ALBERT

Description: Transformation of refractory cratonic mantle into more fertile lithologies is the key to the fate of cratonic lithosphere. This process has been extensively studied in the eastern North China Craton (NCC) while that of its western part is still poorly constrained. A comprehensive study of newly-found pyroxenite xenoliths from the Langshan area, in the northwestern part of this craton is integrated with a regional synthesis of pyroxenite and peridotite xenoliths to constrain the petrogenesis of the pyroxenites and provide an overview of the processes involved in the modification of the deep lithosphere. The Langshan pyroxenites are of two types, high-Mg# [Mg2+/(Mg2++Fe2+)*100 = ∼ 90, atomic ratios] olivine-bearing websterites with high equilibration temperatures (880 ∼ 970 oC), and low-Mg# (70 ∼ 80) plagioclase-bearing websterites with low equilibration temperatures (550 ∼ 835 oC). The high-Mg# pyroxenites show trade-off abundances of olivine and orthopyroxene, highly depleted bulk Sr-Nd (ƐNd = +11.41, 87Sr/86Sr = ∼0.7034) and low clinopyroxene Sr isotopic ratios (mean 87Sr/86Sr = ∼0.703). They are considered to reflect the reaction of mantle peridotites with silica-rich silicate melts derived from the convective mantle. Their depletion in fusible components (e.g., FeO, TiO2 and Na2O) and progressive exhaustion of incompatible elements suggest melt extraction after their formation. The low-Mg# pyroxenites display layered structures, convex-upward rare earth element patterns, moderately enriched bulk Sr-Nd isotopic ratios (ƐNd = -14.20 ∼ -16.74, 87Sr/86Sr = 0.7070 ∼ 0.7078) and variable clinopyroxene Sr-isotope ratios (87Sr/86Sr = 0.706-0.711). They are interpreted to be crustal cumulates from hypersthene-normative melts generated by interaction between the asthenosphere and heterogeneous lithospheric mantle. Combined with studies on regional peridotite xenoliths, it is shown that the thinning and refertilization of the lithospheric mantle was accompanied by crustal rejuvenation and that such processes occurred ubiquitously in the northwestern part of the NCC. A geodynamic model is proposed for the evolution of the deep lithosphere, which includes long-term mass transfer through a mantle wedge into the deep crust from the Paleozoic to the Cenozoic, triggered by subduction of the Paleo-Asian ocean and the Late Mesozoic lithospheric extension of eastern Asia.

Description: The orogenic development after the continental collision between Laurussia and Gondwana, led to two contrasting associations of mantle-derived magmatic rocks on the territory of the Bohemian Massif: (i) a 340–310 Ma lamprophyre-lamproite orogenic association and (ii) a 300–275 Ma lamprophyre association of anorogenic affinity. Major types of potassic mantle-derived magmatic rocks recognised in the orogenic and anorogenic associations include: (i) calc-alkaline to alkaline lamprophyres, (ii) alkaline “orthopyroxene minettes” (and geochemically related rocks), and (iii) peralkaline lamproites. These three types significantly differ with respect to mineral, whole-rock and Sr–Nd–Pb–Li isotope composition, and spatial distribution. The calc-alkaline lamprophyres occur throughout the entire Saxo-Thuringian and Moldanubian zones, whereas the different types of malte-derived potassic rocks are spatially restricted to particular zones. Rocks of the Carboniferous lamprophyre-lamproite orogenic association are characterised by variable negative εNd(i) and variably radiogenic Sr(i), whereas the rocks of the Permian lamprophyre association of anorogenic affinity are characterised by positive εNd(i) and relatively young depleted-mantle Nd-model ages reflecting increasing input from upwelling asthenospheric mantle. The small variation in the Pb isotopic composition of post-collisional potassic mantle-derived magmatic rocks (of both the orogenic and anorogenic series) implies that the Pb budget of the mantle beneath the Bohemian Massif is dominated by the same crust-derived material, which itself may include material derived from several sources. The source rocks of “orthopyroxene minettes” are characterised by isotopically light (“eclogitic”) Li and strongly radiogenic (crustal) Sr and may have been metasomatised by high-pressure fluids along the edge of a subduction zone. In contrast, the strongly Al2O3 and CaO depleted mantle source of the lamproites is characterised by isotopically heavy Li and high SiO2 and extreme K2O contents. This mantle source may have been metasomatised predominantly by melts. The mantle source of the lamprophyres may have undergone metasomatism by both fluids and melts.

Description: The Lu–Hf isotope system and Sr–Nd–Hf–Os isotope systematics of mantle rocks are capable of unravelling the early processes in collision belts, especially in a hot subduction context where the Sm–Nd and U–Pb systems in crustal rocks are prone to resetting owing to high temperatures and interaction with melts during exhumation. To improve models of the Devonian–Carboniferous evolution of the Bohemian Massif, we investigated in detail mafic and ultramafic rocks (eclogite, pyroxenite, and peridotite) from the ultrahigh-pressure and ultrahigh-temperature Kutná Hora Crystalline Complex (KHCC: Úhrov, Bečváry, Doubrava, and Spačice localities). Petrography, multiphase solid inclusions, major and trace element compositions of rocks and minerals, and radiogenic isotopic data document contrasting sources and protoliths as well as effects of subduction-related processes for these rocks. The Úhrov peridotite has a depleted composition corresponding to the suboceanic asthenospheric mantle, whereas Bečváry and Doubrava peridotites represent lithospheric mantle that underwent melt refertilization by basaltic and SiO2-undersaturated melts, respectively. Multiphase solid inclusions enclosed in garnet from Úhrov and Bečváry peridotites represent trapped H2O ± CO2-bearing metasomatizing agents and Fe–Ti-rich melts. The KHCC eclogites either formed by high-pressure crystal accumulation from mantle-derived basaltic melts (Úhrov) or represent a fragment of mid-ocean ridge basalt-like gabbroic cumulate (Spačice) and crustal-derived material (Doubrava) both metamorphosed at high P–T conditions. The Lu–Hf age of 395 ± 23 Ma obtained for the Úhrov peridotite reflects garnet growth related to burial of the asthenospheric mantle during subduction of the oceanic slab. By contrast, Spačice and Doubrava eclogites yield younger Lu–Hf ages of ∼350 and 330 Ma, respectively, representing mixed ages as demonstrated by the strong granulite-facies overprint and trace element zoning in garnet grains. We propose a refined model for the Early Variscan evolution of the Bohemian Massif starting with the subduction of the oceanic crust (Saxothuringian ocean) and associated oceanic asthenospheric mantle (Úhrov) beneath the Teplá–Barrandian at ≥380 Ma, which was responsible for melt refertilization of the associated mantle wedge (Bečváry, Doubrava). This was followed by continental subduction (∼370–360 Ma?) accompanied by the oceanic slab break-off and incorporation of the upwelling asthenospheric mantle into the Moldanubian lithospheric mantle and subsequent coeval exhumation of mantle and crustal rocks at ∼350–330 Ma.

Description: The Changning–Menglian orogenic belt (CMOB) in the southeastern Tibetan Plateau is an important link between the Longmu Co–Shuanghu suture (LCSS) in the northern Tibetan Plateau and the Chiang Mai–Inthanon and Bentong–Raub sutures in Thailand and Peninsular Malaysia. These belts and sutures are generally regarded as containing the remnants of the oceanic crust of the Palaeo-Tethys that formed by seafloor spreading as a result of the separation of Gondwana- and Eurasia-derived blocks during the Middle Cambrian. In this paper we report the first discovery of abundant unaltered and retrograde eclogites that occur as irregular lenses and blocks in metasedimentary rocks of the CMOB, and these eclogites form an elongate and almost north–south-trending high-pressure (HP)–ultrahigh-pressure (UHP) metamorphic belt that is ∼200 km long and ∼50 km wide. The newly discovered phengite/talc/epidote–glaucophane eclogites, lawsonite–talc–phengite eclogites, dolomite/magnesite–kyanite eclogites and phengite–kyanite-bearing retrograde eclogites have enriched (E-) and normal mid-ocean ridge basalt (N-MORB)-like affinities and mainly positive as well as some negative whole-rock εNd values (–4·34 to +7·89), which suggest an enriched and depleted oceanic lithosphere source for their protoliths. Magmatic zircons separated from the epidote–glaucophane, magnesite–kyanite and (phengite–kyanite-bearing) retrograde eclogites gave protolith ages of 317–250 Ma, which fit well within the time frame of the opening of the Palaeo-Tethys during the Middle Cambrian and its closure during the Triassic. Abundant metamorphic zircons in the eclogites indicate a Triassic metamorphic event related to the subduction of the Palaeo-Tethys oceanic crust from 235 to 227 Ma. Taking into account previous isotopic age data, we now establish the periods of Early–Middle Triassic (246–227 Ma) and Late Triassic (222–209 Ma) as the ages of subduction and exhumation of the Palaeo-Tethyan oceanic crust, respectively. Thermodynamic modelling revealed that the eclogites record distinct HP–UHP peak metamorphic conditions of 23·0–25·5 kbar and 582–610 °C for the phengite–glaucophane eclogites, 24·0–25·5 kbar and 570–586 °C for the talc–glaucophane eclogites, 29·0–31·0 kbar and 675–712 °C for the dolomite–kyanite eclogites, and 30·0–32·0 kbar and 717–754 °C for the magnesite–kyanite eclogites. These P–T estimates and geochronological data indicate that the Palaeo-Tethys oceanic slab was subducted to different mantle depths from 75 km down to 95 km, forming distinct types of eclogite with a variety of peak eclogite-facies mineral assemblages. The eclogites consistently record clockwise metamorphic P–T–t paths characterized by a heating–compression prograde loop under a low geothermal gradient of 5–10 °C km–1, indicating the rapid subduction of cold oceanic crust at a rate of 4·5–6·0 km Ma–1, followed by isothermal or cooling–decompressive retrogression and exhumation at an average rate of 3·2–4·2 km Ma–1. The newly discovered eclogites of the CMOB with their signatures of ocean-crust subduction are petrologically, geochemically and geochronologically comparable with those of the LCSS, providing powerful support for the idea that a nearly 2000 km long HP–UHP eclogite belt extends from the northern Tibetan Plateau to the southeastern Tibetan Plateau, and that it represents the main boundary suture of the Palaeo-Tethyan domain. These results have far-reaching implications for the tectonic framework and complex metamorphic evolution of the Palaeo-Tethyan domain.

Description: Replacive symplectites (vermicular intergrowths of two or more minerals) are an important feature of layered igneous intrusions, recording evidence of late-stage reactions between interstitial liquid and crystals. They are common throughout the Layered Series of the 564 Ma Sept Iles layered intrusion in Quebec, Canada, and fall into three types: oxy-symplectites, ‘Type I’ symplectites, and ‘Type II’ symplectites. Oxy-symplectites are comprised of magnetite and orthopyroxene, nucleate on olivine primocrysts, and form via the reaction Olivine + O2 → Orthopyroxene + Magnetite; Type I symplectites (of which there are 3 distinct categories) are comprised of anorthitic plagioclase with pyroxene, amphibole, or olivine vermicules, grow from primocryst oxide grains, and replace primary plagioclase; and Type II symplectites (of which there are 2 distinct categories) are comprised of anorthitic plagioclase with orthopyroxene ± amphibole vermicules, grow from primocryst olivine grains, and replace primocryst plagioclase. Rare symplectites composed of biotite and plagioclase are also present. Symplectite growth occurred at 700-1030 °C with pressure constraints of 1-2 kbar. We propose that Type I symplectites, and some Type II symplectites, formed from interaction of primocrysts with residual Fe-rich liquid as a consequence of differential loss of an immiscible Si-rich liquid conjugate from the crystal mush. However, redistribution and concentration of hydrous fluids in incompletely solidified rock, or an increase in water activity of the interstitial melt, may be more plausible processes responsible for the formation of replacive symplectites comprising abundant hydrous mineral assemblages.

Description: We present experimental data on the partitioning of Li, Be, B, K, Mg, Sr, Ga, Rb, Cs, Ba, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, Th, U, Hf, Zr, Nb and Ta between lawsonite and fluid, and zoisite and fluid at 3·0–3·5 GPa and 650–850°C. The aim is to provide data bearing on the trace element contents of fluids released during dehydration of subducting oceanic crust. Experimental trace element partition coefficients for lawsonite indicate a preference for the light rare earth elements (LREE) over the heavy REE (HREE) and for Be. These characteristics are consistent with the chemical composition of lawsonite in natural rocks. Experimental trace element partition coefficients for zoisite indicate a preference for HREE relative to LREE. This observation, consistent with earlier experimental data, is the reverse of the observed trace element compositions of natural zoisites, indicating the influence of other factors on the trace element contents of this phase. Lattice strain theory explains well the experimentally derived partitioning of divalent cations in the Ca-site between lawsonite and fluid. However, the weak relative fractionation of REE between lawsonite and fluid cannot be explained by lattice strain theory, as previously observed for zoisite–fluid REE partitioning. We combine our experimental data with thermodynamic models of mineral stability to model the compositions of fluids released during subduction of altered normal mid-ocean ridge basalt. The low La/Sm ratio associated with very high Ba/Th in arc magmas can be explained only if allanite is stable in the subducting oceanic crust. This suggests that the crustal fluid component involved in arc magma petrogenesis results from processes occurring in the warm, top part of the subducting slab. Decreasing lawsonite modal proportion with depth is associated with a large release of fluid characterized by low B/Be ratios that could explain the decreasing B/Be ratios in arc magmas with increasing distance from the trench. This implies that an important Be input in arc magma originates from the fluid generated during oceanic crust dehydration.

Description: To contribute to our understanding of the mechanisms and pathways of fluid movement through deeply subducted crust, we investigate high-pressure veins cutting eclogite-facies (~2·0 GPa and ~600°C) metagabbros of the Monviso Ophiolite, Italian Western Alps. The veins consist mainly of omphacite with minor garnet, rutile, talc and accessory zircon. Most of the vein minerals have major and trace element compositions that are comparable with the host-rock minerals, and vein and host-rock zircons have similar Hf isotopic compositions. These observations support the conclusions of previous studies that these veins largely formed from a locally sourced hydrous fluid during prograde or peak metamorphism. However, the bulk-rock Cr and Ni contents of the veins are significantly higher than those of the surrounding host eclogites. We also document distinct Cr-rich (up to weight per cent levels) zones in omphacite, garnet and rutile in some vein samples. Vein garnet and talc also have relatively high MgO and Ni contents. X-ray maps of vein garnet and rutile grains reveal complex internal zoning features, which are largely defined by micrometre-scale variations in Cr content. Some grains have concentric and oscillatory zoning in Cr, whereas others feature a chaotic fracture-like pattern. These Cr-rich zones are associated with high concentrations of Ni, B, As, Sb, Nb, Zr and high ratios of light rare earth elements (LREE) to middle REE (MREE) compared with low-Cr vein and host-rock minerals. Petrological and mass-balance constraints verify that the Cr-rich zones in the veins were not derived from internally sourced fluids, but represent precipitates from an external fluid. The external source that is consistent with the distinctive trace element characteristics of the vein components is antigorite serpentinite, which forms the structural basement of the high-pressure metagabbros. We propose at least two separate growth mechanisms for the Monviso veins. Most vein infillings were formed during progressive prograde metamorphism from locally derived fluid. Influx of the serpentinite-derived or other external fluid was transient and episodic and was probably achieved via brittle fractures, which preferentially formed along the pre-existing vein structures. The dehydration of serpentinite at high pressures in subduction zones may provide crucial volatiles and trace elements for arc magmas. Our results indicate that the movement of these fluids through subducted oceanic crust is likely to be highly channeled and transient so the progressive development of vein systems in mafic rocks may also be crucial for forming channelways for long-distance fluid flow at depth in subduction zones.

