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Home , Crystal growth, Enthalpy of fusion, Single crystal, Skaergaard intrusion

American Mineralogist, Volume 96, pages 682–689, 2011

The fractional of crystallizing

S.A. Mo r S e *

Department of Geosciences, University of Massachusetts, 611 North Pleasant Street, Amherst, Massachusetts 01003-9297, U.S.A.

Ab S t r A c t The fractional latent heat of a crystallizing system is simply the latent heat of fusion divided by the total . When plotted against , this function displays robust pulses with the successive saturation of each incoming . Each pulse endures for tens of degrees, cor- responding to tens to hundreds of thousands of years in large bodies. From an original 1963 development by P.J. Wyllie using a synthetic system, I show that the liquidus slopes (degrees per gram of produced) decrease discontinuously at the arrival of a new phase, and their inverse, the crystal productivity, undergoes sharp upward pulses at the same time. The overall liquidus slope increases continuously, interrupted by small downward jumps, with the evolution of the multicomponent melt. The crystal productivity pulses feed the fractional latent heat pulses, which dominate the crystalliza- tion history of a melt. These elementary relationships govern the near-solidus growth of dihedral angles in cumulates as they relate to the liquidus events. The latent heat ordinarily approximates to about 80% of the total enthalpy budget, but jumps to 100% (by definition) when solidification oc- curs by isothermal adcumulus growth and approaches 100% when mafic phases such as augite and Fe-Ti oxides are over-produced (relative to their equilibrium saturation) in layered intrusions. The feedback of latent heat to a self-regulating cooling history of large magma bodies is deduced here in principle. The overall results help to clarify the history of mafic magmas. In particular, they support the solidification of floor cumulates by interchange with parent magma and without the help of compaction. Keywords: Latent heat, enthalpy, solidification, mafic magma, layered intrusions, crystal produc- tivity, Skaergaard intrusion

In t r o d u c t I o n occurs or negative when crystallization occurs. In common prac- This paper is a study of the heat effects encountered during tice we may use the term latent heat of fusion without regard to the crystallization of a mafic magma. It begins with a simple sign and understand the process from the context. The fractional synthetic ternary system, diopside-anorthite-albite, (Bowen latent heat is the enthalpy of fusion evolved during crystallization 1915) within which the crystallizing melt at first runs away divided by the total enthalpy of the system (which is the sum of from a single-crystal phase, here diopside. From the path thus the sensible heat, i.e., the multiplied by the change followed, the crystal productivity is calculated as the mass of in temperature, and the latent heat). Over the history of crystal- diopside per degree of temperature. Then the is followed lization, within a system of constant bulk composition cooling along the cotectic with , and the crystal productivity at steady state, the fractional latent heat is released in durable calculated again. It turns out that the crystal productivity jumps pulses that coincide with the arrival of each new phase on the discontinuously to a much higher value when all three phases are liquidus. In large magma bodies, these pulses may endure for present. Wyllie (1963) discussed this sequence in terms of the thousands of years and they have, therefore, the ability to prolong changes in liquidus slope from steep to shallower as the number the textural maturation of the cumulate near the solidus. It is this of solid phases increases. He also emphasized two important periodic effect of the durable release of latent heat that exactly features of such multiphase equilibria. First, although the slope correlates with the jumps in dihedral angle with each incoming of the liquidus at first flattens onto what he called shelves, within liquidus phase in the Skaergaard intrusion (Holness et al. 2007) the shelf history the liquidus steepens again. This tendency is and therefore defines a cause-and-effect of textural maturity. repeated in successive shelves. Second, he emphasized that: “the This interpretation of the Skaergaard dihedral angles has been latent heat of fusion is an important item in the thermal budget” hotly contested (McBirney et al. 2009) and as strongly defended (italics in the original). That emphasis is the goal of this paper, (Holness et al. 2009, including the present author), and one pur- in which the path from the crystal productivity to the fractional pose of this contribution is to clarify the origin of the pulses of latent heat is derived. latent heat that may have been obscure in Holness et al. (2007, The heat absorbed in a chemical reaction is taken as positive 2009). However, the results of this exercise also ramify to issues and hence the enthalpy of fusion is taken as positive when fusion of isothermal solidification by chemical exchange (adcumulus growth) and of compaction in igneous cumulates. The magma types considered here are primarily those that * E-mail: [email protected] crystallize the sequence -plagioclase-augite-(ilmen-

0003-004X/11/0004–682$05.00/DOI: 10.2138/am.2011.3613 682 MOrSE: FrAcTIONAL LATENT HEAT 683

1 ite)-magnetite-apatite (and possibly ternary ) such as the liquid evolves. At the end of stage 2, all of the remaining those at Skaergaard and the Kiglapait intrusion (Morse 1969) liquid from stage 1 has produced of plagioclase and and many other layered intrusions (cawthorn 1996), but also, diopside, again with a drop of 35 °c, but with much greater and interestingly, some varieties of primitive MOrB as in the crystal productivity (2.37 g/°c). The first slope (35/17.2 = 2.0 Ghiorso (1997) MELTS exercise from which this treatment of °c/g) has now given way to the shallower mean slope (35/82.8 the fractional latent heat is taken. = 0.42 °c/g), permitting a much greater yield of crystals per degree, and hence, a much greater ratio of latent heat to sensible th e Wy l l I e S e q u e n c e In dI-An-Ab heat released.

