusional processes have modified the zoning profile.

Finally, consider the case in which cooling was rapid following the production of staurolite by reaction (2) such that diffusional processes do not greatly modify the zoning profile. Depending on the sue of the garnet rim, it might not be possible to analyse points (C) or (D). Instead, it might appear to the analyst that point (E) was the true garnet rim. Thermobarometry on point (E) and the matrix biotite (D) gives P-T conditions of approximately 430" C, 4.2kbar (Fig. 7b). over 150°C below the peak metamorphic conditions. Even if diffusion modifies the profile somewhat, the mistaken identification of point (E) as the rim composition will result in a computed temperature that is 50-100"C lower than the peak metamorphic conditions.

The volumes of minerals as a function of the P-T path are shown in Fig. 6(d). Volumetric changes are rather gradual until the nucleation of staurolite at 577°C. Of particular note is the large change in the modal proportions of muscovite and biotite. This rapid

A SO0 A

10 I I I I 1 I I

t / Peak:

Garnet (D) Biotite (D)

core: Gamet (A ) Biotite ( A ) \

0

0

"400 SO0 600 700 800 T ("C)

Fig. 7. (a) AFh4 diagram showing evolution of mineral assemblages depicted in Fig. 6. Points labelled (A)-(D) correspond to points in the evolution of the garnet as were disarscd in Fig. 6. Fc/(Fc + Mg) of phases is schematic. (b) P-T diagram showing P -T conditions computed from thennobarometry for different garnet-biotite-plagioclasc analyses in the modelled rock.

production of 'second-generation' biotite will be easily observed petrographically and should serve as a time marker along with staurolite, which might be useful for correlation of mineral growth with fabric development.

The effect of diffusion on P-T path calculations

One effect of diffusional homogenization of garnet is to alter the effective bulk composition of the rock by the process of 'internal metasomatism', which will drive chemical reactions (Spear, 1988b). This may lead to reaction textures indicative of progressive metamorphism in situations where the temperature is not changing or even as the rock is cooling.

A second effect will be to modify the P-T path calculated from either thennobarometry on inclusion suites or the Gibbs method (e.g. Florence & Spear, 1989). The reconstruction of a P-T path from chemical zoning in garnet, be it using the Gibbs method (Spear '% Selverstone, 1983) or thermobarometry on inclusion suites (e.g. St-Onge, 1987). requires that the garnet has not

634 F . S. SPEAR ET AL.

altered composition substantially by diffusional homogeni- zation. Unfortunately, it is impossible to determine from examination of a garnet zoning profile the extent to which the zoning has been modified by diffusion. It is possible, however, to predict the effect that diffusional re- equilibration will have on the computation of P-T paths in modelling experiments, which can then be used to evaluate the significance of a path inferred from natural data.

The degree to which diffusional processes alter the zoning profile depends on the T-t history of the rock and the size of the garnet. Spear (1989) has estimated that for typical T-t histories of me tamoqhc terranes, garnets with radii of lcm will suffer no significant modification of zoning if the thermal maximum is less than approximately 600°C and the core composition will not be modified unless the thermal maximum is greater than approximately 725°C. Conversely, for P-T paths with temperature maxima of less than 585"C, complete homogenization will occur in garnets with radii of less than 0.01 cm whereas the zoning in garnets with radii greater than approximately 0.5 cm will not be seriously affected. In summary, for rocks with peak metamorphic conditions in the middle amphibolite fades (T,,<6oooC), large garnets ( r > 1.0 cm) will preserve most of the growth zoning.

To evaluate the effect of diffusional homogenization on P-T paths computed from thennobarometry, consider the garnet grown along an isobaric heating path between 500 and 600°C (Figs 4c & 6). Presume for the sake of discussion that inclusions of biotite and plagioclase are present from the core to the rim of the garnet. The plagioclase in the core of the garnet will be more anorthitic than that at the rim or in the matrix, as discussed above. Biotite in the core has a higher Fe/(Fe + Mg) than biotite at the rim and in the matrix.