Description: Siwi caldera, in the Vanuatu arc (Tanna island), is a rare volcanic complex where both persistent eruptive activity (Yasur volcano) and rapid block resurgence (Yenkahe horst) can be investigated simultaneously during a post-caldera stage. Here we provide new constraints on the feeding system of this volcanic complex, based on a detailed study of the petrology, geochemistry and volatile content of Yasur–Siwi bulk-rocks and melt inclusions, combined with measurements of the chemical composition and mass fluxes of Yasur volcanic gases. Major and trace element analyses of Yasur–Siwi volcanic rocks, together with literature data for other volcanic centers, point to a single magmatic series and possibly long-lived feeding of Tanna volcanism by a homogeneous arc basalt. Olivine-hosted melt inclusions show that the parental basaltic magma, which produces basaltic-trachyandesites to trachyandesites by ~50–70% crystal fractionation, is moderately enriched in volatiles (~1 wt % H 2 O, 0·1 wt % S and 0·055 wt % Cl). The basaltic-trachyandesite magma, emplaced at between 4–5 km depth and the surface, preserves a high temperature (1107 ± 15°C) and constant H 2 O content (~1 wt %) until very shallow depths, where it degasses extensively and crystallizes. These conditions, maintained over the past 1400 years of Yasur activity, require early water loss during basalt differentiation, prevalent open-system degassing, and a relatively high heat flow (~10 9 W). Yasur volcano releases on average ≥ 13·4 x 10 3 tons d –1 of H 2 O and 680 tons d –1 of SO 2 , but moderate amounts of CO 2 (840 tons d –1 ), HCl (165 tons d –1 ), and HF (23 tons d –1 ). Combined with melt inclusion data, these gas outputs constrain a bulk magma degassing rate of ~5 x 10 7 m 3 a –1 , about a half of which is due to degassing of the basaltic-trachyandesite. We compute that 25 km 3 of this magma have degassed without erupting and have accumulated beneath Siwi caldera over the past 1000 years, which is one order of magnitude larger than the accumulated volume uplift of the Yenkahe resurgent block. Hence, basalt supply and gradual storage of unerupted degassed basaltic-trachyandesite could easily account for (or contribute to) the Yenkahe block resurgence.

Description: Primitive basalts are rarely found in arcs. The active NW Rota-1 volcano in the Mariana arc has erupted near-primitive lavas, which we have sampled with ROV Hyper-Dolphin (HPD). Samples from the summit (HPD480) and eastern flank (HPD488) include 17 magnesian basalts (51–52 wt % SiO 2 ) with 7·5–9·5 wt % MgO and Mg-number of 61–67, indicating little fractionation. Olivine phenocrysts are as magnesian as Fo 93 and contain 0·4 wt % NiO; the Cr/(Cr + Al) values of spinels are mostly 0·5–0·8, indicating equilibrium with depleted mantle. There are three petrographic groups, based on phenocryst populations: (1) cpx–olivine basalt (COB); (2) plagioclase–olivine basalt (POB); (3) porphyritic basalt. Zr/Y and Nb/Yb are higher in POB (3·1–3·2 and 1·2–1·5, respectively) than in COB (Zr/Y = 2·8–3·0 and Nb/Yb = 0·7–0·9), suggesting that POB formed from lower degrees of mantle melting, or that the COB mantle source was more depleted. On the other hand, COB have Ba/Nb (70–80) and Th/Nb (0·4–0·5) that are higher than for POB (Ba/Nb = 30–35 and Th/Nb = 0·1–0·2), and also have steeper light rare earth element (LREE)-enriched patterns. Moreover, COB have enriched 87 Sr/ 86 Sr and 143 Nd/ 144 Nd, and higher Pb isotope values, suggesting that COB has a greater subduction component than POB. 176 Hf/ 177 Hf between COB and POB are similar and Hf behavior in COB and POB is similar to that of Zr, Y and HREE, suggesting that Hf is not included in the subduction component, which produced the differences between COB and POB. The calculated primary basaltic magmas of NW Rota-1 volcano (primary COB and POB magmas) indicate segregation pressures of 2–1·5 GPa (equivalent to 65–50 km depth). These magmas formed by 24–18% melting of mantle peridotite having Mg-number ~89·5. Diapiric ascent of hydrous peridotite mixed heterogeneously with sediment melts may be responsible for the NW Rota-1 basalts. These two basalt magma types are similar to those found at Sumisu and Torishima volcanoes in the Izu–Bonin arc, with COB representing wetter and POB representing drier magmas, where subduction zone-derived melt components are coupled with the water contents.

Description: The origin of andesite is an important issue in petrology because andesite is the main eruptive product at convergent margins, corresponds to the average crustal composition and is often associated with major Cu–Au mineralization. In this study we present petrographic, mineralogical, geochemical and isotopic data for basaltic andesites of the latest Pleistocene Pilavo volcano, one of the most frontal volcanoes of the Ecuadorian Quaternary arc, situated upon thick (30–50 km) mafic crust composed of accreted Cretaceous oceanic plateau rocks and overlying mafic to intermediate Late Cretaceous–Late Tertiary magmatic arcs. The Pilavo rocks are basaltic andesites (54–57·5 wt % SiO 2 ) with a tholeiitic affinity as opposed to the typical calc-alkaline high-silica andesites and dacites (SiO 2 59–66 wt %) of other frontal arc volcanoes of Ecuador (e.g. Pichincha, Pululahua). They have much higher incompatible element contents (e.g. Sr 650–1350 ppm, Ba 650–1800 ppm, Zr 100–225 ppm, Th 5–25 ppm, La 15–65 ppm) and Th/La ratios (0·28–0·36) than Pichincha and Pululahua, and more primitive Sr ( 87 Sr/ 86 Sr ~0·7038–0·7039) and Nd ( Nd ~ +5·5 to +6·1) isotopic signatures. Pilavo andesites have geochemical affinities with modern and recent high-MgO andesites (e.g. low-silica adakites, Setouchi sanukites) and, especially, with Archean sanukitoids, for both of which incompatible element enrichments are believed to result from interactions of slab melts with peridotitic mantle. Petrographic, mineral chemistry, bulk-rock geochemical and isotopic data indicate that the Pilavo magmatic rocks have evolved through three main stages: (1) generation of a basaltic magma in the mantle wedge region by flux melting induced by slab-derived fluids (aqueous, supercritical or melts); (2) high-pressure differentiation of the basaltic melt (at the mantle–crust boundary or at lower crustal levels) through sustained fractionation of olivine and clinopyroxene, leading to hydrous, high-alumina basaltic andesite melts with a tholeiitic affinity, enriched in incompatible elements and strongly impoverished in Ni and Cr; (3) establishment of one or more mid-crustal magma storage reservoirs in which the magmas evolved through dominant amphibole and clinopyroxene (but no plagioclase) fractionation accompanied by assimilation of the modified plutonic roots of the arc and recharge by incoming batches of more primitive magma from depth. The latter process has resulted in strongly increasing incompatible element concentrations in the Pilavo basaltic andesites, coupled with slightly increasing crustal isotopic signatures and a shift towards a more calc-alkaline affinity. Our data show that, although ultimately originating from the slab, incompatible element abundances in arc andesites with primitive isotopic signatures can be significantly enhanced by intra-crustal processes within a thick juvenile mafic crust, thus providing an additional process for the generation of enriched andesites.

Description: Mid-ocean ridge basalts (MORB) from the Arctic Ocean have been significantly less studied than those from other oceans. The Arctic ridges (Gakkel Ridge and Lena Trough) are ultraslow-spreading ridges with low melt productivity and are thus the best locations to investigate mantle heterogeneity. We report the major and trace element and Sr–Nd–Pb–Hf isotope compositions of basalts generated along the Lena Trough and the westernmost part of the Gakkel Ridge in the Arctic Ocean. Basalts from the northern Lena Trough and westernmost Gakkel Ridge (NLT–WGR) have compositions close to normal MORB. The geochemical composition of the NLT–WGR lavas confirms a binary mixing model involving melts from a depleted MORB mantle source and a Spitsbergen amphibole-bearing subcontinental lithospheric mantle (SCLM) source. In contrast, in the central part of the Lena Trough (CLT), the basalts are alkalic with relatively high Mg-number (60–65), high SiO 2 (51·0–51·6 wt %), Al 2 O 3 (18·1–18·4 wt %), Na 2 O (4·0–4·2 wt %), K 2 O (1·0–1·6 wt %), K 2 O/TiO 2 (0·6–0·9) and (La/Sm) PM (1·4–1·8), and low FeO (6·5–6·8 wt %) contents. These basalts display isotope variations with 87 Sr/ 86 Sr ranging from 0·70361 to 0·70390, 143 Nd/ 144 Nd from 0·51283 to 0·51290 ( Nd + 3·7 to +5·2), 176 Hf/ 177 Hf from 0·28313 to 0·28322 ( Hf + 11·6 to +14·9) and 206 Pb/ 204 Pb from 17·752 to 17·884, 207 Pb/ 204 Pb from 15·410 to 15·423 and 208 Pb/ 204 Pb from 37·544 to 37·670. These isotope compositions clearly distinguish the CLT lavas from those generated along the Gakkel Ridge. For the CLT lavas, involvement of a phlogopite- or amphibole- and (possibly garnet)-bearing SCLM source component is proposed. Owing to SCLM contamination along the entire length of the Lena Trough, we classify the Lena Trough as an ocean–continent transition boundary. Magmatism similar to that observed in the Lena Trough would be expected to occur wherever ocean spreading initiates.

Description: The Western Alpine Sesia–Lanzo Zone (SLZ) is a sliver of eclogite-facies continental crust exhumed from mantle depths in the hanging wall of a subducted oceanic slab. Eclogite-facies felsic and basic rocks sampled across the internal SLZ show different degrees of retrograde metamorphic overprint associated with fluid influx. The weakly deformed samples preserve relict eclogite-facies mineral assemblages that show partial fluid-induced compositional re-equilibration along grain boundaries, brittle fractures and other fluid pathways. Multiple fluid influx stages are indicated by replacement of primary omphacite by phengite, albitic plagioclase and epidote as well as partial re-equilibration and/or overgrowths in phengite and sodic amphibole, producing characteristic step-like compositional zoning patterns. The observed textures, together with the map-scale distribution of the samples, suggest open-system, pervasive and reactive fluid flux across large rock volumes above the subducted slab. Thermodynamic modelling indicates a minimum amount of fluid of 0·1–0·5 wt % interacting with the wall-rocks. Phase relations and reaction textures indicate mobility of K, Ca, Fe and Mg, whereas Al is relatively immobile in these medium-temperature–high-pressure fluids. Furthermore, the thermodynamic models show that recycling of previously fractionated material, such as in the cores of garnet porphyroblasts, largely controls the compositional re-equilibration of the exhumed rock body.

Description: We consider two algorithms to approximate the solution Z of a class of stable operator Lyapunov equations of the form AZ + ZA * + BB * = 0. The algorithms utilize time snapshots of solutions of certain linear infinite-dimensional differential equations to construct the approximations. Matrix approximations of the operators A and B are not required and the algorithms are applicable as long as the rank of B is relatively small. The first algorithm produces an optimal low-rank approximate solution using proper orthogonal decomposition. The second algorithm approximates the product of the solution with a few vectors and can be implemented with a minimal amount of storage. Both algorithms are known for the matrix case. However, the extension of the algorithms to infinite dimensions appears to be new. We establish easily verifiable convergence theory and a priori error bounds for both algorithms and present numerical results for two model problems.

Description: We study several discontinuous Galerkin methods for solving the Signorini problem. A unified error analysis is provided for the methods. The error estimates are of optimal order for linear elements. A numerical example is reported to illustrate numerical convergence orders.

Description: This paper considers the stability of both continuous and discrete time-varying linear systems. Stability estimates are obtained in either case in terms of the Lipschitz constant for the governing matrices and the assumed uniform decay rate of the corresponding frozen time linear systems. The main techniques used in the analysis are comparison methods, scaling and the application of continuous stability estimates to the discrete case. Counterexamples are presented to show the necessity of the stability hypotheses. The discrete results are applied to derive sufficient conditions for the stability of a backward Euler approximation of a time-varying system and a one-leg linear multistep approximation of a scalar system.

Description: The classical error analysis for Nédélec edge interpolation requires the so-called regularity assumption on the elements. However, in Nicaise (2001, SIAM J. Numer. Anal. , 39 , 784–816) optimal error estimates were obtained for the lowest order case under the weaker hypothesis of the maximum angle condition. This assumption allows for anisotropic meshes that become useful, for example, for the approximation of solutions with edge singularities. In this paper we prove optimal error estimates for the edge interpolation of any order under the maximum angle condition. We also obtain sharp stability results for that interpolation on appropriate families of elements. mixed finite elements; edge elements; anisotropic finite elements.

Description: This paper deals with a fluid–solid interaction problem inspired by a biomechanical brain model. The problem consists of determining the response to prescribed static forces of an elastic solid containing a barotropic and inviscid fluid at rest. The solid is described by means of displacement variables, whereas displacement potential and pressure are used for the fluid. This approach leads to a well-posed symmetric mixed problem, which is discretized by standard Lagrangian finite elements of arbitrary order for all the variables. Optimal-order error estimates in the H 1 - and L 2 -norms are proved for this method. A residual a posteriori error estimator is also proposed, for which efficiency and reliability estimates are proved. Finally, some numerical tests are reported to assess the performance of the method and that of an adaptive scheme based on the error estimator.

Description: A numerical method of second order of accuracy for computing conditional Wiener integrals of smooth functionals of a general form is proposed. The method is based on the simulation of a Brownian bridge via the corresponding stochastic differential equations (SDEs) and on ideas of the weak-sense numerical integration of SDEs. A convergence theorem is proved. Special attention is paid to integral-type functionals. A generalization to the case of pinned diffusions is considered. Results of some numerical experiments are presented.

Description: In this paper we utilize affine biquadratic elements and a two-step temporal discretization to develop a finite volume element method for parabolic problems on quadrilateral meshes. The method is proved to have an optimal order convergence rate in L 2 (0, T ; H 1 ()) under the ‘asymptotically parallelogram’ mesh assumption. Numerical experiments that corroborate the theoretical analysis are also presented.

Description: An error analysis is given for a discretization of the Gross–Pitaevskii equation by Strang splitting in time and Hermite collocation in space. A second-order error bound in L 2 for the semidiscretization error of the Strang splitting in time is proven under suitable regularity assumptions on the exact solution. For the semidiscretization in space, high-order convergence is shown, depending on the regularity of the exact solution. The analyses of the semidiscretizations in time and space are finally combined into an error analysis of the fully discrete method.

Description: In this paper we consider the finite-volume-element method for general second-order quasilinear elliptic problems over a convex polygonal domain in the plane. Using reasonable assumptions, we show the existence and uniqueness of the finite-volume-element approximations. It is proved that the finite-volume-element approximations are convergent with , where r 〉 2, and in the H 1 -, W 1, - and L 2 -norms, respectively, for u W 2, r () and u W 2, () W 3, p (), where p 〉 1. Moreover, the optimal-order error estimates in the W 1, - and L 2 -norms and an estimate in the L -norm are derived under the assumption that u W 2, () H 3 (). Numerical experiments are presented to confirm the estimates.

Description: We present guaranteed , robust and computable a posteriori error estimates for nonconforming approximations of elliptic problems. Our analysis is based on a Helmholtz-type decomposition of the error expressed in terms of fluxes. Such a decomposition results in a gradient term and a divergence-free term, that are the exact solutions of two auxiliary problems. We suggest a new approach to deriving computable two-sided bounds of the norms of these solutions. The a posteriori estimates obtained in this paper differ from those that are based on projections of nonconforming approximations to a conforming space. Numerical experiments confirm that these new estimates provide very accurate error bounds, and can be efficiently exploited in practical computations.

Description: The numerical solution for a class of sub-diffusion equations involving a parameter in the range – 1 〈 α 〈 0 is studied. For the time discretization, we use an implicit finite-difference Crank–Nicolson method and show that the error is of order k 2+ α , where k denotes the maximum time step. A nonuniform time step is employed to compensate for the singular behaviour of the exact solution at t = 0. We also consider a fully discrete scheme obtained by applying linear finite elements in space to the proposed time-stepping scheme. We prove that the additional error is of order h 2 max(1, log k –1 ), where h is the parameter for the space mesh. Numerical experiments on some sample problems demonstrate our theoretical result.