Equilibrium crystallization Fractional crystallization The quasi-ternary system Di-An-Ab (Fig. 1) furnishes a In the more interesting case of fractional crystallization, the straightforward opportunity to grasp the basic principles of liquid will continue to move along the cotectic to the Di-Ab Wyllie’s (1963) discussion of a sequence of slopes and shelves. sideline, upon which it leaves the ternary system into quater- The figure shows a simple equilibrium crystallization path from nary space. One may ignore its further history and leave the the diopside field to the plagioclase cotectic and then along the plagioclase composition stalled at An9. Accordingly, consider cotectic until the total solid composition has reached the bulk only the path of the liquid to the sideline and the crystals to their composition, at which point by definition, the liquid is consumed. limiting composition. The T-X plot (Fig. 2a) illustrates the liquid The two segments of this path are labeled 1 and 2. In segment and crystal paths as well as the total solid composition path.2 1, diopside crystals have formed and the liquid has been driven The calculation gives a residue of 13% liquid at the sideline to the cotectic over a range of temperature from 1270 to 1235 and hence, a total fraction solidified equal to 0.87, or 87%. It is °c. By means of the lever rule, 17.2% of the system has now of critical importance to note the steepening of the liquidus as crystallized as diopside, so 100Fs = 17.2, where Fs is the fraction it approaches the limit. solid. From these relations the mean crystal productivity has been No single value of crystal productivity is of interest with re- 0.49 g of diopside per degrees celsius. gard to this fractional crystallization path but instead, a sampling At this point, the liquid turns sharply away from the Di path of 18 individual stages from the fractionation output is more and runs along the cotectic (stage 2 of the process), crystalliz- informative, as shown in Figure 2b. Productivity is calculated ing large quantities of plagioclase along with diopside. Ideally from the rayleigh fractionation output as the stepwise difference the plagioclase crystals react perfectly to maintain pervasive in the fraction solidified divided by the stepwise temperature equilibrium with liquid, adjusting their entire composition as difference. In the figure, it is clear that there is a discontinuous jump in productivity when the liquid reaches saturation with Di 1391.5 plagioclase, and that this new maximum is followed by a steep decline over a range of 50 °c to values below the pre-cotectic level. One may presume that this behavior is characteristic of

1350 all such new saturations in crystal species that a fractionating magma encounters. 2 1 100 Fs = 82.8 100 F = 17.2 St u d I e S u SI n g MeltS o n Morb 1300 s T = 35deg T = 35deg xl Prod = 2.37 g/deg xl Prod = 0.49 g/deg Thermal slopes of evolving magmas 1 Wyllie (1963, p. 209) emphasized abundant evidence that 1250 the liquidus slopes within the shelves of evolved magmas tend 2 to become steeper all the way to a limit. Since plagioclase is 1300 1200 1350 a ubiquitous phase in such an evolution of a common mafic 1250 1400 1 1133 1450 This would be Bowen’s (1913) hypothetical “perfect equilib- 1200 1500 1150 rium.” In reality we know that this equilibration is impossible Ab An 1100 Weight percent 1553 in a dry system because it would take all of geologic time to equilibrate a 1 mm grain of plagioclase internally at 1000 °c FI g u r e 1. System diopside-anorthite-albite from Bowen (1915) (e.g., Morse 1984; Grove et al. 1984). The ensuing discussion as modified by Kushiro (1973) and calibrated by Morse (1997). The here of fractionation will avoid this problem. 2 path of equilibrium crystallization is shownMorse for a Figurebulk composition 1 on The diopside-saturated plagioclase loop is a parameterization the altitude of the triangle at 1270 °C. In the first part of the path Di using KD = 0.26 and the relation for the partition coefficient D S L S S crystallizes and sends the liquid to the cotectic with plagioclase, with = X Ab/X Ab = KD∙X An + 0.975 X Ab (Morse 1997) with the experi- a crystal productivity of 0.49 g/°c. In the cotectic part of the path, mental data of Bowen (1915) and Kushiro (1973) as described the crystal productivity has jumped to approximately 2.37 g/°c. This in the referenced paper. The reaction progress is calculated with diagram illustrates Wyllie’s (1963) insight about the importance of a rayleigh fractionation routine operating on the weight frac- changes of liquidus slope when a system becomes saturated with a new tion liquid. In Figure 2a, the temperature T °c on the cotectic is S crystalline phase. Symbols: FS = fraction solidified; xl Prod = crystal given for plagioclase composition x = X An by the polynomial T productivity. = 200.85x3 – 336.76x2 + 299.15x + 1108.3. 684 MOrSE: FrAcTIONAL LATENT HEAT