Fe-Mg diffusion will operate in such a garnet on two length scales. First, Fe-Mg exchange will occur on a local scale between biotite inclusions and the surrounding garnet host. Secondly, the growth zoning profile over the entire garnet will relax as d i h i o n proceeds. The final result of such a process is a homogeneous garnet with homogeneous biotite inclusions such that all garnet-biotite pairs lie on the same K, line. Temperatures computed from garnet-biotite thermometry will all give identical values. If the process is incomplete so that some zoning is preserved, the computed A T will be smaller than the true value. That is, diffusion will serve to compress the computed AT of the P- T path.

The effect on geobarometry may be evaluated if it is reco@ that plagioclase inclusions in garnet cannot change composition and will therefore preserve the original growth compositions (anorthitic in the core, albitic in the rim). Geobarometry on a homogeneous garnet (the end-member case) with these plagioclase inclusions would reveal higher pressures for the rim than the core. That is, the end-member case for ahomogeneous garnet is a P-T path that displays nearly isothermal compression (path 2, Fig. 8).

The evolution of the computed P-T path from the initial ('true') path to the end-member path depends on the

10 I I I I I I

I I V / A I

8 -

- - 6 -

n 0

d - - &.I-

- 2 -

-

,

/

0 ' I I I I I I I 400 500 600 700 800

r ("C)

m. 8. P- T diagrams showing how P - T paths will change as garnet homogenizes by -ion. Lines are isopleths of K, for the garnet-biotite geothermometer (steep lines) and the garnet-plagioclasc geobarometer (sloped Lines). Dots show hypothetical P-T points determined from the garnet core, middle and rim. Path 1: initial (true) P-Tpath is one of isobaric heating. Path 2: path calculated from homogeneous garnet. Path 3: path calculated if < D(Fe, = D(Mg). Patb 4: path calculated if D,,) ' b e ) = D(M8) .

relative dif€usivities (D) of Ca, Fe and Mg in garnet. Examination of the relative Len@ scales of natural diffusion profiles in garnet for Ca, Fe and Mg suggests that Dcl<&==D- (Loomis, 1978; Spear, 1988a). If this is the case, then K, for the thermometer will change more than the Kq for the barometer and the computed P-T path will rotate from a true path of isobaric heating towards heating with decompression (path 3, Fig. 8). If, alternatively, D,>DFc= Dk(. then the Kcq for the barometry will change faster than the K, for the thermometry and the P-T path will rotate from a true path of isobaric heating through heating with compression (path 4, Fig. 8) to the end-member case of, again, nearly isothermal cornpression (path 2).

Assuming D,<4..= DMg, the consequence of diffusional homogenization, therefore, should be to rotate P-T paths in a clockwise manner relative to their initial (true) orientations. This same result holds whether the P-T path is computed using thennobarometry on inclusions or the Gibbs method (Florence & Spear, 1989).

DISCUSSION

The models presented above make predictions about the compositions and textures of medium-grade pelitic schists that can be tested by petrographical observation and detailed microprobe analysis. Whereas it is beyond the scope of this paper to present an exhaustive review of pelitic schist parageneses, a few references will suggest the types of observations that support these models.

MINERAL COMPOSITIONAL CHANGE ALONG P-r PATHS ex

Garnet-phgiodase equilibria

A number of studies have presented data on metamorphic plagioclase in which the plagioclase is zoned in a 'normal' sense (i.e. towards albite on the rim) (e.g. Crawford, 1974, 1977; Orange, 1985; Spear & Rumble, 1986; Staubli, 1989; Treloar et d., 1989), which supports the mass balance considerations of the model described here. In some cases plagioclase inclusions in coexisting garnet also support the sense of plagioclase evolution.

A number of studies, however, report plagioclase that is zoned in a 'reverse' sense (i.e. towards more anorthitic rims). In many cases reverse zonation can be directly related to the presence of additional Ca-bearing phases in the matrix or as inclusions in garnet (e.g. Crawford, 1974, 1977; Fletcher & Greenwood, 1979). In some cases, however, there is no reported extra calcic phase to account for the plagioclase zonation (e.g. Ghent, 1975; St-Onge, 1987; Hubbard, 1989; Stowell, 1989). The only explana- tions we can offer for this behaviour is that either additional calcic phases existed in the early parageneses of these rocks but have now completely disappeared (e.g. carbonate, epidote. margarite). or the rock was open to Ca, Na or both (see Crawford, 1974, 1977, for additional discussion).