Description: The inf–sup stability and optimal convergence of an isogeometric C 1 discretization for the Stokes problem are shown. In this discretization the velocities are the pushforward through the geometrical map of cubic C 1 non-uniform rational B-spline (NURBS) functions and the pressures are the pushforward of quadratic C 1 NURBS. This paper follows the work in Bazilevs et al. (2006, Math. Models Methods Appl. Sci. , 16 , 1031–1090) where the authors showed the numerical result of this discretization and proved the inf–sup stability for C 0 NURBS functions. The use of more regular functions is useful to decrease the degrees of freedom and thus the computational cost. The analysis is performed by means of the Verfürth trick, the macro-element technique, some approximation properties and the inf–sup condition for tensor products of B-spline spaces.

Description: The question in the title is answered using tools of potential theory. Convergence and divergence rates of interpolants of analytic functions on the unit interval are analysed. The starting point is a complex variable contour integral formula for the remainder in radial basis function (RBF) interpolation. We study a generalized Runge phenomenon and explore how the location of centres affects convergence. Special attention is given to Gaussian and inverse quadratic radial functions, but some of the results can be extended to other smooth basis functions. Among other things, we prove that, under mild conditions, inverse quadratic RBF interpolants of functions that are analytic inside the strip |Im ( z )| 〈 (1/2 ), where is the shape parameter, converge exponentially.

Description: We address in this paper the approximation of functions in an equilateral triangle by a linear combination of Laplace–Neumann eigenfunctions. The Laplace–Neumann basis exhibits a number of advantages. The approximations converge fairly fast and their speed of convergence can be much improved by using techniques familiar in Fourier analysis and spectral methods, in particular, the hyperbolic cross and polynomial subtraction. Moreover, expansion coefficients can be computed rapidly by a mixture of asymptotic methods and Birkhoff–Hermite quadratures.

Description: This paper analyses a globalized inexact smoothing Newton method for the numerical solution of optimal control problems subject to mixed control-state constraints. The method uses the smoothed Fischer–Burmeister function to reformulate first-order necessary conditions and aims at minimizing the squared residual norm using Newton steps and gradient-like steps. Numerical experiments are provided to illustrate the convergence results.

Description: Perhaps, the most astonishing idea in eigenvalue computation is Rutishauser's idea of applying the LR transform to a matrix for generating a sequence of similar matrices that become more and more triangular. The same idea is the foundation of the ubiquitous QR algorithm. It is well known that this idea originated in Rutishauser's qd algorithm, which precedes the LR algorithm and can be understood as applying LR to a tridiagonal matrix. But how did Rutishauser discover qd and when did he find the qd–LR connection? We checked some of the early sources and have come up with an explanation.

Description: In this article we prove the convergence of adaptive finite-element methods for Steklov eigenvalue problems under very general assumptions for simple as well as multiple eigenvalues starting from any initial triangulation. We also prove the optimality of the approximations, assuming Dörfler's strategy for marking when we consider simple eigenvalues.

Description: We propose, analyse and demonstrate a discontinuous Galerkin method for fractal conservation laws. Various stability estimates are established along with error estimates for regular solutions of linear equations. Moreover, in the nonlinear case and whenever piecewise constant elements are utilized, we prove a rate of convergence towards the unique entropy solution. We present numerical results for different types of solutions of linear and nonlinear fractal conservation laws.

Description: The Idaho batholith and spatially overlapping Challis intrusive province in the North American Cordillera have a history of magmatism spanning some 55 Myr. New isotopic data from the ~98 Ma to 54 Ma Idaho batholith and ~51 Ma to 43 Ma Challis intrusions, coupled with recent geochronological work, provide insights into the evolution of magmatism in the Idaho segment of the Cordillera. Nd and Hf isotopes show clear shifts towards more evolved compositions through the batholith's history and Pb isotopes define distinct fields correlative with the different age and compositionally defined suites of the batholith, whereas the Sr isotopic compositions of the various suites largely overlap. The subsequent Challis magmatism shows the full range of isotopic compositions seen in the batholith. These data suggest that the early suites of metaluminous magmatism (98–87 Ma) represent crust–mantle hybrids. Subsequent voluminous Atlanta peraluminous suite magmatism (83–67 Ma) results primarily from melting of different crustal components. This can be attributed to crustal thickening, resulting from either subduction processes or an outboard terrane collision. A later, smaller crustal melting episode, in the northern Idaho batholith, resulted in the Bitterroot peraluminous suite (66–54 Ma) and tapped different crustal sources. Subsequent Challis magmatism was derived from both crust and mantle sources and corresponds to extensional collapse of the over-thickened crust.

Description: In this paper we discuss a numerical solution of the delay differential equation where a , b != 0 are complex numbers, 0 〈 q 〈 1 is a real number and f is a complex-valued function satisfying as t -〉 for a suitable real scalar β . After a brief review of the equation’s basic stability and asymptotic properties, we analyse these characteristics for its -methods discretizations. Doing this, we consider meshes with a constant step size and a variable step size proposed by other authors for theoretical as well as computational reasons. Further, we discuss the consequences of our results for numerical investigations of the pantograph equation, especially with respect to a possible correspondence between the asymptotics of exact and numerical solutions. Some illustrating examples and calculations conclude the paper.

Description: This paper is concerned with the cyclic block coordinate descent method, also known as the nonlinear Gauss–Seidel (GS) method, where the solution of an optimization problem is achieved by partitioning the variables in blocks and successively minimizing with respect to each block. The properties of the objective function that guarantee the convergence of such alternating scheme have been widely investigated in the literature and it is well known that without suitable convexity hypotheses, the method may fail to locate the stationary points when more than two blocks of variables are employed. In this paper the general constrained nonconvex case is considered, presenting three contributions. First, a general method allowing an approximate solution of each block minimization subproblem is devised and the related convergence analysis is developed, showing that the proposed inexact method has the same convergence properties as the standard nonlinear GS method. Then a cyclic block gradient projection method is analysed, proving that it leads to stationary points for every number of blocks. Finally, the cyclic block gradient method is applied to large-scale problems arising from the non-negative matrix factorization approach. The results of a numerical experimentation on image recognition problems are also reported.

Description: This paper deals with a finite element approximation of the vibration modes of fluid–structure systems coupled on curved interfaces. It is based on a displacement formulation for both the fluid and the solid. Lowest order Raviart–Thomas elements are used for the fluid and standard continuous linear elements for the solid. Compatibility conditions are weakly imposed at a polygonal approximation of the fluid–solid interface by means of a Lagrange multiplier. Convergence, nonexistence of spurious or circulation nonzero frequency modes and optimal order error estimates for eigenfunctions/eigenvalues are proved. To do this, we use recent results about spectral approximations for noncompact operators to nonconforming hybrid finite element methods. The validity of a discrete compactness property for the discrete spaces considered here is also discussed.

Description: The Lavrentiev gap phenomenon is a well-known effect in the calculus of variations, related to singularities of minimizers. In its presence, conforming finite-element methods are incapable of reaching the energy minimum. By contrast, it is shown in this work that for convex variational problems the nonconforming Crouzeix–Raviart finite-element discretization always converges to the correct minimizer and that the discrete energy converges to the correct limit.

Description: We derive a variational characterization of the exact discrete Hamiltonian, which is a Type II generating function for the exact flow of a Hamiltonian system, by considering a Legendre transformation of Jacobi’s solution of the Hamilton–Jacobi equation. This provides an exact correspondence between continuous and discrete Hamiltonian mechanics, which arise from the continuous- and discrete-time Hamilton’s variational principle on phase space, respectively. The variational characterization of the exact discrete Hamiltonian naturally leads to a class of generalized Galerkin Hamiltonian variational integrators that includes the symplectic partitioned Runge–Kutta methods. This extends the framework of variational integrators to Hamiltonian systems with degenerate Hamiltonians, for which the standard theory of Lagrangian variational integrators cannot be applied. We also characterize the group invariance properties of discrete Hamiltonians that lead to a discrete Noether’s theorem.

Description: We propose a numerical method for computing transport and diffusion on a moving surface. The approach is based on a diffuse interface model in which a bulk diffusion–advection equation is solved on a layer of thickness containing the surface. The conserved quantity in the bulk domain is the concentration weighted by a density which vanishes on the boundary of the thin domain. Such a density arises naturally in double obstacle phase field models. The discrete equations are then formulated on a moving narrow band consisting of grid points on a fixed mesh. We show that the discrete equations are solvable subject to a natural constraint on the evolution of the discrete narrow band. Mass is conserved and the discrete solution satisfies stability bounds. Numerical experiments indicate that the method is second-order accurate in space.

Description: We prove a priori error estimates for a family of Eulerian–Lagrangian methods for time-dependent convection–diffusion equations with degenerate diffusion. The estimates depend only on certain Sobolev norms of the initial and right side data of the problem but not on the lower bound of the diffusion or any norms of the true solution. Thus these estimates hold uniformly with respect to the degenerate diffusion. On a general unstructured mesh, these estimates are suboptimal but sharp when the Courant number is less than unity and are optimal otherwise. We further prove an optimal-order error estimate and a superconvergence estimate for a special case of d -linear approximations on a d -dimensional rectangular domain with a uniform rectangular partition. We then use the interpolation of spaces and stability estimates to derive an estimate for problems with minimal or intermediate regularity, where the convergence rates are proportional to certain Besov norms of the initial and right side data. Numerical experiments are presented to confirm the theoretical results.

Description: We analyse the adaptive finite-element approximation to solutions of partial differential equations in variational formulation. Assuming well-posedness of the continuous problem and requiring only basic properties of the adaptive algorithm, we prove convergence of the sequence of discrete solutions to the true one. The proof is based on the ideas by Morin, Siebert and Veeser but replaces local efficiency of the estimator by a local density property of the adaptively generated finite-element spaces. As a result, estimators without a discrete lower bound are also included in our theory. The assumptions of the presented framework are fulfilled by a large class of important applications, estimators and adaptive strategies.

Description: Methods for the numerical evaluation of the Weber parabolic cylinder functions W ( a , ± x ), which are independent solutions of the inverted harmonic oscillator y '' + ( x 2 /4 – a ) y = 0, are described. The functions appear in the solution of many physical problems and notably in quantum mechanics. It is shown that the combined use of Maclaurin series, Chebyshev series, uniform asymptotic expansions for large a and/or x and the integration of the differential equation by local Taylor series are enough for computing the functions accurately in a wide rage of parameters. Differently from previous methods, the computational scheme is stable in the sense that high accuracy is retained: only two or three digits may be lost in double precision computations.

Description: Higham (2002, IMA J. Numer. Anal. , 22 , 329–343) considered two types of nearest correlation matrix problems, namely the W -weighted case and the H -weighted case. While the W -weighted case has since been well studied to make several Lagrangian dual-based efficient numerical methods available, the H -weighted case remains numerically challenging. The difficulty of extending those methods from the W -weighted case to the H -weighted case lies in the fact that an analytic formula for the metric projection onto the positive semidefinite cone under the H -weight, unlike the case under the W -weight, is not available. In this paper we introduce an augmented Lagrangian dual-based approach that avoids the explicit computation of the metric projection under the H -weight. This method solves a sequence of unconstrained convex optimization problems, each of which can be efficiently solved by an inexact semismooth Newton method combined with the conjugate gradient method. Numerical experiments demonstrate that the augmented Lagrangian dual approach is not only fast but also robust.

Description: This work is devoted to obtaining optimal error estimates (for the velocity and the pressure) for a first-order time-discrete splitting scheme (using decomposition of the viscosity) for solving the incompressible time-dependent Navier–Stokes equations in three-dimensional domains. This scheme has been previously studied by other authors (Blasco et al. 1997 Int. J. Numer. Methods Fluids , 28 , 1391–1419; Blasco & Codina, 2004, Appl. Numer. Math. , 51 , 1–17), but the main novelty of this paper is to establish optimal error estimates for the pressure. This behaviour has been numerically observed, but never hitherto proved. Moreover, owing to the introduction of a weight for the initial steps, these optimal error estimates are obtained without imposing either constraints on the time step or global compatibility conditions for the pressure at the initial time (related to further regularity hypotheses on the exact solution).

Description: An initial boundary-value problem for a semilinear reaction–diffusion equation is considered. Its diffusion parameter 2 is arbitrarily small, which induces initial and boundary layers. It is shown that the conventional implicit method might produce incorrect computed solutions on uniform meshes. Therefore we propose a stabilized method that yields a unique qualitatively correct solution on any mesh. Constructing discrete upper and lower solutions, we prove existence and investigate the accuracy of discrete solutions on layer-adapted meshes of Bakhvalov and Shishkin types. It is established that the two considered methods enjoy second-order convergence in space and first-order convergence in time (with, in the case of the Shishkin mesh, a logarithmic factor) in the maximum norm, if ≤ C ( N –1 + M –1/2 ), where N and M are the numbers of mesh intervals in the space and time directions, respectively. Numerical results are presented that support the theoretical conclusions.

Description: In this paper we study an accuracy-increasing postprocessing technique for full discretizations of nonlinear convection–diffusion problems. The spatial discretization is based on finite elements. Once the fully discrete standard Galerkin approximation is computed at any fixed time, a discrete stationary convection–diffusion problem with data based on the Galerkin approximation is solved. The semidiscrete in space case was first considered in de Frutos, García-Archilla and Novo (2009, IMA J. Numer. Anal. , 30, 2010, 1137–1158). A posteriori error estimates, based on this postprocessing technique, are obtained both for the semidiscrete in space and for the fully discrete cases. We prove that the semidiscrete error estimator proposed in this paper is efficient and asymptotically exact. In the fully discrete case the estimates have the property of giving a measure of the spatial errors that is independent of temporal errors. Some numerical experiments are provided.

Description: Regionally metamorphosed pelitic rocks at Campolungo, Central Alps, contain biotite, muscovite, garnet, staurolite, kyanite, and quartz, and the minor minerals tourmaline, plagioclase, chlorite, rutile, and ilmenite. Accessory allanite, apatite, monazite, potassium feldspar, xenotime, and zircon have also been identified. The bulk-rock chemical composition is similar to that of shales, and indicates that the protolith was deposited in an active continental margin setting. Element distribution maps, electron microprobe analyses and in situ UV–laser ablation inductively coupled plasma mass spectrometry data document a pronounced zoning in garnet and tourmaline porphyroblasts. Garnet displays a typical bell-shaped MnO zoning profile, with a maximum (~3 wt %) in the euhedral core. Cores are also rich in Y and heavy rare earth elements (HREE; e.g. 2150 ppm Y). In their broad rim, all garnet crystals display a subhedral annulus (10–15 µm wide), which is distinctly enriched in Ca, Sr, Y, and HREE, and which probably resulted from the breakdown of allanite (at ~550°C, ~6·4 kbar). Another characteristic feature of garnet rims is their sinusoidal chondrite-normalized REE pattern, which may represent partial equilibration with a light REE-enriched medium, probably generated through the breakdown of metamorphic allanite. Similar REE patterns are exhibited by a Ca-poor internal zone (inside the annulus), which may represent an earlier partial equilibration following the breakdown of detrital monazite. The large tourmaline crystals exhibit an optically visible three-stage zoning, which comprises: a euhedral core; a continuously zoned inner rim with a prominent euhedral Ca-rich annulus; and an outer rim, which also displays a distinct Ca-rich annulus and is separated from the inner rim by a sutured boundary. This boundary represents a marked chemical discontinuity, characterized for example by a decrease in the Zn concentration from 250 ppm (inner rim) to 20 ppm (outer rim). This change in Zn content reflects staurolite growth, which started after resorption of the inner rim of tourmaline and after a major deformation event. This chemical and textural discontinuity coincides with a marked shift in 18 O, which increases by ~0·8 across the inner rim–outer rim boundary. Our thermodynamic models suggest that resorption of the inner rim of tourmaline may be associated with small amounts (5–7 vol. %) of melt formed at ~650°C and 8·5 kbar. By using detailed textural observations, major and trace element zoning patterns and thermodynamic data, it was possible to model the metamorphic evolution of these rocks in considerable detail and, specifically, to correlate the growth and breakdown of major and accessory minerals.