1280 System Di-An-Ab MORB Slopes L(Di) 30 1240 FS = 0.17 0.4 0.6 L(Di,Pl) 1200 0.7

T, deg C. 20 0.75 1160 0.8 ISC TSC Ap 0.87 a

Deg C/ gram solid 1120 10 0 0.2 0.4 0.6 0.8 1.0 XAn Mt 5 Ol System Di-An-Ab Aug b 0 Pl 4 850 950 1050 1150 1250 Temperature, deg. C 3 FI g u r e 3. changes in slope (in degrees per gram crystallized) during 2 Di+Pl+L the crystallization of a damp MOrB compositionMorse Figure at 1.0 3 kbar pressure Xl prod, g/deg as modeled by the MELTS program of Ghiorso and Sack (1995) and 1 as used by Ghiorso (1997). A local decrease in slope occurs at every Di+L addition of a saturating crystal phase, and although these changes are small, their influence is large. The overall slope in the crystallizing 0 magma increases steadily with falling temperature, in keeping with the 1120 1160 1200 1240 1280 common observation of steepening liquidus slopes on thermal shelves T, deg C in evolved magmas (e.g., Wyllie 1963, p. 209). FI g u r e 2. (a) Fractional crystallization in the system Di-An-Ab in a Morse Figure 2 combined plot of the liquid path from diopside, L(Di) (bold arrow) onto Crystal productivity in MELTS the T-X profile of the cotecticL (Di, Pl). The liquid and crystal paths are calculated with a rayleigh fractionation routine. The liquid leaves the Figure 4a shows the crystal productivity, the counterpart to ternary plane near the Ab-Di sideline and the plagioclase composition Figure 3, being the inverse of the liquidus slopes in Figure 3, i.e., stops at An9, leaving a fraction 0.13 of the mass of the system yet to units of g/°c, using, for convenience, the grams of solid in each crystallize and ignored here. Key: FS = fraction solid; L = liquid; Di = 2-degree step size of the MELTS experiment directly without diopside; Pl = plagioclase; ISC = instantaneous solid composition; TSC = dividing by 2. As in Figure 2b for the synthetic system, the jumps total solid composition. (b) crystal productivity for the paths illustrated in productivity are instantaneous and discontinuous within the above in a, calculated from the output of the rayleigh fractionation sensitivity of the 2-degree step size. The peaks are sharp, dis- program. When the liquid reaches saturation with plagioclase, there is a tinct, and followed by smooth declines as for Di-An-Ab. They discontinuous jump in the crystal productivity, which then decays with scale with the individual latent heat multiplied by the intrinsic time and temperature. This λ-like discontinuity is characteristic of when they become saturated with an added crystal phase. abundance of the incoming phase, augite being the chief of these because of its high-mass concentration at saturation, as can be seen in the familiar ternary system Fo-Di-An and its natural magma, it is reasonable to conclude that the steepening cur- cousins Ol-Aug-Pl (e.g., Morse et al. 2004). The apatite peak is vature seen in Figure 2a for the plagioclase-saturated cotectic barely discernible, though present. This diagram almost exactly (and in the system An-Ab from which that slope derives) is mimics the total latent heat of fusion diagram of Ghiorso (1997) an important contributing factor in such intrinsic steepening. and its repetition in Figure 1a of Holness et al. (2009). This is The curious student must then wonder how this steepening not surprising, because the latent heat of a crystallizing system squares with the Wyllie principle of flatter liquidus slopes bears a direct relation to the masses of crystals being produced. when a new phase saturates the melt. The answer is found in However, this figure is a map of crystal productivity, not latent Figure 3, derived from the MELTS spreadsheet for MOrB at heat, and serves to bridge the Wyllie principles of slope changes 1 kbar used by Ghiorso (1997). This spreadsheet compiles the to the inevitable principle of the fractional latent heat. evolving free of all components and the system in 2-degree intervals, hence any instantaneous event is smeared Fractional latent heat over two degrees in temperature. In Figure 3, it is observed The fractional latent heat is that portion of the total enthalpy that the liquidus slope (expressed in °c/g) does indeed drop of the system that is not the sensible heat. The total enthalpy is discontinuously (allowing for the 2-degree step size) at each the sum of the sensible heat evolved, essentially the sum of the incoming phase, after which it recovers to the overall rising term for the heat capacity multiplied by the change in tempera- trend with falling temperature. The local effects are small but ture, CPdT, and the latent heat of fusion, ∆Hf. The fractional distinct; however, their accompanying effects on the crystal latent heat is simply the ratio of ∆Hf to the total enthalpy of the productivity are anything but small. liquid, which is a sum routinely calculated in the MELTS free MOrSE: FrAcTIONAL LATENT HEAT 685

3.0 Aug Finally, the calculated fractional latent heat goes to 1.0 at the a MORB Crystal Productivity end of crystallization, as it must, recalling that it also goes to 1.0 2.5 Ol at an invariant point such as an isothermal eutectic.

2.0 Ap p l I c A t I o n S t o I g n e o u S c u M u l A t e S General remarks 1.5 The principle of the fractional latent heat has, until now, been

1.0 Mt used primarily to understand the maturation of the dihedral angle

Grams / 2 deg C Pl “cpx-pl-pl” in igneous cumulates where this angle is observed 0.5 to jump with the incoming of a new phase (e.g., Holness et al. Ap 2007). The fact that the fractional latent heat jumps just where 0 the dihedral angle jumps strongly suggests a cause-and-effect 850 900 950 1000 1050 1100 1150 1200 1250 relationship. In retrospect it becomes obvious that the jump in Temperature, deg. C latent heat maintains the cumulate hotter for longer and thereby allows the subsolidus or near-solidus growth of dihedral angles 1.0 to proceed further than before the new phase arrived. Much con- Fractional Latent Heat b fusion has arisen about this suggestion, which I hope to clarify 0.9 below. In addition, there are many other features of the fractional

ys Ap S Mt Aug latent heat that confuse our understanding of magmatic processes H attending the formation of igneous cumulates.