Thennobarometry in staurolite-bearing rocks

A number of studies have pointed to the discrepancy between temperatures inferred from garnet-biotite ther- mometry and those inferred from staurolite phase equilibria experiments. Hodges & Spear (1982) found a difference of greater than lWC, which they attributed to P(H20) << P(total). Pigage & Greenwood (1982) found a similar difference in temperature but were unable to ascribe it to low P(H20) because this value was constrained from other equilibria. Holdaway et d. (1988) found temperatures that differ by 60" C, which they suggest can be partially eliminated by introduction of non-ideality in staurolite.

We propose that these discrepancies may, in part, be due to the difficulty in analysing the peak garnet rim composition that was in equilibrium with the biotite in the matrix. In this regard we note the following: (i) garnet in the samples studied by Hodges & Spear (1982) is generally xenoblastic, suggesting resorption of the rim; and (ii) in the study of Pigage & Greenwood (1982) garnet cores generally have a higher Mg/Fe than garnet rims. suggesting garnet resorption. One test of our suggestion might be if a decrease in temperature between the upper garnet zone and the lower staurolite zone were calculated using garnet-biotite thermometry on garnet rims.

Dihsional readjustment of P-T paths

A rigorous model for the effect of diffusional homogeniza- tion in garnet on P-T paths inferred from that zoning is beyond the scope of this paper and is treated elsewhere (Florence & Spear, 1989). However, simple diffusion

calculations can provide limits to the conditions for which diffusional homogenization is a problem. As discussed above, the fractional change of a zoning profile relative to an unmodified proiile is a function of the T-I history and the size of the garnet. According to Spear (1989) and Florence & Spear (1989), significant modification of garnet zoning is not observed in crystals with radii of lcm or greater if the thermal maximum is less than approximately 600-6WC. Smaller garnet crystals will experience significant modification at lower temperature.

From the above analysis, it is clear that many P-T paths reported in the literature are unlikely to have been affected by diffusional re-equilibration in garnet (e.g. Selverstone et al., 1984; Selverstone & Spear, 1985; Spear & Rumble, 1986) because the garnets analysed are large and the peak temperatures below 600" C. In contrast, some of the paths presented by St-Onge (1987) may have been somewhat modified because garnet radii are on the order of 2mm and the peak temperatures range from 625 to 650" C. It is interesting to note that in the study of St-Onge (1987) plagioclase displays reverse zoning in these samples, opposite to that predicted by our model and suggesting that other calcic phases may have been present during garnet growth or that the system may not have been closed. In such a rock, homogenization of garnet will increase the magnitude of decompression that is computed because an increase towards the rim in X.. at constant X, implies a decrease in pressure.

CONCLUSIONS Numerical experiments such as those presented here are useful means of testing models of metamorphic paragen- eses. These types of forward modelling experiments are necessary because metamorphism is a pathdependent, irreversible process. The results of these models are specific to the bulk composition and the partially closed system assumptions, but some. useful generalizations can be made. (i) P -T paths computed from garnet zoning must be consistent with garnet growth vectors, or the results are self contradictory. (ii) There are systematic relations between garnet and plagioclase composition and growth that predict observable textures and measurable composi- tional variations. (iii) Changes in mineral composition from zoning studies should not be used to infer P-T paths using thermobarometry unless direct calculation of the equilibrium anstant can be made. A more reliable procedure is to use isopleth diagrams. (iv) Thermo- barometry on garnet rims in rocks in which garnet has been consumed near the peak of metamorphism may render it impossible to recover the peak metamorphic temperatures from garnet-biotite thermometry. (v) Diffusional homogenization of garnet will rotate the P -T path, most likely in a clockwise direction, relative to the true value.

One final point should be emphasized. m e modelling experiments presented here involve numerous assumptions concerning the metamorphic process and do not consider

696 F. S . SPEAR € 7 AL.

other possible processes such as infiltration. n e s e models should best be considered as hypotheses that can be tested by field observations. The real value of such models is that they can indicate which observations are the critical ones to make.

ACKNOWLEDGEMENTS

This work was supported, in part, by grants from the National Science Foundation EAR-8514659, EAR- 8708609 and EAR-8903820. Constructive reviews from H. W. Day, B. R. Frost and J . A. Grambling are gratefully appreciated.

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Received 24 October 1989; revision accepted 24 March l m .