Description: The well-preserved, 6 km thick mantle section of the Voykar ophiolite in the Polar Urals contains numerous dunite bodies as well as dunite and pyroxenite veins within the host harzburgites. These rocks provide evidence of a composite asthenosphere–lithosphere history of partial melting, plastic deformation, multi-stage melt migration and melt–rock interaction. We investigated the petrology and geochemistry of multiple samples of the different mantle lithologies to define the sequence of mantle melting and melt migration events, as well as the composition of the percolating melts. Spinel harzburgites sampled far from dunite bodies and pyroxenite veins have fairly homogeneous bulk-rock, olivine and Cr-spinel compositions and are interpreted as residues after 14–16% of partial melting, most probably at a mid-ocean ridge. Near the contacts with the dunite bodies and pyroxenite veins, spinel peridotites demonstrate distinct compositional changes marking different stages of melt migration in a supra-subduction environment. At the earliest stage, which probably took place in the lithosphere–asthenosphere boundary of the forearc mantle at temperature between 1050 and 1200°C and a pressure of 1–1·7 GPa, the dunite bodies formed as a result of stress-driven focused melt flow. The latest stage melts moved in cracks under a conductive cooling regime within the lithospheric mantle section when it was horizontally displaced towards the trench. The trace element composition of the melts that migrated through the mantle section during dunite formation have geochemical characteristics like those of high-Ca boninites. The role of the slab-derived component progressively increased through time and late-stage, pyroxenite-forming melts were conspicuously rich in SiO 2 and H 2 O. These low-viscosity melts impregnated the surrounding harzburgites, modifying or obliterating their primary composition.

Description: We consider a Clenshaw–Curtis–Filon-type method for highly oscillatory Bessel transforms. It is based on a special Hermite interpolation polynomial at the Clenshaw–Curtis points that can be efficiently evaluated using $$\hbox{ O }\left(NlogN\right)$$ operations, where N is the number of Clenshaw–Curtis points in the interval of integration. Moreover, we derive corresponding error bounds in terms of the frequency and the approximating polynomial. We then show that this method yields an efficient numerical approximation scheme for a class of Volterra integral equations containing highly oscillatory Bessel kernels (a problem for which standard numerical methods fail), and it also allows the study of the asymptotics of the solutions. Numerical examples are used to illustrate the efficiency and accuracy of the Clenshaw–Curtis–Filon-type method for approximating these highly oscillatory integrals and integral equations.

Description: We study the numerical approximation to the solution of the steady convection–diffusion equation. The diffusion term is discretized by using the hybrid mimetic method (HMM), which is the unified formulation for the hybrid finite-volume (FV) method, the mixed FV method and the mimetic finite-difference method recently proposed in Droniou et al. (2010, Math. Models Methods Appl. Sci. , 20 , 265–295). In such a setting we discuss several techniques to discretize the convection term that are mainly adapted from the literature on FV or FV schemes. For this family of schemes we provide a full proof of convergence under very general regularity conditions of the solution field and derive an error estimate when the scalar solution is in H 2 (). Finally, we compare the performance of these schemes on a set of test cases selected from the literature in order to document the accuracy of the numerical approximation in both diffusion- and convection-dominated regimes. Moreover, we numerically investigate the behaviour of these methods in the approximation of solutions with boundary layers or internal regions with strong gradients.

Description: In this paper we present a numerical scheme to solve nonlinear Maxwell’s equations based on backward Euler discretization in time and curl-conforming finite elements in space. The nonlinearity is due to a field-dependent conductivity in the form of a power law. The system under study is hyperbolic and due to the nonlinear conductivity it lacks strong estimates of the second time derivative. We are able to prove convergence of our numerical scheme based on boundedness of the second derivative in the dual space. Convergence of the nonlinear term is based on the Minty–Browder technique. We also present the error estimate for the fully discretized problem and support the theory by some numerical experiments.

Description: We derive and study a interior penalty method for a sixth-order elliptic equation on polygonal domains. The method uses the cubic Lagrange finite-element space, which is simple to implement and is readily available in commercial software. After introducing some notation and preliminary results, we provide a detailed derivation of the method. We then prove the well-posedness of the method as well as derive quasi-optimal error estimates in the energy norm. The proof is based on replacing Galerkin orthogonality with a posteriori analysis techniques. Using this approach, we are able to obtain a Cea-like lemma with minimal regularity assumptions on the solution. Numerical experiments are presented that support the theoretical findings.

Description: We consider the so-called Krylov–Gottlieb–Eckhoff (KGE) approximation of a function f with a discontinuity at a known point. This approximation is based on certain corrections associated with the jumps in the first q derivatives of f . The approximation of the exact jumps is accomplished by the solution of a system of linear equations. We show that, in the regions where the period-2 extension of the approximated function is smooth, the KGE method with approximate values of the jumps converges faster compared with the case where the exact values are used. We call this accelerated convergence the auto-correction phenomenon, which was discovered in the past by numerical experiments. The paper presents a theoretical explanation of the phenomenon with numerical illustrations.

Description: A commonly used algorithm for generating adaptive meshes for a given adaptation function in one dimension is due to de Boor. In its original form the algorithm produces a sequence of meshes upon using piecewise constant interpolation for the adaptation function on the current mesh and generating a new mesh that exactly equidistributes the interpolant. In this paper we present a proof for the existence of a limit mesh and for the convergence of de Boor's algorithm. Numerical results are given to illustrate the theoretical findings, and stopping criteria necessary for the implementation of the algorithm are examined.

Description: The numerical solution of the wave equation on three-dimensional domains is calculated using the convolution quadrature method for the time discretization and a Galerkin boundary element method for the space discretization. A computation-reduction strategy is developed whose parameters are given by an a priori error analysis. This gives a maximum for the number of discrete convolution matrices that must be computed when a particular time step is employed. Numerical examples are then presented to illustrate the predicted convergence results and the practicality of the methods.

Description: In this paper we analyse a mixed finite-element method and boundary-element method coupling for a time-dependent eddy current problem posed in the whole space and formulated in terms of the electric field Ee . The coupled problem is obtained by first proposing a mixed formulation of the interior problem in order to handle efficiently the divergence-free constraint satisfied by E in a dielectric material. Next we incorporate the far-field effect in the latter formulation through boundary integral equations defined on the coupling interface. We show that the resulting degenerate parabolic problem (with saddle point structure) is well-posed and use Nédélec edge elements and standard nodal finite elements to define a semidiscrete Galerkin scheme. Furthermore, we introduce the corresponding backward Euler fully discrete formulation and analyse the asymptotic behaviour of the error in terms of the discretization parameters for both schemes.

Description: We study the existence, uniqueness and regularity properties of solutions for the functional equation y ( t ) = b ( t ) y ( ( t )) + f ( t ), t [0, T ], where the delay function ( t ) vanishes at t = 0. Functional equations corresponding to the linear delay function ( t ) = qt (0 〈 q 〈 1) represent an important special case. We then analyse the optimal order of convergence of piecewise polynomial collocation approximations to solutions of these functional equations. The theoretical results are illustrated by extensive numerical examples.

Description: The linearized Cahn–Hilliard–Cook equation is discretized in the spatial variables by a standard finite-element method. Strong convergence estimates are proved under suitable assumptions on the covariance operator of the Wiener process, which is driving the equation. Backward Euler time stepping is also studied. The analysis is set in a framework based on analytic semigroups. The main effort is spent on proving detailed error bounds for the corresponding deterministic Cahn–Hilliard equation. The results should be interpreted as results on the approximation of the stochastic convolution, which is a part of the mild solution of the nonlinear Cahn–Hilliard–Cook equation.

Description: By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, their asymptotic expressions and bilinear generating functions, we obtain quadrature formulas for the integral transforms generated by the classical orthogonal polynomials. These integral transforms, related to the so-called Poisson integrals, correspond to a fractional Fourier transform in the case of Hermite polynomials, a Bessel transform in the case of Laguerre polynomials and an Appell transform in the case of Jacobi polynomials. We also give a nonclassical example in which a quadrature formula for an integral transform generated by Bateman–Pasternak polynomials is obtained. The kernel of this transform is related to a product of Gauss hypergeometric functions.

Description: We discuss the numerical approximation of the solution to the multi-configuration time-dependent Hartree–Fock (MCTDHF) equations in quantum dynamics. The MCTDHF method approximates the high-dimensional wave function of the time-dependent electronic Schrödinger equation by an antisymmetric linear combination of products of functions depending only on three-dimensional spatial coordinates. The equations of motion, obtained via the Dirac–Frenkel time-dependent variational principle, consist of a coupled system of three-dimensional nonlinear partial differential equations and ordinary differential equations. We investigate the convergence properties of a time integrator based on a splitting of the Hamiltonian directly in the variational principle. First-order convergence in the H 1 -Sobolev and second-order convergence in the L 2 -norm are established under a solution regularity of H 2 . As a pre-requisite, we show that the MCTDHF equations have a solution in this Sobolev space if the initial data have such regularity.

Description: In this paper we are interested in finding approximate solutions to a Volterra integral equation of the first kind with a highly oscillatory Bessel kernel. We begin our analysis by showing the relationship between the integral of a highly oscillatory Bessel function and the order of the integrand Bessel function. We then establish the asymptotic expansion of the exact solution of the Volterra integral equation in inverse powers of the frequency and develop a Filon-type method to provide approximations. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

Description: We prove a posteriori error estimates of optimal order in the L ( L 2 )-norm for time-splitting spectral methods applied to the linear Schrödinger equation in the semiclassical regime. The a posteriori error estimates are obtained by considering an appropriate extension in time of the numerical schemes and using energy techniques. Numerical experiments are presented that confirm our theoretical results.

Description: The Isle au Haut Igneous Complex provides a unique opportunity to examine in detail the in situ physical and chemical interactions between contemporaneously emplaced mafic and silicic magmas. The complex contains a 600 m thick sequence of 11 alternating layers of gabbro and diorite (typically 15–40 m thick). Purely on the basis of density contrasts (2·65 g cm – 3 gabbro vs 2·55 g cm – 3 diorite), the entire system should have undergone wholesale instability and mixing; it is instead arrested in a grossly unstable state of interaction while molten. Chilled margins along the lower contacts of the gabbros and structural integrity of the diorite layers indicate that near-liquidus gabbroic magma invaded partly crystalline, cooler diorite. Mineral assemblages, chemical analyses, and phase equilibria calculations indicate initial temperatures during emplacement of ~1180°C (gabbro) and ~1000°C (diorite). Conductive thermal models yield solidification timescales of 15–60 years for single gabbro layers and about a thousand years for the entire complex. There is ample evidence for two phases of small-scale interfacial Rayleigh–Taylor type instabilities of dioritic melt into the gabbros. Phase I occurred immediately upon gabbro emplacement whereas evenly spaced, slender more silicic pipes represent a much later stage (Phase II). Pipe geometry and spacing, estimated viscosities of the gabbroic magma and silicic melt, and the sudden increase in silica near the upper contact of the diorite, all indicate a thin (~18–53 cm) buoyant layer at the upper contact of the diorite as the source of the pipes. Compaction of the diorite produced this layer over a period of about 10 years. Simultaneous solidification along the lower contact of the overlying gabbro, thickening inwards, increased viscosity enough to arrest pipe ascent after a few meters. Crystal size distribution analyses of the gabbro layers yield crystal growth rates [ G o = (2 – 4) x 10 – 10 cm s – 1 ] and nucleation rates ( J o = 10 –5 –10 –6 cm –3 s –1 ) indicative of conductive cooling coupled with some sluggish convective stirring owing to collapse of the roof-ward gabbro solidification fronts. Were the complex larger, with a much longer solidification time, all this evidence would have been lost, thus suggesting that in larger systems similar processes may commonly take place leaving little direct evidence of their operation apart from the ultimate final petrological product.

Description: In this paper we obtain new results on Filon-type methods for computing oscillatory integrals of the form $${\displaystyle {\int }_{-1}^{1}}f\left(s\right)exp\left(\hbox{ i }ks\right)\hbox{ d }s$$ . We use a Filon approach based on interpolating f at the classical Clenshaw–Curtis points $$cos(j\pi /N),j=0,\dots ,N$$ . The rule may be implemented in $$O\left(NlogN\right)$$ operations. We prove error estimates that show explicitly how the error depends both on the parameters k and N and on the Sobolev regularityof f . In particular we identify the regularity of f required to ensure the maximum rate of decay of the error as k -〉 . We also describe a method for implementing the method and prove its stability both when N ≤ k and N 〉 k . Numerical experiments illustrate both the stability of the algorithm and the sharpness of the error estimates.

Description: In this paper, based on a line search technique proposed by Solodov and Svaiter (1998, Reformulation: Nonsmooth, Piecewise Smooth, Semismooth, and Smoothing Methods (M. Fukushima & L. Qi eds). Dordrecht: Kluwer, pp. 355–369), we propose a class of derivative-free methods for solving nonlinear monotone equations. These methods can be regarded as an extension of the spectral gradient method and some recently developed modified conjugate gradient methods for solving unconstrained optimization problems. Due to their lower storage requirement, these methods can be applied to solve large-scale nonlinear equations. We obtain global convergence of our methods without requiring differentiability, provided that the equation is Lipschitz continuous. Moreover, the whole sequence generated by the method converges to a solution of the equation even if the solution set is not a singleton. Preliminary numerical results show that the proposed methods are efficient.

Description: For the steady-state solution of a differential equation from a one-dimensional multistate model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B – – XF – – F + X + XB + X = 0, where F ± ( I – F ) D ± and B ± BD ± with positive diagonal matrices D ± and possibly low-ranked matrices F and B . We prove the existence of the minimal positive solution X * under a set of physically reasonable assumptions and study its numerical computation by fixed-point iteration, Newton’s method and the doubling algorithm. We shall also study several special cases. For example when B and F are low ranked then $${X}^{*}=\Gamma \circ \left({\displaystyle {\sum }_{i=1}^{4}{U}_{i}}{V}_{i}^{\hbox{ T }}\right)$$ with low-ranked U i and V i that can be computed using more efficient iterative processes. Numerical examples will be given to illustrate our theoretical results.

Description: Prior to the recent development of symplectic integrators, the time-stepping operator $${\hbox{ e }}^{h(A+B)}$$ was routinely decomposed into a sum of products of $${\hbox{ e }}^{hA}$$ and $${\hbox{ e }}^{hB}$$ in the study of hyperbolic partial differential equations. In the context of solving Hamiltonian dynamics, we show that such a decomposition gives rise to both even - and odd -order Runge–Kutta and Nyström integrators. By the use of Suzuki’sforward-time derivative operator to enforce the time-ordered exponential, we show that the same decomposition can be used to solve nonautonomous equations. In particular, odd-order algorithms are derived on the basis of a highly nontrivial time-asymmetric kernel. Such an operator approach provides a general and unified basis for understanding structure nonpreserving algorithms and is especially useful in deriving very high-order algorithms via analytical extrapolations. In this work algorithms up to 100th order are tested by integrating the ground-state wave function of the hydrogen atom. For such a singular Coulomb problem, the multi-product expansion shows uniform convergence and is free of poles usually associated with structure-preserving methods. Other examples are also discussed.

Description: We propose a new finite volume numerical scheme for the approximation of regularized mean curvature flow level set equations, which ensures the maximum principle, and which is shown to converge to the solution of the problem. The convergence proof uses the monotonicity of the operator in order to get the strong convergence of the approximation of the gradient. Numerical examples provide indications about the accuracy of the method. Applications to noisy image filtering show less diffusive behaviour than the classical finite difference scheme.

Description: We derive a robust a posteriori error estimate for hp -adaptive discontinuous Galerkin discretizations of stationary convection–diffusion equations. We consider one-irregular meshes consisting of parallelograms. The estimate yields global upper and lower bounds of the errors measured in terms of the natural energy norm associated with the diffusion and a seminorm associated with the convection. The ratio of the constants in the upper and lower bounds is independent of the local mesh sizes and weakly dependent on the local polynomial degrees. Moreover, it is also independent of the magnitude of the Péclet number of the problem, and hence the estimate is fully robust for convection-dominated problems. We apply our estimator as an error indicator in an hp -adaptive refinement algorithm and illustrate its practical performance in a series of numerical examples.