/ 0.8

f Ol There is one very important general point yet to be made H before laying the groundwork for the discussion of cumulates. As 0.7 Figure 4b shows, the latent heat of fusion absolutely dominates the heat budget of a crystallizing magma. It approximates closely Pl to 80% of the heat budget during most of the magmatic history, 0.6 850 900 950 1000 1050 1100 1150 1200 1250 and goes to 100% at the end. It also reaches 100% at any time Temperature, deg. C that isothermal solidification takes place, and that includes all occasions in which adcumulus growth, however incomplete or FI g u r e 4. (a) crystal productivity in the MOrB experiment of complete, occurs. It also approaches 100% whenever a mafic Ghiorso (1997 and spreadsheet furnished to Morse in 2008), stated Morse Figure 4 phase overshoots its cotectic proportion and causes overproduc- in grams of solid per 2 °c, the step size at which the thermodynamic properties are calculated in the numerical experiment. compared with tion of the mafic crystals. In this light, the word “cooling” must the sharp discontinuity shown in Figure 2b, the steps here appear slightly be used with great caution. smeared owing almost entirely to the 2-degree step size. The quasi-λ shape holds in principle. Peaks of productivity are obscured for the first Boundary conditions crystal species, plagioclase, but are pronounced for olivine, augite, and We consider some of the actions in a mature magma body magnetite, and just perceptible for apatite. (b) Fractional latent heat for that meets the following idealized criteria. In calling a magma the same experiment, after Holness et al. (2009). Here the heat effects are body mature, it is specified that its residence time has been long similar to each other in magnitude and pronounced even for apatite. enough to reach a secular thermal equilibrium with its surround- ings. During the term of this discussion there will be no seismic events, nor foundering of roof rocks, nor of the roof, nor minimization routine. Take special note of the fact that slumping at the walls, nor injections of fresh magma: in other the sensible heat involves a change in temperature, whereas the words, no extraneous disturbances. It is assumed that the magma latent heat does not; we return to this in due course. has been emplaced within the upper crust of the Earth, thereby Figure 4b is a slightly re-scaled version of Figure 1b of Hol- splitting the geotherm and causing the roof to be a more important ness et al. (2009), and is the goal of this discussion in developing cooling surface than the floor. Following field evidence, a mafic the underlying roots of the fractional latent heat path of crystal- floor cumulate that serves as a latent heat source as it solidifies is lizing mafic magma. Of special interest are the sharpness of hypothesized, thereby rendering any transfer of heat out through the jumps, which are by their very nature truly discontinuous, the floor a Stefan problem3 with limited heat flux. Since melting and the sustained elevation of the fractional latent heat for each of the roof is forbidden, the formation of an Upper Border Zone phase, even while declining, over an interval of temperature. is presumed. Lateral dimensions are unspecified, and the magma The effect for olivine lasts for at least 20 °c, its tail being hid- thickness is of kilometers in scale. This depiction could apply to den under the augite peak. The effect for augite lasts for at least dozens of well-known layered intrusions in the literature (e.g., 50 °c and for magnetite at least 40 °C; for apatite, the effect is cawthorn 1996). of indeterminate, perhaps permanent, duration. These numbers The magma loses heat to its surroundings, and it is assumed are of great significance for large magma bodies, in which the (e.g., Morse 1986) that the main heat loss is through the thermal cooling rate may be as low as 10−3 to 10−4 °c/year, hence the boundary layer at the roof: another Stefan problem. The heat loss effect of enhanced fractional latent heat (Fig. 4b) might last for is at a steady state at any given time; for any moment of this tens to hundreds of thousands of years. consideration, the heat loss is taken as fixed. It may or may not 686 MOrSE: FrAcTIONAL LATENT HEAT be attended by cooling, according to whether or not the fractional heat contribution is low. latent heat has a value less than 1.0; the normal expectation In the matter of cumulate maturation, the peaks at augite, will be, consonant with Figure 4b, a heat transfer of about 20% magnetite, and apatite all lie in the range of the fraction of latent sensible heat and 80% latent heat. cooling indeed causes crystal heat F(lh) = 0.83–0.86. For a nominal mean cooling rate of 1 growth, but can occur isothermally as well. degree per thousand years with a large value of F(lh), it is clear When crystals grow, they accrue compatible components that a cumulate at a peak could remain nearly isothermal for from the surrounding medium and reject the incompatible well over 1000 years. To refine such estimates, one may choose solute to the surroundings (e.g., Tiller 1991). For felsic crystals a crystallization history of 200 000 years for cooling with root in mafic magma, the rejected solute (rS) is mafic and dense4, time from 1200 to 1000 °c. At the F(lh) peak for augite, and whereas for mafic crystals the rS is felsic and buoyant and may supposing that a small amount of liquid is trapped so that solidi- easily escape by compositional convection, carrying with it the fication is effected only after the system has cooled 2 (or 5) °c, latent heat and felsic components. Therefore, all mafic layers in then the time spent near the peak of fractional latent heat is 4200 a cumulate will release less dense rejected solute on solidifica- (or 104) years. For magnetite, the numbers are 2230 and 5500 tion, and this solute will stream upward along an adiabat that years, respectively. If the presence of melt fosters an increase in instantly becomes superheated (and evolved) relative to the the dihedral angle, it also increases the duration of a high value ambient liquidus. In this manner, the latent heat of fusion from of F(lh). A correlation between the increase of dihedral angle the solidification front at the floor is carried in a gravitational and the residual porosity would then result. “short circuit” to the upper thermal boundary layer where heat can be removed from the system. The effect is particularly strong Overproduction of mafic crystals for Fe-Ti oxide-rich cumulates. The rS from sloping felsic layers Overproduction is defined as the sustained formation of a may also drain away, but all flat felsic layers will solidify without crystal phase in excess of its nominal cotectic ratio. The nominal the help of compositional convection, perhaps by diffusion if the ratio can be determined experimentally, or inferred by examina- sedimentation rate is low enough (Morse 1986). tion of modal abundances after the overproduction has relaxed to a stable value. Cooling rate The augite and Fe-Ti oxide peaks of modal abundance in This is an ambiguous and possibly misleading term in the layered intrusions are well known for a tendency to overshoot context of igneous cumulates. In terms of an entire magma sys- the nominal saturation abundance, particularly in the Kiglapait tem, it is useful for describing one aspect of the rate of heat loss, intrusion (Morse 1979b, Figs. 12–16) and in the Skaergaard intru- since the change of temperature over time is an indicator of the sion (Tegner et al. 2009, Fig. 3). When this metastable overshoot evolution of the condition from all liquid to all solid. But because occurs, the system is overloaded with the new phase and its latent only ∼20% of the heat loss may be ascribed to actual cooling, the heat effect is driven to high levels. The result gives an extended more general term heat loss or transfer is more precise. lifetime to a pulse of fractional latent heat and an elevation of the That said, the peaks of the fractional latent heat array describe fraction toward 1.0. Such modal excesses guarantee the longer intervals of time in which the CPdT cooling of the system actu- duration of a latent heat pulse, in some cases likely to extend to ally slows down in response to competition from the evolution many thousands of years at nearly isothermal conditions. of latent heat from the growing crystals, all the while assuming Two examples of overproduction are worth describing, for a constant secular heat loss from the system. It is therefore they are not trivial. For the Skaergaard intrusion (Tegner et al. precisely correct, but unfortunately confusing, to say that these 2009), augite is overproduced for ∼280 m (nominally from 500 peaks represent times of a lower cooling rate. It is more helpful to 784 m), which at a mean accumulation rate of 1.6 cm/year to say that they represent an enhanced contribution of latent heat would endure for ∼18 000 years. The Fe-Ti oxide overproduction to the system, with the understanding that this increase of latent runs for ∼100 m above the 784 m stratigraphic level, enduring for heat displaces sensible heat in the total heat budget. The effect, ∼6400 years. For the Kiglapait intrusion (Morse 1979b, 1988) the as used to explain the observed variation in dihedral angles, is corresponding values are 1375 m and 105 years for augite, and to keep the cumulate hot for a longer time than when the latent 600 m and 40 000 years for the oxides. During these episodes the fractional latent heat must have averaged out to higher values 3 The Stefan effect arises when the heat to be extracted from a than 0.83 and plausibly approached 1.0 at times. system through a boundary layer encounters a latent heat source Effect of latent heat on dihedral angles within the boundary layer. Such a source impedes the loss of sensible heat from the system. The classic case addressed by This effect is a matter of proximity. In many, or perhaps Stefan (see Turcotte and Schubert 2002) is the of most, igneous cumulates the dihedral angle population is far on the sea or a pond. The ice- boundary and its latent heat from equilibrium and therefore sensitive to its thermal history must be cooled through the ice itself, and the thicker the ice, the (Holness 2010). The maximum temperature in the magma col- worse the problem. The upper and lower boundaries of magma umn occurs in the cotectic assemblage above the solidification bodies that crystallize at both floor and roof are always Stefan front. In case 1, assume that the residual porosity will be zero. problems. Assume the assemblage olivine + plagioclase + liquid, for which 4 Micrometer-scale streamers of mafic solute rejected from the fractional latent heat is about 0.72 (Fig. 4b). The release of plagioclase in an experimental run can be seen clearly in the this latent heat pins the temperature at the solidification front to backscattered electron image in Morse et al. (2004, Fig. 14). the cotectic temperature. Assume that the local system cools 1 MOrSE: FrAcTIONAL LATENT HEAT 687

°c in a thousand years. For simplicity, partition the latent and accumulation and stable cotectic production of all phases (with sensible heat over time and thereby assume that the system lower latent heat and elevated sensible heat). At the very least, remains isothermal at the cotectic for 720 years, and then cools we do know that the overproduction of mafic phases must have 1 °c over the next 280 years. resulted in two things: the faster depletion of magma and the Now assume that augite joins the assemblage in the mushy greater release of latent heat. The diagram shows a predicted zone, causing the fractional latent heat to jump to 0.83 (Fig. 4b). result, hence the curved fit is non-trivial. The equivalent history of the enthalpy could then be isothermal for 830 years and thereafter cooled 1 °c over the next 170 The effect of isothermal solidification years. The solidification front would then remain at the cotectic The study of residual porosity (reported as trapped liquid) temperature for 110 years longer than the pre-augite condition. in layered cumulates shows a general tendency to lower values Beneath the solidification front, the system cools by conduction up-stratigraphy (e.g., Wager 1963; Morse 1979b, 2009; Tegner through the floor. For a thermal gradient of 0.03 °c/cm, the et al. 2009), thereby expressing a tendency to more and more cumulate temperature at 10 cm depth would be within 0.3 °c of complete adcumulus growth. In turn, this secular tendency to the interface temperature, a small difference. better adcumulates is inferred logically to reflect a decreasing In the more general case 2, the residual porosity is nonzero accumulation rate with time, allowing more time for adcumulus and a capture front must be specified beneath which the en- growth (Morse 1986). trained liquid will be trapped and solidified only by equilibrium Since adcumulus growth is effected by an isothermal exchange crystallization on a long time scale. During this time, the latent couple, and because even mesocumulates have, by definition, heat of fusion will be evolved within the cumulate at decreasing achieved some measure of adcumulus growth, then the body of , and again lost through the floor. The effect will cumulates as a whole has experienced intervals of F(lh) = 1.0, and be to bring hot melt into closer proximity to solidified parts of more generally so with stratigraphic height. This effect is further the assemblage, and the growth of dihedral angles may take enhanced with the accumulation of augite, which rejects a less place in the crystallized edifice within very short distances of dense solute, and even more strongly with Fe-Ti oxides, which the crystallizing melt residing in pore spaces. In this manner, maximally reject a less dense solute, so that adcumulus growth is the dihedral angles mature at temperatures above the eventual enhanced by compositional convection. solidus. A rough calculation suggests that, for a residual porosity of 0.3, the contribution of fractional latent heat might lower the 4 thermal gradient near the floor to <1 °c/m. 8400 PCS = 82 97 99.7 The latent heat effect is also locally enhanced by the modal 3.8 overshoot of an incoming phase. Suppose that augite is over- Kiglapait magma depth produced relative to cotectic saturation, as at Skaergaard and 3.6 Kiglapait. Now the fractional latent heat is augmented by the 3600 amount of extra augite and, in our presumed partitioning, the 3.4 2500 isothermal history is extended beyond the nominal 830 years. 2100