Description: This paper is concerned with solving numerically the Dirichlet boundary value problem for Laplace's equation in a nonlocally perturbed half-plane. This problem arises in the simulation of classical unsteady water wave problems. The starting point for the numerical scheme is the boundary integral equation reformulation of this problem as an integral equation of the second kind on the real line in Preston et al. (2008, J. Int. Equ. Appl. , 20 , 121–152). We present a Nyström method for numerical solution of this integral equation and show stability and convergence, and we present and analyse a numerical scheme for computing the Dirichlet-to-Neumann map, i.e., for deducing the instantaneous fluid surface velocity from the velocity potential on the surface, a key computational step in unsteady water wave simulations. In particular, we show that our numerical schemes are superalgebraically convergent if the fluid surface is infinitely smooth. The theoretical results are illustrated by numerical experiments.

Description: We consider the time discretization of the equation where A is a self-adjoint densely defined linear operator on a Hilbert space H with a complete eigen-system and β ( t ) is completely monotonic and locally integrable, but not constant. The equation is discretized in time using second-order difference in combination with second order convolution quadrature for the memory term. The stability properties of the discretization in time are derived in the l t 1 (0, ; H ) l t (0, ; H ) norm.

Description: Modified Fourier expansion is a powerful means for the approximation of nonperiodic smooth functions in a univariate or multivariate setting. In the current paper we consider further enhancement of this approach by two techniques familiar from conventional Fourier analysis: the polynomial subtraction and the hyperbolic cross. We demonstrate that, judiciously subtracting simpler functions dependent on linear combinations of derivatives along boundaries, it is possible to accelerate convergence a great deal and this procedure is considerably more efficient than in the case of conventional Fourier expansion. Moreover, examining the pattern of decay of coefficients in a multivariate setting, we demonstrate that most of them can be disregarded without any ill effect on the quality of approximation.

Description: We consider block LDL T factorization for symmetric indefinite matrices in the form LDL T , where L is unit lower triangular and D is block diagonal with each diagonal block having dimension 1 or 2. The stability of this factorization and its application to solving symmetric indefinite linear systems has been well studied. On the other hand, while all rounding error analysis of block LDL T factorization in the literature relies on the outer product form, this paper gives a novel componentwise backward error analysis based on the inner product form. The new results include a condition under which block LDL T factorization in inexact arithmetic is guaranteed to preserve the inertia and a reliability analysis of rank estimation and inertia estimation of symmetric indefinite matrices by block LDL T factorization.

Description: Prior to the ad 1902 Plinian eruption of 8 km 3 of dacite and subsequent growth of the 〉1 km 3 Santiaguito dacite dome complex, Santa María volcano grew into an 8 km 3 composite cone over ~75 kyr in four phases (at 103–72, 72, 60–46, and 35–25 ka). The 1902 eruption occurred after an ~25 kyr period of repose in growth of the composite cone. To provide context for processes that ultimately led to the 1902 eruption, we present geochemical and isotopic (Sr, Nd, Pb, U-series) data from lavas of the composite cone for which ages are constrained by 40 Ar/ 39 Ar dating. The four cone-building phases comprise basaltic to basaltic-andesite lava (51·4–56·1% SiO 2 ) whose major- and trace-element compositions suggest that crystallization was important in differentiation. Relative to other Central American arc volcanoes, these lavas also have large 238 U excesses and high 207 Pb/ 204 Pb ratios that imply melting of a mantle wedge modified to an unusual extent by fluid from subducted crust and sediment of the Cocos plate. Major- and trace-element and isotopic variations over time imply that mafic recharge and magma mixing were prevalent during early phases of cone-building, whereas assimilation processes were more dominant during the latest stage of cone growth. Indeed, some early erupted basalts have lower 143 Nd/ 144 Nd and higher 87 Sr/ 86 Sr ratios than more SiO 2 -rich basaltic andesites that erupted during the final phase of cone-building. These features point to an assimilant that is not typical continental crust and instead may be more like mid-ocean ridge basalt with respect to major- and trace-element composition and Sr, Nd, Pb, and U–Th isotope ratios. Energy-constrained modeling of a parental basalt that undergoes crystal fractionation, assimilation and periodic recharge with basalt in the lower crust can reproduce lava compositions erupted during phases I–III and the early part of phase IV. Modeling further indicates that assimilation within the lower crust of partially melted garnet-amphibolite metabasalt, without basaltic recharge, may produce the youngest cone-forming lavas in phase IV. These models link the 8 km 3 of cone growth over 75 kyr to the mass flux of magma into the crust. Our findings suggest an along-arc magma flux into the lower crust beneath Santa María of 〉20 km 3 km – 1 Myr – 1 , which is higher than anticipated in recent numerical–thermal approaches to basalt–crust interaction. Consequently, the thermal incubation period needed to produce hybrid basaltic-andesite magma may be only a few tens of thousand years.

Description: More than 1·5 million people live in or near the Phlegrean Volcanic District (PVD) in southern Italy, which represents one of the most carefully studied volcanic hazard areas in the world. Throughout its history, the style of volcanic activity has varied greatly, from relatively quiescent lava flows to explosive phreatomagmatic eruptions. The goal of this study is to develop a more detailed understanding of the physical and chemical processes associated with the Solchiaro eruption in the PVD. The PVD includes three volcanic fields: the Campi Flegrei (CF) caldera and the volcanic islands of Ischia and Procida. The Solchiaro eruption on the island of Procida is one of the few primitive (less evolved) eruptions in the PVD and can provide information on the source of the more evolved magmas associated with this volcanic system. One of the more important chemical parameters that determine the style of volcanic eruptions is the volatile budget of the magma before and during eruption. Melt inclusions (MI) provide the most direct information on the volatile contents of the pre-eruptive melt in the source region for the PVD. The composition of the melt phase before eruption was determined by analyzing the major, minor and trace element and volatile contents of 109 MI in olivine from four samples of the Solchiaro eruption, representing different stratigraphic heights in the deposits and, therefore, different relative times of eruption. Olivine compositions vary from Fo 82 to Fo 88 , with one maximum value of Fo 90 . The compositions of the MI in olivine were corrected for post-entrapment crystallization (PEC) and for Fe loss by diffusion. Most (97 out of 109) of the MI studied are classified as ‘normal’ MI because they show chemical evolution trends consistent with that of bulk-rocks from the PVD. Two types of anomalous MI were also recognized based on their major and trace element compositions: (1) Sr-rich MI, and (2) enriched MI that are variably enriched in TiO 2 , K 2 O, P 2 O 5 , large ion lithophile elements, high field strength elements and rare earth elements relative to ‘normal’ MI. These MI probably originated from dissolution–reaction–mixing processes in the mush zone of the magma body. ‘Normal’ MI include both bubble-bearing and bubble-free (containing only glass ± trapped chromite) types. Bubble-free MI most closely record the pre-eruptive volatile content of the melt over a range of temporal and spatial conditions. The observed trends in CO 2 contents of MI versus crystallization indicators (e.g. Al 2 O 3 /CaO) support the interpretation that variations in the volatile contents of bubble-free MI reflect real variations in the volatile budget of the melt during the evolution of the magma. The correlation between CO 2 contents of MI and the relative stratigraphic position of each sample is consistent with eruption of a volatile-saturated magma that initially ascended through the crust from an original depth of at least 8 km. The magma ponded at 4–2 km depth prior to eruption and crystallization and the concomitant volatile exsolution from the saturated melt in the shallow chamber triggered the Solchiaro eruption. As the eruption proceeded, the Solchiaro magma continued to ascend through the crust to a final storage depth of about 1 km.

Description: Melt inclusions (MI) represent the best source of information concerning the pre-eruptive volatile contents of magmas. If the trapped melt is enriched in volatile species, following trapping the MI may generate a vapor bubble containing volatiles that have exsolved from the melt. Thermodynamic modeling of vapor-saturated albitic composition (NaAlSi 3 O 8 ) MI shows that the CO 2 content of the melt phase in the MI is sensitive to small amounts of post-entrapment crystallization (PEC), whereas the H 2 O content of the melt is less sensitive to PEC. During PEC, CO 2 is transferred from the melt to the vapor phase and the vapor bubble may contain a significant amount, if not most, of the CO 2 in the MI. The contrasting behaviors of H 2 O and CO 2 during PEC lead to H 2 O–CO 2 trends that are similar to those predicted for open-system degassing during magma ascent and decompression. Thus, similar H 2 O–CO 2 trends may be produced if (1) vapor-saturated MI are trapped at various depths along a magmatic ascent path, or (2) MI having the same volatile content are all trapped at the same depth, but undergo different amounts of PEC following trapping. It is not possible to distinguish between these two contrasting interpretations based on MI volatile data alone. However, by examining the volatile trends within the context of other geochemical monitors of crystallization or magma evolution progress, it may be possible to determine whether the volatile trends were generated along a degassing path or if they reflect various amounts of PEC in an originally homogeneous melt inclusion assemblage. The volatile trends resulting from PEC of MI described in this study are directly applicable to silica-rich (granitic) MI trapped in non-ferromagnesian host phases, and are only qualitatively applicable to more mafic melt compositions and/or host phases owing to modifications resulting from Fe exchange with the host and to post-entrapment re-equilibration processes.

Description: The presence of volatile species in the Earth's upper mantle drives the formation of low-degree melts at pressures and temperatures at which volatile-free mantle rocks would be subsolidus. The two most abundant volatile species, given the oxidation state of the Earth's upper mantle, are carbon dioxide and water; each species has a distinct effect on the melting process. We present experimental melting results from 3 GPa and 1375°C on hydrous systems with controlled water contents and rigorously minimized carbon contamination that constrain the independent effects of these volatiles. The hydrous melts in these experiments are in equilibrium with garnet peridotite at pressures reasonable for hydrous melting under mid-ocean ridges. Compared with anhydrous experiments or carbon-rich silicate melting, the addition of water produces a melt with increased SiO 2 content relative to MgO and FeO, tantamount to an increase in the stability of olivine at the solidus relative to the other crystalline phases. We also report a substantial and unexpected change in the composition of clinopyroxene in equilibrium with the melt; the clinopyroxene stability field contracts when water is added to the system, producing clinopyroxenes with higher CaO and lower Al 2 O 3 than found at the same pressure without water. The contraction of the clinopyroxene field decreases the bulk partition coefficients of TiO 2 , Na 2 O, heavy rare earth elements, U, and H 2 O, with important implications for hydrous melting of the mantle; for example, initiating hydrous melting deeper in the garnet lherzolite stability field.

Description: The Taupo Volcanic Zone (TVZ), located in the North Island of New Zealand, represents part of a magmatic arc that is at present undergoing active extension. Around 0·9 Myr ago, an acceleration in rifting was followed by a progressive transition in the composition of volcanic products (until ~0·7 Ma) from typical arc-type andesite into overwhelmingly large, caldera-forming rhyolitic eruptions with subordinate basalt and dacite in the Central TVZ. Despite an obvious compositional gap in the erupted products in the Central TVZ within the last 0·7 Myr (little to no erupted products with SiO 2 contents between 55 and 65 wt %), phenocryst minerals (plagioclase, amphibole, pyroxene) show an uninterrupted compositional record that suggests crystallization from a continuum of melt compositions. Coupled with radiogenic isotope evidence, the whole-rock and mineral chemistry data are consistent with magmatic differentiation controlled by crystal fractionation of primary mantle-derived magmas accompanied by some assimilation of local wall-rocks. In the Southern TVZ and in the early part of the Central TVZ, magmatic differentiation was dominated by the lower crustal evolution of relatively dry (~1 wt % H 2 O) arc basalts, crystallizing a pyroxene–plagioclase-dominated assemblage. However, the conditions of crystallization in the lower crust appear to have changed within the last million years in the Central TVZ, with amphibole and oxides appearing earlier in the crystallization sequence. In this framework and using numerical simulations coupling crystallization kinetics and multiphase fluid dynamics of magma reservoirs, we show that melts extracted from crystal mushes within an optimal ‘extraction window’ (~50 and 80 vol. % crystals) match those erupted at the surface. Lower crustal mushes fed by basalt with 1 wt % H 2 O (dominated by a pyroxene–plagioclase assemblage) release andesitic melts at the extraction window. These melts then erupt at the surface to form the observed andesitic part of the arc. With a slightly higher water content (~2 wt %) in the basalt, the melt composition at the extraction window from lower crustal mushes is dacitic rather than andesitic. Although some dacitic melts will reach the surface, most will be trapped in the upper crust and crystallize to form a silicic mush. Extraction of the interstitial liquid after 〉50% crystallization from this upper crustal reservoir produces the large volumes of rhyolitic magma erupted over the past 0·7 Myr (〉4000 km 3 from ignimbrite-forming eruptions).

Description: The Miocene Ksieginki nephelinite in SW Poland is part of the Central European Volcanic Province and occurs at the NE termination of the Cenozoic Ohre (Eger) Rift in the NW part of the Bohemian Massif. It contains numerous pyroxenite and peridotite xenoliths and clinopyroxene megacrysts. Pyroxenites (mostly olivine clinopyroxenites with no primary orthopyroxene) commonly have cumulate textures. Wehrlite and websterite xenoliths, texturally and chemically related to the pyroxenite suite, contain volcanic-textured intercumulus material similar to that of the host nephelinite. The Mg/(Mg + Fe) ratios of clinopyroxenes from the pyroxenite suite are 0·81–0·93; these are low-calcium]Ca 0·72–0·82 atoms per formula unit (a.p.f.u.)] and contain 0·22–0·37 a.p.f.u. Al. Clinopyroxene rare earth element (REE) patterns are slightly enriched in middle REE (MREE) and light REE (LREE) relative to heavy REE (HREE) (Sm N /Yb N = 2·4–4·3, La N /Yb N = 1·7–3·0) and are deflected downwards from Pr/Nd to La. The whole-rock REE patterns of the pyroxenite xenoliths are identical and show slight enrichment in LREE relative to HREE (La N /Yb n = 3–5). The wehrlite cumulates are more LREE-enriched (La N /Yb n = 13) and their REE patterns are identical to those of the host nephelinite. The peridotite xenoliths exhibit a range of textures from protogranular to mylonitic. Typically the peridotites (mostly harzburgites) consist of domains of olivine I (Fo 88·9–91·9 ) and orthopyroxene I plus interstitial, texturally later clinopyroxene I [both with Mg/(Mg + Fe) = 0·89–0·93] and rare spinel [Cr/(Cr + Al) = 0·35–0·62]. The REE patterns of clinopyroxene I are similar to those of the pyroxenite suite. Fine grains of olivine II, commonly with forsterite contents identical to those of olivine I, but enriched in Ca, occur at the contacts between olivine I grains and in interstices or pools with glass. Clinopyroxene I commonly has spongy rims of clinopyroxene II, which are richer in Mg and poorer in Na. The peridotite xenoliths contain sparse fine-grained micrometre- to millimetre-scale aggregates consisting of (1) spinel, olivine II, clinopyroxene III and altered glass ± titanian biotite or (2) plagioclase, olivine II and clinopyroxene III ± glass. Olivine I in xenoliths containing these aggregates is typically Fo 85·6–89·4 , and the Mg/(Mg + Fe) ratio of pyroxenes is similar to that of olivine. The lithospheric peridotitic mantle beneath Ksieginki was infiltrated by alkaline silicate melts similar to the nephelinite lava prior to and during the volcanism. Pyroxenite cumulates crystallized from the melt in places where the flow was channelized. Locally melt infiltration was pervasive, leading to ‘Fe-metasomatism’ of the peridotites, which lowered the Fo content of the olivine to c. 86 %. The metasomatism was cryptic, and only the clinopyroxene trace element content was modified in peridotites not affected by pervasive melt infiltration or not in contact with pyroxenites. The mantle section affected by the metasomatism was located at a depth of ~35–50 km and was thermally equilibrated (temperatures of 1060–1120°C are recorded in both the pyroxenite and peridotite suites). The fine-grained aggregates are interpreted as remnants of the interstitial melt frozen after entrainment of the xenoliths during the eruption of the host nephelenite; they show that some of the peridotites resided at pressures allowing plagioclase crystallization, whereas the others were brought to the surface directly from spinel peridotite facies depths. The Ksieginki xenoliths provide a complete snapshot of the various effects of alkaline melt migrating through the lithospheric mantle during the initial stages of continental rifting.