If the feed of augite remains in excess, so does the isothermal log (depth) 3.2 history. In principle, the partitioned isothermal history could last for the whole 1000 years, postponing the actual cooling to some 3 1200 later history. Indeed, that could be the general expectation for the meters overproduction of augite (or Fe-Ti oxides, or apatite). 2.8

First-order feedback 2.6 0 0.5 1 1.5 2 2.5 3 Such an effect of increased latent heat production should - log (FL) have impeded the sensible cooling of the intrusions temporarily. FI g u r e 5. Sheet cooling in the Kiglapait intrusion, with alternative Indeed, for the Kiglapait there is suggestive evidence that this was treatment of the data for magma depth in the range 82–99.7 PcS (percent the case. An enlargement of Figure 1 of Morse (1988) is shown in solidified), after Figure 1 of Morse (1988). The figure is a log-log plot for Figure 5. This figure is a log-log treatment of mean stratigraphic which the dotted line is a regression on all the dataMorse (early Figurevalues shown 5 height in meters (converted to magma depth) plotted against the as small circles) with slope –0.5 on log FL and so with depth scaling 1/2 fraction of liquid remaining, equal to 1 – (PcS/100) where PcS with FL . The solid line is a third-order polynomial fit to the data points is percent solidified. A dotted line describes the overall fit to all shown, with R2 = 0.995. It runs below the mean curve where augite and the data that scales with the square root of the fraction of liquid oxide are over-produced, and above the main curve where they remaining. The points were picked off the cross sections of the have their stable cotectic proportions. This behavior is as expected for intrusion. The region 82–97 PcS is the region in which augite the excess latent heat evolved when mafic minerals are overproduced, followed by a period of increased release of sensible heat. Error limits: and oxide minerals begin to increase and are overproduced. The On the original points 0–84.5 PcS the regression has a standard error of data points in the region of interest fit a third-order polynomial 0.0024 on log (depth) with correlation coefficientR 2 = 0.9998 and slope 2 curve with correlation coefficient R = 0.995, in which the values –0.5000 ± 0.0026. The linear regression for all data shown without the below the dotted line represent faster than normal reduction of polynomial has a standard error of 0.03 on log (depth) and R2 = 0.9936. magma depth, i.e., overproduction of augite and oxide miner- For the region 82–99.8 PcS covered by the polynomial the linear als (and latent heat), followed by a recovery period of slower regression has R2 = 0.9877 compared to 0.995 with the polynomial fit. 688 MOrSE: FrAcTIONAL LATENT HEAT

Every orthocumulate retains a fractionating and reacting expected for Racc up to 1 cm/year and mesocumulates for Racc > trapped liquid that solidifies only several tens of degrees below 1 cm/year (Morse 1986 Fig. 7). These limits are now considered the temperature of accumulation (recall 35 °c in Di-An-Ab, Fig. too small. The backscattered electron image of Figure 6 in Morse 1). This trapped liquid locks in latent heat that is therefore given et al. (2004) shows euhedral plagioclase crystals of uniform off slowly for a long time with cooling. Orthocumulates are latent composition up to 50 µm long grown at 1200 °c at 5 kbar for 6.5 heat emitters that durably impede solidification. In contrast, ad- h. The inferred adcumulus/isothermal growth by diffusion alone cumulates promote solidification isothermally. As a consequence, was then effected by rates as high as 4–6 cm/year. By implication, igneous cumulates solidify most efficiently by adcumulus growth then, values of Racc up to at least 4 cm/year should be considered and least efficiently by sequestering trapped liquid. adequate for adcumulus growth even for felsic cumulates on a flat floor. The estimated mean accumulation rates for Skaergaard Second-order feedback and Kiglapait are ∼1.6 and ∼0.8 cm/year, respectively (Nielsen Efficiency would not seem to be an objective incentive in 2004; Morse 1979a). The failure of adcumulus growth in the a physical process, but in fact it actually is, by virtue of the Skaergaard LZa then should imply early accumulation of crystals following feedback cycle. Isothermal crystal growth releases at a rate greater than 4 cm/year. only latent heat into the adiabat that carries it efficiently to the The notion that compositional convection can occur only within site of heat removal. Polythermal growth obviously requires mushes hundreds of meters thick (Tegner et al. 2009, p. 836) is a cooling the whole body in addition to removing latent heat. misconception due to a particular experimental design studied by The accumulation of latent heat near the roof minimizes the Tait and Jaupart (1989) and not to the cumulate interface of a large chance for that is needed to nucleate new crystals. intrusion (Morse 1969, p. 74) or the growth of crystals at the It therefore slows the nucleation and accumulation rate, thereby surface of the Earth’s Inner core (Braginski 1963). allowing more general adcumulus growth (assuming that rapid accumulation impedes adcumulus growth, which the observed IM p l I c A t I o n S F o r SolIdIFIcAtIon stratigraphy proclaims is true). The remarks below may help to develop my philosophy of And so, in fact, the more adcumulus growth the less need to cumulates. cool the body of magma, and the higher the fractional latent heat, First, the saturation of a mafic magma in a new crystal phase the less frequently crystals need to nucleate, in turn permitting is a permanent condition except in the case of olivine, which is more adcumulus growth. And adcumulus growth involves F(lh) vulnerable to and silica activities that may result in an = 1.0, not 0.8. olivine hiatus, and except in the case of low-ca pyroxene, which is vulnerable to the low-pressure equivalent stability of fayalite Compaction, compositional convection, and diffusion plus a silica . Mechanical compaction has been invoked as an alternative Second, an extended opportunity for increasing the dihedral or complement to adcumulus growth for the solidification of angle of cpx-pl-pl in a solidifying crystal edifice is afforded by igneous cumulates (e.g., Tegner et al. 2009, p. 833 ff and refer- the extra latent heat released in a durable pulse upon the entry ences therein). One trouble with compaction is that mush thick- of each new crystal phase at saturation. nesses of tens to hundreds of meters are needed to compress the Third, in the absence of any other mechanism for discon- crystals into a cumulate and expel the parent melt upward. Such tinuously and permanently increasing the dihedral angle, then dimensions seem incompatible with the limited thicknesses of the fractional latent heat pulse attendant upon saturation of a disturbed layering suggested by the work of Irvine et al. (1998) new crystal phase is uniquely the cause of the increase in the at Skaergaard and the observations of Morse (1969) at Kigla- dihedral angle. pait. Moreover, the possibilities for solidification very near the Fourth, the benefit of the fractional latent heat release is accumulation surface are enhanced by adcumulus growth with scale dependent, requiring proximity of the new crystal growth compositional convection and consistent with the field evidence within the solidifying cumulate and its immediately overlying in many cases. That neither compaction nor much adcumulus saturated liquid. growth occurred in the Skaergaard LZa (Tegner et al. 2009) is Fifth, this proximity relationship, combined with the near- demonstrated by the large residual porosities encountered there, solidus growth of the dihedral angle, renders difficult any alterna- and accordingly the conclusion is that accumulation rates were tive history of dispersed crystals within a mushy layer. too high to permit efficient solidification. The orthocumulates Sixth, because the fractional latent heat-dihedral angle rela- in LZa (and HZ) record cooling and chilling of intercumulus tionship demonstrably occurs within a nearly solid cumulate, the melt through the floor below the level of maximum latent heat process of solidification is dominantly that envisioned by Hess release at the mush/magma interface. Elsewhere, as at Kiglapait (1939), Wager et al. (1960), and others who saw compelling field and above LZa/b in Skaergaard, the aids to solidification from evidence for solidification at and near the floor of an intrusion by compositional convection and latent heat release considered here interchange of crystalline floor-builders with fresh magma. can account for the low residual porosities encountered. Seventh, the role of compaction in the solidification of Skaer- In a study of solidification aided by compositional convection, gaard cumulates is therefore small or nil, as the field evidence Morse (1986) considered the process of adcumulus solidification of currents, blocks, rhythmic layering, cross-bedding, graded for thin (centimeter scale) mush thicknesses to be plausible by layering, and trough banding all abundantly testify to the conclu- diffusion alone if the accumulation rate (Racc) and characteristic sion of a thin mushy zone compatible with low accumulation diffusion distance were near 0.5 cm/year. Adcumulates were rates above LZa/b. MOrSE: FrAcTIONAL LATENT HEAT 689