Description: The temporal evolution of erupted magma compositions at Paricutin Volcano (Mexico) is often cited as a classic example of assimilation–fractional crystallization processes with significant progressive changes in major element, trace element, and isotopic compositions occurring over the relatively short 9 year lifespan of the volcano. In this study, major and trace element compositions of olivine- and orthopyroxene-hosted melt inclusions are integrated with new trace element analyses of the erupted lavas and data for entrained xenoliths and xenolith glasses to provide a more comprehensive evaluation of the evolution of Paricutin Volcano that questions this view. Melt inclusion compositions are bimodal with an undegassed, low-Si population (Type I) similar in composition to the whole-rock samples and a degassed, high-Si population (Type II) recording late-stage degassing and crystallization of the magma. Despite the rapid changes in lava composition, melt inclusions hosted in both olivine and orthopyroxene do not record any progressive contamination or mixing of magmas. Homogeneity of Type I melt inclusions within single lava samples implies significant contamination prior to crystallization and potentially a decoupling of assimilation–fractional crystallization processes. Pre-existing models of magma evolution at Paricutin Volcano are not consistent with the melt inclusion results or new trace element whole-rock data. Whole-rock and melt inclusion trace element analyses corroborate previous studies, which have suggested that the early erupted material (Phase 1; February–July 1943) was of a compositionally distinct magma compared with the bulk of the erupted material during Phase 2 (July 1943–1946). There is a second compositional transition between the Phase 2 and Phase 3 (1947–1952) lavas, marked by a sudden change in Zr/Nb despite similar MgO values, that is consistent with the arrival of a new magma batch. This transition occurs prior to the major compositional change from basaltic andesite to andesite magmas in the waning stages of the eruption that is consistent with progressive crustal assimilation within this latest magma batch. These data demonstrate that the petrogenetic evolution of magmas at Paricutin is more complex than simple progressive assimilation and fractional crystallization and requires the presence of three compositionally distinct magma batches at shallow levels.

Description: The Cretaceous Mont Saint-Hilaire complex (Quebec, Canada) comprises three major rock units that were emplaced in the following sequence: (I) gabbros; (II) diorites; (III) diverse partly agpaitic foid syenites. The major element compositions of the rock-forming minerals, age-corrected Nd and oxygen isotope data for mineral separates and trace element data of Fe–Mg silicates from the various lithologies imply a common source for all units. The distribution of the rare earth elements in clinopyroxene from the gabbros indicates an ocean island basalt type composition for the parental magma. Gabbros record temperatures of 1200 to 800°C, variable silica activities between 0·7 and 0·3, and f O2 values between –0·5 and +0·7 (log FMQ, where FMQ is fayalite–magnetite–quartz). The diorites crystallized under uniform a SiO2 ( a SiO2 = 0·4–0·5) and more reduced f O2 conditions (log FMQ ~ –1) between ~1100 and ~800°C. Phase equilibria in various foid syenites indicate that silica activities decrease from 0·6–0·3 at ~1000°C to 〈0·3 at ~550°C. Release of an aqueous fluid during the transition to the hydrothermal stage caused a SiO2 to drop to very low values, which results from reduced SiO 2 solubilities in aqueous fluids compared with silicate melts. During the hydrothermal stage, high water activities stabilized zeolite-group minerals. Fluid inclusions record a complex post-magmatic history, which includes trapping of an aqueous fluid that unmixed from the restitic foid syenitic magma. Cogenetic aqueous and carbonic fluid inclusions reflect heterogeneous trapping of coexisting immiscible external fluids in the latest evolutionary stage. The O and C isotope characteristics of fluid-inclusion hosted CO 2 and late-stage carbonates imply that the surrounding limestones were the source of the external fluids. The mineral-rich syenitic rocks at Mont Saint-Hilaire evolved as follows: first, alkalis, high field strength and large ion lithophile elements were pre-enriched in the (late) magmatic and subsequent hydrothermal stages; second, percolation of external fluids in equilibrium with the carbonate host-rocks and mixing processes with internal fluids as well as fluid–rock interaction governed dissolution of pre-existing minerals, element transport and precipitation of mineral assemblages determined by locally variable parameters. It is this hydrothermal interplay between internal and external fluids that is responsible for the mineral wealth found at Mont Saint-Hilaire.

Description: Phase equilibrium data pertaining to melting of simplified carbonated peridotite in the systems CaO–MgO–SiO 2 –CO 2 and CaO–MgO–Al 2 O 3 –SiO 2 –CO 2 at pressures of 10–26 GPa, corresponding to ~300–750 km depths in the Earth, are presented. In both the studied systems, liquid compositions, with changing crystalline phase assemblage, are carbonatitic throughout the studied pressure range. In the system CMS–CO 2 , melting phase relations are isobarically invariant; liquid is in equilibrium with forsterite + clinoenstatite + clinopyroxene + magnesite, forsterite + majorite + clinopyroxene + magnesite, wadsleyite + majorite + clinopyroxene + magnesite, ringwoodite + majorite + calcium-silicate perovskite + magnesite, magnesium-silicate perovskite + periclase + calcium-silicate perovskite + magnesite at 12, 14, 16, 20, and 26 GPa, respectively. In the system CMAS–CO 2 , a phase assemblage consisting of forsterite + orthopyroxene + clinopyroxene + magnesite + garnet + melt from 10 to 14 GPa is isobarically invariant. However, owing to the disappearance of orthopyroxene at pressures greater than 14 GPa, from 16 and up to at least 26 GPa, the solidus of simplified carbonated peridotite spans a divariant surface in pressure–temperature space. The liquid coexists with wadsleyite + clinopyroxene + garnet + magnesite, ringwoodite + calcium-silicate perovskite + garnet + magnesite, and magnesium-silicate perovskite + periclase + calcium-silicate perovskite + magnesite at 16, 20, and 26 GPa, respectively. A curious, and as yet unexplained, feature of our study is an abrupt drop in the solidus temperature between 14 and 16 GPa that causes a small amount of melting of carbonated mantle in the Transition Zone of the Earth. In the systems CMS–CO 2 and CMAS–CO 2 liquid compositions at 16 and 20 GPa are highly calcic bona fide carbonatites; however, these liquids revert to being magnesiocarbonatites at 10–14 and 26 GPa. In the system CMS–CO 2 , at 16 GPa we locate an isobaric invariant point consisting of wadsleyite + clinopyroxene + anhydrous B + magnesite + melt. The presence of anhydrous B at 16 GPa and 1475°C is interesting, as it lies outside the composition space of the mantle peridotite analog we have studied. However, despite the presence of two highly magnesian silicate crystalline phases, wadsleyite and anhydrous B, at 16 GPa and 1475°C, the liquid composition remains calcic with molar Ca-number [Ca/(Ca + Mg) x 100] of about 63. The melting reactions at 16 and 20 GPa (with or without anhydrous B) show that lime-bearing crystalline silicates play a fairly large part in generating and controlling the composition of the liquids. At 16 GPa, in the system CMS–CO 2 , we also report an experimental run at 1575°C, in which liquid coexists with only wadsleyite and majorite. The liquid composition is less calcic (Ca-number 54) than that for other runs at lower temperatures, but is still more calcic than liquids at 10–14 and 26 GPa in both the studied systems. At present, the likely cause for these changes in the reported phase relations is not known. For normally assumed mantle temperatures, melting in the Transition Zone of the Earth, owing to the presence of carbonate, is probably unavoidable. The depth range of the drop in the carbonated peridotite solidus closely matches that of commonly observed low seismic velocities at ~400–600 km depth in the Earth.

Description: The 2·05 Ma Bushveld magmatic event culminated in the production of 〉90 000 km 3 of granite and granophyres. In these granitic rocks, high-temperature equilibrium O-isotope fractionations are generally preserved between quartz and zircon, but not between quartz and feldspar, or between biotite and amphibole. Quartz separated from four granite samples shows no significant difference in core and rim 18 O values, which indicates that quartz is not significantly zoned, and provides further evidence that it is unaffected by alteration. Quartz can, therefore, be used as a proxy for the magma 18 O value, leading to estimates of 6·9 for both the granites (assuming quartz – magma = 1·11) and granophyres (assuming quartz – magma = 0·62). Similar magma 18 O values (6·6) were obtained using zircon 18 O values, assuming zircon – magma = –1·3. The initial Nd-isotope ratio of the granitic rocks ranges from 0·509676 to 0·509822, with an average value of 0·509655 ( n = 12). This corresponds to average Nd values of –5·9 and –4·8 for the granites and granophyres, respectively. The similarity in isotope composition between the granites and granophyres, and between the granitic rocks from each of the three major lobes of the Bushveld complex, is consistent with a common origin. The 18 O values of the granitic rocks suggest derivation from mantle-derived magmas by either fractional crystallization or partial melting, but this hypothesis is incompatible with their crustal Nd values (average –5·5). The associated Rustenburg Layered Suite (RLS) rocks have average 18 O values of 7·1, which is within error of the average estimate for the Bushveld granitic rocks, and similar Nd values. However, granitic magma derived from the same paretal magmas that produced the RLS would have had an average magma 18 O of about 7·9, 1 higher than observed. We therefore suggest that the granitic magmas were produced by fractional crystallization of RLS magma (or by partial melting of solidified RLS magma at depth) followed by assimilation, at a shallower level, of a significant quantity of hydrothermally altered low 18 O material from the since eroded volcanic edifice.

Description: The uppermost mantle as sampled by xenoliths, peridotite massifs and primitive basaltic melts appears to be relatively oxidized, with oxygen fugacities between the magnetite–wüstite and fayalite–ferrosilite–magnetite equilibria. Whether this range in oxygen fugacity is a shallow mantle signature or representative of the entire upper mantle still is unclear and a matter of debate because mantle regions deeper than 200 km are not well sampled. To constrain the redox state of the deeper upper mantle, we performed experiments from 1 to 14 GPa and 1220 to 1650°C on a model peridotite composition, encompassing the convecting asthenospheric mantle down to the Transition Zone at 410 km depth. The experiments were run in iron metal capsules to buffer f O 2 close to an oxygen fugacity about 0·5 log units below the iron–wüstite equilibrium. Analysis of the experimental phases for ferric iron using electron energy loss spectroscopy reveals that at pressures higher than 7 GPa, subcalcic pyroxene and majoritic garnet incorporate appreciable amounts of ferric iron, even though at the experimental conditions they were in redox equilibrium with metallic iron. The major ferric iron carrier in the upper mantle is majoritic garnet, followed by subcalcic pyroxene. At around 8 ± 1 GPa, corresponding to ~250 ± 30 km depth in the upper mantle, sufficient quantities of subcalcic pyroxene and majoritic garnet are stabilized that all the ferric iron thought to be present in fertile upper mantle (i.e. ~2000 ppm) can be accommodated in solid solution in these phases, even though they were synthesized in redox equilibrium with metallic Fe. Based on the results of the experiments, it can be stated that, on a global scale, an oxidized upper mantle near the fayalite–ferrosilite–magnetite equilibrium is the exception rather than the rule. More than 75 vol. % of the Earth’s present-day mantle is likely to be saturated with metallic iron.

Description: Incipient-stage alteration products in relatively fresh oceanic gabbros from deep boreholes provide critical information on hydration processes in the oceanic lower crust and their effect on lithosphere dynamics. We present the results of a petrographic study on the alteration of olivine-bearing gabbroic rocks recovered from the deeper parts of Integrated Ocean Drilling Program (IODP) Hole U1309D in the Atlantis Massif near the Mid-Atlantic Ridge at 30°N. In these rocks, alteration is localized in proximity to fluid-infiltration veins or igneous contacts. It is most conspicuous in halos surrounding amphibole + chlorite veins or leucocratic veins in olivine-bearing gabbros, where coronitic fringes of tremolite, chlorite and talc occur around discrete olivine grains. Many of the halos exhibit a zonal pattern with systematic changes in mineral assemblage, generally consisting of three zones: tremolite + chlorite around relict olivine–plagioclase contacts; talc pseudomorphs after olivine; and tremolite pseudomorphs after olivine. The tremolite + chlorite assemblage appears in increasing amounts and talc grows unevenly with increasing thickness toward the veins. The alteration minerals have highly magnesian compositions, reflecting the compositions of the precursor igneous phases. Within the zone closest to the veins, green hornblende with a relatively high Al content occurs, showing textures suggestive of its later formation than the coronitic tremolite and chlorite. Considering the mode of occurrence and chemical composition of the minerals combined with thermodynamic calculations of silica and water activities in a simplified system, we conclude that the zoned halos were caused by metasomatism owing to protracted or sequential infiltration of hydrothermal fluids at amphibolite-facies conditions (450–750°C, 1·5–2 kbar). Textural relationships clearly indicate that zoned halos formed earlier than serpentinization and clay mineral formation, and suggest that the high-temperature, amphibolite-facies alteration took place in a near-axis region before the exhumation of the lower crustal rocks. Recent results of seafloor drilling have provided supporting evidence for the predominance of gabbroic rocks in oceanic core complexes. The similarity in mineral association between zoned halos and schistose fault rocks suggests that preferential formation of talc and/or chlorite, rather than serpentine, at contacts between gabbroic rocks and peridotite plays an essential role in detachment faulting and tectonic exhumation of oceanic core complexes from lower crustal levels.

Description: Geochemically, the large family of alkaline plutonic rocks (both Qtz-undersaturated and -oversaturated compositions) can be subdivided into metaluminous [(Na 2 O + K 2 O) 〈 Al 2 O 3 ] and peralkaline [(Na 2 O + K 2 O) 〉 Al 2 O 3 ] types. In this paper, we discuss two important aspects of the mineralogical evolution of such rocks. With respect to their Fe–Mg phases, a major mineralogical transition observed is the precipitation of arfvedsonite or aegirine instead of fayalite or magnetite (± ilmenite). The relative stability of these phases is controlled by oxygen fugacity and Na activity in the crystallizing melts. If Na activity in the melt is high enough, arfvedsonite + aegirine form a common assemblage in peralkaline rocks under both reduced and oxidized conditions. Major mineralogical differences within this rock group exist with respect to their high field strength element (HFSE)-rich minerals: most syenitic rocks, known as miaskites, contain zircon, titanite or ilmenite as HFSE-rich minerals, whereas in agpaites complex Na–K–Ca–(Ti, Zr) silicates incorporate the HFSE. Similarly, only a small group of peralkaline granites are found to lack zircon, titanite or ilmenite but instead contain Na–K–Ca–(Ti, Zr) silicates. Here, we present a detailed phase petrological analysis of the chemical parameters (µNa 2 O, µCaO, µK 2 O) that influence the transition from miaskitic to agpaitic rocks. Based on the occurrence of Ti and Zr minerals, several transitional mineral assemblages are identified and two major evolution trends for agpaites are distinguished: a high-Ca trend, which is exemplified by the alkaline rocks of the Kola Province, Russia, and a Ca-depletion trend, which is displayed by the alkaline rocks of the Gardar Province, South Greenland. Both trends show significant Na-enrichment during magmatic evolution. High-Ca agpaites evolve from nephelinitic parental melts that did not crystallize large amounts of plagioclase. In contrast, agpaites showing Ca-depletion originate by extensive fractionation of plagioclase from basaltic parental melts. In some peralkaline granites evolutionary trends are observed that culminate in agpaite-like HFSE-mineral associations in the most evolved rocks.