co n c l u d I n g r e MA r k S Ghiorso, M.S. and Sack, r.O. (1995) chemical mass transfer in magmatic pro- cesses IV. A revised and internally consistent thermodynamic model for the The Wyllie (1963) sequence of changing liquidus slopes with interpolation and extrapolation of liquid-solid equilibria in magmatic systems the addition of new crystal phases saturating a melt is shown to at elevated temperatures and pressures. contributions to Mineralogy and Petrology, 119, 197–212. be of general importance in the evolution of magmas and their Grove, T.L., Baker, M.B., and Kinzler, r.J. (1984) coupled caAl-NaSi diffusion in crystalline products. The overall liquidus slope always increases plagioclase feldspar: Experiments and applications to cooling rate speedometry. Geochimica et cosmochimica Acta, 48, 2113–2121. with magma evolution but is interrupted by small decreases when Hess, H.H. (1939) Extreme fractional crystallization of basaltic magma: The new crystal phases appear. crystal productivity is defined as the Stillwater igneous complex. Transactions American Geophysical Union, inverse of the liquidus slope, and it peaks discontinuously and 430–432. Holness, M.B. (2010) Decoding dihedral angles in melt-bearing and solidified rocks. strongly upon arrival of each new phase, then slowly declines. In M.A. Forster and J.D. FitzGerald, Eds., The Science of Microstructure—Part Because the mass of crystals being formed is directly related I. Journal of the Virtual Explorer, Electronic Edition, 35, paper 2. Download to the latent heat released, the total latent heat of the system from: http://virtualexplorer.com.au/article/2010/265/decoding-dihedral-angles- in-melt-bearingand-solid. follows the crystal productivity exactly when plotted against Holness, M.B., Tegner, c., Nielsen, T.F.D., Stripp, G., and Morse, S.A. (2007) A temperature. By isolating the latent heat from the sensible heat textural record of solidification and cooling in the Skaergaard Intrusion, East Greenland. Journal of Petrology, 48, 2359–2377. due to cooling, the fractional latent heat is formed, which also Holness, M.B., Morse, S.A., and Tegner, c. (2009) response to comment by peaks discontinuously but declines slowly enough to keep the McBirney, Boudreau and Marsh. Journal of Petrology, 50, 97–102. crystal mass relatively hot for long periods of time, even as Irvine, T.N, Andersen, J.c.Ø., and Brooks, c.K. (1998) Included blocks (and blocks within blocks) in the Skaergaard Intrusion: Geologic relations and long as tens to hundreds of thousands of years. This sustained the origins of rhythmic modally graded layers. Geological Society America delivery of local latent heat facilitates the attainment of textural Bulletin, 110, 1398–1447. maturity in the cumulate rocks, thereby logically and surpris- Kushiro, I. (1973) The system diopside-anorthite-albite: determination of com- positions of coexisting phases. carnegie Institution of Washington Yearbook, ingly linking a subsolidus activity to the liquidus history of a 72, 502–507. crystallizing magma. McBirney, A.r., Boudreau, A.E., and Marsh, B.D. (2009) comments on “Textural maturity of cumulates: a record of chamber filling, liquidus assemblage, In addition to this local effect in the cumulate, the effect of cooling rate and large-scale convection in mafic layered intrusions” and “A isothermal adcumulus growth is to generate spikes of latent heat textural record of solidification and cooling in the Skaergaard intrusion, East that is transferred efficiently to the roof cooling site, thereby ef- Greenland.” Journal of Petrology, 50, 93–95. Morse, S.A. (1969) The Kiglapait , Labrador. Geological Society fecting solidification more efficiently because it occurs without of America Memoir 112, 204 pp. cooling the magma body. The accumulation of latent heat at the ——— (1979a) Kiglapait geochemistry I: Systematics, sampling, and density. roof site of nucleation tends to slow the accumulation rate and Journal of Petrology, 20, 555–590. ——— (1979b) Kiglapait geochemistry II: Petrography. Journal of Petrology, therefore enhances the chance for isothermal adcumulus growth 20, 591–624. and release of latent heat. ——— (1984) cation diffusion in plagioclase feldspar. Science, 225, 504–505. ——— (1986) convection in aid of adcumulus growth. Journal of Petrology, The latent heat budget of crystallizing mafic magma is by 27, 1183–1215. far the most important part of the entire enthalpy budget and ——— (1988) Motion of crystals, solute, and heat in layered intrusions. canadian it controls the timing of solidification and therefore the cool- Mineralogist, 26, 209–244. ——— (1997) Binary solutions and the lever rule revisited. Journal of Geology, ing history of the magma body. Its influence on the subsolidus 105, 471–482. growth of the near-solidus dihedral angle to the conclusion ——— (2009) Pre-cumulus zoning, residual porosity, and impermeable barriers that the action occurs in a dense solid. If instead the saturation in igneous cumulates. Eos Transactions, AGU, 90(52), Fall Meeting Supple- ment, Abstract V13F-03. with a new phase occurred in a dispersed mushy zone of many Morse, S.A., Brady, J.B., and Sporleder, B.A. (2004) Experimental petrology of meters’ thickness, only the final compacting layer could have the Kiglapait intrusion: cotectic trace for the Lower Zone at 5kb in graphite. Journal of Petrology, 45, 2225–2259. been involved in the growth of the dihedral angle, as the main Nielsen, T.F.D. (2004) The shape and volume of the Skaergaard Intrusion: Im- release of latent heat was itself dispersed above. The most effec- plications for mass balances and bulk composition. Journal of Petrology, tive release of latent heat from the new phase will occur where 45, 507–530. Tait, S.r. and Jaupart, c. (1989) compositional convection in viscous melts. the cumulate is physically closest to the parent liquid. The parent Nature, 338, 571–574. (not the evolved) liquid is also needed for the most efficient, Tegner, c., Thy, P., Holness, M.B., Jakobsen, J.K., and Lesher, c.E. (2009) Dif- isothermal, adcumulus growth. ferentiation and compaction in the Skaergaard Intrusion. Journal of Petrology, 50, 813–840. Tiller, W.A. (1991) The science of crystallization: microscopic interfacial phenom- Ac k n o W l e d g M e n t S ena. cambridge University Press, 391 pp. Turcotte, D.L. and Schubert, G. (2002) Geodynamics, 2nd Ed., 456 p. cambridge I thank Marian Holness for helpful suggestions, and find her blameless for any- University Press, U.K. thing she may not care for in these pages. Mark Ghiorso and John Brady kindly assisted Wager, L.r. 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