Description: The discovery of chemically and isotopically enriched mid-ocean ridge basalts (E-MORB) has offered substantial insight into the origin, time scales, and length scales of mantle heterogeneity. However, the exact processes involved in producing this E-MORB enrichment are vigorously debated. Additionally, because the ages of E-MORB are not well constrained, the petrogenetic, temporal, and geological relationships between E-MORB and normal (N)-MORB are not known. To investigate these relationships and to explore how melting and melt transport processes contribute to or modify enriched mantle source compositions and generate E-MORB melts beneath mid-ocean ridges, we measured major and trace elements, and Sr, Nd, Hf, Pb, and U–Th–Ra isotopes for a suite of lavas that were collected off-axis, including several E-MORB, at 9–10°N along the East Pacific Rise (EPR). These data show coherent mixing trends among long-lived radiogenic isotopes, U-series nuclides, and incompatible trace elements, implying that mixing of melts from different sources occurs at different depths. Our results are consistent with previous studies that show that melting occurs in a two-porosity melting regime, with high-porosity channels forming deeply in the presence of garnet and transporting enriched melts with large 230 Th excesses to the crust, whereas low-porosity channels transport melts more slowly, allowing them to equilibrate at shallow depths and develop large 226 Ra excesses at the expense of diminished 230 Th excesses. Forward modeling of the trace element data also is consistent with mixing of melts in a two-porosity melting regime. U-series age constraints suggest that E-MORB neither erupt at systematically different times from N-MORB, nor necessarily through different pathways. Previous studies of E-MORB at 9–10°N have suggested that E-MORB compositions could be explained by off-axis eruption. However, when considered in light of previously published magnetic paleointensity and U-series age constraints, recent geological studies, and the most widely accepted contemporary understanding of volcanic construction at 9–10°N EPR, the asymmetric, off-axis distribution of E-MORB at 9–10°N EPR is consistent with, and more simply explained by, a model in which E-MORB erupted within the axial summit trough (AST) and flowed down the ridge flanks (~0–3 km). These E-MORB subsequently spread away from the AST, and, finally, were preserved on the seafloor through asymmetric construction of the extrusive layer. Taken together, the range of ages of E-MORB at 9–10°N EPR and the geochemical and isotopic mixing trends suggest that enriched melts are continuously supplied to the ridge axis, but because of their small proportions relative to the volumetrically and volcanically dominant N-MORB, E-MORB preservation and exposure is comparatively scarce.

Description: The deep crustal magmatic history of arc volcanoes is obscured by diversity in mantle inputs, modest isotopic contrast between magma and wall-rock, and overprinting processes in the middle and upper crust. To identify and quantify processes in the deep arc crust, we investigated the evolution of the mafic composite North Sister Volcano, the oldest and most mafic of the Three Sisters Volcanic Field of the central Oregon Cascade arc. Here, intra-arc extension limits the degree of magma interaction with the mid- to upper crust and the range in primitive magmas delivered from the mantle is known. North Sister Volcano has produced low-K basaltic andesitic magmas (0·5–0·8 wt % K 2 O) for ~400 kyr during four central-vent eruptive stages and along the late, 11 km long Matthieu Lakes Fissure. Although restricted in bulk composition (53–55 wt % SiO 2 ), North Sister basaltic andesites from different stages cluster into elemental and isotopic groups. Over time, North Sister basaltic andesites generally have decreasing compatible elements, such as Ni (from 112 to 40 ppm), and increasing Al 2 O 3 and TiO 2 . Concurrently, incompatible elements remain the same or decrease (e.g. from 302 to 247 ppm Ba). Isotopic variations at North Sister are small, but systematically progress toward more mantle-like ratios with time; 87 Sr/ 86 Sr decreases (from 0·70369 to 0·70356), and 144 Nd/ 143 Nd increases (from 0·51285 to 0·51292). We present a multi-stage petrological model for the evolution of North Sister magmas to account for: (1) the generation of low-K basaltic andesite; (2) geochemical variations within the eruptive stages; (3) evolution of the magma system over time to more mantle-like compositions. The earliest and most isotopically ‘crust-like’ (highest 87 Sr/ 86 Sr and lowest 143 Nd/ 144 Nd) North Sister magma is consistent with two-component mixing of regionally typical mantle-derived, low-K tholeiites with partial melts of the crust. Crustal melts must be high in SiO 2 and Al 2 O 3 , and most probably result from low-degree melting of plagioclase–clinopyroxene amphibole-bearing gabbro at high pressure. Variations in highly compatible elements within compositional groups (e.g. 60 ppm Ni within a single group) reflect fractionation of plagioclase, olivine, and clinopyroxene and recharge by more primitive basaltic andesite that overprint longer-term variations between groups. To understand the evolution of the North Sister basaltic andesite magmas through time, we use an energy-constrained model that balances assimilation of refractory gabbroic wall-rocks and abundant recharge by mantle-derived low-K tholeiites. These complementary processes allow Sr and Nd isotopic ratios to become more like those of the regional basalts while maintaining high Ni concentrations. Low-K basaltic andesites like those of North Sister Volcano are found along the Oregon Cascade arc and they imply that low-K tholeiitic magmas interact with a refractory mafic underplate along its length. Dominantly basaltic andesite volcanoes are common in arcs and provide insight into the extensive, albeit compositionally cryptic mafic underplating and intraplating that affects arc crust.

Description: The effects of metamorphic reactions occurring during decompression were explored to understand their influence on the 40 Ar– 39 Ar ages of micas. Monometamorphic metasediments from the Lepontine Alps (Switzerland) reached lower amphibolite facies during the Barrovian metamorphism related to the collision between European and African (Adria) continental plates. Mineral assemblages typically composed of garnet, plagioclase, biotite, muscovite and paragonite (or margarite) were screened for petrological equilibrium, to focus on samples that record a minimum degree of retrogression. X-ray diffraction data indicate that some mineral separates prepared for 40 Ar– 39 Ar stepwise heating analysis are monomineralic, whereas others are composed of two white micas (muscovite with paragonite or margarite), or biotite and chlorite. In monomineralic samples 37 Ar/ 39 Ar and 38 Ar/ 39 Ar (proportional to Ca/K and Cl/K ratios) did not change and the resulting ages can be interpreted unambiguously. In mineral separates containing two white micas, Ca/K and Cl/K ratios were variable, reflecting non-simultaneous laboratory degassing of the two heterochemical Ar reservoirs. These ratios were used to identify each Ar reservoir and to unravel the age. In a chlorite–margarite–biotite calcschist equilibrated near 560°C and 0·65 GPa, biotite, margarite, and muscovite all yield ages around 18 Ma. At slightly higher grade (560–580°C, 0·8–0·9 GPa), the assemblage muscovite–paragonite–plagioclase is in equilibrium and remains stable during retrogression. In this case, muscovite and paragonite yield indistinguishable ages around 16·5 Ma. Above 590°C, paragonite was mostly consumed to form plagioclase 〉590°C, whereby the relict mica yields an age up to 5·6 Ma younger than muscovite. This partial or total resetting of the Ar clock in paragonite is interpreted to reflect plagioclase growth during decompression. Where biotite is present within this same assemblage, it systematically yields a younger age than muscovite, by 0·5–2 Ma. However, these biotites all show small amounts of retrograde chlorite formation. We conclude that even very minor chloritization of biotite is apparently a more effective process than temperature in resetting the Ar clock, as is the formation of plagioclase from paragonite decomposition. Multi-equilibrium thermobarometry is an excellent means to ensure that equilibrium in investigated samples is preserved, and this helps to obtain geologically meaningful metamorphic ages. However, even samples passing such equilibrium tests may still show retrograde effects that affect the Ar retention of micas. A more robust interpretation of such 40 Ar– 39 Ar results may require use of a second geochronometer, such as U–Pb on monazite.

Description: To gain insights into the long-standing problem of marginal reversals, we have studied in detail the ~200 m thick marginal zone of the ~3 km thick, mafic Koitelainen Layered Intrusion (KLI). This marginal zone consists of pigeonite gabbros that are chilled against gneisses of the underlying basement but reveals a non-chilled upper contact with orthopyroxenites of the overlying Layered Series. There is a sharp textural and compositional break between the rocks of the marginal zone and the Layered Series of the KLI. The marginal zone is remarkable in showing systematic reverse fractionation trends from the bottom to the very top. These are exemplified by a significant upward increase in whole-rock and pyroxene Mg-number along with normative and actual An content of plagioclase, as well as by an upward decrease in all incompatible components (e.g. TiO 2 , Zr, Y, REE). This is accompanied by a systematic upward decrease in ratios of highly incompatible elements, such as Zr/Y and La/Yb and in initial whole-rock Nd . The uppermost rocks of the marginal zone are very fine-grained and show well-developed plagioclase lamination and ultra-depletion in all incompatible elements (e.g. Zr = 0·5 ppm; Ce = 0·11 ppm). The marginal zone is interpreted as having developed by in situ crystallization of magma in a slowly opening fissure. The origin of reverse compositional trends is attributed to three principal factors, which we refer to as the ‘three- increase model’: (1) an increase in compositional primitivity of the magma that gradually filled the chamber; (2) an increase in the degree of chemical equilibrium among phases associated with increasing distance from the cold country rocks; (3) an increase in the proportion of cumulus minerals in response to more effective removal of evolved liquid from in situ growing crystals with distance from the intrusive contact. The marginal reversal of the KLI is distinct from typical reversals in layered intrusions in that its development was interrupted by the emplacement of a large volume of new magma that was parental to the overlying Layered Series. This new, hot magma caused significant textural and compositional reconstitution of the uppermost gabbroic rocks of the marginal reversal by their partial melting, compaction and recrystallization. This gave rise to a considerable decrease in grain size, appearance of plagioclase lamination, and ultra-depletion in highly incompatible elements. The anomalous nature of these rocks is thus a manifestation of their restitic rather than primary magmatic origin. The proposed ‘three-increase model’ may represent a general explanation for the origin of marginal reversals in many mafic sills and layered intrusions.

Description: Phase assemblages, melting relations and melt compositions of a dry carbonated pelite (DG2) and a carbonated pelite with 1·1 wt % H 2 O (AM) have been experimentally investigated at 5·5–23·5 GPa and 1070–1550°C. The subsolidus mineralogies to 16 GPa contain garnet, clinopyroxene, coesite or stishovite, kyanite or corundum, phengite or potassium feldspar (≤8 GPa with and without H 2 O, respectively), and then K-hollandite, a Ti phase and ferroan dolomite/Mg-calcite or aragonite + ferroan magnesite at higher pressures. The breakdown of clinopyroxene at 〉16 GPa causes Na-rich Ca-carbonate containing up to 11 wt % Na 2 O to replace aragonite and leads to the formation of an Na-rich CO 2 fluid. Further pressure increase leads to typical Transition Zone minerals such as the CAS phase and one or two perovskites, which completely substitute garnet at the highest investigated pressure (23·5 GPa). Melting at 5·5–23·5 GPa yields alkali-rich magnesio-dolomitic (DG2) to ferro-dolomitic (AM) carbonate melts at temperatures 200–350°C below the mantle geotherm, lower than for any other studied natural composition. Melting reactions are controlled by carbonates and alkali-hosting phases: to 16 GPa clinopyroxene remains residual, Na is compatible and the magnesio- to ferro-dolomitic carbonate melts have extremely high K 2 O/Na 2 O ratios. K 2 O/Na 2 O weight ratios decrease from 26–41 at 8 GPa to 1·2 at 16 GPa when K-hollandite expands its stability field with increasing pressure. At 〉16 GPa, Na is repartitioned between several phases, and again becomes incompatible as at 〈3 GPa, leading to Na-rich carbonate melts with K 2 O/Na 2 O ratios 1. This leaves the pressure interval of c . 4–15 GPa for ultrapotassic metasomatism. Comparison of the solidus with typical subducting slab-surface temperatures yields two distinct depths of probable carbonated pelite melting: at 6–9 GPa where the solidus has a negative Clapeyron slope between the intersection of the silicate and carbonate melting reactions at ~5 GPa, and the phengite or potassium feldspar stability limit at ~9 GPa. The second opportunity is related to possible slab deflection along the 660 km discontinuity, leading to thermal relaxation and partial melting of the fertile carbonated pelites, thus recycling sedimentary CO 2 , alkalis and other lithophile and strongly incompatible elements back into the mantle.

Description: We present major element, trace element, and petrographic data on alkali basalts from St. Helena, and examine the geochemical characteristics of a recycled component involved in the source of HIMU ( 206 Pb/ 204 Pb 〉20·5) ocean island basalts. Petrographic and compositional variations in the St. Helena basalts are best explained by the combined effect of fractional crystallization and accumulation of phenocrysts. Primary melt compositions are estimated by correcting for the effects of crystal–liquid differentiation by reconstructing the order of crystallization and the relative amount of fractionated phases. This calculation indicates that the St. Helena alkali basalts are derived from a common primary magma with 14–20 wt % MgO. Simple partial melting of fertile mantle peridotite, depleted mid-ocean ridge basalt (MORB)-source mantle, or garnet pyroxenite fails to produce the St. Helena primary melt. Instead, this primary melt can be reproduced if there are contributions from ancient recycled oceanic crust and depleted peridotite [(Rb/Nb) PM = 0·38–0·80]. Subducted sediment can be excluded to explain the low (Rb, Ba, U)/Nb and Ce/Pb of St. Helena basalts. Geochemical modeling using major and trace element abundances, together with Sr, Nd, Pb, and Hf isotope ratios, indicates that the St. Helena primary melt can be formed by 1–2% melting of a peridotitic source that was refertilized by a small amount (8–18%) of melt derived from recycled oceanic crust. This source has a similar trace element pattern to modern normal (N)-MORB, but element abundances are 0·1–0·2 times N-MORB values. The calculated recycled crust has a wide range of present-day Pb isotopic ratios ( 206 Pb/ 204 Pb of 21·7–79·3 and 208 Pb/ 204 Pb of 40·8–89·3), 87 Sr/ 86 Sr of 0·7018–0·7028, 143 Nd/ 144 Nd of 0·51274–0·51285, and 176 Hf/ 177 Hf of 0·28262–0·28293 after a residence time of 1·2–2·8 Gyr. Rb, Ba, Pb, Sr, and light rare earth element abundances in the recycled crust are depleted compared with modern N-MORB, whereas Th, U, Sm, and Nd abundances fall within the range of compositional variations in modern N-MORB. The trace element compositions of the recycled oceanic crust can be explained by element behavior during seafloor alteration and subduction zone dehydration of oceanic crust. Therefore, recycling of ancient subducted oceanic crust is a potential process for producing the St. Helena HIMU basalts.

Description: Pyrometamorphic ejecta erupted during recent paroxysmal explosions and hydrothermally altered volcanic and subvolcanic rocks from Stromboli volcano (Aeolian Islands, Italy) were studied by combined polarizing microscopy, electron microprobe, scanning electron microscope, X-ray diffraction, transmission electron microscope and whole-rock inductively coupled plasma optical emission spectrometry–mass spectrometry analysis. The pyrometamorphic ejecta are compositionally heterogeneous at the macro- and micro-scale, and are classified as buchites; that is, glass coupled with a high-temperature, low-pressure modal mineralogy. Three distinct facies (i.e. cordierite–plagioclase, mullite–plagioclase palisade and tridymite–clinopyroxene) are recognized among the buchites, whose bulk-rock chemistry is compatible with aluminous- to silica-rich protoliths. Abundant mineral growth (e.g. plagioclase palisades and clinopyroxene coronae) occurs at the contact between the ejecta and the crystal-rich host shoshonitic basalt, which typically coats the pyrometamorphic blocks. Mineral constraints, pseudoporphyritic microstructures, plagioclase relict phenocrysts and major and trace element compositions strongly support the origin of the buchite ejecta as hydrothermally altered rocks from the Vancori edifice (a period of activity spanning 〉10 kyr), which underwent high-temperature, low-pressure contact metamorphism. In the uppermost magmatic system of an active island arc volcano, such as Stromboli, volcanic and subvolcanic rocks usually undergo alteration processes induced by acid–sulphate hydrothermal systems. The circulation of acidic fluids results in leaching of the Stromboli rocks (in the upper part of the volcano edifice) and, depending on the pH, temperature and exposure time to the hydrothermal fluids, production of different alteration facies. The hydrothermally altered Stromboli rocks show facies consistent with argillic to silicic alteration, containing abundant hydrous sulphate and hydroxy-sulphate minerals such as the alunite group minerals, hexahydrite, millosevichite, pickeringite and kieserite. The hydrothermally altered wall-rocks, heated by contact with the basaltic magma of the feeder dyke system of Stromboli, can easily undergo pyrometamorphic processes, inducing partial melting and subsolidus crystallization of the volcanic protoliths. This results in the development of buchite rocks at the magma–hydrothermal system interface.

Description: The undeformed 564 Ma Sept Iles layered intrusion (Quebec, Canada) is a large igneous body of c . 20 000 km 3 . From the base to the top, it consists of a Layered Series dominated by troctolite and gabbro, an anorthositic Upper Border Series and a dominantly granitic Upper Series. The parent magma of the Layered Series is inferred to be an iron-rich tholeiitic basalt (48 wt % SiO 2 ; 15 wt % FeO t ). Whole-rock compositions from the chilled margin, dykes cross-cutting the Layered Series and silicic rocks from the Upper Series display continuous major and trace element geochemical trends ranging from basalts to ferroan metaluminous A-type granites (77 wt % SiO 2 ). Initial 143 Nd/ 144 Nd (0·51201–0·51207) and 87 Sr/ 86 Sr (0·70353–0·70548) indicate a juvenile-mantle source and minimal contamination by old crust (1–2%) during crystallization. Geochemical modeling, using the MELTS thermodynamic calculator combined with equations predicting mineral–melt equilibria from experiments on tholeiitic basalts, indicate that basaltic to monzonitic melt compositions are in equilibrium with the troctolites and gabbros of the Layered Series. Fe–Ti oxides saturate early in the Layered Series, after 14% fractionation of plagioclase–olivine cumulates. Further fractionation of Fe–Ti oxide-bearing gabbros drives the residual liquids toward silica enrichment and iron depletion. Major and trace element modeling indicates that the A-type granites from the Upper Series were produced by protracted fractional crystallization of an iron-rich basaltic parent magma, at a fraction of residual liquid of only 8%. The observed relative volumes of mafic cumulates and silicic rocks in the intrusion are in agreement with the calculations. Most of the intermediate compositions correspond to magmatic mafic enclave-bearing granitoids and display geochemical evidence of hybridization. Intermediate compositions produced by fractional crystallization are scarce and a Daly gap occurs from 57 to 67 wt % SiO 2 . This gap could result either from the fractional crystallization process or from silicate–liquid immiscibility during that compositional interval.

Description: We have developed density–viscosity–composition (–µ– X ) models for natural aqueous fluids and hydrous melts, based on experimental data for silicate + H 2 O, especially for the pressure ( P ) and temperature ( T ) conditions above subduction zones. We examine hydrothermal and melt pathway systematics above subducting slabs into the Earth’s mantle, back up along the top-of-slab, and downward with the subduction. Aqueous slab fluids and hydrous mantle melts show distinct flow properties (as observed in activation energy in viscosity data) despite continuity in solute-polymerization characteristics. Buoyancy changes are small for fluids except in the localized vicinity of critical behaviour and at solidi where H 2 O partitions also into melt. Our model predicts dilute high- PT potassic haplogranite fluids to be less viscous than sodic varieties whereas for concentrated fluids a deep viscosity minimum occurs in mixed K/Na ( c . 1:1 molar) compositions. Higher dissolved silicate concentrations increase fluid density and viscosity leading to slower less-buoyant flow with increasing PT . Thus ascent rates of slab fluid increase by about an order of magnitude (from c . 10 –3·5 to 10 –4·3 m s –1 for porous flow; c . 1 to 7 m s –1 for flow through 1 mm wide fractures) with decompression from 5 to 3 GPa, as a result of decreasing solute loads, and µ. Mantle fluid viscosities are predicted (10 –4 to 10 –3·7 Pa s) to be approximately half those of crustal fluids (10 –3·9 to 10 –3·1 Pa s) and of lower density (e.g. 1·4 compared to 1·6 g cm –3 ), reflecting their compositional differences (here mainly SiO 2 ). Thus, ascending slab fluids tend to accelerate as they move back up the slab and also moving from slab to porous mantle. Slab melts are up to c . 6 orders of magnitude less viscous (e.g. c . 10 –0·5 to 10 2·5 Pa s) and therefore faster flowing than hydrous deep crustal granitoids (e.g. c . 10 6·5 to 10 3·5 Pa s), reflecting higher water contents of the former (e.g. 30 vs 10 wt %). Concentrated crustal fluids migrate 5–6 orders of magnitude faster than hydrous melt, mostly because of calculated viscosity differences. We find that fluids flow faster in the mantle than in the crust, and that most of the mass transfer through the mantle occurs via hydrous melt.

Description: Redox melting refers to any process by which melt is generated by the contact of a rock with a fluid or melt with a contrasting oxidation state. It was originally applied to melting owing to the oxidation of reduced CH 4 - and H 2 -bearing fluids in contact with more oxidized blocks in the mantle, particularly recycled crustal blocks. This oxidation mechanism causes an increase in the activity of H 2 O by the reaction of CH 4 with O 2 , and the increased a H 2 O causes a rapid drop in the solidus temperature, and is here termed hydrous redox melting (HRM). Recently, a second redox melting mechanism (carbonate redox melting; CRM) has been discovered that operates in more oxidized conditions, and may post-date the first mechanism in the same geographical area, explaining the sequence of igneous rock types from lamproites to ultramafic lamprophyres that occurs during the development of rifts through cratons. The CRM mechanism relies on the oxidation of solid carbon as graphite or diamond that has accumulated in the lithosphere over time. The solidus temperature for rocks with both CO 2 and H 2 O is lower than in conditions with H 2 O alone; it does not occur at depths less than 65 km, but has recently been confirmed experimentally to depths of at least 200 km. Melts produced by HRM are not SiO 2 -undersaturated, even at depths of 200 km, and may often resemble lamproites or SiO 2 -rich picrites, whereas melts produced by CRM are always SiO 2 -undersaturated and range from carbonatitic to ultramafic lamprophyric or melilititic with increasing degree of melting. The operation of redox melting may be more common than has been recognized because the oxidation state of the upper mantle is not uniform as a function of depth, geodynamic setting or geological time. The general decrease of oxygen fugacity ( f O 2 ) of c . 0·7 log units per 1 GPa pressure increase dictates that rapidly subducted oceanic lithosphere will be considerably more oxidized than ambient mantle peridotite at depths of 200–300 km. Hydrothermal alteration (serpentinization), addition of continental or carbonate sediments, and dehydration reactions during subduction all contribute to the heterogeneity of oxidation states in the subducted slab, which may vary over 6 log units; this raises the potential for redox reactions on local and regional scales. The oceanic lithosphere has a lower average f O 2 than either continental or cratonic mantle lithosphere at a given depth, so that the HRM mechanism dominates in recycled blocks and at the base of the continental lithosphere. The higher thermal gradients dictate that HRM is more common in the modern Earth beneath ocean islands and in upwelling mantle currents than in subduction zones. The oxidation state of the mantle is often described as having been constant since 3·5 Ga, but this overlooks the bias towards continental samples. Redox melting of oxidized recycled blocks (at approximately the fayalite–magnetite–quartz buffer) in the mantle was not important in the Hadean and Archaean, as it had to await the gradual oxidation of the mantle and the establishment of the subduction process, as well as the stabilization of the continents. The lack of CRM explains the lack of carbonatites before 2·7 Ga. However, the lower f O 2 of the Archaean asthenosphere and higher volatile contents caused more prevalent HRM in the Hadean and Archaean mantle. Degassing is controlled by solubility of volatile species in melts, which are H 2 O-rich but C-poor in reducing conditions. Silicate melts under reduced conditions contain much less carbon but more nitrogen than melts in the modern mantle, arguing for a nitrogen-rich, CO 2 -poor early atmosphere.

Description: Igneous phenocrysts commonly exhibit zoning in major and trace element composition, reflecting (and potentially constraining) the differentiation and/or mixing histories of their parent melts. To date, little work has been done characterizing zonation of oxygen isotopes in minerals from mafic and ultramafic rocks. We present 259 ion probe (CAMECA ims-1280) measurements of 18 O in 34 natural magmatic and mantle olivines and pyroxenes from five hand samples from diverse igneous environments. We compare 18 O variations with zonation in other elements [especially P; analyzed by electron microprobe analysis (EMPA) and nano-secondary ionization mass spectrometry (nanoSIMS)]. There is generally a close (average within ~0·1–0·2 ) agreement between average 18 O values of olivines measured by SIMS (standardized against San Carlos olivine) and independently known values for bulk separates from the same samples measured by laser fluorination. These data demonstrate that current ion microprobe techniques are not only precise but also accurate enough for study of sub-per-mil oxygen isotope variations in silicates (within ~0·2 ), provided samples are prepared and analyzed following strict guidelines. All but one of the 34 studied grains are homogeneous in 18 O within a small multiple of analytical precision [estimated ±0·2, 1 for most data; poorer for a subset of measurements made on small (~5 µm) spots]. This population of isotopically homogeneous grains includes some with oscillatory micrometer-scale P banding. The lack of 18 O variations suggests that whatever factors lead to this common mode of trace element zonation have no detectable effect on melt–crystal partitioning of oxygen isotopes. Large (2) oxygen isotope variations are observed in one olivine glomerocryst from Mauna Kea, Hawaii. This glomerocryst contains P-rich domains that are either equant or skeletal or feathery in outline, and these P-rich domains are systematically low in 18 O compared with adjacent, later-grown, P-poor olivine. This unusual oxygen isotope zonation pattern might reflect a kinetic fractionation during nucleation and growth of the cores of some olivine phenocrysts. We tested this hypothesis through measurements of 18 O distributions in synthetic olivines grown at a range of rates and exhibiting diverse patterns of P zoning. These synthetic olivines are homogeneous in 18 O, within the limits of our analyses (± 0·3–0·4 in this case) and show no connection between P zonation and oxygen isotope heterogeneity. We therefore think it more plausible that unusual O isotope zonation in the Mauna Kea glomerocryst reflects addition of a low- 18 O component to some Hawaiian magmas just before nucleation of olivine. More generally, this study demonstrates the utility of modern SIMS techniques for in situ study of the subtle (~1 range) oxygen isotope variations characteristic of common mafic and ultramafic rocks.

Description: One hypothesis for the origin of alkaline lavas erupted on oceanic islands and in intracontinental settings is that they represent the melts of amphibole-rich veins in the lithosphere (or melts of their dehydrated equivalents if metasomatized lithosphere is recycled into the convecting mantle). Amphibole-rich veins are interpreted as cumulates produced by crystallization of low-degree melts of the underlying asthenosphere as they ascend through the lithosphere. We present the results of trace-element modelling of the formation and melting of veins formed in this way with the goal of testing this hypothesis and for predicting how variability in the formation and subsequent melting of such cumulates (and adjacent cryptically and modally metasomatized lithospheric peridotite) would be manifested in magmas generated by such a process. Because the high-pressure phase equilibria of hydrous near-solidus melts of garnet lherzolite are poorly constrained and given the likely high variability of the hypothesized accumulation and remelting processes, we used Monte Carlo techniques to estimate how uncertainties in the model parameters (e.g. the compositions of the asthenospheric sources, their trace-element contents, and their degree of melting; the modal proportions of crystallizing phases, including accessory phases, as the asthenospheric partial melts ascend and crystallize in the lithosphere; the amount of metasomatism of the peridotitic country rock; the degree of melting of the cumulates and the amount of melt derived from the metasomatized country rock) propagate through the process and manifest themselves as variability in the trace-element contents and radiogenic isotopic ratios of model vein compositions and erupted alkaline magma compositions. We then compare the results of the models with amphibole observed in lithospheric veins and with oceanic and continental alkaline magmas. While the trace-element patterns of the near-solidus peridotite melts, the initial anhydrous cumulate assemblage (clinopyroxene ± garnet ± olivine ± orthopyroxene), and the modelled coexisting liquids do not match the patterns observed in alkaline lavas, our calculations show that with further crystallization and the appearance of amphibole (and accessory minerals such as rutile, ilmenite, apatite, etc.) the calculated cumulate assemblages have trace-element patterns that closely match those observed in the veins and lavas. These calculated hydrous cumulate assemblages are highly enriched in incompatible trace elements and share many similarities with the trace-element patterns of alkaline basalts observed in oceanic or continental setting such as positive Nb/La, negative Ce/Pb, and similiar slopes of the rare earth elements. By varying the proportions of trapped liquid and thus simulating the cryptic and modal metasomatism observed in peridotite that surrounds these veins, we can model the variations in Ba/Nb, Ce/Pb, and Nb/U ratios that are observed in alkaline basalts. If the isotopic compositions of the initial low-degree peridotite melts are similar to the range observed in mid-ocean ridge basalt, our model calculations produce cumulates that would have isotopic compositions similar to those observed in most alkaline ocean island basalt (OIB) and continental magmas after ~0·15 Gyr. However, to produce alkaline basalts with HIMU isotopic compositions requires much longer residence times (i.e. 1–2 Gyr), consistent with subduction and recycling of metasomatized lithosphere through the mantle. EM magmas cannot readily be explained without appealing to other factors such as a heterogeneous asthenosphere. These modelling results support the interpretation proposed by various researchers that amphibole-bearing veins represent cumulates formed during the differentiation of a volatile-bearing low-degree peridotite melt and that these cumulates are significant components of the sources of alkaline OIB and continental magmas. The results of the forward models provide the potential for detailed tests of this class of hypotheses for the origin of alkaline magmas worldwide and for interpreting major and minor aspects of the geochemical variability of these magmas.

Description: Isothermal decompression experiments were performed to simulate magma ascent at Unzen volcano from the depths of magma storage to shallow crustal levels, corresponding to pressure decrease from 300 to 50 MPa. A partially crystallized synthetic rhyodacitic magma (representing equilibrium conditions at 850°C and 300 MPa) was used as a starting material; this has a composition identical to the groundmass of Unzen rocks erupted in 1991–1995. Decompression rates were varied from 0·0002 to 20 MPa s –1 . Experiments conducted with decompression rates ≥0·1 MPa s –1 were decompressed continuously; a multi-step decompression approach was used at decompression rates ≤0·1 MPa s –1 . The experiments were fluid-saturated, either containing only water as a fluid component (H 2 O-bearing) or containing a water and carbon dioxide mixture (H 2 O + CO 2 ; initial mole fraction of H 2 O in the fluid ~0·6). The experimental products of the H 2 O-bearing experiments consist of amphibole, pyroxenes, oxides and glass. Plagioclase microlites nucleated and grew only in experiments with the two lowest decompression rates of 0·0005 and 0·0002 MPa s –1 . The length of those plagioclases is up to 200–250 µm, which is consistent with the size of plagioclase microlites observed in the natural samples. The experimental products of the H 2 O + CO 2 -bearing system are composed of pyroxenes, oxides, glass and plagioclase. Plagioclase microlites in the H 2 O + CO 2 -system were already present in the starting assemblage and grew to a maximum size of ~80 µm. Equilibrium concentrations of water in the residual glasses at the final pressure of 50 MPa are reached at decompression rates ≤1 MPa s –1 for the H 2 O + CO 2 -bearing system and ≤0·1 MPa s –1 for the H 2 O-bearing system. The bubble number density (BND) values range from 10 3·7 to 10 5·6 mm –3 in the H 2 O-bearing system and from 10 4·6 to 10 6·4 mm –3 in the H 2 O + CO 2 -bearing systems. In both systems, BND values decrease with decreasing decompression rate from 20 to 0·01 MPa s –1 , and increase with decreasing decompression rates 〈 0·01 MPa s –1 , which is interpreted to reflect predominant bubble growth and bubble nucleation, respectively. The onset of crystallization, observed from changes in the chemical composition of the residual melt, occurs at decompression rates 〈 0·1 MPa s –1 . At the lowest decompression rate (0·0002 MPa s –1 ) the chemical composition of the residual melt in the H 2 O + CO 2 -bearing system becomes similar to the natural matrix glass composition. There is no significant variation of the microlite number density (MND) value as a function of the decompression rate. The MND values for plagioclases-only range from 10 5·4 to 10 5·7 mm –3 , whereas the MND values for the other phases range from 10 5·3 to 10 5·9 mm –3 . Our experimental MND Pl values are in the range of those from natural samples (10 5 –10 6 mm –3 ). We show that the size of microlites nucleating and crystallizing during decompression (plagioclase in our experimental dataset) is useful to constrain magma ascent rates at the onset of the crystallization of the corresponding phase. Based on the size of plagioclase microlites and on the composition of the residual melts, the average magma ascent rates of Unzen magmas in the pressure range 200 to 50 MPa is estimated to be 10–50 m h –1 .